International Workshop on High-Order Finite Element Methods · 11:40-12:05 Acoustic scattering...

International Workshop on High-Order Finite Element Methods

17-19 May 2007

Haus der bayerischen Landwirtschaft Herrsching, Herrsching am Ammersee (near Munich), Germany

sponsored by

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Workshop Chairmen Prof. Dr. rer. nat. Ernst Rank Phone: +49 89 289 23048 Lehrstuhl für Bauinformatik Fax: +49 89 289 25051 Technische Universität München E-Mail: [email protected] Arcisstr. 21 D-80333 München Prof. Zohar Yosibash Phone: +972 8 647710 3 Dept. of Mechanical Engineering Fax +972 8 647710 1 Ben-Gurion University E-Mail: [email protected] PoBox 653 Beer-Sheva 84105, Israel PD Dr.-Ing. Alexander Düster Phone: +49 89 289 25060 Lehrstuhl für Bauinformatik Fax: +49 89 289 25051 Technische Universität München E-Mail: [email protected] e Arcisstr. 21 D-80333 München Venue Haus der bayerischen Phone: +49 8152 938 000 Landwirtschaft Herrsching Fax: +49 8152 938 222 Rieder Str. 70 82211 Herrsching am Ammersee

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Programme Wednesday May, 16th

Day 1 Thursday May, 17th

Day 2 Friday May, 18th

Day 3 Saturday May, 19th

Opening 08:45-09:00 Session 1 09:00 – 10:25

Session 1 08:30 – 10:20

Session 1 08:30 – 10:20

Coffee break / Poster session 10:25 – 10:50



Session 2 10:50 – 12:30

Session 2 10:50 – 12:30

Session 2 10:50 – 12:30

Lunch 12:30 – 14:15

Lunch 12:30 – 14:15

Lunch 12:30 – 14:15

Session 3 14:15 – 16:30

Session 3 14:15 – 16:30

Session 3 14:15 – 16:50 Closing



Coffee break 16:50 – 17:15

Arrival at Haus der bayerischen Landwirtschaft Herrsching

Session 4 17:00 – 18:15

Session 4 17:00 – 18:15

Dinner 19:30-21:00

Dinner 19:30-21:00

Bavarian Evening 19:30-23:00

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Day 1 - Thursday, May 17th

08:45 – 09:00 Opening 09:00 – 10:25 Session 1 Keynote Lecture 09:00 – 09:35 High Order Finite Element Methods for Solving Problems with Microstructures Ivo Babuška 09:35 – 10:00 High-order FEM vs classical engineering models for a

shell roof J. Pitkäranta*, I. Babuška, B. Szabó

10:00-10:25 Asymptotic analysis of shell vibration and related

numerical hazards L. Beirão da Veiga*, H. Hakula, C. Lovadina, J. Pitkäranta

10:25 – 10:50 Coffee break / Poster session 10:50 – 12:30 Session 2 10:50-11:15 Higher-Order Dual-Mixed Finite Element Models in

Elasticity Edgár Bertóti

11:15-11:40 Mixed Finite Element Methods for Linear Elasticity –

Part I Joachim Schöberl*, Astrid Sinwel 11:40-12:05 p-FEMs for a class of finite deformation pressure

dependent plasticity models validated by experimental observations W. Bier, M. Dariel, A. Düster, N. Frage, S. Hartmann, U. Heisserer, S. Holzer, E. Rank, M. Szanto, Z. Yosibash*

12:05-12:30 High-order DIRK/MLNA method applied to finite

strain viscoplasticity S. Hartmann*, K. Quint

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12:30 – 14:15 Lunch 14:15 – 16:30 Session 3 14:15-14:50 Keynote Lecture Fully Automatic hp-Adaptive Finite Elements Where are we going? Leszek Demkowicz 14:50-15:15 Fast and Exact Projected Convolution of hp-Functions

W. Hackbusch 15:15-15:40 hp-Adaptive discontinuous Galerkin methods for saddle

point problems Dominik Schötzau

15:40-16:05 An hp finite element method for singularly perturbed

systems of reaction-diffusion equations Christos Xenophontos*, Lisa Oberbroeckling

16:05-16:30 Multi-Mesh hp-FEM and Selected Applications

Pavel Solin* , Ivo Dolezel, Jakub Cerveny, Lenka Dubcova

16:30 – 17:00 Coffee break / Poster session 17:00 – 18:15 Session 4 17:00-17:25 An hp-BEM for scattering by convex polygons J. M. Melenk*, S. Langdon 17:25-17:50 Fast solvers for hp-FEM using hexahedral elements Sven Beuchler 17:50-18:15 Piecewise Defined Basis Functions for hp-FEM Using

Object-oriented Software Concepts Matthias Baitsch*, Dietrich Hartmann

19:30 – 21:00 Dinner

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Day 2 - Friday, May 18th

8:30 – 10:20 Session 1 08:30-09:05 Keynote Lecture On the role of hierarchic spaces and models in

validation Barna Szabó*, Ricardo Actis 09:05-09:30 Sparse, stabilized high order tensor FEM for high-

dimensional transport-dominated diffusion problems Christoph Schwab*, Endre Suli, Radu-Alexandru Todor 09:30-09:55 Solution of 3D contact problems using a B-spline

approximation A. Baksa, I. Páczelt*, T. Szabó 09:55-10:20 High Order BEM for Contact Problems Alexey Chernov*, Matthias Maischak, Ernst P. Stephan 10:20 – 10:50 Coffee break / Poster session 10:50 – 12:30 Session 2 10:50-11:15 Polynomial extension operators for spaces H1, H(curl)

and H(div) on a cube M. Costabe1, M. Dauge*, L. Demkowicz 11:15-11:40 Incorporation of contact for high order FEM in

covariant form A. Konyukhov*, K. Schweizerhof 11:40-12:05 Acoustic scattering computations in littoral setting

using p-finite elements and perfectly-match-layers Saikat Dey 12:05-12:30 Visualization of High-Order Finite Element Methods Robert M. Kirby

12:30 – 14:15 Lunch

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14:15 – 16:30 Session 3 Keynote Lecture 14:15-14:50 Isogeometric Higher Order Methods Thomas J.R. Hughes 14:50-15:15 Mechanical contact for the solution of problems in the

aerospace industry Ricardo L. Actis*, Barna A. Szabó 15:15-15:40 Least-squares spectral element methods for

compressible flows Marc Gerritsma

15:40-16:05 The Discontinuous Enrichment Method for Multiscale

Wave Propagation, Flow, and Transport Problems Charbel Farhat, Radek Tezaur*

16:05-16:30 On the Application of Filters for Discontinuity

Capturing with High Order Discontinuous Galerkin Discretizations

Ioannis Toulopoulos, John A. Ekaterinaris* 16:30 – 17:00 Coffee break / Poster session 17:00 – 18:15 Session 4 17:00-17:25 Improving the CFL Condition for Discontinuous

Galerkin Methods T. Warburton*, T. Hagstrom 17:25-17:50 A discontinuous Galerkin approach for fluid flow

simulation based on the discrete Boltzmann equation A. Düster*, L. Demkowicz, E. Rank

17:50-18:15 A Review of Discrete Maximum Principles for Higher-

Order Finite Elements Tomáš Vejchodský*, Pavel Šolín

19:30 – 23:00 Bavarian Evening

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Day 3 - Saturday, May 19th 8:30 – 10:20 Session 1 Keynote Lecture 08:30-09:05 Goal-oriented combined adaptivity of mathematical modeling and FE-discretization for elastic systems Erwin Stein 09:05-09:30 A Posteriori Error Estimation for Lowest Order Raviart

Thomas Mixed Finite Elements Mark Ainsworth

09:30-09:55 A Posteriori Error Estimates without Generic Constants

Dietrich Braess*, Joachim Schöberl 09:55-10:20 Computation of the band structure of two-dimensional

Photonic Crystals with high order Finite Elements Kersten Schmidt*, Peter Kauf

10:20 – 10:50 Coffee break / Poster session 10:50 – 12:30 Session 2 10:50-11:15 Progress using Higher Order Finite Elements for

Electromagnetic Scattering A. Zdunek*, W. Rachowicz 11:15-11:40 Spectral Elements for the Integral Formulation of

Maxwell’s Equations in the Time-Harmonic Domain E. Demaldent*, G. Cohen, D. Levadoux 11:40-12:05 Certified DG-FEM Reduced Basis Methods and Output

Bounds for the Harmonic Maxwell’s Equations J.S. Hesthaven, Y. Maday, J. Rodriguez* 12:05-12:30 High Order Finite Elements for vector-valued Function

Spaces S. Zaglmayr*, J. Schöberl

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12:30 – 14:15 Lunch 14:15 – 16:50 Session 3 14:15-14:40 Higher Order Finite and Infinite Elements for the Solution of Helmholtz Problems Jan Biermann*, Otto von Estorff, Steffen Petersen 14:40-15:05 Stabilization techniques for inverse problems of nested

analysis and design Kai-Uwe Bletzinger*, Matthias Firl 15:05-15:30 Customized High-Order Enrichment Functions for the

Generalized Finite Element Method C.A. Duarte*, Dae-Jin Kim

15:30-15:55 An hp-adaptive multilevel particle-partition of unity method Marc Alexander Schweitzer*, Michael Griebel

15:55-16:20 The cell-based partition-of-unity method

Stefan M. Holzer*, Carsten Riker 16:20-16:45 A High-Order Embedded Domain Method

E. Rank*, J. Parvizian, Z. Yang, A. Düster

Closing 16:50 – 17:15 Coffee break

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Poster Sessions

Stochastic Galerkin method with sparse polynomial chaos expansion M. Bieri*, Ch. Schwab

A novel High-Order Finite Element Approach to

Modeling Localization and Failure Holger Heidkamp*, A. Düster, E. Rank p-FEM is free from volumetric locking under finite

deformations U. Heisserer*, S. Hartmann, Z. Yosibash, A. Düster, E. Rank

Force transfer for higher-order finite element methods using intersected meshes

S. Kollmannsberger*, A.. Düster, E. Rank Investigation of some steady state wear problems

I. Páczelt, T. Szabó, A. Baksa* Failure criteria in brittle elastic V-notched structures

under mixed mode loading: p-FEM & experiments Elad Priel*, Zohar Yosibash

High-order solid finite elements applied to the

computation of the impact sound level from lightweight floors

A. Rabold*, A. Düster, E. Rank Mixed Finite Element Methods for Linear Elasticity –

Part II Joachim Schöberl, Astrid Sinwel*

Nonlinear computation of foam-like structures based on

anisotropic high-order finite elements H.-G. Sehlhorst A. Düster, E. Rank

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p-FE analysis of the proximal femur Nir Trabelsi*, Zohar Yosibash Nektar++ : An Object-Oriented Spectral/hp Element

Library P.E.J. Vos*, B. Nelson, J. Frazier, R.M. Kirby, S.J. Sherwin

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Keynote Lectures

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany High Order Finite Element Method for Solving Problems with Microstructures

Ivo Babuska

Institute for Computational Engineering and Sciences, The University of Texas at Austin, TX 78712, U.S.A. e-mail: [email protected]

Abstract

The presentation will address the numerical solution of strictly elliptic boundary value problems with rough coefficients

,on / Ron 0 grad )( div- 2

Ω∂=∂∂⊂Ω=

gnuuxa

c

where Ω∂ is regular, )(2 Ω∂∈ Lg and ∫ Ω∂ dsg ,

)(xa is measurable on Ω and Ω∈∞<≤≤≤ xxa ,)(0 21 αα .

The solution 1Hu∈ exists, is unique up to an additive constant and is locally Holder continuous in Ω . This problem is a typical problem with microstructure. The talk will address the following questions 1. What is Finite Element Method, (approximation, stability) for solving the problem with microstructure? 2. How to construct shape functions which approximate well the solution u(x)? 3. What does it mean higher order FEM, especially the p and h-p FEM versions? 4. If the coefficient a(x) is a periodic function, what is the relation to the homogenization approach? 5. What are the implementational difficulties?

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany Fully Automatic hp-Adaptive Finite Elements Where are we going?

Leszek Demkowicz

Institute for Computational Engineering and Sciences, The University of Texas at Austin, TX 78712, U.S.A. e-mail: [email protected]

Abstract

In the early 2000, shortly after Waldek Rachowicz and I put together our first 3D hp code for Maxwell equations, I had a chance to present our work at several Air Force labs. The audience was receptive and open to the idea of high order methods and excited about the exponential convergence, error control and high accuracy simulations. At the same time, though, I clearly saw that the complexity of the code was scary to the engineers and, especially, their managers. I understood that without an automatic hp meshing strategy, the multitude of meshes, one can create with hp-refinements, was more of a nuisance that a clear advantage. After six years of the work on the hp strategy, four 2D versions written by myself and two 3D versions completed by Waldek Rachowicz, and Jason Kurtz, we feel very confident to claim that our hp algorithm is very robust (stable) and it works for the class of problems for which the hp elements provide a stable discretization. The claim has been documented with multiple of 2D and 3D examples reported in our two volumes monograph [1, 2], for both elliptic and Maxwell problems. The goal-driven hp-adaptivity has been proved to be a invaluable tool for problems with high contrast and high accuracy requirements. Equally important, the underlying theory of Projection Based Interpolation for spaces H1

, H(curl) and H(div), forming the exact sequence,

is now complete and free of unproved conjectures annoying for mathematicians.

So where do we go now ? My presentation will focus on two aspects of our current work on hp methods. In the first part, I will report several challenges that we have faced in three space dimensions: geometry modeling, conditioning and convergence of iterative methods, direct multi-frontal solvers, parallel, large scale implementations. In the second part, I will present new classes of problems that we are trying to explore: coupled multi-physics problems and the full potential equation. The presented work represents a team effort and has been done with many colleagues. Reported computational work was done in collaboration with Jason Kurtz, David Pardo, Maciek Paszynski, Waldek Rachowicz, Andrzej Safjan, and Adam Zdunek. Theory of projection-Based Interpolation was developed with the help of Ivo Babuska, Annalisa Buffa, Martin Costabel, Monique Dauge, Jay Gopalakrishan and Joachim Schoeberl. References [1] L. Demkowicz. Computing with hp Finite Elements. I. One- and Two-Dimensional

Elliptic and Maxwell Problems. Chapman & Hall/CRC Press, Taylor and Francis, 2006.

[2] L. Demkowicz, J. Kurtz, D. Pardo, M. Paszynski, W. Rachowicz, and A. Zdunek. Computing with hp Finite Elements. I. Frontiers: Three-Dimensional Elliptic and Maxwell Problems with Applications. CRC Press, Taylor and Francis, 2007. in preparation.

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany

Isogeometric Higher Order Methods Thomas J.R. Hughes 1 Institute for Computational Engineering and Sciences, The University of Texas at Austin, TX 78712, U.S.A. e-mail: [email protected]

Abstract

Geometry is the foundation of analysis yet modern methods of computational geometry have until recently had very little impact on computational mechanics. The reason may be that the Finite Element Method (FEM), as we know it today, was developed in the 1950’s and 1960’s, before the advent and widespread use of Computer Aided Design (CAD) programs, which occurred in the 1970’s and 1980’s. Many difficulties encountered with FEM emanate from its approximate, polynomial based geometry, such as, for example, mesh generation, mesh refinement, sliding contact, flows about aerodynamic shapes, buckling of thin shells, etc. It would seem that it is time to look at more powerful descriptions of geometry to provide a new basis for computational mechanics. The purpose of this talk is to explore the new generation of computational mechanics procedures based on modern developments in computational geometry. The emphasis will be on the Isogeometric approach in which basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite element h- and p-refinement schemes are presented and a new, more efficient, higher-order concept, k-refinement, is described. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD (Computer Aided Design) description. In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. Extraordinary accuracy is noted for k-refinement in structural vibrations and wave propagation calculations. Surprising robustness is also noted in fluid mechanics problems. It is argued that Isogeometric Analysis is a viable alternative to standard, polynomial-based, finite element analysis and possesses many advantages. In particular, k-refinement seems to offer a unique combination of attributes, that is, robustness and accuracy, not possessed by classical p-methods.

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Goal-oriented combined adaptivity of mathematical modeling and FE-discretization for elastic systems Erwin Stein

IBNM, Gottfried Wilhelm Leibniz Universität Hannover, Appelstr. 9A, 30167 Hannover, Germany e-mail: [email protected]

Abstract

Error-controlled hp-adaptive finite element methods are presented for goal-oriented combined model and discretization error estimates and mesh adaptivity for BVPs of elliptic PDEs for structural problems in non-linear and linear elasticity, like 2D plates as the ‘coarse’ model and the 3D-C1 point continuum. The methodology is aiming at combined verification and validation, i.e. required accuracy of quantities of interest for discretization and mathematical modeling during the loading process of a structure, especially influenced by boundary layers and various other disturbances of strain and stress fields. As a consequence, the ‘coarse’ and the fine model are defined in varying subdomains. A major problem is the necessary prolongation of finite element approximations of the ‘coarse’ 2D model to the corresponding solution space of the ‘fine’ 3D model in order to admit a proper definition of the modeling error. By this, the orthogonality property of the used implicit error estimates (with improved element interface tractions by solving local Neumann problems) gets lost. However, discretization error estimates of the ‘coarse’ model have not to be prolongated to the fine model due to the fact that goal-oriented total error estimates with associated dual problems are used, yielding bilinear variational forms with Betti-Maxwell symmetry property, [1]. Implicit error estimates based on equilibrated residua with improved element interface tractions as well as of averaging type are used. FE meshes of the dual problems can be chosen much coarser than those of the related primal problems in order to improve the computational efficiency without noteworthy loss of accuracy. Examples are presented for crack propagation and for plates in bending with boundary layers. References

[1] Stein, E., Rüter, M., Ohnimus, S.: Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity, CMAME, in print: 32 pages

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany On the role of hierarchic spaces and models in validation Barna Szabó1* and Ricardo Actis2

1 Department of Mechanical and Aerospace Engineering, Washington University, Campus Box 1129, St. Louis, MO 63130, USA 2 Engineering Software Research & Development, Inc., 111 West Port Plaza, Suite 825, St. Louis, MO 63146, USA e-mail: [email protected], [email protected]

Abstract

A fundamental question of computational engineering is whether it is possible to predict the response of some physical system or process to some form of excitation by numerical simulation with sufficiently high degree of reliability to justify basing engineering decisions on the predictions. This problem is receiving a great deal of attention today. It is well understood that numerical simulation involves the formulation of a mathematical model and its numerical solution and that both the model and the approximate solution must be shown to satisfy certain necessary conditions. This means that certain metrics, the selection of which depends on the intended use of the model, must satisfy pre-determined criteria. Validation involves the comparison of the predicted values of one or more functionals that can be observed and measured in a physical experiment with their measured values. More often than not, the data of interest cannot be observed in physical experiments. Therefore, in the majority of the cases, what is observed in a validation experiment and what the mathematical model is called upon to predict are not the same functionals. In other words, the prediction of the data of interest generally involves extrapolation. We denote the exact solution of a mathematical model by uEX and its numerical approximation by uFE. We denote the measured data by Φi

* (i=1,2,…). The errors | Φi* - Φi(uEX) | associated

with the choice of the mathematical model are called errors of idealization. When we compare predictions based on a mathematical model with experimental measurements then we are interested in the errors of idealization. However, since we generally do not know uEX, we compare Φi

* with Φi(uFE). Clearly, | Φi* - Φi(uFE) | cannot be a close approximation to | Φi

* -

Φi(uEX) | unless it can be shown that | Φi(uFE) - Φi(uEX) | is small (not larger than the experimental error associated with the measurement of Φi

*). In other words, validation of a mathematical model is possible only if it can be verified that the errors in the numerical solution are small. The hierarchic structure of finite element spaces associated with the p-version of the finite element method is ideally suited for the purposes of verification, since it is easy to produce converging sequences of functionals and verifying that Φi(uFE) are substantially independent of the polynomial degree of elements. It is also necessary for the computer implementation to support hierarchic sequences of models, allowing investigation of the sensitivities of the data of interest and the data measured in validation experiments to the various assumptions incorporated in the model [1]. Aspects of implementation are discussed and specific examples are presented.

Reference

[1] B. Szabó and R. Actis, On the importance and uses of feedback information in FEA. Applied Numerical Mathematics 52 (2005) 219-234.

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Lectures

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany Mechanical contact for the solution of problems in the aerospace industry Ricardo L Actis1*, Barna A. Szabo2 1 Engineering Software Research & Development, Inc., 111 West Port Plaza, Suite 825, St. Louis, MO, 63146, USA 2 Center for Computational Mechanics, Washington University, Campus Box 1129, St. Louis, MO, 63130, USA e-mail: [email protected], [email protected]

Abstract

The contact implementation addressed here is based on a modified augmented Lagrangian method within the context of the p-version of the FEM. There is a large literature on the application of this method to mechanical contact, see, for example, [1], [2]. One of the challenging aspects of implementation is that in most aerospace applications the geometric description of the contacting solid bodies is available in some computer-aided design (CAD) format. The finite element meshes are usually generated by automatic generators independently for each of the contacting bodies. Therefore the vertices, edges and faces of surface elements generally do not coincide in the contact zone. This complicates efficient and reliable computation of the gap function.

To overcome these difficulties we implemented an approach that does not require mesh conformity. Each side of the declared contact pair is independently connected to grounded Winkler springs. The solution is obtained by an iterative procedure in which the gap (g) between the bodies is computed and corrective tractions are applied where g>0 to cancel the effects of the spring. In the region where g<0 (penetration) the corrective traction is determined from the gap function. Iteration continues until no significant change in the maximum contact pressure is detected in consecutive iterations. In this implementation the stiffness matrices of the contacting bodies are computed only once and the effect of contact is incorporated in the load vectors by updating the tractions in the contact zones.

To account for material nonlinearities, a dual iteration process is required. During the first iterative procedure, the tractions in the contact regions are determined assuming that the material remains linear. Once convergence was achieved, the second iterative procedure is initiated by modifying the stiffness matrices of those elements which are affected by plasticity, while the tractions in the contact regions are kept fixed. At the end of the second iterative procedure, a new contact analysis is performed to account for changes in the stiffness in the contact region. This dual iteration process is repeated until convergence in both the contact and plasticity conditions is realized. Examples of applications in the aerospace industry involving metallic and composite materials will be presented. References [1] J.C. Simo, T.A. Laursen, An augmented Lagrangian treatment of contact problems

involving friction, Computers and Structures, 42 (1992) 97-116. [2] I. Páczelt., B. Szabó, T. Szabó, Solution of elastic contact problems by the p-version of

the finite element method, Computers and Mathematics with Applications, 38 (2000) 49-69.

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A Posteriori Error Estimation for Lowest Order Raviart Thomas Mixed Finite Elements Mark Ainsworth

Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, Scotland e-mail: [email protected]

Abstract

For the lowest order Raviart-Thomas mixed finite element, we derive an a posteriori error estimator that provides actual, guaranteed computable upper bounds on the error in the flux variable regardless of jumps in the material coefficients across interfaces. Moreover, the estimator is efficient in that it provides a local lower bound on the error up to a constant that is independent of the solution and the local mesh-size. The estimator may be evaluated at virtually no additional cost compared to the evaluation of the finite element approximation itself.

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Piecewise Defined Basis Functions for hp-FEM Using Object-oriented Software Concepts M. Baitsch1*, S. Worapittayaporn, D. Hartmann1 1 Department of Civil Engineering, Ruhr-University of Bochum, Universitätsstraße 150, D-44780 Bochum, Germany e-mail: [email protected], [email protected], [email protected]

Abstract

The hp-version of the finite element method is considered to be very efficient due to its potential of unconditional exponential convergence [1]. In an hp-adaptive analysis both the polynomial degree and the mesh are adapted to the actual problem. However, the corresponding mesh patterns for the p-version are highly specific to the geometry of the discretized domain. Therefore, it is difficult to generate and refine such meshes automatically. In the first part of the present paper, high-order basis functions for quadrilaterals are introduced that simplify h-refinement for high-order elements substantially. These basis functions are constructed by the tensor product of piecewise hierarchical one-dimensional basis functions. Using this new type of basis functions, an element edge can have an arbitrary number of intermediate nodes. Thus, mesh refinement can be achieved simply by inserting additional nodes on edges where h-refinement is required. Moreover, the proposed approach represents a natural solution to the problem of managing hanging nodes on non-matching meshes [2]. Inter-element continuity of the approximate solution is ensured purely on the element level without any need for the introduction of dependent degrees of freedom. In the second part of the contribution, an object-oriented finite element software system is presented that makes it easy to incorporate new types of basis functions or geometrical mappings. This is achieved by strictly separating aspects of physical modeling from the finite element approximations using the concept of cells. A cell comprises both the geometry of an element as well as the element basis functions. Because cells encapsulate all information required for numerical integration, an element formulation does not depend on the type of basis functions used. Finally, several application examples illustrate the applicability of our approach to engineering problems. References [1] Ch. Schwab, p- and hp-Finite element methods, Oxford University Press, New York, 1998. [2] Y.S. Cho and S. Im, MLS-based variable-node elements compatible with quadratic interpolation. Part I: Formulation and application for non-matching meshes, International journal for numerical methods in engineering, 65(4):494–516, 2006.

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Asymptotic analysis of shell vibration and related numerical hazards E. Artioli 1, L. Beirão da Veiga 2*, H. Hakula 3, C. Lovadina 4, J. Pitkäranta 3

1 IMATI-CNR, University of Pavia, Via Ferrata 1, Pavia, 27100, Italy 2 Department of Mathematics, University of Milan, Via Saldini 50, Milan, 20133, Italy 3 Department of Mathematics, Helsinki University of Technology, P.O.Box 1100, Helsinki, 02015 TKK, Finland 4 Department of Mathematics, University of Pavia, Via Ferrata 1, Pavia, 27100, Italy email:[email protected],[email protected],[email protected],[email protected], [email protected]

Abstract

In the present talk we address the problem of shell vibration from the asymptotic point of view, and discuss the related numerical difficulties. After a brief review of the classical asymptotic analysis of shells (source problem), we consider the asymptotic study of the first eigenvalue and related eigenmodes. In particular, we show how in general the latter problem is undoubtedly more complex, and that the so called "mixed bending-membrane" cases are often encountered. As a consequence, in the finite element analysis of shell vibrations the locking pathology must in general be dealt with even in shells with a "non-flexural" geometry. In order to obtain a wider understanding, the particular case of a clamped cylindrical shell is analyzed more in deep. Finally, a set of numerical tests is shown; such tests confirm the above theory and underline the effects of the locking phenomenon. References [1] L. Beirão da Veiga, C. Lovadina, An Interpolation Theory approach to Shell Eigenvalue Problems, submitted. [2] L. Beirão da Veiga, H. Hakula, J. Pitkäranta, Asymptotic and numerical analysis for a clamped cylindrical shell, submitted.

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Higher-Order Dual-Mixed Finite Element Models in Elasticity Edgár Bertóti

Department of Mechanics, University of Miskolc, H-3515 Miskolc-Egyetemváros, Hungary e-mail: [email protected]

Abstract

Dual-mixed variational formulations and the related finite element models assuming not a priori symmetric stresses can be grouped into two broad categories: (1) Translational equilibrium, i.e., equation of motion, is enforced in a weak sense; the developments are based on a three-field Hellinger-Reissner-type variational principle in terms of displacements, non-symmetric stresses and rotations [1]. (2) Translational equilibrium is satisfied in the strong sense by introducing first-order stress functions; the starting point of the developments is Fraeijs de Veubeke's two-field variational principle in terms of the non-symmetric first Piola-Kirchhoff stress tensor and the orthogonal rotation tensor [2]. In both cases, rotational equilibrium, i.e., the symmetry of the stress tensor, is enforced in a weak sense using the rotations as Lagrangian multipliers. This paper presents a first-order stress function approach for the development of higher-order (hp-version) finite element models for elasticity problems, using the dual-mixed variational principle of Fraeijs de Veubeke [2]. The continuity of the surface tractions at the element interfaces are a priori satisfied through C0 continuous approximations for the first-order stress functions. The rotations are approximated independently on each element. It will be shown that the dual-mixed approach and the related elements lead to robust, reliable and accurate stress computations for not only higher order p-, but also for low order h-type approximations. The locking-free property of the dual-mixed elements developed is numerically justified for problems possessing different limiting constraint, like the constraint of incompressibility. It is pointed out that the main advantage in using higher-order dual-mixed elements is the much higher rates of convergence, just like in the case of pure displacement elements, as compared to low-order elements. References [1] E. Reissner, A note on variational principles in elasticity, Int. J. Solids Struct., 1

(1965) 93-95. [2] B.M. Fraeijs de Veubeke, A new variational principle for finite elastic

displacements, Int. J. Engng. Sci. 10 (1972) 745-763.

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Fast solvers for hp-FEM using hexahedral elements Sven Beuchler1*

1 Institute of Computational Mathematics, University of Linz, Altenberger Strasse 69, Linz, A-4040, Austria e-mail: [email protected]

Abstract

In this talk, we investigate the hp-FEM discretization of an elliptic boundary value problem in 3D. The corresponding linear system is solved by a preconditioned conjugate gradient method with domain decomposition (DD) preconditioners. The key ingredient of the preconditioner is a polynomial basis which is stable in L_2(-1,1) and H^1(-1,1). In [1], a stable polynomial basis in L_2 and H^1 has been developed for the interior bubbles in [-1,1]. In this presentation, we generalize this result to the case of all polynomials. Using the tensor product structure of hexahedral elements, an efficient DD-preconditioner can easily be derived. Some numerical experiments show the efficiency of the proposed method. This is a joint work with J. Schoeberl (Aachen). References [1] S. Beuchler, R. Schneider, C. Schwab, Multiresolution weighted norm equivalences and applications, Numerische Mathematik, 98 (2004) 67-97.

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany Higher Order Finite and Infinite Elements for the Solution of Helmholtz Problems Jan Biermann 1*, Otto von Estorff 1, Steffen Petersen2 1 Institute for Modelling and Computation, Hamburg University of Technology, Denickestraße 17, Hamburg, D-21073, Germany 2 Novicos GmbH,Kasernenstraße 12, Hamburg, D-21073, Hamburg e-mail: [email protected], [email protected], [email protected]...

Abstract

In recent years the focus of research with respect to acoustic simulations shifted to the range of higher wave numbers [1]. In the mid- and high frequency range, however, the conventional Finite Elements with linear shape functions fail to provide reliable results due to so-called pollution effects [2]. These gave motivation for using higher order shape functions and related p-FEM concepts. The solution of large systems of equations arising from engineering problems often involves the use of iterative solution procedures, mostly by means of Krylov subspace methods. The performance of these methods, however, strongly depends on the spectrum of the resulting system matrices, which is affected by the polynomial approximation function space. The current work shows that finite elements based on Bernstein polynomials yield a quite good performance in combination with commonly employed Krylov solvers. The improved eigen value distribution of the resulting system matrices does not only lead to a reduction of the computation time but also to a higher robustness of the iterative solution process. Comparative numerical examples with elements based on Legendre Polynomials reveal that the elements based on Bernstein polynomials become even more predominant with increasing frequencies. The same reasoning is true for the simulation of unbounded domains with infinite elements. Here, an increased effectiveness and robustness for Astley-Leis infinite elements was achieved by using Jacobi polynomials for the radial approximation [3,4]. Now, the utilization of Bernstein polynomials for the basis of those improved infinite elements even leads to a further enhancement of the performance. Numerical examples show a gain of computation speed of a factor of two compared to infinite elements with standard Lagrange polynomials. References [1] F. Ihlenburg, The medium-frequency range in computational acoustics: practical and numerical aspects, J. Comp. Acoust. 11(2) (2003) 175-193. [2] F. Ihlenburg, I. Babuška, Finite element solution of the Helmholtz equation with high wave number part II: the h-p version of the FEM, SIAM J. Numer. Anal. 34(1) (1997) 315-358. [3] D. Dreyer, O.v. Estorff, Improved conditioning of infinite elements for exterior acoustics, Int. J. Numer. Methods Eng. 58 (2003) 933-953 [4] D. Dreyer, S. Petersen, O.v. Estorff, Effectiveness and robustness of improved infinite elements for exterior acoustics, Comput. Methods Appl. Mech. Eng. 195 (2006) 3591-3607

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Stabilization techniques for inverse problems of nested analysis and design Kai-Uwe Bletzinger*, Matthias Firl

Chair of Structural Analysis, Faculty of Civil Engineering and Geodesy, Technische Universität München, Arcisstr. 21, 80290 München, Germany e-mail: [email protected], [email protected]

Abstract

In shape optimal design there are good reasons to distinguish between design and analysis models although both of them treat structural shape as basic information. Besides procedural arguments regarding the interaction of CAE/CAD and FEM the optimization procedure itself as a mechanically inverse and typically non-convex problem adds a further important aspect not to take geometrical parameters of FEM meshes directly as design parameters: Shape optimization needs careful mesh control to prevent the optimizer to run into artificial, local solutions characterized by heavy mesh distortions and/or shape oscillations which most often are amplified by deficiencies of FEM formulations of all present types. For the time being CAD related techniques are the preferred mean to solve all the mentioned aspects in the context of shape optimal design. On the other hand actual research is concerned with shape optimization methods directly using the FE mesh and related parameters as design model which, as a consequence, raise fundamental mathematical and numerical questions of how to deal with large number of optimization parameters, adjoint sensitivity analysis, mesh stabilization and mesh independence of solutions. As a remedy we apply filter techniques as known from signal processing to decompose shape modification into significant scales independently from the chosen discretization. Optimal shapes are determined by modification of surface at the right position, the right amplitude and wave length. High frequency waves which appear as geometrical noise on the surface are damped out. The updated reference strategy (URS) is a method to determine generalized stable minimal surfaces as a generator of surfaces for membrane structures such as tents. Also, the method can be used as a general technique to stabilize tangential mesh movement during shape optimization or other problems where the geometry of FE meshes must be updated. Most recent work uses an extension of this method for inplane stabilization of free form shape optimal design directly on a FEM model. By this approach it is possible to use all three spatial coordinates at a FEM node as design variable without linking these into the surface normal direction. Therefore we get extra design freedom which is used in the context of free form bead optimization of thin metal sheets as one of several practical applications.

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A Posteriori Error Estimates without Generic Constants Dietrich Braess1*, Joachim Schöberl2 1 Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany 2 Center for Computational Engineering Sciences, RWTH Aachen, Pauwelstr. 19, Aachen, 52047, Germany e-mail: [email protected], [email protected]

Abstract

Reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem. We separate the data oscillation and obtain a cheap equilibration by local problems on patches. The method is also applicable to the hp-method. For a verification that the efficiency can be bounded independently of p, a corresponding a priori estimate for Raviart-Thomas elements of higher order is required.

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany High Order BEM for Contact Problems Alexey Chernov1*, Matthias Maischak2, Ernst P. Stephan3

1 Seminar for Applied Mathematics, ETH Zurich, Rämistrasse 101, Zurich, 8092, Switzerland 2 Department of Mathematical Sciences, Brunel University Uxbridge, Middlesex, UB8 3PH , UK 3 Institute for Applied Mathematics, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany e-mail: [email protected], [email protected], [email protected]

Abstract

A new high order mortar Boundary Element Method is designed and applied for two-body contact problems with Tresca friction in linear elasticity. The method is based on a symmetric primal boundary integral formulation, given by a variational inequality of the second kind involving the Steklov-Poincaré operator. The set of admissible solutions contains unilateral non-penetration constraints. In the framework of BEM only boundary of the solids have to be discretized, while the interior linear elastic behavior is provided by the properties of the corresponding nonlocal boundary integral operators. This is very natural for the contact problems, since here nonlinearity comes only through the contact interface. The contacting bodies are discretized independently, i.e. the meshes do not fit in general on the contact interface. The hp-mortar projection, introduced originally for domain decomposition methods, see e.g. [3], is used to define the discrete unilateral contact conditions in a convenient way. An upper bound for the convergence rate in the energy norm is derived under additional

assumptions on regularity of the exact solution u H 3 2 Î“ . The error between the exact solution of the original variational formulation and the BE solution is decomposed into three parts, approximation error, consistency error in the contact discretization, and consistency error in the discretization of the Steklov-Poincaré operator. These three error sources are estimated

independently. Finally, the convergence rate O h p 1 4is obtained under additional

assumptions on the discretization parameters [1], [2]. The discrete problem is solved by the Dirichlet-to-Neumann algorithm. The original two-body formulation is rewritten as a one-body frictional contact problem and a one-body Neumann problem and is solved with the fix point iterations. The Uzawa algorithm is applied to solve the one-body frictional contact subproblem. References A. Chernov, Nonconforming boundary elements and finite elements for interface and contact

problems with friction – hp-version for mortar, penalty and Nitsche's methods, PhD thesis, Institute of Applied Mathematics, University of Hanover, 2006

A. Chernov, M. Maischak, E. P. Stephan, hp boundary element method for two-body contact problems with friction, submitted in Math. Meth. Appl. Sci.

P. Seshaiyer, M. Suri, Uniform hp convergence results for the mortar finite element method, Math. Comp., 69 (2000) 521-546.

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Polynomial extension operators for spaces )(,1 curlHH and )div(H on a cube

M. Costabel1, M. Dauge1*, L. Demkowicz2 1 IRMAR, Université de Renne 1, 35042 Rennes, France 2 Institute for Computational Engineering and Sciences, The University of Texas at Austin, TX 78712, U.S.A. e-mail: monique.dauge, martin.costabel @univ-rennes1.fr, [email protected]

Abstract

We address the construction of continuous trace lifting operators compatible with the de Rham complex on the reference hexahedral element 3)1,1(−=Ω . We consider three trace

operators:

1. The standard trace 0γ from )(1 ΩH into )(2

1

Ω∂H ,

2. The tangential trace tγ from )( Ωcurl,H into )(2

1

Ω∂−

curl,H ,

3. The normal trace nγ from )( Ωdiv,H into )(2

1

Ω∂−

H .

For each of them we construct a continuous right inverse by separation of variables. More importantly, we consider the same trace operators acting from the polynomial spaces forming the exact sequence corresponding to Nédélec's hexahedron of the first type of degree p .

The core of this analysis is the construction of polynomial trace liftings with operator norms bounded independently of the polynomial degree p . This construction relies on a spectral

decomposition of the trace data using discrete Dirichlet and Neumann eigenvectors on the unit interval, in combination with a result on interpolation between Sobolev norms in spaces of polynomials [1]. Details can be found in [2]. References [1] C. Bernardi, M. Dauge, and Y. Maday. Polynomials in the Sobolev world. Preprint 07-14, IRMAR Rennes, 2007. [2] M. Costabel, M. Dauge, and L. Demkowicz, Polynomial extension operators for

)(,1 curlHH and )div(H - spaces on a cube. Preprint 07-15, IRMAR Rennes, 2007.

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Spectral Elements for the Integral Formulation of Maxwell’s Equations in the Time-Harmonic Domain E. Demaldent1*, G. Cohen2, D. Levadoux1 1 DEMR, ONERA, Le Fort de Palaiseau, Palaiseau, 91120, France 2 Projet POEMS, INRIA, Domaine de Voluceau-B.P. 105 Rocquencourt, Le Chesnay Cedex, 78153, France e-mail: [email protected], [email protected], [email protected]

Abstract

Spectral elements were successfully used in their mixed formulation for solving Maxwell’s equations both in the time and the time-harmonic domain [1]. These formulations are based on the first family of hexahedral or quadrilateral Nédélec’s edge elements in the time-harmonic domain and the second family in the time domain and use Gauss and Gauss-Lobatto quadrature points. The efficiency of these elements is based on their ability of providing space matrices and low storage, even in their high-order formulation. These properties are derived from the orthogonal character of the basis functions induced by their geometrical definition and the discrete scalar product derived from the quadrature rules. The purpose of our talk is to show how these techniques can be applied to integral formulations of Maxwell’s equations in the time-harmonic domains. For these formulations, the main advantage of spectral elements is to provide a cheap way to compute the double integral of the Green’s function by applying adequate quadrature rules. However, this good behaviour is useful for far interactions but one must carefully deal with close interactions. In a first step, we show the positive effect of curved quadrilateral low-order non-spectral elements in terms of accuracy and storage versus classical triangular methods for CFIE approach. Unfortunately, the use of higher-order approximation increases dramatically the assembly time of the integral and make useless high-order approximations. In a second step, we show how the use of quadrature points and quadrature rules induces a substantial gain of assembly time both for far and close interactions. The storage globally increases in a linear way when one uses spectral elements versus a quartic increase for classical quadrilateral elements. In fact, this method combines the advantages of MOM and Nyström’s method. This behaviour can be particularly useful for direct resolution of the problem and could be efficiently adapted to FMM in order to obtain substantial gain of time and accuracy. References [1] G. Cohen, M. Duruflé, Non Spurious Spectral-Like Element Methods for Maxwell's Equations, to appear in J. of Comp. Math., 2007.

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Acoustic scattering computations in littoral setting using p-finite elements and perfectly-match-layers$ Dr. Saikat Dey1*

1 SFA Inc./NRL, 2200 Defense Hwy. Suite 405, Crofton, MD 21114, USA e-mail: [email protected]

Abstract

In this presentation we discuss numerical solution of acoustic scattering from elastic objects near the water-sediment interface in a littoral environment. We model the sediment as another (damped) fluid medium and discretize the pressure fields in the fluid and sediment using p-version finite element approximations. Advantages of using high-order approximations for wave-dependent problems is well established in terms of better control of dispersion error and increased rate of error-convergence. However, the inhomogeneous nature of the exterior fluids requires special consideration in the truncation of the infinite domain as well as the computation of the so-called far-field projection of the scattered pressure. We utilize the perfectly-matched-layer (PML) technique to truncate the infinite exterior and compute the far-field pressure projections based on Green's function suitable for a two-fluid exterior domain. Three-dimensional examples are presented based on scattering from elastic objects in the vicinity of water-sediment interface. $ Research funded by ONR.

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Customized High-Order Enrichment Functions for the Generalized Finite Element Method C. A. Duarte*, Dae-Jin Kim

Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Lab, 205 N. Mathews Av., Urbana, IL 61801, USA e-mail: [email protected], [email protected]

Abstract

The effectiveness of generalized finite element methods (GFEM) [1,2] relies, to a great extent, on the proper selection of enrichment functions. Customized enrichment functions can be used to model local features in a domain, like cracks, boundary layers, inclusions, voids, microstructures, etc., instead of a strongly refined mesh, as required in the finite element method. This has lead to a growing interest on this class of methods by the engineering community and to the solution of problems difficult to handle with other methods. Custom enrichment functions for two-dimensional linear problems are amenable to analytical derivation. However, this is not the case for most non-linear and many three-dimensional linear elasticity problems of practical relevance. In this talk, we present recent advances in a global-local procedure to numerically construct high-order enrichment functions for the generalized finite element method [3]. The procedure involves the solution of local boundary value problems in the neighborhood of local features and the enrichment of the global solution space with local solutions through the partition of unity framework of the GFEM. High-order piece-wise polynomial approximations and graded meshes are used to solve the local problems and build local-global enrichment functions. Our analysis demonstrates that the accuracy of a global problem can be controlled using only a few global-local enrichment functions on a coarse global mesh, regardless of the size of the local problems. Applications of the procedure to three-dimensional fracture mechanics problems are presented. References [1] Babuska and J.M. Melenk. The partition of unity finite element method. International Journal for Numerical Methods in Engineering, 40:727--758, 1997. [2] C.A. Duarte, I. Babuska, and J.T. Oden. Generalized finite element methods for three dimensional structural mechanics problems. In S.N. Atluri and P.E. O'Donoghue, editors, Modeling and Simulation Based Engineering, volume I, pages 53-58. Tech Science Press, October 1998. [3] C.A. Duarte and I. Babuska. A global-local approach for the construction of enrichment functions for the generalized fem and its application to propagating three-dimensional cracks. In V.M.A. Leitao, C.J.S. Alves, and C.A. Duarte, editors, ECCOMAS Thematic Conference on Meshless Methods, Lisbon, Portugal, 11-14 July 2005. 8 pages.

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A discontinuous Galerkin approach for fluid flow simulation based on the discrete Boltzmann equation A. Düster1*, L. Demkowicz2, E. Rank1 1Lehrstuhl für Bauinformatik, Faculty of Civil Engineering and Geodesy, Technische Universität München, 80290 München, Germany 2 Institute for Computational Engineering and Sciences, The University of Texas at Austin, TX 78712, U.S.A. e-mail: [email protected], [email protected], [email protected]

Abstract

A discontinuous Galerkin (dG) method for solving the discrete Boltzmann equation is presented, allowing to compute approximate solutions for two-dimensional fluid flow problems. Based on piecewise discontinuous polynomials on unstructured meshes and an explicit time stepping scheme, the discrete Boltzmann equation is discretized in space and time. Contrary to the Lattice Boltzmann method (LBM), where the discrete Boltzmann equation is discretized using the finite difference method, the proposed dG approach enables to accurately account for curved boundaries by applying the blending-function method and it provides a straight-forward approach for mesh refinement and increase in polynomial degree. Several numerical examples will be presented, including stationary and transient problems with curved boundaries. The results are compared to the exact solution of the Navier-Stokes equations and it is demonstrated that the proposed method allows to obtain the desired, highly efficient exponential convergence rate. References [1] A. Düster, L. Demkowicz, E. Rank. High-order finite elements applied to the discrete Boltzmann equation. International Journal for Numerical Methods in Engineering, 67:1094-1121, 2006. [4] R. Tezaur, C. Farhat, Three-dimensional discontinuous Galerkin elements

with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems, Int. J. Numer. Meth. Engng., 66 (2006) 796-815.

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On the Application of Filters for Discontinuity Capturing with High Order Discontinuous Galerkin Discretizations Ioannis Toulopoulos1, John A. Ekaterinaris2* 1 School of Mathematics, University of Athens, FORTH/IACM,P.O. Box 1385, 71110 Heraklion, Crete, GREECE. 2 School of Mechanical and Aerospace Engineering, University of Patras, FORTH/IACM, P.O. Box 1385, 71110 Heraklion, Crete, GREECE. e-mail: [email protected], [email protected]

Abstract

The discontinuous Galerkin (DG) method [1] gained popularity because it makes possible high order accurate discretizations of compressible flows with a compact stencil. Resolution of strong shocks with the DG discretizations requires application of limiters. However development of limiters for flows with discontinuities that computed with a high order DG method is difficult. A new approach is proposed to overcome difficulties associated with limiters of high-order accurate DG discretizations. High resolution computation of flows containing both smooth but complex flow features and discontinuities is obtained by applying explicit filtering to DG discretizations. Filter operators enforce monotonicity, remove spurious oscillations of the numerical solution, and retain high accuracy at the smooth parts of the flow without excessively smearing discontinuities and increasing computing cost. These filters are based on the artificial compression method (ACM) of Harten. They are an extension of the characteristic base filters applied for finite differences by Yee et al. [2] to the DG finite element context. The efficiency and accuracy of the proposed method is investigated in a series of one- and two-dimensional test problems for the Euler equations of gasdynamics and ideal magnetohydrodynamics (MHD). Numerical solutions of the one-dimensional compressible Euler equations demonstrated that sharp capturing of discontinuities can be achieved with the ACM filters. Applications of ACM filters to the one-dimensional MHD equations were obtained for high order accurate DG discretizations. Shock capturing with the use of ACM filters for two-dimensional problems with triangular meshes demonstrated that high-order DG discretizations remain stable in regions of discontinuities, while at the same time complex flow features at the smooth regions of the flow are well resolved. References [1] B. Cockburn, S. Hou, C.W. Shu, “TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element method for Conservation Laws IV,” Journal of Computational Physics, 54 (1990) 545-581. [2] H.C. Yee, N.D. Sandham, M.J. Djomenhri, “Low Dissipative High Order Shock-Capturing Methods Using Characteristic Based Filters” Journal of Computational Physics, 162 (2000), 33-81.

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Least-squares spectral element methods for compressible flows Marc Gerritsma*1 1 Department of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, Delft, 2629 HS, The Netherlands e-mail: [email protected]

Abstract

The application of higher order methods to hyperbolic, compressible flow problems is usually considered to be difficult. Stabilization terms are required to render a stable scheme. But these additional terms also sacrifice higher order approximation. Within a proper functional setting the least-squares formulation is capable of approximating hyperbolic equations without the need to use user-defined stabilization terms. The method converges exponentially fast away from discontinuities and display optimal (algebraic) convergence in the vicinity of discontinuities. In the talk the appropriate functional setting will be discussed as well as the conditions one needs to impose on the conforming finite-dimensional subspaces, [4]. In addition, the convex least-squares functional which is minimized serves as an excellent error indicator, [1,2,3]. Based on the error indicator, the decay of the Legendre coefficients provide a good way of choosing between mesh refinement or polynomial enrichment, [5].

Supersonic flow, M=1.4, over circular bump, 3x10 elements, Polynomial degree N=6

References [1] A. Galvão, M. Gerritsma, B. De Maerschalck, hp-adaptive least squares spectral element method for hyperbolic partial differential equations, J. Comput. Appl. Math. (2007). [2] M. Berndt, T.A. Manteuffel, S.F. McCormick, Local error estimation and adaptive refinement for first order system least squares (FOSLS), Electron. Trans. Numer. Anal. 6 (1997), 35-43. [3] J.-L. Liu, Exact a posteriori error analysis of the least-squares finite element method, Appl. Math. Comput. 116 (2000) 239-257. [4] B. De Maerschalck, M.I.Gerritsma, Least-squares spectral element method for non-linear hyperbolic differential equations, J. Comput. Appl. Math. (2007). [5] P. Houston, B. Senior, E. Süli, Sobolev regularity estimation for hp-adaptive finite element methods, Report NA-02-02, University of Oxford, UK, 2002.

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Fast and Exact Projected Convolution of hp-Functions W. Hackbusch Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstr. 22, D-04103 Leipzig, Germany e-mail: [email protected]

Abstract

Usually, the fast evaluation of a convolution integral f*g=∫f(y)g(x-y)dy requires that the functions f,g are discretised on an equidistant grid in order to apply the fast Fourier transform. Here we discuss the case that f and g are hp-functions, i.e., they are defined on a locally refined grid with piecewise variable polynomial order. The locally refined grid stems from a uniform grid, where in a recursive way certain subintervals are divided regularly. The exact convolution f*g cannot be represented efficiently. Instead we use another hp-space of the same kind and perform the L2-orthogonal projection of f*g into the given space. It is to be emphasised that the Projection is computed exactly. There are no quadrature or interpolation errors except the unavoidable projection error. Under certain conditions, the overall costs are still O(N*log(N)), where N is the sum of the dimensions of the subspaces containing f, g and the resulting function. References [1] W. Hackbusch, On the efficient evaluation of coalescence integrals in population

balance models. Computing 78 (2006), 145-159. [2] W. Hackbusch, Fast and exact projected convolution for non-equidistant grids. Extended

version. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Preprint 102/2006, Leipzig 2006.

[3] W. Hackbusch, Fast and exact projected convolution of piecewise linear functions on non-equidistant grids. Extended version. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Preprint 110/2006, Leipzig 2006.

[4] W. Hackbusch, Approximation of coalescence integrals in population balance models with local mass conservation. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Preprint 111/2006, Leipzig 2006.

[5] D. Potts, Schnelle Fourier-Transformationen für nichtäquidistante Daten und Anwendungen. Habilitation thesis, Universität zu Lübeck, 2003.

[6] D. Potts, G. Steidl, and A. Nieslony, Fast convolution with radial kernels at nonequispaced knots. Numer. Math. 98 (2004) 329-351.

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High-order DIRK/MLNA method applied to finite strain viscoplasticity1 S. Hartmann1*, K. Quint1 1 Institute of Mechanics, University of Kassel, Moenchebergstr.7, 34125 Kassel, Germany e-mail: [email protected]

Abstract

The treatment of inelastic constitutive equations in geometrically non-linear structural problems using finite elements has a long tradition. Commonly, on element (i.e. Gauss-point) level various stress algorithms are developed in order to compute the internal variables and, accordingly, the stresses. A different approach looks at the global structure of all equations yielding a system of differential-algebraic equations after the spatial discretization within the vertical line method, see [1-2]. In this presentation high-order diagonally implicit Runge-Kutta methods (DIRK) in combination with the Multilevel-Newton algorithm (MLNA), which preserves the structure of current versions of finite element programs, see [1-2] – however, it offers the possibility of time-adaptive computations – are applied to rate-independent and rate-dependent plasticity models (Perzyna-type plasticity) for compression dependent, see [4], and von Mises based yield function concepts (see [5]). The influence of plastic incompressibility (in the latter case), the viscosity parameter of the Perzyna-type models, which regularizes the problem under consideration, as well as the order reduction phenomena are studied in several examples. References [1] P. Ellsiepen, S. Hartmann: Remarks on the interpretation of current non-linear finite element analyses as differential-algebraic equations, International Journal for Numerical Methods in Engineering 51 (2001) 679-707. [2] S. Hartmann: A remark on the application of the Newton-Raphson method in non-linear finite element analysis, Computational Mechanics, 36 (2005) 100-116. [3] W. Bier, S. Hartmann: A finite strain constitutive model for metal powder compaction using a unique and convex single surface yield function, European Journal of Mechanics, Series A/Solids, (2006), in press. [4] S. Hartmann, W. Bier: High-order time integration applied to metal powder plasticity, submitted for publication to International Journal of Plasticity, (2006) [5] C. Tsakmakis, A. Willuweit: A comparative study of kinematic hardening rules at finite deformations, International Journal of Non-Linear Mechanics 39 (2004) 539-554

1 This presentation is based on investigations of the collaborative research center SFB/TR

TRR30, which is kindly supported by the German Research Foundation (DFG).

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The cell-based partition-of-unity method Stefan M. Holzer1*, Carsten Riker1 1,2 Department of “Mathematik und Bauinformatik”, University of the Federal Armed Forces Munich, Werner-Heisenberg-Weg 39, Neubiberg, 85577, Germany e-mail: [email protected], [email protected]

Abstract

We describe a novel discretization technique for elliptic, parabolic, and hyperbolic problems which is a kind of blend between classical Galerkin Finite Element methods and meshfree Galerkin methods. The method starts from a set of arbitrarily scattered nodes or particles; however, the nodal support associated with each of these is derived from a mixed cell complex. The mixed cell complex is a transitional structure between Voronoi and Delaunay tesselations. Our method keeps most of the attractive features of meshless Galerkin methods, such as arbitrary local enrichment of the trial space, while facilitating numerical quadrature and efficient solving, as compared to classical MFG methods. The method is used in conjunction with higher-order local trial functions, as well as local enrichments. Computational studies demonstrate exponential convergence of the error in energy norm both for smooth problems and for problems with non-analytic solution under the condition that the trial function space is adequately enriched by singular expansion terms.

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany Visualization of High-Order Finite Element Methods Robert M. Kirby1* 1 School of Computing, University of Utah, 50 S. Central Campus Dr., Salt Lake City, UT 84112 USA e-mail: [email protected]

Abstract

Visualization of high-order methods is an area which is heavily used but often scientifically marginalized. The aim of this talk is to present new research on visualization techniques which both respect and exploit the mathematical structure of high-order methods.

Over the last forty years, tremendous effort has been exerted in the pursuit of numerical methods which are both flexible and accurate, hence providing sufficient fidelity to be employed in the numerical solution of a large number of models and sufficient quantification of accuracy to allow researchers to focus their attention on model refinement and uncertainty quantification. High-order finite element methods (also known as spectral/hp element methods) using either the continuous Galerkin or discontinuous Galerkin formulation have reached a level of sophistication such that they are now commonly applied to a diverse set of real-life engineering problems in computational solid mechanics, fluid mechanics, acoustics and electromagnetics. Unfortunately, there has been little emphasis by the scientific community and, in particular, the visualization community, on providing visualization algorithms and the corresponding software solutions tailored to high-order methods; in particular, almost no research has been done to develop visualization methods based on the a priori knowledge that the data was produced by a high-order finite element simulation.

Visualization of computed results is often used as a means of understanding and evaluating the numerical approximation of the mathematical model, and it provides a means of ``closing the loop'' – that is, of critically evaluating the computational results for refinement of the model and/or numerics or for interpretation of the physical world. Visualizations of high-order finite element results which do not respect the a priori knowledge of how the data were produced and which do not provide a quantification of the visual error produced undermine the scientific process just described.

In this talk we describe several efforts at creating accurate and efficient visualization methods for high-order methods. We will present the mathematical foundations and algorithms for ray-tracing high-order methods [1] and for particle-based methods for isosurface extraction [2]. References [1] B. Nelson and R.M. Kirby, “Ray-Tracing Polymorphic Multi-Domain Spectral/hp Elements for Isosurface Rendering”, IEEE Transactions on Visualization and Computer Graphics, Vol. 12, Number 1, pages 114-125, 2006. [2] M. Meyer, B. Nelson, R.M. Kirby and R. Whitaker, “Particle Systems for Efficient and Accurate Finite Element Visualization”, IEEE Transactions on Visualization and Computer Graphics, Submitted for Publication, 2006.

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Incorporation of contact for high order FEM in covariant form. A. Konyukhov1*, K. Schweizerhof1*

1 Institute of Mechanics, University of Karlsruhe, Englerstrasse 2, Karlsruhe, 76131, Germany e-mail: [email protected], [email protected]

Abstract

One of the big advantages of high order finite elements along with high accuracy is the possibility to describe the given geometry of surfaces exactly. The high quality for shell structure analysis has been shown e.g. in [1]. In a similar fashion so-called iso-geometrical analysis is developed for arbitrary 3D structures in [2]. The exact geometry naturally leads also to improved results for contact problems, however, some difficulties occur due to the nonlinear nature of contact problems. The covariant description of contact problems [3] allows to describe the contact interaction independently of approximations of surfaces, namely, all necessary contact parameters such as contact tractions, sliding path etc. are given in the surface coordinate system in a covariant form. Also tangent matrices, necessary for the nonlinear iterative solution, are given in the covariant form containing only geometrical parameters of the surfaces. The full incorporation of contact is provided then by combination of the penalty method with the Mortar method leading to, so-called, contact layer elements with arbitrary order of approximation which can be used to cover FE meshes with different order of approximation. This approach is illustrated for anisotropically refined contact layer elements covering a standard linear finite element mesh. A good correlation with the classical Hertz contact problem is found with only a few contact elements. References [1] Rank E., Düster A., Nübel V., Preusch K., Bruhns O.T. High order finite

elements for shells, Comput. Methods Appl. Mech. Engrg., 194 (2005) 2494-2512.

[2] Hughes T.J.R., Cottrell J.A., Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comp. Meth. Appl. Mech. Eng., 194 (2005) 4135-4195.

[3] Konyukhov, A., Schweizerhof, K. Covariant description for frictional contact problems, Comput. Mech., 35 (2005) 190-213.

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an hp-BEM for scattering by convex polygons J. M. Melenk1, S. Langdon2

1 Institute for Analysis and Scientific Computing, Vienna University of Technology Vienna 2 Department of Mathematics, University of Reading, Reading, RG6 6AX, United Kingdom e-mail: [email protected], [email protected]

Abstract

Time harmonic accoustic scattering by a convex polygon is considered. S. Chandler-Wilde and S. Langdon have recently proposed an integral equation based method for the high frequency scattering problem. Using a detailed regularity analysis of the solution, they were able to design an h-version trial space that has approximation properties that depend only logarithmically on the wave number. The key features are a) the ability to identify the leading order (in the wave number) behavior of the solution and b) a precise characterization of the solution behavior near the vertices of the polygon. Since the approximation order is fixed, the achievable convergence rate is algebraic. In this talk, we extend their work to the hp-version of the BEM. It is shown that the solution can be approximated at an exponential rate from the trial space; the problem size required to achieve a given accuracy grows only logarithmically with the wave number. The talk will also address the possibility of setting up the stiffness matrix with work independent of the wave number. References S. Chandler-Wilde, S. Langdon, A Galerkin boundary element method for high frequency

scattering by convex polygons, to appear in SIAM J. Numer. Anal.

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany Solution of 3D contact problems using a B-spline approximation A. Baksa, I. Páczelt* and T. Szabó

Department of Mechanics, University of Miskolc, H-3515 Miskolc-Egyetemvaros, Hungary e-mail: [email protected], [email protected], [email protected]

Abstract

In our investigation of 3D contact problems the bodies are discretized by p-extension finite elements using penalty technique on the base of the principle of virtual power. When the p-version is used the accuracy is typically high enough for the singularities to induce oscillation in the numerical solution. In the contact region both the normal stress and the tangential stresses may have singularities, as well as jumps in their derivatives. The boundary of the contact zone is unknown apriori. Therefore, the locations of the singularities in the normal and tangential stresses and jumps in their derivatives are also unknown. In order to treat such contact problems an adaptive method is required to localize them. Similar singularities may arise in case of elasto-plastic problems [3]. It is assumed that the displacements and deformations are small, the material of the contacting bodies are elastic. In normal direction the Signorini contact conditions are valid. The dry friction is modeled by the Coulomb’s rule. The convergence of the solution of a FEM program is greatly influenced by the alignment of singular points, i.e. whether they are located on the border of the elements or not. If they are on the border the singularity can be represented properly and it is a problem of Category B, otherwise it is a Category C, according to [1]. Generally, in case of contact problems the border of the contact zone is not aligned to the border of the elements. Accurate solution requires that the border of elements to be positioned on the border of the contact zone, which makes a problem of Category B. B-splines give us a sufficient tool to approximate the border of contact zone with high. The border points are searched along some prescribed directions from which the interpolating splines are generated. The formulas for closed B-splines are summarized in [2]. During the solution of the problem finding the mapping of the elements and to rebuild the system of algebraic equations for solving the discretized contact problem are the main steps. The modifications of the contact and adhesion borders are continued until the approximated zones are changing. Effectiveness of the proposed algorithms is demonstrated by numerical examples. In these examples the contact and adhesion domains are single connected. References [1] B. Szabó, I. Babuška, Finite Element Analysis, John Wiley & Sons, New York, 1991. [2] A. Baksa, Numerical investigation of contact problems, PhD dissertation, University of Miskolc, (in Hungarian) 2005. [3] A. Düster, Andreas Niggl, Vera Nübel, Ernst Rank, A Numerical Investigation of High-Order Finite Elements for Problems of Elastoplasticity, J. Sci. Comput., 17(1-4) (2002) 397-404.

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High-order FEM vs classical engineering models for a shell roof J. Pitkäranta1*, I. Babuška2, B. Szabó3

1 Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland 2Institute for Computational Engineering Sciences, The University of Texas at Austin, TX 78712, U.S.A. 3Department of Mechanical Engineering, Washington University, Campus Box 1129, St. Louis, MO 63130, U.S.A. e-mail: [email protected], [email protected], [email protected]

Abstract

We consider a benchmark shell roof problem introduced originally by Girkmann [1]. The roof consists of a spherical shell with a stiffening foot ring, and the goal is to determine the resultant shear force and moment acting at the junction of the shell and the ring when the roof is loaded by its own weight. We study the accuracy of classical engineering models where asymptotic membrane and bending theories are applied to approximate the deformations of the shell and the ring. We present a posteriori error estimates based on the assumed known solution according to the engineering model and compare with exact reference values obtained by p-FEM. The comparison is somewhat surprising. References [1] K. Girkmann, Flächentragwerke, Third edition, Springer, 1954.

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A High-Order Embedded Domain Method E. Rank1*, J. Parvizian1,2, Z. Yang1, A. Düster1 1 Lehrstuhl für Bauinformatik, Faculty of Civil Engineering and Geodesy, Technische Universität München, 80333 München, Germany. 2 also: Isfahan University of Technology, Isfahan, 84156 83111, Iran. e-mail: rank, parvizian, yang, [email protected]

Abstract

A smooth integration of geometric models and numerical simulation has been in the focus of research in computational mechanics for long, as the classical transition from CAD-based geometric models to finite element meshes is, despite all support by sophisticated preprocessors, very often still error prone and time consuming. High-order finite element methods bear some advantages for a closer coupling, as much more complex surface types can be represented by p-elements than by the classical low order approach [1]. Significant progress in the direction of model integration has recently been made with the introduction of the ‘isogeometric analysis’ concept [2], where the discretisation of surfaces and the Ansatz for the shape functions is based on a common concept of a NURBS-description. In this paper we discuss a recently proposed different approach, the Finite Cell Method (FCM) [3], which combines ideas from meshless and embedded domain methods with high-order approximation techniques. The basic idea is an extension of a partial differential equation beyond the physical domain of computation up to the boundaries of an embedding domain, which can easier be meshed. The actual domain is only taken into account using a precise integration technique of ‘cells’ which are cut by the domains’ boundary. If this extension is smooth, the solution can be well approximated by high-order polynomials. The method shows exponential rate of convergence for smooth problems and good accuracy even in the presence of singularities. The formulation in this paper is applied to linear elasticity in two and three dimensions, although the concepts are generally valid. The method seems to be especially promising for domains of very complex shape or very inhomogeneous material distribution, which arise e.g. from CT-scans of human bone. Whereas generation of flawless meshes may take hours, an FCM grid can be generated within seconds. Numerical results are presented and compared to finite element results with respect to accuracy, computational resources and engineering effort for data preparation. References [1] B. Szabo, A. Düster, E. Rank, The p-version of the finite element method. In: Encyclopedia of Computational Mechanics, E. Stein, R. de Borst, T.J.R. Hughes (eds.), John Wiley and Sons, 2004 [2] T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 193 (39-41): 4135-4195, 2005 [3] J. Parvizian, A. Düster, E. Rank, Finite Cell Method: h- and p-extension for embedded domain problems in Solid Mechanics, submitted to: Computational Mechanics, 2006

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany Certified DG-FEM Reduced Basis Methods and Output Bounds for the Harmonic Maxwell’s Equations J.S. Hesthaven1, Y. Maday2, and J. Rodriguez3* 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA 2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris, France 3 Laboratoire Poems, École Nationale Supérieure de Techniques Avancées, 75739 Paris, France e-mail: [email protected]; [email protected]; [email protected]

Abstract

In many applications one is interested in computing a functional of interest depending on the solution of a partial differential equation as a variation of one or several parameters. The large number of evaluations needed in many applications, e.g., control, design, large scale parameter scans, requires the use of numerical methods that are not only accurate, but also very fast. A natural, and widely used, approach for dealing with this is to create a problem specific basis for representing the solution. Under the assumption that the solution of the PDE depends smoothly on the input parameters this is reasonable and can lead to dramatic reductions in the computational effort required to evaluate the output functional at a new value of the parameter. This approach, while often used, leaves many questions open, in particular regarding the accuracy of the reduced basis approximation and techniques for properly constructing the reduced basis. In this presentation we shall discuss how to combine discontinuous Galerkin finite element methods (DG-FEM) with an error certified reduced basis methods to enable the very fast solution of the harmonic Maxwell’s equations under parameter variations. The reduced basis enables a dramatic dimension reduction. By assuming that the parameter variation is affine in both the operator and the output functional (assumptions that can be overcome if needed) this leads to a significant computational gain through an off-line on-line computational strategy. The reduced basis enables a dramatic dimensional reduction with significant computational advantages through an online-offline procedure. An essential element of the procedure is the formulation of an a posteriori error-estimator for the reduced basis approximations. Without this, we cannot determine whether the reduced basis is chosen appropriately, i.e., with enough elements in the basis to represent the solution adequately without overresolving, resulting in an excessive computational cost. In more pragmatic terms, without error bounds one cannot rapidly and rigorously determine if critical design or safety conditions and constraints are satisfied. The rigorous a posteriori error estimator furthermore enables an efficient approach for building the reduced basis. We shall discuss the fundamental issues related to the reduced basis method for problems associated with the harmonic Maxwell’s equations and demonstrate the use of DG-FEM to solve both the primal and dual problem to build the basis and the error estimators. Time permitting we shall discuss specific problems associated with parameter variations through resonances and techniques to overcome these problems as well as the treatment of non-affine output functionals.

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Computation of the band structure of two-dimensional Photonic Crystals with high order Finite Elements Kersten Schmidt1*, Peter Kauf 1 1 Seminar for Applied Mathematics, ETH Zurich, Rämistrasse 101, 8050 Zürich, Switzerland e-mail: [email protected], [email protected]

Abstract

Photonic crystals are refractive objects with a certain periodic structure. One can construct them such that the behaviour of light is different for different wavelengths. This makes applications in photonics like optical mirrors and switches possible. Many properties of electrons in semiconductors, which are well explained by quantum mechanics, translate to photonic crystals. Hence, models for the description of photonic crystals are similar to those in solid state physics. For the prediction of photonic crystal properties one relies on a model of an infinte crystal with ideal periodicity in two or three directions. By the Floquet-Bloch transformation [1] the eigenvalue problem for the propagating frequencies in an infinite domain is reformulated into a set of eigenvalue problems in a periodicity cell, ordered by the quasi-momemtum k. The relation between quasi-momentum and frequency is the well-known band structure. The eigenvalue problems are mostly solved via a plane wave [2] or finite element (FE) approximation [3], where the latter can better resolve the material interfaces. In this talk we will show the application of p-FE and hp-adaptive FE [4] of our software package Concepts [5] for two-dimensional photonic crystals. We refine the mesh geometrically towards corners of material interfaces and increase the polynomial degrees away from them. The numerical examples exhibit exponential convergence of the eigenvalues both for smooth and polygonal interfaces. With the implemented quasi-periodic boundary conditions the system matrices for different k only vary in entries belonging to the boundary, and we can reuse the majority of the entries for all the eigenvalue problems. References [1] Peter Kuchment. Floquet Theory for Partial Differential Equations. Birkhäuser Verlag, Basel, 1993. [2] R.D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, and O.L. Alerhand. Accurate theoretical-analysis of photonic band-gap materials. Phys. Rev. B, 48 (1992) 8434–8437. [3] David C. Dobson. An Efficient Method for Band Structure Calculations in 2D Photonic Crystals. J. Comp. Phys., 149 (1999) 363–376. [4] Christoph Schwab. p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Oxford University Press, New York, 1998. [5] Philipp Frauenfelder and Christian Lage. Concepts – An Object-Oriented Software Package for Partial Differential Equations. Math. Model. Numer. Anal., 36 (2002) 937–951.

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hp-Adaptive discontinuous Galerkin methods for saddle point problems Dominik Schötzau1 1 Department of Mathematics, University of British Columbia, Mathematics Road, Vancouver, BC V6T 1X7, Canada e-mail: [email protected]

Abstract

We propose and analyze hp-adaptive discontinuous Galerkin finite element methods for the discretization of saddle point problems arising in incompressible fluid flow and in nearly incompressible elasticity. The main advantages of these methods in comparison with standard conforming finite element approaches lie in their robustness in transport-dominated regimes, their flexibility in the mesh-design, and their exact satisfaction of the incompressibility constraints. We first present an a-priori error analysis that shows that the methods yield exponential rates of convergence for saddle point problems with piecewise analytic data. We then develop the energy norm a-posteriori error estimation, and derive computable upper and lower bounds on the errors measured in terms of a natural energy norm. The bounds are explicit in the local mesh sizes and approximation orders. A series of numerical experiments illustrate the performance of the proposed estimators within automatic adaptive refinement procedures, and show that the estimators can resolve solution singularities at exponential rates of convergence. References [1] P. Houston, D. Schötzau, T. Wihler, Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems, Math. Models Meth. Appl. Sci., to appear. [2] P. Houston, D. Schötzau, T. Wihler, An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity, Comput. Methods Appl. Mech. Engrg., 195:2006, 3224-3246. [3] D. Schötzau, T. Wihler, Exponential convergence of mixed hp-DGFEM for Stokes flow in polygons, Numer. Math., 96:2003, 339-361.

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany Sparse, stabilized high order tensor FEM for high-dimensional transport-dominated diffusion problems Christoph Schwab1*, Endre Suli2, Radu-Alexandru Todor1 1 SAM, ETH Zurich, Switzerland 2 Oxford University Computing Laboratory, Oxford, UK e-mail: [email protected]

Abstract

Partial differential equations with nonnegative characteristic form arise in numerous mathematical models in science. One source of such equations are Fokker-Planck Equations corresponding to transport-diffusion models. In problems of this kind, the exponential growth of computational complexity as a function of the dimension d of the problem domain, the so-called ``curse of dimension'', is exacerbated by the fact that the problem may be transport-dominated. We develop the numerical analysis of stabilized sparse tensor product finite element methods for such high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations, using piecewise polynomials of degree 1≥p . Our convergence analysis is based on new high-dimensional, hp-type approximation results in sparse tensor-products of hierarchical Finite Element spaces. By tracking the explicit dependence of the constants in our error bounds on the dimension d, the meshwidth h and on the polynomial degree p , we show in the case

of elliptic transport-dominated diffusion problems that for any 1≥p the error

constant exhibits exponential decay as ∞→d . In the general case when the characteristic form of the partial differential equation is non-negative, under a mild condition relating the polynomial degree p to d , the error constant is shown to

grow no faster than )( 2dO . In any case, the sparse stabilized finite element

method exhibits an optimal rate of convergence with respect to the mesh size h , up to a factor that is polylogarithmic in Lh .

Other applications of the approximation results, e.g. for Galerkin approximation of random fields, are briefly mentioned.

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An hp-adaptive multilevel particle-partition of unity method Michael Griebel1, Marc Alexander Schweitzer1*

1 Institute for Numerical Simulation, Universität Bonn, Wegelerstraße 6, D-53115 Bonn, Germany e-mail: [email protected], [email protected]

Abstract

We consider the adaptive multilevel solution of a second order elliptic partial differential equation using the particle-partition of unity method (PUM). We focus on the construction of an automatic hp-refinement procedure and the efficient solution of the arising linear system. The PUM [1,2,5] is a meshfree Galerkin method for the numerical treatment of partial differential equations (PDE). In essence, it is a generalized finite element method (GFEM) which employs piecewise rational shape functions rather than piecewise polynomial functions. The PUM shape functions, however, make up a basis of the discrete function space unlike other GFEM approaches which allows us to construct fast multilevel solvers in a similar fashion as in the finite element method (FEM). An appropriate error estimator/indicator for the PUM was presented in [1,4]. Here, we employ this approach to steer a local hp-refinement procedure, i.e., we allow for the incrementation of the local polynomial degree (p-refinement) as well as the generation of new particles (h-refinement) in a single refinement step. We furthermore allow for the use of singular/discontinuous enrichment functions in the approximation. The arising linear system is solved iteratively using an appropriate multilevel solver [3,5] as the inner solver of a nested iteration loop. The overall computational efficiency of the presented approach is demonstrated considering several model problems in two and three space dimensions. References [1] I. Babuska, J. M.Melenk, The Partition of Unity Method, Int. J. Numer. Meth. Engrg., 40 (1997), pp. 727–758. [2] M. Griebel, M. A. Schweitzer, A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic and Hyperbolic PDE, SIAM J. Sci. Comput., 22 (2000), pp. 853–890. [3] M. Griebel, M. A. Schweitzer, A Particle-Partition of Unity Method-Part III: A Multilevel Solver, SIAM J. Sci. Comput., 24 (2002), pp. 377–409. [4] M. Griebel, M. A. Schweitzer, A Particle-Partition of Unity Method-Part VII: Adaptivity, Meshfree Methods for Partial Differential Equations III (M. Griebel and M. A. Schweitzer, eds.), LNCSE, vol. 57, Springer, 2006, pp. 123–150. [5] M. A. Schweitzer, A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations, vol. 29 of LNCSE, Springer, 2003

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Mixed Finite Element Methods for Linear Elasticity – Part I Joachim Schöberl1, Astrid Sinwel2*

1 Center for Computational Engineering Sciences, RWTH Aachen, Pauwelstr. 19, Aachen, 52047, Germany 2 Radon Institute for Computational and Applied Mathimatics, Austrian Academy of Science, Altenbergerstr. 69, Linz, 4030, Austria e-mail: [email protected], [email protected]

Abstract

We introduce a new finite element method of arbitrary order to approximate the Hellinger-Reissner mixed formulation of elasticity. For the displacements, we use tangential-continuous vector-valued Nedelec elements. To approximate the stresses, we propose symmetric-tensor-valued finite elements. These elements are normal-normal continuous. This formulation is suitable for nearly incompressible materials, where the Poisson ratio tends to ½. Also, the elements do not suffer from shear locking when anisotropic elements are used. We present shape functions of arbitrary order for both the displacement and stress space. We see that the solution satisfies optimal order error estimates. We discuss the implementation of the new elements, and give numerical results.

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Multi-Mesh hp-FEM and Selected Applications Pavel Solin1,2*, Ivo Dolezel1, Jakub Cerveny1,2, Lenka Dubcova1,2 1 Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejskova 5, Praha 8, CZ 18200, Czech Republic 2 Department of Mathematics, University of Texas at El Paso, 500 West University Avenue, El Paso, TX 79968, U.S.A. e-mail: [email protected], [email protected], [email protected], [email protected]

Abstract

Most finite element computations rely on a single mesh to discretize a PDE or a PDE system. This traditional approach may become cumbersome in more complex problems where various solution components exhibit qualitatively different behaviors. Consider, for example, a fluid-thermal- electromagnetic coupled problem where the flow velocity has boundary layers near solid walls, electric field strength has singularities at reentrant corners or edges, and the temperature and pressure fields are very smooth everywhere in the domain. In such case, a single-mesh approach would force us to treat all fields as if they had singularities and boundary layers. In this presentation we introduce a new multi-mesh hp-FEM approach where every solution component can be discretized on an independent mesh and equipped with a separate automatic hp-adaptive algorithm. We explain basic ideas of the multi-mesh assembling algorithm and show that it may save a lot of degrees of freedom and CPU-time compared to single-mesh hp-FEM. Time-permitting, we will mention some other advanced applications of the multi-mesh hp-FEM such as, e.g., multi-goal-oriented adaptivity, or a new approach to space-time adaptivity based on a multi-mesh Rothe’s method. References [1] P. Karban, I. Dolezel, P. Solin, Computation of General Nonstationary 2D Eddy Currents in Linear Moving Arrangements Using an Integro-Differential Approach, COMPEL 25, No. 3, 2006, pp. 635 - 641. [2] P. Solin, J. Cerveny, I. Dolezel, Arbitrary-Level Hanging Nodes and Automatic Adaptivity in the hp-FEM, Math. Comp. Sim., accepted, November 2006. [3] P. Solin, L. Demkowicz: Goal-Oriented hp-Adaptivity for Elliptic Problems, Comput. Methods Appl. Mech. Engrg. 193 (2004), 449 - 468. [4] P. Solin, L. Dubcova, Higher-Order Finite Elements for Maxwell’s equations Based on Generalized Eigenfunctions of the Curl-Curl Operator, submitted, January 2007. [5] P. Solin, K. Segeth, I. Dolezel, Higher-Order Finite Element Methods, Chapman & Hall/CRC Press, Boca Raton, 2003. [6] P. Solin, T. Vejchodsky, Higher-Order Finite Elements Based on Generalized Eigenfunctions of the Laplacian, submitted, August 2006.

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The Discontinuous Enrichment Method for Multiscale Wave Propagation, Flow, and Transport Problems Charbel Farhat1, Radek Tezaur1*

Department of Mechanical Engineering and Institute for Computational and Mathematical Engineering, Stanford University, Building 500, Room 501G, 488 Escondido Mall, Mailcode 3035, Stanford, CA 94305, USA

e-mail: [email protected], [email protected]

Abstract

Wave propagation problems pertain to many technologies including sonar, radar, geophysical exploration, medical imaging, nondestructive testing, and structural design. As technology expands in the nano-world, evanescent waves are playing a major role in the design various optical, chemical, and biological sensors. Advection-diffusion arises in many important flow and transport problems such as contaminant transport through aquifers in hydromechanics, heat and mass transport in chemical engineering, and filtration processes and transport of drugs in biomedical engineering. Two important attributes that are shared by all the aforementioned problems and applications are: (a) their multiscale nature, and (b) the computational complexity required to solve them numerically by standard approximation methods. Indeed, the analysis by the standard finite element method of wave propagation problems in the medium frequency regime is either computationally unfeasible or simply unreliable, particularly in the presence of evanescent waves. Similarly, high Reynolds number flows are well beyond the current reach of standard approximation methods and numerical considerations continue to limit the size of the Peclet number one can simulate using the standard finite element method. The higher-order (or p-type) finite element method alleviates some of these problems, but only to some extent. Alternative approximation methods based on the idea of using special, problem dependent approximation functions have recently emerged to to address these issues [1-2]. The Discontinuous Enrichment Method (DEM) is such an alternative [3-4]. It distinguishes itself from other enrichment methods by its ability to evaluate the important system matrices analytically, thereby bypassing the typical accuracy and cost issues associated with high-order quadrature rules. DEM also provides a unique

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multiscale approach to computation by employing fine scales that contain solutions of the underlying homogeneous partial differential equation in a discontinuous framework. The theoretical and computational underpinnings of this method will be overviewed and specialized to wave propagation, flow, and transport problems. Then, recent applications to underwater acoustic scattering problems in the medium frequency regime, geo-acoustic scattering at interfaces between fluid and solid media, and various high Peclet number advection diffusion problems will be discussed. One to two orders of magnitude accuracy and/or CPU time improvement over the standard higher-order finite element method will be demonstrated in three dimensions. References [1] C. Daux, N. Moes, J. Dolbow, N. Sukumar, T. Belytschko, Arbitrary branched and

intersecting cracks with the extended finite element method, Int. J. Numer. Meth. Engng., 48 (2000) 1741-1760.

[2] T. Strouboulis, L. Zhang, I. Babuska, p-version of the Generalized FEM using mesh-based handbooks with applications to multiscale problems, Int. J. Numer. Meth. Engng., 60 (2004) 1639-1672.

[3] C. Farhat, I. Harari, L.P. Franca, The discontinuous enrichment method, Comput. Methods Appl. Mech. Engrg., 190 (2001) 6455-6479.

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A Review of Discrete Maximum Principles for Higher-Order Finite Elements Tomáš Vejchodský1*, Pavel Šolín2

1 Institute of Mathematics, Academy of Sciences, Zitna 25, CZ-11567, Prague 1, Czech Republic 2 Department of Mathematical Sciences, University of Texas at El Paso, 500 W. University Ave., El Paso, TX 79968, USA e-mail: [email protected], [email protected]

Abstract

The discrete maximum principles (DMP) guarantee certain qualitative properties of approximate solutions. A typical example is the nonnegativity of temperature, concentration, density, and other naturally nonnegative quantities. It is well known that the finite element approximations do not satisfy the DMP in general. Thus, for example, the finite element solution of the heat conduction problem can result in negative temperatures in some regions. Conditions that guarantee the DMP for various problems and settings are studied since 1970s. The known results deal almost exclusively with the lowest-order (linear) approximations. Historically, an important paper [1] states that a stronger version of the DMP is not valid for higher-order finite elements. This result has probably discouraged the further research. However, the resent results of the authors [2-4] show that it is possible to find simple conditions on the mesh which guarantee the (classical) DMP even for higher-order elements. This contribution surveys the recent results in the research of DMP for hp-version of the finite element method (hp-FEM). The results are nontrivial and the known techniques for the lowest-order elements cannot be straightforwardly generalized. We developed a new technique of proofs based on the discrete Green's function. We show the conditions for DMP for the one-dimensional Poisson problem with various boundary conditions and we mention a few generalizations. References W. Höhn, H.D. Mittelmann, Some remarks on the discrete maximum principle for finite

elements of higher-order, Computing 27 (1981) 145-154. T. Vejchodský, P.Šolín, Discrete maximum principle for higher-order finite elements in 1D,

Math. Comp., 2006 (accepted). T. Vejchodský, P. Šolín, Discrete maximum principle for Poisson equation with mixed

boundary conditions solved by hp-FEM, Research Report No. 2006-09, Department of Math. Sciences, University of Texas at El Paso, 2006.

T. Vejchodský, P.Šolín, Discrete maximum principle for a problem with piecewise-constant coefficients solved by hp-FEM, Research Report No. 2006-10, Department of Math. Sciences, University of Texas at El Paso, 2006.

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Improving the CFL Condition for Discontinuous Galerkin Methods T. Warburton1*, T. Hagstrom2 1 Department of Computational and Applied Mathematics, Rice University, 6100 Main Street, MS-134, Houston, Texas, 77005, USA. 2 Department of Mathematics and Statistics, MSC03 2150, 1 University of New Mexico, Albuquerque, New Mexico, 871310-0001, USA. e-mail: [email protected], [email protected]

Abstract

The upwind discontinuous Galerkin method is an attractive method for solving time-dependent hyperbolic conservation laws. It is possible to use high-order explicit time-stepping methods and high-order spatial approximations without incurring heavy numerical linear algebra overheads. However, the Courant-Friedrichs-Lewy (CFL) condition for these methods depends on the polynomial order used and there is a somewhat excessive cost for using very high order spatial approximation. We will discuss the impact of a staggered grid based filter on the CFL number for these methods and present an algorithm which has CFL number independent of the spatial order of approximation. We will also present computational results for advection, advection-diffusion, and the wave equation on one- and two-dimensional meshes using up to tenth order in space and time.

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An hp finite element method for singularly perturbed systems of reaction-diffusion equations Christos Xenophontos1*, Lisa Oberbroeckling2 1 Department of Mathematics and Statistics, University of Cyprus, P.O. BOX 20537, Nicosia, 1678, Cyprus 2 Department of Mathematical Sciences ,Loyola College, 4501 N. Charles Street, Baltimore, MD 21210, U.S.A. e-mail: [email protected], [email protected]

Abstract

We consider the approximation of a coupled system of two singularly perturbed linear reaction-diffusion equations, with the finite element method. The solution to such problems contains boundary layers which overlap and interact in a manner described in Ref. [1], and the numerical approximation must take this into account in order for the resulting scheme to converge uniformly with respect to the singular perturbation parameters. We propose and analyze an hp finite element method which contains thin elements of size O(pε) and O(pµ) near the boundary, with ε and µ the singular perturbation parameters and p the degree of the approximating polynomials. We show that, under the assumption of analytic input data, the method converges at an exponential rate, as p → ∞, independently of the singular perturbation parameters. Numerical results which validate the analysis and compare the proposed method with other schemes found in the literature will also be presented [2]. References [1] N. Madden, M. Stynes, A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems, IMA J. Numer. Anal. 23 (2003) 627 – 644. [2] C. Xenophontos, L. Oberbroeckling, A numerical study on the finite element solution of singularly perturbed systems of reaction-diffusion problems, in press in Appl. Math. Comp. (2006).

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p-FEMs for a class of finite deformation pressure dependent plasticity models validated by experimental observations2 W. Bier1, M. Dariel2, A. Düster3, N. Frage2, S. Hartmann1, U. Heisserer3, S. Holzer4, E. Rank3, M. Szanto5, Z. Yosibash5* 1Institute of Mechanics, University of Kassel, Mönchebergstr.7, Kassel, 34109, Germany 2Dept. of Material Engrg, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel 3Lehrstuhl für Bauinformatik, Technische Universität München, Arcisstr. 21, München, 80290, Germany 4Dept. of “Mathematik und Bauinformatik”, Federal Armed Forces Univ, Neubiberg, 85577, Germany 5Dept. of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel

E-mail: 1bier/[email protected], 2dariel/[email protected], 3duester/heisserer/[email protected], [email protected], 5mszan/zohary @bgu.ac.il

Abstract

The p-version of the finite element method (p-FEM) performs very well for linear elastic problems. Recently highly non-linear problems, as cold isostatic pressing (CIP) of metal powders (finite deformations and complex constitutive models) have became of interest. These problems require enhancement of p-FEMs to geometrically and physically non-linear capabilities, where good performance is expected, and results should be validated by experimental observations. This task is being performed by designing a set of experimental techniques for a) adoption of a constitutive model that describes qualitatively the experimental observation and b) identification of the material parameters. This talk presents the results of a 3-years project by five different groups in

§ Research Supported by the German Israeli Foundation for Sci. Research and Development, under grant number I-700-26.10/2001

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developing a p-FEM simulation tool for CIP processes validated by experimental observation. The p-FEM academic code AdhoC has been extended to incorporate geometrical and physical non-linear constitutive models of evolutionary type. This includes stress computation and consistent linearization of the inelastic model. On the basis of experimental observations (for particular powder metals) a finite strain viscoplasticity model was developed [1]. This constitutive model was incorporated both in AdhoC (implicit) and in the commercial h-FE code HKS/Abaqus/Explicit. The material parameters for the specific constitutive model were identified on the basis of the designed experimental observations. Experiments on relatively complex geometries were then performed in parallel to p- and h-FEM simulations, and results are compared for validation purposes. The excellent results obtained by p-FEM simulations in comparison to experiments and h-FEMs will be presented [2-4]. References

[1] W. Bier, M.P. Dariel, N. Frage, S. Hartmann, and O. Michailov. Die compaction of copper powder designed for material parameter identification. International Journal of Mechanical Science, In press.

[2] Z. Yosibash, S. Hartmann, U. Heisserer, A. Duester, and E. Rank. Axisymmetric pressure boundary loading for finite deformation analysis using p-fem. Computer Methods in Applied Mechanics and Engineering, 196 (2007) 1261 –1277

[3] U. Heisserer, S. Hartmann, W. Bier, A. Duester, Z. Yosibash, & E. Rank. p-FEM for finite deformation powder compaction. Submitted.

[4] M. Szanto, W. Bier, N. Frage, S. Hartmann, and Z. Yosibash. Experimental based finite element simulation of cold isostatic pressing of metal powders. International Journal of Mechanical Science, Submitted.

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High Order Finite Elements for vector-valued Function Spaces S. Zaglmayr 1*, J. Schöberl2

1 Radon Institute for Computational and Applied Mathematics (RICAM), Altenbergerstr. 69, Linz, 4040, Austria 2 Department for Mathematics CCES, RWTH Aachen, Pauwelstrasse 19, Aachen, 52074, Germany e-mail: [email protected], joachim.schö[email protected]

Abstract

The vector-valued function spaces H(curl) and H(div) occur in many practical problems, e.g. naturally in electromagnetics and fluid mechanics, as well as in various mixed formulations of elasticity. In this talk, we present a general, unified construction principle for H(curl)- and H(div)-conforming finite elements of variable and arbitrary order. In order to allow for geometric h-refinement, we consider unstructured hybrid meshes involving hexahedral, tetrahedral, prismatic and pyramidal elements. The keypoint of our framework is to explicitly respect the exact de Rham sequence already in the construction of the FE basis functions. A short outline of the construction is as follows. Concerning the H(curl)-conforming basis, we start with the classical lowest-order shape functions. Then we take the gradients of edge-based, face-based and cell-based shape functions of the higher-order H^1-conforming FE-space. Finally, we hierarchically extend these sets of functions to a conforming basis of the desired polynomial space. The same principle works for the H(div)-space. Here, we start with the lowest-order Raviart-Thomas shape functions and explicitly use the curl-fields of the high-order face-based and irrotational cell-based shape functions of the H(curl)-conforming FE-space. In the end we add cell-based non-div-free basis functions to span the desired polynomial space. By our separate treatment of the edge-based, face-based, and cell-based functions, and by explicitly including the corresponding natural differential fields we can establish the local exact sequence property: the subspaces corresponding to a single edge, a single face or a single cell already form an exact sequence. A main advantage is that we can choose an arbitrary polynomial order on each edge, face, and cell independently, without destroying the global exact sequence property. Further on, simple block ASM-preconditioning of curl-curl- and div-div-problems get efficient. Moreover, the implementation of the natural discrete differential-operators requires no additional computational costs.

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Progress using Higher Order Finite Elements for Electromagnetic Scattering A. Zdunek1*, W. Rachowicz2

1 Computational Physics, Systems Technology, Swedish Defence Research Agency, FOI, SE-164 90 Stockholm, Sweden 2 Institute for Computer Modelling, Cracow University of Technology, ul. Warszawska 24, Cracow, Poland e-mail: [email protected], [email protected]

Abstract

One of the most challenging problems in computational applied electromagnetics is the scattering characterisation of open-ended aircraft air-inlets and exhaust outlets. It is known that the hp-adaptive finite element method with error control offers optimal convergence rates. It is attractive, since a requested accuracy can be obtained spending the degrees of freedom in an optimal fashion. For the electrically large three dimensional scattering problems we consider the wave resolution requirement implies that afordable meshes rarely reach into the exponential convergence range. This contribution presents recent progress made trying to alleviate the computational burden for electrically large scattering problems by using reduced models without compromising the inherent reliabily of the hp-FEM. For open-ended cavities it is the interior irradiation contribution to the scattering cross section (RCS) that is the prime quantity of interest. The first progress we report on concerns a multi-level domain decomposition eigen-space based method for efficient solution of time-harmonic frequency response problems. We present the non-trivial generalisation of the so-called Automatic Multi-Level Substructuring method (AMLS) in Acoustics and Structural Dynamics [1] to electromagnetics. Half space based models relying on an infinite perfectly conducting ground plane are often used in finite element approaches for determination of RCS for open-ended cavities [2]. A convergence study using p-type extensions provided converged solutions that could not be validated, indicating a modelling error. The second progress we report on concerns verification and validation of a new, full space based, efficient domain truncation method using higher order finite elements for the determination of RCS for open-ended cavities. References [1] J. K. Bennighof, R. B. Lehouc, An Automated multilevel Substructuring Method for eigenspace computation in linear elastodynamics, SIAM, J. Sci. Comp., 25, (2004), 2048-2106. [2] Jianming Jin, Electromagnetic Scattering from Large, Deep, and Arbitrarily Shaped Open Cavities, Electromagnetics, 18, (1998), 3-34.

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Posters

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Stochastic Galerkin method with sparse polynomial chaos expansion M. Bieri1, Ch. Schwab2

1 Seminar for applied mathematics, ETH Zürich, [email protected], Switzerland 2 Seminar for applied mathematics, ETH Zürich, [email protected], Switzerland

Abstract

We consider a deterministic finite element (FE) solution algorithm for solving stochastic elliptic boundary value problems. A sparation of deterministic and stochastic variables is achieved via a Karhunen-Loève expansion, which leads to a high dimensional problem in the stochastic part. We propose a sparse polynomial chaos (SPC) expansion to handle this part whereas for the deterministic part a generic FE-solver can be used. This approach allows us to easily handle KL-expansions up to a truncation order 80. Numerical examples will be given to show results and convergence of the proposed algorithm. References R.A. Todor and Ch. Schwab, Convergence rates for sparse chaos appoximaions of elliptic

prolems with stochastic coefficients, Technical Report 2006-05, Seminar for applied mathematics, ETH Zurich, 2006.

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A novel High-Order Finite Element Approach to Modeling Localization and Failure Holger Heidkamp1*, A. Düster2, E. Rank2 1 SOFiSTiK AG, Bruckmannring 38, 85764 Oberschleißheim, Germany 2 Lehrstuhl für Bauinformatik, Faculty of Civil Engineering and Geodesy, Technische Universität München, Arcisstr. 21, 80333 München, Germany e-mail: [email protected], duester, [email protected]

Abstract

This paper presents a consistent modeling approach to the phenomenon of deformation localization and material failure, based on high-order finite elements. With particular focus on large scale analyses, it adopts a macroscopic perception of the failure process. Viewing the typically small failure zones from the level of practical interest, continuous fields with a steep gradient appear discontinuous; this constitutes the notion of strong discontinuities, i.e., jumps in the displacement field. Accounting for the possible occurrence of strong discontinuities, the pathological mesh sensitivity exhibited by classical continuum softening approaches is overcome. Discontinuities are incorporated into the finite element formulation in an embedded manner, avoiding the need of additional global degrees of freedom and thus, giving rise to an efficient discretization and numerical treatment of the problem. As opposed to previous concepts, the presented approach is consistently deduced regarding its possible application in the context of high-order finite elements. Put forth by a novel reassessment of the strong discontinuity kinematics, the extended p-adaptive formulation is established. Three dimensional numerical investigations show, that in contrast to commonly adopted low-order finite element approximations the proposed p-adaptive high-order approach facilitates a minimization of potential locking effects while at the same time the algorithmic implementation efficiently preserves a high degree of locality. References [1] J.C. Simo, J. Oliver, F. Armero, An analysis of strong discontinuities induced by strain softening in rate-independent inelastic solids, Computational Mechanics, 12 (1993) 277-269. [2] J. Oliver, Modelling strong discontinuities in solid mechanics via strain softening constitutive equations part 1: Fundamentals. part 2: Numerical simulations. Int. J. for Num. Meth. in Engrg., 39 (1996) 3575-3623. [3] J. Mosler, G. Meschke, 3D modeling of strong discontinuities in elastoplastic solids: Fixed and rotating localization formulations, Int. J. for Num. Meth. in Engrg., 57 (2003) 1553-1576. [4] H. Heidkamp, Modeling Localization and Failure with High-Order Finite Elements, PhD thesis, Lehrstuhl für Bauinformatik, Technische Universität München, 2007 (to be published).

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p-fem is free from volumetric locking under finite deformations§ U. Heisserer1*, S. Hartmann2, Z. Yosibash3, A. Düster1 , E. Rank1 1 Lehrstuhl für Bauinformatik, Technische Universität München, Arcisstrasse 21, München, 80290, Germany 2 Institute of Mechanics, University of Kassel, Mönchebergstr.7, Kassel, 34109, Germany 3 Pearlstone Center for Aeronautical Studies, Dept. of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel 1heisserer, duester, [email protected], [email protected], [email protected]

Abstract

We demonstrate the locking-free properties of the displacement-formulation of p-finite elements when applied to nearly incompressible Neo-Hookean material under finite deformations. For an axisymmetric model problem we provide semi-analytical solutions for a nearly incompressible Neo-Hookean material exploited to investigate the robustness of p-FEM with respect to volumetric locking. An analytical solution for the incompressible case is also derived to demonstrate the convergence of the compressible numerical solution towards the incompressible case when the compression modulus is increased. To support this analysis the formulation of displacement-dependent loading (follower loads) in the framework of the p-version is presented and applied to some examples in the plain strain and axisymmetric case. References [1] Heisserer, U.; Hartmann, S.; Yosibash, Z.; Düster, A.: On volumetric locking-free behavior of p-version finite elements under finite deformations. Communications in Numerical Methods in Engineering. accepted for publication, 2007. [2] Yosibash, Z.; Hartmann, S.; Heisserer, U.; Düster, A.; Rank, E.: Axisymmetric pressure boundary loading for finite deformation analysis using p-fem. Computer Methods in Applied Mechanics and Engineering, 196 (7), 1261 - 1277, 2007. [3] Heisserer, U.; Düster, A.; Rank, E.: Follower loads for high order finite elements. Proceedings in Applied Mathematics and Mechanics, 5 (1). 405-406, 2005

§ Research Supported by the German Israeli Foundation for Sci. Research and Development, under grant number I-700-26.10/2001

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Force transfer for higher-order finite element methods using intersected meshes S. Kollmannsberger1*, A. Düster1, E. Rank1 1 Lehrstuhl für Bauinformatik, Faculty of Civil Engineering and Geodesy, Technische Universität München, Arcisstr. 21, 80333 München, Germany e-mail: kollmannsberger, duester, [email protected]

Abstract

Higher-order Finite Element Methods have been shown to be an efficient approach for computing the behavior of fluids and structures alike. However the coupling of such methods in a framework for a partitioned fluid-structure-interaction is still in its early stages. A difficulty hereby is a conservative transfer of the loads from the fluid to the solid and an appropriate transfer of the structural displacements back onto the boundary of the fluid. This contribution focuses on the force transfer. For this purpose, the fluid mesh and the structural mesh are intersected. The force acting on the solid is then computed by a composed integration scheme performed on the intersected mesh. The approach can be interpreted as a L2-projection. This projection method automatically takes into account the discretization on both sides, i.e. fluid and solid. Within the proposed framework an explicit solution of a system of equations for the load transfer does not have to be performed. Two numerical examples will demonstrate the accuracy of this approach. One example will treat the coupling of a high order structural solver with a commercial finite volume method based solver. A second example will demonstrate a fluid structure interaction where the geometrically nonlinear structure discretized by high order FEM interacts bidirectionally with a fluid domain computed by means of the Lattice Boltzmann [3] method. The results will be compared with a recently established FSI-Benchmark. References [1] D. Scholz, S. Kollmannsberger, A. Düster, R. Rank, Thin Solids for Fluid-Structure Interaction, Lecture Notes in Computational Science and Engineering 53, pages 294-335, Springer 2006. [2] S. Kollmannsberger, A. Düster, R. Rank, Force Transfer for high order finite element methods using intersected meshes, in Proceedings of “Pressures Vessels and Pipes” Conference , San Antonio, Texas, 2007. [3] S. Geller, J. Tölke und M. Krafczyk, Fluid-Structure Interaction, Springer Lecture Notes in Computational Science and Engineering 53, pages 270-294, Springer 2006.

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Investigation of some steady state wear problems I. Páczelt, T. Szabó, A. Baksa

Department of Mechanics, University of Miskolc, H-3515 Miskolc-Egyetemvaros, Hungary e-mail: [email protected], [email protected], [email protected]

Abstract

In many practical industrial applications it is very important to predict the form of wear shape and contact stresses. There are two ways to analyze the process of wear. The first one is to treat the problem as a transient process and the second one is a steady state one which is analyzed by variational principles. In the later case the stationary of the functional gives the pressure distribution and the rate of the rigid body movement. In the work of Páczelt and Mróz [1, 2] optimal contact shape is studied by formulating several classes of shape optimization problems, namely minimization of generalized wear volume rate, friction dissipation power and wear dissipation rate. It is shown that the contact shape evolution tends to a steady state, satisfying the minimum principle of the wear dissipation rate. The specific modified Archard (isotropic) wear rule is assumed. It is assumed that the displacements and deformations are small, the material of the contacting bodies are elastic. In the normal direction the Signorini contact conditions are valid. The Coulomb dry friction models are investigated. The temperature effects and heat generated at the frictional interface in our investigation is considered. The discretization of the contacting bodies is performed by the displacement based p-extension finite elements assuring fast convergence of the numerical process and accurate specification of geometry for shape optimization. The contact conditions are checked at the Lobatto integration points of the contact elements during the solution process. In this presentation, we will considering four cases. Firstly, the wear of drum braking, secondly, the wear of a rotating tubular punch disk brake, thirdly, 3D bodies wear problem, and fourthly a plane strain punch problem is analyzed. Both regular and singular optimal solutions are discussed and the physical relevance of minimum principle of the wear dissipation power is emphasized. The coupled thermo-contact-wear problems are solved by operator split technique. References [1] I. Páczelt and Z. Mróz, On optimal contact shapes generated by wear, Int. J. Numer. Meth. Engng., 63 (2005) 1310-1347. [2] I. Páczelt and Z. Mróz, Optimal shapes of contact interfaces due to sliding wear in the steady relative motion, Int. Journal of Solids and Structures, 44(2007) 895-925.

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany Failure criteria in brittle elastic V-notched structures under mixed mode loading: p-FEM & experiments Elad Priel* and Zohar Yosibash

Department of Mechanical Engineering, Ben-Gurion university of the Negev, Beer-Sheva, 84105, Israel e-mail: prielel/[email protected]

Abstract

Brittle elastic components containing V-notches fail at significantly lower loads than the material strength would suggest. In the past 10 years, failure criteria for such components under mode I loading have been proposed and validated by experiments. For a more realistic mixed mode loading the number of failure criteria is much smaller and no agreement exists in the scientific community on the “best criterion”. Herein we present the results of a recent research project in which three new mixed mode failure criteria for brittle elastic V-notched components were investigated [2], two of which are generalizations of known mode I failure criteria [1] and the third is being introduced for the first time. To validate the criteria, mixed mode loading experiments on PMMA and MACOR (glass ceramics) V-notched bar specimens were conducted. The parameters that govern failure initiation were computed by the p-version of the finite element method (p-FEM) from models representing the experimental specimens. The p-FE analyses for the prediction of failure loads are outlined and results comparing prediction to experimental observations are presented. A typical example of a V-notch specimen and its p-FE model and the predicted vs experimental failure load for different mode mixity values is shown below. References [1] Z. Yosibash, E. Priel and D. Leguillion, "A failure criterion for brittle elastic materials under mixed mode loading", International Journal of Fracture, 141 (2006), pp. 291-312 [2] E. Priel, A. Bussiba, I. Gilad and Z. Yosibash, "Mixed mode failure criteria for brittle elastic V-notched structures", International Journal of Fracture, (Submitted)

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High-order solid finite elements applied to the computation of the impact sound level from lightweight floors A. Rabold1*, A. Düster1, E. Rank1 1Lehrstuhl für Bauinformatik, Faculty of Civil Engineering and Geodesy, Technische Universität München, Arcisstr. 21, 80333 München, Germany e-mail: rabold, duester, [email protected]

Abstract

Up to now the research and development in the field of building acoustics is based mainly on experiments. The consequence of this approach is that the development and optimization of a new building component is a very tedious and expensive task. A considerably reduction of these costs could be achieved, if the optimization relying on experiments would be replaced – at least to some extent – by a computational approach. Common prediction models of the impact sound level from lightweight floors are based on measured impact sound level data for the component parts under consideration [1]. These models are very useful for the evidence of performance of known building components, but they are not suited for the development of new components. An alternative approach in this context is the application of the finite element method (FEM). Based on a FE model, a modal- and spectral analyses provides the framework for an optimization of the building component. To this end, the thin-walled structure, i.e. the lightweight floor, consisting of plates and beams is discretized with a fully three-dimensional approach, where anisotropic high-order solid finite elements [2] are applied. This contribution will present the overall approach consisting of the three-dimensional modeling of the structure and the excitation source (tapping machine), the subsequent modal- and the spectral- analyses and the computation of the radiated sound from the floor. References [1] A. Rabold, Schallschutz – Theorie und Praxis am Beispiel MFH Ottostraße, D – Ottobrunn Tagungsband, Internationales Holzbau-Forum, 2005. [2] A. Düster, H. Bröker, E. Rank. The p-version of the finite element method for three-dimensional curved thin walled structures. International Journal for Numerical Methods in Engineering, 52:673-703, 2001.

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Nonlinear computation of foam-like structures based on anisotropic high-order finite elements H.-G. Sehlhorst1*, A. Düster1, E. Rank1 1Lehrstuhl für Bauinformatik, Faculty of Civil Engineering and Geodesy, Technische Universität München, Arcisstr. 21, 80333 München, Germany e-mail: sehlhorst, duester, [email protected]

Abstract

Higher-order finite element methods have turned out to be highly efficient for many problems in computational mechanics. We present a two-dimensional high-order finite element formulation for the discretization of thin-walled structures. The implementation is based on a quadrilateral element, allowing for an anisotropic Ansatz of the displacement field. The polynomial degree of each component of the displacement field can be chosen individually and may also be varied in the two spatial directions of the element. Thus this element formulation is very well suited for the discretization of beam-like structures. The main focus of this contribution lies on the efficient computation of geometrically nonlinear foam-like structures. We will present several numerical examples demonstrating the efficiency of the proposed approach, taking buckling effects of thin-walled structures into account. References [1] B. Szabó, I. Babuška, Finite Element Analysis, John Wiley & Sons, New York, 1991. [2] B. Szabó, A. Düster, E. Rank, The p-version of the finite element method. Encyclopedia of Computational Mechanics, E. Stein, R. de Borst, T.J.R. Hughes (eds.), John Wiley & Sons, 2004. [3] G. Sehlhorst, Implementation and numerical investigation of a two-dimensional anisotropic higher-order finite element formulation, Master Thesis, Lehrstuhl für Bauinformatik, TU München, 2006.

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Mixed Finite Element Methods for Linear Elasticity – Part II Joachim Schöberl1, Astrid Sinwel2*

1 Center for Computational Engineering Sciences, RWTH Aachen, Pauwelstr. 19, Aachen, 52047, Germany 2 Radon Institute for Computational and Applied Mathimatics, Austrian Academy of Science, Altenbergerstr. 69, Linz, 4030, Austria e-mail: [email protected], [email protected]

Abstract

We introduce a new finite element method of arbitrary order to approximate the Hellinger-Reissner mixed formulation of elasticity. For the displacements, we use tangential-continuous vector-valued Nedelec elements. To approximate the stresses, we propose symmetric-tensor-valued finite elements. These elements are normal-normal continuous. This formulation is suitable for nearly incompressible materials, where the Poisson ratio tends to ½. Also, the elements do not suffer from shear locking when anisotropic elements are used. We present shape functions of arbitrary order for both the displacement and stress space. We see that the solution satisfies optimal order error estimates. We discuss the implementation of the new elements, and give numerical results.

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High-Order Finite Element Methods. May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany p-FE analysis of the proximal femur Nir Trabelsi* and Zohar Yosibash

Dept. of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel E-mail: nirtr/[email protected] Abstract Accurate methods for predicting and monitoring in-vivo bone strength are of major importance in clinical applications. These are nowadays unavailable due to difficulties in obtaining individual's bone geometry and mechanical heterogeneous and anisotropic properties, and the thin 3-D layers that preclude the use of automatic mesh generators. Numerous studies describe the generation of the femur's 3D model based on quantitative computerized tomography (QCT) scans. In these past works, the voxel-based and structural-based have been widely. Both methods use discrete values for the inhomogeneous mechanical elastic coefficients describing the average property of voxels inside each element. An improved structure based method is suggested so that the geometry is represented by smooth surfaces accurately and the inhomogeneous elastic properties are evaluated according to a similar volume as the test specimens used for their estimation. For that purpose, the QCT data were processed following several steps, starting from bone borders detection at each CT slice, trough surface approximation, to solid body representation, and finally to mesh generation (see right figure). An internal smooth surface is used to separate the cortical and trabecular regions upon which a p-FE auto-mesh is constructed. Within each region (cortical or trabecular) the QCT Housefeld Unit (HU) values are recalculated using a moving average method regardless of the FE mesh and inhomogeneous mechanical properties assigned by LMS approximations. To validate the FE results we performed mechanical in-vitro experiments on freshly frozen femurs measuring head deflection and strains at several points (see right Figure). Two FE model were created and compared to the in-vitro experiments. In the first FE model we created isoparametric mapping using an auto-mesher. In the second model we increased the mesh density and used blending mapping instead of the isoparametric method. The generation of the FE model, the isotropic and anisotropic material properties influence, and the comparison to the experimental observations will be presented [1]. References [1] Z. Yosibash, R. Padan, L. Joskowicz & C. Milgrom, "A CT-based high-order finite element analysis of the human proximal femur compared to in-vitro experiments", J. Biomechanical Engrg, In Press.

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High-Order Finite Element Methods (Abstracts) May 17-19, 2007, Herrsching am Ammersee (near Munich), Germany Nektar++ : An Object-Oriented Spectral/hp Element Library P.E.J. Vos1*, B. Nelson2, J. Frazier2, R.M. Kirby2 and S.J. Sherwin1 1 Department of Aeronautics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK 2 School of Computing and Scientific Computing and Imaging Institute, University of Utah, 50 S. Central Campus Drive, Salt Lake City, UT 84112, USA e-mail: [email protected], [email protected], [email protected], [email protected], [email protected]

Abstract

There are at least two major challenges which arise when one presents high-order finite element methods to the engineering community: i) presenting the mathematical structure of the methods in a digestible and coherent manner, and ii) presenting the data structures and algorithms in a manner that high-order methods can be applied in day-to-day scientific and engineering practice. Nektar++ is an open source software library currently being developed and designed to provide a bridge to the community – to provide a toolbox of data structures and algorithms which implement the spectral/hp element method, a high-order numerical method yielding fast error-convergence [1]. Nektar++ is the continuation and adaptation of the Nektar flow solver [1,2]. As opposed to its predecessor which focused on solving fluid dynamics problems, Nektar++ is implemented as a C++ object-oriented toolkit which allows developers to implement spectral element solvers for a variety of different engineering problems. The typical elemental decomposition of a global approximation using a spectral/hp discretization involves parametrically mapping from a standard elemental region (reference space) to a local elemental region (world space). The local regions are then collected to form a global approximation. This structure, supplemented with building blocks such as block matrix linear algebra routines and automatic data coordinating objects, can be encapsulated in an efficient object-oriented C++ implementation. Different sub-libraries, employing this characteristic pattern, are provided in the full library and can then be used as ingredients for the driving application, for example, a Navier-Stokes solver. This conceptual approach of the software leads to a high user-flexibility, including the selection of the preferred expansion basis, order and the preferred numerical quadrature. Currently, the different libraries are in their last stage of development. After the integration of these libraries with a standard scalar and vector solvers, we intend to demonstrate the efficiency of the Spectral/hp library for both high and low polynomial order Galerkin approximations. References [1] G.E. Karniadakis, S.J. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd ed., Oxford University Press, Oxford, 2005. [2] http://www.nektar.info

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Participants

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Dr. Ricardo L Actis Tel.: +1 314 744 5021 ESRD, Inc., Fax: +1 314 935 4014 111 West Port Plaza, Suite 825, E-Mail: [email protected] St. Louis, MO, 63146, USA Prof. Mark Ainsworth Tel.: +44 141 548 4535 Department of Mathematics, Fax: +44 141 548 3345 University of Strathclyde, E-Mail: [email protected] Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, Scotland. Prof. Dr. Ivo Babuska Tel.: +1 512 471 2156 Institute for Computational Engineering and Fax: +1 512 471 8694 Sciences E-Mail: [email protected] The University of Texas at Austin Austin, TX 78712 USA Dr.-Ing. Matthias Baitsch Tel.: +49 234 32 26175 Ruhr-Universität Bochum Fax: +49 234 32 06175 Lehrstuhl für Ingenieurinformatik E-Mail: [email protected] im Bauwesen Gebäude IA 44780 Bochum Dr. Attila Baksa Tel.: +36 46 565 111 1643 University of Miskolc, E-Mail: [email protected] Departement of Mechanics , Miskolc-Egyetemváros, Miskolc, HUNGARY 3515

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Prof. Dr. Edgar Bertoti Tel.: +36 46 565 111 Department of Mechanics, Fax: +36 46 565 163 University of Miskolc, E-Mail: [email protected] Miskolc-Egyetemvaros, Miskolc, H-3515, Hungary Dr. Sven Beuchler Tel.: +43 732 2468 5214 Institute of Computational Mathematics, Fax: +43 732 2468 5212 University of Linz, E-Mail: [email protected] Altenberger Strasse 69, Linz, A-4040, Austria Marcel Bieri Tel.: +41 44 632 3112 Seminar für Angewandte Mathematik, Fax: +41 44 632 1104 HG G 56.1, E-Mail: marcel.bieri Rämistrasse 101, @sam.math.ethz.ch 8092 Zürich, Switzerland Dipl.-Ing. Jan Biermann Tel.: +49 40 428 78 4480 Institute for Modelling and Computation, Fax: +49 40 428 78 4353 Hamburg University of Technology, E-Mail: [email protected] Denickestraße 17, Hamburg, D-21073, Germany Prof. Dr.-Ing. Kai-Uwe Bletzinger Tel.: +49 89 289 22422 Chair of Structural Analysis, Fax: +49 89 289 22421 Faculty of Civil Engineering and Geodesy, E-Mail: [email protected] Technische Universität München, 80333 München, Germany

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Prof. Dr. Dietrich Braess Tel.: +49 234 322 3405 Fakultät für Mathematik, Fax: +49 234 321 4750 Ruhr-Universität Bochum, E-Mail: Dietrich.Braess@ruhr-uni- 44780 Bochum bochum.de Germany Dr. Alexey Chernov Tel.: +41 44 632 3459 Seminar for Applied Mathematics, E-Mail: alexey.chernov ETH Zurich, @sam.math.ethz.ch Rämistrasse 101, Zurich, 8092, Switzerland Prof. Monique Dauge Tel.: +33-2 23 23 60 51 IRMAR, Fax: +33-2 23 23 67 90 Université de Rennes 1 E-Mail: Monique.Dauge Campus de Beaulieu, @univ-rennes1.fr 35042, Rennes Cedex, France Dr. Edouard Demaldent Tel.: +33 1 69 93 86 35 DEMR, ONERA, Fax: +33 1 69 93 62 20 Le Fort de Palaiseau, E-Mail: edouard.demaldent Palaiseau, 91120, @onera.fr France Prof. Dr. Leszek F. Demkowicz Tel.: +1 512 471 4199 Institute for Computational Engineering Fax: +1 512 471 8694 and Sciences, E-Mail: [email protected] The University of Texas at Austin, TX 78712, U.S.A.

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Dr. Saikat Dey E-Mail: [email protected] SFA Inc./NRL, 2200 Defense Hwy. Suite 405, Crofton, MD 21114, USA Prof. C. Armando Duarte E-Mail: [email protected] Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Lab, 205 N. Mathews Av., Urbana, IL 61801, USA PD Dr.-Ing. Alexander Düster Tel.: +49 89 289 25060 Lehrstuhl für Bauinformatik, Fax: +49 89 289 25051 Faculty of Civil Engineering and Geodesy, E-Mail [email protected] Technische Universität München, 80290 München, Germany Prof. John A. Ekaterinaris Tel.: +30 2810 391773 School of Mechanical Fax: +30 2810 391761 and Aerospace Engineering, E-Mail: [email protected] University of Patras, FORTH/IACM, P.O. Box 1385, 71110 Heraklion, Crete, Greece Prof. Dr.-Ing. Otto von Estorff Tel.: 040 428 78 - 3032 Modellierung und Berechnung, Fax. 040 428 78 - 4353 Hamburg University of Technology E-Mail: [email protected] Denickestraße 17, 21073 Hamburg, Germany

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Dipl.-Ing. David Franke Tel.: +49 89 289 25058 Lehrstuhl für Bauinformatik Fax: +49 89 289 25051 Faculty of Civil Engineering and Geodesy, E-Mail: [email protected] Technische Universität München Arcisstraße 21 D-80333 München, Germany Dr. Marc Gerritsma E-Mail: [email protected] Department of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, Delft, 2629 HS, The Netherlands Prof. Dr. Dr. h.c. Wolfgang Hackbusch Tel.: +49 (0)341 9959 752 Max-Planck-Institut für Mathematik Fax: +49 (0)341 9959 999 in den Naturwissenschaften, E-Mail: [email protected] Inselstr. 22-26, 04103 Leipzig, Germany PD Dr.-Ing. Stefan Hartmann Tel.: +49 561 804 2719 Institute of Mechanics, Fax +49 561 804 2720 University of Kassel, E-Mail: stefan.hartmann@uni- Moenchebergstr.7, kassel.de 34125 Kassel, Germany Dipl.-Ing. Holger Heidkamp E-Mail: [email protected] SOFiSTiK AG, Bruckmannring 38, 85764 Oberschleißheim, Germany

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Dipl.-Ing. Ulrich Heisserer Tel.: +49 89 289 25062 Lehrstuhl für Bauinformatik, Fax: +49 89 289 25051 Technische Universität München, E-Mail: [email protected]. Arcisstrasse 21, München, 80290, tu-muenchen.de Germany Univ.-Prof. Dr.-Ing. Stefan M. Holzer E-Mail: [email protected] Department of “Mathematik und Bauinformatik”, University of the Federal Armed Forces, Werner-Heisenberg-Weg 39, Neubiberg, 85577, Germany Prof. Dr. Thomas J.R. Hughes Tel.: +1 512 232 7774 Institute for Computational Engineering Fax: +1 512 232 7508 and Sciences, E-Mail: [email protected] The University of Texas at Austin, TX 78712, U.S.A. Dipl.-Ing. Ralf Jänicke Tel.: +49 681 302 2157 Universität des Saarlandes, Fax: +49 681 302 3992 Lehrstuhl für Technische Mechanik, E-Mail: r.jaenicke Postfach 151150, @mx.uni-saarland.de 66041 Saarbrücken, Germany Dr. Christos Katsis Tel.: +1 408 764 8557 Parametric Technology Corporation, Fax: +1 408 764 8700 2730 San Tomas Expwy., Suite 100, E-Mail: [email protected] Santa Clara, CA, 95051, USA

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Prof. Robert M. Kirby Tel.: +1 801 585 3421 School of Computing, University of Utah, Fax: +1 801 585 6513 50 S. Central Campus Dr., E-Mail: [email protected] Salt Lake City, UT 84112 USA Dipl.-Ing. Stefan Kollmannsberger Tel.: +49 89 289 25021 Lehrstuhl für Bauinformatik, Fax: +49 89 289 25051 Faculty of Civil Engineering and Geodesy, E-Mail: [email protected] Technische Universität München, Arcisstr. 21, 80333 München, Germany Dr. Alexander Konyukhov Tel.: +49 721 608 3715 Institute of Mechanics, Fax: +49 721 608 7990 University of Karlsruhe, E-Mail: [email protected] Englerstr. 2, karlsruhe.de Karlsruhe, 76131, Germany Dr. Raoul van Loon Tel.: +44 20 759 45129 Department of Aeronautics, E-Mail: [email protected] Imperial College, Room 454, ACE Extension South Kensington Campus, London, SW7 2AZ, UK Univ.-Prof. Dipl.-Ing. Dr. Ulrich Langer Tel.: +43 732 2468 9168 Institute of Computational Mathematics Fax: +43 732 2468 9148 Johannes Kepler University Linz, E-Mail: ulanger Altenberger Strasse 69, @numa.uni-linz.ac.at A-4040 Linz, Austria

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Dmitry Ledentsov MSc Tel.: +49 89 289 25116 Lehrstuhl für Bauinformatik Fax: +49 89 289 25051 Technische Universität München E-Mail: [email protected] Arcisstraße 21 80290 München, Germany Prof. Jens Markus Melenk Tel.: +43 1 58801 10165 Computational Mathematics Fax: +43 1 58801 10196 TU Wien E-Mail: [email protected] Wiedner Hauptstrasse 8-10 A-1040 Wien, Austria Prof. Istvan Paczelt Tel.: +36 46 565-111 18-77 Department of Mechanics, Fax: +36 46 565-163 University of Miskolc, E-Mail: [email protected] Miskolc-Egyetemváros, Miskolc, 3515 Hungary Prof. Jamshid Parvizian Tel.: +49 (0)89 289-25061 Lehrstuhl für Bauinformatik, Fax: +49 (0)89 289-25051 Technische Universität München, E-Mail: [email protected] Arcisstraße 21, 80290 München, Germany Prof. Juhani Pitkäranta Tel.: +358 9 451 3024 Institute of Mathematics, Fax: +358 9 451 3016 Helsinki University of Technology, E-Mail: [email protected] P.O.Box 1100, FI-02015 TKK, Finland

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Elad Priel Tel.: +972 8 6477054 Department of Mechanical Engineering Fax: +972 8 6472813 Ben-Gurion university of the Negev, E-Mail: [email protected] Beer-Sheva, 84105, Israel Dipl.-Ing. Andreas Rabold Tel.: +49 89 289 25063 Lehrstuhl für Bauinformatik, Fax: +49 89 289 25051 Faculty of Civil Engineering and Geodesy, E-Mail: [email protected] Technische Universität München, Arcisstr. 21, 80333 München, Germany Prof. Dr. rer.nat. Ernst Rank Tel.: +49 89 289 23047 Lehrstuhl für Bauinformatik, Fax: +49 89 289 25051 Faculty of Civil Engineering and Geodesy, E-Mail: [email protected] Technische Universität München, 80333 München, Germany Prof. Jeronimo Rodriguez Garcia Tel.: +33 1 45 52 52 25 Laboratoire POEMS Fax: +33 1 45 52 52 82 UMR 2706 CNRS-ENSTA-INRIA, E-Mail: Jeronimo.Rodriguez 32 Boulevard Victor, @ensta.fra 75015 Paris, France Dr.-Ing. Martin Ruess Tel.: +49 89 289 22425 Lehrstuhl für Bauinformatik, Fax: +49 89 289 25051 Technische Universität München, E-Mail: [email protected] Arcisstraße 21, 80290 München, Germany

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Dipl.-Ing. Kersten Schmidt Tel.: +41 44 632 6038 Seminar for Applied Mathematics, E-Mail: kersten.schmidt@ ETH Zurich, sam.math.ethz.ch Rämistrasse 101, 8050 Zürich, Switzerland Prof. Dr. Joachim Schöberl Tel.: +49 241 80 28466 RWTH Aachen University, E-Mail: joachim.schoeberl Pauwelstr. 19, @rwth-aachen.de 52074 Aachen, Germany Dr. Dominik Schötzau Tel.: +1 604 822 4346 Mathematics Department, Fax. +1 604 822 6074 University of British Columbia, E-Mail: [email protected] 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada Prof. Dr. Christoph Schwab Tel.: +41 44 632 3595 Seminar for Applied Mathematics, Fax: +41 44 632 1104 ETH Zurich, E-Mail: [email protected] Rämistrasse 101, 8050 Zürich, Switzerland Dr. Marc Alexander Schweitzer Tel.: +49 228 733174 Institute for Numerical Simulation, Fax: +49 228 737527 Universität Bonn, Wegelerstraße 6, E-Mail: [email protected] D-53115 Bonn, Germany

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Prof. Dr.-Ing. Karl Schweizerhof Tel.: +49 721 608 2070 Institute of Mechanics, Fax: +49 721 608 7990 University of Karlsruhe, E-Mail: [email protected] Englerstrasse 2, karlsruhe.de Karlsruhe, 76131, Germany Dipl.-Ing. Hans-Georg Sehlhorst MSc Tel.: +49 89 289 25115 Lehrstuhl für Bauinformatik, Fax: +49 89 289 25051 Faculty of Civil Engineering and Geodesy, E-Mail: [email protected] Technische Universität München, Arcisstr. 21, 80333 München, Germany DI Astrid Sinwel Tel.: +43 732 2468 5299 Radon Institute for Computational and E-Mail: [email protected] Applied Mathimatics, Austrian Academy of Science, Altenbergerstr. 69, Linz, 4030, Austria Dr. Pavel Solin Tel.: +420 2 6605 2058 Institute of Thermomechanics, Fax: +420 2 8689 0433 Academy of Sciences of the Czech Republic, E-Mail: [email protected] Dolejskova 5, Praha 8, CZ 18200, Czech Republic Prof. em. Dr.-Ing. Dr.-Ing. E.h. Tel.: +49 511 762 4290 Dr. h.c. mult. Erwin Stein Fax: +49 511 762 19053 IBNM, Universität Hannover, E-Mail. [email protected] Appelstr. 9A, hannover.de 30167 Hannover, Germany

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Prof. Dr. Barna A. Szabo Tel.: +1 314 935 6352 Center for Computational Mechanics, Fax: +1 314 935 4014 Washington University, Campus Box 1129, E-Mail: [email protected] St. Louis, MO, 63130, USA Dr. Tamas Szabo Tel.: +36 96 503 400 Dept. of Machine Structures and Mechanics, Fax: +36 96 503 491 Széchenyi István University, E-Mail: [email protected] Egyetem tér 1., Györ, H-9026, Hungary Nir Trabelsy Tel.: +972 8 6477053 Dept. of Mechanical Engineering, Fax: +972 8 6472813 Ben-Gurion University of the Negev, E-Mail: [email protected] Beer-Sheva, 84105, Israel Dr. Radek Tezaur Tel.: +1 650 725 9646 ICME, Building 500, Fax: +1 650 725 3525 Stanford University, E-Mail: [email protected] 488 Escondido Mall, MC 3035 Stanford, CA 94305 U.S.A Prof. L. Beirão da Veiga Tel.: +39 2503 16081 Department of Mathematics, E-Mail: :[email protected] University of Milan, Via Saldini 50, Milan, 20133, Italy

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Dr. Tomas Vejchodsky, Ph.D. Tel.: +420 222 090 713 Institute of Mathematics, Fax: +420 222 211 638 Academy of Sciences, E-Mail: [email protected] Zitna 25, CZ-11567, Prague 1, Czech Republic Dr. Peter Vos Tel.: +44 20 759 45103 Department of Aeronautics, Fax: +44 20 7584 8120 Imperial College London, E-Mail: [email protected] South Kensington Campus, London, SW7 2AZ, UK Prof. Timothy Warburton Tel.: +1 713 348 5666 Department of Computational and Fax: +1 713 348 5318 Applied Mathematics, E-Mail: [email protected] Rice University, 6100 Main Street, MS-134, Houston, Texas, 77005, USA. Dipl.-Ing. Ziad Wassouf MSc Tel.: +49 89 289 25023 Lehrstuhl für Bauinformatik, Fax: +49 89 289 25051 Technische Universität München, E-Mail: [email protected] Arcisstraße 21, 80290 München, Germany Prof. Dr. Christian Wieners Tel.: +49 721 608 2063 Institut für Angewandte und Fax: +49 721 608 3197 Numerische Mathematik, E-Mail: [email protected] Universität Karlsruhe, karlsruhe.de Englerstraße 2, 76128 Karlsruhe, Germany

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Prof. Christos Xenophontos Tel.: +357 22 89 Department of Mathematics and Statistics, Fax: +357 22 89 University of Cyprus, E-Mail: [email protected] P.O. BOX 20537, Nicosia, 1678, Cyprus Zhengxiong Yang Msc Tel.: +49 89 289 25065 Lehrstuhl für Bauinformatik, Fax: +49 89 289 25051 Technische Universität München, E-Mail: [email protected] Arcisstraße 21, 80290 München, Germany Prof. Zohar Yosibash Tel.: +972 8 6477103 Dept. of Mechanical Engineering, Fax: +972 8 6477101 Ben-Gurion University of the Negev, E-Mail: [email protected] Beer-Sheva, 84105, Israel Dr. Sabine Zaglmayr Tel.: +43 732 2468 5236 Radon Institute for Computational and Fax: +43 732 2468 5212 Applied Mathematics (RICAM), E-Mail: sabine.zaglmayr Altenbergerstr. 69, @oeaw.ac.at Linz, 4040, Austria Dr. Adam Zdunek Tel.: +46 8 555 43 72 Computational Physics, Fax: +46 8 25 89 19 Systems Technology, E-Mail: [email protected] Swedish Defence Research Agency, FOI, SE-164 90 Stockholm, Sweden

International Workshop on High-Order Finite Element Methods · 11:40-12:05 Acoustic scattering...

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Transcript of International Workshop on High-Order Finite Element Methods · 11:40-12:05 Acoustic scattering...