Book of Abstracts · plasticity with small deformations 09:25 { 09:50 Bla z Hudobivnik...

Book of Abstracts

Welcome 1

Welcome

We welcome everybody to the workshop

Current Trends and Open Problems in Computational Solid MechanicsLeibniz Universitat Hannover, October 8 – 9, 2018

The workshop provides a forum for the discussion of recent results in the simulation ofsolids. Of particular interest are problems related to phase-field models for damage andfracture, virtual element methods, extended finite elements, as well as modeling and sim-ulation of incompressible (solid) materials.

A specific wish would be to put some emphasis in each talk on current open questionsrather than ‘only’ reporting on methods that already work.

On the following pages, more information on the location, the schedule, the participants,and details on presentations including their abstracts are provided.

We are very grateful to Ulla Fleischhauer and Carmen Gatzen who helped us tremendouslywith the administration and we wish to thank our main sponsors

• IfAM, Leibniz Universitat Hannover

• DFG-SPP 1748

who provided the basic funding of this event.

We look very much forward to meeting all participants and wish everybody an enjoyableand stimulating conference.

OrganizersFadi Aldakheel (Leibniz Universitat Hannover)Mirjam Walloth (Technische Universitat Darmstadt)Thomas Wick (Leibniz Universitat Hannover)Winnifried Wollner (Technische Universitat Darmstadt)Peter Wriggers (Leibniz Universitat Hannover)

Local organizing committeeFadi AldakheelKatrin MangThomas Wick


2 Welcome


Conference location 3

Conference location

1 Leibniz Universitat Hannover, Welfengarten 1, 30167 Hannover, main building 1101/Room b302

2 Tram station Leibniz Universitat (U-Bahn), lines 4 & 5

3 Restaurant Meiers LebenslustThe conference dinner will take place on Monday evening (7.30 pm) at the restaurantMeiers Lebenslust, Restauration & Hausbrauerei in Osterstraße 64, 30159 Hannover.Next tram station: Aegidientorplatz (lines 1, 2, 4, 5, 6, 8, 11,18).


4 Conference location

1 Leibniz Universitat Hannover, Welfengarten 1, 30167 Hannover, main building 1101/Room b302

2 Tram station Leibniz Universitat (U-Bahn), lines 4 & 5

3 Restaurant Meiers LebenslustThe conference dinner will take place on Monday evening (7.30 pm) at the restaurantMeiers Lebenslust, Restauration & Hausbrauerei in Osterstraße 64, 30159 Hannover.Next tram station: Aegidientorplatz (lines 1, 2, 4, 5, 6, 8, 11,18).


Time table 5

Time table

Monday, October 08

10:45 – 11:15 Registration

11:15 – 11:25 Opening

11:25 – 12:10 Roberto Alessi Modeling fatigue phenomena inbrittle and ductile materialswith a variational phase-field

approach

12:10 – 12:35 Michele Marino Virtual Element Method for theHomogenization of

Polycrystalline Materials

12:35 – 13:00 Andro Mikelic A phase-field model of afluid-driven fracture in a

nonlinear poroelastic medium

13:00 – 13:45 Lunch break

13:45 – 14:10 Stefan Frei Fluid-structure interactionswith contact using Nitsche’s

method

14:10 – 14:35 Rolf Krause An Immersed approach toContact in Fluid Structure

Interaction

14:35 – 15:00 Fabian Castelli The Numerical Simulation ofPhase Segregation in Electrode

Particles of Lithium IonBatteries

15:00 – 15:20 Coffee break 1

15:20 – 16:05 Anna Pandolfi A Class of Electro-ActiveAnisotropic Material Models

16:05 – 16:30 David Neusius Evolutional problem for knittedrotationally symmetric shellreduced to a periodic strip of

meshes

16:30 – 16:55 Marijo Milicevic Discretization ofBV -regularized damage modeland convergence to semi-stable

energetic solution


6 Time table

16:55 – 17:15 Coffee break 2

17:15 – 17:40 Bernhard Endtmayer Adjoined Based ErrorEstimation for Nonlinear

Problems

17:40 – 18:05 Marco Favino Multigrid methods fornon-linear incompressible

elasticity

18:05 – 18:30 Timo Heister A parallel, adaptive software forphase-field fracture problems

19:30 – 21:00 Conference dinner


Time table 7

Tuesday, October 09

09:00 – 09:25 Nicolas Pignet A Hybrid High-Order methodfor incremental associative

plasticity with smalldeformations

09:25 – 09:50 Blaz Hudobivnik Gradient-Thermo-Plasticityusing an Efficient VirtualElement Method (VEM)

09:50 – 10:15 Nima Noii A Variationally ConsistentPhase-Field Modeling of BrittleFracture Applied to Pressurized

and Non-isothermal ElasticMedia

10:15 – 10:35 Coffee break 3

10:35 – 11:20 Gregoire Allaire Optimal design of latticematerials

11:20 – 11:45 Bo Yin Fracture Simulation ofAnisotropic Media by a

Phase-Field Method

11:45 – 12:10 Christian Steinke On Energetically motivatedBranching characterized by aDirectional Phase-field Split

12:10 – 12:55 Lunch break

12:55 – 13:20 Maria Laura De Bellis A VEM formulation of Scalardamage in 2D solids

13:20 – 13:45 Fadi Aldakheel A Virtual Element Method forCrack Propagation

13:45 – 13:55 Coffee break 4

13:55 – 14:20 Marreddy Ambati A phase-field model for brittlefatigue fracture

14:20 – 14:45 Daniel Jodlbauer Matrix-Free Solvers forPhase-Field Fracture Problems

14:45 – 15:10 Katrin Mang A Phase-field Model forFractures in Incompressible

Solids

15:10 – 15:15 Closing


8 Time table


Invited speakers 9

Invited speakers

Our invited speakers are distinguished researchers who are briefly introduced here:

Roberto Alessi

He is a researcher at Sapienza, University of Rome. His research activity mostly focuseson the constitutive modeling of engineering materials.

Gregoire Allaire

He is a professeur of Applied Mathematics at Centre de Mathematiques Appliquees (CMAP)at Ecole Polytechnique in Palaiseau near Paris in France. His main research fields arescientific computing, multiscale modelling, partial differential equations, homogenization,composite materials, shape and topology optimization, porous media, two-phase flows, andneutron transport.

Anna Pandolfi

She is a professor of Structural Mechanics at Politecnico di Milano in Italy and a visit-ing Associate in Aerospace at the California Institute of Technology, Pasadena CA in theUnited States. Her main research topics are space and time discretization methods forcoupled problems, biomechanics of biological tissues, in healthy conditions and sufferingfrom diseases. Furthermore, mechanical and chemo-mechanical behavior of biocompatibleelastomers, metaconcretes, simulations of the propagation of fractures with various appli-cations, constitutive models in finite deformations, configurational distortions of nematicliquid crystals in the presence of electric and magnetic fields, rate-sensitivity of fracturepropagation in polymer composites, mechanics of non-smooth contact with and withoutfriction.


10 Invited speakers


Abstracts 11

Abstracts

The talks are listed in chronologic order.

1. Modeling fatigue phenomena in brittle and ductilematerials with a variational phase-field approachRoberto Alessi,Dipartimento di Ingegneria Strutturale e Geotecnica, SAPIENZA Universita di Roma

Fatigue is a key phenomenon in mechanics and is responsible for most of structuralfailures. Yet, the prediction of fatigue failures still represents an open challenge formodelling and simulation at the cutting edge of mechanics. Current available fatigueanalysis tools are characterised by some limitations since they are often based onempirical laws and strongly rely on expensive experimental tests. Furthermore theydescribe each fatigue failure stage separately. As a consequence, they lack generalityand are not sufficiently flexible to describe complicated fatigue crack scenarios.

This presentation will discuss a new possible way to extend variational phase-fieldmodels to account for fatigue effects in both brittle and ductile materials. So far,phase-field models have been successfully applied in a great variety of fracture prob-lems but limited to monotonic loads. The key modelling idea is to reduce the materialfracture toughness as one or more history variable increase. The resulting model isexpected to capture, in a unified manner, both fatigue crack initiation and fatiguecrack propagation, which are commonly described as distinct processes. Concerningductile materials, the model can also describe both low-cycle and high-cycle fatigueregimes. In addition, the variational structure of the problem suggests a possiblenumerical implementation for the model.


12 Abstracts

2. Virtual Element Method for the Homogenizationof Polycrystalline MaterialsMichele Marino, Blaz Hudobivinik, Peter Wriggers,Institute of Continuum Mechanics, Leibniz Universitat Hannover

The computational homogenization of polycrystalline materials is associated withrequests of flexibility with regards to mesh generation and element shapes. Forinstance, polygonal or polyhedral Voronoi tessellation might be effectively exploited.In this framework, the virtual element method (VEM) has been recently introduced[1] and it has been attracting attention in the framework of advanced modelingtechniques for anisotropic material behavior [2] and homogenization of compositematerials [3]. This method permits the use of polygonal/polyhedral elements wherethere is no need for a restriction to convex elements, nor it is necessary to avoiddegeneracies such as element sides having an interior angle close to π-radians intwo dimensions [1]. This flexibility is paid in terms of a rank-deficiency of the energyfunctional due to the projection of the primary variable on a polynomial space. Hence,a stabilization term is conveniently introduced [4].

This work addresses the extension of VEM applications to anisotropic mechanicalproperties in finite elasticity, in particular for the computational homogenization ofpolycrystalline materials. By the use of the representation theorem for tensor func-tions, suitable structural tensors allow to define a minimal basis of tensor invariantsfor the definition of constitutive equations under the respect of given material sym-metries. Moreover, a constrained variational formulation is introduced in order toaccount for strong anisotropies associated with element locking. The formulation,already well-established addressing a transversely isotropic behavior [2], is here alsoenriched in order to account for an orthotropic response. To reach this goal, non-standard invariants that control the area change of elements in the direction along orperpendicular to material preferred direction are introduced. The proposed formu-lation is tested in benchmark tests involving regular, distorted and Voronoi meshes.The Lagrange multiplier associated with the constrained anisotropic formulation isdescribed by means of a special ansatz that allows to obtain a locking free behavior.

[1] Beirao da Veiga, L., Brezzi, F., Marini, L.D. and Russo, A., The Hitchhiker’s Guide to theVirtual Element Method. Mathematical Models and Methods in Applied Sciences, Vol 24(08),page 1541-1573 (2014).

[2] Wriggers, P., Reddy, B.D., Rust, W. and Hudobivnic, B., Efficient Low Order Virtual El-ements for Anisotropic Materials at Finite Strain. Computational Methods in AppliedSciences, Vol 46, page 417-434 (2018).

[3] Artioli, E., Asymptotic homogenization of fibre-reinforced composites: a virtual element methodapproach. Meccanica, doi:10.1007/s11012-018-0818-2 (2018).

[4] Wriggers, P., Reddy, B.D., Rust, W. and Hudobivnic, B., Efficient virtual element formula-tions for compressible and incompressible finite deformations. Computational Mechanics,Vol 70, page 253-268 (2017).


Abstracts 13

3. A phase-field model of a fluid-driven fracturein a nonlinear poroelastic mediumAndro Mikelic,Univ Lyon, Universite Claude Bernard Lyon 1, CNRS UMR 5208,Institut Camille Jordan, 43 blvd. du 11 novembre 1918,F-69622 Villeurbanne cedex, FranceE-mail: [email protected]

In this talk we present a phase field model for a fluid-driven fracture in a nonlinearporoelastic medium. We consider a fully coupled system where the pressure field isdetermined simultaneously with the displacement and the phase field. The mathe-matical model consists of a linear elasticity system with fading elastic moduli as thecrack grows, which is coupled with an elliptic variational inequality for the phasefield variable and with the mass conservation equation for the pressure containingthe phase field variable in its coefficients. The porosity is calculated from the massconservation for the solid skeleton and the permeability depends on porosity. Theconvex constraint of the variational inequality assures the irreversibility and entropycompatibility of the crack formation. The phase field variational inequality containsquadratic pressure and strain terms, with coefficients depending on the phase fieldunknown. We establish existence of a solution to the incremental problem throughconvergence of a finite dimensional approximation. Furthermore, we construct thecorresponding Lyapunov functional that is linked to the free energy. Computationalresults are provided that demonstrate the effectiveness of this approach in treatingfluid-driven fracture propagation.

This is a work in collaboration with C.J. van Duijn (Darcy Center Eindhoven-Utrecht)and T. Wick (University of Hannover).


14 Abstracts

4. Fluid-structure interactions with contactusing Nitsche’s methodStefan Frei,University College London

In this presentation we develop a Nitsche-based contact formulation for fluid-structureinteraction (FSI) problems with contact. Our approach is based on the works ofChouly and Hild [SIAM Journal on Numerical Analysis. 2013;51(2):1295–1307] forcontact problems in solid mechanics. Using a suitable extension of the fluid equa-tions below the contact surface, we are able to formulate the FSI interface and thecontact conditions simultaneously in equation form on a joint interface-contact sur-face Γ(t). Due to the continuous switch between interface and boundary conditions,the so-called “chattering” phenomenon known in the engineering literature, is pre-vented. We show a stability result and present numerical examples to investigate theperformance of the method.


Abstracts 15

5. An Immersed approach to Contactin Fluid Structure InteractionRolf Krause,Universita della Svizzera italiana

We present a completely parallel approach for contact between the elastic structuresinside a fluid. Our approach exploits the mortar method for variational coupling ofthe contact surfaces between the solids [1], as well as the fictitious domain methodfor the fluid structure interaction (FSI).

The dynamic contact problem is treated with a contact-stabilized Newmark methodwhich is proven to be dissipative hence does not allow energy blow-ups [3].

The FSI problem is modelled by embedding the solid in the background flow. Thefluid is described in an Eulerian fashion (fixed mesh), while the solid in a Lagrangianone. The coupling between the fluid and the solid is constructed by means of L2-projections for weakly enforcing the velocity vector constraint along the FSI-interface.

Note that meshes are in general non-matching on both the predicted area of contactand in intersection between fluid and solid. Moreover, in parallel computing envi-ronments such meshes are arbitrarily distributed among processes. For handling thischallenge we perform a scalable parallel assembly of the coupling operators by meansof a tree-based search and load-balancing algorithm [2].

The presented methodologies are validated by means of 2D and 3D benchmarks.

[1] T. Dickopf and R. Krause. Efficient simulation of multi-body contact problemson complex geometries: A flexible decomposition approach using constrainedminimization. International journal for numerical methods in engineering, 77(13):1834-1862, 2009.

[2] R. Krause and P. Zulian. A parallel approach to the variational transfer ofdiscrete fields between arbitrarily distributed unstructured finite element meshes.SIAM Journal on Scientific Computing, 38(3): C307-C333, 2016.

[3] Krause R, Walloth M. Presentation and comparison of selected algorithms fordynamic contact based on the Newmark scheme. Applied Numerical Mathemat-ics. 2012 Oct 1; 62(10):1393-410.


16 Abstracts

6. The Numerical Simulation of Phase Segregationin Electrode Particles of Lithium Ion BatteriesFabian Castelli and Willy Dorfler,Institute of Applied and Numerical Mathematics,Karlsruhe Institute of Technology (KIT)

Lithium ion batteries are key technologies for mobile power devices. For the betterunderstanding of the electrode materials (e. g. LiMn2O4) we investigate the concen-tration profile of lithium ions in a single particle of the electrode material during the(dis-)charge process. Due to changes in the host material a separation into lithiumrich and lithium poor phases is possible. This behavior can be modeled with the wellknown Cahn–Hilliard equation, see [1, 2].

Our focus is on the numerical treatment of this fourth-order PDE, which resultsfrom underlying the phase-field model. The main challenges are on the one hand thenon-linearities due to the non-constant mobility as well as the logarithmic free energydensity and on the other hand, depending on the electrode material, the almost sharpinterface between the lithium rich and the lithium poor phase.

We split up the forth-order Cahn–Hilliard equation and consider the two resultingsecond-order PDEs. To solve the arising system numerically we employ a standardfinite element method together with an appropriate time integrator. The numericalalgorithm was implemented with the open-source finite element library deal.II [3].

[1] M. Huttin, M. Kamlah, Phase-field modeling of stress generation in electrodeparticles of lithium ion batteries, Applied Physics Letters 101(13) (2012) 133902-1–133902-4.

[2] M. Huttin, Phase-field modeling of the influence of mechanical stresses on charg-ing and discharging processes in lithium ion batteries, PhD thesis, KarlsruherInstitut fur Technologie (KIT), Karlsruhe, 2014.

[3] W. Bangerth, R. Hartmann, G. Kanschat, deal.II—A General Purpose ObjectOriented Finite Element Library, ACM Transactions on Mathematical Software(TOMS) 33(4) (2007) 24/1–24/27.


Abstracts 17

7. A Class of Electro-Active AnisotropicMaterial ModelsAnna Pandolfi,Politecnico di Milano, Civil and Environmental Engineering Department

In the last decade the interest towards the modeling of the active behavior of de-formable biological tissues (heart, muscles, intestines) has been growing progres-sively. The basic features of the mechanical response of biological active tissues issufficiently well described by hyperelastic models, disregarding in first approximationmore complicated effects related to growth and remodeling. In particular, for heartand muscles, mathematical models of passive elasticity have been proposed includ-ing isotropic, transversely isotropic and, more recently, orthotropic models. In thisrespect, the introduction of the active stress component in the formulation encom-passes the key aspects related to the description of the subcellular electrophysiologicaldynamics of the tissue.

In order to account for the strong contraction induced by the electromechanical cou-pling, the behavior of cardiac tissue must be described in terms of finite deformations.The standard approach for modeling biological tissues is based on a passive responseexpressed in terms of hyperelastic weakly compressible or incompressible materials,combined to an active response (or active stress) formulated in an independent way,in several cases as a phenomenological description of some inelastic action. As op-posed to the concept of active stress, alternative approaches that rely on the conceptof active strain and are based on the multiplicative decomposition of the deforma-tion gradient into active part and passive part have appeared recently in the cardiacliterature.

Here we discuss a class of constitutive models for electro-active viscous materialsbased on the multiplicative decomposition of the deformation gradient in passive,active, and viscous parts, adopting a fully thermodynamical approach, able to con-ciliate the contrasting concepts of active stress and active strain, and consideringadvanced anisotropic material models, in view of numerical simulations of the be-havior of muscles and heart. A particular viscous doubly fibrous electromechanicalmodel derived from this general framework has been implemented in a finite elementcode. The code is employed in the numerical simulation of the complex peristalsismotion in a portion of the human colon.

The study has been developed in collaboration with Alessio Gizzi and Marcello Vasta,from the University of Chieti-Pescara.


18 Abstracts

8. Evolutional problemfor knitted rotationally symmetric shellreduced to a periodic strip of meshesJulia Orlik, David Neusius,Fraunhofer-Institut fur Techno- und Wirtschaftsmathematik, Kaiserslautern

A stretching behavior of knitted or woven textile rotationally symmetric shell is mod-eled. The yarns are modeled as one-dimensional hyperelastic strings with frictionalcontact. Capstan law known for Coulomb’s friction of yarns is extended to an addi-tional adhesion.

We use the rotational symmetry of the shell (developable surface) and reduce itsevolutional boundary value problem to an ODE on the vertical lateral line of theshell. This can be done only for homogeneous effective shell. Therefore, we replacethe longitudinal string by an equivalent knitted strip with periodic conditions in thecircular direction and couple its longitudinal and circular forces w.r.t. the stringequilibrium or evolution equation. In this way the number of degrees of freedom isreduced dramatically.

A two-step Newton’s method is applied for the solution of the large stretching withsliding evolution in the contact nodes. The approach is illustrated on a simulatedknitting process for the compression stockings.

Due to the simplicity of the reduced geometry, knitted structure of the periodicand rotationally symmetric strip can be easily parametrized for optimization. E.g.optimizing machine parameters for a compressible stocking, given a leg shape andtarget pressure profile along the leg axis. This has been validated by producing andtesting these compression stockings.


Abstracts 19

9. Discretization of BV -regularized damage modeland convergence to semi-stable energetic solutionMarijo Milicevic,Department of Applied Mathematics (University of Freiburg)

We consider damage processes in the framework of (semi-stable) energetic solutionswith an energy functional that involves a nonsmooth BV -regularization of the dam-age variable and a nonsmooth dissipation functional that ensures the unidirectional-ity of the damage evolution. Stability of an alternate minimization in time is shownwhere the nonsmooth subproblems are solved with an operator splitting approach.Numerical experiments are presented in which the nonsmooth BV -regularization iscompared to a smooth regularization of the damage variable by the H1-seminorm.


20 Abstracts

10. Adjoined Based Error Estimationfor Nonlinear ProblemsBernhard Endtmayer,RICAM, Austrian Academy of Sciences

In this presentation, we design a posteriori error estimates and mesh adaptivity formultiple goal functionals defined on the solutions to certain nonlinear problems basedon results from [2] in the single-goal case. We use a dual-weighted residual approachin which localization is achieved in a variational form using a partition-of-unity. Thekey advantage is that the method is simple to implement. Backward integration byparts is not required. For the treatment of multiple goals, we use the ideas from [1]where linear problems are treated, but we do not have to solve an additional problem.Our adaptive algorithm has been developed for a wide class of non-linear problemssuch like optimal control or the p-Laplace problem for which we run numerical tests.

This work has been supported by the Austrian Science Fund (FWF) under the grantP 29181 ‘Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multi-physics Problems’.

[1] B. Endtmayer and T. Wick A Partition-of-Unity Dual-Weighted Residual Ap-proach for Multi-Objective Goal Functional Error Estimation Applied to EllipticProblems. Comput. Methods Appl. Math. 17 (2017), no. 4, 575-599.

[2] Rannacher, R. and Vihharev, Adaptive finite element analysis of nonlinear prob-lems: balancing of discretization and iteration errors. J. Numer. Math.21(2013), no. 1, 23-61.

[3] B. Endtmayer, U. Langer and T. Wick, Multigoal-Oriented Error Estimatesfor Non-linear Problems. J. Numer. Math, accepted for publication (2018),published online.


Abstracts 21

11. Multigrid methodsfor non-linear incompressible elasticityMarco Favino,Universita della Svizzera italiana

We present efficient solution strategies for discrete large scale problems arising frommixed finite element discretizations in incompressible non-linear mechanics. Dueto the non-linear material law, the elasticity operator exhibits a non-trivial kernel,which prevents the straight-forward application of existing preconditioning strategiesfor saddle point problems. In order to overcome this problem, here, we present twoaugmentation strategies, which in combination with multigrid-methods allow for thedesign of optimal segregated preconditioners. Exploiting the saddle-point structureof the resulting linear problem, a Schur-complement-based segregated approach isconsidered for the solution of incompressible elasticity. Considering standard bench-marks in cardiac mechanics, we show that our segregated multigrid preconditionedapproaches scales well with the problem size and number of cores, allowing for tack-ling large problems in a reasonable time.


22 Abstracts

12. A parallel, adaptive softwarefor phase-field fracture problemsTimo Heister,SCI Institute, University of Utah

We present a scalable, parallel implementation of a solver for the solution of a phase-field model for quasi-static brittle fracture.

Numerical solutions in 2d and 3d with adaptive mesh refinement show optimal scal-ing of the linear solver based on algebraic multigrid to thousands of processors, andconvergence of the phase-field model towards exact values of functionals of interestssuch as the crack opening displacement or the total crack volume. In contrast touniform refinement, adaptive mesh refinement allows us to recover optimal conver-gence rates for the non-smooth solutions encountered in typical test problems. Wealso present numerical studies of the influence of the finite domain size on functionalevaluations used to approximate the infinite domain in one of the examples.

[1] T. Heister, T. Wick, https://github.com/tjhei/cracks

[2] T. Heister, T. Wick, Parallel solution, adaptivity, computational convergence,and open-source code of 2d and 3d pressurized phase-field fracture problems.2018. arXiv:1806.09924

[3] T. Heister, M. Wheeler, T. Wick, A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-fieldapproach. CMAME, Volume 290, 15 June 2015, Pages 466-495.doi:10.1016/j.cma.2015.03.009


Abstracts 23

13. A Hybrid High-Order methodfor incremental associative plasticitywith small deformationsMickael Abbas1, Alexandre Ern2, Nicolas Pignet1,2,1 EDF R&D, Palaiseau, France and IMSIA, UMR EDF/CNRS/CEA/ENSTA 9219,Palaiseau, FranceE-mail: [email protected], [email protected],2 Universite Paris-Est, CERMICS (ENPC), Champs-sur-Marne, France and INRIA,FranceE-mail: [email protected], [email protected]

We devise and evaluate numerically a Hybrid High-Order (HHO) method for incre-mental associative plasticity with small deformations. HHO methods were initiallyintroduced in [1] for linear elasticity and in [2] for diffusion problems. For a recentwork on HHO methods for hyperelasticity, we refer to [3]. The HHO method usesas discrete unknowns piecewise polynomials of order k ≥ 1 on the mesh skeleton,together with cell-based polynomials that can be eliminated locally by static conden-sation. The HHO method supports polyhedral meshes with non-matching interfaces,is free of volumetric locking, and the integration of the behavior law is performedonly at cell-based quadrature nodes. Moreover, the principle of virtual work is sat-isfied locally with equilibrated tractions. We evaluate the proposed HHO methodon two- and three-dimensional test cases from the literature: a sphere under inter-nal pressure, a quasi-incompressible Cook’s membrane, a perforated strip subjectedto uniaxial extension, and a cube under compression, and we compare the resultsproduced by the HHO method to the analytical solution whenever available or tonumerical results obtained using the industrial software code aster with conformingand mixed finite elements.

[1] D. A. Di Pietro, and A. Ern, A Hybrid High-Order locking-free method forlinear elasticity on general meshes. Comput. Methods Appl. Mech. Engrg.,283:1–21, (2015).

[2] D. A. Di Pietro, A. Ern, and S. Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstructionoperators. Comput. Methods Appl. Math., 14(4):461–472, (2014).

[3] M. Abbas, A. Ern, and N. Pignet, Hybrid High-Order methods for finitedeformations of hyperelastic materials. Comput. Mech.published online http://dx.doi.org/10.1007/s00466-018-1538-0 (2018).


24 Abstracts

14. Gradient-Thermo-Plasticityusing an Efficient Virtual Element Method (VEM)Blaz Hudobivnik, Fadi Aldakheel, Peter Wriggers,Institute of Continuum Mechanics, Leibniz Universitat Hannover

The coupled thermo-mechanical strain gradient plasticity theory, based on an efficientlow order virtual element formulations, introduced in [1] is outlined within this work.To this end, we introduce a micromorphic approach to gradient thermo-plasticitymodel at finite strains. The key point is the introduction of dual local-global fieldvariables via a penalty method, where only the global fields are restricted by boundaryconditions. Hence, the problem of restricting the gradient variable to the plasticdomain is relaxed, which makes the formulation very attractive for implementation,see [2, 3].

In the presented contribution, the recently developed virtual element method (VEM)will be used, because of the flexible choice of nodes number in an element which canbe changed easily during the simulation process, as addressed in [4–8]. Thus, thepotential energy is formulated in terms of suitable polynomial functions, instead ofcomputing the unknown shape functions for complicated element geometries, e.g.arbitrary convex or concave polygonal elements.

The modeling capabilities and algorithmic performance of the proposed formulationis demonstrated by a number of numerical examples.

[1] L. Beirao da Veiga, , F. Brezzi , L.D. Marini and A. Russo, The Hitchhiker’s Guide to theVirtual Element Method, Mathematical Models and Methods in Applied Sciences, 24(8):1541–1573, 2014.

[2] F. Aldakheel, C. Miehe, Coupled thermomechanical response of gradient plasticity, Int. J.Plasticity 91: 1-24, 2017.

[3] F. Aldakheel, Micromorphic approach for gradient-extended thermo-elastic-plastic solids inthe logarithmic strain space Continuum Mech. Thermodyn. 29(6): 1207-1217, 2017.

[4] P. Wriggers, B. D. Reddy, W. Rust, and B. Hudobivnik, Efficient virtual element formulationsfor compressible and incompressible finite deformations Computational Mechanics 60:253–268, 2017.

[5] P. Wriggers and B. Hudobivnik, A low order virtual element formulation for finite elasto-plastic deformations Computer Methods in Applied Mechanics and Engineering, 327:459–477,2017.

[6] M. L. De Bellis, P. Wriggers, B. Hudobivnik, and G. Zavarise, Virtual element formulationfor isotropic damage Finite Elements in Analysis and Design, 144:38–48, 2018.

[7] B. Hudobivnik, F. Aldakheel, and P. Wriggers, A low order 3d virtual element formulationfor finite elasto-plastic deformations, Computational Mechanics, 1–17, 2018.

[8] F. Aldakheel, B. Hudobivnik, A. Hussein, and P. Wriggers, Phase-field modeling of brittlefracture using an efficient virtual element scheme, Computer Methods in Applied Mechanicsand Engineering, 2018.


Abstracts 25

15. A Variationally Consistent Phase-Field Modelingof Brittle Fracture Applied to Pressurizedand Non-isothermal Elastic MediaNima Noii, Thomas Wick,Institute of Applied Mathematics, Leibniz Universitity Hannover

The variational approach to fracture by Francfort and Marigo and the related regular-ized formulation of Bourdin et al. which is also commonly referred to as a phase-fieldmodel of fracture is a widely accepted framework for modelling and computing thefracture failure phenomena in elastic solids.

This paper aims to extend the recently introduced phase-field formulation for a pres-surized fracture to a non-isothermal setting. An augmented regularized energy func-tional while describing a crack propagation in elastic media for a non-isothermaland pressurized material modelling is described. Primal-dual active set framework isformulated, to enforce crack dissipation inequality as a constraint, to the energy min-imization problem. Due to the extreme dependency of a phase-field model to the finemesh, i.e. accurately capturing the propagation of the crack, a predictor-correctorscheme for local mesh adaptivity is employed.

We verify our proposed non-isothermal and pressurized phase-field modelling withTran’s et. al test based on Sneddon’s setting. Later, we demonstrate the performanceof the proposed model by means of representative numerical examples. The proposedformulation is considered to be canonically consistent and robust scheme for treatingpressurized fractures and non-isothermal setting in an elastic media.


26 Abstracts

16. Optimal design of lattice materialsGregoire Allaire,CMAP, Ecole Polytechnique, France

This work is concerned with the topology optimization of so-called lattice materials,i.e., porous structures made of periodically perforated material, where the microscopicperiodic cell can be macroscopically modulated and oriented. Lattice materials arebecoming increasingly popular since they can be built by additive manufacturingtechniques. The main idea is to optimize the homogenized formulation of this prob-lem, which is an easy task of parametric optimization, then to project the optimalmicrostructure at a desired length-scale, which is a delicate issue, albeit computa-tionally cheap. The main novelty of our work is, in a plane setting, the conformaltreatment of the optimal orientation of the microstructure. In other words, althoughthe periodicity cell has varying parameters and orientation throughout the compu-tational domain, the angles between its members or bars are conserved. Severalnumerical examples are presented for compliance minimization in 2-d. This is a jointwork with Perle Geoffroy-Donders Olivier Pantz.


Abstracts 27

17. Fracture Simulation of Anisotropic Mediaby a Phase-Field MethodBo Yin, Michael Kaliske,Institute for Structural Analysis, TU Dresden

Since fracture plays a particularly important role in industrial application, not onlyexperiments are adopted to investigate the complex fracture patterns, but also severalnumerical methodologies are intensively studied to simulate crack initiation, direc-tion, propagation as well as branching. The phase-field method has been provedto satisfy Γ-convergence of fracture, and is widely employed for structural analy-sis during the last decade. The fracture evolution does not depend on any explicitcriterion and does not require to trace the discontinuous displacement neither. Mean-while, simulation results show good agreement with other numerical strategies andexperiments. Regarding composite material, both the mechanical properties and frac-ture evolution influence the crack patterns. With respect to the fracture part, sincethe fracture resistance along the different directions is not identical, an anisotropicphase-field model is formulated by defining the fracture energy density function asdirectional dependent quantity. For the mechanical response, the fibre reinforcementinduces an anisotropic behaviour. Therefore, a modified portion of the potential re-lated to the fibre has to be considered. Coupling those two aspects, the driving forceto evolve the phase-field is discussed in details in this contribution. This proposal isformulated and implemented in the context of Finite Element (FE) framework. Forverification and evaluation, numerical examples are performed and discussed. Lastbut not least, conclusions and future perspectives close this presentation.


28 Abstracts

18. On Energetically motivated Branchingcharacterized by a Directional Phase-field SplitChristian Steinke, Michael Kaliske,Institute for Structural Analysis, TU Dresden, Germany

The phase-field method is a promising approach for the simulation of fracture pro-cesses in solid materials due to its continuous and mesh-independent description ofthe crack’s topology, that is combined with an energetic evaluation of the strain en-ergy density to drive the crack’s evolution. It is shown to cover all relevant aspectsof crack evolution, e.g. initiation, propagation, kinking, branching and arrest, whichcan be simulated both for static and dynamic problems, see e.g. [1] and [2], respec-tively. A basic assumption within the phase-field framework is the split of the strainenergy density, that leads to the decomposition of the strain and the stress tensorinto crack driving and persistent components. Most of the applications of the phase-field method are based on either a spectral or a volumetric-deviatoric split, despitetheir fundamental deficiencies in the correct approximation of basic crack boundaryconditions as discussed in [4]. Approving the necessity of a discrete orientation ofthe continuously approximated crack, a novel split is introduced, that is based on adirectional decomposition of the stress tensor, see [3]. It is shown, that elementarycrack kinematics are fulfilled and a thermodynamically consistent driving force forthe phase-field evolution results in plausible results for crack evolution benchmarksimulations. Based on an evaluation of the phase-field driving force, an alternativeexplanation for the branching phenomenon is discussed.

[1] M. Ambati, T. Gerasimov, L. De Lorenzis, A review on phase-field models ofbrittle fracture and a new fast hybrid formulation. Computational Mechanics55 (2015), pp. 383-405.

[2] M. J. Borden, Isogeometric Analysis of Phase-Field Model for Dynamic Brittleand Ductile Fracture PhD Thesis, The University of Texas at Austin (2012).

[3] M. Strobl, T. Seelig, A novel treatment of crack boundary conditions in phasefield models of fracture. Proceedings in Applied Mathematics and Mechanics1(2015), pp. 155-156.

[4] C. Steinke, M. Kaliske, A phase-field crack model based on directional stress de-composition. Computational Mechanics, accepted 2018.


Abstracts 29

19. A VEM formulation of Scalar damage in 2D solidsMaria Laura De Bellis,University of Salento, Lecce, Italy

This work is devoted to the study of a first order virtual element approach for model-ing the strain-softening response of quasi-brittle materials, such as concrete, rocks orceramic materials. In the context of a 2D formulation, virtual elements of arbitraryshapes, including concave geometries and hanging nodes, are implemented. We aimto test the effectiveness of the VEM in dealing with highly localized strains due tomaterial instabilities, typically exhibited by such types of materials undergoing severeloading conditions. The method is based on minimization of an incremental energyexpression, with a novel construction of the stabilization energy properly modifiedto deal with the isotropic elasto-damage model. Two regularization techniques, ei-ther based on local or non- local approaches, are used to overcome the well-knownspurious mesh sensitivity problems, that occurs in numerical computations when, inthe presence of softening behavior, the governing differential equations may looseellipticity thus, resorting to ill-posed boundary value problems. Numerical examplesinclude a non-uniform tensile test, a three-point bending test on a concrete beam,and, finally, a splitting test of a granite cylindrical specimen.


30 Abstracts

20. A Virtual Element Method for Crack PropagationFadi Aldakheel, Blaz Hudobivnik, Ali Hussein, Peter Wriggers,Institute of Continuum Mechanics, Leibniz Universitat Hannover

In this contribution crack propagation within brittle materials is presented by meansof the novel developed technique, the virtual element method (VEM). The method,which is introduced in [1], can be seen as a generalization of the classical finite el-ement method. In comparison to FEM, VEM allows usage of arbitrary polygonal(2D) and polyhedral (3D) elements due to its applicability beyond the isoparametricconcept. In addition to usage of arbitrary number of nodes in the elements, it iseasily possible to add extra nodes to existing elements during the computation with-out the need to change the element formulation. Such a method can be extremelyuseful for simulation of crack propagation problems. Further information about theimplementation of the method used in this contribution can be found in the work of[2,3].

[1] L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, Basicprinciples of virtual element methods., Mathematical Models and Methods inApplied Sciences.23(01), 199-214 (2013).

[2] B. Hudobivnik, F. Aldakheel, and P. Wriggers, A low order 3D virtual elementformulation for finite elasto-plastic deformations., Computational Mechanics(2018): DOI: 10.1007/s00466-018-1593-6.

[3] F. Aldakheel, B. Hudobivnik, A. Hussein and P. Wriggers, Phase-field modelingof brittle fracture using an efficient virtual element scheme. Computer Methodsin Applied Mechanics and Engineering, 2018.


Abstracts 31

21. A phase-field model for brittle fatigue fractureMarreddy Ambati1, Pietro Carrara1, Roberto Alessi2 and Laura De Lorenzis1,1 Institute of Applied Mechanics, Technische Universitat Braunschweig, Germany2 Dip. Matematica, Universita Roma La Sapienza, Italy

Fatigue is a key phenomenon in mechanics, and is responsible for more than 75%of structural failures. Despite the significance of the problem, most existing fatiguetheories are based on empirical laws that lack of generality and predictive capabilities.Hence, the development of mathematically sound and reliable fatigue models is stillan open issue.

Recently, Alessi et al. [1] proposed a new variational fatigue phase-field model wherethe formulation relies on the introduction of a suitable internal (strain) history vari-able. The basic idea of the model is to let the fracture energy decrease as theaccumulated strain measure increases, which is obtained by introducing a dissipationpotential which explicitly depends on the history. Therein the analysis is limitedto a simple case, namely: one-dimensional, linear elasticity, brittle material behav-ior and symmetric response in tension and compression. In this work, we extendthis model to higher dimensions including the unsymmetric response in tension andcompression for the evolution of both phase-field and internal strain variable. Thetension-compression splits proposed by Amor et al. [2], Miehe et al. [3] and Freddiand Royer-Carfagni [4] are adopted. To show the predictive capability of the pro-posed model, classical benchmark problems, such as single edge tension/shear testand compact tension (CT) specimen for cyclic loadings are investigated. The effectsof load amplitude and fatigue material parameters on the fatigue life are studied.Also, for the CT specimen, the Paris law curves relating the fatigue crack growthrate obtained numerically ( ∆a

∆N) vs. the stress intensity factor (∆K) deduced by avail-

able semi-empirical relationships are presented. It is shown that the model is ableto describe all three typical fatigue fracture regimes, namely the elastic, the stablefatigue crack propagation and the final abrupt fracture stages.

[1] R. Alessi, S. Vidoli, L. De Lorenzis, Variational approach to fatigue phenomena witha phase-field model: the one-dimensional case., Engineering Fracture Mechanics 190(2018) 53-73.

[2] H. Amor, J.J Marigo, C. Maurini, Regularized formulation of the variational brittlefracture with unilateral contact: Numerical experiments., Journal of the Mechanicsand Physics of Solids 57 (2009) 1209-1229.

[3] C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically consistent phase-fieldmodels of fracture: variational principles and multi-field FE implementations., Inter-national Journal of Numerical Methods in Engineering83 (2010) 1273-1311.

[4] F. Freddi, G. Royer-Carfagni, Regularized variational theories of fracture: A unifiedapproach. Journal of the Mechanics and Physics of Solids58 (2010) 1154-1174.


32 Abstracts

22. Matrix-Free Solversfor Phase-Field Fracture ProblemsDaniel Jodlbauer,RICAM, Linz

The non-linear and non-convex nature of the energy functional combined with thevariational inequality associated to phase-field fracture models puts a great challengeto most optimization algorithms. Dealing with possibly many iterations of the non-linear solver (e.g. active set), a fast method for solving the linearized problemsis essential for a good performance. In this talk, we take a look at a matrix-freegeometric multigrid implementation, which gains additional speed by eliminatingthe need to update the Jacobian every time when either the grid, linearization point(within Newton’s method) or active set changes.


Abstracts 33

23. A Phase-field Model for Fracturesin Incompressible SolidsKatrin Mang1, Thomas Wick1, Winnifried Wollner2,1Institute of Applied Mathematics, Leibniz Universitat Hannover2Technische Universitat Darmstadt

In this talk, a quasi-static phase-field fracture model in incompressible solids is devel-oped. Due to the incompressible material behavior, we adopt a mixed system for thesolid displacement equation resulting in two unknowns: a displacement field and a hy-drostatic pressure variable. The fracture path is described with a phase-field functionthat is a smoothed indicator variable. The final system consists of three variables. Inorder to fulfill the stability of inf-sup type, we employ a Taylor-Hood element for thedisplacement-pressure system. The phase-field variable is spatially discretized withbilinear functions. The resulting discrete system is treated in a monolithic way. Twonumerical tests substantiate our model and algorithmic developments. These testsinclude spatial mesh refinement studies and variations in Poisson’s ratio approachingthe incompressible limit.

[1] B. Bourdin, G. Francfort, and J.-J. Marigo. Numerical experiments in revisited brittlefracture. J. Mech. Phys. Solids , 48(4):797–826, 2000.

[2] The Differential Equation and Optimization Environment: DOpElib. http://www.

dopelib.net.

[3] G. Francfort and J.-J. Marigo.Revisiting brittle fracture as an energy minimizationproblem. J. Mech. Phys. Solids, 46(8):1319–1342, 1998.

[4] C. Goll, T. Wick, and W. Wollner. DOpElib: Differential equations and optimiza-tion environment; A goal oriented software library for solving pdes and optimizationproblems with pdes. Archive of Numerical Software, 5(2):1–14, 2017.

[5] T. Wick.An error-oriented Newton/inexact augmented Lagrangian approach for fullymonolithic phase-field fracture propagation. SIAM Journal on Scientific Computing,39(4):B589–B617, 2017.

[6] T. Wick.Modified Newton methods for solving fully monolithic phase-field quasi-staticbrittle fracture propagation. Computer Methods in Applied Mechanics and Engineer-ing, 325:577 – 611, 2017.


http://www.dopelib.net.

http://www.dopelib.net.

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