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COMPONENT-BASED REDUCED BASIS SIMULATIONS

2015

AKSELOS S.A. EPFL Innovation Park 1015 Lausanne, Switzerland

Akselos Inc. 15 River St. Boston, MA 02108, USA

www.akselos.com info@akselos.com community.akselos.com

Abstract

Akselos offers the possibility to perform extremely fast simulations of large and com-plex parametrized systems. The two main ingredients to achieve these fast simulationsare (1) the Reduced Basis (RB) Method, a powerful model order reduction techniquefor parameterized systems; and (2) the component based approach, allowing engineersto create and modify fast simulation models in a very flexible and easy-to-use way. Inthis whitepaper we give an overview of the mathematical principles behind Akselos’scomponent-based RB simulation technology.

We start by recalling the principles of the CAE gold standard technology, FiniteElement Analysis (FEA), and we then explain how RB simulations are built on top ofFEA in order to accelerate the simulations without loosing accuracy.

In the second section, we first explain why “substructuring” or “superelement” tech-nologies are relevant for the simulation of large industrial problems, but still suffer fromhigh computational costs. We then describe how RB can be combined with substruc-turing in order to allow fast and accurate simulations of large systems.

Subsequently, we summarize the advantages of Akselos’s technology in the thirdsection and give an overview over the types of problems that can currently be modeledwith Akselos software. We conclude with a demonstration of the performance of thenew technology for a structural analysis, a modal analysis and a transient problemsetting in Section 5 and summarize in Section 6.

Contents

1 How do RB simulations relate to FEA? 21.1 The current FEA principles and weaknesses . . . . . . . . . . . . . . . . 21.2 The Reduced Basis technology accelerates FEA while maintaining accuracy 3

2 Key to success: Combining RB with acomponent-based approach 52.1 Basic principles of component-based approaches . . . . . . . . . . . . . . 52.2 Speeding up component approaches with Reduced Basis methods . . . . 7

3 Technology differentiators 8

4 Solver capabilities 11

5 Numerical examples 125.1 Example: structural analysis of a shiploader . . . . . . . . . . . . . . . . 125.2 Example: modal analysis of a shiploader shuttle . . . . . . . . . . . . . . 145.3 Example: transient analysis of a shuttle . . . . . . . . . . . . . . . . . . 14

6 Summary 16

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1 How do RB simulations relate to FEA?

1.1 The current FEA principles and weaknesses

FEA follows three major steps: modeling of the system geometry, meshing of thegeometry and then computational approximation of the Partial Differential Equation(PDE) solution. The modeling of the geometry is performed using CAD software,and the meshing is then produced semi-automatically using computational algorithms;however for complex geometries it is not uncommon for the mesh generation process torequire significant human intervention in order to develop high-quality meshes, whichtypically involves many days of work. Finally, based on this mesh, one chooses a discretefunction space that will serve to approximate the PDE solution – typically functionsthat are piecewise linear or quadratic on each element of the mesh.

Figure 1: Current workflow for system design using FEA. Design is performedwith CAD software and FE Analysis is used for the final validation.

This leads to a linear algebraic system of size N – with N dependent on the numberof mesh nodes:

[A]︸︷︷︸N×N

[x]︸︷︷︸N×1

= [f ]︸︷︷︸N×1

.

(If the PDE is non-linear, then we obtain a nonlinear algebraic system, which is solved

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by first linearizing using Newton’s method. As a result, in either the linear or nonlinearcase we must deal with large linear systems of equations as described above.)

FEA is very powerful because any geometry – even the most complex ones – canbe represented using a mesh, and the methodology applies to most classes of PDEs.FEA is used to simulate many types of physics, e.g. elasticity, thermal, acoustics,electromagnetics, fluid dynamics. FEA can also deal with multi physics, in whicha simulation model couples multiple different types of physics together into a singlemodel.

However, FEA software has some significant drawbacks:

• Steep learning curve: Its use requires expertise in computational engineering.

• Labor intensive: Building computational models is very time-consuming andtedious, as pointed out above the meshing step alone often takes several days.

• Computationally intensive: Requires powerful, expensive hardware and sim-ulations can take up several hours or even days. The cost of solving the linearalgebraic system is about O(N 1+α), (α > 0, depending on sparsity and condi-tioning), and for large scale problems it is very common to have N ≥ 107. Thematrix inversion process then takes several hours, for one solution.

• Simulations are one-shot: Back to the CAD and meshing process each timeyou change a design.

• System size is limited, since solving time and memory requirements grow super-linearly with system size.

1.2 The Reduced Basis technology accelerates FEA whilemaintaining accuracy

To address these shortcomings, there has been a significant research effort on ReducedBasis (RB) technology since 2000, with leading research being done on this topic atEPFL and MIT. The RB technology is a subclass of Model Order Reduction (MOR)algorithms.

Akselos is bringing this new technology to market. RB algorithms use FEAas the underlying technology, and adds an extensive layer of mathematics andcomputational algorithms on top of FEA in order to:

• accelerate the simulation time by orders of magnitude without compromisingaccuracy

• simplify the model-construction process so that it is very easy to modifysimulations models; this is crucial in practice since engineers and scientists gener-ally need to simulate many different system configurations in order to gain insightinto a problem.

The key observation that leads to RB is that the discrete function space XN ofsize N that is used in FEA to approximate the solution of a PDE is much larger thannecessary. Actually, for a given PDE, if we vary some parameters µ of the PDE, weobserve that the solution uN (µ) lives on a manifold of much smaller dimension than

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XN . For instance, if the PDE depends on a one dimensional parameter, then thesolution uN (µ) lives on a one dimensional (nonlinear, but often very smooth) filamentas represented in the figure below.

uN (µ1)uN (µ2)

uN (µ3)

uN (µN )

uN (µ)XN

As a consequence, it is sufficient to approximate this filament instead of the wholefunction space, and this can be achieved using an approximation space of much smallerdimension N N . This approximation space is built up using FEA computationsof N “snapshots” uN (µi) of the filament for various parameter values µi, 1 ≤ i ≤ N .Hence, RB algorithms use FEA as the underlying technology. Eventually one arrivesat a linear algebraic system of size N

[A]︸︷︷︸N×N

[u]︸︷︷︸N×1

= [f ]︸︷︷︸N×1

.

for which the solution complexity is O(N3).In practice, for many problems of commercial interest, we can achieve N ∼ 10 (com-

pared to N ≥ 107 for FEA), hence the solution computation is orders-of-magnitudefaster with RB than FEA, while typically retaining accuracy within 1% with respectto the FEA solution. It is important to note that there is no free lunch here: this hugecomputational speed up is obtained at the expense of prior “Offline” computation ofN snapshots, of complexity O(NN 1+α). Hence the RB approach is interesting in twocontexts: 1) the many-query context and 2) the real-time context.

Many-query context: In this context, an engineer or scientist needs to solve aPDE many times, but with different configurations. This occurs in design optimizationworkflows, for example. In this case the Offline cost is redeemed by the large number ofsolutions required. For instance, assuming that N = 10 and each FE solution requires1 hour, if we need to compute 1000 different solutions, it would take:

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• 1000 hours with FEA

• 10 hours with RB to perform the Offline computations (and the subsequent RBonline computations are extremely fast)

Real-time context: In this context, an engineer or scientist wants to get simulationresults in real-time, e.g. in order to use simulations in conjunction with an infrastructureassessment device that detects cracks in infrastructure. For large-scale problems, it isnot possible to get real-time response with FEA, whereas it is possible with RB, andhence this is another area where RB is very attractive.

However, the RB technology also presents some limitations:

• N becomes too large for problems that are not smooth enough, in particular whenthe parameter dimension is too large, or when considering complex geometricalvariations.

• Complicated changes of the geometry can’t be handled (they necessitate the con-struction of a new RB).

• RB can only be applied on very specific and well identified problems. This is oftennot the case in industrial problems, where engineers need considerable flexibility inmodifying or calibrating simulation models in order to match the physical systemsthey deal with.

To overcome these limitations and to turn the RB model order reduction techniqueinto a flexible and powerful tool for engineers in industry, Akselos combines the RBtechnology with a parametrized component-based approach. This combination is de-scribes in the next Section.

2 Key to success: Combining RB with a

component-based approach

2.1 Basic principles of component-based approaches

In the framework of FEA, many companies (e.g. MSC Nastran, ANSYS, Dassault Sys-temes) provide “superelement” technologies, which consist in a subdivision of complexsystems into simpler components in order to facilitate the design and meshing steps,which are potentially laborious and computationally intensive. For instance, a structuresuch as a shiploader shuttle can be broken down into several substructures.

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This divide and conquer strategy allows for:

• Division of labor: different substructures are taken care of by different designteams, which can work independently and in parallel as long as the interfacebetween substructures remain compatible.

• Taking advantage of repetition: many structures are built of many similar com-ponents, in that case it is only necessary to deal with each archetypal component.

• Parallel computation: the computation can be seamlessly run in parallel followingthe substructuring.

• Modularity makes it easier to modify systems: Change a global system by modi-fying individual components.

Let us present for instance the simple case for which a system is broken down into twocomponents corresponding to subdomains Ω1 and Ω2 with interface P .

P

Ω1 Ω2

We can group the degrees of freedom into degrees of freedom internal to each subdomain(uΩ1 , uΩ2) and degrees of freedom U belonging to the interface. The FE linear systemthen reads AP AT

P,Ω1AT

P,Ω2

AP,Ω1 AΩ1 0AP,Ω2 0 AΩ2

UuΩ1

uΩ2

=

fP

fΩ1

fΩ2

.

We then follow a static condensation procedure that allows elimination of the degreesof freedom internal to each subdomain: we compute the Schur complement matrix and

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right hand side

A = AP −ATP,Ω1

A−1Ω1AP,Ω1 −A

TP,Ω2

A−1Ω2AP,Ω2

F = fP −ATP,Ω1

A−1Ω1

fΩ1 −ATP,Ω2

A−1Ω2

fΩ2

which yields the static condensation system

[A]︸︷︷︸Np×Np

[U]︸︷︷︸Np×1

= [F]︸︷︷︸Np×1

.

This system is now of size Np, the number of degrees of freedom on the interface P ,which is much smaller than N . Hence this system is much faster to solve than theoriginal FE system, with complexity O(N 3

p ) instead of O(N 1+α). However, assembling

the static condensation system is very costly: computation of A−1Ω1AP,Ω1 corresponds

to Np FE solutions on the subdomain Ω1, of complexity O(NpN 1+α1 ), where N1 is the

size of the FE discretization on Ω1. So the total complexity of the static condensationprocedure is O(Np(N 1+α

1 +N 1+α2 )+N 3

p ), which is in practice often larger than the costof the original FE system of complexity O(N 1+α).

Hence the flexibility of design offered by a substructuring technique comes with anincreased computational cost if it is used in a regular FE setting.

2.2 Speeding up component approaches with Reduced Ba-sis methods

To ameliorate this large computational cost issue, several strategies have been developedusing model order reduction (MOR) locally in each component. Up to this day, themost commonly used method was the component mode synthesis (CMS).

With CMS, an eigenmodal basis truncation is used locally in each component, sothat the computation of A−1

Ω1AP,Ω1 is dramatically reduced: using a small number

N ∼ 10 of eigenmodes, computation of A−1Ω1AP,Ω1 is of complexity O(NpN3) instead of

O(NpN 1+α1 ). The total complexity of a CMS approach is O(NpN3 + N 3

p ) instead of

O(Np(N 1+α1 +N 1+α

2 )+N 3p ) for a FE static condensation, and in practice leads to large

computational gains, while retaining the design flexibility.Unfortunately, CMS approaches show poor convergence on some problems, hence

requiring a large N which in turn decreases the speed up. Moreover, CMS does notenable parameter variations: if a component is modified (material properties, geome-try), its eigenmodes have to be recomputed, corresponding to the solution of an FEeigenproblem of dimension N1.

Last but not least, the areas of the interfaces between components is actually verylimited with CMS, since the number Np is directly related to the number of mesh nodeson the interfaces. If the components have too many interface nodes, Np becomes toobig and the reduced model becomes computationally too expensive.

The technology commercialized by Akselos overcomes the issues raised with CMSby using RB for MOR, and coupling with the static condensation method used in superelements:

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• For smooth enough problems, RB provides optimal exponential convergence, henceguarantees a minimal N ,

• Since RB is used locally on a component, the local problem is expected to besmooth (simple geometry, few parameters), hence guarantees the previous point,

• RB offers parametric flexibility, since it is by nature built to approximate a PDEsolution over a range of parameters. There is no need to rerun the costly Offlinecomputations or remesh when parameters change.

• RB guarantees the accuracy of the MOR, which is not the case for CMS,

• Finally Akselos technology uses optimal port spaces that guarantees a minimal Npthat is independent on the number of mesh nodes on the component interfaces.

This approach is described in the key scientific paper “A Static Condensation Re-duced Basis Element Methods: Approximation and A Posteriori Error Estimation” byDr. Phuong Huynh, Dr. David Knezevic and Prof. A.T. Patera (April 2011).

3 Technology differentiators

In summary the advantages of Akselos’s simulation technology are the following:

• Large systems Akselos technology can produce simulation results in a few min-utes for systems with more than 1 billion degrees of freedom.

• Simulation speed Simulations are typically a thousand times faster than whenusing FEA. (An overview for a few example application for different physics canbe seen in Table 1) Akselos technology unlocks the full power of simulations byallowing real-time queries, multiple scenarios and iterative design.

• Parametrized geometries Every time a modification of the geometry is de-sired, one has to go back to the CAD model and perform a re-meshing and acostly re-computation of the modified model. Even if using Craig-Bampton CMS,geometrical changes require to rerun costly computations. With Akselos technol-ogy, one can modify the geometry of individual components without re-meshing– within defined parameter ranges – and rerun lightning fast simulations.

• Flexible component decomposition and large port areas With Craig-Bampton CMS, the component decomposition is limited by a strong constraint:the areas of the interfaces between components must be small so that they con-tain only a little number of mesh nodes in order for the reduction to be efficient.In contrast, Akselos technology uses optimal port representations that are notaffected by the port areas, and so any component decomposition is possible.

These advantages pay-off especially during the early design phase. The current de-sign workflow for engineers is shown in Figure 1. The work starts with many iterationsof the CAD (drawing only) and relies on the engineer’s experience and intuition. Anal-ysis (simulations) are kept out of the early design and used later to validate the design,once the most important decisions have been taken.

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Physics Number of dof Speed up Accuracy(Akselos vs FEA) (Akselos vs FEA)

Acoustics (Helmholtz) 3 million 5000x < 1%Linear Elasticity 5.5 million 900x < 1%

Modal analyis (elasticity) 1.3 million 700x < 3%Transient analyis (elasticity) 1.3 million 700x < 3%

Table 1: Examples of simulation speed up obtained with Akselos versus FEA.

Accuracy Large systems Speed Parametrized geometriesStandard FEA 3 7 7 7

FEA substructuring 3 3 7 7

Craig-Bampton CMS 7 3 3 7

Akselos 3 3 3 3

Table 2: Comparison between CAE simulation technologies.

In contrast, Akselos proposes a simplified workflow (Figure 2) whereby pre-computed,parameterized simulation components are used to perform simulations early in the de-sign process, when 90% of the decisions are taken.

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Figure 2: Workflow with component-based RB simulations. Pre-computedparametrized components embed the simulation expertise (meshes, physics, etc.)and enable the engineer to focus on designing robust and efficient systems.

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4 Solver capabilities

Table 3 summarizes the physics that can be handled with Akselos software. Note,that not all types of physics can be accelerated using the Reduced Basis technology.However, it is always possible to represent particular phenomenas using FEA withinthe software.

Akselos constantly works on extending the range of physics for RB-modeling, e.g.creep and other plastic deformations. If a customer requires a particular kind of physicsmodeled in the RB framework, we are happy to discuss the possibilities to enable theRB modeling of this type of physics within our software.

Analysis type Reduced Basis FEA-Enabled features

- steady-state linear - buckling analysis

- transient linear - nonlinear steady stateand transient

Structural- modal analysis

- contact analysis

- frequency-domain - time-domain acousticsacoustics

- modal analysis

Acoustics

- Steady-state linear - nonlinear (temperature-dependent materials)

- transient linear

Thermal

Table 3: Overview over the types of analysis that can be performed using Akselossoftware.

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5 Numerical examples

In this section we will demonstrate the capabilities of Akselos’s simulation approach for(1) a structural analysis computing stresses and strains, (2) a fast modal analysis and(3) a time-dependent analysis of a structure.

5.1 Example: structural analysis of a shiploader

A shiploader is a large steel structure that supports a belt and is used to carry min-erals from trucks on the ground to ships on the water. The complete 3D model of ashiploader is shown in Figure 3 (a). Using FEA, the mesh for the complete model has1.8 million nodes, 6.3 million elements (tetrahedral elements), and the final finiteelement system has 5.5 million degrees of freedom. In order to use Akselos sim-ulation technology, this model is decomposed in 323 components, with 65 differentcomponent types. Some of these components are shown in figure (c). Some componenttypes are reused several times in the model, such as trusses. The total data footprintfor Akselos precomputed components is 5.1 GB.

With Akselos, the structural analysis results (displacements) are obtained within2 seconds, whereas it takes 30 minutes with a standard FEA. Akselos technologyallows to obtain a massive speed up of 900 while keeping the accuracy (maximumerror 1%). After a postprocessing of the displacement solution field, we can displaythe stresses in the structure as shown in figure (b). Some of the components in theAkselos model are parametrized, such as a truss with a crack that can be movedlaterally. The user can change the location of the crack without remeshing, recomputethe global displacement solution also in 2 seconds, and then visualize the stresses asshown in figure (d).

Speed up to FEA 900

Maximum error to FEA 1%

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(a) Shiploader model. (b) Stress visualization.

(c) Some shiploader components. (d) Parametrized crack component.

Figure 3: The figures show the model and the result for the structural analysisof a shiploader from Section 5.1.

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5.2 Example: modal analysis of a shiploader shuttle

Here we consider a subpart of the shiploader structure presented on the previous page:the shuttle is the part of the shiploader that can slide in and out in order to adjustthe length of the structure. The full shuttle model with its components is shown onpage 6. This model is decomposed in 69 components, with 31 different componenttypes. Regarding FEA, the mesh for the complete model has 440 thousands nodes,1.6 million elements (TETRA), and the final finite element system has 1.3 milliondegrees of freedom.

With Akselos simulation technology, the modal analysis results (first five displace-ment eigenmodes) are obtained within 0.5 seconds, whereas it takes 6 minutes with astandard FEA. Akselos technology allows to obtain a speed up of 700. Fig-ure 4 (a) shows the shuttle model with the locations (indicated by lock symbols) whereclamping boundary conditions have been imposed. Figures 4 (b) and (c) show the firstand fifth eigenmodes respectively.

(a) Shuttle model with clamp-ing locations. (b) First mode. (c) Fifth mode.

Figure 4: Model and exemplary eigen modes of the shiploader shuttle.

Speed up to FEA 700

Maximum error to FEA 3%

5.3 Example: transient analysis of a shuttle

Here we consider the same shuttle structure as in the previous page, but we introducea small modification: we removed the diagonal trusses in order to decrease the lateralstiffness of the structure. This modification of the geometry doesn’t requireremeshing since we only need to remove a few components in the model. This illus-trates one of the advantages of Akselos component-based models, namely to modify thetopology of simulation models without the need to remesh.

We then perform a transient analysis of the structure after a lateral shock on theend of the structure (normal load applied at t=0s), with the same clamping boundary

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conditions as in Figure 4 (a) in the previous page. Figures 5 (a) to (d) show theevolution of the displacement with respect to time, where we observe the structureoscillating around its resting position. The transient analysis is computed using modalsuperposition, hence the speed up obtained with respect to FEA is the same as the oneobtained for modal analysis.

(a) t=0.1s (b) t=0.3s

(c) t=0.6s (d) t=1s

Figure 5: A local lateral shock is applied at initial time t=0s.

Speed up to FEA 700

Maximum error to FEA 3%

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6 Summary

In this paper we outlined the mathematical background of Akselos’s simulations tech-nology. The main ingredients are the Reduced Basis Method in combination with acomponent-based approach. This allows engineers to build up complex and large modelsin a very easy way and subsequently empowers them to perform fast simulations.

The simulation models can be equipped with thousands of parameters, which allowsto decouple the design process from CAD and to perform design optimizations usingfast parametrized simulations.

In exemplary numerical simulation we demonstrated for a structural analysis, amodal analysis and a transient analysis the large speed-up of Akselos’s simulation tech-nology compared to conventional FEA.

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