ANR SeARCH – D 1.3.1

New simulation method of deformable metallic objects

ESTIAESTIANicolas Verdon, Pierre Joyot

Date : septembre 2012

ii

Contents

1 Scientific Report 11.1 Objectives and accomplished work . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 ANM-POD approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 POD or Karhunen-Loève decomposition . . . . . . . . . . . . . . . . . . 71.3.2 Asymptotic Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.2.1 Main ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2.2 Padé approximants . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2.3 Application for hyperelastic materials . . . . . . . . . . . . . . 91.3.2.4 Application for 1D plasticity . . . . . . . . . . . . . . . . . . . . 151.3.2.5 Application for 3D plasticity . . . . . . . . . . . . . . . . . . . . 20

1.3.3 Combination of the two approaches . . . . . . . . . . . . . . . . . . . . . 231.4 An alternative: PGD-ANM approach . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4.1 Proper Generalized Decomposition . . . . . . . . . . . . . . . . . . . . . 241.4.2 PGD-ANM algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.5.1 Hyperelastic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.5.1.1 Traction test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.5.1.2 Bending test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.5.1.3 Computational costs . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5.2 Results for plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.5.2.1 Imposed deformation for a 1D example . . . . . . . . . . . . . 301.5.2.2 Plastic deformation of a 3D beam . . . . . . . . . . . . . . . . . 30

1.5.2.2.a ANM results in traction . . . . . . . . . . . . . . . . . . . 311.5.2.2.b Computing the POD for the traction case . . . . . . . . . 321.5.2.2.c ANM results in bending . . . . . . . . . . . . . . . . . . 341.5.2.2.d Computing the POD for the bending case . . . . . . . . 351.5.2.2.e Performances of the ANM-POD approach . . . . . . . . 35

iv

1.5.2.3 Conclusions and perspectives for plasticity . . . . . . . . . . . 371.5.3 PGD-ANM approach for the heat equation (voir Pierre) . . . . . . . . . . 38

1.6 Conclusions and perspectives of the work . . . . . . . . . . . . . . . . . . . . . . 38References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Appendices 43

A Quelques mots sur les optimisations dans FEniCS 45A.1 Numérotation des noeuds et stockage numpy . . . . . . . . . . . . . . . . . . . . 45

A.1.1 Vecteur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.1.2 Tenseur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

A.2 Produit tensoriel sans compilation . . . . . . . . . . . . . . . . . . . . . . . . . . 46A.2.1 Syntaxe classique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A.2.2 Syntaxe numpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

A.3 “Produit mixte” sans compilation . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A.3.1 Syntaxe classique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48A.3.2 Syntaxe numpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

B Users guide of ANM-POD FEniCS code and its GUIs 49B.1 The different steps of a platic computation . . . . . . . . . . . . . . . . . . . . . 49

B.1.1 Organisation of the folder . . . . . . . . . . . . . . . . . . . . . . . . . . . 49B.1.2 From launching a computation to the post-processing . . . . . . . . . . . 50

B.2 GUI for monitoring the computation . . . . . . . . . . . . . . . . . . . . . . . . . 50B.3 Scripts for post-processing datas . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

B.3.1 Generating pdf files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52B.3.2 Creating the stress-strain curves . . . . . . . . . . . . . . . . . . . . . . . 53B.3.3 Computing the POD basis and the reconstruction solution for each load

parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

List of Figures

1.1 3D beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.2 Evolution of the beam deformation with increasing the load λ (traction) . . . . 271.3 Evolution of the maximal displacement with λ for the traction test . . . . . . . . 281.4 Evolution of the beam deformation with increasing the load λ (bending) . . . . 291.5 Evolution of the maximal displacement with λ for the bending test . . . . . . . 291.6 Comparison of the full and reduced models in terms of CPU time for different

mesh sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.7 Stress-strain response for the 1D problem with deformation theory. Computa-

tional parameters: N = 15, δ = 10−2, η1 = 10−4 . . . . . . . . . . . . . . . . . . . 311.8 Stress-strain response for the 3D beam in traction using ANM. Computational

parameters: N = 5 and η1 = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.9 Comparison between ANM results in the case of traction on the 3D beam for

different values of δ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.10 Evolution of the displacement with the load parameter λ for 3D plastic compu-

tations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.11 Error fields computed between reference solutions and solutions reconstructed

using the POD basis for different load parameters λ. Parameters are: Nsnap = 10,NPOD = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.12 Error fields computed between reference solutions and solutions reconstructedusing the POD basis for traction at different load parameters λ. Parameters are:Nsnap = 100, NPOD = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.13 Energy contribution of the 10 first modes of the POD basis from a sample of 100snapshots for the traction case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.14 Stress-strain response for the 3D beam in bending using ANM. Computationalparameters: N = 10 and η1 = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.15 Error fields computed between reference solutions and solutions reconstructedusing the POD basis for bending at different load parameters λ. Parameters are:Nsnap = 10, NPOD = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vi LIST OF FIGURES

1.16 Energy contribution of the 10 first modes of the POD basis from a sample of 100snapshots for the bending case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.17 Stress-strain response for the 3D beam in traction and bending using the ANM-POD approach. The computational parameters are the same as those used forthe respective full order computations. . . . . . . . . . . . . . . . . . . . . . . . . 36

1.18 Comparison between PGD direct and PGD-ANM for different values of k(u). . 38

B.1 Structure of the reference folder that contains files for plastic computations andthe corresponding post-processing scripts. . . . . . . . . . . . . . . . . . . . . . . 49

B.2 Selecting the monitoring LOG file . . . . . . . . . . . . . . . . . . . . . . . . . . . 50B.3 Sequence of screenshots during the monitoring phase . . . . . . . . . . . . . . . 51B.4 Example of a resulting pdf file that contains all the informations of an ANM

elastoplastic computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

List of Tables

1.1 Classification of ROMs following Antoulas et al [7] . . . . . . . . . . . . . . . . 51.2 Sum up of the different methods proposed by the computer graphics and me-

chanics communities for the three mechanical fields of interest: linear elasticity,hyperelasticity and elastoplasticity. The red + and - indicate the relevancy of themethod to the considered application, whereas the red + and - precise the CPUefficiency. The gray cells precise that using the methods for such examples arenonsense. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 ANM parameters used for plastic computations for the traction case. . . . . . . 31

viii LIST OF TABLES

1Scientific Report

1.1 Objectives and accomplished work

Concerning the mechanical part of the ANR project SeARCH, the work was divided into twodifferent tasks:

1. development of a new simulation method of deformable metallic objects

2. computation of inverse simulations of eroded stones

For the first task, the main objective is to determine the effects of any arbitrary mechanicalloading to a metallic object, from the mechanical point of view. In other words, we would liketo be able through computations to access to the following datas:

• the displacements

• the strain and stress fields in the structure

• the deformed state: elastic, plastic, etc...

The challenging part of this work is the physical determination of the structural deforma-tions. As the project involves different communities, namely computational mechanics andcomputer graphics, it is important to distinguish the criteria of accuracy that are required toassert the validity of the computations. In most of the works achieved in the field of computergraphics, the most important aspect is the visual rendering. The computations are aimed to bevisually realistic, but realistic renderings do not assure a correct physical behaviour. In con-trary, from the mechanical point of view, the correct description of stresses and strains in thematerial is the principal criterion of validity of the model. In consequence, it is important toconsider the equations and assumptions that correspond to the physics we want to properlydescribe. For instance, when determining the deformation of a structure, we can assume thedeformations of the material to be:

• elastic: it describes materials that return to their rest shape after applied stresses areremoved. The corresponding hypothesis is that we consider small displacements and

2 CHAPTER 1. SCIENTIFIC REPORT

small deformations. Such conditions are generally rarely observed in realistic configu-rations.

• hyperelastic: it describes a nonlinear elasticity of materials. For such a case, small dis-placements and deformations are no more required. The involved equations are nownonlinear and hence more difficult to solve numerically.

• elastoplastic: it describes materials that permanently deform after a sufficient appliedstress. This mechanical behaviour is strongly nonlinear and its numerical solving isnowadays always a challenging task: plasticity requires incremental procedures that arecomputationally expensive and their improvements are not easy. Hence, performingreal-time plastic deformations of metallic structures is a difficult task, even impossiblewith the actual computational power.

According to the modelisation of our problem, we have to deal with systems of equations thatare of varying complexity. In the framework of the archeological application using tangibleinterfaces, another constraint is imposed to the mechanical simulations: the CPU costs have tobe very low, ideally we would like to be able to perform real-time simulations. This additionalconstraint was a motivation for getting into the problematic step by step. First with a linearapproach, we could easily implement the coupling between the mechanical simulations andthe TUI aspects. Then, we chose to increase the difficulty by dealing first with the hyperelas-ticity before plasticity.

The second task of the project concerns the modelisation of the erosion with reduced or-der models such as POD or PGD, these methods will be described after. Even if equationsdescribing the erosion phenomenon are available in the literature, the main problem of itsmodelisation relies on the evolving boundaries that are involved in the process. As far as weknow, this problem, which can also be seen as an evolving wave/discontinuity, remains un-solved. For this reason, and because the first task was enough difficult to solve, we focusedour work on the reduced-order modeling applied to the elastoplastic deformations of metallicobjects.

The results of the accomplished work in the framework of the ANR project is summed uphere. We present synthetically what has been achieved, the main advantages and limitationsof the approaches, and what we will propose for the future works. So, up to now, the followingtasks have been carried out:

• the theoretical ideal case of linear elasticity has been used to test the coupling betweenthe TUI and the mechanical solve.

• the hyperelastic case has been then intensively studied. It is for our purpose a goodcandidate for testing algorithms on nonlinear equations. First computations have beentreated using the classical Newton approach in order to have reference solutions. Then,the combined ANM-POD approach – initially developped in [24, 29] – has been imple-mented in the finite elements library FEniCS/Dolfin we used throughout the project, see[18, 19]. The CPU gains are indeed very interesting for real-time applications, that iswhy its coupling with the TUI is currently in progress.

1.2 Literature review 3

• the elastoplastic case is finally considered. Its solving is achieved using the ANM ap-proach. The method has been tested for the simplest case of the 1D perfect plasticitywith imposed deformation before being extended to the 3D geometry. The 3D solutionobtained with the ANM has then been sampled in order to build a POD basis. The inter-polation of the solution – i.e. without solving reduced equations – over this basis leadsto informations that can be used for coupling with a tangible interface because it doesnot require large CPU costs. The ANM-POD method used for the hyperelasticity is alsoimplemented for the plasticity equations. Its computational efficiency is not so impres-sive as the CPU gains obtained with the hyperelastic example. Indeed, whereas only oneANM continuation step is needed for computing the whole solution for the hyperelasticcase, the strong nonlinearity involved in the plasticity equations requires much morecontinuations with the ANM-POD approach (as many continuations as for the ANMalone) which limits the CPU gains we could obtain. This formulation can however beinteresting for problems with far more finite elements.

• an alternative to the ANM-POD approach has been explored: it consists in couplingANM and PGD. Because this method is still in development, it has been applied to astandard nonlinear example that does not correspond to the final objectives of the ANRproject, namely it has been test for the nonlinear heat equation.

In the following, in order to explain with more details the context and the problematicof the mechanical part in the ANR project, a short literature review presents some works inthe fields of computer graphics as well as in the reduced-order modelling community. Afterhaving explained the reasons of our choice, we will explain the different ingredients of thenumerical methods that are employed in this work, respectively the POD, the ANM and thePGD, and how these methods can be combined. Detailled theoretical developments are givenas well as the main important results that can be used for further applications.

1.2 Literature review

With the increasing computational power, both researchers and engineers are nowadays ableto compute the solutions of complex physical configurations in times that are more and moresatisfactory, at least for an industrial point of view. Nevertheless, achieving real-time simu-lations of concrete physical problems remains a challenging task so that it can be most of thetime considered as the Grail quest for the community of computational physics people. Thetask of fast computing phenomenon that are described by complex mathematics – most ofthe time Partial Differential Equations (PDEs) – is obviously not easy. As this problem arises inmany different fields of research, the solutions or at least suggestions that have been proposedmay be completely different regarding to the goal of the computations. We will try in the fol-lowing to distinguish the different approaches corresponding to the fields of application byprecising their range of validity and accuracy.In a first part of our investigation, we restrict the studied problem to the deformation, possi-bly nonlinear and plastic, of materials. This topic has been obviously intensively studied inthe field of computational mechanics where it is commonly treated with the use of the Finite

4 CHAPTER 1. SCIENTIFIC REPORT

Element Method, see [10] and [30] for precise and complete descriptions of the method, butthe community of computer graphics also developped methods for this purpose. Fast com-puting of such class of problems involves a reduction of the complexity due to the physicsand/or mathematics. In the field of computer graphics, it is usually done by considering ge-ometric deformations that are assumed to represent naturally the deformations. For instance,spring-mass based solutions are often used, due to their easy implementations and their lowcomputational costs, see for example [22] where this approach is used in combination withan haptic device. Such approaches are interesting in terms of CPU costs, but the results focusonly on 3D rendering as realistic as possible without validation of the complex mechanicalbehaviour, so that mechanical equations have to be included into the models. Unfortunately,even in the linear elastic assumption, the use of mechanical equations for describing the de-formation is seen unappropriate for real-time applications, because of the amount of largesystems to solve ; it is hence common to approximate the theoretical solution “by an heuristicgeometrical transformation which creates an intuitive and natural deformation” [26].Our task in the ANR SeARCH project is related to the understanding of the history of a staplemade of iron by means of mechanical computations, as accurate and fast as possible, becausethese computations are drived by an interaction device (here a ShapeTape). This device shouldallow to archeologists to easy simulate what happened to the staple, by defining intuitivelythe boundary conditions and exterior loads submitted to the structure. It is straightforwardthat computations for this purpose have to be really accurate, so that geometrical approachestypically used for real-time applications in computer graphics are nonsense. Even if effortsare made in the physically plausible rendering, the order of accuracy required for determiningthe history of what mechanically happened to the staple is far much higher than what canbe obtained with geometrical assumptions or multi rigid-bodies like computations as in [9].Other works like [21] may include more physics into the computations so that the renderingof the results are improved but as specified in the table that specify the CPU times required inthe computations, the need of precomputations for achieving real-time simulations is alwaysimportant. Changing any conditions in the configuration, such as the load(s) or the bound-ary conditions, would then require another precomputation phase which is too much timeconsuming for our imposed application. The key point lies in the fact that the goals of thecomputations are really different: whereas the works such as [21] for example aim to performthe best realistic 3D renderings, we are much more interested in determining: 1) the physicallyaccurate deformation of the staple in accordance with the real observed staple, and 2) the cor-responding loads that led to this deformed state. This could be achieved by running lots ofaccurate FE computations, for many possible boundary conditions and loads, but it wouldrequire too much CPU time for a real time application. This class of problems – parametricstudies – as well as optimisation and control problems are nowadays challenging tasks for fastand efficient computations.In this context, Reduced Order Models (ROMs) can be seen as the most appropriate candidatefor fast computations that must keep the physical meaning present in the PDEs. They havebeen indeed intensively used in many different research fields where such high-dimensionalityproblems occurs, for example face recognition [1], image analysis [20] or engineering prob-lems, covering control and systems theory [7, 16] and more generally multidimensional physics[14]. ROMs allow to compute very accurate solutions of complex physical problems with low

1.2 Literature review 5

computational times, without loosing the physical sense of the equations because they arebuilt in a way they must preserve some important properties from the physics, so that accu-rate solutions can be obtained in a range of parameters which is not necessarily the range ofthe precomputed solutions. Antoulas et al. [7] point out the important characteristics thatmust be satisfied when constructing a low order model:

1. the approximation error is small regarding the full-order system, and this error is bounded

2. system properties must be preserved (in control system stability, passivity, in fluid me-chanics divergence-free, etc...)

3. the numerical procedure must be stable and efficient

Many reduced order models have been developped regarding the field of activity, but An-toulas et al [7] classified them into two large branches: the SVD-based (Singular Value Decomposition)and Krylov-based methods, where another subdivision between linear and nonlinear applica-tions exits, such as depicted in table 1.1. In the field of mechanics, especially fluid mechanics,

SVD-based methods Krylov-based methodsNonlinear systems Linear systems

POD methods Balanced Truncation LanczosEmpirical Grammians Hankel Approximation Arnoldi

Interpolation

Table 1.1: Classification of ROMs following Antoulas et al [7]

POD is actually the most used ROM method, since it finds its (numerical) origins in the late80’s [28], and due to its good properties(optimal basis in an energetic sense, the basis func-tions are divergence-free and respect the boundary conditions). Whereas it is a common toolfor fluid mechanics, let us just mention one relevant reference in the field of aeroelasticityfrom the works of Charbel Fahrat’s team [6], it has not been used so far for solid mechanics,excepted in some marginal works, for example for information reduction purposes as we canfind in the works of Ryckelynck, see [27]. This can be easily explained with the degree ofcomplexity of the equations that are solved: whereas solutions of highly nonlinear problemsin the field of solid mechanics can be obtained with reasonnable CPU costs1, solving accuratelyNavier-Stokes equations in the turbulent domain leads to prohibitive CPU times, due to themultiple time and size scales that are involved in these phenomena. But now that the needof real-time applications is increasingly important, ROMs have been used also for solid me-chanics. One of the late applications concerns real-time surgery using a POD-ANM approach,see the works of Niroomandi et al [23, 24, 25]. The approach combining both POD and ANMtechniques proved to be very efficient in terms of accuracy and CPU costs, allowing real-timeapplications for surgerical applications with an haptic device.In addition, motivated by high-dimensional problems arising from kinetic theory and quan-tum chemistry, Chinesta and coworkers revisited the decomposition method called LATIN

1reasonnable for engineering purposes

6 CHAPTER 1. SCIENTIFIC REPORT

method [17], developped by Ladevèze in the 80’s. The resulting method, PGD for ProperGeneralized Decomposition has been widely applied, see for example the works of Ammar etal.[3, 4, 5], Chinesta et al. [12, 13] or González et al [15] where the authors successfully solvedproblems up to 109 degrees of freedom. For more detailed informations about the variety ofworks in this domain, a recent review [14] gives a quasi-exhaustive view of what has beenachieved until now with the PGD.

According to the references that have been collected in this short literature review, we cantry to sum up the main important features of the methods presented in this section for possibleapplication to our problematics. Table 1.2 can be hence seen as an attempt of classifying themost used methods in the two fields of activity mentionned before, using two simple criteria.

Computer Graphics Mechanics

spring-massrigid elastic

elastonsPOD1 PGDmulti-body deformation

[22] [9] [26] [21]

Linear elasticity++

+++++

+++++++

+++-

++++++

++++++

Hyperelasticity-/+

(rendering)+++

-/++(rendering)

++

-/++(rendering)

+

+++- - -

++++++

++ (?)+++

Elastoplasticity+++- - -

+++(++)

++ (?)++(+)

Table 1.2: Sum up of the different methods proposed by the computer graphics and mechanics communitiesfor the three mechanical fields of interest: linear elasticity, hyperelasticity and elastoplasticity. The red + and- indicate the relevancy of the method to the considered application, whereas the red + and - precise the CPUefficiency. The gray cells precise that using the methods for such examples are nonsense.

The first line of each cell, in red, corresponds to the relevancy of the method to representthe good physical behaviour relatively to the mechanical phenomenon we want to describe,namely the linear elasticity, the hyperelasticity or the elastoplasticity ; whereas the secondline, in blue, indicates the CPU efficiency of the method for the considered case. We tryed toestablish a hierarchy between the different methods thanks to their ability to describe well thephysics, in terms of the accurate determination of the efforts, coupled to the computationalefficiency. Some additional informations are also given: for the hyperelastic case for instance,it is precised that the methods are only able to produce visual rendering of large deformationswhereas the models used are obviously not valid anymore (nonlinear deformations with lin-ear assumptions) and hence the efforts cannot be described accurately. At last, some questionmarks have been added to precise that the methods have not been used yet for these purposeand then some questions remain.

1POD for linear elasticity, ANM-POD for the two other applications.

1.3 ANM-POD approach 7

So, in the context of the present work, namely nonlinear elastoplastic deformations of ametallic staple, both POD and PGD methods have been retained (as mentionned in green intable 1.2) due to their validated efficiency and accuracy for many applications. The followingwill present these methods combined with the ANM, an efficient method for solving stronglynonlinear problems that contains advantages for increasing the gain in CPU cost of the ROMs.

1.3 ANM-POD approach

Whereas the two different methods are well known and studied in many field of mechanicssince the late 80’s, the combination between POD and ANM technique is a quite recent pro-posed method. It has been first introduced in the paper of Yvonnet et al. [29] for studyingpost-buckling of cellular microstructures, and Niroomandi et al. [23, 24, 25] used the same ap-proach for hyperelastic deformations. In the following sections, we present the POD and theANM methods, and especially the formulation of the combined method for the deformationof an hyperelastic material.

1.3.1 POD or Karhunen-Loève decomposition

The idea of this technique is to build a basis of an evolution process based on the knowledge ofthe solution. For instance, let us assume that the solution of the studied equation is computedfor different loads. With this set of snapshot, we are able to describe the phenomenon with aproper basis, that is assumed to catch the solution for other loads.The basis is obtained as the solution of the following equation:

C φ = λ φ (1.1)

where Cis the two-point correlation matrix obtained through the matrix Q that contains the

M snapshots ui =

ui1ui2...uiN

, with i = 1, · · · ,M and N is the number of nodes of the completemodel. The matrix Q can hence be expressed as:

u11 u21 . . . u

M1

u12 u22 . . . u

M2

...... . . .

...u1N u

2N . . . u

MN

(1.2)so that the correlation matrix C is defined with:

c = QQT (1.3)

The resolution of (1.1) leads to M eigenmodes that are sorted thanks to their energetic con-tribution. By assuming that the m < M first modes φi are able to catch the most energeticcontribution, we retain these modes for building the corresponding low-order model, as ex-plain in the following sections for the examples of the linear and non-linear beam.

8 CHAPTER 1. SCIENTIFIC REPORT

1.3.2 Asymptotic Numerical Method

The Asymptotic Numerical Method (ANM) has been introduced in the 90’s to deal with non-linear problems. It consists in transforming the nonlinear problem into a recursive sequenceof well posed linear problems having the same tangent operator. In comparison with classicalmethods that are used for solving nonlinear problems, namely the Newton-Raphson methods,the ANM has proved to be much more efficient in terms of computational costs. More detailsabout the wide range of applications can be found in [2].

1.3.2.1 Main ideas

Let us describe the physical nonlinear problem by the following equations:

f(u,E) = 0 (1.4)

andg(S) = 0 (1.5)

which respectively represent the compatibility and equilibrium relations, and where u, E andS are respectively the displacement, the Green-Lagrange strain tensor and the second Piola-Kirchhoff stress tensor. The function f is defined according to the behaviour of the material(elastic, hyperelastic, viscoplastic, ... ) whereas g takes the body forces into account.

The main idea of this method is the expansion around an equilibrium point of U and thecontrol parameter λ into power series with respect to a path parameter a:{

U(a) = U0 + aU1 + a2U2 + · · ·

λ(a) = λ0 + aλ1 + a2λ2 + · · ·

(1.6)

By substituting the relations (1.6) into the governing equations of the problem (1.4) and(1.5) we obtain the recursive sequence of linear problems.

1.3.2.2 Padé approximants

Instead of using a polynamial expansion, it is also common to use Padé approximants, whichhave been proved to be more computationally efficient. The expansion (1.6) is then rewrittenas:

U(a) = U0 + aDn−2Dn−1

U1 + a2Dn−3Dn−1

U2 + · · ·+ an−11

Dn−1Un−1

λ(a) = λ0 + aDn−2Dn−1

λ1 + a2Dn−3Dn−1

λ2 + · · ·+ an−11

Dn−1λn−1

(1.7)

where Di(a) are polynomials of degree (i) with real coefficients di, i = 1, · · · , n− 1:

Di(a) = 1 + ad1 + a2d2 + · · ·+ aidi (1.8)

1.3 ANM-POD approach 9

1.3.2.3 Application for hyperelastic materials

Here, we write the ANM development of equations (1.4) and (1.5) for the case of hyperelasticmaterials, typically we will consider a Saint-Venant material as it is done in [24, 25]. If X arethe material coordinates, the deformed configuration is given, in a Lagrangian way, by thevector x such as:

x = X + u (1.9)

where u is the displacement field, as defined in section 1.3.2.1. As it is usually done for non-linear behaviours, the Green-Lagrange strain tensor E is expressed as a sum of linear andnon-linear contributions, as follows:

E =1

2(FTF− 1) = γl(u) + γnl(u) (1.10)

where F = ∇u + I is the deformation tensor, γl and γnl are defined as follows:

γl(u) =1

2(∇uT +∇u) (1.11)

γnl(u) =1

2∇uT∇u (1.12)

The problem to be solved can be hence be written as:

∇T + B = 0T = (I +∇u) · SS = C : E

u(X) = u for X ∈ ∂ΩuT · n = λt for X ∈ ∂Ωt

(1.13)

When rewriting the weak formulation of the constitutive equation in (1.13), we obtain:∫Ω

T : ∇δu dΩ = λ∫∂Ωt

t · δu dΓ ∀δu ∈ H1(Ω) (1.14)

that is: ∫Ω

S : δE dΩ = λ

∫∂Ωt

t · δu dΓ ∀δu ∈ H1(Ω) (1.15)

Here we briefly demonstrate using index notations how (1.15) is obtained from (1.14). Re-calling that T = (I +∇u)S, we will develop the expression T : ∇δu. First the tensor T writtenin index notations, gives:

Tij = Sij + ui,kSkj

so that T : ∇δu gives:Tijδuj,i = Sijδuj,i + ui,kSkjδuj,i (1.16)

10 CHAPTER 1. SCIENTIFIC REPORT

The second Piola-Kirchhoff tensor is known to be symmetric so that we can write the followingidentity:

Sijδuj,i + Sjiδui,j = 2Sijδuj,i = Sij(δuj,i + δui,j)

Hence, the first term in (1.16) is in tensor notation1

2S : (∇δuT +∇δu) = S : γl(δu) following

the notations introduced in (1.11).Doing the same for the second term in (1.16), we obtain:

ui,kSkjδuj,i + δui,jSjkuk,i = 2ui,kSkjδuj,i = Skj(ui,kδuj,i + δui,juk,i)

that can be rewritten in terms of tensors as follows:

1

2S : (∇uT∇δu +∇δuT∇u) = γnls(u, δu)

when we introduced the function γnls(u,v) = γnl(u,v) + γnl(v,u). At last we obtain:

T : ∇δu = S : (γl(δu) + γnls(u, δu)) = S : δE

Now equation (1.15) is solving by using the ANM approach, that is by developping thevariables (u,S, λ) in series. In other word, assuming that the solution is known in an equilib-rium point (un,Sn, λ

n), we will determine the solution of the following problem: un+1Sn+1

λn+1

= unSn

λn

+ N∑p=1

ap

upSpλp

(1.17)where the unknows of the problem are (up,Sp, λp) and a is the so-called path parameter thatdescribe the solution (un+1,Sn+1, λ

n+1) over a branche. In this study, we consider the Saint-

Venant Kirchhoff’s model that is expressed using the fourth-order constitutive tensor C:

S = C : E (1.18)

Now we construct the quadratic form, which is known to be well suited for using the ANM.For this purpose, the variables will be written in term of series. First, we rewrite the expressionof δEn+1 and En+1:

δEn+1 = γl(δu) + γnls(un+1, δu)

δEn+1 = γl(δu) + γnls

(un +

N∑p=1

apup, δu

)

δEn+1 = γl(δu) + γnls (un, δu) +

N∑p=1

apγnls (up, δu)

(1.19)

1.3 ANM-POD approach 11

and

En+1 = γl(un+1) + γnl(u

n+1,un+1)

En+1 = γl

(un +

N∑p=1

apup

)+ γnl

(un +

N∑p=1

apup,un +

N∑p=1

apup

)

En+1 = γl(un) +

N∑p=1

apγl(up) + γnl

(un +

N∑p=1

apup,un +

N∑p=1

apup

)

En+1 = γl(un) +

N∑p=1

apγl(up) + γnl (un,un) + γnl

(un,

N∑p=1

apup

)+ γnl

(N∑p=1

apup,un

)

+ γnl

(N∑p=1

apup,N∑p=1

apup

)

By rearranging the sums with respect to a, we obtain:

a[γl(u1) +∇un

T∇u1 +∇uT1∇un]

+ a2[γl(u1) +∇un

T∇u2 +∇uT2∇un +∇uT1∇u1]

+ a3[γl(u1) +∇un

T∇u3 +∇uT3∇un +∇uT1∇u2 +∇uT2∇u1]

+ a4[γl(u1) +∇un

T∇u4 +∇uT4∇un +∇uT1∇u3 +∇uT2∇u2 +∇uT3∇u1]

+ · · ·

+ ak

[γl(u1) +∇un

T∇uk +∇uTk∇un +k−1∑j=1

∇uTj ∇uk−j

]which allows to write finally:

En+1 = γl(un) + γnl (u

n,un) +N∑p=1

ap

[γl(up) + γnls(u

n,up) +

p−1∑i=1

γnl(ui,up−i)

](1.20)

The equation (1.20) will be then useful for writing Sn+1 as a serie Sn+1 = Sn +N∑p=1

apSp using

(1.18). Hence we have:

Sn+1 = C : En+1 = C :

[γl(u

n) + γnl (un,un) +

N∑p=1

ap

(γl(up) + γnls(u

n,up) +

p−1∑i=1

γnl(ui,up−i)

)](1.21)

that is to say:

Sn+1 = C : En + C :

[N∑p=1

ap

(γl(up) + γnls(u

n,up) +

p−1∑i=1

γnl(ui,up−i)

)](1.22)

12 CHAPTER 1. SCIENTIFIC REPORT

or

Sn+1 = Sn +N∑p=1

apSp (1.23)

with

Sp = C :

(γl(up) + γnls(u

n,up) +

p−1∑i=1

γnl(ui,up−i)

)(1.24)

With relations (1.19), (1.23) and (1.24), we are then able to rewrite the weak formulation (1.15):

∫Ω

[(Sn +

N∑p=1

apSp

):

(γl(δu) + γnls (u

n, δu) +N∑p=1

apγnls (up, δu)

)]dΩ =(

λn

+N∑p=1

apλp

)∫∂Ωt

t · δu dΓ(1.25)

Now we develop the expressions in equation (1.25) in order to rearrange the terms with thesame power of a, particularly by using (1.24). So we can formally write:

1.3 ANM-POD approach 13

a0 {Sn : γl(δu) + Sn : γnls(un, δu)}

a1 {[γl(δu) + γnls(un, δu)] : C : [γl(u1) + γnls(un,u1)]+ Sn : γnls(u1, δu)}

+ a2 {[γl(δu) + γnls(un, δu)] : C : [γl(u2) + γnls(un,u2) + γnl(u1,u1)]+ [γnls(u1, δu)] : C : [γl(u1) + γnls(un,u1)]+ Sn : γnls(u2, δu)}

+ a3 {[γl(δu) + γnls(un, δu)] : C : [γl(u3) + γnls(un,u3) + γnl(u1,u2) + γnl(u2,u1)]+ [γnls(u1, δu)] : C : [γl(u2) + γnls(un,u2) + γnl(u1,u1)]+ [γnls(u2, δu)] : C : [γl(u1) + γnls(un,u1)]+ Sn : γnls(u3, δu)}

+ a4 {[γl(δu) + γnls(un, δu)] : C : [γl(u4) + γnls(un,u4) + γnl(u1,u3) + γnl(u2,u2) + γnl(u3,u1)]+ [γnls(u1, δu)] : C : [γl(u3) + γnls(un,u3) + γnl(u1,u2) + γnl(u2,u1)]+ [γnls(u2, δu)] : C : [γl(u2) + γnls(un,u2) + γnl(u1,u1)]+ [γnls(u3, δu)] : C : [γl(u1) + γnls(un,u1)]+ Sn : γnls(u4, δu)}

+ · · ·

+ ak

{[γl(δu) + γnls(u

n, δu)] : C :

[γl(uk) + γnls(u

n,uk) +k−1∑i=1

γnl(ui,uk−i)

]+ Sn : γnls(up, δu)

+k−1∑i=1

[γnls(uk−i, δu)] : C :

[γl(ui) + γnls(u

n,ui) +i−1∑j=1

γnl(uj,ui−j)

]}

So the development leads to the following expressions of the different orders. If we write theweak formulation as:

∫Ω

(T0 +

N∑p=1

apT p)dΩ =

(λn

+N∑p=1

apλp

)∫∂Ωt

t · δu dΓ (1.26)

we have then the following expressions of T0 and T p for each p = 1, · · · , N :

T0 = Sn : γl(δu) + Sn : γnls(un, δu) (1.27)

14 CHAPTER 1. SCIENTIFIC REPORT

andT p = [γl(δu) + γnls(un, δu)] : C : [γl(up) + γnls(un,up)]

+

p−1∑i=1

{[γnl(ui,up−i)] : C : [γl(δu) + γnls(un, δu)]}

+

p−1∑i=1

Si : γnls(up−i, δu)

+ Sn : γnls(up, δu)

(1.28)

By identifying terms with the same power, we find from (1.26) and (1.27) that:∫Ω

T0 dΩ =∫

Ω

Sn : [γl(δu) + γnls(un, δu)] dΩ =

∫Ω

Sn : δE dΩ = λn∫∂Ωt

t · δu dΓ (1.29)

which corresponds to the weak formulation at state n. The second expression (1.28) associatedto (1.26) allows to write the problem (1.13) as a succession of linear systems:

L(δu,un) = λp∫∂Ωt

t · δu dΓ + Fp(δu) (1.30)

where the up are the unknowns and

L(δu,un) =∫

Ω

{Sn : γnls(up, δu) + [γl(δu) + γnls(un, δu)] : C : [γl(up) + γnls(un,up)]} dΩ

(1.31)and

Fp(δu) = −∫

Ω

{p−1∑i=1

Si : γnls(up−i, δu) +

p−1∑i=1

[γnl(ui,up−i)] : C : [γl(δu) + γnls(un, δu)]

}dΩ

(1.32)which leads after discretisation to the following sequence of linear systems:

order 1

{Ktu1 = λ1f

(u1,u1) + λ2

1 = 1(1.33)

order p

{Ktup = λpf + fp

(up,u1) + λpλ1 = 0(1.34)

where (X,Y) = (X,Y)L2(Ω) =∫

Ω

XY dΩ is the L2− scalar product between X and Y, Kt is

the tangent stiffness matrix obtained by discretizing equation (1.31), f is the loading vectorand fp is the second term vector obtained from the discretisation of (1.32). These problems aresolved using a prediction-correction approach, their solutions are obtained as follows:

order 1

û1 = K

−1t f

λ1 =1√

(û1, û1) + 1

u1 = λ1û1

(1.35)

1.3 ANM-POD approach 15

order p

ûp = K

−1t fp

λp = −λ1(ûp,u1)

up =λp

λ1u1 + ûp

(1.36)

Indeed, at order 1, we obtain obviously the relation between û1 and u1. For determining λ1,we develop:

(u1,u1) + λ2

1 = (λ1û1, λ1û1) + λ2

1 = λ2

1[(û1, û1) + 1] = 1 (1.37)

For the orders p, we use the facts that Ktu1 = λ1f and Ktûp = fp. Hence, we have :

Ktup = λpf + fp

=λp

λ1λ1f + Ktûp

=λp

λ1Ktu1 + Ktûp

(1.38)

which leeds to the expression of up in (1.36). Then we can write the expression of λp from(1.34):

λp = −1

λ1(up,u1)

= − 1λ1

(λp

λ1u1 + ûp,u1)

= − 1λ

2

1

λp(u1,u1)−1

λ1(ûp,u1)

= −1− λ2

1

λ2

1

λp −1

λ1(ûp,u1)

(1.39)

which obviously leeds to the expression of λp in (1.36).

1.3.2.4 Application for 1D plasticity

In this section, we briefly present the application of the ANM to the example of the 1D plas-ticity. Hence, we consider the model of plasticity as explained in [8]:

σ = E(ε− εP )f =

σ − σeσe

εPf = −η1(1 + f)σe = σy + hε

P

(1.40)

where σ, σe, σy are respectively the total stress, the effective stress and the yeld stress ; ε is thetotal strain and εP the plastic part of this strain. At last, E, h and η1 are respectively the Youngmodulus, the plastic modulus and a regularization parameter.

16 CHAPTER 1. SCIENTIFIC REPORT

The aim of the ANM method is to decompose the five variables σ, ε, εP , f and σe as an asymp-totic expansion:

σ = σ0 +N∑n=1

(t− t0)nσn (1.41)

ε = ε0 +N∑n=1

(t− t0)nεn (1.42)

εP = εP0 +N∑n=1

(t− t0)nεPn (1.43)

f = f0 +N∑n=1

(t− t0)nfn (1.44)

σe = σe0 +N∑n=1

(t− t0)nσen (1.45)

By rewriting the relations of the model (1.40) using the expressions (1.41-1.45), we obtain therelations of recurrence that allow to compute te coefficients of the asymptotic expansions. Forinstance, the first relation gives:

σ0 +N∑n=1

(t− t0)nσn = E

[ε0 +

N∑n=1

(t− t0)nεn −

(εP0 +

N∑n=1

(t− t0)nεPn

)]

or by grouping the terms of different orders, we obtain:

σ0 +N∑n=1

(t− t0)nσn = E(ε0 − εP0 ) + EN∑n=1

(t− t0)n(εn − εPn )

which allows to identify the different terms:{σ0 = E(ε0 − εP0 ) at order 0σn = E(εn − εPn ) at order n

(1.46)

In a similar way, we can rewrite the second relation of (1.40):

f0 +N∑n=1

(t− t0)nfn =σ0 +

N∑n=1

(t− t0)nσn −

(σe0 +

N∑n=1

(t− t0)nσen

)

σe0 +N∑n=1

(t− t0)nσen

that is:

1.3 ANM-POD approach 17

f0σe0+f0

N∑n=1

(t−t0)nσen+σe0N∑n=1

(t−t0)nfn+N∑n=1

(t−t0)nfn·N∑n=1

(t−t0)nσen = (σ0−σe0)+N∑n=1

(t−t0)n(σn−σen)

We note that, according to the second relation of the model, f0σe0 = (σ0− σe0) which allows towrite:

f0

N∑n=1

(t− t0)nσen + σe0N∑n=1

(t− t0)nfn +N∑n=1

(t− t0)nfn ·N∑n=1

(t− t0)nσen =N∑n=1

(t− t0)n(σn − σen)

By ordering the powers of a = (t− t0), we can write:

a [f0σe1 + σe0f1 − (σ1 − σe1)] +a2 [f0σe2 + σe0f2 + f1σe1 − (σ2 − σe2)] +a3 [f0σe3 + σe0f3 + f1σe2 + f2σe1 − (σ3 − σe3)]+ · · ·+

ak

[f0σek + σe0fk +

k−1∑i=1

fiσek0

]+ · · ·

This relation is true for each a, which allows us to write the following recurrence formulae:f1 =

σ1 − σe1(1 + f0)σe0

fn =σn − σen(1 + f0)

σe0− 1σe0

n−1∑i=1

fiσen−i

(1.47)

We can then rewrite the third equation in (1.40) using the asymptotic expansion, we obtain:

(ε0 +

N∑n=1

(t− t0)nεPn

)(f0 +

N∑n=1

(t− t0)nfn

)= −η1

[1 +

(f0 +

N∑n=1

(t− t0)nfn

)](1.48)

Developping (1.48) and introducing a = (t− t0) leads then to the following relation:

εP0 f0 + εP0

N∑n=1

anfn + f0

N∑n=1

anεPn +

(N∑n=1

anεPn

)(N∑n=1

anfn

)+ η1

N∑n=1

anfn = −η1 − η1f0 (1.49)

18 CHAPTER 1. SCIENTIFIC REPORT

By rearranging the powers of a, one obtains:

a0[εP0 f0 + η1(1 + f0)

]+

a1[εP0 f1 + f0ε

P1 + η1f1

]+

a2[εP0 f2 + f0ε

P2 + ε

P1 f1 + η1f2

]+

a3[εP0 f3 + f0ε

P3 + ε

P2 f1 + ε

P1 f2 + η1f3

]+ · · ·+

ak

[εP0 fk + f0ε

Pk +

k−1∑i=1

εPi fk−i + η1fk

]+ · · ·

That allows to write the following recurrence formulae:εP1 = −

(η1 + εP0 )

f0f1

εPn = −(η1 + ε

P0 )

f0fn −

1

f0

n−1∑i=1

εPi fn−i

(1.50)

At last, write the ANM expansion for the relation σe = σy + hεP leads to:

σe0 +N∑n=1

(t− t0)nσen = σy + h

(εP0 +

N∑n=1

(t− t0)nεPn

)(1.51)

that leads to:

a0[σe0 − σy − hεP0

]+

a1[σe1 − hεP1

]+

a2[σe2 − hεP2

]+

a3[σe3 − hεP3

]+ · · ·+ak

[σek − hεPk

]+ · · ·

so the recurrence formulae can be written:{σe1 = hε

P1

σen = hεPn

(1.52)

Finally we obtain the following systems:

1.3 ANM-POD approach 19

σ1 = E(ε1 − εP1 )

f1 =σ1 − σe1(1 + f0)

σe0

εP1 = −(η1 + ε

P0 )

f0f1

σe1 = hεP1

at order 1 (1.53)

and

σn = E(εn − εPn )

fn =σn − σen(1 + f0)

σe0− 1σe0

n−1∑i=1

fiσen−i

εPn = −(η1 + ε

P0 )

f0fn −

1

f0

n−1∑i=1

εPi fn−i

σen = hεPn

at order n > 1 (1.54)

Assuming that we know the deformation ε as a linear variation in time, we can obtainthe ANM coefficients σn, fn, εPn and σen as functions of ε. Practically they can be obtained asfollows:

f1 =σ1 − σe1(1 + f0)

σe0

=E(ε1 − εP1 )− hεP1 (1 + f0)

σe0

=Eε1 − εP1 (E + h(1 + f0))

σe0

=Eε1 +

(η1+εP0 )

f0f1 (E + h(1 + f0))

σe0

that is

f1σe0 = Eε1 +(η1 + ε

P0 )

f0f1 (E + h(1 + f0))[

σe0 −(η1 + ε

P0 )

f0(E + h(1 + f0))

]f1 = Eε1

or

f1 =

[σe0 −

(η1 + εP0 )

f0(E + h(1 + f0))

]−1Eε1 (1.55)

20 CHAPTER 1. SCIENTIFIC REPORT

1.3.2.5 Application for 3D plasticity

Here we present the extension of the ANM development for 3D plasticity. Writing of theANM decomposition for plasticity is not really different as for hyperelasticity. Indeed, theequilibrium equation is the same as for hyperelasticity, equation (1.15) we rewrite here:∫

Ω

S : δE dΩ = λ

∫∂Ωt

t · δu dΓ ∀δu ∈ H1(Ω) (1.56)

The only difference, but really important difference, lies on the behaviours’ law that must beconsidered for elastoplasticity. Here the relation between S and E (1.18) can be expressed inthe following manner:

CE = (1 + ν)Sd − (1− 2ν)PI +ησ2y

σ2y − S2eqSd (1.57)

where we introduced the following notations:

• P = −13S : I is the trace of the stress tensor

• Sd = S + PI is the deviatoric part of the stress tensor

• Seq =√

3

2Sd : Sd is the Von Mises equivalent stress

As for hyperelasticity, equation (1.10), the deformation E, as well as δE, can be written as asum of a linear contribution and a nonlinear one:

E = γl(u) + γnl(u)

δE = γl(δu) + γnls(u, δu)

Introducing the following classical notations seq and ζ :

seq = S2eq (1.58)

and

ζ =ησ2y

σ2y − seq(1.59)

solving the elastoplastic problem is then equivalent to solving the following set of equations:∫Ω

S : δE dΩ = λ

∫∂Ωt

t · δu dΓ (1.60a)

CE = (1 + ν)Sd − (1− 2ν)PI + ζSd (1.60b)ζ(σ2y − seq) = ησ2y (1.60c)

seq = S2eq =3

2Sd : Sd (1.60d)

1.3 ANM-POD approach 21

In the framework of the ANM, we consider the global unknown U =

uSζseq

and we seek Uunder the asymptotic decomposition U = U0 +

N∑p=1

apUp, that is to say we will rewrite equa-

tions (1.60a)-(1.60d) using the following decompositions:

u = u0 +N∑p=1

apup

S = S0 +N∑p=1

apSp

ζ = ζ0 +N∑p=1

apζp

seq = seq0 +N∑p=1

apseqp

Additionnaly, we will also use the asymptotic decompositions of E, Sd and P:

E = E0 +N∑p=1

apEp

Sd = Sd0 +N∑p=1

apSdp

P = P0 +N∑p=1

apPp

Rewriting (1.60b) with these decompositions leads then to the following relation:

C

(E0 +

N∑p=1

apEp

)= (1+ν)

(Sd0 +

N∑p=1

apSdp

)−(1−2ν)

(P0 +

N∑p=1

apPp

)I+

(ζ0 +

N∑p=1

apζp

)(Sd0 +

N∑p=1

apSdp

)

By developping this relation and rearranging the powers of a, we obtain:

a0[CE0 − (1 + ν)Sd0 + (1− 2ν)P0I− ζ0Sd0

]+ a

[CE1 − (1 + ν + ζ0)Sd1 + (1− 2ν)P1I− ζ1Sd0

]+ a2

[CE2 − (1 + ν + ζ0)Sd2 + (1− 2ν)P2I− ζ2Sd0 − ζ1Sd1

]+ a3

[CE3 − (1 + ν + ζ0)Sd3 + (1− 2ν)P3I− ζ3Sd0 − ζ2Sd1 − ζ1Sd2

]+ a4

[CE4 − (1 + ν + ζ0)Sd4 + (1− 2ν)P4I− ζ4Sd0 − ζ3Sd1 − ζ2Sd2 − ζ1Sd3

]+ · · ·

+ ak

[CEk − (1 + ν + ζ0)Sdk + (1− 2ν)PkI− ζkSd0 −

k−1∑i=1

Siζk−i

]

22 CHAPTER 1. SCIENTIFIC REPORT

which leads to the following relation, for each order p:

CEp = (1 + ν + ζ0)Sdp − (1− 2ν)PpI + ζpSd0 +

p−1∑i=1

Siζp−i (1.61)

Rewriting (1.60c) and (1.60d) in the same way leads to the following relations:

ζp(σ2y − s

eq0 ) = ζ0s

eq0 +

p−1∑i=1

ζiseqp−i (1.62)

and

seqp = 3S0 : S0 +3

2

p−1∑i=1

Sdi : Sdp−i (1.63)

The deformation is written here in the same way as for the hyperelastic case, see (1.20) sothat the ANM decomposition for equation (1.60a) is the same as (1.30) where the terms areexplicitely given by relations (1.31) and (1.32). As already noticed, the only difference here isthe behaviours’ law that must be considered. In order to explicitely write the terms Sp of theANM decomposition of the stress tensor, we have to rewrite (1.57) to express the terms Sp. So,we first use the expressions of (1.62) and (1.63) into (1.61), written here for simplicity just forp = 1:

CE1 = (1 + ν + ζ0)Sd1 − (1− 2ν)P1I +

3ζ0σ2y − S2eq

[Sd0 : Sd1]S

d0 (1.64)

After simplifications, we obtain:

CE1 = (1 + ν + ζ0)S1 −1

3(ζ0 + 3ν)[S1 : I]I +

3ζ0σ2y − S2eq

[Sd0 : Sd1]S

d0 (1.65)

Then, we right multiply (in the sense of tensor product) (1.65) by the identity tensor, that gives:

CE1 : I = (1 + ν + ζ0)S1 : I−1

3(ζ0 + 3ν)[S1 : I]I : I +

3ζ0σ2y − s

eq0

[Sd0 : Sd1]S

d0 : I (1.66)

As Sd1 : I = 0 (by definition of the deviatoric part of a tensor), we obtain the expression ofS1 : I:

S1 : I =C

1− 2νE1 : I (1.67)

In the same way, right multiplying (1.65) by Sd1, we obtain the expression of S1 : Sd1:

S1 : Sd1 = (1 + ν + ζ0) +

3ζ0Sd0 : S

d0

σ2y − seq0

CE1 : Sd0 (1.68)

1.3 ANM-POD approach 23

Introducing (1.67) and (1.68) into (1.65) allows to express S1:

S1 =E

1 + ν + ζ0E1−

C(ν + ζ03

)

(1 + ν + ζ0)(1− 2ν)E1 : I+

−3ζ0C(1 + ν + ζ0)2(σ2y − s

eq0 [S

d0 : S

d0]) + 3ζ0

[E1 : Sd0]S

d0

(1.69)So that finally, the coefficients Sp can be computed as follows:

S1 = Dt : E1 and Sp = Dt : Ep + Sresp ∀p > 1 (1.70)

Finally, the sequence of solving the elastoplastic problem with the ANM is the following:

• Solving the equation (1.56) leads to up

• With up we can compute the coefficients of the deformation Ep thanks to its ANM de-composition

• With Ep and using (1.70) we can compute Sp, and hence seqp with (1.63)

• With seqp and (1.63) we obtain the coefficients ζp

So that finally all the coefficients are known which allows to reconstruct all the variables withthe ANM decompositions for the next continuation steps.

1.3.3 Combination of the two approaches

The main idea proposed and developped in [25, 29] is to use the POD basis (φi)i=1,··· ,N – ob-tained by applying a POD on a set of snapshots of the solution – into the asymptotic expansionof the equations.Rewriting the pth term of the asymptotic expansion of the problem using the POD decompo-sition leads to:

up =N∑n=1

φnξnp = Φξp (1.71)

which allows to obtain the reduced systems of linear equations to solve. For example, intro-ducing (1.71) into the systems (1.33) and (1.34), and multiplying at the left by ΦT we have:

order 1

{ΦTKtΦξ1 = λ1Φ

T f

(ξ1, ξ1) + λ2

1 = 1(1.72)

order p

{ΦTKtΦξp = λpΦ

T f + ΦT fp

(ξp, ξp) + λpλ1 = 0(1.73)

24 CHAPTER 1. SCIENTIFIC REPORT

So that, as the same manner we obtained equations (1.35) and (1.36), we can write the finalsuite of reduced systems to solve:

order 1

ξ̂1 = K−1r fr

λ1 =1√

(ξ̂1, ξ̂1) + 1

ξ1 = λ1ξ̂1

u1 = Φξ1

(1.74)

for the first order and

order p

ξ̂p = K−1r fpr

λp = −λ1(ξ̂p, ξ1)

ξp =λp

λ1ξ1 + ξ̂p

up = Φξp

(1.75)

where Kr = ΦTKtΦ, fr = ΦT f , and fpr = ΦT fp are respectively the reduced tangent matrix –common at each ANM order, the reduced linear second member and the reduced non-linearsecond member.

1.4 An alternative: PGD-ANM approach

Even if the POD is quite efficient for many classes of problems, it has some drawbacks whichare limiting. First the sampling phase necessary to build the POD basis can be quite expensivein term of both storage and CPU costs. Furthermore, it is well known that this basis has a va-lidity range which is limited when the computation is achieved with parameters far differentfrom those in the sampling phase. To circumvent these problems we chose to look at the PGDwhich will be described in the next section.

1.4.1 Proper Generalized Decomposition

In this section, we present the PGD with the example of the nonlinear heat equation. Here,the nonlinearity comes from the thermal conductivity function which depends on the temper-ature. Hence, the problem we treat is recalled here:

u,t − (k(u)u,i),i = b in Ω× τu = 0 in ∂Ω× τu(t = 0) = 0 in Ω

(1.76)

where Ω is the spatial domain, ∂Ω its boundary, τ the time interval, u the temperature field,b the source term and k the thermal conductivity function.The main idea of the PGD is to decompose the solution of the problem (1.76) as a sum of prod-ucts of spatial functions Xα(x) and temporal functions Tα(t), namely we seek u as follows:

1.4 An alternative: PGD-ANM approach 25

u(x, t) ≈N∑α=1

Xα(x)Tα(t) (1.77)

In order to explain what we will call in the following the direct PGD, let us assume thep − 1 first couples of XαTα known. The solution u(p) at iteration p is then expressed u(p) =u(p−1) + XT and the functions X and T are obtained as solutions of the following nonlinearsystem:

∫τ

(u

(p),t T

)dτ −

∫τ

((k(u(p))u

(p),i

),iT

)dτ =

∫τ

b T dτ (1.78)∫Ω

(u

(p),t X

)dx−

∫Ω

((k(u(p))u

(p),i

),iX

)dx =

∫Ω

b X dx (1.79)

Equation (1.78) is a PDE in space that can be solved for instance using the BEM [11] whereasequation (1.79) is an ODE in time.

1.4.2 PGD-ANM algorithm

There are obviously several ways of coupling the PGD and the ANM. A first strategy consistsin applying the PGD on the suite of linear systems that would be obtained by the ANM onthe nonlinear heat equation. The other approach takes advantage of the ANM as a nonlinearsolver, so that the order of the steps is different: the ANM is applied inside the PGD algorithm.The first approach is a work currently in progress and we will here briefly present the mostimportant features of the second approach we have implemented.The first steps of the algorithm remain unchanged, so that we deal with the system of coupledequations (1.78)-(1.79) that are obtained thanks to the PGD procedure. Here, instead of sepa-rating the nonlinear term k(u) into a couple of space-time functions, we propose to considerit as a new unknown that will be expressed as an asymptotic expansion. If we note v = k(u),the expansion will be now written around the known solution (un, vn, λn) at step n:

un+1vn+1λ

= unvn

0

+ N∑p=1

ap

upvpλp

(1.80)so that the PGD method will determine at the same time the separated representation of u andk(u). Typically, knowing u and v at iteration p− 1, we seek their expression at iteration p as:

u(p) = u(p−1) +XT (1.81)v(p) = v(p−1) +RS (1.82)

26 CHAPTER 1. SCIENTIFIC REPORT

and the ANM will be applied inside the PGD step over X and R so that u(p) and v(p) will bethen expressed as:

u(p) = u(p−1) +

(N∑α=1

aαXα

)T (1.83)

v(p) = v(p−1) +

(N∑α=1

aαRα

)S (1.84)

Rewriting these expressions into the direct PGD algorithm leads finally to the PGD-ANMmethod.

1.5 Results

In this section, we will present some results for the different approaches we presented. Firstwe focus on the results obtained by the POD-ANM approach before turning into the appli-cation of the ANM for plastic problems. As last results, we will present the efficiency of thePGD-ANM for the nonlinear heat equation.

1.5.1 Hyperelastic example

We first present the results of the POD-ANM method applied to the example of a 3D beamunder traction or bending forces. The non-linear model used here is the hyperelastic modelas developped in the section 1.3.2.3. The beam is discretised using 40 Lagrange quadratictetrahedral elements. The length of the beam is 400mm and the cross section is a squarewith sides of length 40mm. The Young modulus is E = 1MPa and the Poisson coefficient isν = 0.25. The beam is depicted in figure 1.1.

1.5.1.1 Traction test

In this first test, we consider the beam clamped at its left end and submitted to a traction forceimposed to the right face. The magnitude of this force increases with the load parameter λthat varied from 0 to 1. The solutions used for building the POD basis were obtained with theclassical Newton-Raphson scheme, and M = 10 snapshots were used here. Then the POD-ANM method is used with a number of POD modes equal to m = 3. The evolution of thesolution is shown in figure 1.2 whereas the evolution of the maximum displacement versusthe load for different numbers of ANM modes is represented in figure 1.3.In figure 1.3, we recover the behaviour which is emphasized in the equations, namely thePOD-ANM with NANM = 1 leads to a linearised solution. Then, when increasing the numberof ANM modes, we observe that the reduced solution exactly fits the full one. The results weobtained are in agreement with those observed in [25] that validated our code.

1.5 Results 27

Figure 1.1: 3D beam

(a) λ = 0.1 (b) λ = 0.7

(c) λ = 1 (d) λ = 2

Figure 1.2: Evolution of the beam deformation with increasing the load λ (traction)

28 CHAPTER 1. SCIENTIFIC REPORT

0.0 0.5 1.0 1.5 2.0λ

0

50

100

150

200

Umax

NR solutionPOD-ANM solution NANM=1

POD-ANM solution NANM=2

POD-ANM solution NANM=3

Figure 1.3: Evolution of the maximal displacement with λ for the traction test

1.5.1.2 Bending test

Now we extend our tests for the 3D beam to the bending example. Keeping the same meshand the same parameters as mentioned before for the traction example, we impose now avertical force at the right side of the beam whereas it remains clamped at its left end. Figure1.4 presents the evolution of the deformation when the load is increased whereas figure 1.5shows the maximal deformation obtained with the reduced model in comparison with thefull Newton-Raphson scheme. The results that are presented here where obtained by keepingmost of the snapshots in the POD basis, namely m = 8 whereas the number of ANM modesis NANM = 14 in order to have an accurate solution, namely a maximal error at the maximalload less than 3%.As it was also observed for the traction example, whereas the basis is construct over the loadinterval λ ∈ [0, 1], the POD-ANM method is able to reproduce accurately the full solution evenfor a load outside the range of the snapshots , typically we observed very good results untilλ = 2.

1.5.1.3 Computational costs

In the past sections, we have proved the accuracy of the method even for loadings biggerthan those computed in the sample phase, here we analyse the computational efficiency of themethod for its possible integration in the software that uses the ShapeTape. Figure 1.6 sumsup all the computational times necessary to solve both full and reduced models. It is obviousthat the POD-ANM algorithm requires more ANM and POD modes for the bending case asfor the traction because bending is known to be much more nonlinear, which is confirmedwhen looking at figures 1.3 and 1.5. So the CPU times for the bending cases are bigger thanfor the traction ones. Nevertheless, figure 1.6 allows us to realize of efficient the POD-ANMis in comparison with the classical Newton-Raphson scheme. Even this method cannot beemployed for real-time applications, especially for moves that involve bending, it can be veryuseful for applications that requires fast computations (quasi real-time). As the main objectiveof the ANR SeaRCH is to obtain a very accurate mechanical solution as fast as possible, we

1.5 Results 29

(a) λ = 0.1 (b) λ = 0.4

(c) λ = 0.7 (d) λ = 1

Figure 1.4: Evolution of the beam deformation with increasing the load λ (bending)

0.0 0.5 1.0 1.5 2.0λ

100

150

200

250

300

350

400

Umax

NR solutionPOD-ANM solution

Figure 1.5: Evolution of the maximal displacement with λ for the bending test

30 CHAPTER 1. SCIENTIFIC REPORT

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Ndof

100

101

102

103t(s)

NRPOD-ANM

(a) CPU time for the traction test

500 1000 1500 2000 2500 3000 3500Ndof

101

102

103

t(s)

NRPOD-ANM

(b) CPU time for the bending test

Figure 1.6: Comparison of the full and reduced models in terms of CPU time for different mesh sizes

consider this method reliable for this purpose, that is why it will be adapted for the plasticdeformation.

1.5.2 Results for plasticity

This part sums up all the interesting results that have been obtained for plasticity. It willfirst quickly presents the results obtained for a 1D test case, whose aim was just to validate theANM procedure for a simple case of plasticity before going further with the more complicated3D case.

1.5.2.1 Imposed deformation for a 1D example

In this example, we do not solve the constitutive equation but as explained in [8], the defor-mation is imposed in order to compute all the other variables using the ANM expansions. Theresults we obtained for different values of hardening are shown in figure 1.7 and correspondwell to those obtain in the literature. So these results first validated our ANM code that couldtherefore be applied for the more complicated test of 3D plasticity, which will involve now theuse of FEniCS.

1.5.2.2 Plastic deformation of a 3D beam

In this section, we present the different computations that have been carried out in order tobuild the most appropriate framework for real-time coupling with the TUI device. Here weconsider once again a 3D beam with a squared section, as defined in section 1.5.1 in figure 1.1.In the following, we will hence present the results of the following computations:

(i) beam in traction with the ANM for two different levels of accuracy (different values ofthe continuation parameter a).

(ii) the POD basis is computed from samples and used for fast reconstruction of the dis-placement field.

1.5 Results 31

0.00 0.02 0.04 0.06 0.08 0.10ε

0

50

100

150

200

250

σ

(a) h = 0

0.00 0.02 0.04 0.06 0.08 0.10ε

0

50

100

150

200

250

300

350

400

450

σ

(b) h = 2000MPa

Figure 1.7: Stress-strain response for the 1D problem with deformation theory. Computational parameters:N = 15, δ = 10−2, η1 = 10−4

(iii) beam in traction with the ANM-POD method for one value of the continuation parame-ter.

At the end of this section, we will briefly discuss the results and give some advices that couldhelp for further computations, especially concerning the efficiency of the ANM-POD method.

1.5.2.2.a ANM results in traction

The first task of this work about plasticity is clearly to run direct computations using the ANM.Here the parameters that have been used are summed up in table 1.3:

η1 a = δ

(‖u1‖‖uNANM‖

) 1NANM

NANM NCONT

run 1 10 δ = 0.2 5 70run 2 10 δ = 0.01 5 700

Table 1.3: ANM parameters used for plastic computations for the traction case.

Although the first run can reproduce well the elastoplastic behavior of the beam, as repre-sented in figure 1.8(a), we decided to compute another solution firstly to observe the effects ofthe choice of δ and secondly to be able to better describe the whole stress-strain curve, in orderto build a reference solution. The comparison between the two obtained curves is depicted infigure 1.9.

In the following, we will use the results of the full ANM computations with 700 continua-tion steps as the reference results for evaluating the performances of our reduced-order model.In addition, the POD basis will be constructed using snapshots from these solutions, in order

32 CHAPTER 1. SCIENTIFIC REPORT

0 10 20 30 40 50 60 70Strain ε

0

50

100

150

200

250Str

ess σ

(a) δ = 0.2

0 10 20 30 40 50 60 70Strain ε

0

50

100

150

200

250

Str

ess σ

(b) δ = 0.01

Figure 1.8: Stress-strain response for the 3D beam in traction using ANM. Computational parameters: N = 5and η1 = 10.

to have comparable solutions, but in case of a general application, it will be recommanded touse the fastest computation – here the one with 70 continuation steps – for building the PODbasis, especially if we are interested more in fast computing than extreme accuracy2.

0 10 20 30 40 50 60 70Strain ε

0

50

100

150

200

250

Str

ess σ

δ=0.2

δ=0.01

Figure 1.9: Comparison between ANM results in the case of traction on the 3D beam for different values of δ.

1.5.2.2.b Computing the POD for the traction case

Once the ANM computations have given the elastoplastic evolution of the beam, we are ableto compute the POD basis that correspond to a given sample. As already explained in theprevious section, we compute the POD basis as explained in section 1.3.1 thanks snapshotsfrom the fine computation, which also allows to analyze the influence of the number of modesused for building the POD basis. In this study, we have computed the POD basis using snap-shots that belong to the first 500 ANM steps, so that the beginning of the plastic behavior is

2Note also that in this case, the relaxation η1 = 10 has been chosen deliberately large to avoid extreme non-linearities and then speed up the computations

1.5 Results 33

recovered. The number of snaphsots that has been considered varying between Nsnap = 10and Nsnap = 100.There are two different ways of using the POD basis constructed:

(i) we use it as an interpolation basis for computing all the unknown values

(ii) we use this decomposition for writing the ANM-POD scheme such as described in sec-tion 1.3.3. It will be the purpose of the next paragraph.

In this part we focus on the use of the POD as an interpolation basis for reconstructing thedisplacement field of the beam for each value of the load parameter λ.

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007Displacement [m]

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

load

λ

Sample valuesReconstructed valueExact value

(a) λ = 1.3203

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007Displacement [m]

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

load

λ

Sample valuesReconstructed valueExact value

(b) λ = 1.3368

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007Displacement [m]

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

load

λ

Sample valuesReconstructed valueExact value

(c) λ = 1.3562

Figure 1.10: Evolution of the displacement with the load parameter λ for 3D plastic computations.

First we present the results obtained with a POD basis of 5 modes computed from 10 snap-shots of the solution. From this figure we can clearly observe that the maximum displacementsi well recovered using the POD basis. In order to validate the reconstructed solutions, figure?? presents the error on the displacement field between the reference solution and the solutionreconstructed from the reduced model, using the same scale in order to see the influence of λon the error of reconstruction. As expected, the results in terms of error are really good and

(a) λ = 1.3203 (b) λ = 1.3368 (c) λ = 1.3562

Figure 1.11: Error fields computed between reference solutions and solutions reconstructed using the POD basisfor different load parameters λ. Parameters are: Nsnap = 10, NPOD = 5

even if the observed error slightly increases when the interpolation is far from the sampledvalues, it remains in a range which is very acceptable.Now we turn on to the same results obtained with a sample that contains Nsnap = 100 snap-shots of the reference solution. Here we can observe that the error remains under a low value

34 CHAPTER 1. SCIENTIFIC REPORT

(a) λ = 1.3203 (b) λ = 1.3368 (c) λ = 1.3562

Figure 1.12: Error fields computed between reference solutions and solutions reconstructed using the POD basisfor traction at different load parameters λ. Parameters are: Nsnap = 100, NPOD = 5

so is negligeable. That confirms that the quality of the solution is increased by computingthe POD basis with more snapshots. This is due to the fact that the modes that are obtainedcontain more detailed information about the dynamics of the nonlinear problem. Note herethat the number of modes has been chosen in order to ensure the POD basis to catch morethan 99.9999% of the energy of the system. The figure 1.13 confirms our choice.

0 2 4 6 8 10Mode

0

20

40

60

80

100

Ener

gy (

%)

Energy of one modeCumulative Energy

(a) Single and cumulative energy

0 2 4 6 8 10Mode

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Ener

gy (

%)

Energy of one mode

(b) Single mode energy in log scale

Figure 1.13: Energy contribution of the 10 first modes of the POD basis from a sample of 100 snapshots for thetraction case.

1.5.2.2.c ANM results in bending

Here the same analysis as done in section 1.5.2.2.a will be achieved for the bending test case,computationally more difficult. We will successively compute the solutions using the ANMand then use this solution as a sample for building a POD basis. In the case of bending, like fortraction, two degrees of accuracy have been tested, namely δ = 10−2 and δ = 10−1. The resultsare presented in figure 1.14. From these figures it obvious that the ANM is more sensitiveto the choice of the parameters for bending. Especially, in this case, we need more ANMmodes to obtain a correct solution (NANM = 10) and for a too large continuation parameter a,the computation diverges, as he can be observed in figure 1.14(a). In the following, we willcompute the POD basis from the samples with 10 or 100 snapshots.

1.5 Results 35

0.00 0.05 0.10 0.15 0.20Strain ε

0

100

200

300

400

500

600Str

ess σ

(a) δ = 0.2

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45Strain ε

0

50

100

150

200

250

Str

ess σ

(b) δ = 0.01

Figure 1.14: Stress-strain response for the 3D beam in bending using ANM. Computational parameters: N =10 and η1 = 10.

1.5.2.2.d Computing the POD for the bending case

Once again, the POD basis is computing from the ANM solution of the elastoplastic problemin bending. The error fields computed from the reference ANM solution for different loadparameters are presented in figure 1.15. From these results we can draw the same conclusions

(a) λ = 1.7511 (b) λ = 2.3095 (c) λ = 2.4055

Figure 1.15: Error fields computed between reference solutions and solutions reconstructed using the POD basisfor bending at different load parameters λ. Parameters are: Nsnap = 10, NPOD = 5

as for the traction case: even if the computations are more difficult in term of number of ANMmodes, we are able with this approach to represent the whole elastoplastic behaviour. It isalso obvious by analysing the evolution of energy contained by each mode, figure 1.16 thattaking NPOD = 5 is sufficient for our computations.

1.5.2.2.e Performances of the ANM-POD approach

In this part, we quickly discuss the performances of the combined ANM-POD approach forthe elastoplastic problems. The procedure that has been used in this section is the same asdescribed in 1.5.1 and used in 1.3.2.3. In the case of plasticity, the nonlinearity is much moredifficult to numerically handle that is why the computations required so much modes andcontinuations. As expected, the coupled ANM-POD approach is not able to cover the wholeelastoplastic range with only one ANM step, so that we also need here to introduced a contin-uation process. The continuation allows to obtain the solution, with almost the same accuracy

36 CHAPTER 1. SCIENTIFIC REPORT

0 2 4 6 8 10Mode

0

20

40

60

80

100

Ener

gy (

%)

Energy of one modeCumulative Energy

(a) Single and cumulative energy

0 2 4 6 8 10Mode

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Ener

gy (

%)

Energy of one mode

(b) Single mode energy in log scale

Figure 1.16: Energy contribution of the 10 first modes of the POD basis from a sample of 100 snapshots for thebending case.

as for the direct ANM solution, but it drastically limits the computational efficiency of themethod because at each continuation point, we have to reconstruct the fulle solution in orderto recompute the tangent matrix of the new step, which is terrible in terms of computationaltimes. Figures 1.17(a) and 1.17(b) shows the evolution of the stress-strain curve.

0 10 20 30 40 50 60 70Strain ε

0

50

100

150

200

250

Str

ess σ

(a) δ = 0.2

0 10 20 30 40 50 60 70Strain ε

0

50

100

150

200

250

Str

ess σ

(b) δ = 0.01

Figure 1.17: Stress-strain response for the 3D beam in traction and bending using the ANM-POD approach.The computational parameters are the same as those used for the respective full order computations.

Even if this method can be interesting for large meshes, where the computational timefor matrix inversion is not negligeable, in our case, we consider this method unappropriatefor our purpose and we recommand the use of the POD basis as interpolation basis for fastcomputations coupled to the ShapeTape.

1.5 Results 37

1.5.2.3 Conclusions and perspectives for plasticity

Here we try to outline the works that has been carried out and the tasks that could be envis-aged in a near future. So in this work the following has been achieved:

• reference solutions have been computed with the ANM for traction and bending, whereasthe bending case is very sensitive to the choices of the number of ANM modes, the re-sults describe well the perfect elastoplastic law that has been imposed.

• the POD basis has been computed, with different Nsnap and NPOD, and used for recon-structing solutions for every load parameter λ, it leads to very accurate results for verysmall online computational times.

• the ANM-POD method has been tested for this application, the high nonlinearity ofthe plasicity is difficult to handle so that continuations are necessary and decrease thecomputational efficiency of the method. It can however be envisaged for applicationswith large mesh sizes where the tangent matrices are difficult to inverse in the full ordermodel.

38 CHAPTER 1. SCIENTIFIC REPORT

1.5.3 PGD-ANM approach for the heat equation (voir Pierre)

Now, we present the results obtained by applying the PGD-ANM method compared to thedirect PGD, recalling here that this method avoids achieving a large number of SVD which isthe case when using the direct PGD. We keep the same value of the analytic solution but nowwe consider the nonlinear term defined as k(u) = (u + ε)n where n takes the values n = 1or 2. Figure 1.18 presents the evolution of the error in a L2Ω×τ sense with the number of PGDmodes for the direct PGD compared to the PGD-ANM method at respectively order 1 and2. We observe in this figure that the PGD-ANM at first order slightly improves the results incomparison with the direct PGD whereas they are significantly better at second order, even forthe most nonlinear case. For this case, with 5 PGD modes, the error is divided by two whenusing the PGD-ANM instead of the direct PGD, which is already a quite interesting results.

1 2 3 4 5Number of PGD modes

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Err

or in L

2 Ω×τ

nor

m

k(u) =(u+ε)

PGD DirectPGD-ANM order 1PGD-ANM order 2

1 2 3 4 5Number of PGD modes

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Err

or in L

2 Ω×τ

nor

m

k(u) =(u+ε)2

PGD DirectPGD-ANM order 1PGD-ANM order 2

Figure 1.18: Comparison between PGD direct and PGD-ANM for different values of k(u).

1.6 Conclusions and perspectives of the work

Bibliography

[1] ABATE, A., NAPPI, M., RICCIO, D., AND SABATINO, G. 2d and 3d face recognition: Asurvey. Pattern Recognition Letters 28, 14 (Oct. 2007), 1885–1906.

[2] ABICHOU, H., ZAHROUNI, H., AND POTIER-FERRY, M. Asymptotic numerical methodfor problems coupling several nonlinearities. Computer Methods in Applied Mechanics andEngineering 191, 51-52 (Dec. 2002), 5795–5810.

[3] AMMAR, A., CHINESTA, F., DIEZ, P., AND HUERTA, A. An error estimator for sepa-rated representations of highly multidimensional models. Computer Methods in AppliedMechanics and Engineering 199, 25-28 (May 2010), 1872–1880.

[4] AMMAR, A., MOKDAD, B., CHINESTA, F., AND KEUNINGS, R. A new family of solversfor some classes of multidimensional partial differential equations encountered in kinetictheory modelling of complex fluids: Part II: Transient simulation using space-time sepa-rated representations. Journal of non-newtonian fluid mechanics 144, 2 (2007), 98–121.

[5] AMMAR, A., RYCKELYNCK, D., CHINESTA, F., AND KEUNINGS, R. On the reduction ofkinetic theory models related to finitely extensible dumbbells. Journal of Non-NewtonianFluid Mechanics 134, 1-3 (2006), 136–147. 2nd Annual European Rheology Conference.

[6] AMSALLEM, D., AND FARHAT, C. Interpolation Method for Adapting Reduced-OrderModels and Application to Aeroelasticity. AIAA Journal 46 (July 2008), 1803–1813.

[7] ANTOULAS, A., SORENSEN, D., AND GUGERCIN, S. A survey of model reduction meth-ods for large-scale systems. Contemporary Mathematics 280 (2001), 193–219.

[8] ASSIDI, M., ZAHROUNI, H., DAMIL, N., AND POTIER-FERRY, M. Regularization andperturbation technique to solve plasticity problems. Int. J. Mater. Form. 2, 1 (2009), 1–14.

[9] BARBIC, J., AND ZHAO, Y. Real-time Large-deformation Substructuring. 2–8.

[10] BATHE, K.-J. Finite Element Procedures (Part 1-2). Prentice Hall, June 1995.

40 BIBLIOGRAPHY

[11] BONITHON, G., JOYOT, P., CHINESTA, F., AND VILLON, P. Non-incremental boundaryelement discretization of parabolic models based on the use of the proper generalizeddecompositions. Engineering Analysis with Boundary Elements 35, 1 (Jan. 2011), 2–17.

[12] CHINESTA, F., AMMAR, A., AND CUETO, E. Proper generalized decomposition of mul-tiscale models. Int. J. Numer. Meth. Engng. 83, 8-9 (2010), 1114–1132.

[13] CHINESTA, F., AMMAR, A., AND CUETO, E. Recent Advances and New Challenges inthe Use of the Proper Generalized Decomposition for Solving Multidimensional Models.Archives of Computational Methods in Engineering 1, February (2010), 327–350.

[14] CHINESTA, F., LADEVEZE, P., AND CUETO, E. A short review on model order reduc-tion based on proper generalized decomposition. Archives of Computational Methods inEngineering 18, 4 (Nov. 2011), 395–404.

[15] GONZÁLEZ, D., AMMAR, A., CHINESTA, F., AND CUETO, E. Recent advances on the useof separated representations. Int. J. Numer. Methods Eng. 81, 5 (2010), 637–659.

[16] GUGERCIN, S., AND ANTOULAS, A. C. Model reduction of large-scale systems by leastsquares. Linear Algebra and its Applications 415, 2–3 (June 2006), 290–321.

[17] LADEVEZE, P. On algorithm family in structural mechanics. C R Acad Sci Ser 300, 2 (1985),41–44.

[18] LOGG, A., ØLGAARD, K. B., ROGNES, M. E., AND WELLS, G. N. FFC: the FEniCS FormCompiler. Springer, 2012, ch. 11.

[19] LOGG, A., WELLS, G. N., AND HAKE, J. DOLFIN: a C++/Python Finite Element Library.Springer, 2012, ch. 10.

[20] LUTTMAN, A., STONE, E., AND BARDSLEY, J. Numerical analysis of pattern formationon the surface of transpiring leaves. Physica D Nonlinear Phenomena 232 (Aug. 2007), 142–155.

[21] MARTIN, S., KAUFMANN, P., BOTSCH, M., AND GROSS, M. Unified simulation of elasticrods , shells , and solids. Lloydia (Cincinnati) (2007), –.

[22] MCDONNELL, K. T., QIN, H., AND WLODARCZYK, R. A. Virtual clay: a real-time sculpt-ing system with haptic toolkits. In Proceedings of the 2001 symposium on Interactive 3Dgraphics (New York, NY, USA, 2001), I3D ’01, ACM, pp. 179–190.

[23] NIROOMANDI, S., ALFARO, I., CUETO, E., AND CHINESTA, F. Accounting for large de-formations in real-time simulations of soft tissues based on reduced-order models. Com-puter Methods and Programs in Biomedicine In Press, Corrected Proof , –.

[24] NIROOMANDI, S., ALFARO, I., CUETO, E., AND CHINESTA, F. Real-time deformablemodels of non-linear tissues by model reduction techniques. Computer Methods and Pro-grams in Biomedicine 91, 3 (Sept. 2008), 223–231.

BIBLIOGRAPHY 41

[25] NIROOMANDI, S., ALFARO, I., CUETO, E., AND CHINESTA, F. Model order reduction forhyperelastic materials. Int. J. Numer. Methods Eng. 81, 9 (2010), 1180–1206.

[26] PRADOS, F. J. R., AND TORRES, J. C. Interactive elastic deformation of 3D images. Con-tinuum xx (2011), 1–7.

[27] RYCKELYNCK, D., AND MISSOUM BENZIANE, D. Multi-level a priori hyper-reduction ofmechanical models involving internal variables. Computer Methods in Applied Mechanicsand Engineering 199, 17-20 (Mar. 2010), 1134–1142.

[28] SIROVICH, L. Turbulence and the dynamics of coherent structures. I - Coherent struc-tures. II - Symmetries and transformations. III - Dynamics and scaling. Quarterly of Ap-plied Mathematics 45 (Oct. 1987), 561–571.

[29] YVONNET, J., ZAHROUNI, H., AND POTIER-FERRY, M. A model reduction method forthe post-buckling analysis of cellular microstructures. Computer Methods in Applied Me-chanics and Engineering 197, 1-4 (Dec. 2007), 265–280.

[30] ZIENKIEWICZ, O. C., TAYLOR, R. L., AND ZHU, J. Z. The finite element method: its basisand fundamentals. Butterworth-Heinemann, 2005.

42 BIBLIOGRAPHY

Appendices

AQuelques mots sur les optimisations dans FEniCS

Ce document donne quelques indications qui permettent de mieux comprendre et de mieuxutiliser le code FEniCS.

A.1 Numérotation des noeuds et stockage numpy

Dans toute la suite de ce document, on se place dans le cas simple où l’on considère un cubeunitaire, ceci dans le but de simplifier les écritures vectorielles.

1 2

3 4

5 6

7 8

La numérotation des noeuds est effectuée par FEniCS comme indiquésur le schéma ci-contre. On travaille avec des éléments de Lagrange dedimension 1.

A.1.1 Vecteur

Si l’on note xk, yk et zk les coordonnées du noeud k du maillage unitaire, alors FEniCS stocke

le vecteur u =

xyz

de la façon suivante :

uk =

x1...x8y1...y8z1...z8

(A.1)

http://fenicsproject.org/

46 APPENDIX A. QUELQUES MOTS SUR LES OPTIMISATIONS DANS FENICS

A.1.2 Tenseur

Si l’on note le tenseur gradient de déplacement ∇u = (ui,j), i, j = 1, · · · , 3, son expression entableau numpy s’écrit :

∇uk =

u11,1...u81,1u11,2

...u81,2u11,3

...u81,3u12,1

...u82,1

...uki,j

...u13,3

...u83,3

pour le tenseur ∇u =

u1,1 u1,2 u1,3u2,1 u2,2 u2,3u3,1 u3,2 u3,3

(A.2)

A.2 Produit tensoriel sans compilation

Dans cette partie, on cherche à effectuer le calcul suivant :∫Ω

S : T dx (A.3)

où I et J sont deux tenseurs d’ordre 2, définis sur l’espac