Patrick Le Tallec

Numerical Methods in Sensitivity Analysis and Shape Optimization

Sensitivity analysis and optimal shape design are key issues in engineering that have been affected by advances in numerical tools currently available. This book, and its supplementary online files, presents basic optimization techniques that can be used to compute the sensitivity of a given design to local change, or to improve its performance by local optimization of these data. The relevance and scope of these techniques have improved dramatically in recent years because of progress in discretization strategies, optimization algorithms, automatic differentiation, software availability, and the power of personal computers. Numerical Methods in Sensitivity Analysis and Shape Optimization will be of interest to graduate students involved in mathematical modeling and simulation, as well as engineers and researchers in applied mathematics looking for an up-to-date introduction to optimization techniques, sensitivity analysis, and optimal design.

Filtering for distributed mechanical systems using position measurements: Perspectives in medical imaging

We propose an effective filtering methodology designed to perform estimation in a distributed mechanical system using position measurements. As in a previously introduced method, the filter is inspired by robust control feedback, but here we take full advantage of the estimation specificity to choose a feedback law that can act on displacements instead of velocities and still retain the same kind of dissipativity property which guarantees robustness. This is very valuable in many applications for which positions are more readily available than velocities, as in medical imaging. We provide an in-depth analysis of the proposed procedure, as well as detailed numerical assessments using a test problem inspired by cardiac biomechanics, as medical diagnosis assistance is an important perspective for this approach. The method is formulated first for measurements based on Lagrangian displacements, but we then derive a nonlinear extension allowing us to instead consider segmented images, which of course is even more relevant in medical applications.

Deriving Adequate Formulations for Fluid-Structure Interaction Problems: from ALE to Transpiration

ABSTRACT Most formulations describing low speed large displacements fluid-structure interaction problems use a totally lagrangian formulation for the structure, and an Arbitrary Eider Lagrange (ALE) formulation for the fluid. The purpose of the present paper is to review the derivation of such formulations, to describe different time discretisation strategies and to explain the type of numerical problems which arise when implementing these techniques. To overcome all technical difficulties arising when dealing with moving grids, we will also explain how an adequate asymptotic expansion can reduce the problem to a standard problem written on a fixed configuration, but using specific transpiration interface boundary conditions. This last formulation is rather popular in the aeronautical community, and will be illustrated by various numerical experiments.

Influence of bending resistance on the dynamics of a spherical capsule in shear flow

The objective of the paper is to study the effect of wall bending resistance on the motion of an initially spherical capsule freely suspended in shear flow. We consider a capsule with a given thickness made of a three–dimensional homogeneous elastic material. A numerical method is used to model the fluid–structure interactions cou- pling a boundary integral method for the fluids with a shell finite element method for the capsule envelope. For a given wall material, the capsule deformability strongly decreases when the wall bending resistance increases. But, if one expresses the same results as a function of the two–dimensional mechanical properties of the mid–surface, which is how the capsule wall is modeled in the thin–shell model, the capsule deformed shape is identical to the one predicted for a capsule devoid of bending resistance. The bending rigidity is found to have a negligible influence on the overall deformation of an initially spherical capsule, which therefore depends only on the elastic stretching of the mid–surface. Still, the bending resistance of the wall must be accounted for to model the buckling phenomenon, which is observed locally at low flow strength. We show that the wrinkle wavelength is only a function of the wall bending resistance and provide the correlation law. Such results can then be used to infer values of the bending modulus and wall thickness from experiments on spherical capsules in simple shear flow.

Coupled variational formulations of linear elasticity and the DPG methodology

This article presents a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated to a distinct variational formulation coming from a mutually well-posed family of broken variational formulations of the original PDE. It can be exploited to solve challenging problems in a variety of physical scenarios where stability or a particular mode of convergence is desired in a part of the domain. The linear elasticity equations are solved in this work, but the approach can be applied to other equations as well. The broken variational formulations, which are essentially extensions of more standard formulations, are characterized by the presence of mesh-dependent broken test spaces and interface trial variables at the boundaries of the elements of the mesh. This allows necessary information to be naturally transmitted between adjacent subdomains, resulting in coupled variational formulations which are then proved to be globally well-posed. They are solved numerically using the DPG methodology, which is especially crafted to produce stable discretizations of broken formulations. Finally, expected convergence rates are verified in two different and illustrative examples.

Delayed Feedback Control Method For Calculating Space-Time Periodic Solutions Of Viscoelastic Problems

We are interested in fast techniques for calculating a periodic solution to viscoelastic evolution problems with a space-time periodic condition of ”rolling” type. Such a solution is usually computed as an asymptotic limit of the initial value problem with arbitrary initial data. We want to invent a control method, accelerating the convergence. The main idea is to modify our problem by introducing a feedback control term, based on a periodicity error, in order to accelerate the convergence to the desired periodic solution of the problem. First, an abstract evolution problem has been studied. From the analytic solution of the modified (controlled) problem, an efficient control has been found, optimizing the spectrum of the problem. The proposed control term can be mechanically interpreted, and its efficiency increases with the relaxation time. In order to confirm numerically the theoretical results, a finite element simulation has been carried out on a full 2D model for a steady rolling of a viscoelastic tyre with periodic sculpture. It has demonstrated that the controlled solution converges indeed faster than the non-controlled one, and that the efficiency of the method increases with the problem’s relaxation time, that is when the memory of the underlying problem is large.

Numerical Homogeneisation Technique with Domain Decomposition Based a-posteriori Error Estimates

The purpose of the present work is to review some basic numerical homogeneisation techniques for the simulation of multiscale materials and to introduce an error control strategy at the local level. This error control uses an a posteriori error estimate built on a local problem coupling different representative volume elements. It introduces a weakly coupled adjoint problem to be solved say by a direct Schur complement method. Mortar element techniques as introduced in domain decomposition techniques are used to couple in a weak and cheap form the different representative elements in the error analysis. The strategy is numerically assessed on a model two dimensional problem.

Computing Gradients by Adjoint States

We have already observed that the key point in sensitivity analysis or in most optimization algorithms is the accurate calculation of the gradient of the cost functionj.In order to detail this calculation, it is useful to first recall the structure of a cost function in shape optimization. In such problems, the cost function is defined by

Finite Dimensional Optimization

\left\{ {\begin{array}{*{20}{c}} {j:{Z_{zdm}} \subset {\mathbb{R}^n} \to \mathbb{R},} \\ {z \mapsto j\left( z \right) = J\left( {\underset{\raise0.3em\hbox{