Jean-Francois Scheid

A Level Set Method in Shape and Topology Optimization for Variational Inequalities

A Level Set Method in Shape and Topology Optimization for Variational Inequalities The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology variations in the form of small holes. The derivation of topological derivatives is performed within the framework proposed in (Sokołowski and Żochowski, 2003). Numerical results confirm that the method is efficient and gives better results compared with the classical shape optimization techniques.

Convergence of the Lagrange--Galerkin Method for the Equations Modelling the Motion of a Fluid-Rigid System

In this paper, we consider a Lagrange--Galerkin scheme to approximate a two-dimensional fluid-rigid body problem. The equations of the system are the Navier--Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the rigid body. In this problem, the equations of the fluid are written in a domain whose variation is one of the unknowns. We introduce a numerical method based on the use of characteristics and on finite elements with a fixed mesh. Our main result asserts the convergence of this scheme.

Level set method with topological derivatives in shape optimization

A class of shape optimization problems is solved numerically by the level set method combined with the topological derivatives for topology optimization. Actually, the topology variations are introduced on the basis of asymptotic analysis, by an evaluation of extremal points (local maxima for the specific problem) of the so-called topological derivatives introduced by Sokolowski and Zochowski [J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4) (1999), pp. 1251–1272] for elliptic boundary value problems. Topological derivatives are given for energy functionals of linear boundary value problems. We present results, including numerical examples, which confirm that the application of topological derivatives in the framework of the level set method really improves the efficiency of the method. Examples show that the level set method combined with the asymptotic analysis is robust for the shape optimization problems, and it allows us to identify the better solution compared to the pure level set method exclusively based on the boundary variation technique.

A numerical method for shape and topology optimization for semilinear elliptic equation

Shape optimization problem for semilinear elliptic equation is considered. There is an optimal solution which is computed by the Levelset method. To this end the shape derivative of the functional is evaluated. In order to predict the topology changes the topological derivative is employed. Numerical results confirm that the proposed framework for numerical solution of shape optimization problems is efficient.

A posteriori error estimates and adaptive finite elements for a nonlinear parabolic problem related to solidification

A posteriori error estimates are derived for a nonlinear parabolic problem arising from the isothermal solidification of a binary alloy. Space discretization with continuous, piecewise linear finite elements is considered. The L-2 in time H-1 in space error is bounded above and below by an error estimator based on the equation residual. Numerical results show that the effectivity index is close to one. An adaptive finite element algorithm is proposed and a solutal. dendrite is computed

A dissolution-growth problem with surface tension: Local existence and uniqueness

Abstract We consider a one-phase Stefan problem with surface tension in space dimension two. We prove the local existence and uniqueness in time of the solution, in the case where the moving interface is parametrized in the form y = f(x,t).

Regularity and uniqueness results for a phase change problem in binary alloys

An isothermal model describing the separation of the components of a binary metallic alloy is considered. A process of phase transition is also assumed to occur in the solder; hence, the state of the material is described by two order parameters, i.e., the concentration c of the first component and the phase field φ. Existence of a solution to the related initial and boundary value problem has been proved in a former paper, where, anyway, uniqueness was obtained only in a very special case. Here some further regularity and uniqueness results are shown in a more general setting by use of an a priori estimates – compactness argument. A key point of the proofs is the analysis of the fine continuity properties of the inverse map of the solution-dependent elliptic operator characterizing one of the equations of the system.

15. Time-optimal control for a perturbed Brockett integrator

The aim of this work is to compute time optimal controls for a perturbation of a Brockett integrator with state constraints. Brockett integrator and its perturbations appears in many applications fields. One of them described in details in this note is the swimming of microorganisms. We present some key results for a fast and robust numerical method to compute time optimal controls for the perturbation of a Brockett integrator. This numerical method is based on explicit formulae of time optimal controls for the Brockett integrator. The methodology presented in this work is applied to the time optimal control of a micro-swimmer.

The Lagrange-Galerkin method for fluid-structure interaction problems

In this paper, we consider a Lagrange-Galerkin scheme to approximate a two-dimensional fluid-structure interaction problem. The equations of the system are the Navier-Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the solid. We are interested in studying numerical schemes based on the use of the characteristics method for rigid and deformable solids. The schemes are based on a global weak formulation involving only terms defined on the whole fluid-solid domain. Convergence results are stated for both semi and fully discrete schemes. This article reviews known results for rigid solid along with some new results on deformable structure yet to be published.

Convergence of a Discretization Scheme Based on the Characteristics Method for a Fluid–Rigid System

In this chapter, we present our latest results concerning the convergence of a numerical method to discretize the equations modeling the motion of a rigid solid immersed into a viscous incompressible fluid using the characteristics technique.