We present a review of known results in shape optimization from the point of view of Geometric An... more We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical aspects of the shape optimization theory. We focus on the theory of gradient flows of objective functions and their regularizations. Shape optimization is a part of calculus of variations which uses the geometry. Shape optimization is also related to the free boundary problems in the theory of Partial Differential Equations. We consider smooth perturbations of geometrical domains in order to develop the shape calculus for the analysis of shape optimization problems. There are many applications of such a framework, in solid and fluid mechanics as well as in the solution of inverse problems. For the sake of simplicity we consider model problems, in principle in two spatial dimensions. However, the methods presented are used as well in three spatial dimensions. We present a result on the convergence of the shape gradient method for a model p...
In the paper the general method for shape-topology sensitivity analysis of contact problems is pr... more In the paper the general method for shape-topology sensitivity analysis of contact problems is proposed. The method uses the domain decomposition method combined with the specific properties of minimizers for the energy functional. The method is applied to the static problem of an elastic body in frictionless contact with an rigid foundation. The contact model allows a finite interpenetration of the bodies on the contact region. This interpenetration is modeled by means of a scalar function that depends on the normal component of the displacement field on the potential contact zone. We present the asymptotic behavior of the energy shape functional when a spheroidal void is introduced in an arbitrary point of the elastic body. For the asymptotic analysis, we use the domain decomposition technique and the associated Steklov-Poincaré pseudodifferential operator. The differentiability of the energy with respect to the non-smooth perturbation is established. A closed form for the topological derivative is also presented.
Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2021
The topological derivative method is used to solve a pollution sources reconstruction problem gov... more The topological derivative method is used to solve a pollution sources reconstruction problem governed by a steady-state convection-diffusion equation. The inverse problem consists in the reconstruction of a set of pollution sources in a fluid medium by measuring the concentration of the pollutants within some subregion of the reference domain. We rewrite the inverse problem as a topology optimization problem which allows us to solve it by using the concept of topological derivatives. The resulting algorithm is able to reconstruct the pollution sources in one step and is independent of any initial guess. A numerical example is presented to show the effectiveness of our reconstruction method.
Proceedings of the Steklov Institute of Mathematics, 2020
We study an initial-boundary value problem with free boundary for one-dimensional equations of vi... more We study an initial-boundary value problem with free boundary for one-dimensional equations of viscous gas dynamics. The problem models the motion of a crankshaft mechanism under gas pressure. It is assumed that the gas fills a cylinder, which is modeled by the interval [0, 1]. A variable point a(t) ∈ [0, 1] models a piston moving inside the cylinder. The piston is assumed to be connected to a planar three-link crankshaft mechanism. We also assume that a velocity distribution on the boundary of the cylinder and a density distribution on gas inflow segments are given. The gas motion is described by the one-dimensional Navier-Stokes equations of viscous compressible fluid dynamics. It is required to determine the joint motion of the gas and crankshaft mechanism. We prove that this problem has a weak renormalized solution.
In this paper, we are interested in a shape optimization problem for a fluid-structure interactio... more In this paper, we are interested in a shape optimization problem for a fluid-structure interaction system composed by an elastic structure immersed in a viscous incompressible fluid. The cost functional to minimize is an energy functional involving together the fluid and the elastic parts of the structure. The shape optimization problem is introduced in the 2-dimensional case. However the results in this paper are obtained for a simplified free-boundary 1-dimensional problem. We prove that the shape optimization problem is wellposed. We study the shape differentiability of the free-boundary 1-dimensional model. The full characterization of the associated material derivatives is given together with the shape derivative of the energy functional. A special case is explicitly solved, showing the relevancy of this shape optimization approach for a simplified free boundary 1-dimensional problem. The full model in two spatial dimensions is under studies now.
In fracture mechanics, an important question concerns the useful life of mechanical components. S... more In fracture mechanics, an important question concerns the useful life of mechanical components. Such components are, usually, submitted to the actions of external forces and/or degrading agents which can trigger the crack nucleation and propagation process. In particular, when a mechanical component is already partially cracked, the question is how to extend its remaining useful life. In this work, a simple and efficient methodology aiming to extend the remaining useful life of cracked elastic bodies is proposed. More precisely, we want to find a way to retard or even avoid the triggering of the crack propagation process by nucleating hard and/or soft inclusions far from the crack tip. The main idea consists in minimize a shape functional based on the Rice's integral with respect to the nucleation of inclusions by using the concept of topological derivative. The obtained sensitivity, which corroborates with the famous Eshelby theorem, is used to indicate the regions where the controls have to be inserted. According to the Griffith's energy criterion, this simple procedure allows for increasing the remaining useful life of the cracked body. Finally, some numerical experiments are presented showing the applicability of the proposed methodology.
Contact problems with given friction are considered for plane elasticity in the framework of shap... more Contact problems with given friction are considered for plane elasticity in the framework of shape-topological optimization. The asymptotic analysis of the second kind variational inequalities in plane elasticity is performed for the purposes of shapetopological optimization. To this end, the saddle point formulation for the associated Lagrangian is introduced for the variational inequality. The non-smooth term in the energy functional is replaced by pointwise constraints for the multipliers. The one term expansion of the strain energy with respect to the small parameter which governs the size of the singular perturbation of geometrical domain is obtained. The topological derivatives of energy functional are derived in closed form adapted to the numerical methods of shape-topological optimization. In general, the topological derivative (TD) of the elastic energy is defined through a limit passage when the small parameter governing the size of the topological perturbation goes to zero. TD can be used as a steepestdescent direction in an optimization process like in any method based on the gradient of the cost functional. In this paper, we deal with the topological asymptotic analysis in the context of contact problems with given friction. Since the problem is nonlinear, the domain decomposition technique combined with the Steklov-Poincaré pseudodifferential boundary operator is used for asymptotic analysis purposes with respect to the small parameter associated with the size of the topological perturbation. As a fundamental result, the expansion of the strain energy coincides with the expansion of the Steklov-Poincaré operator on the boundary of the truncated domain, leading to the expression for TD. Finally, the obtained TD is applied in the context of topology optimization of mechanical structures under contact condition with given friction.
Resumo. A derivada topológicaé definida através da passagem do limite quando o parâmetro que gove... more Resumo. A derivada topológicaé definida através da passagem do limite quando o parâmetro que governa o tamanho da perturbação tende a zero. Então, ela pode ser usada como uma direção de descida em um processo de otimização como em quaquer método baseado no gradiente do funcional custo. Neste trabalho, lida-se com a análise assintótica topológica no contexto de problemas de contato com atrito dado. Uma vez que o problemaé não linear, a técnica de decomposição de domínio em conjunto com o operador pseudo-diferencial Steklov-Poincaré são utilizados para fins de análise assintótica com respeito ao parâmetro relacionado com o tamanho da perturbação topológica. Finalmente, o resultado da derivada topológica obtidoé aplicado no contexto de otimização de estruturas submetidas a condição de contato com atrito dado.
Informatics, Control, Measurement in Economy and Environment Protection, 2016
In the field of shape and topology optimization the new concept is the topological derivative of ... more In the field of shape and topology optimization the new concept is the topological derivative of a given shape functional. The asymptotic analysis is applied in order to determine the topological derivative of shape functionals for elliptic problems. The topological derivative (TD) is a tool to measure the influence on the specific shape functional of insertion of small defect into a geometrical domain for the elliptic boundary value problem (BVP) under considerations. The domain with the small defect stands for perturbed domain by topological variations. This means that given the topological derivative, we have in hand the first order approximation with respect to the small parameter which governs the volume of the defect for the shape functional evaluated in the perturbed domain. TD is a function defined in the original (unperturbed) domain which can be evaluated from the knowledge of solutions to BVP in such a domain. This means that we can evaluate TD by solving only the BVP in the intact domain. One can consider the first and the second order topological derivatives as well, which furnish the approximation of the shape functional with better precision compared to the first order TD expansion in perturbed domain. In this work the topological derivative is applied in the context of Electrical Impedance Tomography (EIT). In particular, we are interested in reconstructing a number of anomalies embedded within a medium subject to a set of current fluxes, from measurements of the corresponding electrical potentials on its boundary. The basic idea consists in minimize a functional measuring the misfit between the boundary measurements and the electrical potentials obtained from the model with respect to a set of ball-shaped anomalies. The first and second order topological derivatives are used, leading to a non-iterative second order reconstruction algorithm. Finally, a numerical experiment is presented, showing that the resulting reconstruction algorithm is very robust with respect to noisy data.
We study a thermo-mechanical system consisting of an elastic membrane to which a shape-memory rod... more We study a thermo-mechanical system consisting of an elastic membrane to which a shape-memory rod is glued. The slow movements of the membrane are controlled by the motions of the attached rods. A quasi-static model is used. We include the elastic feedback of the membrane on the rods. This results in investigating an elliptic boundary value problem in a domain Ω ⊂ R 2 with a cut, coupled with non-linear equations for the vertical motions of the rod and the temperature on the rod. We prove the existence of a unique global weak solution to this problem using a fixed point argument.
В статье доказано существование ренормализованных решений первой краевой задачи для стационарных ... more В статье доказано существование ренормализованных решений первой краевой задачи для стационарных уравнений Навье-Стокса динамики вязкого газа для всех значений показателя адиабаты из интервала (4/3, 3/2]. Библиография: 22 названия.
The deflection of a linearly elastic, thin plate subjected to a transverse load and supported by ... more The deflection of a linearly elastic, thin plate subjected to a transverse load and supported by a finite number of unilateral point supports depends on the thickness and elastic properties of the plate as well as on the force. From a variational problem statement for the deflection, it is shown that the right and left directional derivatives of the deflection
The model of volumetric material growth is introduced in the framework of finite elasticity. The ... more The model of volumetric material growth is introduced in the framework of finite elasticity. The new results obtained for the model are presented with complete proofs. The state variables include the deformations, temperature and the growth factor matrix function. The existence of global in time solutions for the quasistatic deformations boundary value problem coupled with the energy balance and the evolution of the growth factor is shown. The mathematical results can be applied to a wide class of growth models in mechanics and biology.
The paper deals with topology design of thermomechanical actuators. The goal of shape optimizatio... more The paper deals with topology design of thermomechanical actuators. The goal of shape optimization is to maximize the output displacement in a given direction on the boundary of the elastic body, which is submitted to a thermal excitation that induces a dilatation/contraction of the thermomechanical device. The optimal structure is identified by an elastic material distribution, while a very compliant (weak) material is used to mimic voids. The mathematical model of an actuator takes the form of a semi-coupled system of partial differential equations. The boundary value problem includes two components, the Navier equation for linear elasticity coupled with the Poisson equation for steady-state heat conduction. The mechanical coupling is the thermal stress induced by the temperature field. Given the integral shape functional, we evaluate its topological derivative with respect to the nucleation of a small circular inclusion with the thermomechanical properties governed by two contrast parameters. The obtained topological derivative is employed to generate a steepest descent direction within the level set numerical procedure of topology optimization in a fixed geometrical domain. Finally, several finite element-based examples for the topology design of thermomechanical actuators are presented.
The multiscale elasticity model of solids with singular geometrical perturbations of microstructu... more The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at ...
... Shape Sensitivity Analysis of Optimal Compliance Ifunctionals* Martin P. Bendsae Jan Sokolows... more ... Shape Sensitivity Analysis of Optimal Compliance Ifunctionals* Martin P. Bendsae Jan Sokolowski ABSTRACT ... The analysis provides the necessary 'Communicated by KK Choi 35 Copyright 8 1995 by Marcel Dekker, Inc. Page 2. 36 BENDS0E AND SOKOLOWSKI ...
Activities of the CNRS programme GDR Shape optimization are described. Recent developments in sha... more Activities of the CNRS programme GDR Shape optimization are described. Recent developments in shape optimization for eingenvalues and drag minimization are presented.
We present a review of known results in shape optimization from the point of view of Geometric An... more We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical aspects of the shape optimization theory. We focus on the theory of gradient flows of objective functions and their regularizations. Shape optimization is a part of calculus of variations which uses the geometry. Shape optimization is also related to the free boundary problems in the theory of Partial Differential Equations. We consider smooth perturbations of geometrical domains in order to develop the shape calculus for the analysis of shape optimization problems. There are many applications of such a framework, in solid and fluid mechanics as well as in the solution of inverse problems. For the sake of simplicity we consider model problems, in principle in two spatial dimensions. However, the methods presented are used as well in three spatial dimensions. We present a result on the convergence of the shape gradient method for a model p...
In the paper the general method for shape-topology sensitivity analysis of contact problems is pr... more In the paper the general method for shape-topology sensitivity analysis of contact problems is proposed. The method uses the domain decomposition method combined with the specific properties of minimizers for the energy functional. The method is applied to the static problem of an elastic body in frictionless contact with an rigid foundation. The contact model allows a finite interpenetration of the bodies on the contact region. This interpenetration is modeled by means of a scalar function that depends on the normal component of the displacement field on the potential contact zone. We present the asymptotic behavior of the energy shape functional when a spheroidal void is introduced in an arbitrary point of the elastic body. For the asymptotic analysis, we use the domain decomposition technique and the associated Steklov-Poincaré pseudodifferential operator. The differentiability of the energy with respect to the non-smooth perturbation is established. A closed form for the topological derivative is also presented.
Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2021
The topological derivative method is used to solve a pollution sources reconstruction problem gov... more The topological derivative method is used to solve a pollution sources reconstruction problem governed by a steady-state convection-diffusion equation. The inverse problem consists in the reconstruction of a set of pollution sources in a fluid medium by measuring the concentration of the pollutants within some subregion of the reference domain. We rewrite the inverse problem as a topology optimization problem which allows us to solve it by using the concept of topological derivatives. The resulting algorithm is able to reconstruct the pollution sources in one step and is independent of any initial guess. A numerical example is presented to show the effectiveness of our reconstruction method.
Proceedings of the Steklov Institute of Mathematics, 2020
We study an initial-boundary value problem with free boundary for one-dimensional equations of vi... more We study an initial-boundary value problem with free boundary for one-dimensional equations of viscous gas dynamics. The problem models the motion of a crankshaft mechanism under gas pressure. It is assumed that the gas fills a cylinder, which is modeled by the interval [0, 1]. A variable point a(t) ∈ [0, 1] models a piston moving inside the cylinder. The piston is assumed to be connected to a planar three-link crankshaft mechanism. We also assume that a velocity distribution on the boundary of the cylinder and a density distribution on gas inflow segments are given. The gas motion is described by the one-dimensional Navier-Stokes equations of viscous compressible fluid dynamics. It is required to determine the joint motion of the gas and crankshaft mechanism. We prove that this problem has a weak renormalized solution.
In this paper, we are interested in a shape optimization problem for a fluid-structure interactio... more In this paper, we are interested in a shape optimization problem for a fluid-structure interaction system composed by an elastic structure immersed in a viscous incompressible fluid. The cost functional to minimize is an energy functional involving together the fluid and the elastic parts of the structure. The shape optimization problem is introduced in the 2-dimensional case. However the results in this paper are obtained for a simplified free-boundary 1-dimensional problem. We prove that the shape optimization problem is wellposed. We study the shape differentiability of the free-boundary 1-dimensional model. The full characterization of the associated material derivatives is given together with the shape derivative of the energy functional. A special case is explicitly solved, showing the relevancy of this shape optimization approach for a simplified free boundary 1-dimensional problem. The full model in two spatial dimensions is under studies now.
In fracture mechanics, an important question concerns the useful life of mechanical components. S... more In fracture mechanics, an important question concerns the useful life of mechanical components. Such components are, usually, submitted to the actions of external forces and/or degrading agents which can trigger the crack nucleation and propagation process. In particular, when a mechanical component is already partially cracked, the question is how to extend its remaining useful life. In this work, a simple and efficient methodology aiming to extend the remaining useful life of cracked elastic bodies is proposed. More precisely, we want to find a way to retard or even avoid the triggering of the crack propagation process by nucleating hard and/or soft inclusions far from the crack tip. The main idea consists in minimize a shape functional based on the Rice's integral with respect to the nucleation of inclusions by using the concept of topological derivative. The obtained sensitivity, which corroborates with the famous Eshelby theorem, is used to indicate the regions where the controls have to be inserted. According to the Griffith's energy criterion, this simple procedure allows for increasing the remaining useful life of the cracked body. Finally, some numerical experiments are presented showing the applicability of the proposed methodology.
Contact problems with given friction are considered for plane elasticity in the framework of shap... more Contact problems with given friction are considered for plane elasticity in the framework of shape-topological optimization. The asymptotic analysis of the second kind variational inequalities in plane elasticity is performed for the purposes of shapetopological optimization. To this end, the saddle point formulation for the associated Lagrangian is introduced for the variational inequality. The non-smooth term in the energy functional is replaced by pointwise constraints for the multipliers. The one term expansion of the strain energy with respect to the small parameter which governs the size of the singular perturbation of geometrical domain is obtained. The topological derivatives of energy functional are derived in closed form adapted to the numerical methods of shape-topological optimization. In general, the topological derivative (TD) of the elastic energy is defined through a limit passage when the small parameter governing the size of the topological perturbation goes to zero. TD can be used as a steepestdescent direction in an optimization process like in any method based on the gradient of the cost functional. In this paper, we deal with the topological asymptotic analysis in the context of contact problems with given friction. Since the problem is nonlinear, the domain decomposition technique combined with the Steklov-Poincaré pseudodifferential boundary operator is used for asymptotic analysis purposes with respect to the small parameter associated with the size of the topological perturbation. As a fundamental result, the expansion of the strain energy coincides with the expansion of the Steklov-Poincaré operator on the boundary of the truncated domain, leading to the expression for TD. Finally, the obtained TD is applied in the context of topology optimization of mechanical structures under contact condition with given friction.
Resumo. A derivada topológicaé definida através da passagem do limite quando o parâmetro que gove... more Resumo. A derivada topológicaé definida através da passagem do limite quando o parâmetro que governa o tamanho da perturbação tende a zero. Então, ela pode ser usada como uma direção de descida em um processo de otimização como em quaquer método baseado no gradiente do funcional custo. Neste trabalho, lida-se com a análise assintótica topológica no contexto de problemas de contato com atrito dado. Uma vez que o problemaé não linear, a técnica de decomposição de domínio em conjunto com o operador pseudo-diferencial Steklov-Poincaré são utilizados para fins de análise assintótica com respeito ao parâmetro relacionado com o tamanho da perturbação topológica. Finalmente, o resultado da derivada topológica obtidoé aplicado no contexto de otimização de estruturas submetidas a condição de contato com atrito dado.
Informatics, Control, Measurement in Economy and Environment Protection, 2016
In the field of shape and topology optimization the new concept is the topological derivative of ... more In the field of shape and topology optimization the new concept is the topological derivative of a given shape functional. The asymptotic analysis is applied in order to determine the topological derivative of shape functionals for elliptic problems. The topological derivative (TD) is a tool to measure the influence on the specific shape functional of insertion of small defect into a geometrical domain for the elliptic boundary value problem (BVP) under considerations. The domain with the small defect stands for perturbed domain by topological variations. This means that given the topological derivative, we have in hand the first order approximation with respect to the small parameter which governs the volume of the defect for the shape functional evaluated in the perturbed domain. TD is a function defined in the original (unperturbed) domain which can be evaluated from the knowledge of solutions to BVP in such a domain. This means that we can evaluate TD by solving only the BVP in the intact domain. One can consider the first and the second order topological derivatives as well, which furnish the approximation of the shape functional with better precision compared to the first order TD expansion in perturbed domain. In this work the topological derivative is applied in the context of Electrical Impedance Tomography (EIT). In particular, we are interested in reconstructing a number of anomalies embedded within a medium subject to a set of current fluxes, from measurements of the corresponding electrical potentials on its boundary. The basic idea consists in minimize a functional measuring the misfit between the boundary measurements and the electrical potentials obtained from the model with respect to a set of ball-shaped anomalies. The first and second order topological derivatives are used, leading to a non-iterative second order reconstruction algorithm. Finally, a numerical experiment is presented, showing that the resulting reconstruction algorithm is very robust with respect to noisy data.
We study a thermo-mechanical system consisting of an elastic membrane to which a shape-memory rod... more We study a thermo-mechanical system consisting of an elastic membrane to which a shape-memory rod is glued. The slow movements of the membrane are controlled by the motions of the attached rods. A quasi-static model is used. We include the elastic feedback of the membrane on the rods. This results in investigating an elliptic boundary value problem in a domain Ω ⊂ R 2 with a cut, coupled with non-linear equations for the vertical motions of the rod and the temperature on the rod. We prove the existence of a unique global weak solution to this problem using a fixed point argument.
В статье доказано существование ренормализованных решений первой краевой задачи для стационарных ... more В статье доказано существование ренормализованных решений первой краевой задачи для стационарных уравнений Навье-Стокса динамики вязкого газа для всех значений показателя адиабаты из интервала (4/3, 3/2]. Библиография: 22 названия.
The deflection of a linearly elastic, thin plate subjected to a transverse load and supported by ... more The deflection of a linearly elastic, thin plate subjected to a transverse load and supported by a finite number of unilateral point supports depends on the thickness and elastic properties of the plate as well as on the force. From a variational problem statement for the deflection, it is shown that the right and left directional derivatives of the deflection
The model of volumetric material growth is introduced in the framework of finite elasticity. The ... more The model of volumetric material growth is introduced in the framework of finite elasticity. The new results obtained for the model are presented with complete proofs. The state variables include the deformations, temperature and the growth factor matrix function. The existence of global in time solutions for the quasistatic deformations boundary value problem coupled with the energy balance and the evolution of the growth factor is shown. The mathematical results can be applied to a wide class of growth models in mechanics and biology.
The paper deals with topology design of thermomechanical actuators. The goal of shape optimizatio... more The paper deals with topology design of thermomechanical actuators. The goal of shape optimization is to maximize the output displacement in a given direction on the boundary of the elastic body, which is submitted to a thermal excitation that induces a dilatation/contraction of the thermomechanical device. The optimal structure is identified by an elastic material distribution, while a very compliant (weak) material is used to mimic voids. The mathematical model of an actuator takes the form of a semi-coupled system of partial differential equations. The boundary value problem includes two components, the Navier equation for linear elasticity coupled with the Poisson equation for steady-state heat conduction. The mechanical coupling is the thermal stress induced by the temperature field. Given the integral shape functional, we evaluate its topological derivative with respect to the nucleation of a small circular inclusion with the thermomechanical properties governed by two contrast parameters. The obtained topological derivative is employed to generate a steepest descent direction within the level set numerical procedure of topology optimization in a fixed geometrical domain. Finally, several finite element-based examples for the topology design of thermomechanical actuators are presented.
The multiscale elasticity model of solids with singular geometrical perturbations of microstructu... more The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at ...
... Shape Sensitivity Analysis of Optimal Compliance Ifunctionals* Martin P. Bendsae Jan Sokolows... more ... Shape Sensitivity Analysis of Optimal Compliance Ifunctionals* Martin P. Bendsae Jan Sokolowski ABSTRACT ... The analysis provides the necessary 'Communicated by KK Choi 35 Copyright 8 1995 by Marcel Dekker, Inc. Page 2. 36 BENDS0E AND SOKOLOWSKI ...
Activities of the CNRS programme GDR Shape optimization are described. Recent developments in sha... more Activities of the CNRS programme GDR Shape optimization are described. Recent developments in shape optimization for eingenvalues and drag minimization are presented.
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