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We consider the variational formulation of the problem of elastic shells in the membrane approximation, when the medium surface is hyperbolic. It appears that the corresponding bilinear form behaves as some kind of two-dimensional elasticity without shear rigidity. This amounts to saying that the membrane behaves rather as a net made of elastic strings disposed along the asymptotic curves of the surface than as an elastic two-dimensional medium. The mathematical and physical reasons of this behavior...
The notion of quasiconvex exposed points is introduced for compact sets of matrices, motivated from the variational approach to material microstructures. We apply the notion to give geometric descriptions of the quasiconvex extreme points for a compact set. A weak version of Straszewicz type density theorem in convex analysis is established for quasiconvex extreme points. Some examples are examined by using known explicit quasiconvex functions.
The notion of quasiconvex exposed points is introduced for compact sets of matrices, motivated
from the variational approach to material microstructures.
We apply the
notion to give geometric descriptions of the
quasiconvex extreme points for a compact set. A weak version of Straszewicz type
density theorem in convex analysis is established for quasiconvex extreme points. Some examples
are examined by using known explicit quasiconvex functions.
We consider the question raised in [1] of whether relaxed energy
densities involving both bulk and surface energies
can be written as a sum of two functions, one depending on the net gradient
of admissible functions, and the other on net
singular part.
We show that, in general, they cannot. In particular, if the bulk density
is quasiconvex but not convex, there
exists a convex and homogeneous of degree 1 function of the jump such that
there is no such representation.
A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for...
Multidimensional vectorial non-quasiconvex variational problems are relaxed by means of a generalized-Young-functional technique. Selective first-order optimality conditions, having the form of an Euler-Weiestrass condition involving minors, are formulated in a special, rather a model case when the potential has a polyconvex quasiconvexification.
We apply the method of reliable solutions to the bending problem for an elasto-plastic beam, considering the yield function of the von Mises type with uncertain coefficients. The compatibility method is used to find the moments and shear forces. Then we solve a maximization problem for these quantities with respect to the uncertain input data.
The gradient based topological optimization tools introduced during the last ten years tend naturally to modify the topology of a domain by creating small holes inside the domain. Once these holes have been created, they usually remain unchanged, at least during the topological phase of the optimization algorithm. In this paper, a new asymptotic expansion is introduced which allows to decide whether an existing hole must be removed or not for improving the cost function. Then, two numerical examples...
The gradient based topological optimization tools introduced during the
last ten years tend naturally to modify the topology of a domain by
creating small holes inside the domain.
Once these holes have been created, they usually remain
unchanged, at least during the topological phase of the optimization
algorithm. In this paper, a new asymptotic expansion is introduced which
allows to decide whether an existing hole must be removed or not for
improving the cost function. Then, two numerical...
The goal of this paper is to study the so-called worst-case or robust optimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst, compliance when the loads are subject to some unknown perturbations. We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm to compute it....
The goal of this paper is to study the so-called worst-case or robust
optimal design problem for minimal compliance. In the context of linear
elasticity we seek an optimal shape which minimizes the largest, or worst,
compliance when the loads are subject to some unknown perturbations.
We first prove that, for a fixed shape, there exists indeed a worst
perturbation (possibly non unique) that we characterize as the maximizer
of a nonlinear energy. We also propose a stable algorithm to
compute...
The state problem of elasto-plasticity (for the model with strain-hardening) is formulated in terms of stresses and hardening parameters by means of a time-dependent variational inequality. The optimal design problem is to find the shape of a part of the boundary such that a given cost functional is minimized. For the approximate solutions piecewise linear approximations of the unknown boundary, piecewise constant triangular elements for the stress and the hardening parameter, and backward differences...
A minimization of a cost functional with respect to a part of a boundary is considered for an elasto-plastic axisymmetric body obeying Hencky's law. The principle of Haar-Kármán and piecewise linear stress approximations are used to solve the state problem. A convergence result and the existence of an optimal boundary is proved.
Existence of an optimal shape of a deformable body made from a physically nonlinear material obeying a specific nonlinear generalized Hooke’s law (in fact, the so called deformation theory of plasticity is invoked in this case) is proved. Approximation of the problem by finite elements is also discussed.
We present the combination of a state control and shape design approaches for the optimization of micro-fluidic channels used for sample extraction and separation of chemical species existing in a buffer solution. The aim is to improve the extraction and identification capacities of electroosmotic micro-fluidic devices by avoiding dispersion of the extracted advected band.
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