Numerical Techniques in Relaxed Optimization Problems

Roubíček, Tomáš

doi:10.1007/0-387-28654-3_8

Cited by 6 publications

(4 citation statements)

References 43 publications

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“…Moreover, the author shows that the corresponding sequence of solutions converges to the solution of the relaxed problem, (u, η) as the size of the mesh, d, goes to zero. Results that continue to build in this direction are in [2,4,5,22,39,38].…”

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confidence: 86%

A Modified Split Bregman Algorithm for Computing Microstructures Through Young Measures

Jaramillo,

Venkataramani

2019

Preprint

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The goal of this paper is to describe the oscillatory microstructure that can emerge from minimizing sequences for nonconvex energies. We consider integral functionals that are defined on real valued (scalar) functions u(x) which are nonconvex in the gradient ux and possibly also in u.To characterize the microstructures for these nonconvex energies, we minimize the associated relaxed energy using two novel approaches: i) a semi-analytical method based on control systems theory, ii) and a numerical scheme that combines convex splitting together with a modified version of the split Bregman algorithm. These solutions are then used to gain information about minimizing sequences of the original problem and the spatial distribution of microstructure.

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confidence: 86%

A Modified Split Bregman Algorithm for Computing Microstructures Through Young Measures

Jaramillo,

Venkataramani

2019

Preprint

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“…Finally, the numerical approximation based on gradient Young measures faces the difficulty that one needs to discretize at every Gauss point a measure. See, for example, [7,14,16,33,43,44] and the references therein for detailed information on these aspects.…”

Section: Numerical Analysis Of Nonconvex Variational Problems and Thementioning

confidence: 99%

Convergence of adaptive finite element methods for a nonconvex double-well minimization problem

Carstensen

Dolzmann

2015

Math. Comp.

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This paper focuses on the numerical analysis of a nonconvex variational problem which is related to the relaxation of the two-well problem in the analysis of solid-solid phase transitions with incompatible wells and dependence on the linear strain in two dimensions. The proposed approach is based on the search for minimizers for this functional in finite element spaces with Courant elements and with successive loops of the form SOLVE, ESTIMATE, MARK, and REFINE. Convergence of the total energy of the approximating deformations and strong convergence of all except one component of the corresponding deformation gradients is established. The proof relies on the decomposition of the energy density into a convex part and a null-Lagrangian. The key ingredient is the fact that the convex part satisfies a convexity property which is stronger than degenerate convexity and weaker than uniform convexity. Moreover, an estimator reduction property for the stresses associated to the convex part in the energy is established.

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“…In recent years, there has been an increased interest in solving relaxed problems of this type [6][7][8]. The main challenge is how to compute an optimal Young measure α ∈ Ω.…”

Section: Introductionmentioning

confidence: 99%