Abstract
In this article, we discuss the numerical solution of a constrained minimization problem arising from the stress analysis of elasto-plastic bodies.This minimization problem has the flavor of a generalized non-smooth eigenvalue problem, with the smallest eigenvalue corresponding to the load capacity ratio of the elastic body under consideration.An augmented Lagrangian method, together with finite element approximations, is proposed for the computation of the optimum of the non-smooth objective function, and the corresponding minimizer.The augmented Lagrangian approach allows the decoupling of some of the nonlinearities and of the differential operators.Similarly an appropriate Lagrangian functional, and associated Uzawa algorithm with projection, are introduced to treat non-smooth equality constraints.Numerical results validate the proposed methodology for various two-dimensional geometries.
Highlights
Introduction and MotivationsThe stress analysis of elasto-plastic bodies is a crucial problem for engineers who want to estimate maximal stresses and fractures for a given body and geometry [28, 31]
Finite element methods are commonly used for the numerical solution of such problems, see e.g. [10, 44], and numerous works exist in the literature concerning the numerical approximation of stresses, fractures in elastic or plastic materials [22, 29, 37]
We address the solution of (6) and (7) by an augmented Lagrangian method
Summary
Introduction and MotivationsThe stress analysis of elasto-plastic bodies is a crucial problem for engineers who want to estimate maximal stresses and fractures for a given body and geometry [28, 31]. Non-smooth optimization, Stress analysis, Augmented Lagrangian method, Finite elements approximation, Uzawa algorithm. We present a numerical method to compute finite element approximations of these eigenvalues, together with approximations of the corresponding generalized eigenfunctions (i.e. the minimizers of the finite dimensional approximations of (6) and (7)).
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