Numerical solution of a variational problem arising in stress
analysis: The vector case
Alexandre Caboussat - University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, Texas 77204 - 3008, United States (email)
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.
Keywords: Non-smooth optimization, stress analysis, augmented
Lagrangian method, finite elements approximation, Uzawa algorithm.
Received: August 2008; Revised: October 2009; Published: March 2010.
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