Shape Design Sensitivity Analysis and Optimization of Elasto-Plasticity with Frictional Contact

Abstract

A shape design sensitivity analysis and optimization are proposed for the ine nitesimal elasto ‐plasticity with a frictional contact condition. Rate-independent plasticity is considered with a return mapping algorithm and a von Mises yield criterion. The contact condition is formulated using the penalty method and the modie ed coulomb friction law. A continuum-based shape design sensitivity formulation is developed for structural and frictional contact variational equations. The direct differentiation method is used to compute the displacement sensitivity, and the sensitivities of various performance measures are computed from the displacement sensitivity. Path dependency of the sensitivity equation due to the constitutive relation and friction is discussed. It is shown that no iteration is required to solve the sensitivity equation. Response analysis and the proposed sensitivity formulation are implemented using the mesh-free method where the mesh distortion problem can be resolved. Numerical examples show accurate results of the proposed method compared to the e nite difference method. Dife culties in the sensitivity formulation for the e nite deformation problem are discussed. I. Introduction B ECAUSEoftherecentdevelopmentofcomputationalmechanics, it is now possible to analyze practical examples of complicated structural problems. Many design engineers, who are not satise ed with response analysis alone, have keen interests in the methodology of the design. For more than two decades, signie cant research efforthasbeen focused on the rate of response with respect to the changes in structural shape under shape design sensitivity analysis (DSA). Analysis of the design sensitivity information is the most important and costly procedure in the automated optimum design process. It supplies useful quantitative information to the design engineer about the direction of the desired design change. In a classical linear problem, DSA research has proved the differentiability of the solution of the response analysis using the linear operator theory and has derived specie c sensitivity expressions for variousproblems. 1 Aresultworthy of attention in linear DSAis that theoriginalresponseandthesensitivityoftheresponsebelongtothe samekinematicallyadmissiblespaceandhavethesameregularities. Owing to the development of the response analysis capability, engineers have directed their interest to the nonlinear problems that are dealt with efe ciently. Because many design application problems are accompanied by plastic deformation, the design sensitivity of nonlinear problems has been actively developed, and many research results are reported. In the procedure of nonlinear response analysis, the projection, called a return mapping, of the elastic trial stress is carried out to satisfy the variational inequality (VI)through an iteration in the stress space. 2 The DSA, on the other hand, computes the rate of change of the projected response in the tangential direction of the constraint set without iteration. Note that the sensitivity analysis is linear and can be computed without iteration even thoughtheresponseanalysisisnonlinear. 3 Unlikethenonlinearelastic problem, the sensitivity equation of the plastic problem requires sensitivity of the stress and internal evolution variables at the previous time. The sensitivity equation is solved at each time without iteration, and the sensitivity of the stress and evolution variables are