Sensitivity analysis of non-linear (material and/or geometrical) problems plays an important role in structural optimization, inverse problem and reliability analysis. Both finite element method (FEM) (Arora and Cardoso, 1992, Jao and Arora, 1992a, Jao and Arora, 1992b, Choi and Santos, 1987, Santos and Choi, 1988, Badrinarayanan and Zabaras, 1996, Rojc and Tok, 2003, Kim and Choi, 2001 and Cho and Lee, 2002) and continuum boundary element method (BEM) (Mukherjee and Chandra, 1989, Mukherjee and Chandra, 1991, Zhang and Mukherjee, 1992, Zhang et al., 1992, Wei et al., 1994, Leu and Mukherjee, 1994a, Leu and Mukherjee, 1994b and Leu and Mukherjee, 1995) have been developed for geometry and material non-linear sensitivity analysis by a lot of researchers. Currently, there are three different approaches that are used in sensitivity analysis: the finite difference approach (FDA), the adjoint structure approach (ASA), and the direct differentiation approach (DDA). Among these three differentiation approaches, ASA, similarly as DDA, consists in exact analytical differentiation of primary equations, and for large number of design parameters it is advocated as more efficient than DDA (Haug et al., 1986). However for the non-linear history-dependent problems, the DDA has been seen to be more suitable (Tsay and Arora, 1989 and Kleiber et al., 1997). Note that for non-linear problems, an incremental-iterative numerical method is needed. Therefore, a powerful and high efficiency algorithm for non-linear solver is the cornerstone of a successful non-linear analysis. The concept of consistent tangent operator (CTO), which is first proposed in finite element method by Simo and Taylor (1985), has obtained wide application in sensitivity analysis of non-linear problems. Use of the CTO, as it was pointed out by Vidal et al., 1991, Vidal and Haber, 1993, Kleiber and Hien, 1991, Kleiber et al., 1994, Kleiber et al., 1995 and Michaleris et al., 1994, provides very accurate numerical results in sensitivity analysis; while other approaches (e.g. using the continuum tangent operator) might lead to significant errors. Bonnet and Mukherjee (1996), for the first time, have introduced the CTO concept in boundary element and small strain elastic plastic sensitivity analysis. Later, Poon et al. (1998) have further developed this method into 2D elastoplastic sensitivity problem. However, in these papers of CTO-based BEM (Bonnet and Mukherjee, 1996 and Poon et al., 1998), only elastic plastic material sensitivity parameter is studied. The viscoplastic material sensitivity, geometry sensitivity and boundary condition sensitivity analysis are not considered. Recently, Liang et al. (2004) have solved the viscoplastic material sensitivity problem with CTO-based implicit BEM, but the geometry sensitivity and boundary condition sensitivity have not yet been developed.
Therefore, the goal of this paper is to present a sensitivity analysis method for parameters affecting geometry, elastic–viscoplastic material constant and boundary condition with the CTO-based small strain boundary element. The CTO plays a pivotal role in the present work. The design variables for sensitivity analysis include geometry (shape, dimension and size) parameters, elastic–viscoplastic material parameters and boundary condition parameters. The organization of the paper is arranged as follows: First, based on small strain theory, Perzyna’s elastic–viscoplastic constitutive relation is introduced with the mixed strain-hardening material that includes both isotropic and kinematic cases. Two types of viscoplastic flow functions with exponent-type and power-type are built in the viscoplastic material constitutive relation. Secondly, the elastic–viscoplastic CTO-based boundary element and related radial return algorithm (RRA) are derived with new formulae of RRA and CTO which combine the mixed strain-hardening model and both exponent type and power-type of the flow functions. Then, based on the direct differentiation approach, the fully incremental boundary integral equations of geometry sensitivity, elastic–viscoplastic sensitivity and boundary condition sensitivity are developed together with the new sensitivity formulation of RRA and CTO-based equations. A non-linear algorithm for geometry, elastic–viscoplastic material and boundary condition sensitivities is developed. Finally, four plane strain numerical examples with geometry sensitivity, elastic–viscoplastic material constant sensitivity and boundary condition sensitivity analysis are presented and discussed.
