A Comparison of Displacement and Mixed Finite Element Formulations for Variational Design Sensitivity Analysis

The authors formulate design sensitivity analysis in form of a

variational approach

based on a novel

local representation

of continuum mechanics, see the publications of the first author, e.g. [

1

], for a summarising overview on the concept and the publications of the second author, e.g. [

3

], for the subsequent application to elasto-plastic material behaviour.

The central idea is to trace and separate the influence of geometry mappings from the influence of deformation mappings on all field quantities. Thus, a reformulation of continuum mechanics following the

intrinsic formulation

by Noll [

2

] but using two independent mappings defined on a local parameter space is advocated. Consequently, the

consistent linearisation concept

of computational mechanics used to derive tangent

stiffness matrices

, should also be applied to the geometry mappings, i.e. to compute

tangent geometry sensitivity matrices

.

Firstly, the fundamentals of the advocated treatment are described in general terms.

Secondly, the consequences for the finite element development procedure are outlined on the theoretical as well as computational level. Here, the parallelism of sensitivity and stiffness computation are highlighted for different finite elements, i.e. for standard displacement and mixed formulations.

Thirdly, hints are given to guarantee correct variational sensitivity information by a general comparison strategy using finite differences.

The outlined theoretical and computational framework is seen to be an efficient method to investigate the influence of different element formulations on the solution of the optimisation problem.