A Method of “Exact” Numerical Differentiation for Error Elimination in Finite-Element-Based Semi-Analytical Shape Sensitivity Analyses*

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Abstract
The traditional, simple numerical differentiation of finite-element stiffness matrices by a forward difference scheme is the source of severe error problems that have been reported recently for certain problems of finite-element-based, semi-analytical shape design sensitivity analysis. In order to develop a method for elimination of such errors, without a sacrifice of the simple numerical differentiation and other main advantages of the semi-analytical method, the common mathematical structure of a broad range of finite-element stiffness matrices is studied in this paper. This study leads to the result that element stiffness matrices can generally be expressed in terms of a class of special scalar functions and a class of matrix functions of shape design variables that are defined such that the members of the classes admit “exact” numerical differentiation (exact up to round-off error) by means of very simple correction factors to upgrade standard computationally inexpensive first-order finite differences to “exact” numerical derivatives with respect to shape design variables. The correction factors can be easily computed once and for all as an initial step of the sensitivity analysis. Application of this method eliminates frequently encountered problems of severe dependence of semi-analytical design sensitivities on the size of perturbations of design variables and on finite-element mesh size and refinement, among other factors. The results are equivalent to those that would be obtained by numerical evaluation of corresponding analytical design sensitivities. However, the method is much more problem-independent and is easier to implement than the analytical method. Thus, it is shown in this paper that the new approach to semi-analytical shape sensitivity analysis is easily implemented as an integral part of finite-element analysis. The method of error elimination by “exact” numerical differentiation can be implemented even in connection with existing computer codes for semi-analytical sensitivity analysis, where subroutines for computation of element stiffness matrices are available only in the form of black-box routines. The applicability of the method presented is demonstrated for a broad class of commonly used finite elements. It is also shown that the method is compatible with common methods of design boundary parametrization based on master node techniques. Four numerical examples are presented to illustrate and discuss capabilities of the method.

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