Improved semi-analytical sensitivity analysis using a secant stiffness matrix for geometric nonlinear shape optimization

• Exact semi-analytical sensitivity analysis method is extended to geometric nonlinear case.
• A correction term is constructed in the product spaces of two different sets of zero-eigenvectors.
• The analytical formulas of these vectors are presented with respect to the secant stiffness matrix.
• Numerical results show that the method eliminates semi-analytical sensitivity errors significantly.

This work presents a semi-analytical sensitivity analysis approach for geometric nonlinear shape optimization. A secant stiffness matrix is used in the nonlinear solution procedure. Conditions that an accurate derivative of the matrix should satisfy are determined. Following these conditions, a correction term for the finite differencing approximation is constructed. Due to the asymmetry of the secant stiffness matrix, the correction term is expressed in the product spaces of two sets of zero eigenvectors. The analytical formulas of these vectors are also presented, which increases the computational efficiency. Numerical examples highlight the ability of the technique to effectively eliminate sensitivity analysis errors.