Variational sensitivity analysis of a nonlinear solid shell element

Abstract

SUMMARY The paper is concerned with variational sensitivity analysis of a nonlinear solid shell element, which is based on the Hu–Washizu variational principle. The sensitivity information is derived on the continuous level and discretized to yield the analytical expressions on the computational level. Especially, the pseudo load matrix and the sensitivity matrix, which dominate design sensitivity analysis of shape optimization problems, are derived. Because of the mixed formulation, condensation of the pseudo load matrix on the element level is performed to compute the sensitivity matrix. An illustrative example from the field of geometry‐based shape optimization demonstrates the possible application of the presented formulation. Copyright © 2013 John Wiley & Sons, Ltd.

Gradient‐based determination of principal design influences on composite structures

“…Having solved Equation () for $$\Delta \hat{\bm {{u}}}_e$$, the increments of the strains and stresses can be updated as follows $$$$\begin{equation} \begin{aligned} \Delta \bm {{\alpha }}^1_e &= \bm {{C}}_e^{\mathsf {-1}}\,(\bm {{L}}_e\,\Delta \hat{\bm {{u}}}_e + \bm {{b}}_e), \\ \Delta \bm {{\alpha }}^2_e &= -(\bm {{A}}_e^{22})^{\mathsf {-1}}\,(\bm {{A}}_e^{21}\,\Delta \bm {{\alpha }}^1_e + \bm {{a}}_e^2), \\ \Delta \bm {{\beta }}_e &= \bm {{C}}_e^{\mathsf {-1}}\,(\bm {{A}}_e\,\Delta \bm {{\alpha }}_e^1 + \bm {{a}}_e). \end{aligned} \end{equation}$$$$The interested reader is referred to [3] or [2] for further details on element matrices and shape functions that have been neglected here for reasons of brevity.…”

“…The method for the computation of sensitivity information is based on a variational approach as described in [2] in the context of geometric shape sensitivity analysis. This approach is founded on an enhanced kinematic viewpoint that offers a rigorous separation of geometric and physical effects within a deformation process.…”

“…The utilization of SVD in sensitivity analysis of thin‐walled structures has shown promising results ‐ especially in the context of shape optimization ‐ in enhancing the understanding of complex structural behavior and aiding in the designing process, cf. for example [2].…”

Structural design sensitivity analysis in context of non‐linear buckling

“…The sensitivity analysis provides gradient information to include stability quantities in efficient mathematical optimisation schemes. Variational design sensitivity analysis of this solid shell finite element is performed in [3], providing essential quantities by means of the pseudo load matrix and sensitivity matrix. Imperfections are an important issue in the framework of structural stability.…”

Design space exploration based on variational sensitivity analysis

“…The design of such structures is extremely important for their stability, robustness and load-bearing capacity. The variational design sensitivity analysis for this nonlinear solid shell is performed and especially the pseudo load matrix and the sensitivity matrix are derived in [3]. An illustrative example demonstrates the advocated concept.…”