Shape Sensitivity Analysis using Spatial Gradient Reconstruction with Traction and Equilibrium Enforcement

Abstract

Continuum Sensitivity Analysis (CSA) provides a framework for computing response derivatives with respect to design variables and is therefore useful in structural design and optimization. In practical engineering problems, these derivatives are needed to understand how changes in geometry, material distribution, loading, or boundary shape affect structural response. Such information is important in design studies where repeated finite element analyses are expensive and where finite-difference derivatives may become inefficient, noisy, or dependent on the chosen perturbation size. For shape design problems, accurate sensitivities are especially valuable because small boundary changes can alter stresses, displacements, and load paths in ways that directly affect structural performance. Despite these advantages, the practical use of shape sensitivity analysis has remained difficult. In the boundary-velocity form of CSA, the sensitivity equations require spatial derivatives of the analysis response on the moving boundary. These derivatives are often not reliable when taken directly from finite element results, particularly near boundaries where stress recovery is less accurate. The difficulty becomes more significant when stresses are used, since stress is already obtained from derivatives of displacement and is therefore more sensitive to discretization error, boundary-condition treatment, and numerical precision. As a result, inaccurate boundary stress gradients can affect the sensitivity boundary conditions and loads that drive the CSA solution. This thesis addresses this difficulty through stress-based Spatial Gradient Reconstruction (SGR). The work focuses on recovering boundary stress fields and stress gradients from finite element stress data, with particular attention to traction enforcement, equilibrium enforcement, patch construction, weighting, and regularization. The reconstructed stress information is then used within a CSA framework to evaluate shape sensitivities for the benchmark problem. Verification is carried out using the two-dimensional Timoshenko cantilever beam, which provides closed-form stress, displacement, and selected shape-derivative quantities when the exact benchmark boundary conditions are imposed consistently. Quadrilateral and triangular finite element mesh families are used to study how reconstruction choices affect recovered coefficients, sensitivity loads, and final shape-derivative results for length and height design variables. The results from this study provide guidance for using stress-based SGR as a nonintrusive post-processing step for CSA workflows that rely on stress output from black-box finite element solvers.