Shape optimization with immersed interface finite element method

Abstract

AbstractShape Optimization using conventional interface‐fitted finite element method requires either mesh morphing or remeshing, which is computationally expensive and prone to errors due to suboptimal mesh or solution interpolation. In this article, we present a shape optimization scheme which uses a four‐noded rectangular immersed‐interface element circumventing the need for interface‐fitted mesh for finite element analysis, and mesh morphing or re‐meshing for shape optimization. The analysis problem is solved using a non‐conformal, Petrov–Galerkin (ncPG) formulation. We use non‐conformal trial functions with conformal test functions. A fixed structured grid with a linear approximation for the interface within each element is used. We perform a bi‐material patch test to confirm the consistency and convergence of the immersed‐interface finite element method (IIFEM). The convergence of the displacement and stress error norms are of the same or slightly better order compared to the interface‐fitted FEM. Thus, IIFEM reduces the cost of meshing without compromising the accuracy of the solutions obtained. We then perform analytical design sensitivity analysis to obtain the gradient of the global stiffness matrix with respect to shape design variables. The sensitivities are used in the gradient‐based optimization formulation of the shape design problem. We present several ways to parametrize the design space. Finally, verification and case studies are presented to demonstrate the accuracy and potential of the approach.