Topology Optimization With Quadrilateral Elements: A Comparative Study, Codes, and Tutorials

Abstract

This paper offers a comparative study for topology optimization (TO) with quadrilateral family elements, for example, 4-noded (Q4), 8-noded (Q8), and 9-noded (Q9) elements for diverse problems. MATLAB codes are developed with Q8 and Q9 elements for three distinct design problems involving different physics, wherein we provide straightforward and efficient methods for generating connectivity matrices and for determining elemental stiffness matrices in terms of Poisson's ratio for these elements. To demonstrate the relative performances, the paper presents (i) Compliance minimization for stiff structures subjected to constant forces, (ii) Compliance minimization for load-bearing structures under design-dependent pressure loads, (iii) Compliant mechanism problems focused on maximizing desired output deformation. A volume constraint is applied to all problems. Q8 and Q9 finite elements produce optimized designs free of checkerboard patterns. Sensitivity and density filtering schemes are incorporated to ensure a minimum feature size of the optimized designs. An outline for implementing a Heaviside projection filter is also provided for achieving optimized solutions close to binary (0-1). The presented exhaustive comparative study, accompanied by codes and supporting materials, serves as both an educational tool for academia and a valuable resource for those new to the field. The relevant codes are included in the appendices.