Advanced finite element formulations for accurate and efficient topology optimization of 2D continuum structures

Abstract

Finite element analysis is crucial for topology optimization. Most implementations use 4-noded membrane elements (Q4) despite low accuracy from membrane locking. This highlights the need for alternatives balancing accuracy and computational cost. This work presents algorithms to implement advanced finite elements into topology optimization. The advanced elements include Pian-Sumihara (PS) and Allman elements, and quadratic membrane element (Q8). Three problem types are addressed: volume-constrained compliance minimization, stress-constrained volume minimization, and stress- and volume-constrained compliance minimization. Performance of different finite elements is analyzed and compared using multiple criteria: accuracy, convergence characteristics, and computational cost. For compliance minimization with uniform regular meshes, PS delivers the most accurate layouts. Its solutions closely resemble Mitchell's truss theory and follow load paths effectively. PS converges to the lowest compliance with smooth black-and-white designs. Q8 matches closely but yields slightly higher objective values. For distorted mesh problems, Q8 shows superior performance. For stress-constrained volume minimization, PS elements achieved optimal results with 0.226 volume fraction compared to Q4's 0.245. Allman and Q8 values are 0.241 and 0.237. Similar behavior occurred for volume- and stress-constrained compliance minimization. PS showed the most optimal design, while Q8 produced similar designs. Allman and Q4 failed to reach optimal designs, showing gray regions. Average computation times relative to Q4 vary: 1.06-1.19× for PS, 1.39-1.74× for Q8, and 1.55-2.54× for Allman. Q4 is fastest but provides lower accuracy. With adequate h-refinement, Q4 can achieve results similar to advanced elements. PS elements are recommended for uniform regular meshes requiring high accuracy. Q8 elements are preferred for distorted mesh problems. Allman elements suit moderate accuracy requirements.