A quadratic approximation for volume minimization topology optimization

Abstract

This paper proposes a variable density-based topology optimization method, aiming to minimize the volume fraction subject to multiple constraints. The methods presented in literature have proved successful in a variety of applications, although they are all first-order monotonous schemes, exhibit insufficient convergence properties. Following the SINH scheme, the elemental volume is penalized instead of the stiffness matrix. Through the intervention of reciprocal variables, the topology optimization reformulation introduces the positive-definite and separable Hessian matrix. The initial optimization formulation is decomposed into a sequence of quadratic programming. To resolve the problem efficiently, the quadratic programming can be precast into a dual form, and Lagrange multipliers are updated directly. The corresponding MATLAB code is a built-in quadratic programming subroutine explored to solve the dual problem. Finally, four numerical tests encompassing 2D and 3D structures are presented to illustrate the feasibility and practicality by incorporating the results with existing approaches.