An ODE-driven level-set density method for topology optimization

In this paper, a new topology optimization method is discussed. The basic idea consists of exploring a combination of a density-based method together with a level-set description to form a new optimization frame. Due to the level-set description, solutions show clear and smooth boundaries that are deemed more materially efficient. Additionally, the checkerboarding issue, usually paired with an element-wise description, can be avoided. Thus, the filter scheme to tackle this problem becomes unnecessary, and the design space can be further exploited. The ingredient of material interpolation with penalty is utilized here, and its merit lies in making update information (i.e., sensitivity) more distinguished and in turn driving the optimization process to converge into solid–void solutions stably. An ordinary differential equation (ODE) established from optimal criteria builds the relationship between the level-set description and update information, and the structural updating procedure can be efficiently performed by solving the ODE. A regularization scheme for the level-set is introduced to enhance topological variation ability and address topological evolution defects, which helps deliver reasonable and well-posed topological configurations. The regularization strategy is a simple linear scaling manner that proves to be effective and efficient. This paper investigates three classes of optimization problems: compliance minimization, eigenfrequency maximization, and thermal conduction optimization. To validate the proposed method, both 2D and 3D benchmark examples in comparison with the widely accepted Solid Isotropic Material with Penalization (SIMP) method are tested. By contrast, the solutions resulting from the proposed method show advantageous structural representations and better objective function values under specified conditions. In addition, several other numerical examples considering model parameter influences and some extensions are discussed to systematically demonstrate the proposed method’s characteristics. Finally, the MATLAB codes concerning the above-mentioned three classes of optimization problems are shared for educational purposes. The codes are compact and finely structured with only 58, 62, and 53 lines for 2D cases, and 105, 108, and 80 lines for 3D cases of minimum compliance, maximum eigenfrequency, and minimum thermal compliance problems, respectively.