Shell topology optimization based on level set method

TL;DR

The paper addresses the challenge of optimizing shell structures with large shape and topology changes by introducing a level-set representation of the midsurface, defined as the zero level-set of a function $\phi$ in a 3D domain $D$, with a uniform thickness $t$. A Helmholtz filter and a regularized Heaviside projection stabilize the evolution of $\phi$, and the design sensitivity on the shell is mapped to a level-set sensitivity through an approximate relation $\widehat{\partial F/\partial \phi} \approx - c \; \partial F/\partial z$, enabling mean-compliance minimization under a volume constraint $G \le 0$ using a barrier-based unconstrained formulation $F' = F - \gamma_p/ G$. The method employs a conformal mesh generated by Mmg to bridge the level-set field with shell analysis (MITC6 elements) and demonstrates dome, plate, and cantilever-beam problems that exhibit large shape changes and topology changes, such as pinch-offs and bifurcations. While offering computational efficiency and capability to handle complex shell topologies on coarse meshes, the approach faces stability and local-optimum challenges, with future work focusing on buckling, explicit material-volume constraints, and stiffener layout optimization.

Abstract

This paper proposes a level set-based method for optimizing shell structures with large design changes in shape and topology. Conventional shell optimization methods, whether parametric or nonparametric, often only allow limited design changes in shape. In the proposed method, the shell structure is defined as the isosurface of a level set function. The level set function is iteratively updated based on the shape sensitivity on the surface mesh. Therefore, the proposed method can represent an arbitrary manifold surface while dealing with topological changes, for example, from a spherical surface to a toroidal surface. We applied the proposed method to the mean compliance minimization problems of 3D shell structural designs for dome, bending plate and cantilever beam examples to demonstrate its efficacy of the proposed method.

Figures (13)

• Figure 1: Schematic of design domain $D$, level set function $\phi$, and midsurface of shell $A$
• Figure 2: 1D cut plot of the level set function in the normal direction of the gradient
• Figure 3: Procedure of design sensitivity derivation
• Figure 4: Flowchart of the optimization process
• Figure 5: Problem setting of dome example