TreeTOp: Topology Optimization using Constructive Solid Geometry Trees

TL;DR

This paper introduces TreeTOp, a feature-mapping topology optimization framework that generalizes FMTO by incorporating an expanded set of Boolean operations (union, intersection, subtraction) through a differentiable unified Boolean formulation. The design is represented as a balanced CSG tree of polygonal primitives whose leaf parameters and non-leaf operators are optimized concurrently, with primitives projected onto a differentiable density field via a LogSumExp-based SDF and a sigmoid-threshold density map. Finite element analysis using SIMP and gradient-based optimization via MMA enable efficient discovery of manufacturable designs, while automatic differentiation provides exact sensitivities. The approach demonstrates comparable results to SIMP on benchmark problems, reveals insights into the effects of tree depth, initialization, and mesh resolution, and offers a flexible framework for exploring diverse Boolean structures and manufacturing considerations, with code available for replication.

Abstract

Feature-mapping methods for topology optimization (FMTO) facilitate direct geometry extraction by leveraging high-level geometric descriptions of the designs. However, FMTO often relies solely on Boolean unions, which can restrict the design space. This work proposes an FMTO framework leveraging an expanded set of Boolean operations, namely, union, intersection, and subtraction. The optimization process entails determining the primitives and the optimal Boolean operation tree. In particular, the framework leverages a recently proposed unified Boolean operation approach. This approach presents a continuous and differentiable function that interpolates the Boolean operations, enabling gradient-based optimization. The proposed methodology is agnostic to the specific primitive parametrization and is showcased through various numerical examples.

Figures (18)

• Figure 1: (a) Design domain and boundary conditions. (b) An optimized design obtained through density-based TO.
• Figure 2: (a) Classic FMTO parameterizes designs using a union of primitives (such as bars).
• Figure 3: Given a design domain with boundary conditions, the proposed method optimizes the boolean operations ($\text{union}(\cup), \text{intersection}(\cap), \text{difference}(\mapsto), \text{and negative difference}(\mapsfrom)$), and the parameters associated with primitives (polygon). The resulting shapes are hierarchically combined till the final design is obtained at the root.
• Figure 4: The polygon's parameters include the (a) center coordinates $(c_x, c_y)$, (b) the distances from the center to the half-spaces $(d_1, \ldots, d_6)$, and (c) an angular offset $\theta$.
• Figure 5: (a) A polygon, (b) signed distance field (SDF) of half-spaces (c) SDF of polygon, and (c) its projected density.