Gaussian Ensemble Topology (GET): A New Explicit and Inherently Smooth Framework for Manufacture-Ready Topology Optimization

TL;DR

Gaussian Ensemble Topology (GET) introduces an explicit topology-optimization framework that represents geometry as a superposition of anisotropic Gaussian fields, yielding inherently smooth, manufacturable designs without post-processing. The method couples analytic Gaussian derivatives with a regularized Heaviside projection and a fixed background FE mesh, enabling efficient MMA-based optimization and mesh-independent geometry. Across 2D and 3D compliance and compliant-mechanism benchmarks, GET achieves objective values comparable to MMC while producing smoother, CAD-ready boundaries and reduced intermediate densities. The work highlights controllable geometric complexity, discreteness, and boundary smoothness through parameters n, ε, and T, and demonstrates mesh-independence, efficient computation, and strong potential for manufacturing-ready design automation. Practical impact includes smoother load paths, reduced stress concentrations, and seamless CAD export, with avenues for adaptive Gaussian growth and GPU acceleration in large-scale problems.

Abstract

We introduce the Gaussian Ensemble Topology (GET) method, a new explicit and manufacture-ready framework for topology optimization in which design geometries are represented as superpositions of anisotropic Gaussian functions. By combining explicit Gaussian descriptions with a level-set-like Heaviside projection, GET inherently generates smooth, curvature-continuous designs without requiring post-processing steps such as mesh or corner smoothing and feature extraction. The method is validated on standard compliance-minimization and compliant mechanism benchmarks in two and three dimensions. The optimized designs achieve objective values comparable to those obtained with classical Moving Morphable Component (MMC) approaches, but with geometrically consistent, refined boundaries. Numerical examples demonstrate additional advantages of the GET framework, including mesh independence inherent to explicit parameterizations, strong geometric expressiveness, and effective control over smoothness, discreteness, and structural complexity through parameter tuning. As a robust and manufacture-ready approach to explicit topology optimization, GET opens avenues for tackling advanced and complex design problems.

Figures (17)

• Figure 1: Geometric description and fusion in GET and MMC Method. (a) 2D GET: Ellipses, represented by 2D Gaussian functions, are fused either by summation (smooth transition and union; green circle) or by the maximum (sharp transition, Boolean-like union; red circle). Solid regions are defined by $\Phi^{s} \ge T$. (b) 2D MMC: Super-ellipses, represented by the topology description function, are fused by summation (invalid union; black circle) or by the maximum (sharp transition, Boolean-like union; red circle). Solid regions are defined by $\Phi^{s} \ge 0$. (c) 3D GET: Ellipsoids, represented by 3D Gaussian functions, are fused by summation (smooth transition and union; green circle) or by the maximum (sharp transition, Boolean-like union; red circle). Solid regions are defined by $\Phi^{s} \ge T$.
• Figure 2: (a) Surface plot (height map) of five 2D Gaussian fields. Thresholding at $\phi=0.1,0.5,0.9$ partitions the domain into solid ($\phi \ge T$) and void, thereby defining the geometry. (b) Contour (level-set) maps at the same thresholds showing the resulting structural outlines.
• Figure 3: Density–phase distribution of the Gaussian field after applying the Heaviside function with $\epsilon\in\{0.005,\,0.02,\,0.2\}$, illustrating how $\epsilon$ controls the extent of intermediate densities.
• Figure 4: (a) Design domain and boundary conditions of the 2D cantilever beam; (b) Convergence history of the objective and volume fraction for the cantilever beam obtained by the GET method.
• Figure 5: 2D cantilever beam: (a) MMC method: initial design and optimized structure (yellow: solid; blue: void; red box: sharp connection; grayscale: density field). Final performance ($\epsilon=0$ for comparison): $C=73.7$, $V_f=0.4020$. (b) Proposed GET method: similar initial design and optimized structure (red box: smoother connections with fewer gray elements). Final performance ($\epsilon=0$ for comparison): $C=73.6$, $V_f=0.4025$.