Smooth geometry extraction from SIMP topology optimization: Signed distance function approach with volume preservation

TL;DR

This work tackles the challenge of converting density-based SIMP topology optimization results into high-quality geometries suitable for manufacturing and analysis. It introduces a robust two-stage post-processing framework that first maps densities to a nodal field, constructs a density isocontour-based signed distance function on a regular grid, and then refines the boundary with Gaussian radial basis function smoothing to preserve volume and topology while achieving smooth, CAD-ready boundaries. The approach yields tangible improvements in mechanical performance metrics, including reduced maximum stresses (e.g., ~15% in a cantilever beam case) and deformation energies closer to the original optimization, while enabling seamless export via isosurface stuffing to high-quality tetrahedral meshes. The method demonstrates strong robustness across irregular meshes, higher-order elements, and complex geometries (e.g., robot gripper), offering a practical and scalable path from SIMP optimization to manufacturable designs with enhanced boundary quality and volume fidelity.

Abstract

This paper presents a novel post-processing methodology for extracting high-quality geometries from density-based topology optimization results. Current post-processing approaches often struggle to simultaneously achieve smooth boundaries, preserve volume fraction, and maintain topological features. We propose a robust method based on a signed distance function (SDF) that addresses these challenges through a two-stage process: first, an SDF representation of density isocontours is constructed, which is followed by geometry refinement using radial basis functions (RBFs). The method generates smooth boundary representations that appear to originate from much finer discretizations while maintaining the computational efficiency of coarse mesh optimization. Through comprehensive validation, our approach demonstrates a 18% reduction in maximum equivalent stress values compared to conventional methods, achieved through continuous geometric transitions at boundaries. The resulting implicit boundary representation facilitates seamless export to standard manufacturing formats without intermediate reconstruction steps, providing a robust foundation for practical engineering applications where high-quality geometric representations are essential.

Figures (15)

• Figure 1: An illustration of the procedure for extracting a smooth geometry from topology optimization results, demonstrated on a 2D beam example. (a) Input data: visualization of raw topology optimization result from the SIMP method, represented as an element-wise density field. (b) Mapping elemental densities to nodal values and extracting the geometry using an isocontour technique. (c) Constructing an SDF on a regular Cartesian grid to mathematically define the boundary. (d) Refining the geometry by smoothing the SDF with radial basis functions, resulting in an accurate and smooth boundary. (e) Discretizing the smoothed geometry into a high-quality mesh for subsequent applications.
• Figure 2: Illustration of distance calculation from a Cartesian grid point $\boldsymbol{x}_g$ to the density isocontour $\rho = \rho_t$ within a finite element. The shortest distance is represented by the perpendicular projection onto the isocontour.
• Figure 3: Visualization of different geometric configurations and corresponding projection scenarios in distance function construction. The Figure shows three geometric configurations: an element with internal isocontour (1), a fully solid boundary element (2), and a transitional element with partially solid boundary face (3).
• Figure 4: Geometric representation transition: (a) The continuous SDF representation shown through color-mapped values on a Cartesian grid, with the zero-level contour highlighted in gray, (b) The resulting high-quality tetrahedral mesh generated through isosurface stuffing Labelle2007.
• Figure 5: Visual validation of the signed distance function construction. (a) Two finite elements sharing a common face. The nodal densities of these two elements are intentionally assigned to create a roof-like isocontour at $\rho_t = 0.5$, with dark blue representing $\rho_n = 1$, light blue $\rho_n = 0.5$, and white $\rho_n = 0$. A red dashed line indicates the resulting isocontour. (b) Projection analysis showing the computed shortest distances from regular grid nodes to the material domain (defined by the isocontour where interpolated density equals $\rho_t$, or to element faces where interpolated density is higher than $\rho_t$). Gray lines represent projection paths, and the color scale indicates the computed signed distance values. (c) Continuous SDF field obtained by interpolating the discrete nodal distance values within elements, demonstrating smooth geometric transitions and $C^0$ continuity across element boundaries.