CAD-compatible structural shape optimization with a movable Bézier tetrahedral mesh

TL;DR

This work addresses CAD-integrated 3D structural shape optimization by coupling NURBS-described boundaries with a Bézier tetrahedral interior, enabling direct sensitivity computation with respect to boundary control points. The framework reconstructs NURBS surfaces from boundary meshes, computes analytical sensitivities in the analysis mesh, and uses a pseudo-elastic moving mesh to update interior discretization without remeshing, with data exchange via the IGES neutral format. A gradient-based MMA optimizer is employed to minimize compliance under volume constraints, and four 3D benchmarks (hollow sphere, cantilever beam, cube with a hole, and a hammer) demonstrate accuracy and substantial runtime reductions due to the movable-mesh strategy. The approach provides a practical, CAD-compatible pathway for efficient 3D shape optimization of complex geometries, with potential extensions to $C^1$ elements and stress-constrained formulations.

Abstract

This paper presents the development of a complete CAD-compatible framework for structural shape optimization in 3D. The boundaries of the domain are described using NURBS while the interior is discretized with Bézier tetrahedra. The tetrahedral mesh is obtained from the mesh generator software Gmsh. A methodology to reconstruct the NURBS surfaces from the triangular faces of the boundary mesh is presented. The description of the boundary is used for the computation of the analytical sensitivities with respect to the control points employed in surface design. Further, the mesh is updated at each iteration of the structural optimization process by a pseudo-elastic moving mesh method. In this procedure, the existing mesh is deformed to match the updated surface and therefore reduces the need for remeshing. Numerical examples are presented to test the performance of the proposed method. The use of the movable mesh technique results in a considerable decrease in the computational effort for the numerical examples.

Figures (18)

• Figure 1: Barycentric coordinate system in a tetrahedron.
• Figure 2: Parent element and its mapping into the physical space.
• Figure 3: Node numbering of a a) quadratic, b) cubic and c) quartic Bézier tetrahedron.
• Figure 4: Lagrange and Bézier control points required to model exactly the same tetrahedron.
• Figure 5: Refinement by splitting of a tetrahedron. One tetrahedron is split into eight tetrahedra.