[Paper Review] An Interface-enriched Generalized Finite Element Method for Levelset-based Topology Optimization

Sanne J. van den Boom, Jian Zhang+2 more|arXiv (Cornell University)|Mar 30, 2020

Topology Optimization in Engineering74 references24 citations

TL;DR

This paper proposes an interface-enriched generalized finite element method (IGFEM) coupled with radial basis function (RBF)-parameterized levelset topology optimization to achieve smooth, non-pixelated designs without remeshing. The method enables accurate boundary resolution and direct enforcement of essential boundary conditions, yielding crisp black-and-white topologies with minimal numerical artifacts, as validated on benchmark compliance minimization problems.

ABSTRACT

During design optimization, a smooth description of the geometry is important, especially for problems that are sensitive to the way interfaces are resolved, e.g., wave propagation or fluid-structure interaction. A levelset description of the boundary, when combined with an enriched finite element formulation, offers a smoother description of the design than traditional density-based methods. However, existing enriched methods have drawbacks, including ill-conditioning and difficulties in prescribing essential boundary conditions. In this work we introduce a new enriched topology optimization methodology that overcomes the aforementioned drawbacks; boundaries are resolved accurately by means of the Interface-enriched Generalized Finite Element Method (IGFEM), coupled to a levelset function constructed by radial basis functions. The enriched method used in this new approach to topology optimization has the same level of accuracy in the analysis as standard the finite element method with matching meshes, but without the need for remeshing. We derive the analytical sensitivities and we discuss the behavior of the optimization process in detail. We establish that IGFEM-based levelset topology optimization generates correct topologies for well-known compliance minimization problems.

Motivation & Objective

• Address the limitations of density-based topology optimization, such as gray-scale artifacts and staircasing due to mesh-dependent material interfaces.
• Overcome drawbacks of existing enriched FEM methods like X/GFEM, including ill-conditioning, weak enforcement of boundary conditions, and stress overestimation near interfaces.
• Develop a robust, non-remeshing topology optimization framework that maintains high accuracy in analysis while enabling precise boundary tracking.
• Enable direct enforcement of essential boundary conditions and improve convergence behavior through decoupled design and analysis discretizations.
• Demonstrate the feasibility and effectiveness of IGFEM in levelset-based topology optimization for compliance minimization problems, serving as a proof of concept for more complex applications.

Proposed method

• Employ the Interface-enriched Generalized Finite Element Method (IGFEM) to enrich finite element shape functions only in elements cut by the material interface, ensuring enrichment functions vanish at original mesh nodes.
• Use radial basis functions (RBFs) to parameterize the levelset function, decoupling the design space from the analysis mesh and enabling smooth boundary evolution.
• Construct the levelset function as a signed distance field via RBF interpolation of nodal levelset values, allowing for smooth and continuous boundary representation.
• Apply IGFEM to solve the weak form of the governing equations (e.g., elasticity or heat conduction), with enrichment functions localized to cut elements only.
• Enforce essential boundary conditions strongly and directly on non-matching edges by preserving the physical meaning of original mesh nodes.
• Derive analytical sensitivities of the objective function with respect to RBF control points using the adjoint method, enabling efficient gradient-based optimization.

Experimental results

Research questions

• RQ1 Can IGFEM-based levelset topology optimization produce crisp, non-pixelated designs without requiring post-processing or remeshing?
• RQ2 How does the decoupling of the RBF-based design mesh from the fixed analysis mesh affect convergence and numerical stability in topology optimization?
• RQ3 To what extent does IGFEM mitigate common issues in enriched FEM, such as ill-conditioning, stress overestimation near interfaces, and poor convergence in blending elements?
• RQ4 Can the method accurately resolve narrow features and avoid zigzagging artifacts when the design space is not properly aligned with the analysis mesh?
• RQ5 How does the proposed method compare in performance and solution quality to standard SIMP, Ersatz, and discrete levelset approaches in benchmark problems?

Key findings

• The IGFEM-based levelset topology optimization method produces non-pixelated, black-and-white designs without post-processing, due to the smooth, geometry-fitted levelset representation.
• The method achieves accuracy comparable to standard FEM with matching meshes, without requiring remeshing, even when the interface cuts through elements.
• Analytical sensitivities are derived and computed with low computational overhead, enabling efficient gradient-based optimization.
• Zigzagging artifacts in the design are caused by coarse analysis meshes and approximation errors, which can be mitigated by using RBFs to limit design complexity or by refining the mesh.
• The method successfully generates correct topologies for standard compliance minimization benchmarks, including the MBB beam and 3D cantilever beam, with stable convergence.
• The use of RBFs acts as a filter that reduces numerical artifacts and allows the optimizer to converge faster by enabling larger boundary movements per iteration.