[Paper Review] Dynamic Adaptive Mesh Refinement for Topology Optimization

Shun Wang, Eric de Sturler+1 more|arXiv (Cornell University)|Sep 25, 2010

Topology Optimization in Engineering20 references17 citations

TL;DR

This paper proposes a dynamic adaptive mesh refinement (AMR) strategy for topology optimization that maintains design accuracy equivalent to uniform fine meshes while significantly reducing computational cost. By enabling both refinement and derefinement during optimization, the method achieves optimal designs with up to 52% fewer elements and over 3× faster solution times in 3D problems, without compromising accuracy or convergence.

ABSTRACT

We present an improved method for topology optimization with both adaptive mesh refinement and derefinement. Since the total volume fraction in topology optimization is usually modest, after a few initial iterations the domain of computation is largely void. Hence, it is inefficient to have many small elements, in such regions, that contribute significantly to the overall computational cost but contribute little to the accuracy of computation and design. At the same time, we want high spatial resolution for accurate three-dimensional designs to avoid postprocessing or interpretation as much as possible. Dynamic adaptive mesh refinement (AMR) offers the possibility to balance these two requirements. We discuss requirements on AMR for topology optimization and the algorithmic features to implement them. The numerical design problems demonstrate (1) that our AMR strategy for topology optimization leads to designs that are equivalent to optimal designs on uniform meshes, (2) how AMR strategies that do not satisfy the postulated requirements may lead to suboptimal designs, and (3) that our AMR strategy significantly reduces the time to compute optimal designs.

Motivation & Objective

• Address the high computational cost of 3D topology optimization on uniform fine meshes.
• Ensure AMR-based designs match those from uniform fine meshes in accuracy and optimality.
• Develop a dynamic AMR strategy that supports both refinement and derefinement to improve efficiency.
• Reduce solution time and iteration counts while maintaining design fidelity and sensitivity accuracy.
• Enable practical application of high-resolution topology optimization to complex 3D problems.

Proposed method

• Implement dynamic mesh adaptation with continuous refinement and derefinement throughout the optimization process.
• Use a hierarchical mesh structure where the finest level matches the resolution of a uniform fine mesh.
• Apply mesh refinement and derefinement based on element-level error indicators and design sensitivity information.
• Ensure finite element solutions on AMR meshes are as accurate as those on uniform fine meshes by maintaining local resolution at the finest level.
• Employ iterative solvers (e.g., MINRES, RMINRES) with subspace recycling to accelerate linear system solutions.
• Integrate AMR with topology optimization loops, updating the mesh at each optimization step based on design evolution.

Experimental results

Research questions

• RQ1 Can dynamic AMR produce topology optimization designs equivalent to those on uniform fine meshes in terms of final design and compliance?
• RQ2 How does AMR impact the number of optimization and linear solve iterations compared to uniform meshing?
• RQ3 What is the effect of mesh adaptation (refinement and derefinement) on solution time and memory usage in 3D problems?
• RQ4 Can AMR strategies that lack proper error control or dynamic adaptation lead to suboptimal designs?
• RQ5 To what extent can AMR reduce computational cost without sacrificing design accuracy in complex 3D topology optimization?

Key findings

• The proposed AMR strategy produces designs that are within 2.58% relative difference of those obtained on a uniform fine mesh, confirming design equivalence.
• For a 3D cross-shaped domain problem, the adaptive mesh reduced the number of elements from 40,960 to 19,736, a 52% reduction.
• Optimization converged in 106 steps on the adaptive mesh versus over 200 steps on the uniform mesh, reducing the number of nonlinear iterations.
• Solution time was reduced by a factor of more than three (to ~30% of the uniform mesh time), demonstrating significant efficiency gains.
• The method maintained high accuracy in sensitivities and deformations, ensuring that design evolution was not biased by coarse mesh approximations.
• AMR with dynamic derefinement significantly improved efficiency by reducing elements in void regions, especially in 3D where element count is critical.