[Paper Review] A continuous perspective on modeling of shape optimal design problems

Johannes Haubner, Martin Siebenborn+1 more|arXiv (Cornell University)|Apr 15, 2020

Advanced Numerical Methods in Computational MathematicsEngineering41 references25 citations

TL;DR

This paper proposes a continuous, function space-based approach to shape optimization using the method of mappings, where shape design is recast as optimal control over deformation fields in Sobolev spaces. By enforcing C¹-diffeomorphism constraints via a nonlinear determinant condition and using a Laplace-Beltrami extension for regularity, the method prevents mesh degeneration without parameter tuning, achieving robust drag minimization in Stokes flow with high convergence rates and mesh-independent solutions.

ABSTRACT

In this article we consider shape optimization problems as optimal control problems via the method of mappings. Instead of optimizing over a set of admissible shapes a reference domain is introduced and it is optimized over a set of admissible transformations. The focus is on the choice of the set of transformations, which we motivate from a function space perspective. In order to guarantee local injectivity of the admissible transformations we enrich the optimization problem by a nonlinear constraint. The approach requires no parameter tuning for the extension equation and can naturally be combined with geometric constraints on volume and barycenter of the shape. Numerical results for drag minimization of Stokes flow are presented.

Motivation & Objective

• Address mesh degeneration in transformation-based shape optimization by shifting from discrete to continuous modeling of deformations.
• Ensure local injectivity and smoothness of domain transformations via a nonlinear constraint on the deformation Jacobian.
• Develop a parameter-free framework that avoids tuning of extension equations, unlike prior methods relying on empirical parameters.
• Enable re-meshing and refinement during optimization by working in continuous function spaces with sufficient regularity.
• Achieve mesh-independent convergence and accurate optimal shapes in drag minimization for Stokes flow using a Hilbert space formulation.

Proposed method

• Reformulate shape optimization as an optimal control problem over deformation fields in Sobolev spaces, using the method of mappings with a reference domain.
• Define admissible controls in L²(Γd) or H¹(Γd), mapped via a Laplace-Beltrami equation to ensure high regularity of boundary data.
• Use an elliptic extension equation (e.g., Poisson or elasticity-type) to extend boundary data into the domain, generating the full deformation field w.
• Enforce C¹-diffeomorphism via a nonlinear constraint: det(∇(id + w)) ≥ η₁ > 0 in Ω, ensuring injectivity and avoiding self-intersections.
• Introduce a regularization term α‖c‖²_X and geometric constraints g(w) = 0 (e.g., volume and barycenter) to stabilize the optimization.
• Solve the resulting constrained optimization problem using semismooth Newton methods with continuation on the regularization parameter α.

Experimental results

Research questions

• RQ1 Can a continuous, function space-based formulation of shape optimization avoid mesh degeneration without requiring parameter tuning of extension equations?
• RQ2 How can the set of admissible transformations be constrained to ensure local injectivity and C¹-diffeomorphism in a continuous setting?
• RQ3 What is the impact of different control-to-deformation mappings (e.g., normal vs. tangential) on convergence and optimal shape quality?
• RQ4 Can the proposed method support adaptive mesh refinement and re-meshing during optimization without reinitializing the algorithm?
• RQ5 How does the choice of regularization continuation strategy affect convergence and accuracy in 2D and 3D Stokes flow problems?

Key findings

• The method achieves mesh-independent convergence in 2D Stokes drag minimization, with optimal shapes converging under grid refinement, especially when using the S3 strategy (Laplace-Beltrami + normal extension).
• For αtarget = 1×10⁻¹⁰, the S3 strategy produced comparable optimal shapes even on coarse grids (1601 triangles), while S1 (normal-only) showed slower convergence.
• The number of semismooth Newton iterations increased significantly beyond the 14th optimization problem for S1 (αdec = 1/2), due to activation of the positive-part in the objective function.
• In 3D, convergence required a more careful continuation strategy: αinit = 1×10⁻¹, αdec = 0.5, and αtarget = 1×10⁻⁶ to maintain convergence within 40 iterations.
• The use of a Laplace-Beltrami equation to elevate boundary regularity improved solution quality and stability compared to direct boundary control.
• The method successfully produced a smooth, crinkled optimal shape in 3D Stokes flow with a 0.28×0.28 zoomed slice showing fine geometric detail, confirming robustness and accuracy.