[Paper Review] Pre-Shape Calculus: Foundations and Application to Mesh Quality Optimization

Daniel Luft, Volker Schulz|arXiv (Cornell University)|Dec 16, 2020

Computational Geometry and Mesh Generation48 references4 citations

TL;DR

This paper introduces Pre-Shape Calculus, a unified theoretical framework that extends classical shape optimization by incorporating mesh quality optimization through pre-shape derivatives acting in both normal and tangential directions. It enables simultaneous shape and mesh quality optimization via tangential gradient descent, preserving optimal shapes while improving surface and volume mesh quality with minimal computational overhead.

ABSTRACT

Deformations of the computational mesh arising from optimization routines usually lead to decrease of mesh quality or even destruction of the mesh. We propose a theoretical framework using pre-shapes to generalize classical shape optimization and calculus. We define pre-shape derivatives and derive according structure and calculus theorems. In particular, tangential directions are featured in pre-shape derivatives, in contrast to classical shape derivatives featuring only normal directions. Techniques from classical shape optimization and -calculus are shown to carry over to this framework. An optimization problem class for mesh quality is introduced, which is solvable by use of pre-shape derivatives. This class allows for simultaneous optimization of classical shape objectives and mesh quality without deteriorating the classical shape optimization solution. The new techniques are implemented and numerically tested for 2D and 3D.

Motivation & Objective

• Address the degradation of mesh quality during shape optimization, which can compromise solution reliability in PDE-constrained problems.
• Develop a theoretical framework that unifies classical shape optimization with mesh quality improvement by treating parameterizations (meshes) as pre-shapes.
• Formulate a class of pre-shape parameterization tracking problems that regularize shape optimization while maintaining optimal shapes.
• Enable numerical implementation of mesh quality optimization via pre-shape gradient descent without interfering with the original shape optimization process.
• Demonstrate the feasibility and effectiveness of the framework in 2D and 3D for both surface and volume mesh quality improvement.

Proposed method

• Define pre-shapes as diffeomorphisms of a reference manifold, allowing simultaneous treatment of shape and mesh parameterization.
• Introduce pre-shape derivatives that include both normal and tangential components, generalizing classical shape derivatives.
• Derive structure theorems and calculus rules for pre-shape derivatives, including material derivative formulas and normalization conditions.
• Construct a pre-shape parameterization tracking problem with a target function for desired element volume distributions on surfaces and volumes.
• Implement a gradient descent method using only the tangential component of the pre-shape derivative to avoid shape deformation.
• Use Gram-Schmidt orthogonalization to compute local tangential orthonormal bases and compute Jacobian determinants for mesh quality tracking.

Experimental results

Research questions

• RQ1 How can mesh quality degradation during shape optimization be formally addressed within a unified calculus framework?
• RQ2 What is the mathematical structure of pre-shape derivatives that include both normal and tangential variations?
• RQ3 Can pre-shape derivatives be used to optimize mesh quality without altering the optimal shape of the original shape optimization problem?
• RQ4 How can the pre-shape calculus be implemented numerically to achieve mesh quality improvement with minimal computational cost?
• RQ5 To what extent can tangential components of the pre-shape derivative alone drive convergence to a desired mesh quality while preserving the shape?

Key findings

• The pre-shape calculus framework successfully generalizes classical shape calculus by incorporating tangential variations, enabling mesh quality optimization without shape deformation.
• Numerical experiments show that using only the tangential component of the pre-shape derivative leads to convergence of both the target function and gradient norm, while preserving the original spherical shape.
• For the 3D sphere test case, the algorithm converged after 48 iterations and 1256.78 seconds, achieving the desired non-uniform surface cell volume distribution.
• The method maintains optimal shapes invariant, as confirmed by the preservation of the spherical geometry despite mesh quality optimization.
• The implementation demonstrates minimal additional computational cost, with the pre-shape gradient descent method achieving mesh quality improvement without interference in the shape optimization process.
• The framework supports arbitrary dimensionality and is applicable to both surface and volume meshes, with successful numerical validation in 2D and 3D.