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Enforcing Mesh Quality Constraints in Shape Optimization with a Gradient Projection Method

Sebastian Blauth, Christian Leithäuser

TL;DR

The proposed approach is based on Rosen's gradient projection method, which incorporates mesh quality constraints into the shape optimization problem and treats these constraints by projecting the search direction to the linear subspace of the currently active constraints.

Abstract

For the numerical solution of shape optimization problems, particularly those constrained by partial differential equations (PDEs), the quality of the underlying mesh is of utmost importance. Particularly when investigating complex geometries, the mesh quality tends to deteriorate over the course of a shape optimization so that either the optimization comes to a halt or an expensive remeshing operation must be performed before the optimization can be continued. In this paper, we present a novel, semi-discrete approach for enforcing a minimum mesh quality in shape optimization. Our approach is based on Rosen's gradient projection method, which incorporates mesh quality constraints into the shape optimization problem. The proposed constraints bound the angles of triangular and solid angles of tetrahedral mesh cells and, thus, also bound the quality of these mesh cells. The method treats these constraints by projecting the search direction to the linear subspace of the currently active constraints. Additionally, only slight modifications to the usual line search procedure are required to ensure the feasibility of the method. We present our method for two- and three-dimensional simplicial meshes. We investigate the proposed approach numerically for the drag minimization of an obstacle in a two-dimensional Stokes flow, the optimization of the flow in a pipe governed by the Navier-Stokes equations, and for the large-scale, three-dimensional optimization of a structured packing used in a distillation column. Our results show that the proposed method is indeed capable of guaranteeing a minimum mesh quality for both academic examples and challenging industrial applications. Particularly, our approach allows the shape optimization of complex structures while ensuring that the mesh quality does not deteriorate.

Enforcing Mesh Quality Constraints in Shape Optimization with a Gradient Projection Method

TL;DR

The proposed approach is based on Rosen's gradient projection method, which incorporates mesh quality constraints into the shape optimization problem and treats these constraints by projecting the search direction to the linear subspace of the currently active constraints.

Abstract

For the numerical solution of shape optimization problems, particularly those constrained by partial differential equations (PDEs), the quality of the underlying mesh is of utmost importance. Particularly when investigating complex geometries, the mesh quality tends to deteriorate over the course of a shape optimization so that either the optimization comes to a halt or an expensive remeshing operation must be performed before the optimization can be continued. In this paper, we present a novel, semi-discrete approach for enforcing a minimum mesh quality in shape optimization. Our approach is based on Rosen's gradient projection method, which incorporates mesh quality constraints into the shape optimization problem. The proposed constraints bound the angles of triangular and solid angles of tetrahedral mesh cells and, thus, also bound the quality of these mesh cells. The method treats these constraints by projecting the search direction to the linear subspace of the currently active constraints. Additionally, only slight modifications to the usual line search procedure are required to ensure the feasibility of the method. We present our method for two- and three-dimensional simplicial meshes. We investigate the proposed approach numerically for the drag minimization of an obstacle in a two-dimensional Stokes flow, the optimization of the flow in a pipe governed by the Navier-Stokes equations, and for the large-scale, three-dimensional optimization of a structured packing used in a distillation column. Our results show that the proposed method is indeed capable of guaranteeing a minimum mesh quality for both academic examples and challenging industrial applications. Particularly, our approach allows the shape optimization of complex structures while ensuring that the mesh quality does not deteriorate.

Paper Structure

This paper contains 20 sections, 48 equations, 16 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Gradient Projection Method for Nonlinear Optimization
  4. Shape Calculus
  5. Gradient-Based Descent Algorithms for Shape Optimization
  6. Mesh Quality Constraints for Shape Optimization
  7. Discretization of the Shape Optimization Problem
  8. Constraints on the Mesh Quality
  9. Two-Dimensional Triangular Meshes
  10. Three-Dimensional Tetrahedral Meshes
  11. Implementation Details
  12. The Gradient Projection Method for Mesh Quality Constraints
  13. Additional Constraints for Fixed Boundaries
  14. Choice of the Minimum Angle Threshold
  15. Numerical Implementation
  16. ...and 5 more sections

Figures (16)

  • Figure 1: Illustration of the gradient projection method.
  • Figure 2: Velocity magnitude for the Stokes problem \ref{['eq:stokes']}, computed with the BFGS method with mesh quality constraints.
  • Figure 3: History of the optimization algorithm for problem \ref{['eq:shape_stokes']}.
  • Figure 4: Optimized geometry for problem \ref{['eq:shape_stokes']} -- close up of the front tip.
  • Figure 5: Histograms showing the distribution of the mesh quality after the shape optimization for problem \ref{['eq:shape_stokes']}.
  • ...and 11 more figures

Theorems & Definitions (12)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 4.1
  • ...and 2 more