Abstract
We compare surface metrics for shape optimization problems with constraints, consisting mainly of partial differential equations (PDE), from a computational point of view. In particular, classical Laplace–Beltrami type metrics are compared with Steklov–Poincaré type metrics. The test problem is the minimization of energy dissipation of a body in a Stokes flow. We therefore set up a quasi-Newton method on appropriate shape manifolds together with an augmented Lagrangian framework, in order to enable a straightforward integration of geometric constraints for the shape. The comparison is focussed towards convergence behavior as well as effects on the mesh quality during shape optimization.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: Schu804/12-1
Funding statement: This work has been partly supported by the Deutsche Forschungsgemeinschaft within the Priority program SPP 1648 “Software for Exascale Computing” under contract number Schu804/12-1. Furthermore, we acknowledge support by the Sino-German Science Center (grant id 1228) on the occasion of the Chinese-German Workshop on Computational and Applied Mathematics in Augsburg 2015.
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© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Stability and Experimental Comparison of Prototypical Iterative Schemes for Total Variation Regularized Problems
- A Robust Weak Galerkin Finite Element Method for Linear Elasticity with Strong Symmetric Stresses
- Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems
- A Hybridizable Discontinuous Galerkin Method for the Time-Harmonic Maxwell Equations with High Wave Number
- A Note on Multilevel Based Error Estimation
- Characterization of Extreme Points of Multi-Stochastic Tensors
- An Adaptive Finite Element Multi-Mesh Approach for Interacting Deformable Objects in Flow
- Computational Comparison of Surface Metrics for PDE Constrained Shape Optimization
- Two-Level Spline Approximations for Two-Dimensional Navier–Stokes Equations
- On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations
Articles in the same Issue
- Frontmatter
- Stability and Experimental Comparison of Prototypical Iterative Schemes for Total Variation Regularized Problems
- A Robust Weak Galerkin Finite Element Method for Linear Elasticity with Strong Symmetric Stresses
- Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems
- A Hybridizable Discontinuous Galerkin Method for the Time-Harmonic Maxwell Equations with High Wave Number
- A Note on Multilevel Based Error Estimation
- Characterization of Extreme Points of Multi-Stochastic Tensors
- An Adaptive Finite Element Multi-Mesh Approach for Interacting Deformable Objects in Flow
- Computational Comparison of Surface Metrics for PDE Constrained Shape Optimization
- Two-Level Spline Approximations for Two-Dimensional Navier–Stokes Equations
- On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations