A Continuous Perspective on Shape Optimization via Domain Transformations

Haubner, Johannes; Siebenborn, Martin; Ulbrich, Michael

doi:10.1137/20m1332050

Cited by 21 publications

(24 citation statements)

References 28 publications

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“…Finally, the Lagrangian is used to obtain the sensitivities necessary for the descent direction and for the approximation to the Hessian used. For an in-depth discussion of the underlying theory we refer the reader, as previously mentioned, to [29,3,18], the use of the adjoint method in fluid dynamics can be reviewed in [14,20], and [19] can be consulted for a mathematical review of the theory here used.…”

Section: Problem Formulation and Mathematical Backgroundmentioning

confidence: 99%

“…S denotes a so-called extension operator, which links the boundary control variable u ∈ L 2 (Γ obs ) to a displacement field w : Ω → R d . Examples of possible choices of S are given and investigated in [18]. Particularly, the resulting regularity of the domain transformation F is discussed therein.…”

Section: Problem Formulation and Mathematical Backgroundmentioning

confidence: 99%

“…The objective function is non-differentiable due to the presence of the positive part mapping in R in (40), which is discussed in depth in [18]. Moreover, a discussion on quasi-Newton methods for semi-smooth objective functions and can be found in [24].…”

Section: Algorithm 1 Augmented Lagrange Outer Optimization Algorithmmentioning

confidence: 99%

“…Figure 9 shows accumulated wallclock times, gained speedup relative to 24 cores and iteration counts for the first three optimization steps. These results are shown for the nonlinear extension operator (29), state (14) to (18), and the adjoint displacement (40). A P 1 finite element discretization is used for extension operator and its adjoint equation, while mixed P 1 − P 1 shape functions are used for the state equation Navier-Stokes equations.…”

Section: Algorithmic Scalabilitymentioning

confidence: 99%

“…The approach we propose here is based on the research carried out in [18]. Compared to iteratively updating the shape, it offers the possibility to require properties of the deformation from reference to optimal shape.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Fluid dynamic shape optimization using self-adapting nonlinear extension operators with multigrid preconditioners

Pinzon,

Siebenborn

2021

Preprint

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In this article we propose a scalable shape optimization algorithm which is tailored for large scale problems and geometries represented by hierarchically refined meshes. Weak scalability and grid independent convergence is achieved via a combination of multigrid schemes for the simulation of the PDEs and quasi Newton methods on the optimization side. For this purpose a self-adapting, nonlinear extension operator is proposed within the framework of the method of mappings. This operator is demonstrated to identify critical regions in the reference configuration where geometric singularities have to arise or vanish. Thereby the set of admissible transformations is adapted to the underlying shape optimization situation. The performance of the proposed method is demonstrated for the example of drag minimization of an obstacle within a stationary, incompressible Navier-Stokes flow.

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Section: Problem Formulation and Mathematical Backgroundmentioning

confidence: 99%

Section: Problem Formulation and Mathematical Backgroundmentioning

confidence: 99%

Section: Algorithm 1 Augmented Lagrange Outer Optimization Algorithmmentioning

confidence: 99%

Section: Algorithmic Scalabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fluid dynamic shape optimization using self-adapting nonlinear extension operators with multigrid preconditioners

Pinzon,

Siebenborn

2021

Preprint

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Shape Optimization Algorithms for Fluid Dynamics Applications

Escobar

Siebenborn

2023

Proc Appl Math and Mech

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In this work we present a comparison between shape optimization algorithms in different vector spaces. The main goal is to optimize the surface of an object with respect to a physical quantity. The main focus is on applications that require large element deformations as part of the optimization process, as for instance the removal and creation of geometric singularities such as edges and corners. The algorithms take into account the prevention of element degeneracy and overlapping, for instance by enforcing inequality constraints. For this purpose, an approach in the Hilbert space is compared to another in Banach spaces. The former is based on a nonlinear extension equation, whereas the p‐Laplace operator is used in the latter. Computational results are presented in the context of fluid dynamics applications, where the contour of an object is optimized with respect to the energy dissipation.

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Preserving mesh quality in shape optimization

Onyshkevych,

Siebenborn,

Wollner

2023

Proc Appl Math and Mech

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In PDE‐constrained shape optimization, a lot of computational effort is used to update the geometry at every iteration step. This iterative process can result in loss of element quality and degeneracy of the underlying mesh, which is particularly relevant for applications involving large deformations of some parts of the domain. To avoid remeshing, we model the mesh deformation using the method of mappings. We use an extension equation that maps a boundary control variable to a deformation field defined over the entire domain. By using the nonlinear extension operator proposed in our previous work, we increase the set of reachable shapes, allowing us to model large deformations. We discuss how the choice of parameters of the extension equations affects the mesh quality and we study the influence of the extension factor on the convergence properties of the iterative solvers.

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A Continuous Perspective on Shape Optimization via Domain Transformations

Cited by 21 publications

References 28 publications

Fluid dynamic shape optimization using self-adapting nonlinear extension operators with multigrid preconditioners

Fluid dynamic shape optimization using self-adapting nonlinear extension operators with multigrid preconditioners

Shape Optimization Algorithms for Fluid Dynamics Applications

Preserving mesh quality in shape optimization

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