Review: Wavelet-based numerical analysis: A review and classification

Shape gradients of PDE constrained shape functionals can be stated in two equivalent ways. Both rely on the solutions of two boundary value problems (BVPs), but one involves integrating their traces on the boundary of the domain, while the other evaluates integrals in the volume. Usually, the two BVPs can only be solved approximately, for instance, by finite element methods. However, when used with finite element solutions, the equivalence of the two formulas breaks down. By means of a comprehensive convergence analysis, we establish that the volume based expression for the shape gradient generally offers better accuracy in a finite element setting. The results are confirmed by several numerical experiments.

/pdf/comparison-of-approximate-shape-gradients-4jx5b0pdca.pdf

Comparison of approximate shape gradients

Large-scale three-dimensional aerodynamic shape optimization based on the compressible Euler equations is considered. Shape calculus is used to derive an exact surface formulation of the gradients, enabling the computation of shape gradient information for each surface mesh node without having to calculate further mesh sensitivities. Special attention is paid to the applicability to large-scale three dimensional problems like the optimization of an Onera M6 wing or a complete blended-wing–body aircraft. The actual optimization is conducted in a one-shot fashion, in which the tangential Laplace operator is used as a Hessian approximation, thereby also preserving the regularity of the shape.

/pdf/three-dimensional-large-scale-aerodynamic-shape-optimization-axrs25h497.pdf

Three-Dimensional Large-Scale Aerodynamic Shape Optimization Based on Shape Calculus

http://iepoi.uni-mb.si/upload/editor/ravnik/papers/Ravnik_IJMHT2008_3Dnc.pdf

Velocity-vorticity formulation for 3D natural convection in an inclined enclosure by BEM

We derive expressions for the shape Hessian operator of the data misfit functional corresponding to the inverse problem of inferring the shape of a scatterer from reflected acoustic waves, using a Banach space setting and the Lagrangian approach. The shape Hessian is then analyzed in both Holder and Sobolev spaces. Using an integral equation approach and compact embeddings in Holder and Sobolev spaces, we show that the shape Hessian can be decomposed into four components, of which the Gauss–Newton part is a compact operator, while the others are not. Based on the Hessian analysis, we are able to express the eigenvalues of the Gauss–Newton Hessian as a function of the smoothness of the shape space, which shows that the smoother the shape is, the faster the decay rate. Analytical and numerical examples are presented to validate our theoretical results. The implication of the compactness of the Gauss–Newton Hessian is that for small data noise and model error, the discrete Hessian can be approximated by a low-rank matrix. This in turn enables fast solution of an appropriately regularized inverse problem, as well as Gaussian-based quantification of uncertainty in the estimated shape.

/pdf/analysis-of-the-hessian-for-inverse-scattering-problems-ii-1hdxaobu29.pdf

Analysis of the Hessian for inverse scattering problems: II. Inverse medium scattering of acoustic waves

The present paper aims at analyzing the existence and convergence of approximate solutions in shape optimization. Motivated by illustrative examples, an abstract setting of the underlying shape optimization problem is suggested, taking into account the so-called two norm discrepancy. A Ritz-Galerkin-type method is applied to solve the associated necessary condition. Existence and convergence of approximate solutions are proved, provided that the infinite dimensional shape problem admits a stable second order optimizer. The rate of convergence is confirmed by numerical results.

/pdf/on-convergence-in-elliptic-shape-optimization-2iiyz35p4m.pdf

On Convergence in Elliptic Shape Optimization

For a shape optimization problem second derivatives are investigated, obtained by a special approach for the description of the boundary variation and the use of a potential ansatz for the state. The natural embedding of the problem in a Banach space allows the application of a standard dierential calculus in order to get second derivatives by a straight forward ”repetition of dierentiation”. Moreover, by using boundary value characerizations for more regular data, a complete boundary integral representation of the second derivative of the objective is possible. Basing on this, one easily obtains that the second derivative contains only normal components for stationary domains, i.e. for domains, satisfying the first order necessary condition for a free optimum. Moreover, the nature of the second derivative is discussed, which is helpful for the investigation of sucient optimality

/pdf/boundary-integral-representations-of-second-derivatives-in-3my0p3y1xg.pdf

Boundary integral representations of second derivatives in shape optimization

Optimal Shape Design for Elliptic Equations Via Bie-Methods

The present paper is concerned with the solution of a Bernoulli type free boundary problem by means of shape optimization. Two state functions are introduced, namely one which satisfies the mixed boundary value problem, whereas the second one satisfies the pure Dirichlet problem. The shape problem under consideration is the minimization of the L 2-distance of the gradients of the state functions. We compute the corresponding shape gradient and Hessian. By the investigation of sufficient second order conditions we prove algebraic ill-posedness of the present formulation. Our theoretical findings are supported by numerical experiments.

On a Kohn-Vogelius like formulation of free boundary problems

This present paper is concerned with second-order methods for a class of shape optimization problems. We employ a complete boundary integral representation of the shape Hessian which involves first- and second-order derivatives of the state and the adjoint state function, as well as normal derivatives of its local shape derivatives. We introduce a boundary integral formulation to compute these quantities. The derived boundary integral equations are solved efficiently by a wavelet Galerkin scheme. A numerical example validates that, in spite of the higher effort of the Newton method compared to first-order algorithms, we obtain more accurate solutions in less computational time.

Second-order shape optimization using wavelet BEM

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