Kevin Sturm

Bio: Kevin Sturm is an academic researcher from Vienna University of Technology. The author has contributed to research in topics: Shape optimization & Nonlinear system. The author has an hindex of 9, co-authored 26 publications receiving 301 citations. Previous affiliations of Kevin Sturm include University of Duisburg-Essen.

Shape Optimization of an Electric Motor Subject to Nonlinear Magnetostatics

Abstract: The goal of this paper is to improve the performance of an electric motor by modifying the geometry of a specific part of the iron core of its rotor. To be more precise, the objective is to smooth the rotation pattern of the rotor. A shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired rotation pattern. The magnetic field generated by permanent magnets is modeled by a nonlinear partial differential equation of magnetostatics. The shape sensitivity analysis is rigorously performed for the nonlinear problem by means of a new shape-Lagrangian formulation adapted to nonlinear problems.

Minimax Lagrangian Approach to the Differentiability of Nonlinear PDE Constrained Shape Functions Without Saddle Point Assumption

Abstract: The object of this paper is the computation of the domain or boundary expression of a state constrained shape function without explicitly resorting to the material derivative. Our main theorem bypasses the restrictive saddle point assumption in existing differentiability theorems of minimax by introducing the notion of the averaged adjoint state, thus extending the use of minimax theorems to some classes of nonlinear state equations. As an illustration, the theorem is applied to a shape function that depends on a quasi-linear transmission problem. Using a Gagliardo penalization the existence of optimal shapes is established.

Shape optimization of an electric motor subject to nonlinear magnetostatics

Abstract: The goal of this paper is to improve the performance of an electric motor by modifying the geometry of a specific part of the iron core of its rotor. To be more precise, the objective is to smooth the rotation pattern of the rotor. A shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired rotation pattern. The magnetic field generated by permanent magnets is modeled by a nonlinear partial differential equation of magnetostatics. The shape sensitivity analysis is rigorously performed for the nonlinear problem by means of a new shape-Lagrangian formulation adapted to nonlinear problems.

On shape optimization with non-linear partial differential equations

Abstract: This thesis is concerned with shape optimization problems under non-linear PDE (partial differential equation) constraints. We give a brief introduction to shape optimization and recall important concepts such as shape continuity, shape derivative and the shape differentiability. In order to review existing methods for proving the shape differentiability of PDE constrained shape functions a simple semi-linear model problem is used as constraint. With this example we illustrate the conceptual limits of each method. In the main part of this thesis a new theorem on the differentiability of a minimax function is proved. This fundamental result simplifies the derivation of necessary optimality conditions for PDE constrained optimization problems. It represents a generalization of the celebrated Theorem of Correa-Seeger for the special class of Lagrangian functions and removes the saddle point assumption. Although our method can also be used to compute sensitivities in optimal control, we mainly focus on shape optimization problems. In this respect, we apply the result to four model problems: (i) a semi-linear problem, (ii) an electrical impedance tomography problem, (iii) a model for distortion compensation in elasticity, and finally (iv) a quasi-linear problem describing electro-magnetic fields. Next, we concentrate on methods to minimise shape functions. For this we recall several procedures to put a manifold structure on the space of shapes. Usually, the boundary expression of the shape derivative is used for numerical algorithms. From the numerical point of view this expression has several disadvantages, which will be explained in more detail. In contrast, the volume expression constitutes a numerically more accurate representation of the shape derivative. Additionally, this expression allows us to look at gradient algorithms from two perspectives: the Eulerian and Lagrangian points of view. In the Eulerian approach all computations are performed on the current moving domain. On the other hand the Lagrangian approach allows to perform all calculations on a fixed domain. The Lagrangian view naturally leads to a gradient flow interpretation. The gradient flow depends on the chosen metrics of the underlying function space. We show how different metrics may lead to different optimal designs and different regularity of the resulting domains. In the last part, we give numerical examples using the gradient flow interpretation of the Lagrangian approach. In order to solve the severely ill-posed electrical impedance tomography problem (ii), the discretised gradient flow will be combined with a level-set method. Finally, the problem from example (iv) is solved using B-Splines instead of levelsets.

Fully and semi-automated shape differentiation in NGSolve

Abstract: In this paper, we present a framework for automated shape differentiation in the finite element software NGSolve. Our approach combines the mathematical Lagrangian approach for differentiating PDE-constrained shape functions with the automated differentiation capabilities of NGSolve. The user can decide which degree of automatisation is required, thus allowing for either a more custom-like or black-box–like behaviour of the software. We discuss the automatic generation of first- and second-order shape derivatives for unconstrained model problems as well as for more realistic problems that are constrained by different types of partial differential equations. We consider linear as well as nonlinear problems and also problems which are posed on surfaces. In numerical experiments, we verify the accuracy of the computed derivatives via a Taylor test. Finally, we present first- and second-order shape optimisation algorithms and illustrate them for several numerical optimisation examples ranging from nonlinear elasticity to Maxwell’s equations.

Perturbation Analysis Of Optimization Problems

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High-order essentially non-oscillatory schemes for Hamilton-Jacobi equations

Abstract: Hamilton-Jacobi equations are frequently encountered in applications,e.g.,in control theory and differential games.Hamilton-Jacobi equations are closely related to hyperbolic conservation laws-in one space dimension the former is simply the integrated version of the latter.Similarity also exists for the multidimensional cases,and this is helpful in designing difference approximations.In this paper central weighted essentially non-oscillatory (CWENO) schemes for Hamilton-Jacobi equations are investigated,which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives.The schemes are numerically tested on a variety of one-dimensional problems,including a problem related to control optimization.High-order accuracy in smooth regions,high resolution of discontinuities in the derivatives,and convergence to viscosity solutions are observed.

Distributed shape derivative via averaged adjoint method and applications

Abstract: The structure theorem of Hadamard–Zolesio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. Actually the domain representation, also known as distributed shape derivative, is more general than the boundary expression as it is well-defined for shapes having a lower regularity. It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithms. In this paper we describe several advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We identify a tensor representation of the distributed shape derivative, study its properties and show how it allows to recover the boundary expression directly. We use a novel Lagrangian approach, which is applicable to a large class of shape optimization problems, to compute the distributed shape derivative. We also apply the technique to retrieve the distributed shape derivative for electrical impedance tomography. Finally we explain how to adapt the level set method to the distributed shape derivative framework and present numerical results.