Peter Gangl

Peter Gangl, Ulrich Langer

We present and analyze a new finite element method for solving interface problems on a triangular grid. The method locally modifies a given triangulation such that the interfaces are accurately resolved and the maximal angle condition holds. Therefore, optimal order of convergence can be shown. Moreover, an appropriate scaling of the basis functions yields an optimal condition number of the stiffness matrix. The method is applied to an optimal design problem for an electric motor where the interface between different materials is evolving in the course of the optimization procedure.

Peter Gangl, Mario Gobrial, Olaf Steinbach

In this paper we formulate and analyze a space-time finite element method for the numerical simulation of rotating electric machines where the finite element mesh is fixed in space-time domain. Based on the Babuška--Nečas theory we prove unique solvability both for the continuous variational formulation and for a standard Galerkin finite element discretization in the space-time domain. This approach allows for an adaptive resolution of the solution both in space and time, but it requires the solution of the overall system of algebraic equations. While the use of parallel solution algorithms seems to be mandatory, this also allows for a parallelization simultaneously in space and time. This approach is used for the eddy current approximation of the Maxwell equations which results in an elliptic-parabolic interface problem. Numerical results for linear and nonlinear constitutive material relations confirm the applicability, efficiency and accuracy of the proposed approach.

Peter Gangl, Ulrich Langer, Angelos Mantzaflaris et al.

Future e-mobility calls for efficient electrical machines. For different areas of operation, these machines have to satisfy certain desired properties that often depend on their design. Here we investigate the use of multipatch Isogeometric Analysis (IgA) for the simulation and shape optimization of the electrical machines. In order to get fast simulation and optimization results, we use non-overlapping domain decomposition (DD) methods to solve the large systems of algebraic equations arising from the IgA discretization of underlying partial differential equations. The DD is naturally related to the multipatch representation of the computational domain, and provides the framework for the parallelization of the DD solvers.

Peter Gangl, Bochra Mejri, Otmar Scherzer

This paper focuses on identifying vertex characteristics in 2D images using topological asymptotic analysis. Vertex characteristics include both the location and the type of the vertex, with the latter defined by the number of lines forming it and the corresponding angles. This problem is crucial for computer vision tasks, such as distinguishing between fore- and background objects in 3D scenes. We compute the second-order topological derivative of a Mumford-Shah type functional with respect to inclusion shapes representing various vertex types. This derivative assigns a likelihood to each pixel that a particular vertex type appears there. Numerical tests demonstrate the effectiveness of the proposed approach.

Peter Gangl, Stefan Köthe, Christiane Mellak et al.

This paper deals with the design optimization of a synchronous reluctance machine to be used in an X-ray tube, where the goal is to maximize the torque, by means of gradient-based free-form shape optimization. The presented approach is based on the mathematical concept of shape derivatives and allows to obtain new motor designs without the need to introduce a geometric parametrization. We validate our results by comparing them to a parametric geometry optimization in JMAG by means of a stochastic optimization algorithm. While the obtained designs are of similar shape, the computational time used by the gradient-based algorithm is in the order of minutes, compared to several hours taken by the stochastic optimization algorithm. Finally, we show an extension of the free-form shape optimization algorithm to the case of multiple objective functions and illustrate a way to obtain an approximate Pareto front.

Peter Gangl, Nico Nees, Michael Stingl

Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a long and successful tradition in the design optimization community and have led to powerful optimization tools like the prominently used method of moving asymptotes (MMA). In this paper, we introduce several new separable approximations to a model problem and examine them in terms of accuracy and fast evaluation. The models can, in general, be nonconvex and are based on the Sherman-Morrison-Woodbury matrix identity on the one hand, and on the mathematical concept of topological derivatives on the other hand. We show a surprising relation between two models originating from these two -- at a first sight -- very different concepts. Numerical experiments show a high level of accuracy for two of our proposed models while also their evaluation can be performed efficiently once enough data has been precomputed in an offline phase. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models.

Guilherme Henrique Teixeira, Nepomuk Krenn, Peter Gangl et al.

Topology optimization is a valuable tool in engineering, facilitating the design of optimized structures. However, topological changes often require a remeshing step, which can become challenging. In this work, we propose an isogeometric approach to topology optimization driven by topological derivatives. The combination of a level-set method together with an immersed isogeometric framework allows seamless geometry updates without the necessity of remeshing. At the same time, topological derivatives provide topological modifications without the need to define initial holes [7]. We investigate the influence of higher-degree basis functions in both the level-set representation and the approximation of the solution. Two numerical examples demonstrate the proposed approach, showing that employing higher-degree basis functions for approximating the solution improves accuracy, while linear basis functions remain sufficient for the level-set function representation.

Thomas Gauthey, Peter Gangl, Maya Hage Hassan

Permanent magnet-assisted synchronous reluctance motors (PMSynRM) have a significantly higher average torque than synchronous reluctance motors. Thus, determining an optimal design results in a multi-material topology optimization problem, where one seeks to distribute ferromagnetic material, air and permanent magnets within the rotor in an optimal manner. This study proposed a novel density-based distribution scheme, which allows for continuous magnetization direction instead of a finite set of angles. Thus, an interpolation scheme is established between properties pertaining to magnets and non-linear materials. This allows for new designs to emerge without introducing complex geometric parameterization or relying on the user's biases and intuitions. Toward reducing computation time, Nitsche-type mortaring is applied, allowing for free rotation of the rotor mesh relative to the stator mesh. The average torque is approximated using only four-point static positions. This study investigates several interpolation schemes and presents a new one inspired by the topological derivative. We propose to filter the final design for the magnetization angle using K-mean clustering accounting for technical feasibility constraints of magnets. Finally, the design of the electrical motor is proposed to maximize torque value.

Peter Gangl, Brendan Keith, Dohyun Kim et al.

We introduce an efficient and scalable method for density-based multi-material topology optimization, integrating classical mirror descent techniques with point-wise polytopal design constraints. Such constraints arise naturally in this class of problems, wherein the vertices of convex polytopes correspond to distinct design states, only one of which should be occupied at each point in space. The framework generates a descending sequence of iterates by penalizing the design space around the previous iterate with a generalized distance function tailored to the convex geometry of the $n$-dimensional polytope. This distance function, called a Bregman divergence, smooths the optimization landscape, ensuring that each iterate strictly satisfies the point-wise constraints. Subsequently, global constraints (e.g., bounds on the structural mass) can be enforced easily by solving a small, finite-dimensional dual problem. The resulting method is simple to implement and demonstrates robustness and efficiency when combined with an Armijo-type line search algorithm. We validate the method in structural design problems involving the optimal arrangement of both isotropic and anisotropic materials, as well as magnetic flux optimization in electric motors.