arXiv subjects

Peter Gangl

Publications and source records attributed to Peter Gangl.

At least 19 recordsLinked to original sources

arXiv2026

Efficient Pareto-Front Generation for Electric Machines using IGA and Second Order Derivatives

The multiobjective optimization of electric machines always involves a trade-off caused by various competing objectives such as performance and cost. A suitable design is usually determined by comparing variants from the Pareto front, which has been generated by a large number of simulation runs. This paper addresses the efficient generation of the Pareto front using a continuation method based on a homotopy method that exploits second-order derivative information to achieve superlinear convergence, enabling the fast generation of new Pareto-optimal points within only a few iterations. A key contribution is the derivation of formulas to compute the Hessian with respect to geometry parameters and shape, thus enabling direct modifications of the motor geometry in the context of Isogeometric Analysis. We apply our method to nonlinear 2D magnetostatic simulations of a permanent magnet synchronous motor and demonstrate its effectiveness by optimizing the cost, mean torque and torque ripple of the motor. Compared to a first-order optimization method, this approach reduces the number of iterations and function evaluations needed, making the pareto optimization fast and efficient.

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arXiv2026

The SiMPL Method for Multi-Material Topology Optimization

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.

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arXiv2025

Minimization of eddy currents in permanent magnets of an electric machine with shape derivatives

In this work we deal with the shape optimization of an electric machine considering time-dependent effects such as eddy currents. The considered electric machine is an interior permanent magnet synchronous machine and we minimize the average dissipated power due to the eddy currents in the magnets over a period of time corresponding to a rotation, while at the same time maximizing the average torque. Our approach is based on the computation of the shape derivative which -- beside the computation of a time discretization of time-dependent state problem -- also involves solving a a time discretization of a time-dependent adjoint problem. The challenge of this problem is related to the dependency of each one of the N time steps of the adjoint problem on two different time steps, due to the use of finite difference in the calculation of eddy current losses.

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arXiv2025

Isogeometric Topology Optimization Based on Topological Derivatives

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.

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arXiv2025

Electro-thermal topology optimization of an electric machine by the topological derivative considering drive cycles

We consider a 2d permanent magnet synchronous machine operating in a sequence of static operating points coming from a drive cycle. We aim to find a rotor design which maximizes the efficiency defined as the quotient of input and output energy considering Joule losses in the stator and eddy current losses in the permanent magnets. A coupled electromagnetic-thermal analysis of the rotor considers the eddy current losses as heat source and adds a temperature constraint to avoid damage of the permanent magnets. Additionally we impose Von-Mises stress constraints to maintain the mechanical integrity of the design. To solve the resulting free form topology optimization problem we use a level set description of the design and the topological derivative as sensitivity information. We show the effect of these constraints at very high speeds which is a trend in recent machine development.

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arXiv2025

Multi-material topology optimization of electric machines under maximum temperature and stress constraints

The use of topology optimization methods for the design of electric machines has become increasingly popular over the past years. Due to a desired increase in power density and a recent trend to high speed machines, thermal aspects play a more and more important role. In this work, we perform multi-material topology optimization of an electric machine, where the cost function depends on both electromagnetic fields and the temperature distribution generated by electromagnetic losses. We provide the topological derivative for this coupled multi-physics problem consisting of the magnetoquasistatic approximation to Maxwell's equations and the stationary heat equation. We use it within a multi-material level set algorithm in order to maximize the machine's average torque for a fixed volume of permanent-magnet material, while keeping the temperature below a prescribed value. Finally, in order to ensure mechanical stability, we additionally enforce a bound on mechanical stresses. Numerical results for the optimization of a permanent magnet synchronous machine are presented, showing a significantly improved performance compared to the reference design while meeting temperature and stress constraints.

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arXiv2025

Robust Topology Optimization of Electric Machines using Topological Derivatives

Designing high-performance electric machines that maintain their efficiency and reliability under uncertain material and operating conditions is crucial for industrial applications. In this paper, we present a novel framework for robust topology optimization with partial differential equation constraints to address this challenge. The robust optimization problem is formulated as a min-max optimization problem, where the inner maximization is the worst case with respect to predefined uncertainties, while the outer minimization aims to find an optimal topology that remains robust under these uncertainties using the topological derivative. The shape of the domain is represented by a level set function, which allows for arbitrary perturbation of the domain. The robust optimization problem is solved using a theorem of Clarke to compute subgradients of the worst case function. This allows the min-max problem to be solved efficiently and ensures that we find a design that performs well even in the presence of uncertainty. Finally, numerical results for a two-material permanent magnet synchronous machine demonstrate both the effectiveness of the method and the improved performance of robust designs under uncertain conditions.

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arXiv2024

Vertex characterization via second-order topological derivatives

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.

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arXiv2024

Multi-material topology optimization of an electric machine considering demagnetization

We consider the topology optimization problem of a 2d permanent magnet synchronous machine in magnetostatic operation with demagnetization. This amounts to a PDE-constrained multi-material design optimization problem with an additional pointwise state constraint. Using a generic framework we can incorporate this additional constraint and compute the corresponding topological derivative. We present and discuss optimization results obtained by a multi-material level set algorithm.

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arXiv2024

Tracing Pareto-optimal points for multi-objective shape optimization applied to electric machines

In the context of the optimization of rotating electric machines, many different objective functions are of interest and considering this during the optimization is of crucial importance. While evolutionary algorithms can provide a Pareto front straightforwardly and are widely used in this context, derivative-based optimization algorithms can be computationally more efficient. In this case, a Pareto front can be obtained by performing several optimization runs with different weights. In this work, we focus on a free-form shape optimization approach allowing for arbitrary motor geometries. In particular, we propose a way to efficiently obtain Pareto-optimal points by moving along to the Pareto front exploiting a homotopy method based on second order shape derivatives.

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arXiv2024

Space-time shape optimization of rotating electric machines

This article is devoted to the shape optimization of the internal structure of an electric motor, and more precisely of the arrangement of air and ferromagnetic material inside the rotor part with the aim to increase the torque of the machine. The governing physical problem is the time-dependent, non linear magneto-quasi-static version of Maxwell's equations. This multiphase problem can be reformulated on a 2d section of the real cylindrical 3d configuration; however, due to the rotation of the machine, the geometry of the various material phases at play (the ferromagnetic material, the permanent magnets, air, etc.) undergoes a prescribed motion over the considered time period. This original setting raises a number of issues. From the theoretical viewpoint, we prove the well-posedness of this unusual non linear evolution problem featuring a moving geometry. We then calculate the shape derivative of a performance criterion depending on the shape of the ferromagnetic phase via the corresponding magneto-quasi-static potential. Our numerical framework to address this problem is based on a shape gradient algorithm. The non linear time periodic evolution problems for the magneto-quasi-static potential is solved in the time domain, with a Newton-Raphson method. The discretization features a space-time finite element method, applied on a precise, meshed representation of the space-time region of interest, which encloses a body-fitted representation of the various material phases of the motor at all the considered stages of the time period. After appraising the efficiency of our numerical framework on an academic problem, we present a quite realistic example of optimal design of the ferromagnetic phase of the rotor of an electric machine.

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arXiv2023

A space-time finite element method for the eddy current approximation of rotating electric machines

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 Babu\v{s}ka--Ne\v{c}as 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.

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arXiv2023

Regularization and finite element error estimates for elliptic distributed optimal control problems with energy regularization and state or control constraints

In this paper we discuss the numerical solution of elliptic distributed optimal control problems with state or control constraints when the control is considered in the energy norm. As in the unconstrained case we can relate the regularization parameter and the finite element mesh size in order to ensure an optimal order of convergence which only depends on the regularity of the given target, also including discontinuous target functions. While in most cases, state or control constraints are discussed for the more common $L^2$ regularization, much less is known in the case of energy regularizations. But in this case, and for both control and state constraints, we can formulate first kind variational inequalities to determine the unknown state, from wich we can compute the control in a post processing step. Related variational inequalities also appear in obstacle problems, and are well established both from a mathematical and a numerical analysis point of view. Numerical results confirm the applicability and accuracy of the proposed approach.

math.NA↗

arXiv2023

Efficient and accurate separable models for discrete material optimization: A continuous perspective

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.

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arXiv2022

On the computation of analytic sensitivities of eigenpairs in isogeometric analysis

The eigenmodes of resonating structures, e.g., electromagnetic cavities, are sensitive to deformations of their shape. In order to compute the sensitivities of the eigenpair with respect to a scalar parameter, we state the Laplacian and Maxwellian eigenvalue problems and discretize the models using isogeometric analysis. Since we require the derivatives of the system matrices, we differentiate the system matrices for each setting considering the appropriate function spaces for geometry and solution. This approach allows for a straightforward computation of arbitrary higher order sensitivities in a closed-form. In our work, we demonstrate the application in a setting of small geometric deformations, e.g., for the investigation of manufacturing uncertainties of electromagnetic cavities, as well as in an eigenvalue tracking along a shape morphing.

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arXiv2022

A unified approach to shape and topological sensitivity analysis of discretized optimal design problems

We introduce a unified sensitivity concept for shape and topological perturbations and perform the sensitivity analysis for a discretized PDE-constrained design optimization problem in two space dimensions. We assume that the design is represented by a piecewise linear and globally continuous level set function on a fixed finite element mesh and relate perturbations of the level set function to perturbations of the shape or topology of the corresponding design. We illustrate the sensitivity analysis for a problem that is constrained by a reaction-diffusion equation and draw connections between our discrete sensitivities and the well-established continuous concepts of shape and topological derivatives. Finally, we verify our sensitivities and illustrate their application in a level-set-based design optimization algorithm where no distinction between shape and topological updates has to be made.

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arXiv2021

Complete topological asymptotic expansion for $L_2$ and $H^1$ tracking-type cost functionals in dimension two and three

In this paper, we study the topological asymptotic expansion of a topology optimisation problem that is constrained by the Poisson equation with the design/shape variable entering through the right hand side. Using an averaged adjoint approach, we give explicit formulas for topological derivatives of arbitrary order for both an $L_2$ and $H^1$ tracking-type cost function in both dimension two and three and thereby derive the complete asymptotic expansion. As the asymptotic behaviour of the fundamental solution of the Laplacian differs in dimension two and three, also the derivation of the topological expansion significantly differs in dimension two and three. The complete expansion for the $H^1$ cost functional directly follows from the analysis of the variation of the state equation. However, the proof of the asymptotics of the $L_2$ tracking-type cost functional is significantly more involved and, surprisingly, the asymptotic behaviour of the bi-harmonic equation plays a crucial role in our proof.

math.NA↗

arXiv2021

Automated computation of topological derivatives with application to nonlinear elasticity and reaction-diffusion problems

While topological derivatives have proven useful in applications of topology optimisation and inverse problems, their mathematically rigorous derivation remains an ongoing research topic, in particular in the context of nonlinear partial differential equation (PDE) constraints. We present a systematic yet formal approach for the computation of topological derivatives of a large class of PDE-constrained topology optimization problems with respect to arbitrary inclusion shapes. Scalar and vector-valued as well as linear and nonlinear elliptic PDE constraints are considered in two and three space dimensions including a nonlinear elasticity model and nonlinear reaction-diffusion problems. The systematic procedure follows a Lagrangian approach for computing topological derivatives. For problems where the exact formula is known, the numerically computed values show good coincidence. Moreover, by inserting the computed values into the topological asymptotic expansion, we verify that the obtained values satisfy the expected behaviour also for other, previously unknown problems, indicating the correctness of the procedure. We present a systematic approach for the computation of topological derivatives that is applicable to a large class of problems. Most notably, our approach covers the topological derivative for a nonlinear elasticity problem, which has not been reported in the literature.

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