arXiv subjects

Melina Merkel

Publications and source records attributed to Melina Merkel.

At least 19 recordsLinked to original sources

arXiv2026

Subdivision-based isogeometric analysis for axisymmetric electromagnetic problems

This paper applies a subdivision-based isogeometric method to solve the axisymmetric Maxwell eigenvalue problem. The reduction to an $H^1$-formulation allows to use a Catmull-Clark construction for both geometry and field discretization. The approach yields a numerical solution for the electric field, which is $C^1$-continuous everywhere except at extraordinary vertices. This is demonstrated by computing the eigenmodes of a TESLA 9-cell cavity, showing smoother fields with less numerical noise than conventional methods. The convergence rate of the method is numerically analyzed and is in agreement with rates observed in the literature.

cs.CE↗

arXiv2025

Tree-cotree gauging for two-dimensional hierarchical splines

In magnetostatics and eddy current problems, formulated in terms of the magnetic vector potential, the solution is not unique, because the addition of an irrotational function to the solution remains a valid solution. The tree-cotree decomposition is a gauging technique to recover uniqueness when using finite elements, which consists in considering the mesh as a graph, and building a spanning tree on that graph. The idea has been recently extended to isogeometric analysis, applying the construction of the spanning tree on the control mesh, or equivalently, on the Greville grid. In the present paper we extend the construction to hierarchical splines, a set of splines with multi-level structure for adaptive refinement, by constructing a spanning tree for each single level. Since for degree $p=1$ the spaces of finite elements and hierarchical splines coincide, the presented construction is also valid for quadrilateral finite element meshes with hanging nodes. To assess the correctness of the method, we present numerical results for Maxwell eigenvalue problem.

math.NA↗

arXiv2025

Extending the Numerical Flow Iteration to the multi-species Vlasov-Maxwell system through Hamiltonian Splitting

The Numerical Flow Iteration (NuFI) method has recently been proposed as a memory-slim while accurate in phase-space method for the electro-static Vlasov--Poisson system. It stores the temporal evolution of the electric field, instead of the distribution functions, and reconstructs the solution in each time step by following the characteristics backwards in time and reconstructing the solution from the initial distribution. NuFI has been shown to be more accurate than other state-of-the-art electro-static Vlasov solvers given the same amount of degrees of freedom. In this paper, we build on the Hamiltonian structure of the full Vlasov--Maxwell system to extend NuFI to handle electro-magnetic kinetic plasma dynamics. We show that the structure-preserving properties of the NuFI time-stepping are preserved when extending to the electro-magnetic case. Furthermore we discuss how NuFI can be incorporated into existing Semi-Lagrangian codes as an efficient while accurate subcycling technique.

physics.plasm-ph↗

arXiv2025

A stabilized Two-Step Formulation of Maxwell's Equations in the time-domain

Simulating electromagnetic fields across broad frequency ranges is challenging due to numerical instabilities at low frequencies. This work extends a stabilized two-step formulation of Maxwell's equations to the time-domain. Using a Galerkin discretization in space, we apply two different time-discretization schemes that are tailored to the first- and second-order in time partial differential equations of the two-step solution procedure used here. To address the low-frequency instability, we incorporate a generalized tree-cotree gauge that removes the singularity of the curl-curl operator, ensuring robustness even in the static limit. Numerical results on academic and application-oriented 3D problems confirm stability, accuracy, and the method's applicability to nonlinear, temperature-dependent materials.

math.NA↗

arXiv2025

A Two-Step Formulation of Maxwell's Equations Using Generalized Tree-Cotree Gauges for Low-Frequency-Stability

This paper presents a new low-frequency stabilization for a two-step formulation solving the full set of Maxwell's equations. The formulation is based on a electric scalar and magnetic vector potential equation using the electroquasistatic problem as gauge condition. The proposed stabilization technique consists of an adequate frequency scaling for the electroquasistatic problem, and a tree-cotree decomposition of the magnetic vector potential such that its divergence remains consistent with the partial decoupling of the magnetic and electric potentials. The paper discusses two variants and demonstrates the effectiveness by a few computational examples.

math.NA↗

arXiv2025

An Air-Gap Element for the Isogeometric Space-Time-Simulation of Electric Machines

Space-time methods promise more efficient time-domain simulations, in particular of electrical machines. However, most approaches require the motion to be known in advance so that it can be included in the space-time mesh. To overcome this problem, this paper proposes to use the well-known air-gap element for the rotor-stator coupling of an isogeometric machine model. First, we derive the solution in the air-gap region and then employ it to couple the rotor and stator. This coupling is angle dependent and we show how to efficiently update the coupling matrices to a different angle, avoiding expensive quadrature. Finally, the resulting time-dependent problem is solved in a space-time setting. The spatial discretization using isogeometric analysis is particularly suitable for coupling via the air-gap element, as NURBS can exactly represent the geometry of the air-gap. Furthermore, the model including the air-gap element can be seamlessly transferred to the space-time setting. However, the air-gap element is well known in the literature. The originality of this work is the application to isogeometric analysis and space-time.

math.NA↗

arXiv2025

On Domain Decomposition for Magnetostatic Problems in 3D

The simulation of three dimensional magnetostatic problems plays an important role, for example when simulating synchronous electric machines. Building on prior work that developed a domain decomposition algorithm using isogeometric analysis, this paper extends the method to support subdomains composed of multiple patches. This extension enables load-balancing across available CPUs, facilitated by graph partitioning tools such as METIS. The proposed approach enhances scalability and flexibility, making it suitable for large-scale simulations in diverse industrial contexts.

cs.CE↗

arXiv2024

Tree-Cotree-Based Tearing and Interconnecting for 3D Magnetostatics: A Dual-Primal Approach

The simulation of electromagnetic devices with complex geometries and large-scale discrete systems benefits from advanced computational methods like IsoGeometric Analysis and Domain Decomposition. In this paper, we employ both concepts in an Isogeometric Tearing and Interconnecting method to enable the use of parallel computations for magnetostatic problems. We address the underlying non-uniqueness by using a graph-theoretic approach, the tree-cotree decomposition. The classical tree-cotree gauging is adapted to be feasible for parallelization, which requires that all local subsystems are uniquely solvable. Our contribution consists of an explicit algorithm for constructing compatible trees and combining it with a dual-primal approach to enable parallelization. The correctness of the proposed approach is proved and verified by numerical experiments, showing its accuracy, scalability and optimal convergence.

cs.CE↗

arXiv2024

Spline-Based Rotor and Stator Optimization of a Permanent Magnet Synchronous Motor

This work features the optimization of a Permanent Magnet Synchronous Motor using 2D nonlinear simulations in an Isogeometric Analysis framework. The rotor and stator designs are optimized for both geometric parameters and surface shapes via modifications of control points. The scaling laws for magnetism are employed to allow for axial and radial scaling, enabling a thorough optimization of all critical machine parameters for multiple operating points. The process is carried out in a gradient-based fashion with the objectives of lowering motor material cost, torque ripple and losses. It is shown that the optimization can be efficiently conducted for many optimization variables and all objective values can be reduced.

math.OC↗

arXiv2023

Combined Parameter and Shape Optimization of Electric Machines with Isogeometric Analysis

In structural optimization, both parameters and shape are relevant for the model performance. Yet, conventional optimization techniques usually consider either parameters or the shape separately. This work addresses this problem by proposing a simple yet powerful approach to combine parameter and shape optimization in a framework using Isogeometric Analysis (IGA). The optimization employs sensitivity analysis by determining the gradients of an objective function with respect to parameters and control points that represent the geometry. The gradients with respect to the control points are calculated in an analytical way using the adjoint method, which enables straightforward shape optimization by altering of these control points. Given that a change in a single geometry parameter corresponds to modifications in multiple control points, the chain rule is employed to obtain the gradient with respect to the parameters in an efficient semi-analytical way. The presented method is exemplarily applied to nonlinear 2D magnetostatic simulations featuring a permanent magnet synchronous motor and compared to designs, which were optimized using parameter and shape optimization separately. It is numerically shown that the permanent magnet mass can be reduced and the torque ripple can be eliminated almost completely by simultaneously adjusting rotor parameters and shape. The approach allows for novel designs to be created with the potential to reduce the optimization time substantially.

math.OC↗

arXiv2023

Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems

The eddy current problem has many relevant practical applications in science, ranging from non-destructive testing to magnetic confinement of plasma in fusion reactors. It arises when electrical conductors are immersed in an external time-varying magnetic field operating at frequencies for which electromagnetic wave propagation effects can be neglected. Popular formulations of the eddy current problem either use the magnetic vector potential or the magnetic scalar potential. They have individual advantages and disadvantages. One challenge is related to differential geometry: Scalar potential based formulations run into trouble when conductors are present in non-trivial topology, as approximation spaces must be then augmented with generators of the first cohomology group of the non-conducting domain. For all existing algorithms based on lowest order methods it is assumed that the extension of the graph-based algorithms to high-order approximations requires hierarchical bases for the curl-conforming discrete spaces. However, building on insight on de Rham complexes approximation with splines, we will show in the present submission that the hierarchical basis condition is not necessary. Algorithms based on spanning tree techniques can instead be adapted to work on an underlying hexahedral mesh arising from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.

math.NA↗

arXiv2023

Low-Frequency Stabilization of Dielectric Simulation Problems with Conductors and Insulators

When simulating resistive-capacitive circuits or electroquasistatic problems where conductors and insulators coexist, one observes that large time steps or low frequencies lead to numerical instabilities, which are related to the condition number of the system matrix. Here, we propose several stable formulations by scaling the equation systems. This enables a reliable calculation of solutions for very low frequencies (even for the static case), or large time steps. Numerical experiments underline the findings.

cs.CE↗

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.

cs.CE↗

arXiv2022

How to Build the Optimal Magnet Assembly for Magnetocaloric Cooling: Structural Optimization with Isogeometric Analysis

In the search for more efficient and less environmentally harmful cooling technologies, the field of magnetocalorics is considered a promising alternative. To generate cooling spans, rotating permanent magnet assemblies are used to cyclically magnetize and demagnetize magnetocaloric materials, which change their temperature under the application of a magnetic field. In this work, an axial rotary permanent magnet assembly, aimed for commercialization, is computationally designed using topology and shape optimization. This is efficiently facilitated in an isogeometric analysis framework, where harmonic mortaring is applied to couple the rotating rotor-stator system of the multipatch model. Inner, outer and co-rotating assemblies are compared and optimized designs for different magnet masses are determined. These simulations are used to homogenize the magnetic flux density in the magnetocaloric material. The resulting torque is analyzed for different geometric parameters. Additionally, the influence of anisotropy in the active magnetic regenerators is studied in order to guide the magnetic flux. Different examples are analyzed and classified to find an optimal magnet assembly for magnetocaloric cooling.

math.NA↗

arXiv2022

Torque Computation with the Isogeometric Mortar Method for the Simulation of Electric Machines

In this work isogeometric mortaring is used for the simulation of a six pole permanent magnet synchronous machine. Isogeometric mortaring is especially well suited for the efficient computation of rotating electric machines as it allows for an exact geometry representation for arbitrary rotation angles without the need of remeshing. The appropriate B-spline spaces needed for the solution of Maxwell's equations and the corresponding mortar spaces are introduced. Unlike in classical finite element methods their construction is straightforward in the isogeometric case. The torque in the machine is computed using two different methods, i.e., Arkkio's method and by using the Lagrange multipliers from the mortaring.

cs.CE↗

arXiv2021

On torque computation in electric machine simulation by harmonic mortar methods

The use of trigonometric polynomials as Lagrange multipliers in the harmonic mortar method enables an efficient and elegant treatment of relative motion in the stator-rotor coupling of electric machine simulation. Explicit formulas for the torque computation are derived by energetic considerations, and their realization by harmonic mortar finite element and isogemetric analysis discretizations is discussed. Numerical tests are presented to illustrate the theoretical results and demonstrate the potential of harmonic mortar methods for the evaluation of torque ripples.

math.NA↗

arXiv2021

Tree-Cotree Decomposition of Isogeometric Mortared Spaces in H(curl) on Multi-Patch Domains

When applying isogeometric analysis to engineering problems, one often deals with multi-patch spline spaces that have incompatible discretisations, e.g. in the case of moving objects. In such cases mortaring has been shown to be advantageous. This contribution discusses the appropriate B-spline spaces needed for the solution of Maxwell's equations in the functions space H(curl) and the corresponding mortar spaces. The main contribution of this paper is to show that in formulations requiring gauging, as in the vector potential formulation of magnetostatic equations, one can remove the discrete kernel subspace from the mortared spaces by the graph-theoretical concept of a tree-cotree decomposition. The tree-cotree decomposition is done based on the control mesh, it works for non-contractible domains, and it can be straightforwardly applied independently of the degree of the B-spline bases. Finally, the simulation workflow is demonstrated using a realistic model of a rotating permanent magnet synchronous machine.

math.NA↗

arXiv2020

On the stability of harmonic mortar methods with application to electric machines

Harmonic stator-rotor coupling offers a promising approach for the interconnection of rotating subsystems in the simulation of electric machines. This paper studies the stability of discretization schemes based on harmonic coupling in the framework of mortar methods for Poisson-like problems. A general criterion is derived that allows to ensure the relevant inf-sup stability condition for a variety of specific discretization approaches, including finite-element methods and isogeometric analysis with harmonic mortar coupling. The validity and sharpness of the theoretical results is demonstrated by numerical tests.

math.NA↗