Sebastian Schöps

Dynamic Iteration for Coupled Problems of Electric Circuits and Distributed Devices

Coupled systems of differential-algebraic equations (DAEs) may suffer from instabilities during a dynamic iteration. We extend the existing analysis on recursion estimates, error propagation, and stability to (semiexplicit) index-1 DAEs. In this context, we discuss the influence of certain coupling structures and the computational sequence of the subsystems on the rate of convergence. Furthermore, we investigate in detail convergence and divergence for two coupled problems stemming from refined electric circuit simulation. These are the semiconductor-circuit and field-circuit couplings. We quantify the convergence rate and behavior also using Lipschitz constants and suggest an enhanced modeling of the coupling interface in order to improve convergence.

A Cosimulation Framework for Multirate Time Integration of Field/Circuit Coupled Problems

This paper proposes a framework of waveform relaxation methods to simulate electromagnetic fields coupled to electric networks. Within this framework, a guarantee for convergence and stability of Gauß-Seidel-type methods is found by partial differential algebraic equation (PDAE) analysis. It is shown that different time step sizes in different parts of the model can be automatically chosen according to the problems dynamics. A finite-element model of a transformer coupled to a circuit illustrates the efficiency of multirate methods.

GPU Acceleration of Finite Difference Schemes Used in Coupled Electromagnetic/Thermal Field Simulations

The solution procedure of coupled electromagnetic-/thermal-simulations with high resolution requires efficient solvers. High performance computing libraries and languages like Nvidias CUDA help in unlocking the massively parallel capabilities of GPUs to accelerate calculations. They reduce the time needed to solve real world problems. In this paper, the speed-up is discussed, which is obtained by using GPUs for coupled time domain simulations with finite difference schemes. A tailor-made implementation of the time consuming sparse matrix vector multiplication is shown to have advantages over standard CUDA-libraries like cuSparse.

Winding functions in transient magnetoquasistatic field-circuit coupled simulations

Purpose – The purpose of this paper is to review the mutual coupling of electromagnetic fields in the magnetic vector potential formulation with electric circuits in terms of (modified) nodal and loop analyses. It aims for an unified and generic notation. Design/methodology/approach – The coupled formulation is derived rigorously using the concept of winding functions. Strong and weak coupling approaches are proposed and examples are given. Discretization methods of the partial differential equations and in particular the winding functions are discussed. Reasons for instabilities in the numerical time domain simulation of the coupled formulation are presented using results from differential-algebraic-index analysis. Findings – This paper establishes a unified notation for different conductor models, e.g. solid, stranded and foil conductors and shows their structural equivalence. The structural information explains numerical instabilities in the case of current excitation. Originality/value – The presentat...

Linear Subspace Reduction for Quasistatic Field Simulations to Accelerate Repeated Computations

The design of electrical machines and high-voltage devices requires the simulation of non-linear low frequency electromagnetic problems in time domain. Typically magneto- and electro-quasistatic problems in magnetic vector/electric scalar potential formulation lead after space discretization to differential-algebraic or ordinary differential equations, respectively. Therefore, huge systems of equations have to be solved in both cases. Typically a large number of degrees of freedom (DoF) is in domains with constant material parameters, i.e., linear subdomains, e.g., air or vacuum in exterior domains. In this paper, we present a method for low frequency simulations based on the proper orthogonal decomposition to reduce the DoF in these linear subspaces. The application of the method will be shown for a simple transformer model and within a global sensitivity analysis (uncertainty quantification) of the switching point in a non-linear resistive material used in a 11 kV standard insulator model.

Decomposition and regularization of nonlinear anisotropic curl‐curl DAEs

Purpose – The space discretization of eddy‐current problems in the magnetic vector potential formulation leads to a system of differential‐algebraic equations. They are typically time discretized by an implicit method. This requires the solution of large linear systems in the Newton iterations. The authors seek to speed up this procedure. In most relevant applications, several materials are non‐conducting and behave linearly, e.g. air and insulation materials. The corresponding matrix system parts remain constant but are repeatedly solved during Newton iterations and time‐stepping routines. The paper aims to exploit invariant matrix parts to accelerate the system solution.Design/methodology/approach – Following the principle “reduce, reuse, recycle”, the paper proposes a Schur complement method to precompute a factorization of the linear parts. In 3D models this decomposition requires a regularization in non‐conductive regions. Therefore, the grad‐div regularization is revisited and tailored such that it ...

GPU Acceleration of Algebraic Multigrid Preconditioners for Discrete Elliptic Field Problems

The simulation of coupled electromagnetic/thermal problems with high resolution requires efficient numerical schemes. High-performance computing languages like CUDA help in unlocking the massively parallel capabilities of graphic processor units (GPUs) to accelerate those calculations. This reduces the time needed to solve real-world problems. In this paper, the speedup is discussed, which is obtained using NVIDIAs recently presented Kepler architecture as well as by GPU-accelerated algebraic multigrid preconditioners. In particular, extended memory allows for the solving of larger problems with more degrees of freedom without swapping. We discuss a new host-based multigrid setup for GPU-accelerated iterative solvers.

Quantification of Uncertainty in the Field Quality of Magnets Originating from Material Measurements

A challenge in accelerator magnet design is the strong nonlinear behavior due to magnetic saturation. In practice, the underlying nonlinear saturation curve is modeled according to measurement data that typically contain uncertainties. The electromagnetic fields and in particular the multipole coefficients that heavily affect the particle beam dynamics inherit this uncertainty. In this paper, we propose a stochastic model to describe the uncertainties and we demonstrate the use of generalized polynomial chaos for the uncertainty quantification of the multipole coefficients. In contrast to previous works we propose to start the stochastic analysis from uncertain measurement data instead of uncertain material properties and we propose to determine the sensitivities by a Sobol decomposition.

Progress in industrial mathematics at ECMI 2010

ECMI is the brand associated with European mathematics for industry and organizes successful biannual conferences. In this series, the 16th conference was held in the Historical City Hall of Wuppertal (Germany). It covered mathematics in a wide range of applications and methods, from Circuit and Electromagnetic Device Simulation, Model Order Reduction for Chip Design, Uncertainties and Stochastics, Production, Fluids, Life and Environmental Sciences to Dedicated and Versatile Methods. These proceedings of ECMI 2010 emphasize mathematics as an innovation enabler for industry and business, and as an absolutely essential pre-requiste for Europe on its way to becoming the leading knowledge-based economy in the world.We present a new adaptive circuit simulation algorithm base d on spline wavelets. The unknown voltages and currents are expanded in to a wavelet representation, which is determined as solution of nonlinear equ ations derived from the circuit equations by a Galerkin discretization. The spline wavelet representation is adaptively refined during the Newton iteration. The resulti ng approximation requires an almost minimal number of degrees of freedom, and in additi on the grid refinement approach enables very efficient numerical computation s. Initial numerical tests on various types of electronic circuits show promising resu lts when compared to the standard transient analysis.

Stochastic Modeling and Regularity of the Nonlinear Elliptic curl--curl Equation

This paper addresses the nonlinear elliptic curl--curl equation with uncertainties in the material law. It is frequently employed in the numerical evaluation of magnetostatic fields, where the uncertainty is ascribed to the so-called