Andreas Bartel

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.

Parabolic Differential-Algebraic Models in Electrical Network Design

In refined network analysis, a compact network model is combined with distributed models for semiconductor devices in a multidimensional approach. For linear RLC networks containing diodes as distributed devices, we construct a mathematical model that joins the differential-algebraic initial value problem for the electric circuit with parabolic-elliptic boundary value problems modeling the diodes. For this mixed initial boundary value problem of partial differential-algebraic equations a first existence and uniqueness result is given, and its asymptotic behavior is discussed.

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 ...

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.

Characteristic boundary conditions in the lattice Boltzmann method for fluid and gas dynamics

For numerically solving fluid dynamics problems efficiently one is often facing the problem of having to confine the computational domain to a small domain of interest introducing so-called non-reflecting boundary conditions (NRBCs). In this work we address the problem of supplying NRBCs in fluid simulations in two space dimensions using the lattice Boltzmann method (LBM): so-called characteristic boundary conditions are revisited and transferred to the framework of lattice Boltzmann simulations. Numerical tests show clearly that the unwanted unphysical reflections can be reduced significantly by applying our newly developed methods. Hereby the key idea is to transfer and generalize Thompsons boundary conditions originally developed for the nonlinear Euler equations of gas dynamics to the setting of lattice Boltzmann methods. Finally, we give strong numerical evidence that the proposed methods possess a long-time stability property.

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.

On the convergence rate of dynamic iteration for coupled problems with multiple subsystems

In multiphysical modeling coupled problems naturally occur. Each subproblem is commonly represented by a system of partial differential-algebraic equations. Applying the method of lines, this results in coupled differential-algebraic equations (DAEs). Dynamic iteration with windowing is a standard technique for the transient simulation of such systems. In contrast to the dynamic iteration of systems of ordinary differential equations, convergence for DAEs cannot be generally guaranteed unless some contraction condition is fulfilled. In the case of convergence, it is a linear one. In this paper, we quantify the convergence rate, i.e., the slope of the contraction, in terms of the window size. We investigate the convergence rate with respect to the coupling structure for DAE and ODE systems and also for two and more subsystems. We find higher rates (for certain coupling structures) than known before (that is, linear in the window size) and give sharp estimates for the rate. Furthermore it is revealed how the rate depends on the number of subsystems.

Higher Order Half-Explicit Time Integration of Eddy Current Problems Using Domain Substructuring

In this paper domain substructuring is adapted to the nonlinear transient eddy current problem: conductive and nonconductive domains are separately treated for a more efficient time integration. A matrix factorization of the linear (nonconductive) subproblem, e.g., air, is executed on beforehand and used throughout the simulation. For a general 3D problem these non-sparse factors must be replaced by sparse approximations or explicit time integration must be carried out to increase the efficiency of the solution process. This approach is validated using implicit and half-explicit time integration methods. Numerical results underline the feasibility and the possible gain obtained by higher order half-explicit methods.

Analysis of a PDE Thermal Element Model for Electrothermal Circuit Simulation

In this work we address the well-posedness of the steady-state and transient problems stemming from the coupling of a network of lumped electric elements and a PDE model of heat diffusion in the chip substrate. In particular we consider the thermal element model presented in [1] and we prove that it can be controlled by any combination of voltage sources (imposing the average current in a region of the chip) and current sources (imposing the Joule power per unit area produced in a region) connected to its temperature nodes. This result justifies the implementation of the element as a linear n-port conductance as carried out in [2].