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2024 | Buch

Harmonic Balance with Small Signal Perturbation Kapitel lesen Erstes Kapitel lesen

Scientific Computing in Electrical Engineering

SCEE 2022, Amsterdam, The Netherlands, July 2022

herausgegeben von: Martijn van Beurden, Neil V. Budko, Gabriela Ciuprina, Wil Schilders, Harshit Bansal, Ruxandra Barbulescu

Verlag: Springer Nature Switzerland

Buchreihe : Mathematics in Industry

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This volume comprises selected papers presented at the 14th International Conference on Scientific Computing in Electrical Engineering, SCEE 2022, held in Amsterdam, The Netherlands, in July 2022.

The aim of the SCEE 2022 conference was to bring together scientists – mathematicians, electrical engineers, computer scientists, and physicists, from universities and industry – to have in-depth discussions of the latest scientific results in Computational Science and Engineering relevant to Electrical Engineering and to stimulate and inspire active participation of young researchers.

This extensive reference work is divided into four parts: Part I Circuit Simulation and Design.- Part II Device Simulation.- Part III Computational Electromagnetics.- Part IV Mathematical and Computational Methods. Each part starts with a general introduction, followed by the respective contributions.

The book will appeal to mathematicians and electrical engineers. Further, it introduces algorithm and program developers to recent advances in the other fields, while industry experts will be introduced to new programming tools and mathematical methods.

Inhaltsverzeichnis

Frontmatter

Circuit Simulation and Design

Frontmatter
Harmonic Balance with Small Signal Perturbation
Abstract
Investigating perturbations of a periodic steady state of an electric circuit is of interest e.g. for small signal responses, noise analysis or the generation of X-parameter models. We present a method based on Harmonic Balance, to compute the Fourier coefficients of the circuit response for a small signal perturbation of the input. The relation to two-tone Harmonic Balance is investigated and it is shown that under suitable conditions the perturbation method can approximate the full two-tone solution at extremely lower costs. The method is tested on a Gilbert mixer circuit.
Kai Bittner, Martin K. Steiger, Hans Georg Brachtendorf
A Projective-Based Formalism for Symmetric Modeling of Electrical Circuits
Abstract
We survey in this contribution some recent ideas involving the use of a homogeneous formalism to set up electrical circuit models. Broadly, the goal is to avoid any lack of generality in the modeling process by avoiding unnecessarily restrictive assumptions in the form of the characteristics of the circuit devices. We discuss how to use this approach in the framework of nodal analysis, aiming at the development of computationally efficient models. Except for some minor technicalities arising in the index analysis, the discussion is deliberately kept at a simple level.
Ricardo Riaza
A Port-Hamiltonian, Index , Structurally Amenable Electrical Circuit Formulation
Abstract
We present a recently developed electrical circuit formulation that has port-Hamiltonian (pH) structure and results in a structurally amenable differential-algebraic equation (DAE) system of index \(\le 1\). Being pH assures energy stability—the total energy of the system cannot increase. It also provides compositionality—larger pH models can be assembled from smaller ones in a standard way that facilitates building pH models in software. Structurally amenable and index \(\le 1\) eliminate the phases of DAE index analysis and reduction, which are commonly used in circuit simulation software. Thus, standard numerical solvers can be applied directly to integrate the DAE. In addition, it has a known a priori block-triangular form that can be exploited for efficient numerical solution. A prototype Matlab code shows high potential for development of this “compact port-Hamiltonian” (CpH) methodology.
Lena Scholz, John Pryce, Nedialko Nedialkov

Device Simulation

Frontmatter
Simulation of a GNR-FET
Abstract
A field effect transistor is simulated in the case the active area is made by a single graphene nanoribbon. At variance with large area graphene, an energy gap is present and this should improve the performance of the device as transistor. A drift-diffusion model which includes the degenerate effects, coupled to the Poisson equation for the electrostatic potential, is used. The mobility models are obtained, by a fitting procedure, solving numerically the semiclassical Boltzmann equation for the graphene nanoribbon, including also the edges scattering besides the electron-phonon interactions.
Giovanni Nastasi, Vittorio Romano

Computational Electromagnetics

Frontmatter
Solution of Time-Harmonic Maxwell’s Equations by a Domain Decomposition Method Based on PML Transmission Conditions
Abstract
Numerical discretization of the large-scale Maxwell’s equations leads to an ill-conditioned linear system that is challenging to solve. The key requirement for successive solutions of this linear system is to choose an efficient solver. In this work we use Perfectly Matched Layers (PML) to increase this efficiency. PML have been widely used to truncate numerical simulations of wave equations due to improving the accuracy of the solution instead of using absorbing boundary conditions (ABCs). Here, we will develop an efficient solver by providing an alternative use of PML as transmission conditions at the interfaces between subdomains in our domain decomposition method. We solve Maxwell’s equations and assess the convergence rate of our solutions compared to the situation where absorbing boundary conditions are chosen as transmission conditions.
Sahar Borzooei, Victorita Dolean, Pierre-Henri Tournier, Claire Migliaccio
Validation-Oriented Modelling of Electrical Stimulation Chambers for Cartilage Tissue Engineering
Abstract
The capability of electrical stimulation to enhance cell activity, proliferation, and differentiation makes it an attractive method in cell-based therapies. Due to its biocompatibility, capacitive coupling has emerged as a favourable method to deliver electric fields to cartilaginous cells. Unfortunately, there exists no means to measure the electric field directly. It can solely be inferred from other measurement results. Nonetheless, numerical simulations by the finite element method provide a possibility to estimate the electric field distribution and magnitude. The experimental validation of numerical models, however, receives insufficient attention. This study aims to bridge the gap between theory and experiment by applying validation-oriented modelling. The impact of different uncertain input parameters on relevant observables was assessed to suggest validation experiments using uncertainty quantification. The estimated capacitance was found to be in excellent agreement with the experimental result, indicating that the model is accurate. However, the electric field remains uncertain since the electric field and capacitance are dependent upon different input parameters. The electric field is primarily determined by the conductivity of the medium. Hence, a more precise conductivity measurement will allow for more accurate computation of the electric field magnitude.
Lam Vien Che, Julius Zimmermann, Henning Bathel, Alina Weizel, Hermann Seitz, Ursula van Rienen
Matrix-Free Parallel Preconditioned Iterative Solvers for the 2D Helmholtz Equation Discretized with Finite Differences
Abstract
We present a matrix-free parallel iterative solver for the Helmholtz equation related to applications in seismic problems and study its parallel performance. We apply Krylov subspace methods, GMRES, Bi-CGSTAB and IDR(s), to solve the linear system obtained from a second-order finite difference discretization. The Complex Shifted Laplace Preconditioner (CSLP) is employed to improve the convergence of Krylov solvers. The preconditioner is approximately inverted by multigrid iterations. For parallel computing, the global domain is partitioned blockwise. The standard MPI library is employed for data communication. The matrix-vector multiplication and preconditioning operator are implemented in a matrix-free way instead of constructing large, memory-consuming coefficient matrices. These adjustments lead to direct improvements in terms of memory consumption. Numerical experiments of model problems show that the matrix-free parallel solution method has satisfactory parallel performance and weak scalability. It allows us to solve larger problems in parallel to obtain more accurate numerical solutions.
Jinqiang Chen, Vandana Dwarka, Cornelis Vuik
Implementation and Validation of the Dual Full-Wave E and H Formulations with Electric Circuit Element Boundary Conditions
Abstract
Dual full-wave (FW) frequency-domain \(\textbf{E}\) and \(\textbf{H}\) formulations, with scalar potentials on the boundary, and with electric circuit element boundary conditions are discussed and details about their implementation in the finite element method are given. For some magneto-quasi-static devices this duality frames the exact solution thus allowing the accuracy control. In such cases the geometric mean of the dual solutions exhibits a better accuracy and higher convergence rate than the individual numerical solutions. For FW devices the dual formulations allow a compromise between model accuracy and computational effort, especially if the models are not 3D. Implementation is available for free in onelab. Validation for test cases with analytic solution are provided: a conducting cylinder and a coaxial cable.
Gabriela Ciuprina, Daniel Ioan, Ruth V. Sabariego
A Yee-Like Finite Element Scheme for Maxwell’s Equations on Hybrid Grids with Mass-Lumping
Abstract
A novel finite element method for the approximation of Maxwell’s equations over hybrid two-dimensional grids is studied. The choice of appropriate basis functions and numerical quadrature leads to diagonal mass matrices which allow for efficient time integration by explicit methods.On purely rectangular grids, the proposed schemes coincide with well-established FIT and FDTD methods. Additional internal degrees of freedom introduced on triangles allow for mass-lumping without the usual constraints on the shape of these elements. A full error analysis of the method is developed and numerical tests are presented for illustration.
Herbert Egger, Bogdan Radu
Time-Domain Electromagnetic Modeling and Simulation of a Nonlinear Electro-Optical Mixer
Abstract
A full-wave electromagnetic solver coupled with a Poisson’s solver based on time-domain finite element method (TD-FEM) is developed. This solver aims to simulate the side-band frequency generation on optical signal due to the imposed radio frequency (RF) signal through a nonlinear material. The optical signal propagating within an optical waveguide is simulated in time-domain by solving the electromagnetic wave equation, whereas Poisson’s equation is numerically solved to compute the strength of the slowly-varying RF signal. The applied RF signal changes the permittivity of the nonlinear material BTO, and this changing permittivity affects the transient wave behavior of the light. As opposed to the available frequency-domain Maxwell solvers, this proposed time-domain solver is capable of simulating the nonlinear effects introduced by an electro-optical material, and implemented for the modeling of an application where RF signal is mixed into the optical frequencies. As a result of the simulations, nonlinear dielectric constant of electro-optical material is computed, and resulting side-band frequency generation is observed in the spectrum of the time-domain output signal.
Arif Can Gungor, Hande Ibili, Jasmin Smajic, Juerg Leuthold
Iterative Charge-Update Schemes for Electro-quasistatic Problems
Abstract
The electric scalar potential electro-quasistatic field formulation is commonly employed for simulating nonlinear high-voltage problems. To this end, standard iterative nonlinear solvers, such as fixed-point iterations and Newton’s method, are used. Here, iterative charge update schemes that possess a constant coefficient matrix throughout the nonlinear iterations are developed, abstract convergence conditions are proved and numerically verified for the fundamental frequency, while their suitability and performance is assessed, in terms of accuracy and computational complexity.
Fotios Kasolis, Marvin-Lucas Henkel, Markus Clemens
Electrostatic Forces on Conductors with Boundary Element Methods in 3D
Abstract
In this work we are concerned with computing local/global electrostatic forces and torques on perfect electrical conductors using the boundary element method (BEM). Classical boundary based force functionals are not continuous on energy trace spaces and therefore offer low accuracy and convergence rates. Following the work [P. Panchal and R. Hiptmair, Electrostatic Force Computation with Boundary Element Methods, the SMAI journal of computational mathematics, 8 (2022), pp. 49–74], we derive a similar force expression starting from a floating potential problem for conducting bodies. The computations are done by employing the Virtual Work Principle using shape calculus and the adjoint method. The final expression is structurally simple and can be evaluated without explicitly computing the adjoint solution. It enjoys superior accuracy and convergence rates compared to standard formulas which is demonstrated by means of numerical experiments.
Piyush Panchal, Ralf Hiptmair
25 Years Computational Electromagnetics @ SCEE
Abstract
From the beginning, Computational Electromagnetics (CEM) was among the core themes of the conference series on Scientific Computing in Electrical Engineering (SCEE). This invited contribution to the 25th anniversary of the SCEE sheds light on some selected highlights in Computational Electromagnetics as presented during the past 25 years. In this context, CEM comprises different challenges to applied mathematics, such as the treatment of partial differential equations or the numerical solution of linear systems.
After some overview of the number of CEM contributions over the years and the type of discretisation methods employed, one example contribution is described for each edition of the conference series. Typically, these were invited papers.
Ursula van Rienen

Mathematical and Computational Methods

Frontmatter
Machine Learning Techniques to Model Highly Nonlinear Multi-field Dynamics
Abstract
Modelling the dynamics of the membrane displacement in a micromachined beam fixed at both ends for different applied voltages is important for real applications. The strong nonlinearities involved and the interaction between multiple physical fields make this task challenging for classical modelling and model reduction approaches. In this work we search for a simplified, yet accurate, data-driven models, based on different recurrent neural network architectures, using only peripheral input-output information of the original system. The main goal is to find the most suitable neural network architecture having the smallest number of hidden units that provides low error of the minimum gap dynamics for different applied voltages. We show that these black-box models, with only 4 hidden units, are able to accurately reproduce the original system’s response to a variety of different stimuli, and a strategy to make them parameter aware is proposed.
Ruxandra Barbulescu, Gabriela Ciuprina, Anton Duca, L. Miguel Silveira
Port-Hamiltonian Systems’ Modelling in Electrical Engineering
Abstract
The port-Hamiltonian (pH) modelling framework allows for models that preserve essential physical properties such as energy conservation or dissipative inequalities. If all subsystems are modelled as pH systems and the inputs are related to the output in a linear manner, the overall system can be modelled as a pH system, too, which preserves the properties of the underlying subsystems. If the coupling is given by a skew-symmetric matrix, as usual in many applications, the overall system can be easily derived from the subsystems without the need of introducing dummy variables and therefore artificially increasing the complexity of the system. Hence the framework of pH systems is especially suitable for modelling multiphysical systems.
In this paper, we show that pH systems are a natural generalization of Hamiltonian systems, define coupled pH systems as ordinary and differential-algebraic equations. To highlight the suitability for electrical engineering applications, we derive pH models for MNA network equations, electromagnetic devices and coupled systems thereof.
Andreas Bartel, Markus Clemens, Michael Günther, Birgit Jacob, Timo Reis
Large-Scale Optimization for Thermo-Mechanical Reliability of Electronics
Abstract
Optimization of transient models is required in several domains related to thermo-mechanical reliability of electronics, such as Prognostic Health Monitoring (PHM) and design optimization. A novel framework for efficient (local) parameter optimization of transient models in the \(\mathscr {H}_2\) norm is proposed. The optimization is feasible for large-scale transient models because it approximates the gradient using physics-based model order reduction (MOR), in contrast to existing approaches that typically use data-driven surrogate models such as neural networks. To demonstrate the framework an optimal fixed-order virtual sensor for PHM of a Ball Grid Array (BGA) is numerically determined.
Pascal den Boef, Jos Maubach, Wil Schilders, Nathan van de Wouw
Data-Driven Model Order Reduction of Parameterized Dissipative Linear Time-Invariant Systems
Abstract
We introduce a framework for data-driven model order reduction of parameterized LTI systems with guaranteed uniform dissipativity. The strategy casts the problem into a multivariate rational fitting scheme that formally preserves the bounded realness of the model response. The formulation relies on the solution of a semi-definite program arising from a rational parameterization based on Bernstein polynomials. The models can be employed in system-level simulations both in the frequency and time domain.
Tommaso Bradde, Alessandro Zanco, Stefano Grivet-Talocia
Splitting Methods for Linear Coupled Field-Circuit DAEs
Abstract
The application of operator splitting methods to ordinary differential equations (ODEs) is well established. However, for differential-algebraic equations (DAEs) it is subjected to many restrictions due to the presence of (possibly hidden) constraints. In order to get convergence of the operator splitting for DAEs, it is important to have and exploit a suitable decoupled structure for the desired DAE system. Here we present a coupled field-circuit modeling via a loop-cutset analysis and the choice of a suitable tree that results in a port-Hamiltonian DAE system. Finally, we introduce an operator splitting approach of such linear coupled field-circuit DAEs and present convergence results for the proposed approach.
Malak Diab, Caren Tischendorf
Structure-Preserving Identification of Port-Hamiltonian Systems—A Sensitivity-Based Approach
Abstract
We present a gradient-based calibration algorithm to identify a port-Hamiltonian system from given time-domain input-output data. The gradient is computed with the help of sensitivities and the algorithm is tailored such that the structure of the system matrices of the port-Hamiltonian system (skew-symmetry and positive semi-definitness) is preserved in each iteration of the algorithm. As we only require input-output data, we need to calibrate the initial condition of the internal state of the port-Hamiltonian system as well. Numerical results with synthetic data show the feasibility of the approach.
Michael Günther, Birgit Jacob, Claudia Totzeck
BG Approximations of Multiphysics pH Distributed Systems with Finite Number of Ports
Abstract
This paper proposes a procedure for the modeling of linear passive devices with distributed parameters as Hamiltonian systems with a finite number of ports, in the view of their coupling with external systems with lumped parameters (circuits). To obtain this particular Dirac structure, appropriate boundary conditions (BC) are used for the PDEs of several physical fields. Originally, they are Electric Circuit Element BC, here generalized for multidisciplinary fields such as elastic solids, acoustic and thermal devices Their internal field is discretized by the Finite Element Method, thus obtaining the stiffness, damping and mass matrices of a second order ODEs system, transformed then into a first order pH canonical form, having as interaction variables the flow and effort of each terminal.
Daniel Ioan, Gabriela Ciuprina
Bilinear Realization from I/O Data with NNs
Abstract
We present a method that connects a well-established nonlinear (bilinear) identification method from data in the time domain with the advantages of neural networks (NNs). The main challenge for fitting bilinear systems is the accurate recovery of the corresponding Markov parameters from the input and output measurements. Afterward, a realization algorithm similar to that proposed by Isidori can be employed. The novel step is that NNs are used here as a surrogate data simulator to construct input-output (i/o) data sequences from a single experiment. Then, classical realization theory is used to build an interpretable bilinear model that can further optimize engineering processes through robust simulations and control design.
D. S. Karachalios, I. V. Gosea, K. Kour, A. C. Antoulas
Coupling FMUs to Electric Circuits in Multiphysical System Simulation Software for the Development of Electric Vehicles
Abstract
This work is devoted to the analysis of electric circuits stemming from automated modeling processes in system simulation software. Modern applications such as HEV (hybrid electric vehicle), BEV (battery electric vehicle) and FCEV (fuel cell electric vehicle) require not only couplings of electric networks with mechanical, thermal, fluid and gas systems. In many cases it is necessary to extend or control the physics with grey box models like FMUs (Functional Mock-up Units). In particular, the coupling of electric systems with FMUs can be done on various levels (model exchange, co-simulation) via different interfaces (controller, electric, electric-thermal) and is therefore a challenging task. In this work we concentrate on a co-simulation approch for an electric-thermal coupling in a BEV model.
Michael Kolmbauer, Günter Offner, Ralf Uwe Pfau, Bernhard Pöchtrager
Battery Module Simulation Based on Model Exchange FMU Cell Models and Its Application in Multi-physical System Simulation Software
Abstract
Over the past years, the importance of battery development has increased significantly and will increase further. Consequently, the requirements for the system simulation of electric networks are rising. In modern simulation applications, the battery setup and battery modules can be simulated with different levels of complexity and can be coupled with detailed powertrain and cooling, yielding a multi-physcial problem. For some detailed investigations, e.g. about the thermal behavior, it is needed that in the electrical subsystem one can model and simulate detailed cells. The physical models of the cells can be given via grey-box models such as FMUs (functional mock-up units) (https://​fmi-standard.​org/​). Simulating then combined battery cells increases the complexity of the model. In this case, care must be taken to ensure that the simulation delivers stable results and that the computing time is kept as short as possible. Here we use a hierarchical approach to solve the resulting equations for the electrical system.
Michael Kolmbauer, Günter Offner, Ralf Uwe Pfau, Bernhard Pöchtrager
Sensitivity Analysis of Random Linear Dynamical Models Using System Norms
Abstract
We consider linear dynamical systems with a single output, where the systems include random parameters to perform an uncertainty quantification. Using the concept of polynomial chaos, a linear stochastic Galerkin system of higher dimension with multiple outputs is arranged. Quadratic combinations of the outputs yield approximations of time-dependent indices in global sensitivity analysis, which indicate the influence of each random parameter. We investigate system norms for the quadratic outputs, because these norms generate time-independent sensitivity measures. Numerical results are presented for a model of an electric circuit.
Roland Pulch
Compact Modelling of Wafer Level Chip-Scale Package via Parametric Model Order Reduction
Abstract
The interconnect reliability of a packaged chip on the printed circuit board is a major requirement that should be met for assembling microelectronics. The solder connection fatigue is one of the main failure modes. It is caused by the mechanical stress due to thermal expansion. In this work, the finite element model of a package on the printed circuit board is built and the solder joint analysis is performed within a thermo-mechanical simulation. For efficient studies of the temperature impact on the solder joints, we present a successful application of parametric model order reduction for constructing a compact model starting from the full order finite element model. Temperature dependent Young’s modulus, a parameter, which appears on both the left-hand and the right-hand side of the spatially discretized model, is preserved in the symbolic form within this compact model.
Ibrahim Zawra, Jeroen Zaal, Michiel van Soestbergen, Torsten Hauck, Evgeny Rudnyi, Tamara Bechtold
Backmatter
Metadaten
Titel
Scientific Computing in Electrical Engineering
herausgegeben von
Martijn van Beurden
Neil V. Budko
Gabriela Ciuprina
Wil Schilders
Harshit Bansal
Ruxandra Barbulescu
Copyright-Jahr
2024
Electronic ISBN
978-3-031-54517-7
Print ISBN
978-3-031-54516-0
DOI
https://doi.org/10.1007/978-3-031-54517-7

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