scholarly journals Interpolatory model reduction for quadratic-bilinear systems using error estimators

2018 ◽  
Vol 36 (1) ◽  
pp. 25-44 ◽  
Author(s):  
Mian Ilyas Ahmad ◽  
Peter Benner ◽  
Lihong Feng
Keyword(s):  
Model Reduction ◽  
Error Bound ◽  
Model Order Reduction ◽  
Computational Cost ◽  
Order Reduction ◽  
Bilinear Systems ◽  
Descriptor System ◽  
Model Order ◽  
Content Type ◽  
Projection Matrices

Purpose The purpose of this paper is to propose an interpolation-based projection framework for model reduction of quadratic-bilinear systems. The approach constructs projection matrices from the bilinear part of the original quadratic-bilinear descriptor system and uses these matrices to project the original system. Design/methodology/approach The projection matrices are constructed by viewing the bilinear system as a linear parametric system, where the input associated with the bilinear part is treated as a parameter. The advantage of this approach is that the projection matrices can be constructed reliably by using an a posteriori error bound for linear parametric systems. The use of the error bound allows us to select a good choice of interpolation points and parameter samples for the construction of the projection matrices by using a greedy-type framework. Findings The results are compared with the standard quadratic-bilinear projection methods and it is observed that the approximations through the proposed method are comparable to the standard method but at a lower computational cost (offline time). Originality/value In addition to the proposed model order reduction framework, the authors extend the one-sided moment matching parametric model order reduction (PMOR) method to a two-sided method that doubles the number of moments matched in the PMOR method.

Nonlinear model order reduction for the fast solution of induction heating problems in time-domain

2017 ◽  
Vol 36 (2) ◽  
pp. 469-475 ◽  
Author(s):  
Lorenzo Codecasa ◽  
Federico Moro ◽  
Piergiorgio Alotto
Keyword(s):  
Induction Heating ◽  
Model Order Reduction ◽  
Computing Time ◽  
Computational Cost ◽  
Order Reduction ◽  
Reduced Model ◽  
Problem Solution ◽  
Model Order ◽  
Content Type ◽  
Time Harmonic

Purpose This paper aims to propose a fast and accurate simulation of large-scale induction heating problems by using nonlinear reduced-order models. Design/methodology/approach A projection space for model order reduction (MOR) is quickly generated from the first kernels of Volterra’s series to the problem solution. The nonlinear reduced model can be solved with time-harmonic phasor approximation, as the nonlinear quadratic structure of the full problem is preserved by the projection. Findings The solution of induction heating problems is still computationally expensive, even with a time-harmonic eddy current approximation. Numerical results show that the construction of the nonlinear reduced model has a computational cost which is orders of magnitude smaller than that required for the solution of the full problem. Research limitations/implications Only linear magnetic materials are considered in the present formulation. Practical implications The proposed MOR approach is suitable for the solution of industrial problems with a computing time which is orders of magnitude smaller than that required for the full unreduced problem, solved by traditional discretization methods such as finite element method. Originality/value The most common technique for MOR is the proper orthogonal decomposition. It requires solving the full nonlinear problem several times. The present MOR approach can be built directly at a negligible computational cost instead. From the reduced model, magnetic and temperature fields can be accurately reconstructed in whole time and space domains.

Model of induction brazing of nonmagnetic metals using model order reduction approach

2018 ◽  
Vol 37 (4) ◽  
pp. 1515-1524 ◽  
Author(s):  
Pavel Karban ◽  
David Pánek ◽  
Ivo Doležel
Keyword(s):  
Heat Treatment ◽  
Model Order Reduction ◽  
Computation Time ◽  
Order Reduction ◽  
Nonlinear Problems ◽  
Model Order ◽  
Content Type ◽  
Weakly Nonlinear ◽  
Induction Brazing ◽  
Multiphysics Problems

Purpose A novel technique for control of complex physical processes based on the solution of their sufficiently accurate models is presented. The technique works with the model order reduction (MOR), which significantly accelerates the solution at a still acceptable uncertainty. Its advantages are illustrated with an example of induction brazing. Design/methodology/approach The complete mathematical model of the above heat treatment process is presented. Considering all relevant nonlinearities, the numerical model is reduced using the orthogonal decomposition and solved by the finite element method (FEM). It is cheap compared with classical FEM. Findings The proposed technique is applicable in a wide variety of linear and weakly nonlinear problems and exhibits a good degree of robustness and reliability. Research limitations/implications The quality of obtained results strongly depends on the temperature dependencies of material properties and degree of nonlinearities involved. In case of multiphysics problems characterized by low nonlinearities, the results of solved problems differ only negligibly from those solved on the full model, but the computation time is lower by two and more orders. Yet, however, application of the technique in problems with stronger nonlinearities was not fully evaluated. Practical implications The presented model and methodology of its solution may represent a basis for design of complex technologies connected with induction-based heat treatment of metal materials. Originality/value Proposal of a sophisticated methodology for solution of complex multiphysics problems established the MOR technology that significantly accelerates their solution at still acceptable errors.

Model order reduction of nonlinear eddy-current field using parameterized CLN

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Miwa Tobita ◽  
Hamed Eskandari ◽  
Tetsuji Matsuo
Keyword(s):  
Model Order Reduction ◽  
Dynamical Behavior ◽  
Order Reduction ◽  
Parameter Variation ◽  
Computational Time ◽  
State Equations ◽  
Model Order ◽  
Content Type ◽  
Parameter Variations ◽  
Nonlinear State

Purpose The authors derive a nonlinear MOR based on the Cauer ladder network (CLN) representation, which serves as an application of the parameterized MOR. Two parametrized CLN representations were developed to handle the nonlinear magnetic field. Simulations using the parameterized CLN were also conducted using an iron-cored inductor model under the first-order approximation. Design/methodology/approach This work studies the effect of parameter variations on reduced systems and aims at developing a general formulation for parametrized model order reduction (MOR) methods with the dynamical transition of parameterized state. Findings Terms including time derivatives of basis vectors appear in nonlinear state equations, in addition to the linear network equations of the CLN method. The terms are newly derived by an exact formulation of the parameterized CLN and are named parameter variation terms in this study. According to the simulation results, the parameter variation terms play a significant role in the nonlinear state equations when reluctivity is used, while they can be neglected when differential reluctivity is used. Practical implications The computational time of nonlinear transient analyses can be greatly reduced by applying the parameterized CLN when the number of time steps is large. Originality/value The authors introduced a general representation for the dynamical behavior of the reduced system with time-varying parameters, which has not been theoretically discussed in previous studies. The effect of the parameter variations is numerically given as a form of parameter variation terms by the exact derivation of the nonlinear state equations. The influence of parameter variation terms was confirmed by simulation.

Model order reduction techniques applied to magnetodynamic T-Ω-formulation

2020 ◽  
Vol 39 (5) ◽  
pp. 1057-1069
Author(s):  
Fabian Müller ◽  
Lucas Crampen ◽  
Thomas Henneron ◽  
Stephane Clénet ◽  
Kay Hameyer
Keyword(s):  
Eddy Current ◽  
Model Order Reduction ◽  
Scalar Potential ◽  
Order Reduction ◽  
Computational Effort ◽  
Transient Field ◽  
Model Order ◽  
Field Problem ◽  
Content Type ◽  
Reduction Techniques

Purpose The purpose of this paper is to use different model order reduction techniques to cope with the computational effort of solving large systems of equations. By appropriate decomposition of the electromagnetic field problem, the number of degrees of freedom (DOF) can be efficiently reduced. In this contribution, the Proper Generalized Decomposition (PGD) and the Proper Orthogonal Decomposition (POD) are used in the frame of the T-Ω-formulation, and the feasibility is elaborated. Design/methodology/approach The POD and the PGD are two methods to reduce the model order. Particularly in the context of eddy current problems, conventional time-stepping algorithms can lead to many numerical simulations of the studied problem. To simulate the transient field, the T-Ω-formulation is used which couples the magnetic scalar potential and the electric vector potential. In this paper, both methods are studied on an academic example of an induction furnace in terms of accuracy and computational effort. Findings Using the proposed reduction techniques significantly reduces the DOF and subsequently the computational effort. Further, the feasibility of the combination of both methods with the T-Ω-formulation is given, and a fundamental step toward fast simulation of eddy current problems is shown. Originality/value In this paper, the PGD is combined for the first time with the T-Ω-formulation. The application of the PGD and POD and the following comparison illustrate the great potential of these techniques in combination with the T-Ω-formulation in context of eddy current problems.

scholarly journals Model order reduction for gas and energy networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Christian Himpe ◽  
Sara Grundel ◽  
Peter Benner
Keyword(s):  
Model Reduction ◽  
Model Order Reduction ◽  
Gas Transport ◽  
Order Reduction ◽  
Partial Differential ◽  
Model Order ◽  
Vast Number ◽  
Gas Network ◽  
Gas Networks ◽  
Energy Networks

AbstractTo counter the volatile nature of renewable energy sources, gas networks take a vital role. But, to ensure fulfillment of contracts under these circumstances, a vast number of possible scenarios, incorporating uncertain supply and demand, has to be simulated ahead of time. This many-query gas network simulation task can be accelerated by model reduction, yet, large-scale, nonlinear, parametric, hyperbolic partial differential(-algebraic) equation systems, modeling natural gas transport, are a challenging application for model order reduction algorithms.For this industrial application, we bring together the scientific computing topics of: mathematical modeling of gas transport networks, numerical simulation of hyperbolic partial differential equation, and parametric model reduction for nonlinear systems. This research resulted in the (Model Order Reduction for Gas and Energy Networks) software platform, which enables modular testing of various combinations of models, solvers, and model reduction methods. In this work we present the theoretical background on systemic modeling and structured, data-driven, system-theoretic model reduction for gas networks, as well as the implementation of and associated numerical experiments testing model reduction adapted to gas network models.

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