Error Estimators for Proper Generalized Decomposition in Time-Dependent Electromagnetic Field Problems

Müller, Fabian; Henneron, Thomas; Clénet, S.; Hameyer, Kay

doi:10.1109/tmag.2019.2949094

Cited by 5 publications

(15 citation statements)

References 12 publications

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“…The methods of the POD and the PGD are based upon the theory that unknown potentials can be decomposed into products of functions, which depend on a spatial or temporal variable as shown in equation ( 5) (Henneron and Clenet, 2013), (Nouy, 2010), (Chinesta et al, 2011), (Mueller et al, 2019a(Mueller et al, , 2019b. In equation ( 5), T e , as well as X, are each represented by a vector U, and a finite amount m of terms, which are also frequently referred to as modes (Mueller et al, 2019a(Mueller et al, , 2019b.…”

Section: Model Order Reductionmentioning

confidence: 99%

“…The PGD is an a priori approach to achieve a separated form [equation ( 5)]. To use the general methodology developed in Nouy (2010), Chinesta et al (2011) and Chinesta et al (2014) to the T-X-formulation, the two unknowns T e and X must be substituted by the approach outlined in equation ( 5) (Mueller et al, 2019a(Mueller et al, , 2019b. In case of the weak formulation, the basic functions are substituted by equations ( 11) and ( 12).…”

Section: Proper Generalized Decompositionmentioning

confidence: 99%

“…The number of modes (NoM) is increased until the complete enrichment is converged or a maximum NoM (maxMode) are met. The convergence of the complete enrichment process is given by assessing the information content, which is enriched by the new mode compared to the previous enriched modes (Mueller et al, 2019a(Mueller et al, , 2019b. COMPEL 39,5…”

Section: Model Order Reduction Techniquesmentioning

confidence: 99%

“…It only has to solve NoM•maxNL equation systems in the offline stage and consequently NoT•NoM multiplications in the online stage. For an even larger reduction, the decomposition can be improved leading to a better accuracy with less modes, as well as a better nonlinear convergence in the enrichment process by orthogonalization of the basis (Chinesta et al, 2011), (Henneron and Clenet, 2016), (Mueller et al, 2019a(Mueller et al, , 2019b. The POD nonetheless has fewer operations in the offline stage for sequential snapshot method and more operations in the online stage because the reference load vector has to be built and four Model order reduction techniques projection operations are necessary.…”

Section: Comparison Of Proper Orthogonal Decomposition and Proper Generalized Decompositionmentioning

confidence: 99%

“…The Proper Generalized Decomposition (PGD) originates from the field of mechanical and fluid-dynamic problems (Nouy, 2010) (Chinesta et al, 2011). It has shown its usefulness in facing numerous numerical challenges (Nouy, 2010), (Chinesta et al, 2011), (Chinesta et al, 2014), (Qin et al, 2016a(Qin et al, , 2016b and has recently been applied to electromagnetic field computation in context with the magnetic vector potential formulation, inter alia, by Henneron and Clenet (2013), Qin et al (2016aQin et al ( , 2016b and Mueller et al (2019aMueller et al ( , 2019b. Owing to the advantages in terms of eddy current computation, using the PGD to the T-X method is yet to be done, and a first step toward fast simulation of eddy current problems such as nondestructive eddy current testing.…”

Section: Introductionmentioning

confidence: 99%

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

Müller

Crampen

Henneron

et al. 2020

COMPEL

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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.

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Section: Model Order Reductionmentioning

confidence: 99%

Section: Proper Generalized Decompositionmentioning

confidence: 99%

Section: Model Order Reduction Techniquesmentioning

confidence: 99%

Section: Comparison Of Proper Orthogonal Decomposition and Proper Generalized Decompositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Müller

Crampen

Henneron

et al. 2020

COMPEL

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Solving diffusive equations by proper generalized decomposition with preconditioner

Tang¹,

Guan²,

Liu³

2023

Comput Mech

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A hybrid enrichment strategy for the efficient construction of reduced order models by the proper generalized decomposition

Müller

Baumanns

Hameyer

2023

COMPEL

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Purpose The calculation of electromagnetic fields can involve many degrees of freedom (DOFs) to achieve accurate results. The DOFs are directly related to the computational effort of the simulation. The effort is decreased by using the proper generalized decomposition (PGD) and proper orthogonalized decomposition (POD). The purpose of this study is to combine the advantages of both methods. Therefore, a hybrid enrichment strategy is proposed and applied to different electromagnetic formulations. Design/methodology/approach The POD is an a-priori method, which exploits the solution space by decomposing reference solutions of the field problem. The disadvantage of this method is given by the unknown number of solutions necessary to reconstruct an accurate field representation. The PGD is an a-priori approach, which does not rely on reference solutions, but require much more computational effort than the POD. A hybrid enrichment strategy is proposed, based on building a small POD model and using it as a starting point of the PGD enrichment process. Findings The hybrid enrichment process is able to accurately approximate the reference system with a smaller computational effort compared to POD and PGD models. The hybrid enrichment process can be combined with the magneto-dynamic T-Ω formulation and the magnetic vector potential formulation to solve eddy current or non-linear problems. Originality/value The PGD enrichment process is improved by exploiting a POD. A linear eddy current problem and a non-linear electrical machine simulation are analyzed in terms of accuracy and computational effort. Further the PGD-AV formulation is derived and compared to the PGD-T-Ω reduced order model.

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Error Estimators for Proper Generalized Decomposition in Time-Dependent Electromagnetic Field Problems

Cited by 5 publications

References 12 publications

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

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

Solving diffusive equations by proper generalized decomposition with preconditioner

A hybrid enrichment strategy for the efficient construction of reduced order models by the proper generalized decomposition

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