# Reduction in the Computational Complexity of Calculating Losses on Eddy Currents in a Hydrogen Fuel Cell Using the Finite Element Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geometric and Finite Element Models

^{2}space objects was generated using the MeshAdapt algorithm [8], and the finite element mesh of the R

^{3}space objects was generated using the HXT algorithm [9].

## 3. Mathematical Model

_{Ω}− (J

_{S}, A′)

_{ΩS}+ (n × H

_{S}, A′)

_{ΓΩ}= 0,

_{Ω}= 0, Γ

_{Ω}∉ Ω,

_{ΓΩ}= 0,

_{S}are Neumann boundary conditions along the boundaries Γ

_{Ω}, Ω is a computational domain, and w is a vector field that has no closed lines (applied to the cotree elements).

_{Ω}− (J

_{S}, A′)

_{ΩS}+ (n × H

_{S}, A′)

_{ΓΩ}+ (σ∂

_{t}A, A′)

_{Ωσ}+ (σgradV, A′)

_{Ωσ}= 0,

(σ∂

_{t}A, gradV′)

_{Ωσ}+ (σgradV, gradV′)

_{Ωσ}= ∑ I

_{i}U

_{i}(V′),

_{σ}is a volume domain of electrically conductive materials, V is a scalar field potential within the domain functional space Ω

_{σ}, σ is an electrical conductivity, and I

_{i}and U

_{i}are the current and voltage in each unit circuit of a massive conductor (in this case, the conductive volume domain).

^{2}, and for the corrugated electrode, it was 30,500 A/m

^{2}(conductive pad size 0.3 × 50 mm, 57 conductive pads per electrode). In the finite element mesh and mathematical model for the domains of conducting bodies, thin layers (boundary layers) were not distinguished since the electrode thickness of 0.16 mm did not exceed half the depth of the skin effect [16], 0.396 mm, at a frequency of 100 kHz.

_{e}or domain boundaries of the computational domain Ω without considering the common boundaries with the domain of the conducting materials Ω

_{σ}:

_{ΓΩ}= 0, (A·w)

_{Ωe}= 0, Γ

_{Ωσ}∉ Ω

_{e},

_{ΓΩ}= 0, (Γ

_{Ωσ}, Ω

_{σ}) ∉ Ω.

_{σ}, for which an additional nodal functional space was introduced:

_{r})

_{ΓΩσ}= 0,

_{r}is the voltage potential, and Γ

_{Ωσ}are the boundaries of the domain of conductive materials.

_{t}A + dgradV‖

^{2}.

_{1}and ϖ

_{2}[22], which are computed on the basis of the scaled residual ϖ

_{1i}and ϖ

_{2i}, is a reliable way to evaluate the accuracy of the computed solution:

_{1i}= |b − Ax|

_{i}/(|b| + |A| |x|)

_{i},

_{2i}= |b − Ax|

_{i}/(|A| |x|)

_{i}+ ||A

_{i}||

_{∞}+ ||x||

_{∞},

_{i}is a row i of the matrix A.

_{1i}was calculated for all equations except those with a non-zero numerator and a small value in the denominator. The scaled residual ϖ

_{2i}was calculated for such equations. As a matter of fact, the component-wise errors ϖ

_{1}and ϖ

_{2}were the largest values of the scaled residuals ϖ

_{1i}and ϖ

_{2i}, i.e., the upper limit.

_{∞}/‖x‖

_{∞}≤ ϖ

_{1}cond

_{1}+ ϖ

_{2}cond

_{2},

_{1}is a Manhattan condition number (1-norm) and cond

_{2}is a spectral condition number (2-norm).

^{6}can be considered reliable (Figure 7; the study was performed for a computational problem based on a mesh with 1.21 million tetrahedrons).

^{−5}or less, which reduced the computational complexity of the numerical problems (in terms of CPU time utilization) by 30–40%, as was shown in [25].

## 4. Conclusions

^{−5}, allows a significant reduction in the requirements for the amount of memory and processor time utilization of the computer. In the considered case, the use of second-order finite elements, the formulation of a computational problem with the tree–cotree gauge condition, machine arithmetic of double-precision complex numbers, and BLR-factorization together allowed the fast calculation of the eddy current losses in a hydrogen fuel cell using limited hardware resources: about 10–12 GB of memory and 65–70 s of processor time. The eddy current loss value of 0.52–0.53 W obtained as a result of computer modeling was not more than 3% of the nominal power of the hydrogen fuel cell, 20.5 W. Accordingly, during the consideration of single hydrogen fuel cells, it is not an obligatory condition to take into account the power of eddy currents in steady-state operation. Such an approach to solving a numerical problem will make it possible to obtain a result in a very short time even when using a weak computer, which is especially important when performing optimization simulations and studying large-area fuel cells and even full stacks of hydrogen cells. Easily and quickly obtaining the results of eddy current power calculations can help verify and refine analytical models [30,31].

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Pfeif, E.; Jones, Z.; Lasseigne, A.; Koenig, K.; Krzywosz, K.; Mader, E.; Yagnik, S.; Mishra, B.; Olson, D. Submerged Eddy Current Method of Hydrogen Content Evaluation of ZIRCALOY4 Fuel Cladding. AIP Conf. Proc.
**2011**, 1335, 1168. [Google Scholar] [CrossRef] - Felt, W. Electromagnetic Excitation Methods for Thermographic Defect Detection in Fuel Cell Materials; Technical report; National Renewable Energy Laboratory: Golden, CO, USA, 2012. [Google Scholar]
- Elia, G. Characterization of Voltage Loss for Proton Exchange Membrane Fuel Cell. Bachelor’s Thesis, Polytechnic University of Catalonia (UPC), Barcelona, Spain, 2015. [Google Scholar]
- Gou, B.; Na, W.; Diong, B. Linear and Nonlinear Models of Fuel Cell Dynamics. Dynamic Modeling and Control with Power Electronics Applications; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Hirschfeld, J. Tomographic Problems in the Diagnostics of Fuel Cell Stacks, PGI-1/IAS-1; Berichte des Forschungszentrums Jülich: Jülich, Germany, 2009. [Google Scholar]
- Burtsev, Y.A.; Pavlenko, A.V.; Vasyukov, I.V. A step-down converter of a power plant based on fuel cells for an unmanned aerial vehicle. Electr. Eng.
**2021**, 10, 81–85. [Google Scholar] - Geuzaine, C.; Remacle, J.-F. Gmsh: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng.
**2009**, 79, 1309–1331. [Google Scholar] [CrossRef] - Lu, Q.; Shephard, M.S.; Tendulkar, S.; Beall, M.W. Parallel mesh adaptation for high-order finite element methods with curved element geometry. Eng. Comput.
**2014**, 30, 271–286. [Google Scholar] [CrossRef] - Marot, C.; Pellerin, J.; Remacle, J.F. One machine, one minute, three billion tetrahedra. Int. J. Numer. Methods Eng.
**2018**, 117, 967–990. [Google Scholar] [CrossRef] - Bastos, J.; Sadowski, N. Electromagnetic Modelling by Finite Element Methods, 1st ed.; M. Dekker: New York, NY, USA, 2003. [Google Scholar]
- Geuzaine, C.; Remacle, J.-F. (Eds.) Gmsh Reference Manual; The documentation for Gmsh 4.8.4; Belgium, 2022. [Google Scholar]
- Dular, P.; Geuzaine, C.; Henrotte, F.; Legros, W. A general environment for the treatment of discrete problems and its application to the finite element method. IEEE Trans. Magn.
**1998**, 34, 3395–3398. [Google Scholar] [CrossRef] - Biro, O.; Paul, C.; Preis, K. “The use of a reduced vector potential A-r formulation for the calculation of iron induced field errors”, ROXIE: Routine for the Optimization of Magnet X-Sections, Inverse Field Calculation and Coil End Design. In Proceedings of the First International Roxie Users Meeting and Workshop CERN, Geneva, Switzerland, 16–18 March 1998; pp. 31–47. [Google Scholar]
- Biro, O. Edge element formulations of eddy current problems. Comput. Methods Appl. Mech. Eng.
**1999**, 169, 391–405. [Google Scholar] [CrossRef] - Geuzaine, C. High Order Hybrid Finite Element Schemes for Maxwell Equations Taking Thin Structures and Global Quantities into Account. Ph.D. Thesis, Université de Liège, Liège, Belgium, 2001; pp. 71–80. [Google Scholar]
- Krähenbühl, L.; Muller, D. Thin layers in electrical engineering. Example of shell models in analysing eddy-currents by boundary and finite element methods. IEEE Trans. Magn.
**1993**, 29, 1450–1455. [Google Scholar] [CrossRef] - Khoroshev, A.S.; Pavlenko, A.V.; Batishchev, D.V.; Puzin, V.S.; Shevchenko, E.V.; Bolshenko, I.A. Verification of the GMSH GetDP software package for finite element modeling of electromagnetic fields. Proc. High. Educ. Inst. North Cauc. Region. Ser. Tech. Sci.
**2013**, 6, 74–78. [Google Scholar] - Balay, S.; Abhyankar, S.; Adams, M.F.; Adams, M.-F.; Benson, S.; Brown, J.; Brune, P.; Buschelman, K.; Constantinescu, E.-M.; Dalcin, L.; et al. PETSc Web Page. 2020. Available online: http://www.mcs.anl.gov/petsc (accessed on 1 November 2022).
- Gregoire, R. Coupling MUMPS and Ordering Software; CERF ACS Report; WN/PA/02/24; CERFAC: Toulouse, France, 2002. [Google Scholar]
- Pellegrini, F. SCOTCH and LIBSCOTCH 5.1 User’s Guide; Technical Report; LaBRI, Université Bordeaux I: Talence, France, 2008; p. 128. [Google Scholar]
- Schulze, J. Towards a tighter coupling of bottom-up and top-down sparse matrix ordering methods. Comput. Sci. BIT Numer. Math.
**2001**, 41, 800–841. [Google Scholar] [CrossRef] - Arioli, M.; Demmel, J.; Duff, I.S. Solving sparse linear systems with sparse backward error. SIAM J. Matrix Anal. Appl.
**1989**, 10, 165–190. [Google Scholar] [CrossRef] - Belov, S.A.; Zolotykh, N.Y. Numerical Methods of Linear Algebra; Laboratory practice; Publishing House of the Nizhny Novgorod State University. N.I. Lobachevsky: Nizhny Novgorod, Russia, 2005; p. 235. [Google Scholar]
- MUltifrontal Massively Parallel Solver (MUMPS 5.4.0) User’s Guide; Mumps Technologies SAS; Laboratoire de l’informatique du Parallélisme: Lyon, France, 2021.
- Khoroshev, A.S.; Pavlenko, A.V.; Puzin, V.S.; Shchuchkin, D.A.; Batishchev, D.V.; Bolshenko, I.A.; Kramarov, A.S. The Influence of Gauge Conditions on Process and Result of Solving 3-D Problems of Computational Electromagnetism by the Finite Element Method. IEEE Trans. Magn.
**2022**, 58, 1–11. [Google Scholar] [CrossRef] - Saad, Y. Iterative methods for sparse linear systems. Soc. Ind. Appl. Math
**2003**, 541, 41–42. [Google Scholar] - Khoroshev, A.S. Modeling of nonlinear magnetic systems by the finite element method using BLR factorization. Proc. High. Educ. Institutions. Elektromekhanika
**2021**, 64, 14–19. [Google Scholar] [CrossRef] - Bebendorf, M. Hierarchical Matrices: A Means to Eciently Solve Elliptic Boundary Value Problems; Lect. Notes. Comput. Sci. Eng 63; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Amestoy, P.R.; Ashcraft, C.; Boiteau, O.; Buttari, A.; L’Excellent, J.-Y.; Weisbecker, C. Improving Multifrontal Methods by Means of Block Low-Rank Representations. SIAM J. Sci. Comput. Soc. Ind. Appl. Math.
**2015**, 37, 1451–1474. [Google Scholar] [CrossRef] - Dachuan, Y.; Yuvarajan, S. Electronic circuit model for proton exchange membrane fuel cells. J. Power Sources
**2005**, 142, 238–242. [Google Scholar] - Pavlenko, A.; Burtsev, Y.; Vasyukov, I.; Puzin, V.; Gummel, A. Equivalent Scheme of the Fuel Cell Taking into Account the Influence of Eddy Currents and A Practical Way to Determine Its Parameters. Inventions
**2022**, 7, 72. [Google Scholar] [CrossRef]

**Figure 1.**The geometric model of the fuel cell (1 and 3—electrodes, 2—membrane) and its finite element approximation.

**Figure 2.**The dependence of the number of nodes and tetrahedrons in the finite element mesh on the scaling factor.

**Figure 3.**Dependence of computational complexity (“CC”) and the necessary amount of memory (“Mem”) on the number of elements in the mesh for two variants of the vector magnetic potential gauging (“TCT”—tree–cotree, “COUL”—Coulomb).

**Figure 4.**Dependence of the calculated value of eddy current losses (power) on the number of tetrahedrons in the finite element mesh when using first- and second-order elements.

**Figure 5.**The computational complexity (“CC”) of the numerical problem and the necessary amount of memory (“Mem”) for its solution. The PETSc toolkit was configured to use double-precision real number arithmetic (“REAL”) and double-precision complex number arithmetic (“COMPLEX”).

**Figure 6.**Calculated values of the relative forward error of solving SLAE for meshes with different numbers of finite elements. The PETSc toolkit is configured to use the arithmetic of double-precision real numbers (“REAL”) and double-precision complex numbers (“COMPLEX”); data are given for the first- and second-order finite element meshes (“ORD1” and “ORD2”, respectively).

**Figure 7.**Dependence of the relative forward error value of the calculated SLAE solution and the calculated value of the eddy current power (losses) on the given tolerance of the low-rank SLAE approximation.

**Figure 8.**Dependence of computational complexity of the LU-factorization procedure, the amount of memory, and CPU time required to solve the whole numerical problem when applying low-rank approximation as a percentage of the corresponding value for full-rank factorization.

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**MDPI and ACS Style**

Khoroshev, A.; Vasyukov, I.; Pavlenko, A.; Batishchev, D.
Reduction in the Computational Complexity of Calculating Losses on Eddy Currents in a Hydrogen Fuel Cell Using the Finite Element Analysis. *Inventions* **2023**, *8*, 38.
https://doi.org/10.3390/inventions8010038

**AMA Style**

Khoroshev A, Vasyukov I, Pavlenko A, Batishchev D.
Reduction in the Computational Complexity of Calculating Losses on Eddy Currents in a Hydrogen Fuel Cell Using the Finite Element Analysis. *Inventions*. 2023; 8(1):38.
https://doi.org/10.3390/inventions8010038

**Chicago/Turabian Style**

Khoroshev, Artem, Ivan Vasyukov, Alexander Pavlenko, and Denis Batishchev.
2023. "Reduction in the Computational Complexity of Calculating Losses on Eddy Currents in a Hydrogen Fuel Cell Using the Finite Element Analysis" *Inventions* 8, no. 1: 38.
https://doi.org/10.3390/inventions8010038