Overview of Adaptive and Low-Rank Approximation Algorithms for Modeling of The Influence of Electromagnetic Waves Generated by The Cell Phone Antenna on The Human Head
This paper presents an overview of formulations and algorithms dedicated to modeling the influence of electromagnetic waves on the human head. We start from the three-dimensional MRI scan of the human head. We approximate the MRI scan by the continuous approximation span over three-dimensional h adaptive mesh with quadratic polynomials. Next, we introduce time-harmonic Maxwell equations with a 1.8 GHz cell-phone antenna. We solve the problem of the propagation of electromagnetic waves on the human head. We compute the specific absorption rate used as the heat source for the Pennes bioheat equation. Finally, we introduce the Pennes bio-heat equation modeling the heat generated by the electromagnetic waves propagating through the skull, tissue, and air layers in the human head. We discuss the discretization and time-stepping algorithm for the Pennes equation’s solution over the human head. Namely, we focus on the Crank-Nicolson time integration scheme, to solve the bioheat transfer equations. We employ the hp finite elements with hierarchical shape functions and hp adaptive algorithm in three-dimensions. We propose an adaptive algorithm mixed with time-stepping iterations, where we simultaneously adapt the computational mesh, solve the Maxwell and Pennes equations, and we iterative with time steps. We employ the sparse Gaussian elimination algorithm with low-rank compression of the off-diagonal matrix blocks for the factorization of matrices. We conclude with the statement that 15 minutes of talk with a 1.8 GHz antenna of 1 Wat power results in increased brain tissue temperature up to 38.4 Celsius degree.
Demkowicz, L., Kurtz, J., Pardo, D., Paszyński, M., Rachowicz, W., Zdunek, A. (2007). Computing with hp-Adaptive Finite Elements, Vol. II. Frontiers. Three Dimensional Elliptic and Maxwell Problems with Applications. Chapman and Hall/Crc Applied Mathematics and Nonlinear Science.
Hughues, T. (2000). The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Civil and Mechanical Engineering.
Pilat, A. (2004). FEMLab software applied to Active Magnetic Bearing analysis. International Journal of Applied Mathematics and Computer Science, 14(4), 497-501.
Kyunogjoo, K. (2013). Finite element modeling of the radiation and induced heat transfer in the human body. Ph.D. dissertation, The University of Texas at Austin.
Schaefer, R., Łoś, M., Sieniek, M., Demkowicz, L., Paszyński, M. (2015). Quasi-linear computational cost adaptive solvers for three dimensional modeling of heating of a human head induced by cell phone. Journal of Computational Science, 11, 163-174.
Garcia-Castillo L., Gomez-Revuelto I., Amor-Martin A., Łoś M., Paszyński M. (2017). Algorithm for simultaneous adaptation and time step iterations for the electromagnetic waves propagation and heating of the human head induced by cell phone. Procedia Computer Science, 108, 2448-2452.
Salazar-Palma, M., Sarkar, T., Garcia-Castillo, L., Roy, T., Djordjevic, A. (1998). Iterative and Self-Adaptive Finite-Elements in Electromagnetic Modeling. Artech House Publishers.
Garcia-Dooro, D. (2014). A new software suite for electromagnetics. Ph.D. dissertation, Universidad Carlos III de Madrid
Albers, B., Savidis, S., Tasan, E., Estorff, O., Gehlken, M. (2012). BEM and FEM results of displacements in a poroelastic column. Journal of Applied Mathematics and Computer Science, 22(4), 883896.
Duff, I., Reid, J. (1983). The Multifrontal Solution of Indefinite Sparse Symmetric Linear. ACM Trans. Math. Softw., 9(3), 302325.
Duff, I., Reid, J. (1984). The Multifrontal Solution of Unsymmetric Sets of Linear Equations. Journal on Scientific and Statistical Computing, 5, 633641.
Demkowicz, L. (2006). Computing with hp-Adaptive Finite Elements, Vol. I. One and Two Dimensional Elliptic and Maxwell Problems. Chapman and Hall/Crc Applied Mathematics and Nonlinear Science.
Geng, P., Oden, T., Geijn, R. (2006). A Parallel Multifrontal Algorithm and Its Implementation. Computer Methods in Applied Mechanics and Engineering, 149, 289301.
Paszyński, M. (2016). Fast Solvers for Mesh Based Computations. Taylor & Francis, CRC Press.
Amestoy, P., Buttari, A., L’Excellent, J.Y., Mary, T. (2019). Performance and Scalability of the Block Low-Rank Multifrontal Factorization on Multicore Architectures. ACM Trans. Math. Softw., 45(1).
W. Hackbush, Hierarchical matrices: Algorithms and Applications, (2015) Springer
C.-P. Jeannerod, T. Mary, C. Pernet, D. Roch, (2019) Improving the Complexity of Block Low-Rank Factorizations with Fast Matrix Arithmetic, SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 40(4) 1478-
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