# American Institute of Mathematical Sciences

March  2015, 20(2): 323-337. doi: 10.3934/dcdsb.2015.20.323

## An immersed interface method for Pennes bioheat transfer equation

 1 Department of Mathematics, Miami University, Middletown OH, 45042, United States 2 Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43402-0221

Received  October 2013 Revised  May 2014 Published  January 2015

We consider an immersed finite element method for solving one dimensional Pennes bioheat transfer equation with discontinuous coefficients and nonhomogenous flux jump condition. Convergence properties of the semidiscrete and fully discrete schemes are investigated in the $L^{2}$ and energy norms. By using the computed solution from the immerse finite element method, an inexpensive and effective flux recovery technique is employed to approximate flux over the whole domain. Optimal order convergence is proved for the immersed finite element approximation and its flux. Results of the simulation confirm the convergence analysis.
Citation: Champike Attanayake, So-Hsiang Chou. An immersed interface method for Pennes bioheat transfer equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 323-337. doi: 10.3934/dcdsb.2015.20.323
##### References:
 [1] S. H. Chou and S. Tang, Conservative P1 conforming and nonconforming Galerkin FEMs: effective flux evaluation via a nonmixed method approach, SIAM J. Numer. Anal., 38 (2000), 660-680. doi: 10.1137/S0036142999361517. [2] S. H. Chou, An immersed linear finite element method with interface flux capturing recovery, Discrete and Continuous Dynamical Systems-Series-B, 17 (2012), 2343-2357. doi: 10.3934/dcdsb.2012.17.2343. [3] W. Dai, H. Yu and R. Nassar, A forth order compact finite-difference scheme for solving a 1-D Pennes bioheat transfer equation in a tripple layered skin structure, Numerical Heat Transfer, 46 (2004), 447-461. [4] W. Dai, H. Yu and R. Nassar, Optimal temperature distribution in a three dimensional triple layered skin structure embedded with artery and vein vasculature, Num. Heat Transfer, 50 (2006), 809-834. [5] Z. S. Deng and J. Liu, Mathematical modeling of temperature mapping over skin surface and its implementation in thermal disease diagnostics, Comput. Biol. Med., 34 (2004), 495-521. doi: 10.1016/S0010-4825(03)00086-6. [6] X. He, T. Lin and Y. Lin, Immersed finite element methods for elliptic interface problems with non-Homogeneous jump conditions, Inter. J. Numerical Analysis and Modeling, 8 (2011), 284-301. [7] S. C. Jiang, N. Ma and H. J. Li, Effects of thermal properties and geometrical dimensions on skin burn injuries, Burns, 28 (2002), 713-717. doi: 10.1016/S0305-4179(02)00104-3. [8] Z. Li, The immersed interface method using a finite element formulation, Applied Numerical Mathematics, 27 (1998), 253-267. doi: 10.1016/S0168-9274(98)00015-4. [9] Z. Li, T. Lin, Y. Lin and R. C. Rogers, An immersed finite element space and its approximation capability, Numer. Methods. Partial Differential Equations, 20 (2004), 338-367. doi: 10.1002/num.10092. [10] Z. Li, T. Lin and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math., 96 (2003), 61-98. doi: 10.1007/s00211-003-0473-x. [11] T. Lin, Y. Lin and W. Sun, Error estimation of a class quadratic immersed finite element methods for elliptic interface problems, Discrete and Continuous Dynamical Systems Series-B, 7 (2007), 807-823. doi: 10.3934/dcdsb.2007.7.807. [12] E. H. Liu, G. M. Saidel and H. Harasaki, Model analysis of tissue responses totransient and chronic heating, Ann. Biomed. Eng, 31 (2003), 1007-1048. [13] H. H. Pennes, Analysis of tissue and arterial blood temperature in the resting forearm, J. Appl. Physiol., 1 (1948), 93-122. [14] V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Springer, 2006. [15] D. A. Tori and J. D. Dale, A finite element model of skin subjected to a flasf fire, J. Biomed Eng., 116 (1994), 250-255.

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##### References:
 [1] S. H. Chou and S. Tang, Conservative P1 conforming and nonconforming Galerkin FEMs: effective flux evaluation via a nonmixed method approach, SIAM J. Numer. Anal., 38 (2000), 660-680. doi: 10.1137/S0036142999361517. [2] S. H. Chou, An immersed linear finite element method with interface flux capturing recovery, Discrete and Continuous Dynamical Systems-Series-B, 17 (2012), 2343-2357. doi: 10.3934/dcdsb.2012.17.2343. [3] W. Dai, H. Yu and R. Nassar, A forth order compact finite-difference scheme for solving a 1-D Pennes bioheat transfer equation in a tripple layered skin structure, Numerical Heat Transfer, 46 (2004), 447-461. [4] W. Dai, H. Yu and R. Nassar, Optimal temperature distribution in a three dimensional triple layered skin structure embedded with artery and vein vasculature, Num. Heat Transfer, 50 (2006), 809-834. [5] Z. S. Deng and J. Liu, Mathematical modeling of temperature mapping over skin surface and its implementation in thermal disease diagnostics, Comput. Biol. Med., 34 (2004), 495-521. doi: 10.1016/S0010-4825(03)00086-6. [6] X. He, T. Lin and Y. Lin, Immersed finite element methods for elliptic interface problems with non-Homogeneous jump conditions, Inter. J. Numerical Analysis and Modeling, 8 (2011), 284-301. [7] S. C. Jiang, N. Ma and H. J. Li, Effects of thermal properties and geometrical dimensions on skin burn injuries, Burns, 28 (2002), 713-717. doi: 10.1016/S0305-4179(02)00104-3. [8] Z. Li, The immersed interface method using a finite element formulation, Applied Numerical Mathematics, 27 (1998), 253-267. doi: 10.1016/S0168-9274(98)00015-4. [9] Z. Li, T. Lin, Y. Lin and R. C. Rogers, An immersed finite element space and its approximation capability, Numer. Methods. Partial Differential Equations, 20 (2004), 338-367. doi: 10.1002/num.10092. [10] Z. Li, T. Lin and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math., 96 (2003), 61-98. doi: 10.1007/s00211-003-0473-x. [11] T. Lin, Y. Lin and W. Sun, Error estimation of a class quadratic immersed finite element methods for elliptic interface problems, Discrete and Continuous Dynamical Systems Series-B, 7 (2007), 807-823. doi: 10.3934/dcdsb.2007.7.807. [12] E. H. Liu, G. M. Saidel and H. Harasaki, Model analysis of tissue responses totransient and chronic heating, Ann. Biomed. Eng, 31 (2003), 1007-1048. [13] H. H. Pennes, Analysis of tissue and arterial blood temperature in the resting forearm, J. Appl. Physiol., 1 (1948), 93-122. [14] V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Springer, 2006. [15] D. A. Tori and J. D. Dale, A finite element model of skin subjected to a flasf fire, J. Biomed Eng., 116 (1994), 250-255.
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