Evaluation of a radial basis function node refinement algorithm applied to bioheat transfer modeling

Rafael Pinheiro Amantéa; Robspierre de Carvalho; Luiz Otávio Barbosa; Rafael Pinheiro Amantéa; Robspierre de Carvalho; Luiz Otávio Barbosa

doi:10.3934/bioeng.2021005

AIMS Bioengineering

2021, Volume 8, Issue 1: 36-51. doi: 10.3934/bioeng.2021005

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Research article

Evaluation of a radial basis function node refinement algorithm applied to bioheat transfer modeling

Department of Engineering and Management of Process and Systems, IETEC – Institute of Technological Education, Belo Horizonte, Minas Gerais, 30140-138, Brazil.

Received: 30 October 2020 Accepted: 17 December 2020 Published: 23 December 2020

Many bioheat transfer problems involve linear/non-linear equations with non-linear or time-dependent boundary conditions. For heat transfer problems, the presence of time and space-dependent functions under Neumann and Mixed type boundary conditions characterize trivial applications in bioengineering, such as thermotherapies, laser surgeries, and burn studies. This greatly increases the complexity of the numerical solution in several problems, requiring fast and accurate numerical solutions. This paper has a main objective evaluate an adaptive mesh refinement radial basis function method strategy for the classical Penne's bioheat transfer modeling. Our numerical results had errors of ~0.1% compared to analytical solutions. Thus, the proposed methodology is accurate and has a low computational cost. For step function heating, two RBF shape parameters were applied, again achieving excellent results. The distributions of the nodes in the solution domain show that the primary source of error in the numerical solutions came from the boundary conditions. This finding should arouse the interest of engineers and scientists in the development of new strategies for problems involving boundary conditions with periodic functions.
- Radial Basis Function,
- Spatial Adaptive Algorithms,
- bioheat transfer,
- thermotherapies,
- Meshless Methods
Citation: Rafael Pinheiro Amantéa, Robspierre de Carvalho, Luiz Otávio Barbosa. Evaluation of a radial basis function node refinement algorithm applied to bioheat transfer modeling[J]. AIMS Bioengineering, 2021, 8(1): 36-51. doi: 10.3934/bioeng.2021005

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Abstract

Many bioheat transfer problems involve linear/non-linear equations with non-linear or time-dependent boundary conditions. For heat transfer problems, the presence of time and space-dependent functions under Neumann and Mixed type boundary conditions characterize trivial applications in bioengineering, such as thermotherapies, laser surgeries, and burn studies. This greatly increases the complexity of the numerical solution in several problems, requiring fast and accurate numerical solutions. This paper has a main objective evaluate an adaptive mesh refinement radial basis function method strategy for the classical Penne's bioheat transfer modeling. Our numerical results had errors of ~0.1% compared to analytical solutions. Thus, the proposed methodology is accurate and has a low computational cost. For step function heating, two RBF shape parameters were applied, again achieving excellent results. The distributions of the nodes in the solution domain show that the primary source of error in the numerical solutions came from the boundary conditions. This finding should arouse the interest of engineers and scientists in the development of new strategies for problems involving boundary conditions with periodic functions.

Nomenclature

Symbol	Description	Unit
θ_c	Coarse node parameter	-
θ_r	Refine node parameter	-
ω_b	Blood perfusion	m³/s/m³
Ø	Radial basis function	-
Δt	Numerical time step	s
c	Specific heat of tissue	J/(kg·K)
c_b	Specific heat of blood	J/(kg·K)
h₀	Heat convection coeficient	W/(m²·K)
k	Thermal conductivity of tissue	W/(m·K)
L	Distance between skin surface and body core	m
Q_m	Metabolic rate of tissue	W/m³
Q_r	Spatial heating	W/m³
r	Euclidean distance	-
t	Time	s
T	Tissue temperature	°C
T₀	Steady state temperature	°C
T_a	Arterial temperature	°C
T_c	Blood temperature	°C
T_f	Fluid temperature	°C
x	Spatial coordinate	m
α	Thermal diffusivity	m²/s
ϵ	Multiquadric shape parameter	-
λ	Radial basis function interpolator	-
ρ	Density of tissue	Kg/m³
ρ_b	Density of blood	Kg/m³

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