In this paper, the method of fundamental solution (MFS) coupling with the dual reciprocity method (DRM) is developed to solve nonlinear steady state bioheat transfer problems. A two-dimensional nonlinear skin model with temperature-dependent blood perfusion rate is studied. Firstly, the original bioheat transfer governing equation with nonlinear term induced by temperature-dependent blood perfusion rate is linearized with the Taylor's expansion technique. Then, the linearized governing equation with specified boundary conditions is solved using a meshless approach, in which the DRM and the MFS are employed to obtain particular and homogeneous solutions, respectively. Several numerical examples involving linear, quadratic and exponential relations between temperature and blood perfusion rate are tested to verify the efficiency and accuracy of the proposed meshless model in solving nonlinear steady state bioheat transfer problems, and also the sensitivity of coefficients in the expression of temperature-dependent blood perfusion rate is analyzed for investigating the influence of blood perfusion rate to temperature distribution in skin tissues.

Method of fundamental solutions for nonlinear skin bioheat model

To understand the physiology of tissue burns for successful clinical treatment, it is important to investigate the thermal behavior of human skin tissue subjected to heat injury. In this paper, a fundamental solution-based hybrid finite element formulation is proposed for numerically simulating steady-state temperature distribution inside a multilayer human skin tissue during burning. In the present approach, since only element boundary integrals are involved, the computational dimension is reduced by one as the fundamental solutions used analytically satisfies the bioheat governing equation. Further, in multi-layer skin modeling, the burn is applied via a heating disk at constant temperature on a part of the epidermal surface of the skin tissue. Numerical results from the proposed approach are firstly verified by comparing them with exact solutions of a simple single-layered model or the results from conventional finite element method. Thereafter, a sensitivity analysis is carried out to reveal the effect of biological and environmental parameters on temperature distribution inside the skin tissue subjected to heat injury.

A fundamental solution-based finite element model for analyzing multi-layer skin burn injury

For solving steady-state nonlinear heat conduction problems (HCPs) in arbitrary plane domain enclosed by a complex boundary shape, the meshless methods are convenient to programming, of which the polynomial expansion method seems the simplest one. However, it is seldom used as a major medium to solve nonlinear partial differential equations owing to its highly ill-conditioned behavior. First, we decompose the steady-state nonlinear heat conduction equation into a linear portion on the left-hand side, and leave other terms on the right-hand side as a non-homogeneous portion. Then, we need to solve a steady-state linear heat conduction problem in each iteration, for which we propose a multiple-scale Pascal triangle method to generate the linear algebraic equations, where the multiple scales are automatically decided by the collocation points. We can find that these scales can largely reduce the condition number of the coefficient matrix in each linear system, such that the iteration process is convergent very quickly, and the numerical solutions obtained are very accurate and stable against very large noise to over 50%, no matter for the Dirichlet and mixed-type boundary value problems. Numerical results confirm the validity of the present multiple-scale polynomial expansion method for solving steady-state nonlinear HCPs in arbitrary plane domains.

A new meshless method for solving steady-state nonlinear heat conduction problems in arbitrary plane domain

Summary
The aim of this paper is to present a new semi-analytic numerical method for strongly nonlinear steady-state advection-diffusion-reaction equation (ADRE) in arbitrary 2D domains The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem Each basis function used in the algorithm is a sum of an analytic basis function and a special correcting function which is chosen to satisfy the homogeneous boundary conditions of the problem The polynomials, trigonometric functions, conical radial basis functions (RBFs) and the multiquadric RBFs are used in approximation of the ADRE This allows us to seek an approximate solution in the analytic form which satisfies the boundary conditions of the initial problem with any choice of free parameters As a result, we separate the approximation of the boundary conditions and the approximation of the ADRE inside the solution domain The numerical examples confirm the high accuracy and efficiency of the proposed method in solving strongly nonlinear equations in an arbitrary domain This article is protected by copyright All rights reserved

A semi‐analytic collocation technique for steady‐state strongly nonlinear advection‐diffusion‐reaction equations with variable coefficients

The paper presents a new meshless numerical technique for solving nonlinear Poisson-type equations ∇ 2 u = F ( u , u x 1 , u x 2 , x ) + f ( x ) in irregular domains. Using the dual reciprocity method (DRM) approach, the nonlinear term is represented as a linear combination of basis functions F ≃ ∑ m M q m φ m . We use RBFs as the basis functions φ m . According to the DRM this admits of writing the approximate solution in the form of the linear combination of particular solutions corresponding to these basis functions u M = u f + ∑ m M q m Φ m . We use the particular solution as a sum of two parts Φ = Φ 0 + Φ 1 , where Φ 0 is the well known analytical solution of the Poissone equation with a RBF on the right-hand side. The second term Φ 1 provides that Φ satisfies the homogeneous conditions on the boundary of the domain. As a result, it is possible to isolate the nonlinear part of the equation and the problem is reduced to a system of nonlinear equations F ( u M , u M , x 1 , u M , x 2 , x ) = ∑ m M q m φ m for the unknown coefficients qm. Then the nonlinear system is solved numerically. Numerical experiments are carried out for accuracy and convergence investigations. A comparison of the numerical results obtained in the paper with the exact solutions or other numerical methods indicates that the proposed method is accurate and efficient in dealing with complicated geometry and strong nonlinearity.

Method of particular solutions for nonlinear Poisson-type equations in irregular domains

A hybrid finite element model based on F-Trefftz kernels (fundamental solutions) is formulated for analyzing Dirichlet problems associated with two-dimensional nonlinear Poisson-type equations including nonlinear Poisson–Boltzmann equation and diffusion–reaction equation. The nonlinear force term in the Poisson-type equation is frozen by introducing the imaginary terms at each Picard iteration step, and then the induced Poisson problem is solved by the present hybrid finite element model involving element boundary integrals only, coupling with the particular solution method with radial basis function interpolation. The numerical accuracy of the present method is investigated by numerical experiments for problems with complex geometry and various nonlinear force functions.

http://www.math.nsysu.edu.tw/conference/TrefftzMFS2011/abstracts/026.pdf

Xing-Pei Liang

Papers

Solving the nonlinear Poisson-type problems with F-Trefftz hybrid finite element model