The method of fundamental solutions for nonlinear functionally graded materials

AbstractIn this paper, we investigate the application of the method of fundamental solutions (MFS) to two-dimensional steady-state heat conduction problems for both isotropic and anisotropic, single and composite (bi-materials), nonlinear functionally graded materials (FGMs). In the composite case, the interface continuity conditions are approximated in the same manner as the boundary conditions. The method is tested on several examples and its relative merits and disadvantages are discussed

Determination of an additive time- and space-dependent coefficient in the heat equation

Purpose: The purpose of this study is to provide an insight and to solve numerically the identification of an unknown coefficient of radiation/absorption/perfusion appearing in the heat equation from additional temperature measurements. Design/methodology/approach: First, the uniqueness of solution of the inverse coefficient problem is briefly discussed in a particular case. However, the problem is still ill-posed as small errors in the input data cause large errors in the output solution. For numerical discretization, the finite difference method combined with a regularized nonlinear minimization is performed using the MATLAB toolbox routine lsqnonlin. Findings: Numerical results presented for three examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Research limitations/implications: The mathematical formulation is restricted to identify coefficients which separate additively in unknown components dependent individually on time and space, and this may be considered as a research limitation. However, there is no research implication to overcome this, as the known input data are also limited to single measurements of temperature at a particular time and space location. Practical implications: As noisy data are inverted, the study models real situations in which practical measurements are inherently contaminated with noise. Social implications: The identification of the additive time- and space-dependent perfusion coefficient will be of great interest to the bio-heat transfer community and applications. Originality/value: The current investigation advances previous studies which assumed that the coefficient multiplying the lower-order temperature term depends on time or space separately. The knowledge of this physical property coefficient is very important in biomedical engineering for understanding the heat transfer in biological tissues. The originality lies in the insight gained by performing for the first time numerical simulations of inversion to find the coefficient additively dependent on time and space in the heat equation from noisy measurements

Inverse problems for a model of biofilm growth

A bacterial biofilm is an aggregate of micro-organisms growing fixed onto a solid surface, rather than floating freely in a liquid. Biofilms play a major role in various practical situations such as surgical infections and water treatment. We consider a non-linear PDE model of biofilm growth subject to initial and Dirichlet boundary conditions, and the inverse coefficient problem of recovering the unknown parameters in the model from extra measurements of quantities related to the biofilm and substrate. By addressing and analysing this inverse problem we provide reliable and robust reconstructions of the primary physical quantities of interest represented by the diffusion coefficients of substrate and biofilm, the biomass spreading parameters, the maximum specific consumption and growth rates, the biofilm decay rate and the half saturation constant. We give particular attention to the constant coefficients involved in the leading-part non-linearity, and present a uniqueness proof and some numerical results. In the course of the numerical investigation, we have identified extra data information that enables improving the reconstruction of the eight-parameter set of physical quantities associated to the model of biofilm growth.Comment: 25 pages, 5 figure

Inverse problems of damped wave equations with Robin boundary conditions: an application to blood perfusion

Knowledge of the properties of biological tissues is essential in monitoring any abnormalities that may be forming and have a major impact on organs malfunctioning. Therefore, these disorders must be detected and treated early to save lives and improve the general health. Within the framework of thermal therapies, e.g. hyperthermia or cryoablation, the knowledge of the tissue temperature and of the blood perfusion rate are of utmost importance. Therefore, motivated by such a significant biomedical application, this paper investigates, for the first time, the uniqueness and stable reconstruction of the space-dependent (heterogeneous) perfusion coefficient in the thermal-wave hyperbolic model of bio-heat transfer from Cauchy boundary data using the powerful technique of Carleman estimates. Additional novelties consist in the consideration of Robin boundary conditions, as well as developing a mathematical analysis that leads to stronger stability estimates valid over a shorter time interval than usually reported in the literature of coefficient identification problems for hyperbolic partial differential equations. Numerically, the inverse coefficient problem is recast as a nonlinear least-squares minimization that is solved using the conjugate gradient method (CGM). Both exact and noisy data are inverted. To achieve stability, the CGM is stopped according to the discrepancy principle. Numerical results for a physical example are presented and discussed, showing the convergence, accuracy and stability of the inversion procedure

Uniqueness result for an age-dependent reaction–diffusion problem

This paper is concerned with an age-structured model in population dynamics. We investigate the uniqueness of solution for this type of nonlinear reaction–diffusion problem when the source term depends on the density, indicating the presence of, for example, mortality and reaction processes. Our result shows that in a spatial environment, if two population densities obey the same evolution equation and possess the same terminal data of time and age, then their distributions must coincide therein

Reconstruction of an elliptical inclusion in the inverse conductivity problem

This study reports on a numerical investigation into the open problem of the unique reconstruction of an elliptical inclusion in the potential field from a single set of nontrivial Cauchy data. The investigation is based on approximating the potential fields of a composite material as a linear combination of fundamental solutions for the Laplace equation with sources shifted outside the solution domain and its boundary. The coefficients of these finite linear combinations are unknown along with the centre, the lengths of the semi-axes and the orientation of the sought ellipse. These are determined by minimizing the least-squares objective functional describing the gap between the given and computed data. The extension of the proposed technique for the reconstruction of two ellipses is also considered

A meshless method for an inverse two-phase one-dimensional nonlinear Stefan problem

We extend a meshless method of fundamental solutions recently proposed by the authors for the one-dimensional two-phase inverse linear Stefan problem, to the nonlinear case. In this latter situation the free surface is also considered unknown which is more realistic from the practical point of view. Building on the earlier work, the solution is approximated in each phase by a linear combination of fundamental solutions to the heat equation. The implementation and analysis are more complicated in the present situation since one needs to deal with a nonlinear minimization problem to identify the free surface. Furthermore, the inverse problem is ill-posed since small errors in the input measured data can cause large deviations in the desired solution. Therefore, regularization needs to be incorporated in the objective function which is minimized in order to obtain a stable solution. Numerical results are presented and discussed

Regularized MFS solution of inverse boundary value problems in three-dimensional steady-state linear thermoelasticity

We investigate the numerical reconstruction of the missing thermal and mechanical boundary conditions on an inaccessible part of the boundary in the case of three-dimensional linear isotropic thermoelastic materials from the knowledge of over-prescribed noisy data on the remaining accessible boundary. We employ the method of fundamental solutions (MFS) and several singular value decomposition (SVD)-based regularization methods, e.g. the Tikhonov regularization method (Tikhonov and Arsenin, 1986), the damped SVD and the truncated SVD (Hansen, 1998), whilst the regularization parameter is selected according to the discrepancy principle (Morozov, 1966), generalized cross-validation criterion (Golub et al., 1979) and Hansen's L-curve method (Hansen and O'Leary, 1993)

In celebration of the 60th birthday of Professor Alemdar Hasanoğlu (Hasanov)

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