221 research outputs found

    The Method of Fundamental Solutions for Direct Cavity Problems in EIT

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    The Method of Fundamental Solutions (MFS) is an effective technique for solving linear elliptic partial differential equations, such as the Laplace and Helmholtz equation. It is a form of indirect boundary integral equation method and a technique that uses boundary collocation or boundary fitting. In this paper the MFS is implemented to solve A numerically an inverse problem which consists of finding an unknown cavity within a region of interest based on given boundary Cauchy data. A range of examples are used to demonstrate that the technique is very effective at locating cavities in two-dimensional geometries for exact input data. The technique is then developed to include a regularisation parameter that enables cavities to be located accurately and stably even for noisy input data

    The method of fundamental solutions for the Oseen steady‐state viscous flow past obstacles of known or unknown shapes

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    In this paper, the steady‐state Oseen viscous flow equations past a known or unknown obstacle are solved numerically using the method of fundamental solutions (MFS), which is free of meshes, singularities, and numerical integrations. The direct problem is linear and well‐posed, whereas the inverse problem is nonlinear and ill‐posed. For the direct problem, the MFS computations of the fluid flow characteristics (velocity, pressure, drag, and lift coefficients) are in very good agreement with the previously published results obtained using other methods for the Oseen flow past circular and elliptic cylinders, as well as past two circular cylinders. In the inverse obstacle problem the boundary data and the internal measurement of the fluid velocity are minimized using the MATLAB© optimization toolbox lsqnonlin routine. Regularization was found necessary in the case the measured data are contaminated with noise. Numerical results show accurate and stable reconstructions of various star‐shaped obstacles of circular, bean, or peanut cross‐section

    Determination of time-dependent coefficients for a weakly degenerate heat equation

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    In this paper, we consider solving numerically for the first time inverse problems of determining the time-dependent thermal diffusivity coefficient for a weakly degenerate heat equation, which vanishes at the initial moment of time, and/or the convection coefficient along with the temperature for a one-dimensional parabolic equation, from some additional information about the process (the so-called over-determination conditions). Although uniquely solvable these inverse problems are still ill-posed since small changes in the input data can result in enormous changes in the output solution. The finite difference method with the Crank-Nicolson scheme combined with the nonlinear Tikhonov regularization are employed. The resulting minimization problem is computationally solved using the MATLAB toolbox routine lsqnonlin. For both exact and noisy input data, accurate and stable numerical results are obtained

    Identification of the time-dependent conductivity of an inhomogeneous diffusive material

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    In this paper, we consider a couple of inverse problems of determining the time-dependent thermal/hydraulic conductivity from Cauchy data in the one-dimensional heat/diffusion equation with space-dependent heat capacity/ specific storage. The well-posedness of these inverse problems in suitable spaces of continuously differentiable functions are studied. For the numerical realisation, the problems are discretised using the finite-difference method and recast as nonlinear least-squares minimization problems with a simple positivity lower bound on the unknown thermal/ hydraulic conductivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. Regularization is included wherever necessary. Numerical results are presented and discussed for several benchmark test examples showing that accurate and stable numerical solutions are achieved. The outcomes of this study will be relevant and of importance to the applied mathematics inverse problems community working on thermal/hydraulic property determination in heat transfer and porous media

    Determination of a Time-Dependent Free Boundary in a Two-Dimensional Parabolic Problem

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    The retrieval of the timewise-dependent intensity of a free boundary and the temperature in a two-dimensional parabolic problem is, for the first time, numerically solved. The measurement, which is sufficient to provide a unique solution, consists of the mass/energy of the thermal system. A stability theorem is proved based on the Green function theory and Volterra’s integral equations of the second kind. The resulting nonlinear minimization is numerically solved using the lsqnonlin MATLAB optimization routine. The results illustrate the reliability, in terms of accuracy and stability, of the time-dependent free surface reconstruction

    Time-Dependent Reaction Coefficient Identification Problems with a Free Boundary

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    The determination of time-dependent reaction coefficients in free boundary heat transfer problems is numerically investigated. The additional data which provides a unique solution is given by the Stefan boundary condition and the heat moments. The finite difference method with the Crank-Nicolson scheme combined with a regularized nonlinear optimization is employed. The resulting nonlinear system of equations is solved numerically using the MATLAB toolbox routine lsqnonlin for minimizing the Tikhonov regularization functional. A discussion of the choice of regularization parameters is provided. Numerical results are presented and discussed

    Identification of obstacles immersed in a stationary Oseen fluid via boundary measurements

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    In this paper we consider the interior inverse problem of identifying a rigid boundary of an annular infinitely long cylinder within which there is a stationary Oseen viscous fluid, by measuring various quantities such as the fluid velocity, fluid traction (stress force) and/or the pressure gradient on portions of the outer accessible boundary of the annular geometry. The inverse problems are nonlinear with respect to the variable polar radius parameterizing the unknown star-shaped obstacle. Although for the type of boundary data that we are considering the obstacle can be uniquely identified based on the principle of analytic continuation, its reconstruction is still unstable with respect to small errors in the measured data. In order to deal with this instability, the nonlinear Tikhonov regularization is employed. Obstacles of various shapes are numerically reconstructed using the method of fundamental solutions for approximating the fluid velocity and pressure combined with the MATLAB©© toolbox routine lsqnonlin for minimizing the nonlinear Tikhonov's regularization functional subject to simple bounds on the variables

    Identification of conductivity in inhomogeneous orthotropic media

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    Purpose - The purpose of this paper is to solve numerically the identification of the thermal conductivity of an inhomogeneous and possibly anisotropic medium from interior/internal temperature measurements. Design/methodology/approach - The formulated coefficient identification problem is inverse and ill-posed and therefore, in order to obtain a stable solution, a nonlinear regularized least-squares approach is employed. For the numerical discretisation of the orthotropic heat equation, the finite-difference method is applied, whilst the nonlinear minimization is performed using the MATLAB toolbox routine lsqnonlin. Findings - Numerical results show the accuracy and stability of solution even in the presence of noise (modelling inexact measurements) in the input temperature data. Research limitations/implications - The mathematical formulation uses temporal tem- perature measurements taken at many points inside the sample and this may be too much information that is provided to identify a spacewise dependent only conductivity tensor. Practical implications - Since noisy data are inverted, the study models real situations in which practical temperature measurements recorded using thermocouples are inherently contaminated with random noise. Social implications - The identification of the conductivity of inhomogeneous and orthotropic media will be of great interest to the inverse problems community with applications in geophysics, groundwater flow and heat transfer. Originality/value - The current investigation advances the field of coefficient identification problems by generalising the conductivity to be orthotropic in addition of being heterogeneous. The originality lies in performing, for the first time, numerical simulations of inver- sion to find the anisotropic and inhomogeneous thermal conductivity form noisy temperature measurements. Further value and physical significance is brought in by determining the degree of cure in a resin transfer molding process, in addition to obtaining the inhomogeneous thermal conductivity of the tested material

    Reconstruction of the thermal properties in a wave-type model of bio-heat transfer

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    Purpose: This study aims to at numerically retrieve five constant dimensional thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements. Design/methodology/approach: The thermal-wave model of bio-heat transfer is used as an appropriate model because of its realism in situations in which the heat flux is extremely high or low and imposed over a short duration of time. For the numerical discretization, an unconditionally stable finite difference scheme used as a direct solver is developed. The sensitivity coefficients of the dimensionless boundary temperature measurements with respect to five constant dimensionless parameters appearing in a non-dimensionalised version of the governing hyperbolic model are computed. The retrieval of those dimensionless parameters, from both exact and noisy measurements, is successfully achieved by using a minimization procedure based on the MATLAB optimization toolbox routine lsqnonlin. The values of the five-dimensional parameters are recovered by inverting a nonlinear system of algebraic equations connecting those parameters to the dimensionless parameters whose values have already been recovered. Findings: Accurate and stable numerical solutions for the unknown thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements are obtained using the proposed numerical procedure. Research limitations/implications: The current investigation is limited to the retrieval of constant physical properties, but future work will investigate the reconstruction of the space-dependent blood perfusion coefficient. Practical implications: As noise inherently present in practical measurements is inverted, the paper is of practical significance and models a real-world situation. Social implications: The findings of the present paper are of considerable significance and interest to practitioners in the biomedical engineering and medical physics sectors. Originality/value: In comparison to Alkhwaji et al. (2012), the novelty and contribution of this work are as follows: considering the more general and realistic thermal-wave model of bio-heat transfer, accounting for a relaxation time; allowing for the tissue to have a finite size; and reconstructing five thermally significant dimensional parameters
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