Magdalena Mierzwiczak

Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problem

Abstract The paper considers the determination of heat sources in unsteady 2-D heat conduction problem. The determination of the strength of a heat source is achieved by using the boundary condition, initial condition and a known value of temperature in chosen points placed inside the domain. For the solution of the inverse problem of identification of the heat source the θ -method with the method of fundamental solution and radial basis functions is proposed. Due to ill conditioning of the inverse transient heat conduction problem the Tikhonov regularization method based on SVD decomposition was used. In order to determine the optimum value of the regularization parameter the L-curve criterion was used. For testing purposes of the proposed algorithm the 2-D inverse boundary-initial-value problems in square region Ω with the known analytical solutions are considered. The numerical results show that the proposed method is easy to implement and pretty accurate. Moreover the accuracy of the results does not depend on the value of the θ parameter and is greater in the case of the identification of the temperature field than in the case of the identification of the heat sources function.

The determination of heat sources in two dimensional inverse steady heat problems by means of the method of fundamental solutions

This article deals with the inverse determination of heat sources in steady 2-D heat conduction problems. The problem is described by Poissons equation in which the right-hand side function is unknown. The identification of the strength of a heat source is given by using the boundary conditions for the temperature and heat flux at chosen points on the boundary. For the solution of the inverse problem of identification of the heat source, the method of fundamental solutions with radial basis functions and the Tikhonov regularization technique is proposed. Accurate results have been obtained for six test problems where the analytical solutions are available.

Comparison of different approaches in the Trefftz method for analysis of fluid flow between regular bundles of cylindrical fibres

In the paper three different approaches for the Trefftz method are compared in analysis of the fluid flow between regular bundles of cylindrical fibres. The approximate solution is a linear combination of such trial functions which fulfil exactly the governing equations. The trial functions can be defined in the Cartesian coordinate system (the first approach), in the cylindrical coordinate system (and can fulfil also some boundary conditions - the second approach) or be defined as a fundamental solutions (the third approach - the method of fundamental solutions).The average velocity and the product of the friction factor and the Reynolds number ƒ ⋅ Re are compared for selected parameters of a considered region.

Generalized Newtonian fluid flow through fibrous porous media

The numerical calculations of the velocity field and the component of transverse permeability in the filtration equation for steady, incompressible flow of the generalized Newtonian fluid through the assemblages of cylindrical fibers are presented in this paper. The fibers are arranged regularly in arrays. Flow is transverse with respect to the fibers. The non-linear governing equation in the repeated element of the array is solved using iteration method. At each iteration step the method of fundamental solutions and the method of particular solutions are used. The bundle of fibers is treated as a porous media and on the base of velocity field the permeability coefficients are calculated as a function of porosity.

About some application of special purpose Trefftz function for determination of effective viscosity in filtration equation

Determination of the effective viscosity in the Brinkman’s filtration equation by numerical simulation of an imagined physical experiment in volume flow rate viscometer is presented in this paper. Porous medium is modeled by the bundle of parallel fibers ordered in a regular square array. Such bundle is placed in a square duct. In the first step, a micro-structural flow problem is solved by the Trefftz method using special purpose Trefftz functions. In the second step the bundle of fibers is treated as a porous medium with the Brinkman’s filtration equation as governing equation. Comparison the volume flow rate of these two flows permits as to determine a effective viscosity in filtration equation.

The inverse determination of the thermal contact resistance components of unidirectionally reinforced composite

This article considers the problem of determination of the thermal contact resistance in unidirectionally reinforced composite. The problem formulated in such a way should be treated as an inverse heat transfer problem in a non-homogenous material. The thermal conductivities of constituents (fibres and matrix) and fibre arrangement are known. The calculations are carried out for a composite with the fibres arranged in a matrix in a regular manner by a square grid.