Development Of Numerical Methods For Forward And Inverse Heat Transfer Problems

Heat transfer is a subject closely related to the human's life.Many engineers and researchers have devoted in investigating practical heat transfer problems in industry,architecture,medical treatment and other fields.Recently,using numerical methods to solve heat transfer problems becomes a focus,as the complexity of the engineering problems is greatly increasing.This thesis develops various kinds of numerical techniques to solve different types of complex heat transfer problems based on existing domain discretization methods,including mesh-based methods(like FEM),meshless methods(like RBF)and a unique meshless method based on a background mesh(S-FEM).Complex heat transfer problems studied include the moving boundary problems,the Cauchy problems,and the reconstruction problems with dynamically changing boundaries.This thesis consists of five major technical parts,which are respectively presented in five chapters.In the first technical develop part of the thesis,an effective general formulation is proposed for simulating unsteady state heat transfer problems with moving boundaries.In dealing with the unsteady behavior,forward time marching is performed using the standard finite difference method,where the nodal temperature values at the previous time step are employed to solve the current temperature field.Due to the effect of the moving boundary,the positions of the node changes with time,leading to difficulty in the handling of space-time approximation of the temperature and its gradients.We first apply simple techniques that use an extra interpolation and extrapolation operation to obtain the temperature values at locations needed for the time marching.We then introduce a novel technique with a correction term to effectively handle the moving boundary effects,without the need for the extra interpolation or extrapolation.With the correction term,one can directly use the nodal temperature values at the previous step in the time marching,in the same way as in the standard formulation of FEM for heat transfer problems with fixed boundaries.A mathematical study has also been conducted to examine the theoretical basis for the correction item.Finally,through intensive numerical experiments,it concludes that the results of the correction term are more accurate and its computational efficiency is improved 10% comparing with that of the interpolation or extrapolation method.In the second technical develop part of the thesis,the Cauchy heat transfer problem that is a typical ill-posed inverse problem,is studied in the paper.The unknown nodal heat flux equations are extracted from the full-size FEM system equations through a matrix partitioning operation,which effectively converts the inverse problem to a forward-like problem.In order to mitigate the ill-posedness and to achieve reliable and more accurate solution,we adopt SVD(singular value decomposition method)to put off ?unstable modes? in the system.This is achieved by cutting off some small eigenvalues to avoid magnification of the noises in the input data.An automatic algorithm is also proposed to determine the number of small eigenvalues to be deleted,so that the Cauchy type ill-posed problems can be solved in a systematic manner.In the third technical develop part of the thesis,a meshless method based on the radial basis functions(RBFs)is employed to construct a novel space-time solver for Cauchy inverse problems with composite walls.This is achieved by reconstructing the unknowns on the inner boundary of the composite walls,using Cauchy conditions(where both the temperature and the heat flux are all specified).The presented novel solver uses the approximations in both time and space in a unified fashion.Because the time is also considered as a dimension in our RBF approximation,a parameter is introduced and adjusted to deal with the relationship between the time and the space discretization.In addition,because our unified space-time approximation scheme does not use the usual layer-by-layer recursion procedure,possible error accumulation is naturally avoided.In order to obtain better accuracy,a procedure is adopted to minimize the error on the given boundary by selecting a shape parameter c for the RBF individually in each layer.In order to mitigate the ill-posed inverse problem,we use the Tikhonov regularization technique to obtain a stable and accurate numerical approximation of the moving boundary.In the fourth technical develop part of the thesis,a general solver is proposed based on the latest smoothed finite element methods(S-FEMs)for 2D and 3D heat transfer problems.S-FEM is a new numerical method proposed by G.R.Liu recently.Our work,to develop an algorithm for a more effective implementation of S-FEM,is different from the existing published implementation of S-FEM in terms of computing the smoothed temperature gradient.The existing implementation uses the volume/area-weighted average method to compute the smoothed temperature gradient for smoothing domains,especially for 3D problems.Our present algorithm uses the surface/line integral to compute the smoothed temperature gradient strictly following the general W2 formulation of S-FEM.Therefore,our solver is most general,applicable to any polygon elements,and even to higher order interpolation schemes,including using the radial basis functions.Besides,different smoothing domains for 2D and 3D problems are constructed and at the same time,all the connectivities of smoothing domains are set up and the requested data is calculated and recorded in a database for the later computation.In order to improve the efficiency of the solver,a concise procedure is designed to compute the orientation of the unit normal vectors.In post-processing part,the nodal and element temperature gradients are recovered using also the smoothed temperature gradients.In the fifth or final technical develop part of the thesis,the S-FEM is applied to solve the Cauchy inverse problem.The unknown nodal heat flux equations are extracted from the full-size S-FEM system equations through a matrix partitioning operation,which effectively converts the inverse problem to a forward-like problem.The process is similar to that of FEM,but all the operations are now based on smoothing domains.Both SVD and Tikhonov method are separately used to mitigate the ill-posedness.Besides,the results of ES-FEM and NS-FEM are compared and discussed for a number of numerical examples.