Symplectic Superposition Method For Two-dimensional Transient Heat Conduction Problems

In this dissertation,based on the two-dimensional transient heat transfer theory and the symplectic superposition method,new analytical solutions are derived for two-dimensional transient heat transfer problems considering point heat source and various boundary conditions.First,the Laplace transform is used to introduce the governing equation from the time domain to the frequency domain,and then the dual variables are constructed.The problem is introduced into the Hamilton dual system;which is solved analytically based on the symplectic geometry method.Finally,the time domain solution of the problem is obtained by the Laplace inverse transform.This dissertation discusses the characteristics of temperature and heat flux density distribution at different moments under simple boundary and mixed boundary with heat source.By comparison with the finite element results,the accuracy of the above method in solving transient heat conduction problems is validated.In this dissertation,the transient heat conduction of isotropic materials is studied in a rectangular region with heat source for different basic problems(two opposite sides with zero temperature,one side with zero temperature and its opposite side is adiabatic,two opposite sides are adiabatic),and the new analytical solutions derived can be degraded to those without heat source.When dealing with mixed boundaries,making full use of the advantages of the symplectic superposition method,a complex boundary problem splits into basic problems that can be directly solved by the symplectic method,and the undetermined coefficients are obtained by the continuity conditions,and then the original mixed boundary problem is solved by the superposition of solutions of the sub-problems.Analytical solutions for transient heat conduction problems are derived in a rectangular region under six simple boundary conditions based on combinations of three basic systems,and the distribution form of constant,non-uniform and time-dependent functions are considered in solving the problems with temperature boundary conditions at four sides.In the numerical examples,the analytical solutions at different moments are compared with the results of the temperature field and the heat flux field by the finite element method.The analytical results are accurate and the convergence is fast,which fully reflects applicability and rationality of the new analytical method used in this dissertation to deal with the two-dimensional transient heat conduction problems.Based on the study of simple boundaries,the transient heat conduction problems under temperature-temperature and heat flux-heat flux mixed boundaries are studied.The combinations of temperature and heat flux boundary conditions on other edges are fully considered,and the rectangles under twelve different combinations of boundaries are derived for transient heat conduction solutions.It is found from the accuracy of the derivation process and the results that the method used in this dissertation has great universality when dealing with transient heat conduction problems with complex boundary conditions,where the fast convergence and the rationality are also revealed.