Calculation Method For A Class Of Heat Conduction Problems In Multilayer Media

In this paper,the numerical method for a class of heat conduction problems of multi-layer dielectric cylinder is studied.The main work is as follows:(1)A numerical scheme for the heat conduction equation of a cylinder with multi-layer medium is studied.Firstly,in cylindrical coordinates,considering the heat conduction problem of cylinder cross-section,combined with Fourier heat conduction law,the three-dimensional heat conduction problem is transformed into two-dimensional heat conduction problem.Then,the heat conduction problem of cylinder cross section is considered.Under the condition of symmetry,the two-dimensional heat conduction problem is simplified to one-dimensional heat conduction problem.Next,by introducing the state space theory and applying the difference scheme in time domain,the heat conduction equation is transformed into ordinary differential equations with temperature and heat flow as unknown functions of spatial variables,and the state equation of the heat conduction problem on the cross section of a multi-layer dielectric cylinder is derived.Finally,the numerical scheme and algorithm of temperature and heat flux recursive solution are obtained based on the interlayer connection conditions.(2)In this paper,four numerical methods and numerical examples for the fixed solution of multi-layer heat conduction problems are studied.Firstly,four different boundary conditions are given,and the corresponding four forward problems are analyzed.Then,according to the recursive numerical scheme of temperature and heat flow,combined with the boundary conditions,the equations of temperature and heat flow are solved,and four sets of solutions for the forward problem of temperature and heat flow are obtained.In this paper,a numerical example is given to solve the forward problem of heat conduction in a four layer dielectric cylinder composed of four kinds of metals.(3)In this paper,the regularization method for solving the lateral boundary value problem with two different boundary conditions is studied.In order to overcome the ill posedness of the problem,firstly,the coefficient matrix of the equations is decomposed by singular value decomposition.Then,the optimal parameters are selected according to Tikhonov quasi optimization method,and the temperature and heat flow are solved by Tikhonov regularization method.In this paper,the discretization error of backward Euler scheme is studied,and the local truncation error is O(Δt),and the stability conclusion of the algorithm is given.Finally,a numerical example of the side boundary value problem of heat conduction in four layer medium is given,which shows that the algorithm is effective.