Finite Element Methods For Robin Boundary Classical Thermoelastic Problems And A Class Of L-S Type Generalized Thermoelastic Problems

The study of the coupling behavior of solid material structures under thermal and force loads is the main study of thermoelasticity problems.The classical thermoelasticity problem is based on Fourier’s law,which is mathematically represented as a set of partial differential equations with parabolic heat conduction equations and hyperbolic elastodynamic equations coupled with each other,and can be simplified to a singlecoupling problem when the coupling coefficient of solid materials is small.In recent years,non-Fourier heat transfer effects have been observed in rapid laser heating and heat transfer in porous materials,and then various types of generalized thermoelastic models have been established to describe this phenomenon,and the Lord-Shulman model is a typical class of these problems.The classical and generalized thermoelasticity problems are coupled systems of partial differential equations problems,which are more difficult to solve,and mainly have analytical and numerical solutions.Analytical solutions based on Laplace transform can be used in the one-dimensional semi-infinite region.In two dimensions and above,it is necessary to use numerical approximation methods to solve the problems in complex regions.The development of finite element methods has given a great impetus to the study of this problem,and experts in the field of engineering have proposed a series of effective schemes for finite element calculations.For these schemes,it is valuable to analyze their convergence from the theoretical point of view,but the related results are relatively rare.Robin boundary conditions describe the convective heat transfer by the structural boundary,which is more widely used in practice than the Dirichlet condition with fixed boundary temperature.The literature related to the finite element theory of thermoelastic problems with robin boundary conditions has not been seen,which is one of the research directions in this paper.Numerical methods and theoretical analysis of L-S type generalized thermoelastic problems is an area of more interest to researchers in the last decade,and another research direction of this paper is the theoretical analysis of finite element methods for a class of L-S type generalized thermoelastic problems.First,in this paper,from the finite element theory of parabolic problems and the finite element theory of elastic problems,based on the triangular linear finite element and C-N scheme,the finite element method of classical thermoelastic problems under Robin boundary is studied,and the semi-discrete and fully discrete finite element schemes are given,and the convergence results of the fully discrete and semi-discrete schemes are obtained after analysis,and the numerical experimental results have good stability and the convergence order is consistent with the theoretical analysis.Secondly,based on the mixed finite element theory,a class of L-S generalized thermoelastic problems with coupling behavior is studied in this paper,and its mixed finite element semi-discrete and fully discrete schemes are given,and the error estimates of the semi-discrete and fully discrete solutions are obtained analytically.