Finite Element Method For Thermoelastic Problems

In the research fields of machinery manufacturing and military technology,the thermodynamic coupling behavior analysis of material structures is an important topic,which is generally called the thermoelastic problem.Thermoelastic problems mainly study the distribution of temperature field and stress field and their coupling relationship in solid structures under certain thermal and mechanical loads.Thermoelastic problems are mathematically expressed as partial differential equations with the coupling of parabolic equations and hyperbolic elastodynamic equations.The existence of coupling term brings great complexity to the theoretical analysis and solution of the problem.When the temperature change is not drastic,the influence of coupling term in the heat conduction equation and the second derivative of displacement with respect to time are too small to be ignored,which reduces the difficulty of solving,and the original problem becomes a single coupling problem at this time.In the 1960s,analytical solutions based on Laplace and Fourier transform were developed in special regions such as one-dimensional and semiinfinite,which promoted the study of thermoelasticity to a certain extent.But for the common two-dimensional and three-dimensional bounded areas in engineering,such analytical solution is not applicable.With the emergence and development of finite element method,researchers in the field of engineering first applied finite element method to the thermoelastic problem,and put forward a variety of calculation formats such as alternate method,joint method,etc.,and achieved good results in the finite element calculation of thermoelastic problem.It is meaningful to study the convergence of these schemes from the perspective of finite element theory analysis,which is helpful for us to better understand these schemes and improve them.However,at present,the theoretical analysis results and literature that can be referred to are relatively rare,and this paper makes an exploration in this aspect.In this paper,based on the finite element theory of parabolic problem and the finite element theory of elastic problem,the errors of the thermoelastic finite element algorithm based on linear finite element and Crank-Nicolson scheme are analyzed.In this paper,the thermoelastic single coupling and full coupling problems are considered successively,the corresponding semi-discrete and fully discrete schemes are given respectively,and the corresponding semi-discrete approximate solutions and fully discrete approximate solutions are defined.Then,we prove the convergence of the semi-discrete solution and the fully discrete solution respectively,and obtain their convergence order.The proof process mainly uses the properties of projection operator and interpolation operator.Finally,we carry out numerical experiments for thermoelastic single coupling and full coupling problems,and the convergence order of the numerical solutions is consistent with the theoretical analysis results,which verifies the reliability of the theoretical analysis results.