热弹性动力学耦合问题的微分代数方法

对一般的热机械问题提出了一种有效的数值方法，并对二维的热弹性问题进行了测试．该方法的基本思路是将描述热机械耦合问题的偏微分方程进行降阶，使之成为一组微分代数方程，应力应变关系被写成代数方程．所得到的微分代数系统采用全隐式的向后差分公式进行求解．对该方法进行了详细的说明．为了验证该方法的有效性，将其应用于一个动态非耦合的热弹性问题的求解和一个耦合的二维热弹性问题的求解．

Differential-Algebraic Approach to Coupled Problems of Dynamic Thermoelasticity

An efficient numerical approach for the general thermomechanical problems was developed and it was tested for a two-dimensional thermoelasticity problem. The main idea of the numerical method is based on the reduction procedure of the original system of PDEs describing coupled thermomechanical behavior to a system of Differential Algebraic Equations (DAEs) where the stress-strain relationships are treated as algebraic equations. The resulting system of DAEs were then solved with a Backward Differentiation Formula (BDF) using a fully implicit algorithm. The described procedure was explained in detail.And its effectiveness was demonstrated on the solution of a transient uncoupled thermoelastic problem, for which an analytical solution is known, as well as on a fully coupled problem in the two-dimensional case.