基于自适应交叉近似的边界点法研究及其在非线性热-弹接触问题中的应用

The motion and load is usually transferred between the mechanical components by the contact behavior. The contact behavior always causes stress concentrations, which increase the risk for crack initiations, propagations and fatigue failure. Meanwhile, the thermal load analysis of the contact body working under the high temperature conditions is also one of the frequently encountered problems in engineering. In this case, the thermal and mechanical fields of contact bodies are coupling with each other. The contact state influences the temperature distribution of heat conduction under contact conditions, and the thermal field of the heat conduction influences the contact state as thermal load. Because of the nonlinearity of thermal-elastic contact problems mainly existing on the boundary, a boundary-type numerical method based on boundary integral equation is a natural choice. Therefore, in this project the thermo-elastic contact problems will be solved by a boundary-type meshless method—boundary node method (BNM). However, as with the boundary element method, the meshless BNM is limited not to be used in the large scale engineering problems with the asymmetric full matrix of limits. The hierarchical matrix algorithm based on the adaptive cross approximation (ACA) will be studied to solve the large-scale thermo-elastic contact problems combining with the meshless BNM. The thermo-elastic contact problems in the gear engagement system will be studied as an engineering application, and the goal is to accurately simulate the stress and temperature distribution of gear working under high temperature conditions.

机械部件之间通过接触关系进行运动和载荷的传递。部件之间接触应力的产生，是机械部件出现早期裂纹、裂纹扩展甚至疲劳断裂的重要原因。同时，高温工作条件下的接触体热负荷分析，也是工程中经常遇到的难题之一。这种情况下，接触体的热-力场相互耦合。接触状态影响接触热传导过程，而接触热传导的温度场又作为温度载荷影响接触状态。由于热-弹耦合接触问题的非线性条件主要存在于接触边界上，基于边界积分方程的边界类型数值方法是一个自然地选择。本项目结合无网格边界点法，实现热-弹耦合接触问题的数值分析。然而，与边界元法一样，无网格边界点法的非对称满阵方程组制约了其在大规模工程问题中的应用。本项目将研究基于自适应交叉近似的分级矩阵算法，实现无网格边界点法分析大规模热-弹耦合接触问题。以机械传动中齿轮啮合的热-力耦合接触分析为例展开应用研究，目标是能够精确模拟高温工作环境下齿轮啮合的真实应力场和温度场分布。

机械部件之间通过接触关系进行运动和载荷的传递。尤其是高温工作条件下，部件之间的接触状态影响接触热传导过程，而接触热传导的温度场又作为温度载荷影响接触状态。部件之间接触应力的产生，是机械部件出现早期裂纹、裂纹扩展甚至疲劳断裂的重要原因。同时，高温工作条件下的接触体热负荷分析，也是工程中经常遇到的难题之一。在上述实际工程问题中，部件之间接触热传导和接触应力的产生，均发生于部件之间的边界相互接触区域；针对此类热-弹耦合接触问题，基于边界积分方程的边界类数值分析方法是一个自然地选择，便于处理仅发生于边界上的非线性接触条件，同时具有边界积分方程计算精度高，只需要边界离散的优点。本项目发展了无网格边界点法（基于边界积分方程，采用无网格插值的数值方法），将其用于热-弹耦合接触问题的数值分析。研究了基于无网格插值的接触离散模型（共形和非共形接触离散模型）以及无网格插值函数的性质（Kronecker δ函数性质以及支撑域）等对接触问题分析的计算过程和结果的影响。在此基础上，研究了热-力耦合接触问题分析的边界点算法，对求解过程中的接触热传导分析、热应力分析和弹性接触分析的耦合迭代过程，开展了重点研究工作。由于基于边界积分方程，无网格边界点法的非对称满阵方程组制约了其在大规模工程问题中的应用。因此，本项目也同时对基于自适应交叉近似的快速算法开展了相应的研究工作。