三维弹性体移动接触问题的边界元法研究
摘要
边界元法是在有限元法之后发展起来的一种精确高效的工程与科学问题的数值分析方法，具有便于模拟复杂边界形状、求解精度高、降维等优点。对于弹性接触问题等边界非线性问题而言，由于边界位移和边界面力正是边界积分方程中的基本未知量，边界元法能以更高精度满足接触条件，因此，边界元法能更精确地求解弹性接触问题。本文对三维弹性体接触问题的边界元法做了若干应用基础性研究，其中主要包括以下几个部分：
作者首先将本研究组提出的弹性力学问题边界元解误差的直接估计从二维推广到三维问题，给出了确定与域内解连续的边界位移的一种精确有效的方法。在此基础上提出将接触体接触单元间与域内解连续的边界位移之差的某种度量作为三维弹性体接触问题边界元法的一种误差直接估计，并且提出了三维弹性体接触问题边界元法的一种自适应计算方案。这种方案为确定没有解析解可作比较的复杂接触问题边界元解的精度提供了可能。
对于移动接触问题，边界元法的有关文献中通常采用和有限元法中类似的插值方案，利用形函数来强加界面约束条件（节点到非节点）以避免接触表面之间的贯入。付出的代价是丧失了边界元法原有的一些优良特性，接触边界条件在离散意义下也不能再在整个边界单元上精确满足。本文基于作者研究组对二维弹性体移动和滚动接触边界元法的前期研究，将其中的协调离散方案推广到三维问题，提出了针对移动方向已知的三维弹性体移动接触问题的一种边界元协调离散方案。接触面上的位移和面力边界条件均能在边界元离散意义下精确满足，因此它能够保持边界元法的优良特性。
文中给出了一些算例来验证所提出的移动接触算法的有效性和高精度。这些算例着重于在接触区附近有孔洞型缺陷的三维弹性体的移动接触。对于此类问题，在文献中通常用相应于Hertz解的移动载荷来代替移动接触，这种处理忽略了缺陷与移动接触之间的耦合效应。本文的算例表明，在缺陷离接触表面很近的情况下，这种耦合效应是不能忽略的。
最后，本文还针对移动方向未知或可变的三维弹性体移动接触问题提出了一种边界元协调离散方案，保证了接触面上位移协调性和面力平衡条件的同时满足。
As an effective numerical analysis method of scientific and engineering problems developed following Finite Element Method (FEM), Boundary Element Method (BEM) has some attractive advantages, such as easier simulating complex boundary shape, high accuracy and dimension reduction. For the boundary nonlinear problems, such as elastic contact problem, the boundary displacements and boundary tractions are just the basic variables of the boundary integral equation, and the contact conditions can be satisfied with higher accuracy in BEM; therefore, the BEM can be applied to solve the elastic contact problem more accurately. In this dissertation, some basic investigations on the BEM of 3D elastic contact problem have been carried out, which can be listed as follows:
At first, an early investigation of authors' group, on direct error estimation of BEM solution for elasticity problem, is extended from 2D problem to 3D elastic contact problem. An accurate and efficient algorithm for the determination of boundary displacement, which is continuous with the displacement solution within the domain of an elastic body, is then presented. Based on the difference between the corresponding limit of displacement from both side of contacted bodies, a local direct error estimator of BEM solution for 3D elastic contact problem is presented, and then a scheme of adaptive BEM is suggested. This scheme can be used to estimate the error of the complicated contact problems without corresponding analytical solution.
To solve the moving contact problem, a kind of interpolation schemes, which utilizes shape function to impress interfacial constraint conditions (node to point) to prevent penetration between the contacted surfaces, is adopted generally in the references of BEM, as used in the FEM. Some good characteristics of BEM are lost as a significant cost; the contact boundary conditions can not be satisfied on whole boundary, even in the sense of discretization. In this thesis, based on the previous investigation on 2D moving and rolling contact problem by BEM, the conforming discretization is generalized to 3D cases, and a scheme of BEM for moving contact of 3D elastic solids with prescribed moving direction using conforming discretization is presented. Both the displacement and traction boundary conditions are satisfied on the contacted region in the sense of discretization. In this way the good characteristics of BEM can be preserved.
Some numerical examples are given to show the effectiveness and higher accuracy of the presented scheme of moving contact. It is emphasized to the moving and rolling contact of 3D elastic bodies with hole-type defect in the vicinity of contact region. For such kind of problems, instead of the moving contact it is treated as moving loads corresponding to the Hertz solution in the references. In this way the coupling effect between defects and moving contact has been neglected. But the presented numerical examples show that such coupling effect could not be neglected, provided the defect located in the vicinity of the contact region.
Finally, a scheme of BEM for moving contact of 3D elastic solids with unknown or variable moving direction using conforming discretization is presented. Both the displacement and traction boundary conditions are satisfied on the contacted region in the sense of discretization.
作者首先将本研究组提出的弹性力学问题边界元解误差的直接估计从二维推广到三维问题，给出了确定与域内解连续的边界位移的一种精确有效的方法。在此基础上提出将接触体接触单元间与域内解连续的边界位移之差的某种度量作为三维弹性体接触问题边界元法的一种误差直接估计，并且提出了三维弹性体接触问题边界元法的一种自适应计算方案。这种方案为确定没有解析解可作比较的复杂接触问题边界元解的精度提供了可能。
对于移动接触问题，边界元法的有关文献中通常采用和有限元法中类似的插值方案，利用形函数来强加界面约束条件（节点到非节点）以避免接触表面之间的贯入。付出的代价是丧失了边界元法原有的一些优良特性，接触边界条件在离散意义下也不能再在整个边界单元上精确满足。本文基于作者研究组对二维弹性体移动和滚动接触边界元法的前期研究，将其中的协调离散方案推广到三维问题，提出了针对移动方向已知的三维弹性体移动接触问题的一种边界元协调离散方案。接触面上的位移和面力边界条件均能在边界元离散意义下精确满足，因此它能够保持边界元法的优良特性。
文中给出了一些算例来验证所提出的移动接触算法的有效性和高精度。这些算例着重于在接触区附近有孔洞型缺陷的三维弹性体的移动接触。对于此类问题，在文献中通常用相应于Hertz解的移动载荷来代替移动接触，这种处理忽略了缺陷与移动接触之间的耦合效应。本文的算例表明，在缺陷离接触表面很近的情况下，这种耦合效应是不能忽略的。
最后，本文还针对移动方向未知或可变的三维弹性体移动接触问题提出了一种边界元协调离散方案，保证了接触面上位移协调性和面力平衡条件的同时满足。
As an effective numerical analysis method of scientific and engineering problems developed following Finite Element Method (FEM), Boundary Element Method (BEM) has some attractive advantages, such as easier simulating complex boundary shape, high accuracy and dimension reduction. For the boundary nonlinear problems, such as elastic contact problem, the boundary displacements and boundary tractions are just the basic variables of the boundary integral equation, and the contact conditions can be satisfied with higher accuracy in BEM; therefore, the BEM can be applied to solve the elastic contact problem more accurately. In this dissertation, some basic investigations on the BEM of 3D elastic contact problem have been carried out, which can be listed as follows:
At first, an early investigation of authors' group, on direct error estimation of BEM solution for elasticity problem, is extended from 2D problem to 3D elastic contact problem. An accurate and efficient algorithm for the determination of boundary displacement, which is continuous with the displacement solution within the domain of an elastic body, is then presented. Based on the difference between the corresponding limit of displacement from both side of contacted bodies, a local direct error estimator of BEM solution for 3D elastic contact problem is presented, and then a scheme of adaptive BEM is suggested. This scheme can be used to estimate the error of the complicated contact problems without corresponding analytical solution.
To solve the moving contact problem, a kind of interpolation schemes, which utilizes shape function to impress interfacial constraint conditions (node to point) to prevent penetration between the contacted surfaces, is adopted generally in the references of BEM, as used in the FEM. Some good characteristics of BEM are lost as a significant cost; the contact boundary conditions can not be satisfied on whole boundary, even in the sense of discretization. In this thesis, based on the previous investigation on 2D moving and rolling contact problem by BEM, the conforming discretization is generalized to 3D cases, and a scheme of BEM for moving contact of 3D elastic solids with prescribed moving direction using conforming discretization is presented. Both the displacement and traction boundary conditions are satisfied on the contacted region in the sense of discretization. In this way the good characteristics of BEM can be preserved.
Some numerical examples are given to show the effectiveness and higher accuracy of the presented scheme of moving contact. It is emphasized to the moving and rolling contact of 3D elastic bodies with hole-type defect in the vicinity of contact region. For such kind of problems, instead of the moving contact it is treated as moving loads corresponding to the Hertz solution in the references. In this way the coupling effect between defects and moving contact has been neglected. But the presented numerical examples show that such coupling effect could not be neglected, provided the defect located in the vicinity of the contact region.
Finally, a scheme of BEM for moving contact of 3D elastic solids with unknown or variable moving direction using conforming discretization is presented. Both the displacement and traction boundary conditions are satisfied on the contacted region in the sense of discretization.
引文
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