Analysis Of The Elastic Contact Problems With Boundary Face Method And The Research On Its Fast Method With Low Rank Approximation

In mechanical engineering, the motion and load is usually transferred between the mechanical components by the contact behavior, such as gear transmission and rolling bearing, etc. The contact behavior always causes stress concentrations, which increase the risk for crack initiations, propagations and fatigue failure. Therefore, the research on contact problems has important practical significance.With the development of CAE technology, the Finite Element Method(FEM) and Boundary Element Method(BEM) are used to analyze the contact problems in engineering. However, whether it is in the FEM or BEM, the CAE model only contains geometric information of mesh, the CAE model and CAD model are independent of each other. Therefore, it is inevitable that the discrete geometric error is introduced from the CAD solid model to CAE mesh model. The Boundary Face Method(BFM) is a new boundary type numerical method based on the boundary integral equation(BIE) and the computer graphics. In BFM, the boundary integral equation and the boundary representation(B-Rep) data structure used in CAD modeling are organically combined. The discretization of BIE and the variable interpolation are implemented in the parametric space of boundary surfaces. The geometric data in integration elements are calculated directly from the formula of a parametric surface, not approximated by mesh which avoids the discrete geometric error. Moreover, the contact equations are written explicitly with both tractions and displacements which are retained as unknowns in BFM. In this paper, the BFM is applied to analyze the elastic contact problems. Some key problems about the non-linear in the contact problems are solved. Meanwhile, for the low rank approximation of the matrices in the BFM, this paper has studied the Adaptive Cross Approximation(ACA) and attempts to propose a non-iterative cross approximation algorithm based on the geometric information of the nodes. As a result, the following studies are carried out in this dissertation.(1) The BFM is extended to sovle the multi-domain elastic problems. In contact problems, the contact conditions are characterzed by unilateral inequality constraints. The non-linearities are usually introduced even the contact between the linear elastic materials. However, from the point of view of numerical analysis, the elastic contact problem is a multi-domain elastic problem with special interface conditions. Therefore, as a basis for analyzing the elastic contact problem, this paper firstly extends the BFM to solve the multi-domain elastic problem, and designs an independent C++ data structure for the interface conditions. The data structure can be not only used to treat the interface conditions in multi-domain and contact problems, but also to implement the contact search algorithm conveniently.(2) A contact search algorithm based on the parametric space of surface is proposed. The data structure of BFM contains not only the information of mesh, but also includes the parametric information of boundary surfaces. In the theoretical framework of BFM, this paper implements the contact search algorithm in the parametric space of surface. Firstly, the contact node is projected to the associated contact surface. And the parametric coordinates of the projection point on contact surface can be obtained. Then, a quadtree structure is introduced in the parametric space of the contact surface, and the associated element can be efficiently obtained by the hierarchical characteristic of tree structure.(3) The contact constraint equations are constructed with the Node-to-Surface(NTS) contact discretization for the frictionless and friction contact problems. In the numerical analysis of contact problems, the contact discretization is used to define the transfer mode of motion and load. The Node-to-Node(NTN) contact discretization in the early requires that the contact nodes only come into contact with other nodes by the identical discretization in the contact area. It is difficult to generate mesh in the contact zone. In this paper, the NTS contact discretization widely used in FEM is introduced in the BFM for contact problems. The contact constraint between the projection point and the associated element is defined by the shape function of element on the contact area, and the contact equations can be constructed for the frictionless and frictional contact problems. The two contact bodies can be meshed independently. It reduces the difficulty of meshing.(4) A contact iterative algorithm is introduced to solve the frictionless contact problems with the BFM. Because of not considering the influence of friction, the final state of the displacement and traction fields is independent of the loading history. Therefore, the non-linearity is introduced in frictionless contact problems due to the priori unknown of the contact area. In this paper, the non-linearity is treated by a contact iterative algorithm. First, assume the initial contact area, and establish the contact equations. The whole system of equation can be assembled and solved for the first time. Then, the contact nodes will be decided by the contact inequalities(non-interpenetration and compressive contact traction). When the contact nodes do not satisfy the contact inequalities, the contact equations will be updated, and the solution is restarted. The contact area is revised constantly in the iterative procedure, until all the contact nodes satisfy the contact inequalities.(5) An incremental-iterative algorithm is introduced to solve the frictional contact problems with the BFM. Because of the irreversible process of friction, the final state of the displacement and traction fields is determined by the loading history. The frictional contact problem has to be solved by increments. Due to the priori unknown of the contact area and the friction state, the frictional contact problem has high non-linearity. In this paper, an incremental technique is introduced from the FEM for treating frictional contact problem. An incremental-iterative algorithm is proposed to solve the frictional contact problem. In the normal contact, the contact area is determined by the non-interpenetration and compressive contact traction conditions. In the tangential contact, the coulomb law is introduced to determine the friction state of the contact nodes, and to construct the corresponding contact equations. When the interpenetration occurs in the increment, this increment will be divided by scaling the interpenetration distance, and the solution is restarted. Between two increments, the friction state and slip angel in the previous increment will be taken as the initial assumptions for the next increment, and then the iterative procedure is performed.(6) A non-iterative cross approximation algorithm is proposed based on the geometric information of the nodes. Whether it is in the Fast Multipole Method(FMM) or ACA, the far-field attenuation characteristics of the fundamental solution in the BIE is used to construct the low rank approximations for the far-field matrices. When the high-order elements are used in the BFM, the efficient process of numerical integration is as following: firstly, the data for numerical integration is calculated on an element, such as the coordinates of Gauss integration points, normals, Jacobi coefficient, etc. Then, calculate the numerical integration through all the nodes by an element-to-nodes approach. However, it will be repeatedly calculate the data on the element when calculate the numerical integration by a node-to-elements approach in turn. It will reduce the computational efficiency.It is required to repeatedly calculate the data on the element in the iterative procedure of ACA. And, the low rank approximation error is estimated in each iteration step, which accounts the main part of calculation for the whole iterative process. Obviously, a non-iterative cross approximation algorithm can greatly improve the computational efficiency. Therefore, we attempt to propose a non-iterative cross approximation algorithm based on the geometric information of the nodes. It can determine the skeleton points before the numerical integration. So the element-to-nodes approach can continue to be used to calculate the numerical integration and avoid repeatedly calculating the data on the element.