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一种无摩擦接触问题的有限元方法 摘要文章提出了一种新的解决包括整体可能经历一定的运动和变形的接触性问题的有限元方法。这个方法是基于两结构接触性问题变成两个同时发生的问题的分解，最终结果是在几何学上而不是在离散的接触表面上。一个表面接触单元是特定设计在无条件下，允许在两接触表面间进行一直的牵引动力的传输。关键词：无摩擦性接触，大的变形，有限元1.引言有限元方法广泛应用在解决接触性问题上。从完全的计算角度来看，接触性的检测和随后的强制性约束的实现是有待解决一般算法结构的发展的两个重要问题。从康里和瑟瑞吉，禅和图巴的早期作品以后，很多方法论都在接触机械学的文献中被提出了。一个相当广泛的在题目中的调查在参考书目中发现。目前的工作是参与适合的解决在大的运动和变形的两机构的接触性问题有限元分析方法的发展。这一类问题在许多实际应用中有特别重要的意义，如金属成形过程和车辆事故分析。各种商业和科研计算机编码采用的算法，以解决这种问题。拉格朗日算子方法和它们的规格化（罚数和增广的拉格朗日方法）是典型地被用在执行不可测知中。 工作性条件的集成方法的选择在线性动力平衡方面的微弱形式下与接触牵引动力有联系并在接触原理建设中起着关键的作用。节点正交的使用包括一个（二个）点或一个（两个）基座或点和面的接触。假设存在一些连续的接触表面，另外的整合法则也都是合适的。根据两机构问题接近于连续的两同时发生问题，目前的文章的重要贡献是鉴定一般的程序。像传统的双行程运算法则，两相互作用机构的表面被用来分析而不需要采用中间的接触表面(任意的被选择)。提出的方法的重要优势与双程节点上表面的运算法则相比，如果允许一个一体化规则的简单解释用于接触面，并为可容许领域的适当选择，允许一个物体到另一个物体的连续压力的精确传播。根据源于镣铐的工作中的斑贴试验，及其以后的概括，恒压（在大小和方向）的精确代表性的能力被看做是一个健全的必要条件和总体接触算法的收敛。接触力学的一个简洁的阐述在第2节中出现，特别强调在接连发生的算法的发展中常常用公式表示。在第3节中提出并分析了一个二维接触单元，而在第4节中，提出和讨论了正在使用的这个单元的选择数值模拟的结果。在第5节中，给出了结束的评论。2. 两物体接触问题考虑物体认为等同于线性空间的开连通集，配有标准基(,)和欧几里德准则。至少有一个物体是假定可变的。在参考位形中的一个典型的重要点是在线性构造中重要点用向量数学地说明，对于每一个t给出，移植向量是通过来规定的。假定至少在内，映射在定义域内是连续和可转置的。向量，在线性结构（以t变化）中它的范围是用和来确定的，因此，则有和 。并且的外在的单位标准记作。多个物体（包括一个物体）的任何系统的提议都受制于物质的不可测知性，在引文11（224页）被Truesdell和Toypin规定。这意味着这两机构问题一直有 （1）在任何给定的时间上，所说的这两个机构都与它们的边界子集C有联系，当且仅当 (2)根据上面的定义，每个物体的定义可以被唯一的分为三个相互排斥的区域，根据下面的式子当狄里克莱和诺艾曼边界条件分别地被强加在 上 。虽然没有明白的之处，但是一般依照时间来说，它应该不包含, 和 C。不连续函数，可能是多值的，每个物体的边界按照如下的定义：对每一个，是给定的为 （2a）当是在这样的式子中看做指数是1。的凸面性使的值是惟一的，尽管有如此的一个几何条件的限制也不能强加在开端。的一个完整的相似的定义生成下面的结果： (2b)又有对每一,满足。定义方程式(2a) 和(2b)即指不连续函数 和 在C上是相等的，都等于0.也就是 (3)因此不可测知条件（1）根据上面的不连续函数被重写作。在不存在惯性效应时，方程式的局部形式控制着如下给出的每个物体的运动，div in (4a) on （4b） on (4c) on (4d)当是克西的应力张量，是质量密度，是每单位质量的质量力，是法定的界壁位移，是在上的法定牵引向量。标准加权残值法的应用，与拉格朗日乘数的引入p 0对费解的限制相结合，结果在运动方程的若形式中指出，位移解方程（ 4 ）和拉格朗日乘数外地p满足 (5a) (5b)和q是对所有的允许函数。在大部分的缺失不被用在方程(5a)中时，则不连续函数定义在C上。位移区域属于空间即质量函数也属于空间，定义为允许函数p 0是分段连续的。为了证明方程式(5a)的正确性，把这项工作看做是允许函数和上沿C的触点压力，即为 (6)在C上，在没有摩擦和回顾时，柯西引理应力矢量意味着 (7)借助于方程(7)，方程(6)可写为这表明拉格朗日乘数区域是自然等同于正常的在接触范围内的牵动引力（压力）。压力场p一般假定仅仅是分段光滑的，因此认为每个物体的特征材料的接触面在C的附近。而且由于 on (8)不等式(5b)是从(4d)的形式中得到的，而且假定q是非负的。为了进一步证明拉格朗日乘数在两物体接触问题中的作用，为接着发生的近似数值提供了一些动机，用(2b) 和(8)改写(5b)中在接触面C上构成整体所必需的，即 (9)符号仅仅被用来强调接触面C不普遍的看做的一部分，就像(3)中所表明的。整式(9)表明，在每个物体的范围内，适当的给定一个拉格朗日区域，(5b)可以在每个接触面上分别作用。然而，很显然区域应该满足在（一般地）接触面上的线动量的平衡，即为以上资料数据将被应用在(5a) 和 (5b)的近似解决方案中。.(9)式的推导用下面的程序，等式(5a) 和(5b)被记作 (10a) (10b)当，对所用可容许的和都适用。(5a) 和 (5b)的典型的罚数调整是在的条件下获得。 当 是一个Macauley支架和时，上面的（5a）式的积分关系是一个连续的损失参数。约束条件(4d)是无约束的（因此允许受约束的渗透来替代）替代区域是确定的，则有 (11)在约束条件下，原始的无约束问题等同与一个凸面的最小化问题，序列的解决方案当时表明收敛是约束问题的解决方案(5).实践中，罚数方法成功的用在平衡状态下与最小总势能不相关的情况下。由于罚数参数的有限值的普及率，接触面C的一个独特的定义和缺陷函数并不是容易地应用在(11)中的。在提出的明确表达中关键的一步是(10a) 和 (10b),易于接受的规则化的罚数的引入，例如由(12)，等式(10a)可以写为 (13)上面的(13)中的任意两个积分在形式上是相同的，即积分边界来源于一个新问题的规则化罚数。尽管在目前的背景下这两种积分在缺陷函数的两个定义(2a) 和(2b)的基础上是有联系的，但是这种分解要设计要涉及到重要的计算，这在下一节中再讨论。这里提出的事态发展可以很容易地靠归纳两体问题扩展到多体接触问题。3. 三维接触的两个应用这篇文章的其余部分专门讨论在明确表达罚数的基础上的离散二维接触问题和两表面上的缺失函数的鉴定，像在第二节中所述。这里待确定的两个关键问题是接触单元的几何作图和有限元近似值的可允许区域的选择。3.1 接触面的离散化如果有一个接触面的离散化是连续的，那么接触面 和 是分别在 和的条件下是唯一确定的。撇开推行细节，接触面是被一系列正常的从 到的投影（不定靠近点）所离散，如图表2中所示。的离散化采用相同的步骤。因此每个接触单元在接触面上的立体空间元素与其对立面相联系。显然地，非光滑边界的离散会导致单位法线和缺失函数的不连续。这里目的不是设法回避这个问题，尽管它的结果不可忽略，特别是在滚动问题中。对于一个特殊的离散来说会导出代表两个层面的光滑边界，见参考资料13 。3.2 有限元区域在这节中假设一个具体的接触有限元是建立在方程式(10a) 的(12)的基础上。选择有限维领域的指导方针是以前的作品特别是在青年受理领域位移和压力导致单方面接触单元，能够稳定地复制费解的条件（即满足的LBB条件）是准确的。这种收敛性分析，目前尽管并不适用于这里的两体接触问题的动非线性这方面，但是它为可允许区域的选择提供了一个准则。数值积分典型地把所有边界术语用在接触面上。不考虑离散化使用空间的元素（例如三角形的三个定点和四边形的四个点）的线性规则。在单个接触单元中的缺失函数是关于轨迹边界坐标系的非标准函数。因此，一般情况下在上的数值积分是不准确的，所有的引入错误的积分规则直接影响到用公式的确切表示。所提出的接触单元是建立在从发源到标准Q9元素（等参的九个节点）的移置区域和引用的辛普森标准上的。A NOVEL FINITE ELEMENT FORMULATION FOR FRICTIONLESS CONTACT PROBLEMSSUMMARYThis article advocates a new methodology for the finite element solution of contact problems involving bodies that may undergo finite motions and deformations. The analysis is based on a decomposition of the two-body contact problem into two simultaneous sub-problems, and results naturally in geometrically unbiased discretization of the contacting surfaces. A proposed two-dimensional contact element is specifically designed to unconditionally allow for exact transmission of constant normal traction through interacting surfaces.KEY WORDS frictionless contact; large deformations; finite elements1. INTRODUCTIONFinite element methods are used extensively in the solution of contact problems. From a purely computational standpoint, detection of contact and subsequent satisfaction of the impenetrability constraint are the two key issues to be addressed in the development of a general algorithmic framework. Numerous methodologies have been proposed in the literature of compulational contact mechanics since the early works of Conry and Seircg,and Chan and Tuba.A fairly comprehensive survey on the topic is found in Reference.The present work is concerned with the development of finite element methods suitable for the solution of two-body contact problems in the presence of large motions and deformations. This class of problems is of particular significance in numerous practical applications, such as metal forming processes and vehicular crash analyses. Various commercial and research computer codes employ algorithms for the solution of such problems. Lagrange multiplier method,and their regularizations (penalty and augmented Lagrangian methods) are typically used in enforcing impenetrability.The choice of integration method for the work-like term associated with the contact tractions in the weak form of linear momentum-balance plays a pivotal role in the construction of contact elements. Use of nodal quadrature involving the contacting nodes of one of (resp. both) surfaces yields the standard one-pass (resp. two-pass) node-on-surface algorithms.Other integration rules are also applicable, provided there exists a continuous discretization of the contact interface.The main contribution of the present paper is in the identification of a genneral procedure according to which the two-body problem is approached as a sequence of two simultaneous sub-problems. As in the traditional two-pass algorithms, the surfaces of both interacting bodies are used in the analysis without need for introduction of an (often arbitrarily chosen) intermediate contact surface. The main advantage of the proposed approach over the two-pass node-on-surface algorithms if that it allows for a straightforward interpretation of the integration rules used on the contacting surfaces and, for appropriate choices of admissible fields, permits the exact transmission of constant pressure from one body to another. In the spirit of the patch test originating in the work of Irons,and its subsequent generalizations, capability for exact representation of constant pressure (in both magnitude and direction) is viewed as a necessary condition for robustness and convergence of the overall contact algorithm.A brief exposition to contact mechanics is presented in Section 2, with particular emphasis on formulations to be used in the ensuing algorithmic developments. A two-dimensional contact element is proposed and analysed in Section 3,while the results of selected numerical simulations using this element are presented and discussed in Section 4.Concluding remarks are given in Section 5.2. THE TWO-BODY CONTACT PROBLEMConsider bodiesidentified with open and connected sets in linear space , equipped with canonical basis (,) and the usual Euclidean norm. At least one of the bodies is assumed to be deformable. A typical material point of in the reference configuration is algebraically specified by vector of material point in the current configuration is given byat each time t, and the displacement vector is defined according toThe mapping is assumed smooth throughout its domain and invertible at least on .Body and its boundary in the current configuration (at time t) are identified with and ,respectively, henceand .Also, the outer unit normal to is denoted by .The motion of any system of bodies (including a single body) is subject to the principle of impenetrability of matter, as stated by Truesdell and Toypin in Reference11 (p.244).This implies for the two-body problem that at all times （1）At any given time, the two bodies are said to be in contact along a subset C of their boundaries if, and only if, (2)It follows from the above definetion that the boundary of each body can be uniquely decomposed into three mutually exclusive regions according toWhere Dirichlet and Neumann boundary conditions are enforced on and , respectively. Although not explicitly noted, it should be clear from the above that , and C generally depend on time.Gap functions , possibly multi-valued, can be defined on the boundary of each body as follows: for each , is given by （2a）Where is such that see Fifure 1.Convexity of renders single-valued, although such a restrictive geometric condition will not be imposed at the outset . A completely analogous definition for yields (2b)Where,again, for each , satisfies .Defining equations (2a) and (2b) imply that gap functions and are identically equal to zero on C, namely that (3)Consequently, impenetrability condition (1) can be rewritten in terms of the above gap functions asAt the absence of inertial effects, the local form of the equations governing the motion of each body are given bydiv in (4a) on (4b) on (4c) on (4d)Where is the Cauchy stress tensor, the mass density, the body force per unit mass,the prescribed boundary displacement, and the prescribed traction vector on .Application of the standard weighted-residual method,in conjunction with the introduction of Lagrange multiplier p0 for the impenetrability constraint,results in the weak form of the equations of motion, which states that the displacement solution of equations (4) and the Lagrange multiplier field p satisfy (5a) (5b)For all admissible function and q. Without loss of generality is used in equation (5a) for the definition of gap function on C. Displacement fields belong to spaces withand weight functions belong to spaces defined asThe admissible functions q0 are piecewise continuous.To prove that equation (5a) holds, note that the work done by contact forces along C on admissible functions and is given by (6)At the absence of friction and recalling that on C, Cauchys lemma on the stress vector implies that (7)With the aid of equation (7), equation (6) is written as,Which shows that the Lagrange multiplier field is naturally identified with normal traction (pressure) along the contact region. The pressure field p is generally assumed to be only piecewise smooth, thus allowing each body to feature material interfaces in the neighbourhood of C. Moreover, since on (8)inequalities (5b) follow from (4d) and the assumed non-negativeness of q.In order to further clarify the role of the Lagrange multipliers in the two-body contact problem and provide some motivation for the ensuing numerical approximations, use (2b) and (8) to rewrite the integral in (5b) on surface C as (9)The notation is employed to merely emphasize that the contact surface C can be selectively viewed as part of ,as indicated in (3).The integral expression (9) suggests that, given a properly defined Lagrange multiplier field on the boundary of each body, integration of (5b) can be performed separately on each of the contacting surfaces. However, it is clear that fields should satisfy balance of linear momentum on the (common) contact surface, namely that The above observation will be exploited in the approximate solution of (5a) and (5b). Folowing the procedure used in the derivation of (9), equations (5a) and (5b) are rewritten as (10a) (10b)for all admissible and ,where .The classical penalty regularization of (5a) and (5b) is obtained by settingin the last integral term of (5a),where is the Macauley bracket and is a constant penalty parameter. Constraint conditions (4d) are relaxed (thus permitting controlled penetration to take place) and displacement fields are determined so that (11)for all .Under restrictions such as formal equivalence of the original unconstrained problem to a convex minimzation problem，the sequence of solutions as can be shown to converge to the solution of the constrained problem (5).In practice, penalty methods are used successfully even when the equilibrium state is not associated with minimization of the total potential energy.Due to the occurrence of penetration for finite values of the penalty parameter ,a unique definition of the contact surface C and gap function g is not readily available for use in (11).A crucial step in the proposed formulation is the introduction of a well-defined penalty regularization for (10a) and (10b),such that (12)With the aid of (12),equation (10a) becomes (13)Each of the last two integrals of (13) identical in form to the boundary integral emanating from penalty regularization of a Signorini problem. Although in the present context these two integrals are clearly coupled by the definitions (2a) and (2b) of gap functions , this decomposition has important computational implications, as will be discussed in the next section.The developments presented here can be easily extended to encompass the muli-body contact problem by reducing it to a series of coupled two-body problems.3. APPLICATION TO TWO-DIMENSIONAL CONTACTThe remainder of this article is devoted to the discretization of two-dimensional contact problems based upon the penalty formulation and the identification of gap functions on both surfaces, as suggested in Section 2.The two key issues to be addressed here are the geometric construction of contact elements and the choice of admissible fields for the finite element approximation.3.1. Discretization of the contact surfacesA continuous discretization of the contact surfaces is advocated. Surfaces and are uniquely determined as those on which and, respectively. Setting aside implementational details, contact surface is discretized by a series of normal projections (not necessarily closest-point) from to ,as shown in Figure 2. An analogous procedure is followed for the discretization of .Consequently, each contact element relates a single spatial element edge on surface to the opposite surface.The apparent non-smoothness of the discrete boundaries results in discontinuity of unit normals and gap functions. No attempt is made here towards circumventing this problem, although its effect might not be negligible, especially in problems of rolling contact. For a special discretization that results in smooth boundary representation in two-dimensions, see Reference 13.3.2. Finite element fieldsA specific contact finite element is suggested in this section, based upon equeations (10a) and (12). The choice of finite dimensional fields is guided by previous works especially on the Signorini admissible displacement and pressure fields lead to unilateral contact elements that are able to stably replicate the impenetrability condition (i.e. they satisfy the underlying LBB condition) and are accurate. Such convergence analysis, although not currently available for the kinematically non-linear two-body contact problem addressed here, provides a guideline for the selection of admissible fields.Numerical integration is typically employed for all boundary terms on the contact surface. Discounting discretizations that use straight-edge spatial elements(e.g. three-node triangles and four-node quadrilaterals),the gap functions within a single contact element are non-poly-nomial with respect to local boundary co-ordinate systems. Thus, numerical integration on is generally inexact and all integration rules introduce errors that directly influence the formulation. The proposed contact element is based on displacement fields emanating from standard Q9 (nine-node isoparametric) elements and employs Simpsons integration rule.
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