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广西工学院200届设计说明书 An efficient shape optimization method based on FEM and B-spline curves and shaping a torque converter clutch diskS. Hyun, C. Kim, J. H. Son, S. H. Shin and Y. S. KimDepartment of Mechanical Engineering, Kyungpook National University, 1370 Sankyuk-Dong, Book-Gu, Daegu 702-701, South KoreaReceived 3 July 2003; accepted 18 January 2004. Available online 26 February 2004.AbstractAn efficient shape optimization scheme has been developed for designing axis symmetric structures. The sequential linear programming and Simplex method are coupled with finite element analysis. Selected sets of master nodes on design boundaries are employed as design variables and assigned to move towards their normal directions. By interpolating the repositioned master nodes, B-spline curves are constructed so that the remaining nodes on the design boundaries efficiently settle on the B-spline curves. A mesh smoothing scheme is also applied to interior nodes to maintain the finite elements in good quality. Applying these techniques a numerical implementation is presented to obtain the optimum design of an automobile torque converter clutch disk subjected to a hydraulic pressure and centrifugal force loads. The results give the optimum shape of the disk showing weight saving up to 13% after the shape optimization.Author Keywords:Finite element method; Shape optimization; B-spline curve; Adaptive remeshing; Torque converter lock-up clutch diskArticle Outline1. Introduction2. Shape optimization algorithm3. Shape optimization scheme3.1. B-spline curve3.2. Mesh optimization4. Applications and numerical results4.1. Cantilever4.2. Fillet4.3. Torque converter lock-up clutch disk5. Concluding remarksAcknowledgementsReferences 1. IntroductionA number of studies on the shape optimization of structures have been carried out. For a broad class of shape optimization problems, no additional weight reduction can be achieved without changing the shape and location of the boundary of design domains. To this end B-spline curves and surfaces are widely used to represent the continuous boundary surfaces of design domains during the shape optimization process.The first shape optimization, using the finite element method with iso-parametric elements, is proposed by Zienkiewicz and Campbell 1. They used the sequential linear programming and Simplex method for finding the optimal shape of an arch dam. During the shape optimization process, the structural shape is changed by repositioning the boundary nodes. By the 1980s, most researchers used all the nodes along the boundary surface as design variables. Such a scheme, however, sometimes produces discontinuous or unstable boundary surfaces and is often apt to produce distorted mesh causing less reliable solutions. To cope with this problem, Fleury and Baribant 2, instead of the surface nodes, adopted the control points of B-spline curves as design variables which can construct smooth design boundary surface. Bennett and Botkin 3 suggested an automatic mesh generation method coupled with an adaptive mesh refinement scheme during a shape optimization process.An important issue in automobile design is to reduce the weight of an automobile in order to improve the efficiency of power transfer and fuel consumption (km/l). Recently, most passenger cars were adopting the front-engine and front-driving system that creates additional difficulty in balancing the fore-and-after weights. One of the heaviest parts in an automobile is the engine and power transmission which occupy 2025% of its total weight. The torque converter lock-up clutch disk, a part of the automatic transmission, transfers engine power to a driving shaft by hydraulic pressure. It rotates at various speeds and is subjected to frequent pressure loads. Thus, there are high demands for designing the lock-up clutch disk fitting all the working conditions and maximizing its cost effectiveness.The objective of this study is to propose an effective shape optimization scheme to determine appropriate boundary surfaces of an automobile torque converter lock-up clutch disk. B-spline curves 4 and 5, representing the design boundary, are employed in the optimization process and nodes on the design boundary are allowed to move by a proposed method. Besides, the mesh smoothing technique is applied to improve overall mesh quality. A cantilever and a fillet case are implemented to verify the performance of the proposed optimization scheme. 2. Shape optimization algorithmThe shape optimization problem is to find the design variables X while minimizing the objective function W(X) under the constraint functions gi(X), and can be stated mathematically asto minimize W(X)subject to gi(X)0,XLXXH,where XL and XH are allowable lower and upper limits of the design variables which are introduced to deal with various requirements. Many efficient schemes are suggested achieving optimum solutions for non-linear optimization problems 6. In the present study, the sequential linear programming and Simplex method 7 are used for shape optimization. This method involves linearizing the non-linear objective function W(X) and the non-linear constraint functions gi(X) at a design point, XP. Only the linear terms from Taylors expansion are taken and the higher-order terms are ignored. The linearized objective function and constraints are as follows:minimize W=c+bTXsubject to AXd,X0,wherec=W(XP)TW(XP)XP, The partial derivatives that appear in the equations above are obtained by the semi-analytical sensitivity analysis 8. In the shape optimizations of structures, the objective function W(X) represents volume, the design variables X coordinates of boundary nodes and the constraint functions gi(X) the maximum allowable stresses, respectively. The minimization by simplex method requires design variables to be updated repeatedly until a proper convergence is reached. The determination of convergence is made by a volume change caused by continuous shape optimization. The ratio of a volume change is expressed in two adjacent volume aswhere Vi is volume at iteration i and Vi1 is volume at iteration i1. When the ratio becomes less than 0.001, the optimization process is considered to be converged. 3. Shape optimization schemeThe shape optimization scheme consists of constructing B-spline curves to represent design boundaries and mesh smoothing techniques to improve interior mesh shapes. 3.1. B-spline curveA B-spline curve is represented in terms of a blending function aswhere Nik(u) is the blending function given asandIn the above equation, Pi is a set of control points, ti is knot values and k is the order of the B-spline curve. The knot values for a non-periodic B-spline curve that passes two end points are given byEq. (14) expressed in the matrix form becomesD=NB,B=N1D,where D is the matrix of data points, N is the matrix of a blending function and B is the matrix of control points. By substituting the obtained control points from Eq. (19) into Eq. (18), an arbitrary B-spline curve passing through the data points can be defined. Fig. 1 demonstrates the constructing procedure of a B-spline curve using six given data points 4. The requirement for any B-spline curve is given aspn+1,where p is the order of the B-spline curve and n is the number of the data points. In the present work the fifth order of B-spline curves are employed for each span consisting of six data points. Fig. 1. The construction of a B-spline curve from nodal data points.In the process of shape optimization, taking the coordinates of entire nodes as design variables makes the computation exceedingly long and difficult in keeping a design boundary smooth 6. In the present work, a group of carefully chosen nodes along a design boundary is assigned to master nodes. The master nodes are considered as design variables and are directly repositioned during a shape optimization process. Then the other remaining nodes along the design boundary are interpolated by applying B-spline curves. This implementation is done as follows: (1) All nodes on a design boundary are alternatively grouped into master nodes and mid-nodes. End nodes are always assigned to master nodes (see Fig. 2(a). (2) The new coordinates of the master nodes are determined in a shape optimization process. They are constrained to move along their normal directions on the design boundary (see Fig. 2(b). (3) A B-spline curve is constructed from the master nodes (see Fig. 2(c). (4) The mid-nodes between two master nodes are then moved onto the newly constructed B-spline curve where the distance from the two neighboring master nodes is equal (see Fig. 2(d). As a result, all nodes along the boundary can be repositioned so that smooth and continuous design boundary curves are secured. Fig. 2. Procedure to construct a new design boundary: (a) master and mid-nodes on a design boundary; (b) moving directions at master nodes; (c) movements of the master nodes and forming a B-spline curve; and (d) determination of moving vectors at the mid-nodes.3.2. Mesh optimizationIn a shape optimization process, the shape of a structure is changed due to the movement of the nodal points on a design boundary. Thus, it is necessary to prevent meshes from excessive distortion so that finite element analysis provides continuously accurate solutions during an optimization process. A mesh smoothing technique is applied to improve mesh quality. Fig. 3 illustrates the typical interior meshes where eight neighboring nodes (x,y)i are related at a center node (x,y)0. To reposition a center node (x,y)0 a new position (x,y)new can be determined by manipulating the coordinates of its neighboring nodes and the areas of its neighboring elements. This is expressed as 9(x,y)new=(1) (x,y)new,L+(x,y)new,A,where is an arbitrary value between 0 and 1 to be determined. All nodes which belong to interior mesh are repositioned repeatedly until all movements are less than a specified convergent level. The terms (x,y)new,L and (x,y)new,A in Eq. (21) are expressed aswhere (x,y)i are the directly connected nodal points to the center node (x,y)0 and A and B are the areas of the two neighbor elements which share the two nodes (x,y)0 and (x,y)i on a common edge. Full-size image (3K)Fig. 3. Typical element patch. (x,y)1, (x,y)2, (x,y)3 and (x,y)4 are nodes directly connected to the center node (x,y)0 while (x,y)5, (x,y)6, (x,y)7 and (x,y)8 are neighboring nodes to center node (x,y)0. A and B in Eq. (23) are the areas of the two neighbor elements sharing the two nodes (x,y)0 and (x,y)1 along the common edge. In case of node (x,y)1 A is the area of element I and B is the area of element IV and so on.To achieve optimized meshes, one needs to measure the quality of meshes before and after applying mesh smoothing so that possible mesh improvement is easily predicted. If the quality of the meshes can be expressed as a quantitative value, mesh quality may be well recognized and thus, when to apply mesh smoothing is simply identified. In the present work a mesh distortion metric D 10 is used.The mesh distortion metric D is computed aswhereand Jij is the component of a Jacobian matrix. If the shape of a quadrilateral element is a square, which is the ideal shape, the distortion metric D gives 0. As an element is distorted thus its shape gets away from the square one, the distortion metric gives a non-zero value. The more distorted, the larger D value yields. For more detailed information on the mesh smoothing and the distortion metric technique, 9 and 10 should be consulted.4. Applications and numerical results 4.1. CantileverA cantilever whose dimension is 300 mm30 mm is chosen. Boundary conditions and a point load F are applied as presented in Fig. 4(a). Shape optimization is performed allowing both the upper and lower boundaries to move. Thirty-two master nodes, considered as design variables, are used. The plane stress condition is assumed. The maximum von Mises stress is restricted to 1.8 MPa with minimizing the beam volume. As Fig. 4(b) displays, final volume is reduced to 6396 mm3 from initial volume of 9000 mm3, that is down approximately 28.9%. Fig. 5 shows the history of von Mises stresses at a point A and volume changes during the shape optimization process. Convergence is reached after 20 iterations. The history of the maximum distortion metric D during the optimization process is displayed in Fig. 6. Evidently mesh smoothing plays little roll in this case. Fig. 4. The shape optimization of a cantilever under a tip load: (a) before optimization; and (b) after optimization (F=0.1 N, E=1104 MPa, v=0.3).4.2. FilletAn initial shape of a fillet with a distributed loading and boundary conditions are shown in. The plane stress condition is assumed and the maximum allowable von Mises stress is set to 4.5 MPa. With minimizing its volume the boundary between points A and B is allowed to move . Six master nodes are assigned as master nodes. After shape optimization, volume is reduced from initial 15,525 to 14,168 mm3, that is, down 8.7%. Convergence is made in 20 iterations. An obtained optimum shape is illustrated in and volume changes and von Mises stresses at point C during optimization process are shown in The results presented in this paper are very similar to the study done by Rajan and Belegundu . The improvement of mesh quality by mesh smoothing is seen in The shape optimization of a fillet under a pressure load: (a) dimensions; (b) initial shape and loading; and (c) final optimum shape (P=5 MPa, E=1104 MPa, v=0.3). 5. Concluding remarksAlthough B-spline curves provide much flexibility forming a design boundary, it alone may not be sufficient to construct a design boundary on complex structures. An organized strategy for building a desired design boundary is seemingly of importance; its prime parts include employing proper interpolating curves, selecting design variables, specifying the moving directions of the design variables and preventing mesh distortion. The numerical results indicate that the shape optimization of structures even with complicate shapes can be effectively achieved by properly combining and utilizing these ingredients. In the present work, a shape optimization scheme, using B-spline curves and mesh smoothing techniques, has been developed. The described approach presents an opportunity to improve the optimizing performances.The implemented application of an automobile torque converter lock-up clutch disk exemplifies that the reconstructed boundary is fairly smooth and 13% of weight saving has been made by the present optimization procedure. The obtained results probably provide useful designing considerations for building a more streamlined automobile torque converter system. Furthermore, this scheme can contribute to the investigation and development of a shape optimization process for more complex structures. AcknowledgementsThis work is partially supported by the Ministry of Science & Technology (MOST) and the Korea Science and Engineering Foundation (KOSEF) through the Center for Automotive Parts Technology (CAPT) at Keimyung University. This work is also part of a project supported by BK21 at Kyungpook National University in Korea. Their financial support is gratefully acknowledged.References1. O.C. Zienkiewicz and J.S. Campbell, Shape optimization and sequential linear programming. In: R.H. Gallagher and O.C. Zienkiewicz, Editors, Optimum Structural Design, Wiley, New York (1973), pp. 109126.2. C. Fleury and B. Braibant, Structural optimization: a new dual method using mixed variables. Int. J. Numer. Meth. Eng. 23 (1986), pp. 409428. MathSciNet | Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (131)3. J.A. Bennett and M.E. Botkin, Structural shape optimization with geometric problem description and adaptive mesh refinement. AIAA J. 23 3 (1985), pp. 458464. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (47)4. D.F. Rogers and J.A. Adams. Mathematical Elements for Computer Graphics, McGraw-Hill, New York (1990).5. V.B. Anand. Computer Graphics and Geometric Modeling for Engineers, Wiley, New York (1993).6. Y. Ding, Shape optimization of structures: a literature survey. Comput. Struct. 24 (1986), pp. 9851004. Abstract | PDF (1568 K) | MathSciNet | View Record in Scopus | Cited By in Scopus (78)7. W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery. Numerical Recipes in Fortran, Cambridge University Press, Cambridge (1992).8. E. Atrek, R.H. Gallagher, K.M. Ragsdell and O.C. Zienkiewicz. New Directions in Optimum Structural Design, Wiley, New York (1984).9. S. Hyun, L.-E. Lindgren, Mesh smoothing techniques for graded element, in: Proceedings of NUMIFORM 98 the 6th International Conference on Numerical Methods in Industrial Forming Process, 1998, pp. 109114.10. S. Hyun and L.-E. Lindgren, Smoothing and adaptive remeshing scheme for graded element. Commun. Numer. Meth. Eng. 17 (2001), pp. 117. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (12)11. S.D. Rajan and A.D. Belegundu, Shape optimal design using fictious loads. AIAA J. 27 1 (1988), pp. 102107.Corresponding author. Fax.: +82-53-950-6586基于有限元法和B样条曲线形成的有效优化方法开发了液力变矩器离合器S. Hyun, C. Kim, J. H. Son, S. H. Shin and Y. S. Kim摘要为了设计轴对称结构，一个有效的形状优化方案已经形成。它是与有限元分析相关的序线性规划的单纯形法。选择设计边界的主节点为设计变量，朝着法线方向延伸。通过插值重新定位主节点，构建B样条曲线，使得边界上的其余节点更好的落在B样条曲线上。这种网格平滑方案还适用于内部节点，来保持有限元处在一种良好的范围。应用这些技术,给出了一个数值，实现获得汽车液力变矩离合器承受液压和向心力的最优设计。结果是优化后的磁盘重量比优化前的减少了13%。作者关键词：有限元法;形状优化，B样条曲线;自适应网格划分，液力变矩器锁定 离合器 磁盘文章概要1.导言2.形状优化算法3.形状优化方案3.1. B样条曲线3.2.网格优化4.应用程序和计算结果4.1.悬臂4.2.圆角5.结束语鸣谢参考文献1.导言一种基于形状的一些研究结构的优化已经进行了。对于广泛的一类形状优化问题，没有额外的重量可以减少而不改变形状和设计领域的边界位置。为此，B样条曲线和曲面被广泛用来表示在连续边界曲面的形状优化设计域的过程。使用ISO有限元法参数形成的第一个形状优化，是钦科维奇和坎贝尔提出的1。他们用序列线性规划和单纯形法求解某拱坝的最优形状。在形状优化的过程中，结构形状是随着重新定位的边界节点改变的。到了80年代，大多数研究人员利用边界曲面所有节点作为设计变量。但是这项计划，有时还会产生不连续的或不稳定的界限，这些往往是由于表面容易产生扭曲的网格导致的，引起了不可靠的解决方案。为了解决这个问题，弗莱利和布莱贝尔2，通过利用B曲线上的的控制点为设计变量，而不是表面的节点，设计构造边界表面光滑样条曲线。贝内特和波特金3建议在形状优化的过程中，自动网格生成法结合自适应网格细化法的方案。在汽车设计中，其中一个重要的问题是怎样减少汽车的重量，以改善能量转换率和燃料消耗率（公里/升）。最近，大多数轿车都采用前置发动机和前驱动系统来解决前后平衡的难题。汽车中最重的部分之一是发动机，动力传动时占用20-25的总重量。液力变矩器离合器，自动变速器的一部分，利用液压将发动机功率传送到传动轴。它在不同的速度旋转，并经常受到压力作用。因此，设计液力变钜离合器有很高的要求，要求能适应所有工况，并能有效的减少成本。本研究的目的是要提出一个有效的形状优化计划，以确定汽车液力变矩离合器的适当边界。B样条曲线，代表了设计边界，被用于优化过程，设计的边界节点可以根据一个正确的方法移动。此外，网格平滑技术还被用于提高整体网格质量。一悬臂梁和圆角方案实施用于验证优化方案的性能。2.形状优化算法形状优化问题是找出设计变量 X同时尽量减少目标函数W(X)下的约束函数gi(X)，可用数学式子表式为： 减小到最小： W(X)主要要求： gi(X)0, XLXXH,这里的 XL和 XH是允许的下限和上限的设计变量，介绍如何解决各种需求变量限制。 许多有效的计划是实现最优解非线性优化问题6。在本研究中，连续线性规划及单纯形法7用于形状优化。这种方法是线性化的非线性目标函数W(X)）和在非线性约束函数gi(X)设计要点 XP。只有从泰勒的线性范围扩大采取和高阶条件将会被忽略。线性化的目标函数和约束条件如下：最小化 W=c+bTX承受：AXd,X0，这里：c=W(XP)TW(XP)XP, 由此可主认为，在上述的公式出现偏导数，获得了半解析灵敏度分析8。在结构形状优化的目标函数W(X)代表体积，分别为设计变量 X 协调边界节点和约束函数gi(X)所允许的最大许用应力。 通过单纯形法的最低要求设计变量被更新反复适当收敛。测定方法的收敛性是由一个体积变化引起形状优化。一个体积变化,比例是表现在两个相邻的体积，这里Vi体积是迭代i 或Vi 体积是迭代i-1。当比值小于0.001，优化过程被认为是融合。3.形状优化方案这个形状的优化设计方案，包括兴建B样条曲线来表示设计边界和网格平滑技术，以改善内部网的形状。3.1 B样条曲线B样条曲线代表的融合等功能函数为： 这里Nik(u)是由混合函数给出：和在上面的公式，Pi 是一个控制点，ti 是结值和k在B阶样条曲线上。对于非结值的周期样条曲线，将两个终结点都给出：当量（14），在矩阵形式表达变为D=NBB=N1D,这里D是数据点矩阵，N是一个混合函数矩阵，B是矩阵的控制点。通过以所获得的控制点成任意B样条曲线的数据点，通过传递可以被定义。 如图-1 体现了建设程序的B样条曲线，使用六种数据。对于任何B样条曲线给出pn+1,这里p 是的B阶样条曲线和 n 是该数据点的数量。在目前工作的五阶的B样条曲线的每6个数据点组成的跨度。 图-1 B样条曲线的节点数据点在优化过程中的形状，到整个节点的坐标为设计变量，为设计变量进行极其艰难而漫长的保持边界平滑设计，目前的研究工作，一个精心选择的节点设计沿边界组分配给主节点。主节点被视为设计变量，直接置于形状优化过程中。然后沿边界的设计其余节点运用插值B样条曲线。本实施完成如下：（1）设计边界的所有节点分成主节点和次节点。而末节点总是看成主节点（2）新的主节点坐标确定的形状优化的过程。他们被限制沿着边界制定方向运动（3）AB样条曲线是根据主节点构造成的（4）位于两个主节点间的次节点，移动到新建成的B样条曲线上的距离等于两个相邻的主节点的距离.因此，边界沿线的所有节点可以被重新定位，以便保证能连续曲线设计的边界的平滑。图2 图2程序设计建造一个新的边界：（一）主节点和次节点在一个设计边界上;（二）主节点移动方向;（三）主节点的运动，形成了B样条曲线（d）确定次节点的移动矢量。3.2 网格优化在一个形状优化的过程中，一个结构发生变化归结于在设计边界上的结点的运动。因此，要防止网格过度变形，要使有限元分析在一个优化的过程中不断提供准确的解决方案。一个网格平滑的技术应用于改善网格质量，图3 说明了典型的内部网的相邻节点（x，）与一个中心节点（x，）0有关，可以操作坐标的周边节点和元素的周边地区重新定位一个中心节点（x，）0 以此可以获得一个新的位置（x，）。这表现为： （x，）=（1 -）（x，）+（x，）0，其中是0和1之间任意值待定。所有节点，属于内部网的重新定位，反复直到所有的动作比指定的收敛范围之内。范围（x，）和（x，）0在表示为其中（x，）字母i 是直接连接中心节点结点（x，）0 的，A、B区域是两个相邻元素的节点（x，）0 和（x，）字母i 的一个共同的边缘。 图3 图3 典型元素的修补程序。 （x，）1（x，）2（x，）3 和（x，）4 是直接连接中心节点（x，）0 而（x，）5（x，）6（x，）7 和（x，）8 ，而与中心节点（x，）0相邻。A、B区域在两个相邻点（x，）0 和（x，）1 的共同区域。A属于元素区I和B属于元素区4。为了实现优化的网格，我们需要比较应用平滑网格技术前，后网格的质量，以此预测容易改善的部位。如果网格质量可作为定量值表示，网格质量可能得到公认，因此，何时申请网格平滑只是迟早的问题。在目前工作中已使用网格畸变度量 D。该网格畸变度量 D 计算如下 ij 是一个雅可比矩阵的组成部分。 如果一个四边形的形状是正方形，这是理想的形状，失真度量D为0，然而因为一个形状扭曲的元素偏离了它最初的运动轨迹，这时失真度量为非零值。越扭曲，D值越大。4 .应用程序和计算结果4.1 悬臂选择的悬臂尺寸是300毫米 30毫米。边界条件和载荷F的应用如图所示，形状优化表现为使得上，下边界移动。 假设平面应力状态的32个主节点作为设计变量考虑使用。在约束体积为最小的情况下，冯米塞斯的最大压力为1.8兆帕。同样地，最终的量从9000 mm3降至6396 mm3 即下降约28.9。 图5 显示了在A点冯米塞斯的压力以及在优化过程中的形状变化。收敛后达到20迭代。在优化过重中历史最大变形量如图 6所示，显然在这种情,网格平滑扮演了非常小的角色。图5 流量和冯米塞斯历史压力。图 6 变形的历史最高度量 D。4.2 圆角初始形状的分布式圆角加载和边界条件。平面应力状态的假定和最大允许冯米塞斯应力设置为4.5兆帕斯卡。由于其体积最小点之间的边界A和B是可以移动。6个主节点分配作为主节点。形状优化后，体积减少，从最初的15525至14,168 mm3，即下降8.7。融合是在20迭代。获得的一个最佳的形状，体积的变化和在优化过程中在C点的冯米塞斯压力，本文所提出的结果和拉詹与本利顿所做的研究很相似。该网格平滑质量改善 5结束语虽然B样条曲线的形成提供了更大的灵活性设计的边界，但是它本身可能不足以构建出复杂的结构设计边界。有组织策略是建设一个理想设计的重要性，其采用的主要部分包括适当的插值曲线，选择设计变量，指定的设计变量的移动方向和防止网格畸变。计算结果表明，优化结构形状复杂，利用这些成分取得的形状，即使能有效地结合起来。目前的研究工作，形状优化方案，利用B样条曲线和网格平滑技术，已经研制成功。在原有描述方法基础上,提出了一种改进的优化性能的方法。在汽车液力变矩器锁定实现的应用程序行动， 离合器 磁盘重建的例证是比较平稳的边界和13的重量节约了目前的优化程序。所得结果可能为建立一个更精简的汽车液力变矩器系统有用的设计考虑。此外，这个计划能有助于调查和更复杂的结构形状的优化进程的发展。鸣谢这项工作是部分支持科学与技术部（科技部）和韩国科学与工程基金会（KOSEF）通过在启明大学中心汽车零部件技术（明扬交通器材）。这项工作也是由BK21支持在庆北大学在韩国项目的一部分。对他们的财政支持，深表谢意。参考文献1.O.C.津凯维奇和J.S.坎贝尔，形状优化和连续线性规划。在：罗佩Gallagher和O.C.津凯维奇，编辑， 优化结构设计，威利，纽约（1973年），页。 109-126。2.长弗勒和结构优化二布雷邦：一种新方法，使用混合双变量。Int. J. Numer. Meth. Eng. 23 （1986年），页。 409-428。3.J.A.贝内特和ME波特金，结构形状与几何问题的说明和自适应网格加密优化。 AIAA J. 23 3（1985），页。 458-464。4.D.F.罗杰斯和J.A.亚当斯。 数学计算机图形元素，麦格劳希尔，纽约（1990年）。5.五B阿南德。 计算机图形学和几何建模工程师，威利，纽约（1993年）。6. Y. Ding,.形状结构优化丁：文献调查。 Comput。结构。 24 （1986年），页。 985-1004。7.W.H.出版社，第答科斯基，延胡索Vetterling和南疆弗兰纳里。 在Fortran语言的数值食谱，剑桥大学出版社，剑桥大学（1992年）。8.阿特雷克，罗佩加拉格尔，K.M. Ragsdell和O.C.津凯维奇。 新方向的结构优化设计，威利，纽约（1984年）。9.美国铉，L.-E.林格伦，分级元网格平滑技术，在：对NUMIFORM 98法律程序的第六届国际会议的数值方法在工业的形成过程，1998年，第。 109-114。10.美国铉和L.-E.林格伦，平滑和自适应分级元素网格重划计划。 Commun。 +82-54-852-6800。英格. 17 （2001年），页。 1-17。 11.最高法令拉詹和AD Belegundu，形状优化设计使用虚拟负载。 AIAA J. 27 1（1988），页。 102-107。设计方案论证2.1选定离合器车型本次设计选定车型为奥迪（Audi）A3标准型离合器1作为设计目标，该车主要参数如下表1： 表2-1 奥迪A3标准型主要性能参数2.2离合器设计的基本要求目前，各种汽车广泛采用的摩擦离合器是一种依靠主、从动部分之间的摩擦来传递动力且能分离的装置。离合器的主要功用是切断和实现发动机与传动系平顺的接合，确保汽车平稳起步；在换挡时将发动机与传动系分离，减少变速器中换档齿轮间的冲击；在工作中受到较大的动载荷时，能限制传动系所承受的最大转矩，以防止传动系个零部件因过载而损坏；有效地降低传动系中的振动和噪音。为了保证离合器具有良好的工用性能，设计离合器应满足如下基本要求：1）在任何行驶条件下，既能可靠地传递发动机的最大转矩，并有适当的转矩储备又能防止传动系过载。2）接合时要完全、平顺柔和，保证汽车起步时没有抖动和冲击。分离时要迅速彻底。3）从动部分转动惯量要小，以减轻换挡变速器齿轮间的冲击，便于换挡和减小同步器的磨损。4）应有足够的吸热能力和良好的通风散热效果，以保证工作温度不致过高，延长其使用寿命。5）应能避免和衰减传动系的扭转振动，并具有吸收振动缓和冲击和降低噪声的能力。6）操纵轻便、准确，以减轻驾驶员的疲劳。7）作用在从动盘上的总压力和摩擦材料的摩擦因数在离合器工作过程中变化要尽可能小，以保证有稳定的工作性能。8） 具有足够的强度和良好的动平衡，以保证其工作可靠，使用寿命长。9） 结构应简单、紧凑，质量小，制造工艺性好，拆装、维修、调整方便等。2.3离合器的结构方案分析2.3.1摩擦离合器结构选择汽车离合器有摩擦式、电磁式和液力式三种类型。其中，摩擦式的应用最为广泛。现代汽车摩擦离合器的典型结构型式为单征或双片干式（图2-1），它由从动盘、压盘驱动装置、压紧弹簧、离合器盖等构成，本次设计选用摩擦式离合器。图2-1 从动盘部分分解图1，13摩擦片；2，14，15铆钉；3波形弹簧片；4平衡块；5从动片；6，9减振摩擦；7限位销；8从动盘毂；10调整垫片；11减振弹簧；12减振盘2.3.2从动盘数及干湿式的选择1）单片离合器对乘用车和最大总质量小于6t的商用车而言，发动机的最大转矩一般不大，在布置尺寸容许条件下，离合器通常只设置有一片从动盘。单片离合器结构简单，轴向尺寸紧凑，散热良好，维修调整方便，从动部分惯量小，在使用时能保证分离彻底，采用轴向有弹性的从动盘可保证接合平顺。2）双片离合器双片离合器与单片离合器相比，由于摩擦面数增加一部，因而传递转矩的能力较大；接合更为平顺、柔和；在传递相同转矩的情况下，径向尺寸较小，踏板力较小；中间压盘通风散热性差，容易引起摩擦片过热，加快其磨损甚至烧坏；分离行程较大，不易分离彻底，所以设计时在结构上必须采取相应的措施；轴向尺寸较大，结构复杂；从动部分的转矩较大且径向尺寸受到限制的场合。3）多片湿式离合器摩擦面更多，接合更加平顺；摩擦片浸在油中工作，表面磨损小，但分离行程大，分离也不易彻底，特别是在冬季油液粘度增大时；轴向尺寸大；从动部分的转动惯量大，故过去未得到推广。近年来，由于多片湿式离合器在技术方面的不断完善，重型车上又有采用，并不断有增加趋势。因为它采用油泵对摩擦表面强制冷却，使起步时即使长时间打滑也不会过热，起步性能好，据其使用寿命可较干式高56倍。通过比较，本次设计所选车型适合选用单片干式摩擦离合器。2.3.3压紧弹簧和布置形式离合器压紧弹簧的结构型式有：圆柱螺旋弹簧、矩形断面的圆锥螺旋弹簧和膜片弹簧等。可采用沿圆周布置、中央布置和斜置等布置型式根据其布置离合器可分为：1）周置弹簧离合器周置弹簧离合器的压紧弹簧均采用圆柱螺旋弹簧，并均匀地布置在一个或同心的两个圆周上，其特点是结构简单、制造容易，这去广泛应用于各类汽车上。此结构的弹簧压力直接作用于压盘上，为了保证摩擦片上压力不均匀，压紧弹簧的数目要随摩擦片直径的增大而增多，而且应当是分离杠杆的倍数。因压紧弹簧直接与压盘接触，易受热回火失效。当发动机最大转速很高时，周置弹簧由于受离心力作用而向外弯曲，使弹簧压紧力显著下降，离合器传递转矩的能力也随之降低。此外，弹簧靠在其定位座上，造成接触部位严重磨损，甚至会出现弹簧断裂现象。2）中央弹簧离合器中央弹簧离合器采用一至两个圆柱螺旋弹簧或用一个圆锥弹簧作为压紧弹簧，并且布置在离合器的中心。由于可选较大的杠杆比，因此可得到足够的压紧力，且有利于减小踏板力，使操纵轻便；压紧弹簧不与压盘直接接触，不会使弹簧受热回火失效；通过调整垫片或螺纹容易实现压盘对压紧力的调整。这种结构较复杂，轴向尺寸较大，多用于发动机最大转矩大于400500Nm的商用车上4，以减轻其操纵力。
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