外文翻译--设计变量的分类

1 原文： Optimal Designs 2.2 Design variables The most important classification of design variables is into: SIZE design variables SHAPE design variables TOPOLOGY design variables stated here in order of difficulty to solve but also in order of increasing importance for the obtained objective value. It is therefore not surprising that recent research to some extent concentrates on topology design variables. The notion of size design variable, relates to the thickness of a beam, a plate or a shell (although this is often termed the shape of a beam, a plate or a shell). The area of a bar in a truss is also a size design variable, and the definition of size variable is related to the fact that the modelling domain is not changed. So, the line of the beam, rod or bar is unchanged, just like the reference surface of a plate or a shell is assumed unchanged when the concept of size design variable is used. In 3D-problems the mass density or a relative volume density is size. The orientations of non-isotropic material we also treat as size design variables. The notion of shape design variable, relates to the reference domain of the actual model. For beams, rods and bars we may treat the length as a design variable, which is then a shape design variable. Also the curvature of the reference line for these one-dimensional models is a shape design variable. For 2D-models likewise the boundary curve or the curvature of the reference surface are shape design variables. For 3D-models the boundary surface (including internal boundaries like holes) is a shape design variable. Stress concentration problems are often related to shapes of boundaries. Finally, the notion of topology design variable, relates to presence or absence of a certain design aspect. Should two joints in a truss be connected with a bar, - yes or no ?. Should a continuum like a plate have a hole, - yes or no ?. The complications in treating topology design variables are due to the fact that a change in topology results in a discontinuous change in the design response, while a 2 continuous change in size or shape design variables normally results in continuous change in the design response. Let us exemplify the difference between size, shape and topology design variables. In a truss (2D as well as 3D), the bar areas (uniform or non-uniform) are the sizes, the positions of the joints determine the shape, and the chosen bars (among many possibilities) give the topology. In a shell the thickness and material density distributions are the sizes, the boundaries of the reference surface and its curvatures are the shapes, and the number of holes in the reference surface is the topology. 2.2.1 Alternative classifications Many alternative names to classify design variables can be found in the literature, like cross-sectional, geometrical, configuration, layout etc. We try to avoid these names in order to avoid unnecessary confusion. The design variables may also be classified from other points of view. Let us first discuss the distinction between continuous and discrete design variables. If only a number of specific values for the design variable is acceptable, say when catalog values must be used, then the notion of discrete design variables is used, and procedures related to what is called Integer programming come into focus. This is not covered in the present book that concentrates on continuous descriptions, but we include absolute limitations for the values of the design variables. Another meaning of the ” continuous” and ” discrete” relates to the modelling of the design domain. A complete continuous description in space means design variables related to a point (like a design function) and not to a domain. Often this is termed distributed parameter description, in contrast to say a truss description where each bar is described as a unit. In a finite element modelling of a continuum, the element domains may be related to a number of design values, so in reality this is a discrete description. However, with the extensive number of elements and the fact that everything in a computer is discrete, the distinction between continuous and discrete related to the modelling of the design domain is of no practical importance. For a successful optimization the choice of design parametrization is of vital importance, perhaps the most important decision to take. In the experience of the author it is wise to start with as few design variables as possible. A hierarchical description is suggested, and also it is important to make sure that the design variables serve different purposes. It is asking for practical problems, 3 if the design variables are chosen such that different combinations of design variables can give the same design. The parametrization is also related to the chosen optimization procedure, so with an optimality criterion method large quantities, say 50.000 design variables, can be handled without problems. 2.3 Design objective The design objective is a function or a functional that returns a single value from which different designs can be compared. The optimal design is then the design with a minimum (or maximum) value of the objective. In this book we often use the notation to denote the objective. We shall not treat multi-objective formulations, which in most cases are reformulated into a single objective anyhow. Alternative names for the objective include criterion, cost, merit, goal as well as many others. The name ” criterion” is in this book used extensively in relation to optimality criterion formulations (see chapter 14), so we try only to use the name objective, although a name like cost may be more appealing. In fact, the objective value is often a measure of the cost of the design. A minimum and maximum formulation may be interchanged by simply changing the sign of the objective. However, it is important to notice that many methods just locate a stationary value of the objective, which means that the convergence of the procedure must be followed and the final design justified.A much more severe problem is related to the existence of local stationary solutions,and in reality very few (and often non-practical) methods are able to find a global optimal solution. Starting an optimal design procedure from different initial designs and always ending up in the same optimal design may be the most practical procedure for improving the probability of an obtained solution being the global optimal solution. It should be noticed that a number of optimal design formulations for idealized problems may include a proof of global optimal solution. However, for problems where a large number of practical constraints need to be taken into account, it is more safe to state that we have optimized the design as an alternative to obtaining the optimal design. Furthermore, it is not always easy to see from the formulation whether an optimal design exists. If an optimal design does not exist we talk about a not well formulated problem. 4 Even so a procedure may return an optimized design, and the convergence often reveals the missing aspect(s) in the formulation. An important part of an optimization procedure is to decide when to stop. We talk about convergence tests. Two different aspects of convergence must be clarified, convergence of the design objective and convergence of the design variables. Often the rates of these two convergences are very different. Also the formulation of the specific stop condition can be mathematically formulated more or less complicated. The favourite formulation of the present author is as follows: When the design changes are somewhat smaller than the actual accuracy in the design production, then the design procedure should be stopped. At that instant the design objective is often converged at a much earlier step. 5 翻译译文 2.2 个设计变量 设计变量的分类是最重要的： 尺寸设计变量 形状设计变量 拓扑设计变量 这里所说的在解决难 题 也越来越重视，以便获得客观的价值。因此毫不奇怪，在某种程度上，最近的研究集中在拓扑设计变量。 尺寸设计变量的概念，涉及一种梁的厚度，板或壳（虽然这是通常被称为一束，形状的板或壳）。在桁架杆地区也是一个尺寸设计变量，和尺寸变量的定义是这样的事实，建模领域是没有改变的关系。因此，梁的线，杆或棒是不变的，就像一个板或壳参考表面被假定不变时的尺寸设计变量的概念的使用。在三维问题的质量密度或相对密度的大小。取向的非各向同性材料我们也把尺寸设计变量。形状设计变量的概念， 涉及到实际的模型参考域。梁，棒可以将长度为设计变量，然后一个形状设计变量。也为这些一维模型的参考线的曲率是一个形状设计变量。对于二维模型同样 有 边界曲线或基准表面的曲率形状设计变量。 三维模型的边界表面（包括内部边界像孔）是一个形状设计变量。应力集中问题往往是相关的边界的形状。 最后，拓扑设计变量的概念，涉及到一个特定的设计方面存在或不存在。应在桁架节点连接杆， -是或不是？一个连续的。应该像一个盘子上有一个洞，是还是不是？。在处理拓扑设计变量的并发症是由于拓扑中的变化在设计中的响应不连续变化的结 果，而在大小或形状的设计响应不断变化的设计变量连续变化的一般结果。 让我们举例说明之间的差的大小，形状和拓扑设计变量。 6 在桁架（ 2D 和 3D）， 边界区域 （均匀或不均匀）的大小，关节的位置确定的形状，和所选择的 边界 （其中许多可能性）给拓扑。在壳的厚度和材料密度分布的大小，参考面及其曲率的边界的形状，并在参考表面孔的数量是拓扑。 2.2.1 另一种分类 许多其他的名字将设计变量可以在文献中找到，如横截面，几何，结构，布局等，我们尽量避免以避免不必要的混淆这些名字。 设计变量也可以从其 他角度分类。让我们先讨论连续和离散设计变量之间的区别。如果只为设计变量的特定值的数量是可以接受的，说的时候必须使用目录的值，然后使用离散的设计变量的概念，并称之为整数规划成为关注的焦点，有关的程序。这是不包括在本书致力于不断的描述，但我们将设计变量的值的绝对限制。 另一种意义上的 “ 连续 ” 和 “ 离散 ” 涉及到设计领域的建模。连续空间中的一个完整的描述手段一点相关的设计变量（如设计功能）和不到域。这通常被称为分布参数描述，相反，说一个桁架的描述 ， 描述为一个单元，每一杆。在一个连续的有限元模型，单元域可能要数设计值相关，因此在现实中这是一个离散的描述。然而，随着元素的大量事实和计算