文献翻译-有限元和优化

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Finite Element Method and Optimization

As engineers work with increasingly complex structures ,they need rational ,reliable,fast and economical design tools.Over the past two decades,finite element analysis has proven to be the most frequently used method of identifying and solving the problems associated with these complicated designs.

The finite-element method is an approximation procedure for solving differential equations of boundary and/or initial value type in engineering and mathematical physics.The procedure employs subdivision of the solution domain into many smaller regions of convenient shapes,such as triangles and quadrangles and uses approximation theory to quantize behavior on each finite element.Suitably disposed coordinates are specified for each element,and the action of the differential equation is approximately replaced using values of the dependent variables at these nodes.Using a variational principle or a weighted-residual method,the governing differential equations are then transformed into finite element equations govening the isolated element.These local equations are collected together to form a global system of ordinary differential or algebraic equations including a proper accounting of boundary conditions.The nodal values of the dependent variaanles are determined from solution of this matrix equation system.

The fundamental step underlying the finite-element for solving problems in mechanics is the reduction of the originnal partial differential equations to a set of ordinary differential or algebraic equations which can be solved by straightforward techniqus.Generally,the procedure follows the steps used in the classical analytical methods for soving linear partial differential equations by expansions in a set of functions.

The finite element has another advantage when an irregular nodal arrangement is appropriate for a given problem.This may arise when a solution displays singularities or regions of particularly rapid change.Irregularly shaped domains,especially those with curved boundaries,are effectively accommodated with the finite element techniques.Although the approximating algebraic equations arising from finite element procedures are generally less amenable to efficient algebraic maninpulation,new schemes are being rapidly developed and one would anticipate this computational disadvantage to be less pronounced in the years to com.

Now,many engineers follow a manual trial-and-error approach.Such an approach makes designing-even for seemingly simple tasks,more difficult because it usually takes longer,requires extensive human-machine interaction,and tends to be biased by the design group’s experience.

The advance in computers will promote improvement of computational methods,computers have revolutionized the highly iterative design process,particularly the procedures for quickly finding alternative designs.Design optimization,which is based on a rational mathematical approach to modifying

designs too complx for the engineer to modify,automates the design cycle.If automated optimization can be done on a desktop platfrom,it can save a lot time and money.

The goal of optimization is to minimize or maximize an objective,such as weight or fundamental frequency that is subject to constraints on response and design parameters.The size and/or shape of the design determine the optimzation approach.

Looking at optimzation as part of the design process makes it easier to understand.The frist step includes preprocessing,analysis,and postprocessing,just as in customary finite element analysis(FEA)and computer –aided design(CAD)program applications(the difference in CAD lies in building the problem’s geomtry in terms of the design parameters).In the second step,the optimization objective and response constraints are defined.And in the last step,the repetitive task of design adjustment is automated.Optimization programs should allow engineers to monitor the progress of the design,stop it if necessary,change the design conditions,and restart.The power of an optimization program depends on the avaiable preprocessing and analysis capabilities.Applications for 2-D and 3-D need both automatic and parametric and parametric meshing capabilities.Error estimate and adaptive control must be included because the problem’s geometry and mesh might change during the optimization loops.

Revising remeshing,and reevaluating modes to achieve specific design goals start with preliminary design data input.Next comes the specification of acceptable tolerances and posed constraints to achieve an optimum,or at least improve,solution.To optimize products ranging from simple skeletal structures to complicated three-dimensional solid models,designers need access to wide variety of design objectives and bahavior constraints.Additional capabilities will also be needed for easy definition and use of the following:weights,volumes,displacements,stresses,strains,frequencis,buckling safety factors, temperatures,temperature gradients,and heat fluxes as constraints and objective functions.

Moreover,engineers should be able to combine constraints from different types of analysis in multidisciplinary optimization.For example,designers can perform thermal analysis and transfer temperatures as thermal loads for stress analysis,put constraints over maximum temperature,maximum stress,and deflection,and then specfy a range for the desired fundamental frequency.

The objective function can represent the whole model or only parts of it.Even more important,it should reflect the importance of the different portions of the model by specifying weight or cost factors.

The integration of optimization techniques with Finite Element Analysis (FEA) and CAD is [1]having pronounced effects on the product design process. This integration has the power to reduce design costs by shifting the burden from the engineer to the computer. Furthermore, the mathematical rigor of a properly implemented optimization tool can add confidence to the design process. Generally, an optimization method controls a series of applications, including CAD software as well as FEA automatic solid meshers and analysis processors. This combination

allows for shape optimizations on CAD parts or assemblies under a wide range of physical scenarios including mechanical and thermal effects.

Modern optimization methods perform shape optimizations on components generated within a choice of CAD packages. Ideally, there is seamless data exchange via direct memory transfer between the CAD and FEA applications without the need for file translation. Furthermore, if associativity between the CAD and FEA software exists, any changes made in the CAD geometry are immediately reflected in the FEA model. In the approach taken by ALGOR, the design optimization process begins before the FEA model is generated. The user simply selects which dimension in the CAD model needs to be optimized and the design criterion, which may include maximum stresses, temperatures or frequencies. The analysis process appropriate for the design criteria is then performed. The results of the analysis are compared with the design criterion, and, if necessary without any human intervention, the CAD geometry is updated. [2]Care is taken such that the FEA model is also updated using the principle of associativity, which implies that constraints and loads are preserved from the prior analysis. The new FEA model, including a new high-quality solid mesh, is now analyzed, and the results are again compared with the design criterion. This process is repeated until the design criterion is satisfied. Fig.19.1 shows the procedure of shape optimization. Introduction

The typical design process involves iterations during which the geometry of the part(s) is altered. In general, each iteration also involves some form of analysis in order to obtain viable engineering results. Optimal designs may require a large number of such iterations, each of which is costly, especially if one considers the value of an engineer’s time. The principle behinddesign optimization applications is to relieve the engineer of the laborious task by automatically conducting these iterations. At first glance, it may appear that design optimization is a means to replace the engineer and his or her expertise from the design loop. This is certainly not the case because any design optimization application cannot infer what should be optimized, and what are the design variables, the quantities or parameters that can be changed in order to achieve an optimum design. Thus, design optimization applications are simply another tool available to the engineer. The usefulness of this tool is gauged by its ability to efficiently identify the optimum.

Procedure of the Shape Optimization

Initial Design Preprocessing FE Preprocessor Optimization Preprocessor Optimization Loop FE solver Optimization module Optimized design

Design optimization applications tend to be numerically intensive because they must still [3]perform the geometrical and analysis iterations. Fortunately, most design optimization problems can be cast as a mathematical optimization problem for which there exist many efficient solution methods. The drawback to having many methods is that there usually exists an optimum mathematical optimization method for a given problem. This complexity should be remedied by the design optimization application by giving the engineer not only a choice of methods, but also a suggestion as to which approach is most appropriate for his or her design problem.

In this paper, we focus on the design optimization of mechanical parts or assemblies. In this case, a typical optimized quantity is the maximum stress experienced. Typical design variables include geometric quantities, such as the thickness of a particular part. The design of the part or assembly is initiated within a CAD software application. If the component warrants an engineering analysis, the engineer will generally opt to apply finite element analysis (FEA) in order to model or simulate its mechanical behavior. The FEA results, such as the maximum stress, can be used to ascertain the validity of the design. During the design process, the engineer may alter parameters or characteristics of the CAD and/or FEA models, including some of the physical dimensions, the material or how the part or assembly is loaded or constrained. Associativity between the CAD and FEA software should allow the engineer to alter the model in either application, and have the other automatically reflect these changes. For example, if the thickness of a part is changed or a hole is added in the CAD software, the FEA model’s mesh should automatically reflect those changes. Under most circumstances, engineers will employ linear static FEA to obtain the stresses. This analysis approach has the benefit of yielding a solution for FEA

models with many elements in relatively little time. Obviously, linear static FEA has drawbacks as well. For example, significant engineering expertise may be required when estimating the magnitude and direction of loads that are a consequence of motion.

Background and Theory

In this section, we focus on the theory underlying some of the mathematical methods employed by design optimization procedures. But, first we describe how the optimization problem arises. Consider a three-step process:

(1) generation of geometry of part or assembly in CAD； (2) creation of FEA model of part or assembly； (3) evaluation of results of FEA models.

For now, we limit ourselves to the case of linear static FEA. Therefore, the results are comprised of deflections and stresses at one instance. The manual design process involves all three steps, with the results being used to evaluate whether the design is appropriate. If the design is found inadequate, changes are made to steps (1) or (2) or both. It is clear from this description that the output of the FEA results is what should be optimized, and that any input to the CAD or FEA models can be viewed as a design variable. A design optimization algorithm conducts many FEA runs, each one with a different set of values for the design parameters. Before the manual design approach can be transformed into a design optimization algorithm, there must be associativity between the CAD and FEA applications. The rational behind this requirement is best explained using an example. Consider the initial design stage when the engineer applies constraints on a particular surface of the FEA model; it can be safely assumed that this surface coincides with a surface in the CAD model. Now, if the design optimization algorithm decides to alter the geometry of the CAD surface, then the FEA model must automatically reflect thesechanges, and apply the constraints on the new representation of this surface. Thus, associativity is required in order to achieve this automatic communication between the CAD and FEA models. Having defined the design optimization problem for mechanical systems, we now describe the mathematics used to solve these problems.

Most optimization problems are made up of three basic components.

(1) An objective function which we want to minimize (or maximize). For instance, in designing an automobile panel, we might want to minimize the stress in a particular region.

(2) A set of design variables that affect the value of the objective function. In the automobile panel design problem, the variables used define the geometry and material of the panel.

(3) A set of constraints that allow the design variables to have certain values but exclude others. In the automobile panel design problem, we would probably want to limit its weight. It is possible to develop an optimization problem without constraints. Some may argue that almost all problems have some form of constraints. For instance, the thickness of the automotive panel cannot be negative. Although in practice, answers that make good sense in terms of the underlying physics, such as a positive thickness, can often be obtained without enforcing constraints on the design

variables.

Benefits and Drawbacks

The elimination or reduction of repetitive manual tasks has been the impetus behind many software applications. Automatic design optimization is one of the latest applications used to reduce man-hours at the expense of possibly increasing the computational effort. It is even possible that an automatic design optimization scheme may actually require less computational effort than a manual approach. This is because the mathematical rigor on which these schemes are based may be more efficient than a human-based solution. Of course, these schemes do not [4]replace human intuition, which can occasionally significantly shorten the design cycle. One definite advantage of automated methods over manual approaches is that software applications, if implemented correctly, should consider all viable possibilities. That is, no variable combination of the design parameters is left unconsidered. Thus, designs obtained using design optimization software should be accurate to within the resolution of the overall method.

有限元和优化

在结构日益复杂的情况下，工程师们工作时需要合理的、可靠的、快速而经济的设计工具。过去20多年里，有限元分析法已经成为判别和解决这些复杂设计课题的最常用的方法。

有限元法是解决边界微分方程或数理工程问题初始值的一种近似方法。他把问题有关的区域分成一些更小的便于解决的区间，通常是三角形和四边形区间，然后用近似理论量化每一个有限元的特性。每个元素都有对应的坐标，微分方程近似地用与各个节点有关的变量值代替。应用变分原理或加权残余法，微分方程组转化成与每一个原素相关的有限元方程组。这些局部方程组被汇合成这个系统的微分或代数方程组，包括边界条件的必要说明。变量的边界值取决于方程组矩阵的解。

有限元解决力学问题的基本步骤是把原始偏微分方程组转换成普通的微分或代数方程组，以便用简单德尔方法求解。而古典的分析方法通常还要扩展一些函数的功能去求解这些线性偏微分方程。

有限元的另一个优势是处理不规则的节点问题，例如结果中的奇点或变化急剧而迅速的区域中的点。那些形状不规则的域，特别是边界弯曲的域，非常适合于用有限元技术求解。虽然有限元生成的近似代数方程组不太适合于有效的代数运算，但新的方法正在迅速发展，可以预料不久那些不足将会得到显著的改善。

现在许多的工程师使用人工试凑法。这种方法使得那些看起来很简单的的设计任务也变得十分困难，因为通常他要花很多的时间，先自爱广泛的人机交流，更要依赖设计者的经验。

计算机的发展使得计算方法得到大大改进，计算机使繁琐的重复设计过程发生了深刻的变革，特别是在快速对设计进行比较和选择的时候。优化设计是指以理论数学的方法为基础，改进那些对于工程师来说太复杂的以至于无法修改的设计，使其设计过程自动化。如果自动优化设计在台式计算机上能够实现，那就可以节省大量的时间和金钱。

优化的目的就是要将目标极大化或极小化，例如，受频率响应和设计参数等条件约束的重量或基频等目标对象。设计目标的大小和形状决定优化所采用的方法。

把优化看作设计的一部分可以使我们更容易的理解优化设计。首先，需要对问题进行处理、分析和加工，正像通常有限元分析（FEA）和计算机辅助设计（CAD）应用程序所使用的那样（只不过CAD是根据设计参数建立问题的几何图行）。其次，确定优化目标和反应约束。最后，反复自动修正设计。优化程序应该允许工程师监督设计进程，必要时停止设计，改变设计条件或重新开始。优化程序的功能取决于采用的预处理和分析方法的功能。二维和三维优化设计的应用要求能够自动进行搜索区间的参数调整。在优化过程中，问题的几何条件和搜索区间会改变，所以优化程序必须包含误差估计和自适应控制。

修改、重配网格和重新评估模型以实现特定设计目标要从输入初始设计数据开始。按着规定合适的公差并形成约束条件，已获得最优结果；或者至少改进设计。为了对各种产品进行优化，包括简单的框架结构以及复杂的三维实体模型，设计者必须广泛接触设计目标和约束条件。同时，设计者还应该具有熟练确定并利用下列参数作为约束和目标函数的能力，如重量、体积、位移、应力、应变、频率、翘曲安全系数、温度、温度梯度和热通量。

此外，工程师们应该能够通过多学科的不同类型的优化分析使多种约束相互

结合起来。例如设计者可以把温度作为热载荷，通过热力分析进行应力分析，通过对最高温度，最大应力和变形制定约束条件确定理想的基本频率范围。

目标函数可以代表整体模型或模型的一部分，更重要的是，通过制定重量或成本等因素，目标函数应该能反映模型各个部分的重要性。

优化方法与有限元分析和计算机辅助设计的组合正影响着产品的设计过程。这种组合把工作任务从工程师转移给计算机，从而可以降低设计成本。此外，正确地使用优化方法所带来的数学上的严谨也能提高设计过程的可靠性。通常，优化方法决定了包括CAD软件及有限元分析中三维网格自动划分和分析处理器等在内的一系列的应用。这种组合使得在包含机械效应和热效应的各种实际情况下能够对CAD的零件和装配件形状进行优化。

现代的优化方法可对CAD软件产生的零件形状进行优化。理论上，通过存储内容的直接转储在CAD和FEA应用间无需文件的转换而实现数据的无缝交换(传输)。此外，如果CAD和FEA软件间存在关联，CAD几何图形的任何改变会立即在FEA模型中体现出来。在ALGOR所采用的方法中，有限元模型产生之前，优化设计过程就开始了。用户只需要选择CAD模型中要优化的尺寸及设计准则，包括最大应力，温度或频率，然后执行合乎设计准则的分析过程。分析的结果与设计准则进行比较，如果需要的话，无需任何人工干预就可更新CAD图形。必须注意使用关联的原则使FEA模型得到更新，这就意味着要从前面的分析中保存约束和荷载，现在要分析包含新的高质量的立体网格的新的FEA模型，结果要再一次与设计准则进行比较，这一过程反复进行直到满足设计准则。下图说明了形状优化设计的过程。

原始设计 预处理 有限元前处理程序 优化前处理程序 优化过程 有限元求解程序 优化模块 最优设计

优化设计过程示意图

绪论

典型的设计过程在零件几何形状发生改变时会用到迭代法。通常，每一次的

迭代也会用到某种形式的分析以获得可行的工程结论。优化设计会用到很多次这样的迭代，每次迭代代价很高，尤其在考虑工程师的时间的价值时。优化设计应用软件的本质就是通过自动进行迭代减少工程师的工作量。乍一看，似乎优化设计是把工程师和专家们从设计循环中替换出来的一种工具。当然情况并非如此，因为任何优化设计软件都不能指明要对什么目标进行优化，哪些是设计变量，哪些量或参数可以改变以获得最优设计。因此优化设计软件只是工程师可以使用的另一种工具。其作用由其有效地确定最优解的能力来衡量。

优化设计软件往往会强化数字化处理，因为他们仍然要执行几何分析的迭代。值得庆幸的是，大多数优化设计问题可以看成是数学优化问题，而数学优化问题就有许多的高效求解方法。但具有许多求解方法的缺陷是对于一个特定的问题总有一种最佳的数学优化方法。这种复杂性可以通过在优化设计软件中不仅给工程师提供优化方法的选择，还给出什么方法最适合其设计问题的建议来得到改善。

本文重点在机械零件或装配件的优化设计上。此时，常用的优化目标是能承受最大应力。典型的设计变量包括一些几何量，如特殊零件的厚度。零件或装配件的设计从CAD软件开始。如果(所建)结构能保证工程分析的质量，工程师一般将选择使用有限元分析来模拟分析其机械性能。有限元分析的结果，如最大应力，可以用来验证设计的正确性。设计过程中，工程师可能需要改变CAD和/或FEA模型的参数或特征，包括某些具体尺寸、材料或对零件或装配件的加载或约束方式。CAD和FEA 软件间的关联性使得工程师能在其中之一改变模型，而在另一软件中自动体现这些改变。例如，如果在CAD软件中改变了零件的厚度或增加了一个孔，有限元模型的网格能自动反映这种变化。在大多数情况下，工程师将使用线形静态FEA来获得应力。这种分析方法的优点是用较少的时间得到单元数较多的有限元模型的解。显然线形静态FEA也有缺点。例如，在估计由于运动而引起的载荷的大小和方向时需要丰富的工程专业知识。

基础知识和理论

这部分我们来重点讨论优化设计使用的数学方法的基础理论。但是首先我们描述一下有限元问题是如何提出的。考虑如下3个过程：

(1) 在CAD中产生零件或组件的几何图形； (2) 创建零件或组件的FEA模型； (3) 计算FEA模型的结果。

现在，我们只限于讨论线性静态FEA。因此，这种情况下分析结果由变形和应力组成。手工设计过程包括完整的3个步骤，所得结果用于评价该设计是否合适。如果不合适，就要对步骤(1)或(2)或者(1)和(2)均做出修改。从这个描述中可以清楚地看出，FEA输出的是要优化的目标，而CAD或FEA模型的任何输入都可以看做设计变量。优化设计的算法对许多的FEA运用有指导作用，每一种算法都有一套不同的设计变量值。CAD和FEA应用之间必须建立起关联，才能把手工设计方法转变成优化设计算法。这一要求的合理性可通过一个例子得到最好的解释。在开始设计阶段，工程师在FEA模型的一个特殊表面施加约束时，我们可以很放心地假设这个表面与CAD模型中的面一致。现在如果优化设计的算法决定要改变CAD模型中面的几何形状，那么FEA模型必须自动反映这些变化，并在该表面的新模型上施加约束。因此为了实现CAD与FEA模型间自动通讯，就需要二者之间的关联。在明确了机械系统优化设计的问题后，接下来我们来描述解决这些问题的数学方法。

大部分的优化问题都由如下3个部分组成。 (1) 一个要使其最小化(或最大化)的目标函数。如：在设计汽车的仪表板时，使其在特定区域内的应力最小。

(2) 一组影响目标函数值的设计变量。如：在汽车仪表板设计中，用变量来确定仪表

板的几何形状和材料。

(3) 一组约束。这些约束允许设计变量取某些值而排除另一些值。在汽车仪表板设计中，可能会其重量。 也可以建立无约束优化问题。或许有人会争论说几乎所有问题都应具有某种形式的约束。例如，汽车仪表板的厚度就不能为负。不过实际上，常常无须对设计变量强加约束也可获得从基本物理知识上讲有意义的结果，如正的厚度值。

优缺点

消除或减少人的重复工作已经成为了许多软件应用的推动力。自动化的优化设计是以可能增加计算量为代价来减少人的工作时间的最新应用之一，甚至也可能实际上所需要的计算量比手工设计方法还少。这是因为数学作为这些方法的基础，其严谨性比人工的解决方法效率更有效。当然这些方法也不能取代人的知识，人的知识有时可以大大缩短设计过程。自动化方法与手工方法相比，一个明显的优点是如果软件的应用正确的话，可以把一切可行的可能性考虑进来。也就是说，设计参数的各种可行的组合均会考虑到。因此，用优化设计软件所获得的设计应该是精确到全面的解决方案。