文献翻译-有限元和优化
外文资料译文
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 chang e 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
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.