A Discussion on Modern Design Optimization Tools:

Full Associativity of CAD, FEA and Event Simulators

Abstract

The integration of optimization techniques with Finite Element Analysis (FEA) and CAD is 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. 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. The optimization method also allows for global constraints to enforce weight and volume criteria. These global calculations require little additional effort because of the tight integration among the applications, including the weight, center of gravity and mass moment of inertia processor. Even though global constraints can be used to optimize material usage, engineering expertise is generally required to optimize costs.

A possible peril of design optimization is that its accuracy, and thus success, hinges on assumptions and approximations remaining valid for all of the FEA analyses. For example, a linear static stress analysis may be accurate for a certain value of a dimension, but highly inaccurate for another value for which buckling or material nonlinearities occur. A nonlinear analysis would be required to account for such effects. Concerns about the validity of the FEA method by no means reduce the overall usability of design optimization, but do suggest that many situations warrant more powerful analysis tools. Event simulators, such as ALGOR’s Mechanical Event Simulation (MES), are currently the most powerful means by which to simulate mechanical phenomena. In addition to eliminating our concerns by accounting for nonlinear materials and buckling, event simulators can simulate dynamic effects. One may need to consider such effects, as many components whose design warrants optimization techniques experience large-scale motion during their operation. Furthermore, event simulators can consider the entire process associated with the operation of a component. Such a virtual laboratory adds another dimension to the design optimization process by not only providing the entire stress history, but by graphically demonstrating how the component interacts with its surroundings.

1. 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 behind design 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.

Design optimization applications tend to be numerically intensive because they must still 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; some of these methods are discussed later in this paper. 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, many times eliminating the need for physical prototyping. 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. Such “large” FEA models are common when they represent parts generated in CAD. 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. We shall discuss these drawbacks in greater detail in Section 3 of this paper.

Engineering expertise minimizes the effort involved in the manual design process by eliminating certain parameter combinations or characteristics from consideration. For example, only real materials are considered. The engineer is also limited by certain global constraints such as the need to minimize costs or weight. The goal of a design optimization application is to duplicate the procedure that the engineer conducts without sacrificing accuracy. Actually, the mathematical methods underlying the application should insure that all viable possibilities are considered; this is certainly an advantage of an automatic approach. As mentioned above, engineering expertise is still required in order to identify what should be optimized and what should be the design variables. In the next section of this paper, we discuss mathematical optimization methods. Following this discussion, we return to the mechanical design problem and highlight the benefits and perils of applying design optimization in this setting. Finally, we discuss how some of the perils can be avoided if nonlinear FEA or event simulators are used in place of the standard linear analysis.

2. 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, and
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 these changes, 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.

In the unconstrained optimization problem, we develop a minimizer (or maximizer) function, f(x), where x is a vector representing the design variables. In mathematical terms, we seek a vector x*, such that

(1)

 

for all x close to x*. Note that we are not seeking a global minimum, which can be difficult to obtain even with the latest technology. For most applications, local minima are sufficient, particularly when the user can draw on his or her expertise and provide reasonable initial values for the design variables.

There exist two common classes of methods to solve Equation (1): zero-order methods in which derivatives or gradients of f(x) are avoided, and first-order methods that utilize gradients of f(x). In the case of mechanical design optimization discussed in Section 1, the latter method is usually much more computer intensive as these gradients must be evaluated using finite differences in x-space after having obtained f(x) at different values of x. The numerical effort is compounded by the fact that each evaluation of f(x) requires conducting an FEA analysis. Nevertheless, when accuracy requirements are high, one should opt for the more rigorous first-order methods. Actually, first-order methods should be employed whenever f(x) is highly sensible to the chosen design variables.

Most first-order algorithms are adaptations of Newton’s method [1], which requires the calculation of the gradient vector

(2)

 

 

and the Hessian matrix,

(3)

 

Where N is the number of design variables. Newton's method results in a quadratic model function of f(x), around the iterate x_k. Specifically,

(4)

 

The value of that minimizes g_k is obtained by solving an NN system of equations, and is then used to determine the next iterate,

(5)

 

It should be noted that convergence is guaranteed if the initial values of the design variables are sufficiently near a local minimizer x* at which the Hessian is positive definite. In other words, if the starting point lies on an N-dimensional potential well.

As we mentioned above, some design optimization problems may require the enforcement of constraints. Generally, these constraints take the form of bounds on the design variables,

(6)

 

The approach described by Equations (2) through (5) can be adapted to solve constrained optimization problems by introducing variable transformations. These problems are termed reduced-gradient methods, and their mathematical details are beyond the scope of this discussion. It should also be mentioned that there are other popular methods for constrained optimization problems, including augmented Lagrangian and penalty; these methods still rely on the expensive calculation of gradients.

Up to this point, we have only discussed first-order optimization methods – ones that require the calculation of gradients. As mentioned above, there are circumstances in which it may not be practical to employ such numerically intensive methods. The SIMPLEX method [2] and its heirs require neither gradient nor Hessian evaluations. Instead, they perform a pattern search based only on the values of f(x). Because SIMPLEX-type methods makes little use of topological information about f(x), they may require many iterations to find a minimum. These types of methods can be particularly useful when f(x) is non-smooth or when its derivatives are difficult to find. Both of these situations can occur in the mechanical design optimization problem. For example, if the algorithm changes materials, then there will be abrupt changes in material properties and thus in the results and in f(x).

3. 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 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 viable 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.

In the case of mechanical design optimization, the resolution of the method is determined by how accurately the FEA analysis duplicates actual physical behavior. In the case of a linear static FEA analysis, there are many inherent assumptions and approximations. For instance, all displacements are assumed to be infinitesimal, thus neither motion nor material nonlinearities can occur. Nonlinear FEA is required to account for the effects of either of these phenomena. Engineers may utilize linear static FEA in a design optimization scheme if they do not anticipate motion, or have accounted for its effects through modeling techniques. For example, the effects of an impact can be approximated using a linear analysis as long as valid engineering estimates are made for the impact forces. Significant engineering expertise is typically required to accurately make such estimates. Engineers will rarely consider design optimization problems in which motion occurs for just some combinations of the design variables. In contrast, material nonlinearities can easily occur for just certain combinations of the design variables. Thus, if a linear static FEA analysis is being utilized, care must be taken to insure that during each iteration that material linearity is preserved, i.e., no yielding occurs. An iteration that violates this condition would be meaningless because the linear FEA analysis is unable to duplicate the physics. By applying constraints on the results, the user can avoid such violations. These are not the same types of constraints discussed in Section 2, which were on the input design parameters. Constraining the results requires knowledge of the inner details of the design optimization method, which may not always be available to the user. A properly implemented design optimization application should be transparent to the user, giving him or her all pertinent internal details with which to verify the results.

4. Advanced FEA Methods

In Section 3, we discussed some of the limitations of linear static FEA. When these limitations prevent the analysis from accurately modeling the physics, more powerful tools such as nonlinear FEA should be incorporated into the design optimization loop. Nonlinear FEA is fully capable of simulating motion and material nonlinearities at the cost of greater computing resources. Generally, these nonlinear analyses are no longer limited to considering just an instance in time; they may even consider a simulation of the component’s behavior. Such a time-history is invaluable when the engineer is unsure as to when the part experiences its most critical conditions. In order to obtain the most reliable solutions, this time-history should not just be a sequence of pseudo static steps. Instead, the analysis method should account for the inertia associated with the motion, as does ALGOR’s Mechanical Event Simulation (MES) software. Using MES in a design optimization application is equivalent to considering the results of many virtual experiments, each of which is run using different configurations, including geometrical, material or constraints. This is certainly the next technological step from using single-instance linear static FEA analysis in a design optimization application.

Incorporating time-history results into a design optimization application will require the evaluation of results at all of the considered time instances. Furthermore, the greater availability of results may prompt the engineer to consider the introduction of multiple objective functions. The engineer will typically have to define weighted sums of objective functions. For example, the engineer may want to minimize the maximum stress in distinct components of a mechanism, but assign different weights to minimizing one over the other. This additional effort will only increase the confidence in results obtained from the associated design optimization application.

References

1. Kowalik, J and Osborne, M R - Methods for Unconstrained Optimization Problems, American Elsevier Publishing Co. Inc., New York, 1968.
2. Nelder, J A and Mead, R - SIMPLEX Method for Function Minimization, Computer Journal, 1965 (Vol. 7), pp. 308.