Altair's Student Guides - CAE and Design Optimisation - Basics

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CAE and Design Optimization – Basics Contents

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Introduction

About This Series

To make the most of this series you should be an engineering student, in your third or final year of Mechanical Engineering. You should have access to licenses of HyperWorks, to the Altair website, and to an instructor who can guide you through your chosen projects or assignments.

Each book in this series is completely self-contained. References to other volumes are only for your interest and further reading. You need not be familiar with the Finite Element Method, with 3D Modeling or with Finite Element Modeling. Depending on the volumes you choose to read, however, you do need to be familiar with one or more of the relevant engineering subjects: Design of Machine Elements, Strength of Materials, Kinematics of Machinery, Dynamics of Machinery, Probability and Statistics, Manufacturing Technology and Introduction to Programming. A course on Operations Research or Linear Programming is useful but not essential.

About This Book

Finite Element Analysis has traditionally been used as a design-verification method. Recent developments in Mathematics, Software and Mechanics have led to a dramatic change: Computer Aided Engineering (CAE) is now widely deployed at the concept-design stage itself.

This volume presents these changes in the engineering industry and introduces you to the theory necessary to effectively apply optimization techniques using OptiStruct.

You should read Chapters 2 and 3 in their entirety. Chapter 4 can be skipped if you are familiar with the Finite Element Method, but is essential reading if you’re not. Chapter 5 is best read once, then referred to again when you are working on your assignment. Chapter 6 is essential if you want to work on a project addressing laminated composites, but can be safely omitted if you’re working with other materials. The various references cited in the book will probably be most useful after you have worked through your project and are interpreting the results.

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CAE and Design Optimization - Basics Introduction

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Supporting Material

Your instructor will have the Student Projects and Student Projects

Summaries that accompany these volumes – they should certainly be made use of. Further reading and references are indicated both in this book and in the Projects themselves.

If you find the material interesting, you should also look up the HyperWorks On-line Help System. The Altair website, www.altair.com, is also likely to be of interest to you, both for an insight into the evolving technology and to help you present your project better.

"Mach 2 travel feels no different." a passenger commented on an early Concorde flight. "Yes," Sir George replied. "That was the difficult bit."

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Engineering Design Practice CAE and Design Optimization – Basics

Engineering Design Practice

It’s rare to meet a mechanical engineer who hasn’t, at one time or another, been fascinated by automobiles and aircraft. With attractive looks, complex designs, exciting performance, and a long history of colorful personalities, they embody everything you ever wanted to create. Who’d ever want to design anything else?

Lots of people, it turns out.

While the automotive and aerospace sectors continue to define technology and trends in mechanical engineering even today, these are not the only sectors that make widespread use of design software. Neither are they the only segments where serious sums of money hinge on the efforts of the Design Engineer.

The section below outlines the bases for evaluating designs in various engineering sectors.

Characteristics of Different Sectors

First, let’s specify what we mean by the word “Design”. As engineers, we usually ignore aesthetics, leaving that to Industrial Designers. One of the tasks of an engineering designer is to come up with a design that is

functionally “satisfying”. That’s very often very hard to do, since there are so many ways to define “satisfaction”. To understand how optimization fits into engineering design practice, it’s instructive to look at the different ways in which “satisfaction” is measured by some industry segments,

Cars, motorcycles, trucks, buses and other road-transport vehicles are often grouped in one sector. Appearance, ride quality, safety and fuel economy are the most important factors in design – apart from cost, of course. Plastics and Steel are the most commonly used materials. Most often, aesthetics and cost dictate the visible areas (interiors and body), with the other parts being designed for efficiency in function. Performance and cost, in most cases, determine satisfaction levels.

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CAE and Design Optimization - Basics Engineering Design Practice

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Aircraft and space vehicles are often clubbed in the “aerospace” sector. Cost is rarely an issue, but performance is of paramount importance. Safety regulations in passenger aircraft are among the most stringent in the engineering world. Combat aircraft are subject to harsh environments. Spacecraft can gain useful life with every gram of weight shaved off. Advanced materials – ceramics, composites, honeycombs, and exotic alloys – tend to be widely used, along with advanced manufacturing techniques. Ranging from the cardboard boxes that toothpaste

tubes are sold in to the containers of cars that are loaded on to cargo ships, “packaging” is a multi-billion dollar industry. Not very surprising, in fact, when you stop to consider that products like toothpastes and soft-drinks are literally sold in billions. Packaging materials are often “outside” standard engineering technology – Styrofoam and paper are rarely treated as load-bearing materials in engineering courses!

Products such as cell phones, stereos, watches, washing machines, etc. rarely cause loss of life if they fail. Sometimes called “consumer goods”, these products are usually designed for elegance and cost. Product life is sometimes measured in weeks, translating into extreme time pressures on designers. Plastics are used very widely.

One thing that’s common to all sectors is the fact that

designers are always under pressure to create better products in less time and at a lower price.

And this, of course, is why optimization plays such an important role in product design.

CAE And The Design Cycle

It’s long been recognized that the designer is almost always under pressure to meet time targets, performance targets and cost targets. The figure below shows the “typical” design cycle:

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Engineering Design Practice CAE and Design Optimization – Basics

The design cycle almost always originates with a drawing – a sketch to illustrate a concept - and almost always ends with a drawing – the

manufacturing drawing. This is the biggest problem – how to translate the sketch into an acceptable, manufacturable design. A typical design cycle involves numerous trade-offs: appearance vs. function, cost vs. ease of manufacture, etc. Every trade-off changes the design, and changes are inevitable. One of the rules written by Kelly Johnson, legendary head of Lockheed’s Skunk Works, demands that “A very simple drawing and drawing release system with great flexibility for making changes must be provided”.

The widespread use of 3D CAD software has made it easier for engineers to re-create manufacturing drawings when the design changes. But CAE1 is often viewed as a visit to the dentist: put off as long as possible, and usually painful.

The reason for this is easy to find. Since CAE has traditionally been used to verify the preliminary design, analysts usually bear bad news: that the design has failed the verification and must therefore be changed. If the analyst carries good news, it’s often ignored since it’s too late in the design cycle to implement the changes!

Wouldn’t it be great if the concept-designer had a tool that could help suggest designs that are least likely to get rejected by subsequent CAE?

The Impact of Optimization on CAE

Relatively recent advances in mechanics and software have provided just these capabilities: software can suggest the design that is best suited to the conditions you specify. In other words, an “optimum” design.

1_{Short for Computer Aided Engineering. Usually taken to mean simulation of}

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Design vs. Analysis

As opposed to analysis software, optimization tools are for designers. Put forward your definition of a “satisfactory” design, and the optimization software will suggest to you shapes and sizes that are most likely to pass the subsequent analyst’s verification.

This means a tremendous change in the way we can now view CAE: the visit to the dentist will not be as unpleasant! Since our design has been

“certified” by the optimization tool, we can approach the traditional “verification” stage with a much higher level of confidence.

In the rest of this book you will learn how OptiStruct brings these capabilities to the designer.

Other Designer Issues

This section summarizes some design approaches that are not addressed by linear optimization – which is the focus of this book. If you think the below are relevant to your problem or are of interest to you, you should read the other volumes of this series.

Design-Of-Experiments

Statistics teaches us that in a “Normal Distribution”, a large part of the population lies within 6 standard-deviations of the mean. Engineering industries refer to this as “six-sigma” quality – less than 3.4 failures per million parts produced.

To achieve these levels of quality, designers setup numerical experiments to account for the statistical

variations in various design parameters – the elasticity modulus of bars of steel, the rigidity of supports, the variations in loads, and so on. Designers also use non-linear equations to represent material behaviors.

As a result, optimization tools are often classified as linear-optimization software, such as OptiStruct, and design-of-experiment tools such as HyperStudy that interface with non-linear solvers like Radioss.

Multi-Disciplinary-Optimization

As mechanical engineers, we study the difference between mechanisms and structures, and develop different sets of equations to design these. We study equations that govern the flow of heat, equations that govern the flow

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Engineering Design Practice CAE and Design Optimization – Basics of fluids, equations that govern the stress-distribution, …each as a different

subject, with little if any interaction between the equations.

How would you optimize a product that has some parts that move rigidly (i.e. act as mechanisms) and some that flex (i.e. behave as structures), and that has to withstand stresses and also be aerodynamic?

MotionSolve, used in conjunction with OptiStruct, allows designers to find optimal solutions to some of these problems, while HyperStudy addresses others.

Process Optimization

A product that’s been designed and verified still stands the risk of rejection: by the process engineers, if it’s too expensive to manufacture. How can you design the manufacturing process to be most “satisfactory” – in other words, how can you optimize the manufacturing process?

HyperForm is used to simulate the sheet-metal forming process, and,

together with HyperStudy, can be used to arrive at optimal process designs.

Summary: How Engineers Should Design

All these tools, then, change the way you should address the design cycle. Rather than use CAE to verify (and in all probability reject!) suggested designs, the preferred approach is to use software to suggest a design that is much more likely to work. In many cases, the “redesign” cycle is all but eliminated.

Computer software continues to redefine the way products are designed. But that does not eliminate the need for engineering judgment. In fact, it increases the burden on the engineer, who now has to act both as the investigator and as an impartial and knowledgeable judge.

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First, you phrase the design requirements in as realistic a way as possible. You may even want to try different statements of the different

requirements.

Then you ask the software to come up with its suggestions.

Finally you sit in judgment: which of the statements was the most realistic, and which of the suggestions do you want to adopt?

Engineering problems are under-defined, there are many solutions, good, bad and indifferent. The art is to arrive at a good solution. This is a creative activity, involving imagination, intuition and deliberate choice.

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Optimization Theory CAE and Design Optimization – Basics

Optimization Theory

What is an Optimum Design?

It’s evident from the previous chapter that as a designer, you should search for an optimum design. What is not so clear is how exactly we can recognize the “optimum” design. The dictionary definition is a good place to start. An “optimum”, says the dictionary, is “the greatest degree or best result obtained or obtainable under specific conditions”.

It’s the phrase “specific conditions” that gives you your design freedom. As a designer, you define the conditions that allow you to evaluate your design alternatives. In engineering terms, this means you draw up mathematical equations that quantify the performance of a design. The statement “good ride quality” would translate, for instance, into a specification of the

maximum values of the components of acceleration that the passenger’s seat can experience. The quantitative parameter that you use to evaluate a design is called the objective.

Of course, you may well have multiple objectives. For instance, it’s very likely a car designer would simultaneously want excellent safety and low cost. Unfortunately, in many cases, the objectives are contradictory, making it increasingly difficult for the designer to reach the best compromise2. A working design almost always involves a compromise of some sort or the other.

To make things harder for you, few designers have the luxury of infinite resources in the pursuit of their objectives. Whether the resources are the money you can afford to spend on materials, the amount of fuel the

spacecraft can carry or the maximum drag coefficient permitted for a sports car, there are usually limits you have to work between. These limits, or constraints give rise to the subject named constrained optimization. A solution that satisfies the constraints is called a feasible solution, while one that does not is called an infeasible solution.

It’s important to realize that not all design is done from scratch. In several cases, we have to start from existing designs and improve them to the best extent possible. This could be for various reasons, ranging from the

2_{MOO, or multi-objective optimization, is covered in}

CAE And Design Optimization - Advanced.

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CAE and Design Optimization – Basics Optimization Theory

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necessity to liquidate existing inventory to the

modification of a manufactured design that has failed a test.

If you’re starting from scratch, you can list the objectives and constraints and search for the best solution. If you’re working on modifying an existing design things are

usually a little harder since you have less flexibility to change things.

Mechanical Engineers face one further requirement. Most

components you design have to assemble with other components. They need to fit together. This means you have to work with a package space within which your component needs to fit3, and assembly points that cannot be varied since they’re decided by other components. In mathematics, the package space is referred to as the design space or the optimization domain. Finally, you may not be allowed to change every possible parameter. For example, the material you can work with may be restricted by factors beyond your control: working with sheet steel limits you to commercially available thickness. The parameters that you have the freedom to vary are called design variables.

The dependence of the objective on the design variables is expressed as an equation, called the objective function.

The statement of the design optimization problem then, consists of the package space, the design variables, the constraints and the objectives. If you have any of these wrong, it’s pretty likely your design proposals will be useless!

Analysis and Design

Coming up with a concept involves a synthesis of ideas to suggest different alternatives or proposals. Evaluating the performance of each proposed design involves analysis of function of that particular proposal. As a designer, which should you focus on?

The nice part of using Design Optimization as a part of CAE is that you can simultaneously do both, instead of doing them one after the other. As we

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Optimization Theory CAE and Design Optimization – Basics saw in the previous Chapter, the separation of conceptual-design and

design-verification into distinct steps was one of the main reasons analysis is frowned upon even though it’s essential for good product design.

In the conventional design process, the designer would have to rely on experience or insight to come up with proposals. The analysis tool is then used to evaluate each proposal, with the designer using these analysis results or responses to choose the “best”.

Optimization changes this. The designer outlines the constraints, and leaves it to the optimization tool to come up with proposals. The optimizer uses the analysis tool to decide how to change the initial design to arrive at a better one.

In qualitative terms, an analysis problem has only one correct answer4.

Design, of course, has no single “correct” answer.

There are always a variety of options that can satisfy the same requirements, which is why it is extremely important to search for an optimum design. This is the reason a good analyst often does not make a good designer!

For the designer, then, analysis and optimization are very much complementary functions. They are equally important parts of design

optimization: a design optimization model consists of an analysis model and an optimization model. These are related and dependent but distinct areas, so we will take some care to understand which parts of the design problem will be defined in the analysis model and which in the optimization model. This chapter outlines the background of optimization, while the next outlines the basics of one of today’s most popular analysis methods, Finite Element Analysis. Other analysis methods can also be used, of course, as in multi-disciplinary optimization or non-linear optimization. These are covered in the other volumes of this series.

4_{At least in linear analyses, where uniqueness-of-solution is an important}

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The world of optimization is a hard one to live in. It’s a little like being asked to search for a black cat in a dark room. You know it’s in there somewhere, but have to feel your way forwards, backtracking and changing direction frequently since the cat changes its position every time you move5. In the world of linear equations, at least we’re assured that there’s a cat in the room, and that there’s only one cat to look for. In many real world problems, we cannot always count on this, as we’ll see.

Since a person who analyses is called an analyst, perhaps a person who seeks to optimize should be called an optimist!

Our objective, then, is to find a better design than the one we are starting with. In some cases it will be the best while in other cases it may not.

Finding An Optimum

Since we will be happy to find a better solution even if it’s not the best, we are looking for an optimum solution, not necessarily the optimum solution. Why are we emphasizing this statement?

In optimization theory, by convention, we search for the minimum of the objective function. This is not a limitation since maximization of an objective is equivalent to minimizing its reciprocal6. A function that has only one minimum within the optimization domain is called a convex function. It’s useful to recall the basics of

differential calculus. In calculus, a minimum (as well as any other “turning point”) of a curve is characterized by a zero slope (or first derivative). If the objective function is a quadratic function of the design variables, we are then guaranteed a global minimum. This is because a second order curve has only one turning point – and therefore only one minimum in the design space.

5_{In mathematical terms, this behaviour is a characteristic of}_implicit_{equations. The}

“knowns” and “unknowns” cannot be neatly separated into the right-hand-side and left-hand-side.

6_{Sometimes maximization of}

x is addressed as minimization of the negative value of

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Optimization Theory CAE and Design Optimization – Basics A higher order curve may7 have multiple turning points within the design

space. If it does, then we may have multiple minima8. The turning point at which the objective function has the least value is the global minimum, while the other minima are called local minima.

A real life problem may well have hundreds, if not thousands of design variables. And the objective function may well be a non-convex function, with multiple local minima within the design space.

How does optimization software arrive at a better solution within a reasonable time? How does it interface with or use analysis software?

The Mathematics of Optimization

Note that this book is restricted to “linear” problems only –where there is a linear correlation between inputs and responses. For non-linear problems, an entirely different approach is used, as described in CAE And Design

Optimization - Advanced.

To recap, to define a problem in design optimization you must specify the design space, the design variables, the constraints, and the objectives. The corresponding mathematical statement is:

Minimize f(x) = f(x1, x2, x3, …. xn)

Subject to gj(x) ≤ 0, j = 1, … m

xiL≤ xi≤ xiU

7_{A higher order curve has more than one turning point, but some may lie outside}

the design space.

8_{Recall your calculus: a turning point can be a maximum, an inflection point or a}

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where f(x) is the objective function, g(x) are the constraint functions, and x is a vector of design variables.

An Example

We may be asked to design a light-weight bracket that has to fit in a 300 mm x 300 mm x 600 mm volume. We want the bracket to be made of steel, to carry a load of 100 Kg. The maximum

permissible deflection of the bracket is 0.1mm, and the maximum permissible stress is 20 Kg/mm2. We are allowed to use sheet-steel that can be 1 mm, 2mm or 4 mm thick.

In this case, our design space would be the 300 mm x 300 mm x 600 mm volume. The objective would be to minimize the mass. The constraints would be the permissible stress and deflection. The design variables would be the thickness of the steel, and the layout of the steel (i.e. how the sheet should “flow”– where material should be located - within the design space). To solve a problem like this, the optimizer would start with an initial

configuration or proposal. It would ask the analysis software to evaluate the mass, stress and deformations of this configuration – the values calculated by the analysis package and tracked by the optimizer are called responses. The optimizer would evaluate the sensitivity of the responses to the various design variables, and decide which to change and by how much.

When the design variables change, the responses change too. If the steel thickness changes, the mass of the bracket changes. The displacement would probably change too, as would the stress. So the optimizer would again need to ask the analysis package to evaluate the responses.

This iterative procedure would continue until the optimizer concludes it has found the best possible design for the given constraints and variables.

Evaluating Sensitivity

Evaluating the sensitivity of the responses to changes in design variables is, obviously, a very key part of the optimization process.

If we define the response as a function of the deformation u by the equation

u

g

=

T

then the sensitivity of the responses to the design variables is given by the equation

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Optimization Theory CAE and Design Optimization – Basics

x

u

x

g

T T

∂

+

∂

=

∂

Some design problems have more constraints than design variables, while others have more design variables than constraints. Different algorithms are used by OptiStruct for each case, in order to efficiently arrive at the

optimum solution.

The Optimization Model

Asking the analysis package to evaluate the responses each time a variable is changed can be very expensive in terms of computer time. OptiStruct takes a different approach: the optimizer builds an approximate model, and does most of its work within this approximate model, turning back to the analysis software only when essential. This makes the optimization much faster.

It also has another implication. The analysis model itself is an approximation of the physical behavior of the product. Since the optimization model is an approximation too, the responses evaluated by the optimizer are unlikely to be very precise. They are twice removed from the physical product. This means that as a designer you must subject the final proposal of every optimized design to a verification-analysis.

There are various techniques the optimization model uses to reduce computer-time and still get an accurate solution. Most of these are

programmed into OptiStruct. As a designer you can control these methods, but doing that requires a good understanding of the mathematics. That’s not the intent of this book. Here, we want to develop a good understanding of better product design.

The following sections summarize some of the more important techniques. Remember that the intention is not to be mathematically rigorous. Rather, the intention is to provide you with an overview of the workings of the optimization model. This should help you phrase your design problems correctly.

Managing Local and Global Minima

Even if the analysis-model is linear, the optimization problem is frequently not. Take, for instance, the deflection of a cantilever beam that has a rectangular cross-section. The deflection equation is

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CAE and Design Optimization – Basics Optimization Theory 17

EI

wL

3

=

δ

The analysis model is linear since the equilibrium equation is a linear

function of the state variable δ. If the Elasticity Modulus (E) were a function of the deflection, as in a plastic analysis, the analysis model would be non-linear.

Suppose we want to choose an optimum depth for the cross-section. The Moment of Inertia is given by

12

3

bd

I

=

which is not a linear function of the design variable – d.

Depending on the objective function chosen, the optimizer might have to search for the minimum of a non-convex function.

One of the parameters that determine whether the optimizer finds a global or a local minimum is the starting point of the search – the initial

configuration or proposal. Another is the move size, which is the step that the optimizer takes in the direction dictated by the Gradient Search

algorithm. If the step is too large, the optimizer may overshoot the optimum, which means it will have to reverse its direction in the next iteration. If it’s too small, the optimizer may take too long to locate the optimum.

What does this mean to you as a designer? First, you can vary the move size if the optimizer doesn’t converge. Second, an intelligent choice of the initial configuration and design variables can significantly affect the design

suggested by the optimizer.

Convergence and Iteration Control

As the optimizer searches through the design space, it needs to check whether proposals it comes up with are indeed optimal or not.

Mathematical formulations of the optimization problem usually include Lagrange Multipliers, which can be interpreted as a method to find extrema

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Optimization Theory CAE and Design Optimization – Basics of a bounded surface. The objective function, of course, can be viewed as a

surface, with the constraints as boundaries. An interesting interpretation of the Lagrange Multipliers is provided by S.Jensen9:

the constraint function g(P) can be thought of as "competing" with the desired function f(P) to "pull" the point P to its minimum or maximum. The Lagrange multiplier λ can be thought of as a measure of how hard g(P) has to pull in order to make those "forces" balance out on the constraint surface

The Karush-Kuhn-Tucker conditions, also called the Kuhn-Tucker conditions, are a necessary condition for the solution of an optimization problem to be optimal. The KKT conditions are often used not to find the solution but to obtain information about the solution. This is useful to us, since we are looking for a design solution, not a mathematically precise solution.

From our design perspective, it is important to understand that the search is an iterative procedure. First, we can instruct the optimizer how long to search, by telling it the maximum number of iterations. Further, we can tell it how fine the search should be. If the difference between two successive proposals is less than a convergence tolerance, the optimizer can be asked to conclude that this is acceptable to us from a design perspective.

Gradient Search Methods

Most engineers are familiar with Newton’s method to find the roots of a polynomial. As shown in the figure, this method uses the slope of the curve to guess at which direction the initial guess should be adjusted in – to increase or decrease it. In practice, the gradient is often computed using a finite difference method.

9_{See http://www.slimy.com/~Esteuard/professional.html for an excellent}

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Design Variables

Things that can vary – thickness, density, etc.

Responses

Things calculated by the analysis model, and of interest to the Optimizer. Mass, deflection, stress, etc.

Constraints

Limits on responses or design variables.

Objective

Value that measures quality of your design. Mass, frequency, center of gravity, etc.

The gradient search method, also called the method of steepest descent, is one of the many methods used by the optimizer to move from the initial configuration to the final solution. Non-linear optimization normally uses other methods, as described in CAE And Design Optimization - Advanced.

Workable Implementations

Very often, an exact answer is worthless if it comes too late. An approximate answer that is available in time is much more useful.

In order to speed up the optimization process, the optimization model uses Constraint Screening, Constraint Linking and Constraint Deletion.

The first, constraint screening, is a technique used to identify which of the constraints are critical to the current iteration. In an effort to reduce the number of variables, the optimizer uses one or more criteria to choose a subset of all variables for each iteration. This subset is likely to change from one iteration to another as the optimizer moves through the design space. Constraint linking is when you can use factors such as symmetry to reduce the number of constraints that need to be considered. Suppose you want all beams in a structure to use the same cross-section because it makes the purchase process easier. In this case, it makes sense to link all of them together, thereby reducing the load on the optimizer.

As the optimizer searches through the design space, the current

configuration may violate only 2 of 3 constraints. In this case, the third constraint is not important for the iteration. It can be marked inactive and ignored in other words, the constraint can be deleted for this iteration.

Summary

Part of the challenge of optimizing a product is that designers are not always able to clearly define their design problem or state their definition of “optimum”. Don’t let this deter you too much. Even if you don’t arrive at the “best” design, any improvement over your current proposal is better.

Optimization technology is fairly robust today. Most of the methods outlined above are implemented

intelligently by the software. You can, however, make things easier for the optimizer and reduce your

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Optimization Theory CAE and Design Optimization – Basics design time by intelligent choices in both phases of design optimization: the

Optimization Model and the Analysis Model.

You can also, of course, set the optimizer an impossible task if the statement of your problem is itself wrong.

Therefore O students study mathematics and do not build without foundations.

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CAE and Design Optimization – Basics FEA Essentials

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FEA Essentials

As we’ve seen, design optimization relies on CAE to calculate the response of the product.

Computer Aided Engineering, unfortunately, is a catch-all phrase that’s not very well defined. It can mean just about anything. And often does. In more general usage, CAE, sometimes also called “Simulation”, is the use of

numerical analysis to study various behaviors of products. How they react to forces, what drag or lift they experience in a fluid, how they react to

different thermal conditions, the forces generated as they experience accelerations, and so on. The other volumes of this set10 address some of these aspects.

In the context of our study, however, we’ll focus mainly on the Finite Element Method. Without going into the mathematics of the method, we’ll look at those aspects that let us understand how it fits into our goal: using optimization to enhance product design.

Building models for analysis involves making approximations to the initial geometry to omit irrelevant details, specifying conditions on the boundary or at initial time, specifying solver options, choosing output options, and so on. It can help, therefore, if you are familiar not just with the terms used but also with some of the background. This can help you make intelligent decisions when you prepare your models for optimization.

This chapter is not intended to be rigorous – numerous textbooks are available that do that job admirably well. Rather, this is a quick summary of some of the salient aspects of Finite Element usage and theory.

Why use Numerical Methods at all?

Most engineering problems can be solved using one of two methods: analytical or numerical11. An “analytical solution” is a mathematical expression that gives the values of the desired unknown quantity or quantities at any location in the body, and at any instant of time. But analytical solutions can usually be obtained only for relatively simple

10_See_{CAE And Design Optimization – Advanced}_and_{CAE For Multi Body Dynamics.} 11_{A third method, often neglected by engineering beginners, is Physical Testing, or}

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FEA Essentials CAE and Design Optimization – Basics problems: as the geometries or mechanics or both become complicated, the

effort of finding an analytical solution is often so high that the solution cannot be found at an acceptable cost or in an acceptable time. For “complicated” problems, numerical methods provide approximate

solutions that are usually of adequate accuracy. One way of setting up these numerical solutions is to discretize the original body. This means we break the original geometry into several smaller geometries. We first solve the equations governing the mechanics over these smaller bodies, then piece the results back together to get the complete solution.

The two most widely used methods are Finite Element Analysis and Finite Difference Methods. The latter are used mainly for problems in

Computational Fluid Dynamics (CFD), while the former is used in a wide range of applications.

What is Finite Element Analysis?

Finite Element Analysis (FEA) simulates a physical part or assembly’s behavior by dividing the geometry of the part into a number of elements of standard shapes, applying loads and constraints, then calculating variables of interest – deflections, stresses, temperatures, pressures, etc. The

behavior of an individual element is usually described by a relatively simple set of equations. Just as the set of elements would be joined together to build the whole structure, the equations describing the behaviors of the individual elements are joined into a set of equations that describe the behavior of the whole structure.

One way of looking at it is to recall the approach you studied in Engineering Mechanics12. There, you drew free body diagrams of each member in the structure, wrote equations that related the unknown forces in each member, then wrote equations that had to be satisfied for the forces between

members if equilibrium is to be satisfied. Solving these equations gave you the forces in each member.

Elements themselves are defined by specifying the nodes, which are the vertices of the element. Just as 4 corners define a rectangle, the nodes define the shape of an element.

12_{A more complete discussion is presented in}

A Designer’s Guide To Finite Element Analysis.

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When you choose an element to represent a part of the product, you are also specifying the parameters that define the behavior across the element. For instance, in a stress analysis, if you know the 6 components of

deformation13 at any point, you can calculate the strain from this by taking the first spatial derivative. And once you know the strain, you can use the material properties to calculate the stress. For the Finite Element Method, every node has these parameters associated with it, just as in a truss-structure every member has forces associated with its end-points. From the values at the nodes, you can interpolate for the values between the nodes. Suppose you were asked to digitize a surface, using a Coordinate Measuring Machine. Unless your surface were absolutely flat, you would not space the measurement points evenly. Since you have to interpolate between

measured values, you would naturally choose to have more measurement points at areas where the surface curves sharply. In maths, we say these areas have a high derivative, or rate-of-change.

In a similar fashion, for an FE analysis you would create smaller elements (which means more nodes) at areas where you expect the stress to be high14. The choice of the sizes of elements depends on many things - the anticipated stress levels of a certain area, the detail wanted in the results, the stability of a solution algorithm, the available computational power, and so on.

A Finite Element program takes the elements you have defined, lists the equations for each unknown value, puts them together as a matrix equation, then solves all these for the values of the unknown parameters.

The equilibrium equation is of the form

[ ]

K

{ } { }

u

=

f

Since it’s analogous to the equations of spring-deflection, K is often called the Stiffness Matrix, u is called the deformation vector, and f is called the load vector. K is a square matrix, with one row (and column) for each

13_{The 6 components are the translations along the 3 axes, and rotations about the 3}

axes

14_{A high stress means a high strain, from Hooke’s Law. Strain is the first derivative}

of deformation. Hence a high stress area is one where the deformation has a high derivative. And this, of course, means the rate-of-change of deformation is high in areas of high stress.

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FEA Essentials CAE and Design Optimization – Basics unknown variable in the problem-definition. If, for instance, you have used

100 nodes in your model, and each node has 6 unknowns15, your stiffness matrix would be 600 x 600. u and f are each column-matrices. In our example, each has 1 column and 600 rows.

A computer is required because of the large number of calculations needed to analyze a part or assembly. It is not uncommon for a model to have more than 1,00,000 unknowns (called degrees of freedom). The power and low cost of modern computers has made Finite Element Analysis available to many disciplines and companies.

Finally, remember that most Finite Element Analysis models are applicable only to “structures” – they cannot be applied to “mechanisms”. Components such as the shackles that hold up the leaf-springs of a truck chassis require different treatment. These are not treated in this volume – refer to CAE For Multi Body Dynamics for that.

What are Finite Difference Methods?

In some areas, mainly in fluid flow, analysts often prefer to use a different mathematical approach than FEA. In this approach, there are no elements – the discrete points are referred to as grid points or grids. Some analysis programs call for “structured” grids – the numbering of and positioning of grid points must follow specific patterns. Other analysis programs are less stringent in their requirements – unstructured grids or blocks are supported.

Choosing a Numerical Model

As a designer, you need to anticipate the behavior of the product you’re designing. You will need to guess at the conditions it is likely to be exposed to, and then to predict how it will respond to these conditions.

In some situations, the behavior is independent of time – these are called steady state problems. In others, the solution varies with time – these are called transient problems.

In some situations, the response of the body to stimuli is linear. That is, there is a linear correlation between input and output. Such a model is, obviously, called a linear problem. Other situations are non-linear because there’s no linear dependence between stimulus and response.

15_{The 6 components of deformation are the translations along 3 axes and the}

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25

It’s important to remember that the product you’re analyzing does not know whether it is “linear” or not. You, as the analyst, can choose to model it as linear or as non-linear, depending on which is more likely to give you useful results. Since we are designers, not mathematicians, we are not interested in results that are “exactly correct”. We are willing to settle for

“approximately correct” provided we get the results in time and at a cost we can afford.

As you know from your courses on Linear Algebra and Differential Equations, linear equations are far easier to solve than non-linear equations. Therefore, we very often choose to model behaviors as linear even if a non-linear model is more precise. We tend to choose non-linear models only if there’s no linear model that’s even reasonably accurate.

Non-linear models are of several types – the materials used, the geometry involved, or conditions on the boundary can cause the “non-linear” nature. Examples of material non-linearity are plastic deformation, melting and solidification – the stiffness of the body changes as the material properties change. In other problems, the stiffness changes as the body deforms even if the material’s properties do not change – take for example the reduced rigidity of a plastic bottle as it is crushed. Examples of boundary non-linearities are contact and thermal radiation. In the former, the stiffness of the part or assembly changes as sections come into contact with each other. In the latter, the heat lost is proportional to the 4th power of temperature.

Some models, such as those required to simulate the behavior of a car when it crashes, can involve several of these types of “non-linearities”.

The Role of Physical Testing

Remember that the the terms in the preceding paragraphs describe the mathematical model of the physical behaviour. It’s your responsibility to exercise your engineering judgement to ensure that the model you’ve chosen does a reasonably good job of capturing the physical behavior. In order to make it easier for the design engineer to verify this, in actual engineering design, results of physical tests are used to verify that the numerical model is capable of reproducing conditions obtained under test conditions.

Then why not just test a physical model? Why simulate it at all using a numerical or analytical model?

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FEA Essentials CAE and Design Optimization – Basics For two reasons. First, constructing test models is expensive and time

consuming. In many cases, there’s no way to reduce the time for the test. This is starkly different from computer-methods. In the computer-world, a faster computer produces results faster. This same time-compression effect cannot be obtained in most tests! Second, tests themselves are not very easily controllable.

As a result, most engineering practice requries that the analysis model be validated against a test result. That is, the model is used to simulate performance of the product under conditions similar to an existing test. If the model is capable of doing this, then we assume it is capable of

reproducing behaviour under different conditions too. As a result, we can dispense with the physical test for further studies.

Quick Summary of Analysis Terminology

Linear, Static

This model is used when the response of the body is linear, and there’s no variation with time. In stress analysis, this model is appropriate when operating within the elastic region (i.e. the stress-strain curve is linear) and when the deformations are small16 and when the loads do not vary with time.

This model is used widely since it’s quick to solve and relatively easy to interpret the results. Very often, even if a non-linear model is more realistic, a linear model is used to investigate likely behavior. Once the options have been narrowed, a full non-linear analysis can be used.

The equilibrium equation is

[ ]

K

{ } { }

u

=

f

where K, u and f are functions of x, y and z only – they are independent of t.

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27

Linear, Transient

In stress analysis, this model is appropriate when operating within the elastic region (i.e. the stress-strain curve is linear) and when the

deformations are small but when the external conditions do vary with time. Transient problems themselves are often subdivided into further classes, depending on whether the load varies with time in a periodic fashion or not. If the load is periodic, as for instance if the excitation source is a rotating unbalance, it is called “Harmonic” excitation. This is usually easier to solve than when the load is non-periodic.

It’s important to note that Finite Element mathematics is applicable only to spatial discretization. A Finite Difference method is usually used to step the solution forward in time, from the initial time to the final time. To setup the problem for analysis, then, the values at the boundary are specified at the initial time (often referred to as “t = 0”). Time-variant solutions can also be calculated by representing the solution17 as a weighted sum of the “mode shapes”.

In the equilibrium equation

[ ]

K

{ } { }

u

f

t

u

C

t

u

M

+

=













∂

+













∂

2 2

K is a function of x, y and z only – it is independent of t. F and u, however, vary with t. M represents the mass, and C the damping.

Normal Modes

Sometimes our design problem is not just to calculate stresses or

deformations. We may be interested in identifying the resonance frequencies of the system. In vehicle design, avoidance of resonance enhances ride comfort by cutting out unwanted rattles. When designing a loudspeaker or a megaphone, on the other hand, you may want resonance to occur.

In cases like these, we need to solve the “eigenvalue18” problem and evaluate the natural frequencies of the body.

17_{Look up the use of Rayleigh’s method or Dunkerley’s method. Also look up the}

Rayleigh-Ritz method for a rough idea as to how this works.

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FEA Essentials CAE and Design Optimization – Basics The equilibrium equation is

[ ]

₂

[ ]

{ } { }

0

2

=

+













∂

u

K

t

u

M

where K and u are functions of x, y and z only – they are independent of t. The solutions to this equation are pairs of natural frequencies and the corresponding “mode shapes”.

Random Response

In some situations, we cannot specify the exact value of the loads as a function of time, but can specify the total energy in these loads. An example would be the forces experienced by a plane when its engines are firing. We know the total energy being transferred from the jet engines to the frame, but cannot claim that we know the loads precisely as functions of time. In cases like these, the loads are characterized by Probability Density Functions, and the behavior is called stochastic. The designer’s goal then is to predict a probability of safety.

The several ways to evaluate these responses is beyond the scope of this book.

Inertia Relief

Setting up a Finite Element model for static analysis requires that the structure be supported adequately. Some structures, like aircraft in steady flight, are not supported explicitly but are still best represented by static-analysis models. Inertia Relief is an approach used to model such problems.

Frequency Response

In many designs where vibration is important, and correlation with test-results is essential, designers have to characterize the response of the structure as a function of frequency-of-excitation instead of as a function of time. In these cases a Fourier Transform converts the equilibrium equation from the “time domain” to the “frequency domain”.

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29

In the equilibrium equation, the variables are expressed as functions of ωωωω rather than time. This is called the frequency domain. Of course, the Inverse Fourier Transform can convert the solution back to the time domain.

Linear Buckling

Designers sometimes have to take into account the fact that even if stresses are less than permissible values, the structure may fail if it buckles – like a tall column in compression. The equilibrium equation is similar to that of Normal Modes, but the results are interpreted as a “buckling load factor”. Buckling load factors are often important in the design of aerospace structures, where the quest for a minimal weight and the use of advanced materials leads to the frequent use of thin-walled designs.

Non-linear – Gap / Contact

The terms “gap” and “contact” are often used to mean the same thing – an opening in the body that may close or widen under the influence of external factors.

Clearly, if a gap closes or opens, the stiffness of the body changes. Since the gap opens or closes depending on the deformation, this means the stiffness depends on the deformation. In other words, in the equilibrium equation

[ ]

K

{ } { }

u

=

f

K is a function of u, making the equation non-linear. The force, f, too can be a function of u.

Component Mode Synthesis

When working with large models, resource constraints sometimes force the analyst to break the problem into smaller parts. In static analysis, this approach is called sub-structuring. When used in dynamic analyses, it is called Component Mode Synthesis.

It’s sometimes impossible to treat the product purely as a structure or purely as a mechanism. Consider, for example, the feed mechanism for a high-speed packing machine. The rates of acceleration that the mechanism

experiences may be quite high. High enough that the deflection of the levers is large enough, perhaps, for the feed mechanism to jam because of

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FEA Essentials CAE and Design Optimization – Basics Designing such a product requires that the equations of

rigid-body-mechanics be coupled with the equations of structural deformation. Component Mode Synthesis also provides a way to do this.

What are Elements?

An element is a shape for which the Finite Element program can write out the equations relating the unknown and known quantities. An element is defined by its nodes – the unknowns at each node are called the degrees of freedom.

Shapes that are accepted in most finite element programmes are triangles, quadrilaterals, lines, tetrahedra, pentahedra and hexahedra.

The sizes of and the number of elements usually have a bearing on the accuracy of the solution. As problems become more complex (advancing in complexity from linear-statics to nonlinear-dynamics), the requirements on shapes and sizes of elements become increasingly stringent. These

requirements are often referred to as mesh-specifications, and these are usually strongly analysis-program dependent.

In most analyses, the more the number of elements, the better the results. However, the computer time and disk-space required to solve the equations also goes up. Most analysts have to settle for a quality of results that they can afford, given the available computer resources.

Fortunately for us, optimization methods are less dependent on mesh

quality. As we saw in the previous chapter, the optimization algorithm makes some simplifications and assumptions, so it’s not important that a mesh be “perfect”. Only that it be “adequate”. In your assignments, you will learn how to judge whether your mesh is sufficiently fine.

Element Types

Choosing the element type is an important part of any Finite Element analysis. Elements are categorized based on their shape or topology, the number of nodes needed to define them, and the mechanics or behavior they represent.

Element types are usually solver dependent – they vary based on the solver used. The elements listed below are specific to OptiStruct, but are available in almost every commercially available analysis package.

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Categorization based on Mechanics

Beams and Bars (or rods or trusses) are represented by one-dimensional elements – lines or curves – but can lie in 3D space. Plain Strain, Plane Stress and Axi-symmetric elements are two-dimensional shapes that can be used only if the entire model lies in one plane only. Plates and Shells

represent surfaces that are two-dimensional in the sense that they have no volume, but lie in 3D space. Solid Elements represent volumes.

Categorization based on Topology

Standard 2D Elements (plane strain, plane stress, axi-symmetric, plate and shell) are either triangular or quadrilateral.

Standard 3D elements are either tetrahedral, pentahedral, or hexahedral. A pyramid with a rectangular base is a pentahedron, as is a wedge. However the two are different element types: the pyramid has 5 nodes while the wedge has 6. Not all solvers support pentahedral elements, and some support only one of the two pentahedra.

In most stress-analysis problems, quadrilateral and hexahedral elements are preferred over triangular and tetrahedral elements. For reasons that you can find in the references listed at the end of this volume, they give much better results: more accurate and less CPU intensive.

1D elements are all (topologically) curves – either straight lines or arcs, depending on the number of nodes. Typical applications are as beams, bars, rods, pipes, springs, cold- or hot-runners, and axi-symmetric shells.

Categorization based on Order

The variation of the unknown quantity between nodes is assumed (by the analysis code) to be linear, or quadratic, or cubic, etc. Linear and parabolic elements are the most common. Linear elements have two nodes along each edge, while parabolic elements have three nodes along each edge.

Further refinements do exist – for instance, parabolic quadrilateral elements can have either 8 or 9 nodes.

Steps in FE Modeling

Geometry Preparation

While it is possible to build a model directly using elements and nodes, this is not often done today. The geometry that defines the area to be analyzed

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FEA Essentials CAE and Design Optimization – Basics (also called the “domain”) is usually created first using a CAD program, and

elements are created to encompass that boundary or represent the volume. CAD designers create models for manufacture. As many details are included as possible. For a numerical analysis, we often choose to ignore aspects that we think will not significantly affect the solution. For instance, a single hole of 1 mm radius in a plate that is 2 meters wide can probably be ignored safely when calculating the deformation of the plate.

Therefore the first task that most analysts are faced with is that of preparing the geometry for analysis. This involves tasks like removal of features, extraction of mid-surfaces, extrapolation of surfaces, etc.

Further, the CAD world has an abundance of data exchange formats, since most CAD applications use proprietary data storage formats. A transfer of data from the CAD package to the FE preprocessor sometimes results in a loss of accuracy – gaps are introduced during the import process, for example. Also, CAD assembly models are sometimes made up of parts that were created in different CAD applications.

Therefore a cleaning-up of the geometry is often required. This involves filling gaps, eliminating small edges or surfaces that will mislead the

automatic-mesh-generation routines, eliminating dangling faces, and so on.

Mesh Creation

Once the geometry is more or less ready for discretization, you then start to subdivide the geometry into elements or grid points. The collection of elements is usually referred to as a mesh. Meshes that consist of triangular or quadrilateral elements can often be generated automatically, while tetrahedral or hexahedral meshes usually require considerable manual intervention.

Mesh Editing

Once a mesh has been created, the analyst checks if it meets the

specifications – several measures of quality are checked, depending on the analysis requirements. Usually, some editing of the mesh is required. Depending on the complexity of the mesh, this can be done either semi-automatically or manually.

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33

Preparing for Analysis

Once the mesh is ready, additional data is specified – the properties of the materials used, the thickness or cross-sectional properties of shell or beam elements, the conditions on the boundaries (restraints, loads or excitations), initial conditions, data for the specific solution algorithm to be employed, kind of output required for text and graphics records, and so on.

Once this is done, the data is turned over to the solution program for the next phase – solving. Data is often written out in the form of a text file, which is referred to as a deck. Each line of text in the deck is commonly referred to as a card. A card image is the format followed by the analysis program to interpret the text on the line.

The procedure of building the Finite Element Model is sometimes referred to as FEM – short for Finite Element Modeling. Some books, however, use FEM to refer to the Finite Element Method.

Solving

The model created in the earlier steps is now taken up for solution – the computer program reads the data, calculates matrix entries, solves the matrix equations and writes data out for interpretation.

This task is CPU-intensive, and is often called processing19. Most of the time, very little interaction from the user is required. In some cases, the analyst periodically monitors results to check that they are indeed on the right track. If the solution seems to be evolving in an unexpected direction, the analyst can stop the solver and modify the model, thereby saving valuable time.

Post-Processing

After the program has evaluated the results, the analyst examines and interprets the data – looking for errors or improvements in design.

As with pre-processing, this calls for substantial interaction from the analyst.

19_{Hence the term pre-processing for the preceding steps, and post-processing for}

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FEA Essentials CAE and Design Optimization – Basics

Guidelines on Element Choice

Learning which element to choose is a little like learning driving. Guidelines exist, but can’t be applied blindly. You need to adapt them to specific situations. Remember this warning!

If your product has a region that is long and thin, you can probably model it using beam elements. If this region is connected to the rest of the structure by pin-joints, then you should use truss elements. Regions that are like plates are best modeled using shell elements. Any areas that don’t fall in the earlier categories should be modeled using solid elements.

If you have different element types in your model, there are rules that govern the assemblage. For several models, we choose to use just one element type to avoid these complications.

Engineering is the art of modeling materials we do not wholly understand, into shapes we cannot precisely analyse so as to withstand forces we cannot properly assess, in such a way that the public has no reason to suspect the extent of our ignorance.

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CAE and Design Optimization – Basics OptiStruct

35

OptiStruct

Before We Start

The previous chapters outlined the reasons we want optimization to be a part of the product design cycle, and introduced both the Optimization Model and the Analysis Model.

The procedure, to summarize, is as follows:

1. As the designer, you decide the design variables, the constraints and the objectives. You also choose the design space, the loads and the restraints – usually dictated by other components in the assembly. It’s a good idea to list these as a design-specification document.

2. Next, you prepare the FE model. To do this, you

i. inherit the product definition as a CAD model. If necessary, you must modify it to omit unnecessary details. Note that this step is optional. It makes sense if you are trying to improve an existing design, or if it is easier to build the design-space in a CAD modeler. You may choose to define the design space within the FE pre-processor itself if you’re working on a new concept with a geometrically simple design space.

ii. mesh the product-geometry or the design space, depending on which you are starting with. The design space can, but need not, span the entire product. For instance your design may not allow you to change mounting points. In this case, the restraint-areas will not be a part of the design space, although they will be a part of the analysis model.

iii. specify material data for the elements – Modulus of Elasticity, etc.

iv. specify element properties – the thickness of shell elements or the cross-section for beam elements

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OptiStruct CAE and Design Optimization – Basics v. define the forces acting on the body

vi. specify the restraints on the body – where and how it’s supported

vii. choose the type of analysis you want to perform – linear-static, modal, etc.

3. Before running the optimizer, you should check that the Analysis Model is adequate. A good way to do this is to run the analysis for meshes of different element-sizes. If the reported results

(deformation, stress, frequency, etc., depending on your interest) do not vary with the mesh, it’s reasonable to conclude that it’s adequate.

4. Once the FE model is ready, you prepare the Optimization Model. This means you specify

i. the design variables. Remember that different parts of the design space can have different variables. You may have the freedom to place cutouts in one region, but only to vary the thickness in others.

ii. the responses that the optimization model needs from the analysis model. The optimizer will use these to evaluate sensitivities.

iii. the design constraints. iv. the objective function.

5. Now you are ready to perform the optimization.

6. After the optimization is done, you review the results to check that the optimization has proceeded in line with your design

requirement. You may have to revise or restate the optimization model to better reflect the statement of the design requirements. 7. When you are satisfied with the design configuration proposed by

the optimizer, you take this geometry back to your CAD modeler for further CAD-related work such as drawing generation, etc.

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CAE and Design Optimization – Basics OptiStruct

37

In the subsequent sections we’ll review the specific methods OptiStruct uses20.

Terminology

OptiStruct includes both an FE solver21 and an Optimizer. In other words, it can be used to solve the Analysis Model and the Optimization Model. The models themselves are created using HyperMesh, which is the pre-processor. HyperMesh is used to define both the Analysis Model and the Optimization Model. The table below lists the key terms used by HyperMesh and correlates them with the Analysis and Optimization Models.

Analysis Model

Collector A way to group related items together. For instance all elements that have the same thickness would be in the same collector.

Load External forces acting on the boundary. Includes concentrated forces, moments, pressures, gravity, etc.

SPC Short for Single Point Constraint. Refers to restraints applied to the analysis model at locations where the body is supported22.

Subcase Combination of SPCs and Loads. Since they

represent values on the boundary, these are often clubbed together as Boundary Conditions. A subcase is sometimes called a Load Case.

Card Some data in the analysis model, such as the

material properties, cannot be displayed graphically. Such data is entered as a card image by typing in text or numerical values.

Optimization Model

Response Any quantity calculated by the Analysis Model, and of interest to the Optimization Model. This could include

20_{Some of these features are unique to OptiStruct.} 21_See_{A Designer’s Guide To CAE}

22_{Do not confuse these with design constraints, which are applicable to the}