introduction to engineering design optimization

Dr. Tafesse GebresenbetAAiT, Mechanical Engineering Department

1Dr. Tafesse Gebresenbet, AAiT,

Course Objective

The course introduces :

Understanding of principles and possibilities of optimization in Engineering and in particular in designUnderstand how to formulate an optimum design problem Understand how to formulate an optimum design problem by identifying critical elementsknowledge of optimization algorithms, ability to choose proper algorithm for given problemPractical experience with optimization algorithmsP ti l i i li ti f ti i ti t Practical experience in application of optimization to design problems


Course outlineChapter 1: Introduction to Engineering Optimization of Design

Introduction: Historical background, Definition of terms, Basic concepts, Classification of optimizations problems , Applications : Design optimization, benefits of optimization, automated d i i i i h i i i l design optimization, when to use optimization, examples

Chapter 2: Optimum Design FormulationDesign models, Mathematical models, Defining optimization problem, Multi objective design problems applications of optimization in designdesign problems, applications of optimization in design

Chapter 3 Classical Optimization techniquesSingle variable optimizationMultivariable optimization with equality and inequality constraintsMultivariable optimization with equality and inequality constraints

Chapter 4: One dimensional unconstrained optimization techniquesElimination methods: Exhaustive search Interval halving method Elimination methods: Exhaustive search, Interval halving method, Fibonacci Method, Golden Section method.Interpolation methods: quadratic interpolation, cubic interpolationDirect root methods: Newton's method, Quasi ‐Newton method, Secant Direct root methods: Newton s method, Quasi Newton method, Secant method


Course outlineChapter 5: Unconstrained Optimization techniques p 5 p q

Direct search methods: Random search , Grid search Method, Powell method Indirect search(Descent) methods: Steepest descent (Cauchy) method, Conjugate gradient (Fletcher‐Reeves) method, Newton’s method, U t i d ti i ti i M tl bUnconstrained optimization using Matlab

Chapter 6: Constrained Optimization techniques Direct search methods: Random search complex search Method Quadratic Direct search methods: Random search, complex search Method, Quadratic programming Indirect methods: Penalty function method, Lagrange multiplier methodConstrained optimization using Matlab

Chapter 7: Dynamic Programming Introduction , Multistage decision processes, Applications of dynamic programming ., g p , pp y p g g

Chapter 8: Genetic Algorithm based Optimization Introduction to Genetic Algorithm , Applications of GA based optimization techniques , g pp p qGA based Optimization using Matlab


Reference Materials1 S S Rao Engineering Optimization 3rd edition Wiley Eastern 20091. S.S. Rao, Engineering Optimization, 3rd edition, Wiley Eastern, 20092. Papalambros and Wilde, Principle of optimal Design, modeling and

computation, Cambridge University press, 2000 3. Kalyanmoy Deb, Engineering Design for optimization, PHI, 20054. Fred van Keulen and Matthiis Langelaar, Lecture note s in Engineering

Optimization, Technical University of Delft5. Ravindran, Ragsdell and Rekalaitis, Engineering Optimization Methods and

application, 2nd edition, Willey,2006pp y6. Arora, Introduction to Optimum design, 2nd edition, Elsevier Academic Press,

20047. Forst and Hoffmann, Optimization theory and practice, Springer , 20108 Haftka and Gurdal Elements of Structural Optimization 3rd edition Kluwer8. Haftka and Gurdal, Elements of Structural Optimization, 3rd edition, Kluwer

academic, 19919. Belegundu and Chandrupatla, Optimization concepts and applications in

Engineering, 2nd edition, Cambridge University press, 2011K l D b M l i bj i O i i i i E l i 10. Kalyanmoy Deb, Multi‐objective Optimization using Evolutionary Algorithms, Wiley, 2002

11. Bendose, Sigmund, Topology optimization theory and methods and applications, Springer, 2003


Prerequisites Mathematical and Computer background needed to understand the course:F ili it ith li l b ( t d t i Familiarity with linear algebra (vector and matrix operations) andbasic calculus is essential and Calculus of functions of single and multiple variables must also be understoodFamiliarity with Matlab and EXCEL is also essential


Lecture outline IntroductionHistorical perspectiveWhat can be achieved by optimization?What can be achieved by optimization?Optimization of the design processBasic terminology, notations, and definitionsgy, ,Engineering optimization Popularity and pitfalls of optimizationCl ifi i f i i i bl Classification of optimization problems Design optimization Benefits of design optimization Benefits of design optimization Automated design optimizationExamples


IntroductionOptimization is derived from the Latin word “optimus”, the best.Thus optimization focuses on Thus optimization focuses on

● “Making things better”

“G i fi ”● “Generating more profit”

● “Determining the best”

● “Do more with less ”

The determination of values for design variables which The determination of values for design variables which minimize (maximize) the objective, while satisfying all constraints


Introduction

Optimization is defined as a mathematical process of obtaining the set of conditions to produce the maximum or the minimum value of a functionmaximum or the minimum value of a function

It is ideal to obtain the perfect solution to a design situation situation.

Usually all of us must always work within the constraints of the time and funds available we can only hope for the of the time and funds available, we can only hope for the best solution possible.

Optimization is simply a technique that aids in Optimization is simply a technique that aids in decision making but does not replace sound judgment and technical know‐how


Historical perspectiveAncient Greek philosophers: geometrical optimization Ancient Greek philosophers: geometrical optimization problems

Zenodorus, 200 B.C.:“A sphere encloses the greatestvolume for a given surface area

Newton Leibniz Bernoulli De l’Hospital (1697): Newton, Leibniz, Bernoulli, De l Hospital (1697): “Brachistochrone Problem”:


Historical perspectivePeople have been “optimizing” forever, but the roots for

d d ti i ti b t d t th S d modern day optimization can be traced to the Second World War. Ancient Greek philosophers: geometrical optimization G p p g pproblems

Zenodorus, 200 B.C.:“A sphere encloses the greatestA sphere encloses the greatestvolume for a given surface area”

Newton, Leibniz, Bernoulli, De l’Hospital (1697): “Brachistochrone Problem”:Brachistochrone Problem :Lagrange (1750): constrained minimizationCauchy (1847): steepest descenty ( 47) pDantzig (1947): Simplex method (LP)Kuhn, Tucker (1951): optimality conditionsKarmakar (1984): interior point method (LP)Bendsoe, Kikuchi (1988): topology optimization 11Dr. Tafesse Gebresenbet, AAiT,

Historical perspectiveOne of the first problems posed in the calculus of variations.Galileo considered the problem in 1638 but his answer wasGalileo considered the problem in 1638, but his answer wasincorrect.Johann Bernoulli posed the problem in 1696 to a group ofelite mathematicians:

I, Johann Bernoulli... hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praiseworthy of praise.

Newton solved the problem the very next day, but proclaimed I do not love to be dunned [pestered] and t d b f i b t th ti l thi "teased by foreigners about mathematical things.


What can be achieved by optimization ?

Optimization techniques can be used for:Getting a design/system to workGetting a design/system to workReaching the optimal performanceMaking a design/system reliable and robustg g y

Also provide insight inDesign problemUnderlying physicsModel weaknesses


What can be achieved by optimization ?What can be achieved by optimization ?Engineering design is to create artifacts to perform desired functions under given constraintsdesired functions under given constraintsCommon goals for engineering designFunctionalityy

Better performance: More efficient or effective ways to execute tasksMultiple functions Capabilities to e ecute t o or more Multiple functions: Capabilities to execute two or more tasks simultaneously

ValueHigher perceived value: More features with less priceLower total cost: Same or better ownership and sustainability with lower costsustainability with lower cost


Basic Terminology, notations and definitionsRn n‐dimensional Euclidean (real) space x column vector of variables, a point in Rn

x=[x1,x2,…..,xn]T

f(x), f objective function x* local optimizerf(x*) optimum function value f(x ) optimum function value gj(x), gj jth equality constraint function g(x) vector of inequality constrainth ( ) h jth lit t i t f tihj(x), hj jth equality constraint functionh(h(x) vector of equality constraint function C1 set of continuous differentiable functionsC2 set of continuous and twice differentiable differentiable

continuous functions


Norm/Length of a vectorIf we let x and y be two n‐dimensional vectors, then their dot yproduct is defined as

Thus, the dot product is a sum of the product of corresponding elements of the vectors x and y. Two vectors are said to be orthogonal (normal) if their dot

d t i i d th l if product is zero, i.e., x and y are orthogonal if x ∙ y =0.If the vectors are not orthogonal, the angle between them can be calculated from the definition of the dot product:

here θ is the angle bet een ectors x and y and ||x|| where θ is the angle between vectors x and y, and ||x|| represents the length of the vector x. This is also called the norm of the vector


Norm/Length of a vectorThe length of a vector x is defined as the square root of the sum of squares of the components, i.e.,

The double sum of Eq. (1.11) can be written in the matrix form as f llfollows

Since Ax represents a vector, the triple product of the above Eq. will be also written as a dot product:


Basic Terminology and notations Design variables

Parameters whose numerical values are to be determined h h dto achieve the optimum design.

They include such values such as; size or weight, or the number of teeth in a gear coils in a spring or tubes in a number of teeth in a gear, coils in a spring, or tubes in a heat exchanger, or etc.

Design parameters represent any number of variables the Design parameters represent any number of variables the may be required to quantify or completely describe an engineering system.

The number of variables depends upon the type of design involved. As this number increases, so does the complexity of the solution to the design problems.g p


Basic Terminology and notations ConstraintsNumerical values of identified conditions that must be ti fi d t hi f ibl l ti t i blsatisfied to achieve a feasible solution to a given problem.

External constraintsUncontrolled restrictions or specifications imposed on a Uncontrolled restrictions or specifications imposed on a system by an outside agency.Ex.: Laws and regulations set by governmental agencies, ll bl t i l f h t tiallowable materials for house construction

Internal constraintsRestrictions imposed by the designer with a keen p y gunderstanding of the physical system.Ex.: Fundamental laws of conservation of mass, momentum, and energyand energy


What is mathematical/Engineering Optimization ? Mathematical optimization is the process of 1. The formulation and 2. The solution of a constrained optimization problem of the

general mathematical form general mathematical form Minimize f(x), x =[x1,x2,…,xn]T Єsubject to constraints

gj(x) ≤ 0, j=1,2, … , mhj(x) = 0, j=1, 2, …. ,r

Where f(x) g (x) and h (x) are scalar functions of the real column Where f(x), gj(x) and hj(x) are scalar functions of the real column vectorThe continuous components of xi of x =[x1,x2,…, xn]T are called h (d i ) i bl the (design) variables f(x) is the objective function, gj(x) denotes the respective inequality constraints, and gj( ) p q y ,hj(x) the equality constraint function


What is mathematical/Engineering Optimization ? The optimum vector x that solves the formerly defined problem is denoted by x* with the corresponding optimum function value f(x*). function value f(x ).

If no constraints are specified, the problem is called an unconstrained minimization problemu co s a ed a o p ob e

Other names of Mathematical Optimization

M th ti l i Mathematical programming Numerical optimization


Objective and Constraint functions ( ) ( ) ( )The values of the functions f(x), gj(x), hj(x) at any point x

= [x1,x2,…, xn]T gj(x), may in practise be obtained in different ways

i. From analytically known formulae, e.g., f(x)= x12 + 2x22+Sin x3

ii As the outcome of some complicated computational ii. As the outcome of some complicated computational process e.g., g1(x) = a(x) –amax, where a(x) is the stress, computed by means of a finite element analysis, at some

i t i t t th d i f hi h i ifi d b point in structure, the design of which is specified by x; or

iii. From measurement taken of a physical process, e.g., p y p , g ,h1(x)= T(x)‐To, where T(x) is the temperature measured at some specified point in a reactor, and x is the vector of operational settings.p g


Elements of optimizationElements of optimization•Design space–The total region or domain defined by the design The total region or domain defined by the design variables in the objective functions–Usually limited by constraints•The use of constraints is especially important in restricting the region where optimal values of the d i i bl b h ddesign variables can be searched.

Unbounded design spaceNot limited by constraintsNot limited by constraintsNo acceptable solutions


Optimization in the design process

Conventional design process:Optimization‐based design process:

Collect data to describe the system

Collect data to describe the system

Estimate initial design

Identify:1. Design variables2. Objective function3. Constraints


Analyze the system


Analyze the system

Ch k h iCheck performance

criteria

Is design satisfactory? Done

Check the constraints

Does the design satisfy convergence criteria? DoneIs design satisfactory?

Change design based on experience / heuristics /

ild

Doneg

Change the design using an optimization methodwild guesses


Optimization in the design processIs there one aircraft which is the fastest most efficient Is there one aircraft which is the fastest, most efficient, quietest, most inexpensive?

“You can You can only make one thing best at a time.”


Optimization Methods


Comparison of Conventional and Optimal DesignThe CD process involves the use The OD process forces the designer pof information gathered from one or more trial designs together with the designer’s experience an

The OD process forces the designer to identify explicitly a set of design variables, an objective function to be optimized, and the constraint

intuitionIts advantage is that the designer’s experience and intuition can be

be optimized, and the constraint functions for the system. This rigorous formulation of the design problem helps the designer

used in making conceptual changes in the system or to make additional specifications in the

d

des g p ob e e ps t e des g egain a better understanding of the problem. Proper mathematical formulation

procedureThe CD process can lead to uneconomical designs and can i l l f l d i

pof the design problem is a key to good solutions.

involve a lot of calendar time.


Optimization popularityp p p yIncreasingly popular:Increasing availability of numerical modeling Increasing availability of numerical modeling techniques

Increasing availability of cheap computer powerIncreasing availability of cheap computer power

Increased competition, global markets

Better and more powerful optimization techniquesBetter and more powerful optimization techniques

Increasingly expensive production processes (trial and error approach too expensive)(trial‐and‐error approach too expensive)

More engineers having optimization knowledge


Optimization pitfalls!

Proper problem formulation critical!Choosing the right algorithmg g gfor a given problemMany algorithms contain lots of control parameters Optimization tends to exploit weaknesses in modelsOptimization can result in very sensitive designsSome problems are simply too hard / large / expensive


Structural optimizationStructural optimizationStructural optimization = optimization techniques

li d t t tapplied to structuresDifferent categories:

Sizing optimization tLSizing optimizationMaterial optimizationShape optimization

E, ν R

hShape optimizationTopology optimization r


Structural optimizationInegrated optimal design Inegrated optimal design of a vehicle roadarm.

a) Initial Finite ElementM d l Model, b) topology optimized road arm, ) d l dc) reconstructed solid model, d) Finite Element mesh f h d i for shape design e) Von Mises stress of the shape optimized design

d and f) comparison of the 3D Roadarm before and after shape design shape design

Dr. Tafesse Gebresenbet, AAiT, 31

Sizing optimizationg pIn a typical sizing problem the goal may be to find the optimal thickness distribution of a linearly elastic plate or the optimal member areas in a truss structure plate or the optimal member areas in a truss structure.

The optimal thickness distribution minimizes (or maximizes) a physical quantity such as the mean maximizes) a physical quantity such as the mean compliance (external work), peak stress, deflection, etc. while equilibrium and other constraints on the state

d d i i bl i fi d and design variables are satisfied.

The design variable is the thickness of the plate and the t t i bl b it d fl tistate variable may be its deflection.

Dr. Tafesse Gebresenbet, AAiT, 32

Shape optimization p pShape optimization is part of the field of optimal control theory. y

The typical problem is to find the shape which is optimal in that it minimizes a certain cost pfunctional while satisfying given constraints.

In many cases, the functional being solved In many cases, the functional being solved depends on the solution of a given partial differential equation defined on the variable ddomain.


Shape optimization S ape opt at o

Yamaha R1


Topology optimizationp gy pTopology optimization is, in addition, concerned with the number of connected components/boundaries belonging to the domain. Such us determination of features such as the number and location and shape of holes and the connectivity of the domain.o es a d t e co ect v ty o t e do a .

Such methods are needed since typically shape optimization methods work in a subset of allowable pshapes which have fixed topological properties, such as having a fixed number of holes in them.

Topological optimization techniques can then help work around the limitations of pure shape optimization.


Topology optimizationTopology optimizationTopology optimizationis a mathematical

Using topology optimization, engineers can find the best

d i h h approach that optimizes material layout within a given design space, for a

concept design that meets the design requirements

g ve des g space, o agiven set of loads and boundary conditions such that the resulting such that the resulting layout meets a prescribed set of performance targets.


Topology optimization examples


Why Design Optimization ?

Design Complexity


Classifications P blProblems:

Constrained vs. unconstrainedSingle level vs multilevelSingle level vs. multilevelSingle objective vs. multi‐objectiveDeterministic vs stochasticDeterministic vs. stochastic

Responses:Linear vs. nonlinearConvex vs. nonconvexSmooth vs. nonsmooth

Variables:Continuous vs. discrete (integer, ordered, non‐d dordered)


Typical Design Process

Initial Design ConceptHEEDS

Specific Design Candidate

B ild A l i M d l( )M dif Build Analysis Model(s)

Execute the Analyses

ModifyDesign

(Intuition)

$

Design Requirements Met?No

)Time

Money

Final Design

YesIntellectual Capital

HEEDS (Hierarchical Evolutionary Engineering Design System)


A General Optimization Solution

Automotive Civil Infrastructure

Biomedical Aerospace41Dr. Tafesse Gebresenbet, AAiT,

Automated Design OptimizationBasic Procedure:

Plan Design Study

Create Parameterized Baseline Model

Create HEEDS Design Model

Execute HEEDS Optimization


Automated Design Optimization

Id ifIdentify: Objective(s)ConstraintsDesign Variables

Plan Design Study

gAnalysis Methods

Note: Th d fi iti ff t b t


These definitions affect subsequent stepsCreate HEEDS Design Model




C t CAD/CAE M d l Create CAD/CAE Models for a Representative DesignPlan Design Study

Input File(s)

Execute Solver(s)


Execute Solver(s)

Output File(s)


p ( )

Validate Model



Automated Design OptimizationDefine Batch Execution Commands for Solvers

Define Input Files and Output Files

Plan Design Study

Define Design Variables and Responses


Tag Variables in Input Files and

Responses in Output Files


Define Objectives,

Responses in Output FilesExecute HEEDS Optimization

45

Constraints, and Search Method

Dr. Tafesse Gebresenbet, AAiT,


Plan Design Study Modify Variables in Input File

Execute Solver in Batch ModeCreate Parameterized Baseline

ModelExtract Results from Output File

NewDesign(HEEDS)


NoConverged?


Optimized Design(s)

Yes


CAE Portals

“When”

“What”

“Where”


Tangible Benefits*

Crash rails: 100% increase in energy absorbed20% reduction in mass

Composite wing: 80% increase in buckling load15% increase in stiffness

Bumper: 20% reduction in masswith equivalent performanceq p

Coronary stent: 50% reduction in strain

* Percentages relative to best designs found by experienced engineers


Return on Investment

• Reduced Design Costs• Time, labor, prototypes, tooling• Reinvest savings in future innovation projects

• Reduced Warranty Costsy• Higher quality designs• Greater customer satisfaction

• Increased Competitive Advantage• Innovative designs

k• Faster to market• Savings on material, manufacturing, mass, etc.


Topology Optimization

• Suggests material placement or layout based on load path efficiency

• Maximizes stiffness

• Conceptual design toolp g

• Uses Abaqus Standard FEA solver


When to Use Topology Optimization

Early in the design cycle to find shape conceptsTo suggest regions for mass reduction TopologyTo suggest regions for mass reduction Topology optimization


Design of Experiments

• Determine how variables affect the response of a particular design

Design sensitivities

• Build models relating the B

• Build models relating the response to the variables

S d l Surrogate models, response surface models

A


When to Use Design of Experiments

• Following optimization

• To identify parameters that cause greatest variation in your design


Parameter OptimizationMinimize (or maximize): F(x1,x2,…,xn)

h th t G ( ) isuch that: Gi(x1,x2,…,xn) < 0, i=1,2,…,pHj(x1,x2,…,xn) = 0, j=1,2,…,q

where: (x x x ) are the n design variableswhere: (x1,x2,…,xn) are the n design variablesF(x1,x2,…,xn) is the objective (performance)

functionGi(x1,x2,…,xn) are the p inequality constraintsHj(x1,x2,…,xn) are the q equality constraints


Parameter OptimizationObjective:Search the performance design landscape to find h h h k l ll h h f blthe highest peak or lowest valley within the feasible range

• Typically don’t know the nature of surface before search begins

• Search algorithm choice depends on type of d l ddesign landscape

• Local searches may yield only incremental improvementN b f t b l• Number of parameters may be large


Selecting an Optimization Method

Gradient‐Based

Simplex

Simulated lAnnealing

Response Surface

Design Space depends on:

• Number, type and range

Genetic Algorithm

Evolutionary Sof variables and

responses

Obj ti d

Strategy

Etc.• Objectives and constraints


Design Optimization Procedure Using ANSYSThe optimization module (OPT) is an integral part of the ANSYS p g pprogram that can be employed to determine the optimum design.

While working towards an optimum design the ANSYS While working towards an optimum design, the ANSYS optimization routines employ three types of variables that characterize the design process:

design variables,

state variables, and

h bj i f i the objective function.

These variables are represented by scalar parameters in ANSYS Parametric Design Language (APDL). The use of APDL is an g g g ( )essential step in the optimization process.

The independent variables in an optimization analysis are the d i i bl design variables.


Design Optimization Procedure Using ANSYSO i ANSYS d i filOrganize ANSYS procedure into two files:Optimization file—describes optimization variables, and trigger the optimization runs.and trigger the optimization runs.Analysis file—constructs, analyses, and post‐processes the model.Typical Commands in an Optimization File

0102

/CLEAR ! Clear model database

! Initialize design variables020304050607

... ! Initialize design variables/INPUT, ... ! Execute analysis file once

/OPT ! Enter optimization phaseOPCLEAR ! Clear optimization databaseOPVAR, ... ! Declare design variablesOPVAR, ... ! Declare state variables07

0809101112

OPVAR, ... ! Declare state variablesOPVAR, ... ! Declare objective functionOPTYPE, ... ! Select optimization methodOPANL, ... ! Specify analysis file nameOPEXE ! Execute optimization runOPLIST, ... ! Summarize the results

58/33

13,

... ! Further examining results

Dr. Tafesse Gebresenbet, AAiT,

Design Optimization Procedure Using ANSYS

01 /PREP7

Typical Commands in an Analysis File

01020304

/PREP7

... ! Build the model using the! parameterized design variables

FINISH/05

060708

/SOLUTION

... ! Apply loads and solveFINISH

/POST1 ! or /POST26

09101112

*GET, ... ! Retrieve values for state variables*GET, ... ! Retrieve value for objective

function

... FINISH

59/33Dr. Tafesse Gebresenbet, AAiT,

Design Optimization Procedure Using ANSYS

ANSYS Optimization AlgorithmsTwo built‐in algorithms in ANSYS:Fi t d th dFirst order methodSub problem approximation method (Zero order method)

Other Optimization Tools Provided by ANSYSSingle Iteration Design ToolRandom Design ToolGradient ToolS T lSweep ToolFactorial Tool

60/33Dr. Tafesse Gebresenbet, AAiT,

Summary

Design variables: variables with which the design problem is parameterized:Obj i i h i b i i i d

( )1 2, , , nx x x=x KObjective: quantity that is to be minimized (maximized)Usually denoted by: ( )f xUsually denoted by:( “cost function”)Constraint: condition that has to be satisfied

Inequality constraint:Equality constraint:

( ) 0g ≤x( ) 0h =x( ) 0h =x


SSummaryGeneral form of optimization problem:

xg

xx

≤

f

0)(

)(

:tosubject

min

xhxg

ℜ

=≤

nX0)(0)(:tosubject

( )xxx

x

≤≤

ℜ⊆∈ nX

( )62Dr. Tafesse Gebresenbet, AAiT,

SummaryOptimization problems are typically solved using Optimization problems are typically solved using an iterative algorithm:

ModelConstants Responses

D i i f

hgf ,,

Designvariables

Derivatives ofresponses(design sensi‐tivities)x

Optimizer

)

hgf ∂∂∂ ,,

x

iii xxx ∂∂∂,,


introduction to engineering design optimization

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