Course Objectives Understanding of principles and possibilities of

Course Objectives • Understanding of principles and possibilities of optimization • Knowledge of optimization algorithms, ability to choose proper algorithm for given problem • Practical experience with optimization algorithms • Practical experience in application of optimization to design problems

What is optimization? ● “Making things better” ● “Generating more profit” ● “Determining the best” ● “Do more with less” ● “The determination of values for design variables which minimize (maximize) the objective, while satisfying all constraints”

What can be achieved? • Optimization techniques can be used for: – Getting a design/system to work – Reaching the optimal performance – Making a design/system reliable and robust • Also provide insight in – Design problem – Underlying physics – Model weaknesses

Optimization problem • Design variables: variables with which the design problem is parameterized: • Objective: quantity that is to be minimized (maximized) Usually denoted by: ( “cost function”) • Constraint: condition that has to be satisfied – Inequality constraint: – Equality constraint:

Optimization problem (cont. ) • General form of optimization problem:

Solving optimization problems • Optimization problems are typically solved using an iterative algorithm: Constants Responses Model Derivatives of responses (design sensitivities) Design variables Optimizer

Curse of dimensionality Looks complicated … why not just sample the design space, and take the best one? • Consider problem with n design variables • Sample each variable with m samples • Number of computations required: mn Take 1 s per computation, 10 variables, 10 samples: total time 317 years!

Parallel computing • Still, for large problems, optimization requires lots of computing power • Parallel computing

Optimization in the design process Conventional design process: Optimization-based design process: Identify: 1. Design variables 2. Objective function 3. Constraints Collect data to describe the system Estimate initial design Analyze the system Check performance criteria Check the constraints Does the design satisfy Is design satisfactory? convergence criteria? Change the design using an Change design based on optimization method experience / heuristics / wild guesses Done

Optimization popularity Increasingly popular: • Increasing availability of numerical modeling techniques • Increasing availability of cheap computer power • Increased competition, global markets • Better and more powerful optimization techniques • Increasingly expensive production processes (trial-and-error approach too expensive) • More engineers having optimization knowledge

Optimization pitfalls! • Proper problem formulation critical! • Choosing the right algorithm for a given problem • Many algorithms contain lots of control parameters • Optimization tends to exploit weaknesses in models • Optimization can result in very sensitive designs • Some problems are simply too hard / large / expensive

Structural optimization • Structural optimization = optimization techniques L applied to structures R E, n • Different categories: – Sizing optimization – Material optimization – Shape optimization – Topology optimization r t h

Classification • Problems: – Constrained vs. unconstrained – Single level vs. multilevel – Single objective vs. multi-objective – Deterministic vs. stochastic • Responses: – Linear vs. nonlinear – Convex vs. nonconvex (later!) – Smooth vs. nonsmooth • Variables: – Continuous vs. discrete (integer)

Practical example: Airbus A 380 • Wing stiffening ribs of Airbus A 380: • Objective: reduce weight • Constraints: stress, buckling Leading edge ribs

Airbus A 380 example (cont. ) • Topology and shape optimization

Airbus A 380 example (cont. ) • Topology optimization: • Sizing / shape optimization:

Airbus A 380 example (cont. ) • Result: 500 kg weight savings!

Other examples • Jaguar F 1 FRC front wing: reduce weight constraints on max. displacements 5% weight saved

But also … • Optimization is also applied in: – Protein folding – System identification – Financial market forecasting (options pricing) – Logistics (traveling salesman problem), route planning, operations research – Controller design – Spacecraft trajectory planning

What makes a design optimization problem interesting? • Good design optimization problems often show a conflict of interest / contradicting requirements:

The optimization model Constants Responses Model Derivatives of responses (design sensitivities) Design variables Optimizer

Systems approach Input System function Output Environment • Systematic way of thinking: – What is input / output? – What belongs to system / environment? – What level of detail? – Distinguish sub-systems, hierarchies

Example: cantilever beam E, r h F, U U(t) E, r, h, L F(t) wi U(t) Etc.

Model example L E, r Steel h U(x), M(x), V(x) b Mathematical model: Finite element model: F, U h, b

Model example (2) L E, r F, U h, b Steel h U(x), M(x), V(x) b • System (state) variables: U(x), M(x), V(x) • System parameters: h, b, L • System constants: E, r

Features of computer models • Finite accuracy due to: – Discretization in time and space – Finite number of iterations (eigenvalues, nonlinear models) – Numerical round-off errors, ill-conditioning • Responses can be “noisy”: – Due to different discretization in space and/or time (e. g. remeshing)

Exercises • Exercise 1: Introduction to the valve spring design problem – Study analysis model – Formulation of spring optimization model • Exercise 2: Model behavior / optimization formulation – Study model properties (monotonicity, convexity, nonlinearity) – Optimization problem formulation

Defining a design model and optimization problem 1. What can be changed and how can the design be described? – – Dimensions Stacking sequence of laminates Ply orientation of laminates Thicknesses Bridgestone aircraft tire For structures: distinguish sizing, material and shape variables

Defining the optimization problem 2. What is “best”? Define an objective function: – – Weight Production cost Life-time cost Profits 3. What are the restrictions? Define the constraints: – Stresses – Buckling load – Eigenfrequency

Defining the optimization problem (cont. ) 4. Optimization: find a suitable algorithm to solve the optimization problem. Choice depends on problem characteristics: – Number of design variables, constraints – Computational cost of function evaluation – Sensitivities available? – Continuous / discrete design variables? – Smooth responses? – Numerical noise? – Many local optima? (nonconvex)

Summary Defining an optimization problem: 1. Choose design variables and their bounds 2. Formulate objective (best? ) 3. Formulate constraints (restrictions? ) 4. Choose suitable optimization algorithm

Standard forms • Several standard forms exist: Negative null form: Positive null form: Neg. unity form: Pos. unity form:

Structural optimization examples • Typical objective function: weight Note the scaling! • Typical constraint: maximum stress, maximum displacement Scaled vs. Unscaled

Example: minimum weight tubular column design • Length l given • Load P given • Design variables: – Radius R [Rmin, Rmax] – Wall thickness t [tmin, tmax] • Objective: minimum mass l • Constraints: buckling, stress P R t

Tubular column design Design problem:

Tubular column design (2) • Alternative formulation: P Ro l Ri

Multi-objective problems Vector! • Minimize c(x) s. t. g(x) 0, h(x) = 0 • Input from designer required! Popular approach: replace by weighted sum: ● Optimum, clearly, depends on choice of weights ● Pareto optimal point: “no other feasible point exists that has a smaller ci without having a larger cj”

Multi-objective problems (cont. ) • Examples of multi-objective problems: – Design of a structure for • Minimal weight and • Minimal stresses – Design of reduction gear unit for • Minimal volume • Maximal fatigue life – Design of a truck for • Minimal fuel consumption @ 80 km/h • Minimal acceleration time for 0 – 40 km/h • Minimal acceleration time for 40 – 90 km/h

Pareto set • Pareto point: “Cannot improve an objective without worsening another” c 2 Attainable set Pareto point c 1

Pareto set (cont. ) • Pareto set can be disjoint: Attainable set c 2 Pareto set c 1

The design space • Design space = set of all possible designs • Example: kmax Feasible domain F k 2 k 1 Optimum k 1 kmax

Isolines • Isolines (level sets) connect points with equal function values:

The design space (cont. ) Problem overconstrained: no solution exists. No feasible domain Dominated constraint (redundant)

Problem characteristics • Study of objective and constraint functions: – simplify problem – discover incorrect problem formulation – choose suitable optimization algorithms • Properties: – Boundedness – Linearity – Convexity – Monotonicity

Boundedness • Proper bounds are necessary to avoid unrealistic solutions: – Example: aspirin pill design Objective: minimize dissolving time = maximize surface area (fixed volume) h r

Boundedness (cont. ) • Volume equality constraint can be substituted, yielding: f r

Boundedness • Surface maximization of aspirin pill not well bounded: f r

Linearity “A function f is linear if it satisfies f(x 1+ x 2) = f(x 1)+ f(x 2) and f(a x 1) = a f(x 1) for every two points x 1, x 2 in the domain, and all a”

Linearity (2) • Nonlinear objective functions can have multiple local optima: f x 2 x x 1 ● Challenge: finding the global optimum. x 1

Convexity • Convex function: any line connecting any 2 points on the graph lies above it (or on it):

Convexity (cont. ) • Convex set [Papalambros 4. 27]: “A set S is convex if for every two points x 1, x 2 in S, the connecting line also lies completely inside S”

Convexity (cont. ) • Nonlinear constraint functions can result in nonconvex feasible domains: x 2 x 1 ● Nonconvex feasible domains can have multiple local boundary optima, even with linear objective functions!

Monotonicity • Papalambros p. 99: – Function f is strictly monotonically increasing if: f(x 2) > f(x 1) for – weakly monotonically increasing if: f(x 2) f(x 1) for – Similar for mon. decreasing f 2 f 1 x 2 > x 1 x 1 ● Similar: ● Note: monotonicity convexity! ● Linearity implies monotonicity x 2

Optimization problem characteristics • Responses: – Boundedness – Linearity – Convexity – Monotonicity ● Feasible domain: – Convexity

Example: tubular column design P R l t R g 1 g 3 f g 2 t

Optimization problem analysis • Motivation: – Simplification – Identify formulation errors early – Identify under- / overconstrained problems – Insight • Necessary conditions for existence of optimal solution • Basis: boundedness and constraint activity

Well-bounded functions – some definitions • Lower bound: ● Greatest lower bound (glb): f g ● Minimum: ● Minimizer: x* x

Boundedness checking • Assumption: in engineering optimization problems, design variables are positive and finite • Define • Boundedness check: – Determine g+ for – Determine minimizers – Well bounded if

Air tank design • Objective: minimize mass l r h ● Not well bounded: constraints needed t

Air tank constraints • Minimum volume: ● Min. head/radius ratio (ASME code): ● Min. thickness/radius ratio (ASME code): ● Room for nozzles: min. length ● Space limitations: max. outside radius

Partial minimization & bounding constraints • Partial minimization: keep all variables constant but one. Example: air tank wall thickness t: Conclusion: • f not well bounded from below • g 3 bounds t from below

Constraint activity • Removing constraint = relaxing problem • Solution set of relaxed problem without gi is Xi 1. 2. 3. A A and B active B ● Activity information can simplify problem: ● Active: eliminate variable ● Inactive: remove constraint

Constraint activity checking • Example: f(1, x 2, 5) g 2 Conclusion: • g 1 active • g 2 semiactive • g 3 and g 4 inactive g 3 x 2

Activity and Monotonicity Theorem • “Constraint gi is active if and only if the minimum of the relaxed problem is lower than that of the original problem” ● “If f(x) and gi(x) all increase or decrease (weakly) w. r. t. x, the domain is not well constrained” f(x) f g(x) x g 2 g 1 f(x) f g(x) x g

First Monotonicity Principle • “In a well-constrained minimization problem every variable that increases f is bounded below by at least one non-increasing active constraint” ● This principle can be used to find active constraints. ● Exactly one bounding constraint: critical constraint f(x) f g(x) x g

Air tank design • Monotonicity analysis: Critical w. r. t. r Critical w. r. t. h Critical w. r. t. t What about l? Unclear.

Optimizing variables out • Critical constraints must be active:

Optimizing variables out ● Critical constraints must be active:

Optimizing variables out ● Critical constraints must be active:

Optimizing variables out ● Critical constraints must be active:

Optimizing variables out ● Critical constraints must be active:

Problem! • Length not well bounded: ● Additional constraint from above is needed: ● Maximum plate width:

Air tank solution • Length constraint is critical: must be active! l • Solution: r h t ● Result of Monotonicity Analysis: ● Problem found, and fixed ● Solution found without numerical optimization

Recognizing monotonicity • Some useful properties: – Sums: Sums of similarly monotonic functions have the same monotonicity – Products: Products of similarly monotonic functions have: – same monotonicity if – opposite monotonicity if

Recognizing monotonicity • More properties: – Powers: Positive powers of monotonic functions have the same monotonicity, negative powers have opposite monotonicity – Composites:

Recognizing monotonicity • Integrals: f 1 – w. r. t. limits: 0 – w. r. t. integrand: a b x y f 1 a b x

Criticality Refined definitions: # of variables critically bounded by constraint i 0 1 >1 Uncritical constraint Uniquely critical constraint Multiple critical constraint Conditionally critical constraint # of constraints possibly critically bounding variable j 1 >1

Air tank example Critical w. r. t. r Critical w. r. t. h Critical w. r. t. t Conditionally critical w. r. t. l Multiple critical! Multiple critical constraint can obscure boundedness! Eliminate if possible

Air tank example • Starting with eliminating r:

Air tank example • New problem: ? Critical for h Critical for t

Air tank example • Finally, after also eliminating h and t: Not well bounded! ● Conclusion: multiple critical constraint obscured ill-boundedness in l

Summary • Optimization problem checking: – Boundedness check of objective • Identify underconstrained problems – Monotonicity analysis • • Identify not properly bounded problems Identify critical constraints Eliminate variables Remove inactive constraints

But what about … • Equality constraints: – Active if all constraint variables in objective – Otherwise semi-active • Example: x 2 f 3 Relaxed problem: 1 x 1

More on nonobjective variables • Monotonicity Principle for nonobjective variables: “In a well-constrained minimization problem every nonobjective variable is bounded below by at least one non-increasing semiactive constraint and above by at least one non-decreasing semiactive constraint” g(x) 0 gi gj x

Nonobjective variables (2) • Other options: – Equality constraint – Single nonmonotonic constraint h(x) 0 g(x) hi 0 gi x ● See example in book (Papalambros p. 114) x

Nonmonotonic functions • Monotonicity analysis difficult! – Sometimes regional monotonicity can be used – Concave constraints can split feasible domain: g(x) 0 gj gi x

Model preparation procedure (3. 9) • Remove dominated constraints • Check boundedness for each design variable: – Objective monotonic? Constraints monotonic? – Critical constraints? Uniquely / conditionally / multiply? • If possible, eliminate active constraints, and repeat steps Spending time on model checking usually pays off!