1. Concepts and Applications Engineering Optimization Fred van Keulen Matthijs Langelaar CLA H21.1 A.vanKeulen@tudelft.nl

2. 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:

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

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

5. Responses Derivatives ofresponses (design sensi-tivities) Solving optimization problems • Optimization problems are typically solved using an iterative algorithm: Model Constants Designvariables Optimizer

6. Optimization pitfalls! • Proper problem formulation critical! • Choosing the right algorithmfor 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

7. 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

8. Course overview • General introduction, problem formulation, design space / optimization terminology • Modeling, model simplification • Optimization of unconstrained / constrained problems • Single-variable, zeroth-order and gradient-based optimization algorithms • Design sensitivity analysis (FEM) • Topology optimization

9. 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 For structures: distinguish sizing, material and shape variables Bridgestone aircraft tire

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

11. 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)

12. Summary Defining an optimization problem: • Choose design variables and their bounds • Formulate objective (best?) • Formulate constraints (restrictions?) • Choose suitable optimization algorithm

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

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

15. P R R l t t 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 • Constraints: buckling, stress

16. Design problem: Tubular column design

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

18. Multi-objective problems • Minimize c(x)s.t. g(x)  0, h(x) = 0 • Input from designer required! Popular approach: replace by weighted sum: Vector! • 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”

19. 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

20. Pareto set Pareto point Pareto set • Pareto point: “Cannot improve an objective without worsening another” c2 Attainable set c1

21. Pareto set Pareto set (cont.) • Alternative view: c1 c2 x

22. Pareto set Pareto set (cont.) • Pareto set can be disjoint: Attainable set c2 c1

23. Hierarchical systems • Large system can be decomposed into subsystems / components: • Optimization requires specialized techniques,multilevel optimization

24. Local (rib / stiffner) level: plate thickness, fiber orientation Structural hierarchical systems • Example: wing box • Too many designvariables to treat at once • Global level: global loads, global dimensions

25. Contents • Defining an optimization problem • The design space & problem characteristics • Model simplification

26. Optimum The design space • Design space = set of all possible designs • Example: kmax Feasible domain F k2 k2 k1 k1 kmax

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

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

29. A and B inactive A and B active B active, A inactive Objective function isolines Objective function isolines Objective function isolines Interior optimum Optimum Optimum Design space (cont.) A B

30. F F Active constraint optimization • Idea of constraint activity at boundary optimum sometimes used in intuitive design optimization: • Fully stressed design (sizing / topology optimization) • Simultaneous failure mode theory • Careful: does not always give the optimal solution!

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

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

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

34. Linearity “A function f is linear if it satisfies f(x1+ x2) = f(x1)+ f(x2) andf(ax1) = af(x1) for every two points x1, x2 in the domain, and all a”

35. f x2 x2 x1 x1 Linearity (2) • Nonlinear objective functions can have multiple local optima: f x • Challenge: finding the global optimum.

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

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

38. f x2 x2 x1 x1 Linearity • Nonlinear objective functions can have multiple local optima: f x • Challenge: finding the global optimum.

39. Convexity • Convex function: any line connecting any 2 points on the graph lies above it (or on it): • Linearity implies convexity (but not strict convexity)

40. Convexity (cont.) • Convex set [Papalambros 4.27]: “A set S is convex if for every two points x1, x2 in S, the connecting line also lies completely inside S”

41. Convexity (cont.) • Nonlinear constraint functions can result in nonconvex feasible domains: x2 x1 • Nonconvex feasible domains can have multiple local boundary optima, even with linear objective functions!

42. f2 f1 x1 x2 • Similar: • Note: monotonicity  convexity! • Linearity implies monotonicity Monotonicity • Papalambros p. 99: • Function f is strictly monotonically increasing if:f(x2) > f(x1) for x2 > x1 • weakly monotonically increasing if:f(x2) f(x1) for x2 > x1 • Similar for mon. decreasing

43. Feasible domain: • Convexity Optimization problem characteristics • Responses: • Boundedness • Linearity • Convexity • Monotonicity

44. P R t l Example: tubular column design R g3 g1 f g2 t

45. 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

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

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

48. Examples: Bounded at zero Asymptotically bounded

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

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