1. Engineering Optimization
Concepts and Applications
Fred van Keulen
Matthijs Langelaar
CLA H21.1

2. Delft in The Netherlands

3. Background

4. Overview of research projects
Optimization with Uncertainties
Approximate optimization
Topology Optimization
Multilevel optimization
Fast reanalysis
Buckling of submarine
Impregnation
Shoulder endoprosthesis
SMA actuators
Microactuation for Butterfly
Microactuators (them./electr)
MEMS packaging
MEMS surface effects
MEMS measurement structures
Electronic interface modeling
Modeling of MEMS
MEMS optimization

5. Man-made Insect 5

6. Topology Optimization

7. Submarines

8. Micro actuator 60 um 530 um 13 μm, ie 2.5% longitudinal strain
at 2 V, 27 mW, Tmax = 200C
60 um
530 um

9. Who are you?

10. 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
Focus on engineering applications, not on hard-core formal mathematics! Still, mathematics cannot be avoided for this subject.

11. 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

12. Course material
Main text: “Principles of Optimal Design – Modeling and Computation”, P.Y. Papalambros & D.J. Wilde, Cambridge University Press
Selected topics: “Elements of Structural Optimization”, R.T. Haftka & Z. Gurdal, Kluwer Academic Publishers
Exercises and references
Examples will be primarily on structural optimization, instead of the system problems used in Papalambros.

13. Examination
Report on practical exercises using Matlab and Optimization Toolbox (individual or in groups of 2 students)
Report on optimization project (individual or in groups of 2 students):
Definition of problem, approach (ca. 1 page A4, Deadline March 28, via )
Final report
Oral exam (individual)
Cannot do oral exam before handing in sufficiently reasonable reports!

14. Course Schedule No lectures on: 19-2, 11-3 and 1-4
How to find alternative time slots?
Training lectures?

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

16. Historical perspective
Ancient Greek philosophers: geometrical optimization problems
Zenodorus, 200 B.C.: “A sphere encloses the greatest volume for a given surface area”
Picture is actually Aristotle.
Source for Zenodorus:
Source for Newton/Leibniz/Bernoulli anecdote:
While Newton made a secret of his discovery of fluxions, Leibniz publicized his calculus, and by the year of 1695 he and his student John Bernoulli developed calculus into a magnificent tool for solving a variety of problems.
To find out how much Newton really knew, Leibniz and Bernoulli devised the following test. According to the custom of that time, John Bernoulli, published in June 1696 a challenging problem, which he addressed "to acutest mathematicians of the world''
`To find the curve connecting two points, at different heights and not on the same vertical line, along which a body acted upon only by gravity will fall in the shortest time'.
Leibniz and Bernoulli were confident that only a person who knows calculus could solve this problem. Bernoulli allowed six months for the solutions but no solutions were received during this period. At the request of Leibniz, the time was publicly extended for a year in order that all contestants should have an equal chance. On 29th of January 1697 the challenge was received by Newton from France and on the next day (according to his nephew's memoirs) he sent to Montague, who was then President of the Royal society, his solution. The only other solutions were sent by Leibniz and l'Hôpital. (The latter, another student of Leibniz, was the author of the first calculus textbook). Following Bernoulli's suggestion the curve which solves the problem is called the `brachistochrone', which is the Greek for `the shortest time'. ? g
Newton, Leibniz, Bernoulli, De l’Hospital (1697): “Brachistochrone Problem”:

17. Historical perspective (cont.)
                         
Lagrange (1750): constrained minimization
Cauchy (1847): steepest descent
Dantzig (1947): Simplex method (LP)
Kuhn, Tucker (1951): optimality conditions
Karmakar (1984): interior point method (LP)
Bendsoe, Kikuchi (1988): topology optimization
I just guessed the years, I can’t find them!
Richard Bellman did his PhD in 3 months!

18. 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

19. 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:

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

21. Solving optimization problems
Optimization problems are typically solved using an iterative algorithm:
Responses
Derivatives of responses
(design sensi- tivities)
Model
Constants
Design variables
Optimizer

22. 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!

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

24. Optimization in the design process
Optimization-based design process:
Collect data to describe the system
Estimate initial design
Analyze the system
Check the constraints
Does the design satisfy convergence criteria?
Change the design using an optimization method
Done
Identify:
Design variables
Objective function
Constraints
Conventional design process:
Collect data to describe the system
Estimate initial design
Analyze the system
Check performance criteria
Is design satisfactory?
Change design based on experience / heuristics / wild guesses
Done
Taken from J.S. Arora “Introduction to Optimum Design”, fig. 1-2.

25. Optimization popularity
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
Increasingly popular:

26. 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

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

28. Shape optimization
Yamaha R1

29. Topology optimization examples

30. Classification Problems: Responses: Variables:
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)

31. Practical example: Airbus A380
Wing stiffening ribs of Airbus A380:
Objective: reduce weight
Constraints: stress, buckling
Leading
edge ribs

32. Airbus A380 example (cont.)
Topology and shape optimization

33. Airbus A380 example (cont.)
Topology optimization:
Sizing / shape optimization:

34. Airbus A380 example (cont.)
Result: 500 kg weight savings!

35. Other examples
Jaguar F1 FRC front wing: reduce weight constraints on max. displacements
5% weight saved

36. Other examples (cont.)
Design optimization of packaging products (Van Dijk & Van Keulen):
Objective: minimize material used
Constraints: stress, buckling
Result: 20% saved

37. SMA active catheter optimization

38. 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
This course: focus on (structural) design optimization
Focus on structural design optimization, but methods and gained knowledge applies to many other topics as well.

39. What makes a design optimization problem interesting?
Good design optimization problems often show a conflict of interest / contradicting requirements:
Aircraft wing: stiffness vs. weight
F1 car: idem
Oil bottle: stiffness / buckling load vs. material usage
Otherwise the problem could be trivial!

40. The optimization model
Responses
Derivatives of responses
(design sensi- tivities)
Model
Constants
Design variables
Optimizer

41. Systems approach Systematic way of thinking: Input Output
System function
Environment
Systematic way of thinking:
What is input / output?
What belongs to system / environment?
What level of detail?
Distinguish sub-systems, hierarchies
Good model: as simple as possible, but not simpler!

42. Example: cantilever beamh
F, U
U(t)
F(t)
Same physical structure, but different systems, depending on what is considered input and output.
E, r, h, L wi
F(t)
U(t)
Etc.

43. Model example F, U h, b h b E, r L Steel U(x), M(x), V(x)
Mathematical model:
System variables: M, D, U. Specify the state of the system.
System parameters: h, b, L. Constant when model is applied.
System constants: E, rho (steel).
Finite element model:

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

45. 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)
                                                 

46. Noisy response Example: effect of remeshing
Normalized stress constraint
As an example of the noisy behavior of response functions, consider a simple optimization problem of a square plate with a circular central hole, Toropov et al. (1995). The radius of the hole is treated as a single design variable. The normalized stress constraint (Fig. 1) and normalized stability constraint (Fig. 2) are shown for a coarse and a fine mesh (Fig. 3 and 4). In each case the mesh density was kept constant. It can be seen that the the coarse mesh results possess a significantly higher level of noise. Another obvious observation is that the numerical solution is also characterized by an offset from the exact solution. The magnitude of the offset depends on the specified mesh density and tends to zero when the mesh density is increased.
(Vasili Toropov)
Hole radius

47. Features of computer models (cont.)
Computational models are (very) time consuming
Often design sensitivities can be calculated
Cost of design sensitivity analysis?
Accuracy / consistency of sensitivities
Accuracy refers to how well the computed sensitivity matches the sensitivity of the unknown EXACT solution.
Consistency refers to how well the computed sensitivity matches the slope of the numerical solution.
Response
Design variable
Exact
Numerical model

48. Finite difference sensitivities
Straightforward way to compute sensitivities: finite differences f Dx
Small! x
More later!

49. “Everything should be made as simple as possible, but not simpler”
Einstein’s advice
“Everything should be made as simple as possible, but not simpler”
Model simplification important for optimization! More in next lectures.