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

Delft in The Netherlands

Background

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

Man-made Insect 5

Topology Optimization

Submarines

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

Who are you?

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.

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

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.

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 email) Final report Oral exam (individual) Cannot do oral exam before handing in sufficiently reasonable reports!

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

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”

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: http://www.mlahanas.de/Greeks/Optimization.htm Source for Newton/Leibniz/Bernoulli anecdote: http://www.math.purdue.edu/~eremenko/bernoulli.html 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”:

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!

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: Responses Derivatives of responses (design sensi- tivities) Model Constants 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 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.

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:

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 applied to structures Different categories: Sizing optimization Material optimization Shape optimization Topology optimization R r L h t E, n

Shape optimization Yamaha R1

Topology optimization examples

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)

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

Airbus A380 example (cont.) Topology and shape optimization

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

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

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

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

SMA active catheter optimization

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.

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!

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

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!

Example: cantilever beam h 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.

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:

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

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)                                                  

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

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

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

“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.