Dr. Tafesse Gebresenbet AAiT, Mechanical Engineering Department Course ObjectiveThe course introduces : Understanding of principles and possibilities of optimization in Engineering and in particular in design Understand how to formulate an optimum design problem by identifying critical elements 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 Course outline Chapter 1: Introduction to Engineering Optimization of Design Introduction: Historical background, Definition of terms, Basic concepts, Classificationof optimizationsproblems , Applications :Design optimization,benefits of optimization, automated design optimization, when to use optimization, examples Chapter 2: Optimum Design Formulation Design models, Mathematical models, Defining optimization problem, Multi objective design problems, applications of optimization in design Chapter 3 Classical Optimization techniques Single variable optimization Multivariable optimization with equality and inequality constraints Chapter 4: One dimensionalunconstrained optimization techniques Elimination methods: Exhaustive search, Interval halving method, Fibonacci Method, Golden Section method. Interpolation methods: quadratic interpolation, cubic interpolation Direct root methods:Newton's method, Quasi -Newton method, Secant method Course outline Chapter5: Unconstrained Optimizationtechniques Direct search methods: Random search , Grid search Method, Powell method Indirect search(Descent) methods: Steepest descent (Cauchy) method, Conjugate gradient (Fletcher-Reeves) method, Newtons method, Unconstrained optimization using Matlab Chapter 6: Constrained Optimization techniques Direct search methods: Random search, complex search Method, Quadratic programming Indirect methods: Penalty function method, Lagrange multiplier method Constrained optimization using Matlab Chapter 7:Dynamic Programming Introduction , Multistage decision processes, Applications of dynamic programming . Chapter 8: Genetic Algorithm based Optimization Introduction to Genetic Algorithm , Applications of GA based optimization techniques , GA based Optimization using Matlab Reference Materials 1. S.S. Rao, Engineering Optimization, 3rd edition, Wiley Eastern, 2009 2. PapalambrosandWilde, Principle of optimal Design, modeling and computation, Cambridge University press, 20003. Fred van Keulen and Matthiis Langelaar, Lecture note s in Engineering Optimization, Technical University of Delft 4. Ravindran, Ragsdell and Rekalaitis, Engineering Optimization Methods and application, 2nd edition, Willey,2006 5. Arora, Introduction to Optimum design, 2nd edition, Elsevier Academic Press, 2004 6. Forst and Hoffmann, Optimization theory and practice, Springer , 2010 7. Haftka and Gurdal, Elements of Structural Optimization, 3rd edition, Kluwer academic, 1991 8. Belegundu and Chandrupatla, Optimization concepts and applications in Engineering, 2nd edition, Cambridge University press, 2011 9. Kalyanmoy Deb, Multi-objective Optimization using Evolutionary Algorithms, Wiley, 2002 PrerequisitesMathematical and Computer background needed to understand the course: Familiarity with linear algebra (vector and matrix operations) and basic calculus is essential and Calculus of functions of single and multiple variables must also be understood Familiarity with Matlab and EXCELis also essential Lecture outline Introduction Historical perspective What can be achieved by optimization? Optimization of the design process Basic terminology,notations, and definitions Engineering optimization Popularity and pitfalls of optimization Classification of optimizationproblems Design optimization Benefits of design optimization Automated design optimization Examples Introduction Optimization is derived from the Latin word optimus, the best. Thus optimization focuses on 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 Introduction Optimization is defined as a mathematical process of obtaining the set of conditions to produce the maximum or the minimum value of a function It is ideal to obtain the perfect solution to a design situation. Usually all of us must always work within the constraints of the time and funds available, we can only hope for the best solution possible. Optimization is simply a technique that aids in decision making but does not replace sound judgment and technical know-how Historical perspective Ancient Greek philosophers: geometrical optimization problems Zenodorus, 200 B.C.: A sphere encloses the greatest volume for a given surface area Newton, Leibniz, Bernoulli, De lHospital (1697): Brachistochrone Problem:

Historical perspective People have been optimizing forever, but the roots for modern day optimization can be traced to the Second World War. Ancient Greek philosophers: geometrical optimization problems Zenodorus, 200 B.C.: A sphere encloses the greatest volume for a given surface area Newton, Leibniz, Bernoulli, De lHospital (1697): Brachistochrone Problem: 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

One of the first problems posed in the calculus of variations. Galileo considered the problem in 1638, but his answer was incorrect. Johann Bernoulli posed the problem in 1696 to a group of elite 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 praise. Newton solved the problem the very next day, but proclaimed I do not love to be dunned [pestered] and teased by foreigners about mathematical things." What can be achieved by optimization ? 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 What can be achieved by optimization ? Engineering design is to create artifacts to perform desired functions under given constraints Common goals for engineering design Functionality Better performance: More efficient or effective ways to execute tasks Multiple functions: Capabilities to execute two or more tasks simultaneously Value Higher perceived value: More features with less price Lower total cost: Same or better ownership and sustainability with lower cost Basic Terminology, notations and definitions Rnn-dimensional Euclidean (real) spacexcolumn vector of variables, a point in Rn x=[x1,x2,..,xn]T f(x), fobjective functionx* local optimizer f(x*)optimum function valuegj(x), gjjth equality constraint functiong(x)vector of inequality constraint hj(x), hj jth equality constraint function h(h(x)vector of equality constraint functionC1set of continuous differentiable functions C2 set of continuous and twice differentiable differentiable continuous functions Norm/Length of a vector If we let x and y be two n-dimensional vectors, then their dot product is defined as Thus, the dot product is a sum of the product ofcorresponding elements of the vectors x and y. Two vectors are said to be orthogonal (normal) if their dot 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: where 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 vector The 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 follows Since Ax represents a vector, the triple product of the above Eq. will be also written as a dot product: Basic Terminology and notationsDesign variables Parameters whose numerical values are to be determined to 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 heat exchanger, or etc. 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. Constraints Numerical values of identified conditions that must be satisfied to achieve a feasible solution to a given problem. External constraints Uncontrolled restrictions or specifications imposed on a system by an outside agency. Ex.: Laws and regulations set by governmental agencies, allowable materials for house construction Internal constraints Restrictions imposed by the designer with a keen understanding of the physical system. Ex.: Fundamental laws of conservation of mass, momentum, and energy What is mathematical/Engineering Optimization ?Mathematical optimization is the process of1. The formulation and2. The solution of a constrained optimization problem of the general mathematical formMinimize f(x), x =[x1,x2,,xn]T subject to constraintsgj(x) s 0, j=1,2, , m hj(x) = 0, j=1, 2, . ,r Where f(x), gj(x) and hj(x) are scalar functions of the real column vector The continuous components of xi of x =[x1,x2,, xn]T are calledthe (design) variables f(x) is the objective function, gj(x) denotes the respective inequality constraints, and gj(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*). If no constraints are specified, the problem is called an unconstrained minimization problem Other names of Mathematical Optimization Mathematical programming NumericaloptimizationObjective 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 waysi. From analytically known formulae, e.g., f(x)= x12 + 2x22+Sin x3 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 point in structure, the design of which is specified by x; oriii. From measurement taken of a physical process, e.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. Elements of optimization Design space The total region or domain defined by the design variables in the objective functionsUsually limited by constraints The use of constraints is especially important in restricting the region where optimal values of the design variables can be searched. Unbounded design space Not limited by constraints No acceptable solutions Optimization in the design process 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 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: 1. Design variables 2. Objective function 3. Constraints Optimization in the design process Is there one aircraft which is the fastest, most efficient, quietest, mostinexpensive? You can only make one thing best at a time. Optimization Methods Comparison of Conventional and Optimal Design The CDprocess involves the use of information gathered from one or more trial designs together with the designers experience an intuition Its advantage is that the designers experience and intuition can be used in making conceptual changes in the system or to make additional specifications in the procedure The CD process can lead to uneconomical designs and can involve a lot of calendar time. The OD process forces the designer to identify explicitly a set of design variables, an objective function to be optimized, and the constraint functions for the system. This rigorous formulation of the design problem helps the designer gain a better understanding of the problem. Proper mathematical formulation of the design problem is a key to good solutions. 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 lotsof control parameters Optimization tends to exploitweaknesses in models Optimization can result in very sensitive designs Some problems are simply too hard / large / expensive Structural optimization Structural optimization = optimization techniquesapplied to structures Different categories: Sizing optimization Material optimization Shape optimization Topology optimization t E, v R r L h Shape optimizationYamaha R1 Topology optimization examples Why Design Optimization ? Design Complexity Classifications 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) Typical Design Process Initial Design Concept Specific Design Candidate Build Analysis Model(s) Execute the Analyses Design Requirements Met? Final Design Yes No Modify Design (Intuition) Time Money Intellectual Capital HEEDS $ HEEDS (Hierarchical Evolutionary Engineering Design System) A General Optimization Solution AutomotiveCivil Infrastructure Biomedical Aerospace Automated Design Optimization 37 Create Parameterized Baseline Model Create HEEDS Design Model Execute HEEDS Optimization Plan Design Study Basic Procedure: Automated Design Optimization 38 Identify:Objective(s) Constraints Design Variables Analysis Methods Note:These definitions affect subsequent steps Create Parameterized Baseline Model Create HEEDS Design Model Execute HEEDS Optimization Plan Design Study Automated Design Optimization 39 Input File(s) Execute Solver(s) Output File(s) Validate Model Create CAD/CAE Modelsfor a Representative Design Create Parameterized Baseline Model Create HEEDS Design Model Execute HEEDS Optimization Plan Design Study Automated Design Optimization 40 Define Input Filesand Output Files Define Design Variablesand Responses Define Objectives, Constraints, and Search Method Tag Variablesin Input Files andResponses in Output Files Define Batch Execution Commands for Solvers Create Parameterized Baseline Model Create HEEDS Design Model Execute HEEDS Optimization Plan Design Study Automated Design Optimization Create Parameterized Baseline Model Create HEEDS Design Model Execute HEEDS Optimization Plan Design Study Modify Variables in Input File Execute Solver in Batch Mode Extract Results from Output File Optimized Design(s) Yes New Design (HEEDS) No Converged? CAE Portals When What Where Tangible Benefits* 43 Crash rails: 100% increase in energy absorbed 20% reduction in mass Composite wing:80% increase in buckling load 15% increase in stiffness Bumper:20% reduction in mass with equivalent performance 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 Costs Higher quality designs Greater customer satisfaction Increased Competitive Advantage Innovative designs Faster to market Savings on material, manufacturing, mass, etc. - Suggests material placement or layout based on load path efficiency - Maximizes stiffness - Conceptual design tool - Uses Abaqus Standard FEA solver Topology Optimization When to Use Topology Optimization Early in the design cycle to find shape concepts To suggest regions for mass reduction Design of Experiments Determine how variables affect the response of a particular design Design sensitivities Build models relating the response to the variables Surrogate models, response surface models B A When to Use Design of Experiments Following optimization To identify parameters that cause greatest variation in your design Parameter Optimization Minimize (or maximize): F(x1,x2,,xn) such that:Gi(x1,x2,,xn) < 0, i=1,2,,p Hj(x1,x2,,xn) = 0,j=1,2,,q where:(x1,x2,,xn)are the n design variables F(x1,x2,,xn) is the objective (performance) function Gi(x1,x2,,xn) are the p inequality constraints Hj(x1,x2,,xn) are the q equality constraints Parameter Optimization Objective: Search the performance design landscape to find the highest peak or lowest valley within the feasible range Typically dont know the natureof surface before search begins Search algorithm choice depends on type of design landscape Local searches may yield only incremental improvement Number of parameters may be large Selecting an Optimization Method Design Space depends on: Number, type and range of variables and responses Objectives and constraints Gradient-Based Simplex Simulated Annealing Response Surface Genetic Algorithm Evolutionary Strategy Etc. Design Optimization Procedure Using ANSYS Organize ANSYS procedure into two files: Optimization filedescribes optimization variables, and trigger the optimization runs. Analysis fileconstructs, analyses, and post-processes the model. Typical Commands in an Optimization File 52/33 01 02 03 04 05 06 07 08 09 10 11 12 13/CLEAR! Clear model database ...! Initialize design variables /INPUT, ...! Execute analysis file once /OPT! Enter optimizationphase OPCLEAR! Clear optimization database OPVAR, ...! Declare design variables OPVAR, ...! Declare state variables OPVAR, ...! Declare objective function OPTYPE, ...! Select optimization method OPANL, ...! Specify analysis file name OPEXE! Execute optimization run OPLIST, ...! Summarize the results ...! Further examining resultsDesign Optimization Procedure Using ANSYS 53/33 01 02 03 04 05 06 07 08 09 10 11 12 /PREP7 ...! Build the model using the ! parameterized design variables FINISH /SOLUTION ...! Apply loads and solve FINISH /POST1! or /POST26 *GET, ...! Retrieve values for state variables *GET, ...! Retrieve value for objective function ...FINISH Typical Commands in an Analysis File Design Optimization Procedure Using ANSYS ANSYS Optimization Algorithms Two built-in algorithms in ANSYS: First order method Sub problem approximation method (Zero order method) Other Optimization Tools Provided by ANSYS Single Iteration Design Tool Random Design Tool Gradient Tool Sweep Tool Factorial Tool 54/33 Summary 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: ( ) 0 g s x( ) 0 h = x( ) f x( )1 2, , ,nx x x = x Summary General form of optimization problem: ( ) x x xxx hx gxxs s9 _ e =snXf0 ) (0 ) () (: to subjectmin Summary Optimization problems are typically solved using an iterative algorithm: Model Optimizer Design variables Constants Responses Derivatives of responses (design sensi- tivities) h g f , ,i i ixhxgxfcccccc, ,x