Ken YoussefiMechanical Engineering Dept. 1 Design Optimization Optimization is a component of design process The design of systems can be formulated as problems of optimization where a measure of performance is to be optimized while satisfying all the constraints.

Ken YoussefiMechanical Engineering Dept. 2 Design Optimization Design variables – a set of parameters that describes the system (dimensions, material, load, …) Design constraints – all systems are designed to perform within a given set of constraints. The constraints must be influenced by the design variables (max. or min. values of design variables). Objective function – a criterion is needed to judge whether or not a given design is better than another (cost, profit, weight, deflection, stress, ….).

Ken YoussefiMechanical Engineering Dept. 3 Optimum Design – Problem Formulation The formulation of an optimization problem is extremely important, care should always be exercised in defining and developing expressions for the constraints. The optimum solution will only be as good as the formulation.

Ken YoussefiMechanical Engineering Dept. 4 Problem Formulation Design of a two-bar structure The problem is to design a two-member bracket to support a force W without structural failure. Since the bracket will be produced in large quantities, the design objective is to minimize its mass while also satisfying certain fabrication and space limitation.

Ken YoussefiMechanical Engineering Dept. 5 Problem Formulation In formulating the design problem, we need to define structural failure more precisely. Member forces F 1 and F 2 can be used to define failure condition. Apply equilibrium conditions; Σ F y = 0 Σ F x = 0

Ken YoussefiMechanical Engineering Dept. 6 Problem Formulation Design Variables An important first step in the proper formulation of the problem is to identify design variables for the system. 1.All design variables should be independent of each other as far as possible. 2.All options of identifying design variables should be investigated. 3.There is a minimum number of design variables required to formulate a design problem properly. 4.Designate as many independent parameters as possible as at the beginning. Later, some of the design variables can be eliminated by assigning numerical values.

Ken YoussefiMechanical Engineering Dept. 7 Problem Formulation Represent all the design variables for a problem in the vector x. x 3 = outer diameter of member 1 x 4 = inner diameter of member 1 x 5 = outer diameter of member 2 x 6 = inner diameter of member 2 x 1 = height h of the truss x 2 = span s of the truss

Ken YoussefiMechanical Engineering Dept. 8 Problem Formulation Objective Function A criterion must be selected to compare various designs 1.It must be a scalar function whose numerical values could be obtained once a design is specified. 2.It must be a function of design variables, f (x). 3.The objective function is minimized or maximized (minimize cost, maximize profit, minimize weight, maximize ride quality of a vehicle, minimize the cost of manufacturing, ….) 4.Multi-objective functions; minimize the weight of a structure and at the same time minimize the deflection or stress at a certain point.

Ken YoussefiMechanical Engineering Dept. 9 Problem Formulation Objective Function Mass is selected as the objective function Mass = ( density )( area )( length ) x 1 = height h of the truss x 2 = span s of the truss x 3 = outer diameter of member 1 x 4 = inner diameter of member 1 x 5 = outer diameter of member 2 x 6 = inner diameter of member 2

Ken YoussefiMechanical Engineering Dept. 10 Problem Formulation Design Constraints Feasible Design A design meeting all the requirements is called a feasible (acceptable) design. An infeasible design does not meet one or more requirements Implicit & Explicit Constraints All restrictions placed on a design are collectively called constraints. Some constraints are explicit (obvious, provided) such as min. and max. values of design variables, some are implicit (derived, deduced) which are usually more complex such as deflection of a structure.

Ken YoussefiMechanical Engineering Dept. 11 Problem Formulation Linear and Nonlinear Constraints Constraint functions having only first-order terms in design variables are called linear constraints. More general problems have nonlinear constraint functions as well. A machine component must move precisely by a certain value (equality). Stress must not exceed the allowable stress of the material (inequality) It is easier to find feasible designs for a system having only inequality constraints. Design problems may have equality as well as inequality constraints. A feasible design must satisfy precisely all the equality constraints. Equality and Inequality Constraints

Ken YoussefiMechanical Engineering Dept. 12 Problem Formulation Constraints for the example problem are: member stress shall not exceed the allowable stress, and various limitations on design variables shall be met. Constraint on stress σ (applied) < σ (allowable)

Ken YoussefiMechanical Engineering Dept. 13 Problem Formulation Finally, the constraints on design variables are written as Where x il and x iu are the minimum and maximum values for the ith design variable. These constraints are necessary to impose fabrication and physical space limitations.

Ken YoussefiMechanical Engineering Dept. 14 Problem Formulation The problem can be summarized as follows Find design variables x 1, x 2, x 3, x 4, x 5, and x 6 to minimize the objective function, subject to the constraints of the equations x 1 = height h of the truss x 2 = span s of the truss x 3 = outer diameter of member 1 x 4 = inner diameter of member 1 x 5 = outer diameter of member 2 x 6 = inner diameter of member 2

Ken YoussefiMechanical Engineering Dept. 15 Example – Design of a Beer Can Design a beer can to hold at least the specified amount of beer and meet other design requirement. The cans will be produced in billions, so it is desirable to minimize the cost of manufacturing. Since the cost can be related directly to the surface area of the sheet metal used, it is reasonable to minimize the sheet metal required to fabricate the can.

Ken YoussefiMechanical Engineering Dept. 16 Example – Design of a Beer Can Fabrication, handling, aesthetic, shipping considerations and customer needs impose the following restrictions on the size of the can: 1.The diameter of the can should be no more than 8 cm. Also, it should not be less than 3.5 cm. 2.The height of the can should be no more than 18 cm and no less than 8 cm. 3.The can is required to hold at least 400 ml of fluid.

Ken YoussefiMechanical Engineering Dept. 17 Example – Design of a Beer Can Design variables D = diameter of the can (cm) H = height of the can (cm) Objective function The design objective is to minimize the surface area (Non-linear)

Ken YoussefiMechanical Engineering Dept. 18 Example – Design of a Beer Can The constraints must be formulated in terms of design variables. The first constraint is that the can must hold at least 400 ml of fluid. The problem has two independent design variable and five explicit constraints. The objective function and first constraint are nonlinear in design variable whereas the remaining constraints are linear. (Non-linear) The other constraints on the size of the can are: (linear)

Ken YoussefiMechanical Engineering Dept. 19 Standard Design Optimization Model The standard design optimization model is defined as follows: Find an n-vector x = (x 1, x 2, …., x n ) of design variables to minimize an objective function subject to the p equality constraints and the m inequality constraints

Ken YoussefiMechanical Engineering Dept. 20 Observations on the Standard Model The functions f(x), h j (x), and g i (x) must depend on some or all of the design variables. The number of independent equality constraints must be less than or at most equal to the number of design variables. There is no restriction on the number of inequality constraints. Some design problems may not have any constraints (unconstrained optimization problems). Linear programming is needed If all the functions f(x), h j (x), and g i (x) are linear in design variables x, otherwise use nonlinear programming.

Ken YoussefiMechanical Engineering Dept. 21 Helical Compression Spring Free body diagram of axially loaded helical spring D = coil diameter d = wire diameter Maximum shear stress

Ken YoussefiMechanical Engineering Dept. 22 Helical Compression Spring Shear stress correction factor Spring index Maximum shear stress

Ken YoussefiMechanical Engineering Dept. 23 Helical Compression Spring Deflection The total strain energy, torsional and shear components Total strain energy

Ken YoussefiMechanical Engineering Dept. 24 Helical Compression Spring Castigliano’s theorem – deflection is the partial derivative of the total strain energy. Spring rate Linear relationship

Ken YoussefiMechanical Engineering Dept. 25 Helical Compression Spring Helical spring end conditions N (total) = N a (active number of coils) + N i (inactive number of coils)

Ken YoussefiMechanical Engineering Dept. 26 Example of unconstraint optimization problem Design a compression spring of minimum weight, given the following data:

Ken YoussefiMechanical Engineering Dept. 27 Example – spring design The weight of the compression spring is given by the following equation:

Ken YoussefiMechanical Engineering Dept. 28 Example – spring design The equation can be expressed in terms of the spring index C = D/d The maximum shear stress and the deflection in the spring are given by the following equations

Ken YoussefiMechanical Engineering Dept. 29 Example – spring design Substituting all of the equations into the weight equation, we obtain the following expression in terms of spring index C: The plot of the objective function W vs. C, the spring index

Ken YoussefiMechanical Engineering Dept. 30 Infeasible Problem Conflicting requirements, inconsistent constraint equations or too many constraints on the system will result in no solution to the problem. No region of design space that satisfies all constraints.

Ken YoussefiMechanical Engineering Dept. 31 Optimization using graphical method A wall bracket is to be designed to support a load of W. The bracket should not fail under the load. W = 1.2 MN h = 30 cm s = 40 cm 1 2

Ken YoussefiMechanical Engineering Dept. 32 Problem Formulation Design variables: Objective function: Stress constraints: Where forces on bar 1 and bar 2 are:

Ken YoussefiMechanical Engineering Dept. 33 Graphical Solution

Ken YoussefiMechanical Engineering Dept. 34 Numerical Methods for Non-linear Optimization Graphical and analytical methods are inappropriate for many complicated engineering design problems. 1.The number of design variables and constraints can be large. 2.The functions for the design problem can be highly nonlinear. 3.In many engineering applications, objective and/or constraint functions can be implicit in terms of design variables.

Ken YoussefiMechanical Engineering Dept. 35 Structural Optimization Structural optimization is an automated synthesis of a mechanical component based on structural properties. For this optimization, a geometric modeling tool to represent the shape, a structural analysis tool to solve the problem, and an optimization algorithms to search for the optimum design are needed. Structural Optimization Geometric ModelingStructural AnalysisOptimization Finite element modeling Finite element analysis Nonlinear programming algorithm

Ken YoussefiMechanical Engineering Dept. 36 Structural Optimization Categories Size optimization Deals with parameters that do not alter the location of nodal points in the numerical problem, keeps a design’s shape and topology unchanged while changing dimensions of the design. Shape optimization Deals with parameters that describe the boundary position in the numerical model. Design variables control the shape of the design. The process requires re-meshing of the model and results in freeform shapes Topology optimization The goal of topology optimization is to determine where to place or add material and where to remove material. The optimization could determine where to place ribs to stiffen the structure.

Ken YoussefiMechanical Engineering Dept. 37 Optimization Categories Size and configuration optimization of a truss, design variables are the cross sectional areas and nodal coordinates of the truss. The truss could also be optimized for material. The topology or connectivity of the truss is fixed.

Ken YoussefiMechanical Engineering Dept. 38 Optimization Categories Shape optimization of a torque arm. Parts of the boundary are treated as design variables.

Ken YoussefiMechanical Engineering Dept. 39 Optimization Categories Topology optimization can be performed by using genetic algorithm. Optimum shapes of the cross section of a beam for plastic (b), aluminum (c), and steel (d)