Structural Optimization 01-Oct-13 Dr. Walid Al-Awad

Structural Optimization In the process of designing structures in various fields of engineering, the designers and engineers make their best decisions at every step in view of structural and non-structural aspects such as stiffness, strength, serviceability, constructability, and aesthetic property. In other words, they make their optimal decisions to realize their best designs; hence, the process of structural design may be regarded as an optimum design even though optimality is not explicitly pursued. 01-Oct-13 Dr. Walid Al-Awad

Structural Optimization Structural optimization is regarded as an application of optimization methods to structural design. The typical structural optimization problem is formally formulated to minimize an objective function representing the structural cost under constraints on mechanical properties of the structure. The total structural weight or volume is usually used for representing the structural cost. 01-Oct-13 Dr. Walid Al-Awad

Structural Optimization there are many possible formulations for structural optimization, e.g., minimum weight design and maximum stiffness design, the term structural optimization or optimum design is usually used for representing all types of optimization problems corresponding to structural design. 01-Oct-13 Dr. Walid Al-Awad

Structural Optimization A structure in mechanics is defined as “any assemblage of materials which is intended to sustain loads.” Optimization means making things the best. Thus, structural optimization is the subject of making an assemblage of materials sustain loads in the best way. Structural optimization problem. Find the structure which best transmits the load F to the support 01-Oct-13 Dr. Walid Al-Awad

Structural Optimization The term “best.” make the structure as : light as possible to minimize weight. Another idea of “best” stiff as possible to buckling or instability as possible Quantities that are usually constrained in structural optimization problems are stresses, displacements and/or the geometry. To optimize = to do something as well as is possible. 01-Oct-13 Dr. Walid Al-Awad

Structural Optimization Types of Structural Optimization: - Size Optimization - Shape Optimization - Topology Optimization A sizing structural optimization problem is formulated by optimizing the cross-sectional areas of truss members 01-Oct-13 Dr. Walid Al-Awad

Structural Optimization A shape optimization problem: Find the function η(x), describing the shape of the beam-like structure The optimization consists in choosing the integration domain for the differential equations in an optimal way 01-Oct-13 Dr. Walid Al-Awad

Structural Optimization Topology optimization: This is the most general form of structural optimization. In a discrete case, such as for a truss, it is achieved by taking cross-sectional areas of truss members as design variables, and then allowing these variables to take the value zero, i.e., bars are removed from the truss. Topology optimization of a truss. Bars are removed by letting cross-sectional areas take the value zero Fig. 01-Oct-13 Dr. Walid Al-Awad

Structural Optimization Problem Statement: Structural optimization problems can be deceptively simple to formulate. They can be written as Find x to minimize f(x) subject to g(x) ≤ 0 (1) Here f (the objective function) is a scalar, x is an n-vector (has n components), and g (the constraints) is an m-vector. Problems of this type are called mathematical programming problems. Equation (1) is typically simplified to read 01-Oct-13 Dr. Walid Al-Awad

Structural Optimization min f(x) subject to g(x) ≤ 0 or even min f(x) | g(x) ≤ 0 Note that Design Vector: the quantities are treated as variables in the design process and are called design or decision variables xi , i = 1, 2, . . . , n. The design variables are collectively represented as a design vector X = {x1, x2, . . . , xn}T. 01-Oct-13 Dr. Walid Al-Awad

Design Constraints Design Constraints :In many practical problems, the design variables cannot be chosen arbitrarily. The restrictions that must be satisfied to produce an acceptable design are collectively called design constraints. consider an optimization problem with only inequality constraints gj (X) ≤ 0. The set of values of X that satisfy the equation gj (X) = 0 forms a hyper surface in the design space and is called a constraint surface. 01-Oct-13 Dr. Walid Al-Awad

constraint surface Hypothetical two-dimensional design space where the infeasible region is indicated by hatched lines. A design point that lies on one or more than one constraint surface is called a bound point , and the associated constraint is called an active constraint . Design points that do not lie on any constraint surface are known as free points. Depending on whether a particular design point belongs to the acceptable or unacceptable region, it can be identified as one of the following four types: 1. Free and acceptable point 2. Free and unacceptable point 3. Bound and acceptable point 4. Bound and unacceptable point 01-Oct-13 Dr. Walid Al-Awad

constraint surface All four types of points are shown in Fig. Constraint surfaces in a hypothetical two-dimensional design space 01-Oct-13 Dr. Walid Al-Awad

Structural Optimization For most design optimization problems, we shall use the following five step formulation procedure: problem statement. Data and information collection. definition of design variables. Identification of a criterion to be optimized. Identification of constraints. Design is the process of finding a solution to a problem, with all the constraints and requirements that it presents. 01-Oct-13 Dr. Walid Al-Awad

The End 01-Oct-13 Dr. Walid Al-Awad