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In the design of any component, there are always associate with the design certain desirable and undesirable effects. It is possible to obtain design solutions without paying too much attention to these effects (other than casually checking that the component will perform its required function without failure); such a solution might be termed an adequate design. In many instances, however, it is necessary to give more than casual consideration to the various effects: either to maximize a desirable one or minimize an undesirable one. The design solution may then be termed an optimum design. For example, it may be required to minimize the cost of a component (particularly if the design is for Merrell Shoes on Sale mass production), to minimize weight or deflection, or to obtain maximum power transmission capability or load carrying capacity.

When any component is designed, certain functional requirements must be satisfied, and there are usually many design solutions which will satisfy these requirements. It is the purpose of the optimum design method to present a procedure of design ...
... which will give an optimum solution, taking account of all the factors involved.
Any idealized engineering system can be described by a finite set of quantities. For example, an elastic structure modeled by finite elements is characterized by the node coordinates-"Some of these quantities are fixed in advance and they will not be changed by the redesign process (they are often called prescribed parameters). The others are the design variables; they will be modified during each redesign process in order to gradually optimize the mechanical system. A function of the design variables must be defined, whose value permits selecting different feasible designs; this is the objective function (e. g. the weight of an aerospace structure). A design is said to be feasible if it satisfies all the requirements that are imposed to the mechanical system when performing its tasks. Usually, requiring that a design is feasible amounts to assigning upper or lower limits to quantities characterizing the system behavior (inequality constraints). Sometimes given values, rather than lower or upper bounds, are imposed to these quantities (equality constraints). Taking again the case of structural optimization, the behavior constraints are placed on stresses, displacements, frequencies, buckling loads, etc.
Therefore the optimization problem in minimizing an objective function, which Discount Merrell Shoes represents a cost associated with the mechanical, system, subject to equality and inequality constraints which insure the design feasibility.
To help fix ideas let us consider the minimum weight design of the 3-bar truss shown in Fig. 4. 1. Several types of design variables could be chosen for optimizing this structure: the bar cross-sectional areas [A\, Az, As] > their material properties \_(p\, E\ , a\); (pz, ￡2, 02); (.ps > ￡3? as)]; the coordinates of the free node [or, equivalently, the angles (ai, m , as)]- To simplify the problem, we restrict ourselves to the first class of variables (optimal sizing problem). Adopting the same material for each bar and fixing the prescribed geometrical parameters to the values Note that, for symmetry reasons, only two design variable A\ and A2 define the problem, which admits the geometrical interpretation given in the design space of Fig. 4. 1. Each point in the design space corresponds to a possible structural design. The objective function (4-1) is represented by a set of constant weight planes, and the stress limitations (4-2)-(4-4), by restraint surfaces that permits defining the feasible domain. Clearly the optimal design A * corresponds to the point where a constant weight plane is tangent to the boundary of the feasible domain. At the optimum, only one constraint is satisfied as an equality (active constraint). The others are satisfied as inequalities (inactive constraint), which means that the stress level in bars 2 and 3 is below the allowable upper limit.
In this simple example, it is quite easy to detect that only one constraint is really meaningful. Realistic optimization problems, however, involve many design variables and constraints. Their solution can no longer be obtained analytically or graphically, but they require efficient mathematical tools. The question of finding how many and which constraints are active is especially crucial.