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# Optimal Engine Design Using Nonlinear Programming and the ...

Optimal Engine Design Using Nonlinear Programming and the ...

## of use, its explicit

of use, its explicit algebraic expressions and its continued proliferation into the companyoperations.2. Optimization MethodsThe terms objective, constraint, variable, parameter, vector, and feasible domain haveexplicit definitions given below.The objective is the quantity to be optimized (minimized or maximized). It can be anexplicit algebraic function or it can be an output of another computer program .A (design)variable is any quantity allowed to vary during the search for the optimumobjective. At least the objective function or one of the constraints should depend on a variable;otherwise it is not relevant to the problem statement.A parameter is any quantity appearing in the problem statement which is fixed during theoptimization. For example, the values of the bounds appearing in the constraint set are parameters.A constraint bounds the set of variables in some way. Examples are: upper and lowerbounds on variables, equality relationships among variables, upper and lower bounds on explicitalgebraic expressions relating design variables or upper and lower bounds on outputs of a model.The set of variable values bounded by the set of constraints is called the feasible domain.A vector is simply a set of scalars. The set of variables is a vector; the set of equalityconstraints is a vector; the set of inequality constraints is a vector. Also recall from multivariablecalculus that the gradient of a scalar is a vector; and that the gradient of a vector is a matrix.2.1 Unconstrained OptimizationThe goal of optimization is to minimize or maximize a single function f, which depends onone or more independent variables. The value of those variables at that minimum or maximum andthe value of f is termed the optimal solution. The calculation of gradients in the design variablespace in search of a minimum is the essence of the algorithms of interest here. For acomprehensive, understandable introduction to optimization see Chapter 10 of Numerical Recipes- The Art of Scientific Computing, [Press, et.al.,1987].The classical statement of an unconstrained optimization problem is to minimize (ormaximize) a function f which depends upon a vector, x, of n variables where x = {x 1 ,x 2 , ......,x n ,} . The statement of the problem for x is:minimize f(x)x = {x 1 ,x 2 , ......, x n ,} ε R n (1)For a single variable problem, (x = {x}) recall from calculus that the first order necessarycondition for a minimum is df/dx = 0. Also recall that the value of d 2 f/dx 2 at this value of x2

sufficiently determines whether the function is a minimum or maximum; it is called the secondorder sufficiency condition . Similarly, for a function of n variables the first order necessarycondition is that the gradient of f(x) be equal to zero. That is:∂ f = 0∂x 1∇f(x) = ∂ f = 0 (2)∂x 2. .. .∂ f = 0∂x nThis results in a set of n equations that must be solved. Newton's method is a straightforwardalgorithm to solve such a set of equations. The following steps describe the algorithm .Step 1.Step 2.Pick a starting point (a guess of the solution) x 0 and set an iteration counterk = 0. (Superscript indicates iteration number).Calculate a step size, d k , to move in x whered k = -[D(∇f(x) ) k ] -1 (∇f(x) ) k (3)and D(∇f(x) )is the Jacobian of the gradient of f(x) and the Hessian of f(x).Step 3.Calculate a new value of x usingx k+1 = x k + d k (4)Step 4. If a convergence criteria is satisfied (e.g. || x k+1 - x k || ≤ ε) stop;otherwise increment k by one and go to 2.For quadratic functions, it can be shown that this method converges in one step from the startingpoint [Dennis et al., 1989].2.2 Constrained OptimizationMost design problems have many constraints imposed. For example, engine designproblems include geometric constraints on the engine package, and performance criteria constraintson the vehicle. The constraints can be in the form of an equality ( e.g., π (bore) 2 x (stroke) - 4(volume) = 0) or an inequality (e.g., (maximum piston speed) - 25m/sec ≤ 0). Each equalityconstraint equation is valued at 0 and is usually denoted by h. Similarly, inequalities are3

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