Improved two-point function approximations for design optimization

Two-point approximations are developed by utilizing both the function and gradient information of two data points. The objective of this work is to build a high-quality approximation to realize computational savings in solving complex optimization and reliability analysis problems. Two developments are proposed in the new twopoint approximations: 1) calculation of a correction term by matching with the previous known function value and supplementing it to the first-order approximation for including the effects of higher order terms and 2) development of a second-order approximation without the actual calculation of second-order derivatives by using an approximate Hessian matrix. Several highly nonlinear functions and structural examples are used for demonstrating the new two-point approximations that improved the accuracy of existing first-order methods. EVELOPMENTof new and improved constraint approximations has been an active area of research in the mathematical optimization community over the last two decades.1"3 More accurate function approximations reduce the finite element analysis cost and increase the search domain in each iteration of an optimization run. Recent developments in constraint approximations addressed methods for eigenvalues, dynamic response, static forces, structural reliability, etc. Some of the typical methods are summarized in the following. Kirsch4 developed a scaling factor for a stiffness matrix and approximated the displacements, stresses, and forces with respect to sizing and topology variables. Vanderplaats and Salajegheh5 used two levels of approximations for discrete sizing and shape design of structures, first for solving the continuous problem and next the discrete one using a dual theory. Canfield6 developed an approach for eigenvalues by approximating the modal strain and kinetic energies independently. Thomas et al.7 presented an approximation for the frequency response of damped structures by using the concept of Ref. 6. A multivariate spline approximation was developed by Wang and Grandhi8 making use of the previously generated exact finite element analyses. Similarly, a Hermite approximation for n-dimensional problems used several data points and had the property of reproducing the function and gradient values at the known data points.9 Canfield10 used multipoint data in building an approximate Hessian matrix and constructed a second-order approximation for improving the accuracy. Toropov et al.11 approximated functions as a linear combination or products of variables, and the coefficients of the polynomial were computed by applying a least squared approach on multiple data points. Fadel et al. 12 considered intermediate variables in terms of exponentials and they were computed by matching the approximate function gradients with the previous point's exact values. This approach did not compare the function values of the previous point. Wang and Grandhi13 used a single exponential for all of the variables and calculated it by matching the approximate function with the previous point exact value, and the gradient information of the previous point was not utilized. The proposed paper develops an improved two-point approximation utilizing both function and gradient information of two data points. In the new two-point approximation, two developments are proposed: 1) calculation of a correction term based on two function values and