Differential Equation Based Method for Accurate Approximations in Optimization

This paper describes a method to efficiently and accurately approximate the effect of design changes on structural response. The key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB)method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved pedurbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in most cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacements are used to approximate bending stresses. t_m_J_Jatu_ K stiffness matrix M mass matrix s number of steps v design variable v vector of design vanables _v change in design variable " Research Engineer, interdisciplinary Research Office, Member AHS "* Deputy Head, Interdisciplinary Research Office Member AIAA,ASME o_2 d_ e Superscript T Subscript o i vibration eigenvalue (frequency) vibration eigenvector (mode shape) distance along move direction designates transpose designates nominal value ith component of vector