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Example text
However, if the derivative is obtained using a finite difference method, the method is called a semianalytical method. In the continuum approach, the design derivative of the variational equation is taken before it is discretized. If the structural problem and sensitivity equations are solved as a continuum problem, then it is called the continuum-continuum method. However, only very simple, classical problems can be solved analytically. Thus, the continuum sensitivity equation is solved by discretization in the same way that structural problems are solved.
1, although for the central finite difference method, two performance measure evaluations are involved. 21) is the exact 24 1. 1. 88561 Sensitivity results of finite difference method. 20), there is no concern with numerical noise. That is, as the design perturbation size decreases, the finite difference results will converge to the exact design sensitivities. 4. 23). If the explicit form of the stiffness matrix K(u) and the load vector f(u) are known, and if solution z of matrix equation Kz = f is obtained, then the design sensitivity of the displacement vector can also be obtained, as K (u ) dz du df d K z.
Second, an exact integration (instead of a numerical integration) must be used in the generation of the stiffness matrix and in the evaluation of continuum-based design sensitivity expressions. Third, the exact solution (and not a numerical solution) of the finite element matrix equation and the adjoint equation should be used to compare these two methods. Fourth, the movement of discrete grid points must be consistent with the design parameterization method used in the continuum method. 5 Structural Design Sensitivity Analysis 29 For the sizing design variable, it is shown in [5] that the discrete and continuum-discrete methods are equivalent under the conditions given above, using a beam as the structural component.