A general method for design sensitivity analysis with changing boundary condition is developed based on the variational approach. The change of boundary condition is described using the tangential component of velocity field continuously distributed over a subdomain which is defined around the point separating boundary segments of different conditions. By taking the variational approach, the present method has advantages over that based on the boundary integral equation developed previously. This allows many applications possible including the design sensitivity analysis of eigenvalue responses and nonhomogeneous domain problems.
By the developed method, concrete sensitivity formulas for a potential problem, free vibration problems of a beam and a plate considering shear deformation and rotary inertia effects are obtained. For numerical verification of the derived sensitivity formulas, typical individual problems are considered. As a static case, a heat flow problem is taken and the numerical results are compared with those obtained by the boundary integral equation formulation. When second order elements are used, they show good agreements each other. The eigenvalue problems of a simply supported beam and a partially welded plated are taken next to check the accuracy of the sensitivity formula for simple eigenvalues. For purpose of illustration of convergence, the finite element model of the simply supported beam has been refined. With the partially welded plate problem, it is shown that the sensitivities are insensitive to the size of the subdomain where none zero design velocity is assigned. It is also checked with the simply supported plate that repeated eigenvalues can be well treated and a different choice of the design velocity distribution has no influence on the sensitivity results as should not. All of the sensitivities calculated by the formula are compared with those by finite differences obtaining very good agreements.
Optimum location of supports for vibrating structures is considered on the framework of boundary condition optimization. The fundamental natural frequency is to be maximized for the simply supported beam and the plate by using a gradient based optimization method. For the simply supported plate problem, it is shown that crossing of eigenvalues can occur during the optimization process. The corresponding optimum support locations are exactly calculated by imposing the condition of existence of an eigenvalue multiplicity. The results show that there are infinite number of solutions as an interval.
Finally, the formulation is applied to calculate energy release rate with respect to crack extension. An explicit expression for the energy change due to crack extension is derived based on the sensitivity analysis with changing boundary conditions using the adjoint variable method. Rectangular plates under uniform tension are taken as numerical examples to calculate the energy release rate for various crack sizes and location. And interface crack problem is treated also to assess the capability of the method in the case of mixed mode. The results agree very well with the analytic solutions and those obtained by similar formulas obtained earlier based on boundary integral equations.