A general method for shape design sensitivity analysis is developed using a standard boundary integral equation (BIE) form, for self-adjoint elliptic boundary value problems. The performance functional to be considered involves both the domain and boundary integrals. To describe the shape variation, the material derivative concept is utilized, and the shape variation is completely described by including the tangential as well as the normal component of the velocity field. In order to obtain an explicit expression for the sensitivity, an integral identity is first derived utilizing the relation between the direct and indirect BIE. This is then used in defining an adjoint system, from which the variation in state variables can be eliminated. The adjoint system, which takes a form of indirect BIE, can also be solved using the same direct BIE of the original problem, since the two BIE's are formally equivalent. This is an efficient calculational procedure, since it requires only a change of input for prescribed boundary conditions and domain functions.
The developed method is applied and concrete formulas are obtained for potential, plane and axisymmetric elasticity, and plate bending problems, respectively. All the formulas derived are in the form of boundary integrals, and suitable for computational purpose using the boundary element method. More specific individual problems are taken for illustration and their results are compared with those obtained by a different approach which is based on a variational method over the domain.
Numerical design sensitivity calculations are carried out, and optimum shapes are obtained for a seepage, a fillet and an elastic annular ring problem, respectively. In a seepage problem, which is well known for a free boundary value problem, a proper objective functional is defined on the free boundary. Good accuracy for the sensitivity is obtained even with a simple numerical scheme for calculation of the curvature and normal vector of the boundary. In the fillet and the elastic ring problem, stress constraints are imposed over small parts of the boundary, and accuracies of the design sensitivity analysis are studied. Among the several numerical implementations tested, the second order boundary elements with a cubic spline representation of the moving boundary show the best accuracy. A smooth characteristic function is found better than a plateau function for localization of the stress constraint. Optimal shapes are presented, and the results show a kind of fully-stressed state along the optimized boundary.
The developed method is extended to eigenvalue shape design sensitivity analysis by formulating a proper BIE from the formal eigenvalue problem, and explicit formulas for the eigenvalue sensitivity are obtained for membrane and plate vibration problems.
In summary, the presented method has opened a new unified approach using the BIE to the shape design sensitivity problems, and is shown to be a valuable tool for calculations of the sensitivity for the functionals especially with boundary integrals often encountered in practical problems such as free boundary determination or stress concentration minimization.
