In this thesis, a method of shape design sensitivity analysis of boundary stress constraints of linear elastic systems is presented.
As a theoretical basis, a new unified variational principle, based on a mixed functional obtained by linear combination of the total potential energy functional, the modified Helliger-Reissner functional and the Hu-Washizu functional with two parameters, is proposed and mathematical characteristics of the variational equation of the principle is thoroughly investigated for analysis of the boundary value problems in linear elasticity. It is first proved that the Euler-Lagrange equations of the variational equation is identical with the governing equations for the given problem. Then existence of the unique solution of the variational equation is systematically proved by showing that the energy bilinear form is weakly coercive.
As an application of the proposed variational principle, it is further shown that the stress/strain smoothing mthod already widely adopted for post-processing of analysis results of the displacement based FEM can be obtained as a form of the mixed FEM based on the variational equation.
The constraints of point stress on the boundary and boundary integral stress are transformed into those of unsmoothed stress functionals defined on the domains of the finite elements including the point and the boundary using the local stress smoothing method of which theoretical basis is provided by the new unified variational principle. Then the material derivative idea in continuum mechanics and an adjoint variable technique are employed for shape design sensitivity analysis of the constraints.
Validity of the shape design sensitivity analysis method is tested through three numerical examples and it is shown that the method is stable with accurate shape design sensitivity results and is readily applicable to practical structural shape optimization of linear elastic systems.
For each shape optimization problem, stress constraints are formulated as one of four type functions of local nodal stress, global nodal stress, boundary integral stress and domain integral stress, and optimal solutions are obtained respectively. Through the numerical results, it is demonstrated that the constraint formulation of local nodal stress function is most effective in structural shape optimizations.
