A general method for shape design sensitivity analysis as applied to uncoupled steady-state two-dimensional and axisymmetric thermoelasticity problem is developed using the material derivative concept and the adjoint variable method. The method for deriving sensitivity formula is based on the standard direct thermal and elastic Boundary Integral Equation(BIE) formulation. The sensitivity of a general functional is considered which is composed of thermal and mechanical quantities such as temperature, heat flux, displacement, stress and surface traction. Two adjoint systems are introduced, namely adjoint thermal system and adjoint mechanical system. They take the form of an indirect BIE but can be solved by means of direct BIE which is used to solve the primary problems. It is seen that the results of the adjoint mechanical system are input to the adjoint thermal system through the thermoelastic constant and volumetric strain. The BIE's of the adjoint systems generally include domain integrals, thus diminishing the advantages of Boundary Element Method(BEM). But the domain integrals can be transformed into boundary integrals if the functional under consideration is composed of only boundary integrals. The developed method is then applied to obtain an explicit formula for a representative stress constraint imposed over a small part of the boundary.
Three example problems are considered to illustrate numerical implementations for checking accuracy of the design sensitivity formula of the stress constraint functional. The sensitivity calculated by the formula is compared with that by finite differences and fairly accurate results are obtained. The results of a fillet under thermal loading with different representations of design boundary are similar to those shown in the elasticity problem, indicating that smoother representation is preferable to a less smoother representation. It is demonstrated through a hole problem that the overall accuracy is improved by using refined mesh on the design boundary. Furthermore, a turbine disc with mechanical and thermal loading is treated as an application of the axisymmetric formulation.
Optimum shapes are obtained for the example problems by application of the sensitivity formula to an iterative optimization algorithm. The results have been checked favourable by drawing the effective stress distributions, which show the constrained boundaries are fully stressed. It is demonstrated through the fillet design problem that optimum shape under stress constraints varies with prescribed thermal boundary conditions. For the turbine disc problem, the optimum disc profile under the thermomechanical loading with cooling by heat convection on the disc face is shown to require thicker flange than that under the mechanical loading only. The results indicate that the shape optimization based on thermoelasticity can be crucial when thermal field is sensitive to boundary shape.
Only two-dimensional and axisymmetric problems are dealt with here, but formula derivation and numerical implementations for three-dimensional problems can be done in the exactly same line.
