A general procedure to perform shape design sensitivity analysis for two-dimensional transient thermal diffusion problems is developed using boundary integral equation formulation. The procedure is then applied to obtain optimum shapes of industrial problems. The material derivative concept to describe shape variation and the adjoint variable method are used. Ionescu-Cazimir's reciprocal theorem is introduced to define the adjoint systems systematically.
A shape design sensitivity formula for a general functional defined in domain and boundary for a fixed duration of time is derived by the method. The formula is explicit to shape variation and consists of primal solution, adjoint solutions and a design velocity field. Both normal and tangential component of the design velocity are considered in the description of shape variation. It is shown the boundary conditions of the adjoint problem are dependent on the primal solution with reverse sequence in time. A sensitivity formula for a periodic diffusion problem is also derived. In this problem the temperature is decomposed into a steady state component and a perturbation component. The adjoint variable method is used by introducing integral identities for each component.
A boundary element method with a time-dependent fundamental solution is implemented for the primal and the adjoint solution. It is seen that system matrices of the two problems are identical and only load vectors need be recalculated.
The sensitivity formula derived is checked against the analytic solution in the case of a rod example and by numerical comparisons with the finite differencing for a rectangular block under a thermal shock, a thermal fin problem, a simple plunger model and a modified plunger model. The last example is assumed to be under a periodic circumstance. It is shown by Richardson's extrapolation and a graphical way that two numerical results obtained using different time steps converge to the same solution.
Three optimal design problems are formulated and solved to obtain realistic shapes. The first example is a thermal fin problem. The optimal fin shape maximizing the amount of heat flux under a constant area constraint is obtained through an iterative optimization algorithm. The second one is a simple plunger model. An optimum of a cooling boundary minimizing the variation of temperature along the cavity surface is calculated. The shape optimal design problem of a modified plunger model which operates periodically is solved as the final example. As the results several plunger shapes of cooling boundary under the superposition assumption of steady state and perturbation components are obtained with two objectives under a heat flux constraint. One objective is the square sum of temperature variation in the tangential direction along the cavity surface and the other is a normalized difference between a prescribed function and the temperature response of the cavity surface.
The results show that the proposed method is capable of optimizing complex engineering models. To be more realistic, however, it is necessary to include not only the three dimensional nature of a thermal problem but also nonlinearities such as from thermal radiation and interface conditions.