Sensitivity analysis is the study of how a response (i.e. goal) of a physical system behaves when it is subjected to infinitesimal perturbations in the input parameters (i.e. response function gradient w.r.t. input parameters). This can also be recognized as the uncertainty of a response in a physical system with respect to its input parameters. Sensitivity analysis is used in many fields of research such as optimization, uncertainty quantification, robustness estimation. The primal problems of our interest has branches in fluid dynamics, solid mechanis, coupled problems which has large number of degrees of freedom. All of these primal problems involve solving for pre-defined set of partial differential equations. Hence, the sensitivity problem can be thought of as constraint problem where the constraints are the partial differential equations of the primal problem. These sensitivity problems can be solved mainly via two different approaches. One is using the forward/direct approach, another is to use the adjoint approach as shown in the following figure.
The use of adjoint approach has the advantage that, the sensitivities with respect to all the degrees of freedom can be obtained with one adjoint solution. Hence the computational complexity does not scale with the number of degrees of freedom in the original primal problem. Following figure shows a practical example of sensitivities, where the sensitivities are computed on a bridge Millau bridge protection barriers to minimize the wind loading on the vehicles.
Solving an optimization problem requires to find the best design which gives the best response value while adhering to all the given constraints. There are mainly two approaches to solve an optimization problem. Since these optimization problems also have at least one primal problem with a partial differential equations, then number of degrees of freedom in these optimization problems are also higher. Hence, the adjoint approach is used in these problems as well.
Optimization of transient primal problems are also one of the interests. When dealing with the transient sensitivity problems, one has to properly account for the butterfly effect in the primal problem in the primal problem. This is more prominent in the fluid flow problems when the Reynolds number is high. Hence, proper care should be taken when adjoint sensitivity analysis is used to obtain sensitivities of a transient chaotic primal problem. Following is a transient optimization problem where best shape to reduce the lift force vortex shredding frequency is obtained starting with a cylindrical geometry.
AMR methods can be useful in this case where mesh is adapted based on a given criteria. Two of most common AMR methods are based on flow field error minimization and goal oriented error minimization. In the case of the goal oriented AMR method, it is important to obtain the sensitivity of a goal with respect to discretization points in domain. As stated above, it is of our interest to obtain the sensitivites from the adjoint approach since our primal problems are driven by partial differential equations with large number of degrees of freedom. Following example shows an example of a mesh developed with AMR for a transient flow problem where the goal was to increase the accuracy of the drag computation over the cylindrical solver.