Sensitivity represents the effect of a parameter of a system on the system response. Sensitivity analysis helps us understand the physics underlying the system response, is necessary for gradient-based optimization, and is used for uncertainty quantification and reliability analysis. Our focus is on computational sensitivity analysis of systems modeled using PDEs and solved using FEM. In particular, we are interested in the sensitivity of the system response to shape parameters which represent either the domain boundary or interface in a multi-material, multi-fluid, or multi-phase system.

Computational shape sensitivity analysis, in its typical discrete form, inherently couples sensitivity analysis with mesh generation as any change in the boundary or interface leads to change in the mesh or discretization. Discrete shape sensitivity analysis requires mesh sensitivity and thus the calculation of mesh Jacobian sensitivity. In Continuum Shape Sensitivity Analysis (CSSA), instead of taking the sensitivity of discretized equations, we take the sensitivity of the PDE and then we discretize the sensitivity-PDE. CSSA is conducted using the analysis mesh. We can either take the local derivative or the material derivative of the PDEs and correspondingly write sensitivity equations in terms of the local or material sensitivities. In the local form of the CSSA, mesh Jacobian sensitivity is not needed but spatial gradients of the analysis solution are needed at the boundary or interface.

Consider FEM-based solution of the incompressible Navier-Stokes equations at low Reynolds number. We use Q2-Q1 Taylor-Hood elements within the analysis tool IFISS: Incompressible Flow & Iterative Solver Software from Silvester, Elman, and Ramage.  Given below is an analysis verification solution for the test case from Schafer and Turek. The analysis shows a second order convergence for the pressure.

The linearized sensitivity PDE is driven by a non-homogenous Dirichlet boundary condition on the boundaries affected by the shape change parameter. The non-homogenous forcing is dependent on the spatial gradient of the velocity at the boundary obtained from the analysis solution. The analysis shows a fourth order convergence for the velocity and a third order convergence for the spatial gradient of the velocity.

Solving the sensitivity PDEs with the non-homogenous, Dirichlet boundary conditions gives us the local shape sensitivity. We get second order accuracy for the pressure sensitivity and third order accuracy for the velocity sensitivity.