Potentially implementing a new solver

We implemented six options (two direct and two iterative solvers) for the solution of the total derivative as well as three options for solving the finite element problem. Please see subsection 6 of Steps for solving a topology optimization problem in ATOMiCS for details.

If the user needs to change to a customized solver for solving the total derivative, they need to modify the solve_linear() function in atomics/atomics/states_comp.py The user can refer to OpenMDAO documentation on implicit component for details. If mode is fwd (direct method), the right-hand side vector is d_residuals and the solution vector is d_outputs. If mode is rev (adjoint method), the right-hand side vector is d_outputs and the solution vector is d_residuals.

If the user wants to change to a customized solver for solving the finite element problem, they need to modify the solve_nonlinear() function in atomics/atomics/states_comp.py The user can refer to OpenMDAO documentation on implicit component for details. The options for the solver that we recommend are other FEniCS solvers or PETSc solvers.