 peteroznewman
Subscriber
I was thinking that since stiffness matrix itself depends on the response of the nodal displacements
Stiffness matrix for a single solve does not depend on the response of the nodal displacements. Stiffness matrix [K] is assembled from the elements that connect the nodes in their current location. Nodal deformation {u} are the unknowns, applied forces {F} are known. The matrix equation [K]{u}={F} is solved for the unknown nodal deformations.
But apparently the LU decomposition can also work on a non-linear set of equations where the stiffness matrix itself is a function of the nodal displacements.
The stiffness matrix for a single solve is not a function of nodal displacements. Solving [K]{u}={F} can be done using the Sparse direct method or PCG iterative method.
In a nonlinear problem, on each iteration, the matrix equations [K]{u}={F} are linear in that iteration. It is only after the deformed nodal values from that one iteration are added to the current nodal locations to get the new nodal locations that items that were held constant for that one iteration can pick up new values for the next iteration. That is where the nonlinearity occurs. The stiffness matrix can change between iterations because of a material nonlinearity. Contact nonlinearity can change the {F} vector between iterations.
And also, as you have mentioned that the Iterative solver requires only one iteration to converge at t = 1 sec i.e. the final time step for a linear analysis.
I did not say that the PCG iterative solver requires only one internal iteration to converge. What I said was that in a linear analysis, the matrix equation [K]{u}={F} is solved only once, using either the PCG iterative or direct solver.
Again, there is confusion between convergence of a single PCG iterative solver solving [K]{u}={F} one time, and convergence of a nonlinear model that requires multiple solves of the matrix equation [K]{u}={F}.
The PCG iterative solver might take tens or hundreds of internal iterations to solve [K]{u}={F} one time. It is possible that the internal PCG iteration fails to converge. That is an internal convergence for the PCG solver doing a single matrix solve and has nothing to do with the nonlinear model that does many matrix solves and has a completely separate external convergence criteria of results between solves.
In a linear analysis, the matrix equation [K]{u}={F} is solved only once. If the matrix is singular, there is no solution and both the direct and PCG iterative solvers will fail. A common reason for this is insufficient constraints to prevent rigid body motion.
In a nonlinear analysis the matrix equation [K]{u}={F} is solved over and over, updating the nodal coordinates, which affects the stiffness matrix and the force vector. When the updated system of equations is available, the equilibrium of the forces in the system of equations can be checked. It the equilibrium is sufficiently close to zero, that iteration is called converged. If the equilibrium of forces is not sufficiently close to zero, then another solution is done, which is the next iteration. 