  We present the AMPS algorithm, a finite element solution method that combines
principal submatrix updates and Schur complement techniques, well-suited for
interactive simulations of deformation and cutting of finite element meshes.
Our approach features real-time solutions to the updated stiffness matrix
systems to account for interactive changes in mesh connectivity and boundary
conditions. Updates are accomplished by an augmented matrix formulation of the
stiffness equations to maintain its consistency with changes to the underlying
model without refactorization at each timestep. As changes accumulate over
multiple simulation timesteps, the augmented solution algorithm enables tens or
hundreds of updates per second. Acceleration schemes that exploit sparsity,
memoization and parallelization lead to the updates being computed in
real-time. The complexity analysis and experimental results for this method
demonstrate that it scales linearly with the problem size. Results for cutting
and deformation of 3D elastic models are reported for meshes with node counts
up to 50,000, and involve models of astigmatism surgery and the brain.
