Our objective is to create robust and scalable Variational Inequality (VI) solvers to enable the consistent and efficient modeling of transitional phenomena.
Motivation: Predictive simulations containing phase changes, free boundaries, and hybrid discrete-continuum behavior are common to many applications. These simulations typically consist of a set of partial differential equations with “switches” to model such transition phenomena. Current practice is to smooth the “switches,” resulting in exceedingly short time steps or an ad-hoc active set choice, which leads to physical and numerical inconsistencies. A unitary treatment of these transition phenomena can be accomplished by means of differential variational inequalities (DVIs), roughly speaking, evolution equations whose weak variational form uses test functions over nontrivial convex sets. In turn, the problem can be resolved by time stepping and discretization, which transforms it into finite dimensional variational inequalities (VIs) that can be solved by optimization and complementarity techniques. An immediate benefit of this approach is that time steps can now be orders of magnitude larger, which in turn results in significant computational savings and enables higher fidelity simulations.