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**Our objective is to create robust and scalable Variational Inequality (VI) solvers to enable the consistent and efficient modeling of transitional phenomena. **

**Motivation: **PredictiveÂ simulations containing phase changes, free boundaries, and hybridÂ discrete-continuum behavior are common to many applications. TheseÂ simulations typically consist of a set of partial differentialÂ equations with âswitchesâ to model such transition phenomena. CurrentÂ practice is to smooth the âswitches,â resulting in exceedingly shortÂ time steps or an ad-hoc active set choice, which leads to physical andÂ numerical inconsistencies. Â A unitary treatment of these transitionÂ phenomena can be accomplished by means of differential variationalÂ inequalities (DVIs), roughly speaking, evolution equations whose weakÂ variational form uses test functions over nontrivial convex sets. InÂ turn, the problem can be resolved by time stepping and discretization,Â which transforms it into finite dimensional variational inequalities (VIs)Â that can be solved by optimization and complementarity techniques. AnÂ immediate benefit of this approach is that time steps can now beÂ orders of magnitude larger, which in turn results in significantÂ computational savings and enables higher fidelity simulations.