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We propose In-Timestep Remeshing, a fully coupled, adaptive meshing algorithm for contacting elastodynamics where remeshing steps are tightly integrated, implicitly, within the timestep solve. Our algorithm refines and coarsens the domain automatically by measuring physical energy changes within each ongoing timestep solve. This provides consistent, degree-of-freedom-efficient, productive remeshing that, by construction, is physics-aware and so avoids the errors, over-refinements, artifacts, per-example hand-tuning, and instabilities commonly encountered when remeshing with timestepping methods. Our in-timestep computation then ensures that each simulation step's output is both a converged stable solution on the updated mesh and a temporally consistent trajectory with respect to the model and solution of the last timestep. At the same time, the output is guaranteed safe (intersection- and inversion-free) across all operations. We demonstrate applications across a wide range of extreme stress tests with challenging contacts, sharp geometries, extreme compressions, large timesteps, and wide material stiffness ranges - all scenarios well-appreciated to challenge existing remeshing methods. 

Interval computation is widely used in Computer Aided Design to certify computations that use floating point operations to avoid pitfalls related to rounding error introduced by inaccurate operations. Despite its popularity and practical benefits, support for interval arithmetic is not standardized nor available in mainstream programming languages. We propose the first benchmark for interval computations, coupled with reference solutions computed with exact arithmetic, and compare popular C and C++ libraries over different architectures, operating systems, and compilers. The benchmark allows identifying limitations in existing implementations, and provides a reliable guide on which library to use on each system for different CAD applications. We believe that our benchmark will be useful for developers of future interval libraries, as a way to test the correctness and performance of their algorithms. 

We introduce the firstexactroot parity counter for continuous collision detection (CCD). That is, our algorithm computes the parity (even or odd) of the number of roots of the cubic polynomial arising from a CCD query. We note that the parity is unable to differentiate between zero (no collisions) and the rare case of two roots (collisions).

Our method does not have numerical parameters to tune, has a performance comparable to efficient approximate algorithms, and is exact. We test our approach on a large collection of synthetic tests and real simulations, and we demonstrate that it can be easily integrated into existing simulators.