In multiphysics simulations, the computational domains for each physical component are frequently meshed independently, which in turn requires geometric algorithms to correlate the surface meshes at the common interface between each pair of interacting domains to exchange boundary conditions. These surface meshes in general differ both geometrically and combinatorially, and are also partitioned differently for parallel computation. To correlate such interface meshes, we have developed novel algorithms to constructs a common refinement of two triangular or quadrilateral meshes modeling the same surface, that is, a finer mesh whose polygons subdivide the polygons of the input surface meshes.[11] To resolve geometric mismatch, the algorithm defines a conforming homeomorphism and utilizes locality and duality to achieve optimal linear time complexity. Due to the nonlinear nature of the problem, our algorithm uses floating-point arithmetic, but nevertheless achieves provable robustness by identifying a set of consistency rules and an intersection principle to resolve any inconsistencies due to numerical errors.
After constructing the common refinement, we must transfer data between the nonmatching meshes in a numerically accurate and physically conservative manner. Traditional methods, including pointwise interpolation and some weighted residual methods, can achieve either accuracy or conservation, but none could achieve both simultaneously. Leveraging the common refinement, we developed more advanced formulations and optimal discretizations that minimize errors in a certain norm while achieving strict conservation, yielding significant advantages over traditional methods, especially for repeated transfers in multiphysics simulations.[12] For parallel runs, the common refinement also contains the correlation of elements across partitions of different meshes, and hence provides the communication structure needed for inter-module, inter-process data exchange.