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Finite Element Exterior Calculus With FEniCS
Researchers in higher mathematics have long used supercomputers to handle the huge numbers of calculations necessary for their work. Professor Doug Arnold (Mathematics; MSI Fellow) is using Itasca to solve problems with various aspects of finite-element methods for partial differential equations. Professor Arnold has developed the finite element exterior calculus (FEEC), and FEniCS (fenicsproject.org), which is one of the most impressive open-source projects in numerical partial differential equations, has been designed so that it is well suited to implementing and testing FEEC-based algorithms. FEniCS, which allows one to implement finite elements from a high-level mathematical perspective, includes a Python/C++ problem solving environment which provides access to advanced solver systems like PETSc, uBLAS, and Trilinos, which in turn use MPI, and so are in a position to make efficient use of Itasca.
Professor Arnold and his group are studying such areas as elastrodynamics and dynamic viscoelastic computations and harmonic forms, among others. The image above shows Stokes flow through a double pipe (in Stokes flow, fluid velocities are very slow, viscosities are very large, or length-scales of the flow are very small). The flow enters through the smaller pipe and exits the larger one. Since starting work on Itasca, the Arnold group has been able to implement FEniCS and tune its performance; they have been getting good results for a number of problems.