The discretization of the Euler equations of gas dynamics ("compressible hydrodynamics") in a moving material frame is at the heart of many multi-physics simulation algorithms. The Arbitrary Lagrangian-Eulerian (ALE) framework is frequently applied in these settings in the form of a Lagrange phase, where the hydrodynamics equations are solved on a moving mesh, followed by a three-part "advection phase" involving mesh optimization, field remap and multi-material zone treatment.
This talk presents a general Lagrangian framework [1] for discretization of compressible shock hydrodynamics using high-order finite elements. The novelty of our approach is in the use of high-order polynomial spaces to define both the mapping and the reference basis functions. This leads to improved robustness and symmetry preservation properties, better representation of the mesh curvature that naturally develops with the material motion, significant reduction in mesh imprinting, and high-order convergence for smooth problems. We also discuss ongoing work on the application of the curvilinear technology to the “advection phase” of ALE, including a DG-advection approach for conservative and monotonic high-order finite element interpolation (remap), high-order extensions of classical mesh optimization algorithms, such as harmonic and equipotential smoothing and the use of high-order material indicator function for handling mixed elements in multi-material ALE problems.
In addition to their mathematical benefits, high-order finite element discretizations are a natural fit for future HPC hardware, because their order can be used to tune the performance, by increasing the FLOPs/bytes ratio, or to adjust the algorithm for different hardware. In this direction, we present some of our work on scalable high-order finite element software that combines the modular finite element library MFEM [2], the hypre library of linear solvers [3], and the high-order shock hydrodynamics code BLAST [4]. We explain how the MPI-based version of MFEM uses data structures and kernels from the hypre library to enable scalable finite element assembly in parallel and describe the efficient implementation of high-order force matrices in the MFEM-based BLAST application, where we will also demonstrate the benefits of our approach with respect to strong scaling and GPU acceleration. We also consider general non-conforming high-order adaptive refinement in MFEM with applications to compressible hydrodynamics in BLAST and computational electromagnetic problems.
Finally, we give a brief overview of the related efforts in the co-design Center for Efficient Exascale Discretizations (CEED) in the Exascale Computing Project (ECP) of the DOE [5].
[1] "High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics", V. Dobrev and Tz. Kolev and R. Rieben, SIAM Journal on Scientific Computing, (34) 2012, pp.B606-B641. [2] MFEM: Modular finite element library, http://mfem.org. [3] hypre: Scalable linear solvers library, http://llnl.gov/casc/hypre. [4] BLAST: High-order shock hydrodynamics, http://llnl.gov/casc/blast. [5] Center for Efficient Exascale Discretizations, http://ceed.exascaleproject.org.