____                 __
         / __ \___  ____ ___  / /_  ____  _____
        / /_/ / _ \/ __ `__ \/ __ \/ __ \/ ___/
       / _, _/  __/ / / / / / / / / /_/ (__  )
      /_/ |_|\___/_/ /_/ /_/_/ /_/\____/____/

              High-order Remap Miniapp

Purpose

Remhos (REMap High-Order Solver) is a miniapp that solves the pure advection equations that are used to perform monotonic and conservative discontinuous field interpolation (remap) as part of the Eulerian phase in Arbitrary Lagrangian Eulerian (ALE) simulations.

Remhos combines discretization methods described in the following articles:

R. Anderson, V. Dobrev, Tz. Kolev and R. Rieben
Monotonicity in high-order curvilinear finite element arbitrary Lagrangian-Eulerian remap
International Journal for Numerical Methods in Fluids 77(5), 2015, pp. 249-273.

R. Anderson, V. Dobrev, Tz. Kolev, D. Kuzmin, M. Quezada de Luna, R. Rieben and V. Tomov
High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation
Journal of Computational Physics 334, 2017, pp. 102-124.

R. Anderson, V. Dobrev, Tz. Kolev, R. Rieben and V. Tomov
High-order multi-material ALE hydrodynamics
SIAM Journal on Scientific Computing 40(1), 2018, pp. B32-B58.

H. Hajduk, D. Kuzmin, Tz. Kolev and R. Abgrall
Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations
Computer Methods in Applied Mechanics and Engineering 359, 2020.

H. Hajduk, D. Kuzmin, Tz. Kolev, V. Tomov, I. Tomas and J. Shadid
Matrix-free subcell residual distribution for Bernstein finite elements: Monolithic limiting
Computers and Fluids 200, 2020.

The Remhos miniapp is part of the CEED software suite, a collection of software benchmarks, miniapps, libraries and APIs for efficient exascale discretizations based on high-order finite element and spectral element methods. See http://github.com/ceed for more information and source code availability.

The CEED research is supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of two U.S. Department of Energy organizations (Office of Science and the National Nuclear Security Administration) responsible for the planning and preparation of a capable exascale ecosystem, including software, applications, hardware, advanced system engineering and early testbed platforms, in support of the nation’s exascale computing imperative.

Characteristics

The problem that Remhos is solving is formulated as a time-dependent system of ordinary differential equations (ODEs) for the unknown coefficients of a high-order finite element (FE) function. The left-hand side of this system is controlled by a mass matrix, while the right-hand side is constructed from a advection matrix.

Remhos supports two execution modes, namely, transport and remap, which result in slightly different algebraic operators. The main difference between the two modes is that in the case of remap, the mass and advection matrices change in time, while they are constant for the transport case.

Remhos supports two options for deriving and solving the ODE system, namely the full assembly and the partial assembly methods. Partial assembly is the main algorithm of interest for high orders. For low orders (e.g. 2nd order in 3D), both algorithms are of interest.

The full assembly option relies on constructing and utilizing global mass and advection matrices stored in compressed sparse row (CSR) format. In contrast, the partial assembly option defines only the local action of those matrices, which is then used to perform all necessary operations. As the local action is defined by utilizing the tensor structure of the finite element spaces, the amount of data storage, memory transfers, and FLOPs are lower (especially for higher orders).

Other computational motives in Remhos include the following:

• Support for unstructured meshes, in 2D and 3D, with quadrilateral and hexahedral elements. Serial and parallel mesh refinement options can be set via a command-line flag.
• Explicit time-stepping loop with a variety of time integrator options. Remhos supports Runge-Kutta ODE solvers of orders 1, 2, 3, 4 and 6.
• Discontinuous high-order finite element discretization spaces of runtime-specified order.
• Moving (high-order) meshes.
• Mass operator that is local per each zone. It is inverted by iterative or exact methods at each time step. This operator is constant in time (transport mode) or changing in time (remap mode). Options for full or partial assembly.
• Advection operator that couples neighboring zones. It is applied once at each time step. This operator is constant in time (transport mode) or changing in time (remap mode). Options for full or partial assembly.
• Domain-decomposed MPI parallelism.
• Optional in-situ visualization with GLVis and data output for visualization and data analysis with VisIt.

Code Structure

• The file remhos.cpp contains the main driver with the time integration loop.
• The files remhos_ho.hpp and remhos_ho.cpp contain all methods that are used to obtain high-order (but not bounds-preserving) solutions of the problem.
• The files remhos_lo.hpp and remhos_lo.cpp contain all methods that are used to obtain low-order (but bounds-preserving) solutions of the problem.
• The files remhos_fct.hpp and remhos_fct.cpp contain all methods that combine already computed high-order and low-order solutions, thus obtaining a high-order and bounds-preserving solutions of the problem.
• The files remhos_tools.hpp and remhos_tools.cpp contain helper functions utilized by the main classes of the miniapp.

Building

Remhos has the following external dependencies:

• hypre, used for parallel linear algebra, we recommend version 2.10.0b
  https://computation.llnl.gov/casc/hypre/software.html

• METIS, used for parallel domain decomposition (optional), we recommend version 4.0.3
  http://glaros.dtc.umn.edu/gkhome/metis/metis/download

• MFEM, used for (high-order) finite element discretization, its GitHub master branch
  https://github.com/mfem/mfem

To build the miniapp, first download hypre and METIS from the links above and put everything on the same level as the Remhos directory: sh ~> ls Remhos/ hypre-2.10.0b.tar.gz metis-4.0.tar.gz

Build hypre: sh ~> tar -zxvf hypre-2.10.0b.tar.gz ~> cd hypre-2.10.0b/src/ ~/hypre-2.10.0b/src> ./configure --disable-fortran ~/hypre-2.10.0b/src> make -j ~/hypre-2.10.0b/src> cd ../.. For large runs (problem size above 2 billion unknowns), add the --enable-bigint option to the above configure line.

Build METIS: sh ~> tar -zxvf metis-4.0.3.tar.gz ~> cd metis-4.0.3 ~/metis-4.0.3> make ~/metis-4.0.3> cd .. ~> ln -s metis-4.0.3 metis-4.0

Clone and build the parallel version of MFEM: sh ~> git clone https://github.com/mfem/mfem.git ./mfem ~> cd mfem/ ~/mfem> make parallel -j ~/mfem> cd .. The above uses the master branch of MFEM. See the MFEM building page for additional details.

(Optional) Clone and build GLVis: sh ~> git clone https://github.com/GLVis/glvis.git ./glvis ~> cd glvis/ ~/glvis> make ~/glvis> cd .. The easiest way to visualize Remhos results is to have GLVis running in a separate terminal. Then the -vis option in Remhos will stream results directly to the GLVis socket.

Build Remhos sh ~> cd Remhos/ ~/Remhos> make See make help for additional options.

Running

Sample remap examples

Some remap mode sample runs for in 2D and 3D respectively are: sh mpirun -np 8 remhos -m ./data/inline-quad.mesh -p 14 -rs 2 -rp 1 -dt 0.0005 -tf 0.6 -ho 1 -lo 2 -fct 3 mpirun -np 8 remhos -m ./data/cube01_hex.mesh -p 10 -rs 1 -o 2 -dt 0.02 -tf 0.8 -ho 1 -lo 4 -fct 2 This first of the above runs can produce the following plots (notice the -vis option)

Sample transport examples

Some transport mode sample runs for in 2D and 3D respectively are: sh mpirun -np 8 remhos -m ./data/periodic-square.mesh -p 5 -rs 3 -rp 1 -dt 0.00025 -tf 0.8 -ho 1 -lo 4 -fct 3 mpirun -np 8 remhos -m ./data/periodic-cube.mesh -p 0 -rs 1 -o 2 -dt 0.014 -tf 8 -ho 1 -lo 4 -fct 2 This first of the above runs can produce the following plots (notice the -vis option)

Verification of Results

To perform thorough testing, run the script Remhos\autotest\test.sh and compare its output, out_test.dat, to out_baseline.dat. Alternatively, verify the final mass (mass) and maximum value (max) for the runs listed below:

1. mpirun -np 8 remhos -m ./data/periodic-hexagon.mesh -p 0 -rs 2 -dt 0.005 -tf 10 -ho 1 -lo 2 -fct 2
2. mpirun -np 8 remhos -m ./data/periodic-hexagon.mesh -p 0 -rs 2 -dt 0.005 -tf 10 -ho 1 -lo 4 -fct 2
3. mpirun -np 8 remhos -m ./data/disc-nurbs.mesh -p 1 -rs 3 -dt 0.005 -tf 3 -ho 1 -lo 2 -fct 2
4. mpirun -np 8 remhos -m ./data/disc-nurbs.mesh -p 1 -rs 3 -dt 0.005 -tf 3 -ho 1 -lo 4 -fct 2
5. mpirun -np 8 remhos -m ./data/periodic-square.mesh -p 5 -rs 3 -dt 0.005 -tf 0.8 -ho 1 -lo 2 -fct 2
6. mpirun -np 8 remhos -m ./data/periodic-square.mesh -p 5 -rs 3 -dt 0.002 -tf 0.8 -ho 1 -lo 4 -fct 2
7. mpirun -np 8 remhos -m ./data/periodic-cube.mesh -p 0 -rs 1 -o 2 -dt 0.014 -tf 8 -ho 1 -lo 4 -fct 2
8. mpirun -np 8 remhos -m ../mfem/data/ball-nurbs.mesh -p 1 -rs 1 -dt 0.02 -tf 3 -ho 1 -lo 4 -fct 2
9. mpirun -np 8 remhos -m ./data/inline-quad.mesh -p 14 -rs 1 -dt 0.001 -tf 0.75 -ho 1 -lo 4 -fct 2
10. mpirun -np 8 remhos -m ./data/inline-quad.mesh -p 14 -rs 3 -dt 0.005 -tf 0.75 -ho 1 -lo 5 -fct 4 -ps -s 13
11. mpirun -np 8 remhos -m ./data/cube01_hex.mesh -p 10 -rs 1 -o 2 -dt 0.02 -tf 0.8 -ho 1 -lo 4 -fct 2
12. mpirun -np 8 remhos -m ./data/inline-quad.mesh -p 7 -rs 3 -o 1 -dt 0.01 -tf 20 -mono 1 -si 2
13. mpirun -np 8 remhos -m ./data/inline-quad.mesh -p 6 -rs 2 -o 1 -dt 0.01 -tf 20 -mono 1 -si 1

| run | mass | max | | ----- | ------ | ----- | | 1. | 0.3888354875 | 0.9333315791 | | 2. | 0.3888354875 | 0.9446390369 | | 3. | 3.5982222 | 0.9995717563 | | 4. | 3.5982222 | 0.9995717563 | | 5. | 0.1623263888 | 0.7676354393 | | 6. | 0.1623263888 | 0.7480960657 | | 7. | 0.9607429525 | 0.7678305756 | | 8. | 0.8087104604 | 0.9999889315 | | 9. | 0.08479546709| 0.8156091428 | | 10. | 0.09317738757| 0.9994170644 | | 11. | 0.1197294512 | 0.9990312449 | | 12. | 0.1570667907 | 0.9987771164 | | 13. | 0.3182739921 | 1 |

An implementation is considered valid if the computed values are all within round-off distance from the above reference values.

Performance Timing and FOM

To appear soon.

Versions

To appear soon.

Contact

You can reach the Remhos team by emailing remhos@llnl.gov or by leaving a comment in the issue tracker.

Copyright

The following copyright applies to each file in the CEED software suite, unless otherwise stated in the file:

Copyright (c) 2017, Lawrence Livermore National Security, LLC. Produced at the Lawrence Livermore National Laboratory. LLNL-CODE-734707. All Rights reserved.

See files LICENSE and NOTICE for details.