CUMIN: Curved Meshes In Numerical simulations

Author

Charles Pierre

CUMIN is a finite element library including

• finite elements of high order,
• curved geometries: meshes of high geometrical order,
• solvers for diffusion and linear elasticity PDEs,
• coupling with linear system solvers and eigenvalue problem solvers.

Language: modern Fortran

• API documentation:

CUMIN also includes the package CHORAL: Cardiac electrophysiology High Order Algorithms:

• solvers for parabolic semilinear equations coupled to ODE systems
• high order stabilised time stepping methods.

SOME ILLUSTRATIONS

Poisson problem on a sphere.

\( ~~~~~~~~~~-~~~ {\large \Delta_B u + u= f \qquad {\rm on} \quad \Omega}\) Quadratic mesh of the unit sphere \(\Omega\) with 47 curved triangles, \( P^3 \) finite element, \( \Delta_B \) the Laplace Beltrami operator A posteriori error \( \| u - u_h^l \|_{{\rm L}^2(\Omega)} = 0.02 \), using a lift \(u_h^l\) of the numerical solution \(u_h\) on the sphere.

Poisson problem on a disk with 2nd order boundary conditions.

\( \quad\quad\quad\quad {\large \displaystyle{ \begin{array}{lcl} -\Delta u = f \quad & {\rm on}\quad & \Omega \\ -\Delta_B u + \nabla u\cdot n + u= g \quad & {\rm on}\quad & \partial\Omega \end{array} } }\) Quadratic mesh of the unit disk \(\Omega\) with 15 curved triangles, \( P^3 \) finite element, A posteriori error \( \| u - u_h^l \|_{{\rm L}^2(\Omega)} = 4.1E-3 \), using a lift \(u_h^l\) of the numerical solution \(u_h\) on the disk.

Laplace eignefunction on a torus.

\( ~~~~~~~~~~-~~~ {\large \Delta_B u = \lambda u \qquad {\rm on} \quad \Omega}\) Quadratic mesh of a torus \(\Omega\) \( P^3 \) finite element,

Spiral wave with the monodomain model (arising in cardiac electrophysiology).

1 INSTALLATION

• Get the sources
• Installation requires make and cmake

1. Move to your project directory and configure with

   cd your_cumin_dir && mkdir build && cd build
   cmake ..

2. Build with

   make all 

3. Test with

   ctest 

4. Defince your installation directory and install with

   cmake .. -DCMAKE_INSTALL_PREFIX=your_install_dir
   make install

1.1 Configuration variables

Configuration variables are set to default values, they can be modified wirh

ccmake ..

Variable	Desscription	Default
DEBUG	Level of debug	0
REAL_PRECISION	Real precision	8
INTEG_TESTS	Run integration tests	OFF
INTEG_TESTS_3D	Run 3D integration tests	OFF
COMPIL_APPS	Compile applications	OFF
USER_PREFIX_PATH	User path to dependencies	${HOME}

1.2 Dependances

Dependance
OpenMP	Shared memory parallelism
Blas/lapack
Arpack	Eigenvalue problem solver
SCOTCH	Graph reordering
MUMPS	Direct linear system solver
GMSH	Meshing and visualisation
MMG	Remeshining

• Cumin dependances are searched when configuring, if not found they are disabled.
  Without any dependencies, you still can install, run the unit tests (tutorial examples and integration tests require GMSH).
• You can turn-off any dependance using ccmake ...
• Some dependancies are not available when REAL_PRECISION is not set to 8.

• If not found, you may update your installation general directory

  cmake .. -DUSER_PREFIX_PATH=your_dependancies_install_dir

  which can also be set using ccmake.

• More specifically, each dependance has its own installation directory variable, which can be set with (fer e.g. MUMPS)

  cmake .. -DDIR_MUMPS=your_MUMPS_install_dir

  which procedure can also be done using ccmake in advanced mode.

2 TUTORIAL EXAMPLES

• Tutorial examples are provided in the directory _/examples_ and _/examples/choral_.

• Try e.g.

  cd build
  make poisson_2d_Neumann_0_expl 
  ./examples/poisson_2d_Neumann_0_expl

  that compiles and executes the program poisson_2d_Neumann_0_exp.F90 which solves a Poisson problem and visualises its solution.

• Tutorial examples are documented in the API documentation .

3 UTILISATION

• To use the cumin library, your fortran codes should look like this:

  program my_program
    use cumin
    ...

• Linking at compilation is handled with cmake as follows.
  1. In your own project directory, copy the following lines to your CMakeLists.txt:

     set(CUMIN_INSTALL_DIR "your_cumin_install_dir" CACHE STRING "")
        if(EXISTS "${CUMIN_INSTALL_DIR}/Findcumin.cmake")
          list(APPEND CMAKE_MODULE_PATH ${CUMIN_INSTALL_DIR}) 
          find_package(cumin)
        else()
          message(FATAL_ERROR "Not found: Findcumin.cmake, reset 'CUMIN_INSTALL_DIR'")
        endif()

  2. create a target for your code and link it with cumin:

     add_executable(target_name program_name.F90)
     target_link_libraries(target_name cumin)

  3. compilation is then done as usual with

     make target_name

4 INSTALLATION WITH GUIX

You can avoid to install the dependances using the guix environment. Guix moreover guarantees fully reproducable results.
For this you first need to: , then:

• update your guix deposit,

  guix pull

• open the following guix shell for cumin installation,

  guix shell --pure   arpack-ng mumps openmpi scotch32 mmg mmg:lib gmsh make cmake gcc-toolchain  gfortran-toolchain coreutils sed doxygen -- /bin/bash --norc

• configure, build and install with,

  cd your_cumin_dir && mkdir build && cd build
  cmake ..
  make all install