Geometry, meshes, and numerical integration for finite element analyses

Contents

• Introduction
  • Documentation
• Installation
  • Arch Linux
    • 1. Default
    • 2. Alternative
  • Debian/Ubuntu Linux
    • 1. Default
    • 2. Alternative
• Examples
  • MSH file format
  • Numerical integration
• Roadmap
• Appendix A - Shapes and local numbering of nodes
  • Lines (Lin)
  • Triangles (Tri)
  • Quadrilaterals (Qua)
  • Tetrahedra (Tet)
  • Hexahedra (Hex)
• Appendix B - Geometry versus space dimensions

Introduction

This crate contains structures and functions for geometry computations, generate meshes, and perform numerical integration for finite element analyses (FEM/FEA).

Documentation

Installation

This crate depends on Russell (Rust Scientific Library) and, therefore, has the same dependencies as russell. Among the combinations described in russell website, the following are recommended here:

1. Default - Use OpenBLAS and SuiteSparse
2. Alternative - Use Intel MKL and locally compiled SuiteSparse and the MUMPS solver

Therefore, the following (re-exported) features are available here:

• intel_mkl - Tells russell to use Intel MKL
• local_sparse - Tells russell that the local linear solvers are available locally

Arch Linux

1. Default

Install some dependencies:

  pacman -Syu blas-openblas python-matplotlib suitesparse

2. Alternative

Check Russell (Rust Scientific Library) out for detailed instructions on how to install the optional dependencies.

Debian/Ubuntu Linux

1. Default

Install some dependencies:

sudo apt-get install -y liblapacke-dev libopenblas-dev libsuitesparse-dev python3-matplotlib

2. Alternative

Check Russell (Rust Scientific Library) out for detailed instructions on how to install the optional dependencies.

Examples

MSH file format

The MSH file format contains three mandatory sections and two optional sections. The MSH file is a plain text file where comments are marked with # and empty lines are allowed. The mandatory sections are header, points, and cells. We use "cells" here to refer to 2D polygons or 3D polyhedra (aka Elements in the Finite Element Method). The header section specifies the space dimension (2 or 3), the number of points, the number of cells, and the optional number of marked edges and faces. The optional sections specify the marked edges and faces. An example of MSH file is shown below:

#
#           8-------------11
#          /.             /|
#     {-5}/ .        {-5}/ |
#        /  .   {-9}    /  |{123}
#       /   .          /   |       id = 1
# 2.0  9-------------10    |       marker = 2
#      |    .         |    |
#      |    4---------|----7*
#      |   /.         |   /|
#      |  / .         |  / |
#      | /  .         | /  |{-4}
#      |/   .         |/   |
# 1.0  5--------------6    |       id = 0
#      |    .         |{-8}|       marker = 1
#      |    0---------|----3  0.0
#      |   /          |   /
#      |  /           |  /
#      | /            | /
#      |/             |/
# 0.0  1*-------------2*  1.0
#     0.0            1.0
#
# header
# ndim npoint ncell nmarked_edge nmarked_face
     3     12     2            4            2

# points
# id marker x y z
   0  0 0.0 0.0 0.0
   1 -1 1.0 0.0 0.0
   2 -1 1.0 1.0 0.0
   3  0 0.0 1.0 0.0
   4  0 0.0 0.0 1.0
   5  0 1.0 0.0 1.0
   6  0 1.0 1.0 1.0
   7 -1 0.0 1.0 1.0
   8  0 0.0 0.0 2.0
   9  0 1.0 0.0 2.0
  10  0 1.0 1.0 2.0
  11  0 0.0 1.0 2.0

# cells
# id marker kind points
   0  1  hex8  0 1 2 3 4 5  6  7
   1  2  hex8  4 5 6 7 8 9 10 11

# marked edges
# marker p1 p2
123 7 11
-5 11 10
-4 7 3
-5 8 9

# marked faces
# marker p1 p2 p3 {p4}
-8 3 2 7 6
-9 8 10 9 11

Numerical integration

use gemlab::integ::Gauss;
use gemlab::mesh::{At, Features, Mesh};
use gemlab::shapes::Scratchpad;
use gemlab::StrError;
use std::collections::HashSet;

fn main() -> Result<(), StrError> {
    // Input the raw mesh data using a text file
    //
    // 1.0  5------,6.------7
    //      | [3],'   `.[4] |
    //      |  ,'       `.  |
    //      |,'           `.|
    // 0.5  3      [2]      4
    //      |`.           .'|
    //      |  `.       .'  |
    //      | [0]`.   .'[1] |
    // 0.0  0------`1'------2
    //     0.0     0.5     1.0
    let path = "./data/meshes/four_tri3_one_qua4.msh";
    let mesh = Mesh::read(path)?;

    // Extract features such boundary edges and faces.
    // Search entities along the boundary of the mesh given coordinates.
    // The `At` enum provides an easy way to define the type of the
    // constraint such as line, plane, circle, etc.
    let feat = Features::new(&mesh, false);
    assert_eq!(feat.search_point_ids(At::Y(0.5), |_| true)?, &[3, 4]);
    assert_eq!(feat.search_edge_keys(At::X(1.0), |_| true)?, &[(2, 4), (4, 7)]);

    // Perform numerical integration to compute
    // the area of cell # 2
    let ndim = 2;
    let cell_2 = &mesh.cells[2];
    let mut pad = Scratchpad::new(ndim, cell_2.kind)?;
    mesh.set_pad(&mut pad, &cell_2.points);
    let gauss = Gauss::new(cell_2.kind);
    let mut area = 0.0;
    for p in 0..gauss.npoint() {
        let iota = gauss.coords(p);
        let weight = gauss.weight(p);
        let det_jac = pad.calc_jacobian(iota)?;
        area += weight * det_jac;
    }
    assert_eq!(area, 0.5);
    Ok(())
}

Roadmap

• Implement read/write mesh functions
• Add tests for the numerical integrations
• Implement triangle and tetrahedron generators
• Implement drawing functions

Appendix A - Shapes and local numbering of nodes

Lines (Lin)

Triangles (Tri)

Quadrilaterals (Qua)

Tetrahedra (Tet)

Hexahedra (Hex)

Appendix B - Geometry versus space dimensions

The following table shows what combinations of geometry-number-of-dimensions (geo_ndim) and space-number-of-dimensions (space_ndim) are possible. There are three cases:

1. Case CABLE -- geo_ndim = 1 and space_ndim = 2 or 3; e.g., line in 2D or 3D (cables and rods)
2. Case SHELL -- geo_ndim = 2 and space_ndim = 3; e.g. Tri or Qua in 3D (shells and surfaces)
3. Case SOLID -- geo_ndim = space_ndim; e.g., Tri and Qua in 2D or Tet and Hex in 3D

geo_ndim	space_ndim = 2	space_ndim = 3
1	CABLE	CABLE
2	SOLID	SHELL
3	impossible	SOLID