Geometry, meshes, and numerical integration for finite element analyses

Contents

• Introduction

  • Documentation

• Installation

  • TL;DR (Debian/Ubuntu/Linux)
  • Details
  • Setting Cargo.toml

• 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

At this moment, Gemlab works on Linux (Debian/Ubuntu; and maybe Arch).

TL;DR (Debian/Ubuntu/Linux)

First:

sudo apt-get install -y --no-install-recommends \
    g++ \
    gdb \
    gfortran \
    liblapacke-dev \
    libmumps-seq-dev \
    libopenblas-dev \
    libsuitesparse-dev

Then:

cargo add gemlab

Details

This crates depends on russell_lab and, hence, needs some external libraries. See the installation of required dependencies on russell_lab.

Setting Cargo.toml

👆 Check the crate version and update your Cargo.toml accordingly:

[dependencies]
gemlab = "*"

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