Geometry, meshes, and numerical integration for finite element analyses

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

We use Vector and Matrix from Russell Lab, thus some Debian packages are required.

Documentation:

• API reference (docs.rs)

Installation

Install some libraries:

sudo apt-get install \
    liblapacke-dev \
    libopenblas-dev

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

[dependencies]
gemlab = "*"

Examples

use gemlab::integ::default_integ_points;
use gemlab::mesh::{set_pad_coords, At, Extract, Mesh, Region};
use gemlab::shapes::Scratchpad;
use gemlab::StrError;
use std::collections::HashSet;

fn main() -> Result<(), StrError> {
    // Input the raw mesh data using a text file
    // and compute all derived information such as shapes,
    // and boundary entities. These data area stored in a
    // Region for the sake of convenience.
    //
    // 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::from_text_file(path)?;
    let region = Region::with(&mesh, Extract::Boundary)?;

    // Find entities along the boundary of the mesh
    // by giving coordinates. The `At` enum provides
    // an easy way to define the type of the constraint
    // such as line, plane, circle, etc.
    check(&region.find.points(At::Y(0.5))?, &[3, 4]);
    check(&region.find.edges(At::X(1.0))?, &[(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)?;
    set_pad_coords(&mut pad, &cell_2.points, &mesh);
    let ips = default_integ_points(cell_2.kind);
    let mut area = 0.0;
    for p in 0..ips.len() {
        let iota = &ips[p];
        let weight = ips[p][3];
        let det_jac = pad.calc_jacobian(iota)?;
        area += weight * det_jac;
    }
    assert_eq!(area, 0.5);
    Ok(())
}

fn check<T>(found: &HashSet<T>, correct: &[T])
where
    T: Copy + Ord + std::fmt::Debug,
{
    let mut ids: Vec<T> = found.iter().copied().collect();
    ids.sort();
    assert_eq!(ids, correct);
}

Todo

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

Appendix

Available shapes and local numbering of nodes

Lines -- Lin

Triangles -- Tri

Quadrilaterals -- Qua

Tetrahedra -- Tet

Hexahedra -- Hex

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

Coverage tool

The coverage tool cannot properly handle the macros in russell_chk such as assert_approx_eq! and assert_vec_approx_eq! We could use the #[no_coverage] decorator on the testing function to stop the coverage tool assessing the region coverage within the test function. However, we let the coverage tool report incorrect Region Coverage anyway. Sometimes, the coverage tool also fails to report line coverage even when all lines have been run.