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Geometry, meshes, and numerical integration for finite element analyses <!-- omit from toc -->

Contents <!-- omit from toc -->

• Introduction
  • Documentation
• Installation
  • TL;DR (Debian/Ubuntu/Linux)
  • Details
  • Setting Cargo.toml
• Examples
• 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

use gemlab::integ;
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::from_text_file(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 ips = integ::default_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(())
}

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