cfsem 11.1.0

Quasi-steady electromagnetics including filamentized approximations, Biot-Savart, and Grad-Shafranov.
Documentation 
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//! Reference-element definitions and element-local geometry helpers for 3-node triangles.
//!
//! This is the 3-node triangle interpolation listed in Bower's *Applied Mechanics of Solids*,
//! Section 8.1, Table 8.3.

use crate::math::Scalar;
use crate::math::{add_scaled3, dot3, sub3};

/// Number of nodes in the linear triangular element.
pub const NODES_PER_ELEMENT: usize = 3;

/// Linear triangle shape functions on the reference simplex with vertices
/// `(u, v) = (0, 0), (1, 0), (0, 1)`.
pub fn shape<F: Scalar>(u: F, v: F) -> [F; NODES_PER_ELEMENT] {
    [F::one() - u - v, u, v]
}

/// Shape-function gradients with respect to the reference coordinates `(u, v)`.
pub fn grad_ref<F: Scalar>() -> [[F; 2]; NODES_PER_ELEMENT] {
    [
        [-F::one(), -F::one()],
        [F::one(), F::zero()],
        [F::zero(), F::one()],
    ]
}

/// Largest squared edge length of one triangle.
#[inline]
pub fn max_edge_length_squared<T: Scalar>(n0: [T; 3], n1: [T; 3], n2: [T; 3]) -> T {
    let e01 = sub3(n1, n0);
    let e12 = sub3(n2, n1);
    let e20 = sub3(n0, n2);
    let mut out = dot3(e01, e01);
    let e12_sq = dot3(e12, e12);
    let e20_sq = dot3(e20, e20);
    if e12_sq > out {
        out = e12_sq;
    }
    if e20_sq > out {
        out = e20_sq;
    }
    out
}

/// Closest point on a triangle to an observation point.
#[inline]
pub fn closest_point<T: Scalar>(obs: [T; 3], n0: [T; 3], n1: [T; 3], n2: [T; 3]) -> [T; 3] {
    let ab = sub3(n1, n0);
    let ac = sub3(n2, n0);
    let ap = sub3(obs, n0);
    let d1 = dot3(ab, ap);
    let d2 = dot3(ac, ap);
    if d1 <= T::ZERO && d2 <= T::ZERO {
        return n0;
    }

    let bp = sub3(obs, n1);
    let d3 = dot3(ab, bp);
    let d4 = dot3(ac, bp);
    if d3 >= T::ZERO && d4 <= d3 {
        return n1;
    }

    let vc = d1.mul_add(d4, (T::ZERO - d3) * d2);
    if vc <= T::ZERO && d1 >= T::ZERO && d3 <= T::ZERO {
        return add_scaled3(n0, ab, d1 / (d1 - d3));
    }

    let cp = sub3(obs, n2);
    let d5 = dot3(ab, cp);
    let d6 = dot3(ac, cp);
    if d6 >= T::ZERO && d5 <= d6 {
        return n2;
    }

    let vb = d5.mul_add(d2, (T::ZERO - d1) * d6);
    if vb <= T::ZERO && d2 >= T::ZERO && d6 <= T::ZERO {
        return add_scaled3(n0, ac, d2 / (d2 - d6));
    }

    let bc = sub3(n2, n1);
    let va = d3.mul_add(d6, (T::ZERO - d5) * d4);
    if va <= T::ZERO && (d4 - d3) >= T::ZERO && (d5 - d6) >= T::ZERO {
        return add_scaled3(n1, bc, (d4 - d3) / ((d4 - d3) + (d5 - d6)));
    }

    let denom_inv = T::ONE / (va + vb + vc);
    let v = vb * denom_inv;
    let w = vc * denom_inv;
    add_scaled3(add_scaled3(n0, ab, v), ac, w)
}

/// Split a triangle into three subtriangles sharing an interior point.
#[inline]
pub fn subdivide_about_point<T: Scalar>(
    point: [T; 3],
    n0: [T; 3],
    n1: [T; 3],
    n2: [T; 3],
) -> [[[T; 3]; 3]; 3] {
    [[point, n0, n1], [point, n1, n2], [point, n2, n0]]
}

#[cfg(test)]
mod tests {
    use super::{grad_ref, shape};
    use crate::mesh::elements::tri::mapping::{area, map_point};

    #[test]
    fn shape_functions_sum_to_one() {
        let n = shape(0.2_f64, 0.3_f64);
        let sum: f64 = n.into_iter().sum();
        assert!((sum - 1.0).abs() < 1.0e-12);
    }

    #[test]
    fn gradients_sum_to_zero() {
        let grad = grad_ref::<f64>();
        let su: f64 = grad.iter().map(|g| g[0]).sum();
        let sv: f64 = grad.iter().map(|g| g[1]).sum();
        assert!(su.abs() < 1.0e-12);
        assert!(sv.abs() < 1.0e-12);
    }

    #[test]
    fn map_point_hits_vertices() {
        let coords = [[0.0_f64, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 2.0, 0.0]];
        assert_eq!(
            map_point(coords[0], coords[1], coords[2], [0.0, 0.0]),
            coords[0]
        );
        assert_eq!(
            map_point(coords[0], coords[1], coords[2], [1.0, 0.0]),
            coords[1]
        );
        assert_eq!(
            map_point(coords[0], coords[1], coords[2], [0.0, 1.0]),
            coords[2]
        );
    }

    #[test]
    fn area_matches_right_triangle() {
        let n0 = [0.0_f64, 0.0, 0.0];
        let n1 = [1.0_f64, 0.0, 0.0];
        let n2 = [0.0_f64, 2.0, 0.0];
        assert!((area(n0, n1, n2) - 1.0).abs() < 1.0e-12);
    }
}