rustytorch/crates/specialized/rtx-fea/tests/quadrature_reference_measure.rs

//! Every quadrature rule must integrate 1 to the measure of its reference
//! domain.
//!
//! This is the cheapest possible invariant on a quadrature rule, it holds for
//! every order and every element family, and it is decisive: a rule whose
//! weights do not sum correctly scales every integral computed with it by a
//! constant factor. Nothing downstream can detect that — the stiffness matrix
//! stays symmetric, the mass matrix stays positive definite, rigid-body modes
//! stay exact. Only the total is wrong.
//!
//! Two rules failed it when it was first written: `triangle(3)` summed to 0.25
//! against a reference area of 0.5, and `tetrahedron(3)` to 1/36 against a
//! reference volume of 1/6. Both carried a spurious division by the reference
//! measure applied to weights that were already tabulated for it.

use rtx_fea::elements::quadrature::QuadratureRule;

fn weight_sum(rule: &QuadratureRule) -> f64 {
    rule.points.iter().map(|p| p.weight).sum()
}

fn check(name: &str, rule: &QuadratureRule, expected: f64) {
    assert!(
        !rule.points.is_empty(),
        "{name}: rule has no points — integration loops over it would silently \
         produce zero rather than fail"
    );
    let sum = weight_sum(rule);
    assert!(
        (sum - expected).abs() < 1e-12,
        "{name}: weights sum to {sum}, expected the reference measure {expected} \
         (ratio {:.6})",
        sum / expected,
    );
}

/// The 1-D reference domain is [-1, 1], measure 2.
#[test]
fn line_rules_integrate_the_reference_length() {
    for order in 1..=5 {
        check(
            &format!("line({order})"),
            &QuadratureRule::line(order).unwrap(),
            2.0,
        );
    }
}

/// The reference triangle has area 1/2.
#[test]
fn triangle_rules_integrate_the_reference_area() {
    for order in 1..=3 {
        check(
            &format!("triangle({order})"),
            &QuadratureRule::triangle(order).unwrap(),
            0.5,
        );
    }
}

/// The reference quadrilateral is [-1, 1]^2, area 4.
#[test]
fn quadrilateral_rules_integrate_the_reference_area() {
    for order in 1..=5 {
        check(
            &format!("quadrilateral({order})"),
            &QuadratureRule::quadrilateral(order).unwrap(),
            4.0,
        );
    }
}

/// The reference tetrahedron has volume 1/6.
#[test]
fn tetrahedron_rules_integrate_the_reference_volume() {
    for order in 1..=3 {
        check(
            &format!("tetrahedron({order})"),
            &QuadratureRule::tetrahedron(order).unwrap(),
            1.0 / 6.0,
        );
    }
}

/// The reference hexahedron is [-1, 1]^3, volume 8.
#[test]
fn hexahedron_rules_integrate_the_reference_volume() {
    for order in 1..=5 {
        check(
            &format!("hexahedron({order})"),
            &QuadratureRule::hexahedron(order).unwrap(),
            8.0,
        );
    }
}

/// Gauss-Legendre of `n` points integrates polynomials of degree `2n - 1`
/// exactly, which the weight sum alone does not check.
#[test]
fn gauss_legendre_is_exact_to_its_nominal_degree() {
    for n in 1..=5usize {
        let rule = QuadratureRule::gauss_legendre_1d(n).unwrap();
        let degree = 2 * n - 1;
        // Integral of x^d over [-1, 1] is 0 for odd d, 2/(d+1) for even d.
        for d in 0..=degree {
            let numeric: f64 = rule
                .points
                .iter()
                .map(|p| p.weight * p.coords.xi().powi(d as i32))
                .sum();
            let exact = if d % 2 == 1 {
                0.0
            } else {
                2.0 / (d as f64 + 1.0)
            };
            assert!(
                (numeric - exact).abs() < 1e-12,
                "gauss_legendre_1d({n}) integrating x^{d}: got {numeric}, exact {exact}"
            );
        }
    }
}