116 lines
3.1 KiB
C++
116 lines
3.1 KiB
C++
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#pragma once
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// Axisymmetric finite-element integration kernels.
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#include <array>
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#include <cmath>
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#include "Elasticity.h"
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namespace fem_axisymmetric {
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constexpr double kTwoPi = 6.28318530717958647692;
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struct TriangleQuadraturePoint {
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double weight;
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std::array<double, 3> phi;
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};
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inline const std::array<TriangleQuadraturePoint, 3> kTriangleQuadrature = { {
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{ 1.0 / 3.0, { 1.0 / 6.0, 1.0 / 6.0, 2.0 / 3.0 } },
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{ 1.0 / 3.0, { 1.0 / 6.0, 2.0 / 3.0, 1.0 / 6.0 } },
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{ 1.0 / 3.0, { 2.0 / 3.0, 1.0 / 6.0, 1.0 / 6.0 } }
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} };
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inline double signed_double_area(const Point& p1, const Point& p2, const Point& p3) {
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return (p2.x - p1.x) * (p3.y - p1.y) - (p3.x - p1.x) * (p2.y - p1.y);
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}
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inline void assemble(
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Matrix& A,
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std::vector<double>& F,
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const std::vector<Point>& points,
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const std::vector<Triangle>& triangles,
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double,
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double E,
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double nu,
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const vec_function& f) {
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const double lambda = (E * nu) / ((1.0 + nu) * (1.0 - 2.0 * nu));
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const double mu = E / (2.0 * (1.0 + nu));
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const Matrix C({
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{ lambda + 2.0 * mu, lambda, lambda, 0.0 },
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{ lambda, lambda + 2.0 * mu, lambda, 0.0 },
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{ lambda, lambda, lambda + 2.0 * mu, 0.0 },
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{ 0.0, 0.0, 0.0, mu }
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});
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for (const Triangle& T : triangles) {
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const Point& p1 = points[T.a];
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const Point& p2 = points[T.b];
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const Point& p3 = points[T.c];
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const double two_area = signed_double_area(p1, p2, p3);
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const double area = 0.5 * std::fabs(two_area);
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const Matrix grad_phi({
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{ (p2.y - p3.y) / two_area, (p3.x - p2.x) / two_area },
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{ (p3.y - p1.y) / two_area, (p1.x - p3.x) / two_area },
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{ (p1.y - p2.y) / two_area, (p2.x - p1.x) / two_area }
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});
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Matrix Ae(6ull);
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std::vector<double> Fe(6, 0.0);
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for (const auto& qp : kTriangleQuadrature) {
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const Point q = qp.phi[0] * p1 + qp.phi[1] * p2 + qp.phi[2] * p3;
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const double r_q = q.x;
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Matrix B(4ull, 6ull);
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for (size_t p = 0; p < 3; ++p) {
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const double dphi_dr = grad_phi[p][0];
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const double dphi_dz = grad_phi[p][1];
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const double phi = qp.phi[p];
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B[0][2 * p] = dphi_dr;
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B[1][2 * p + 1] = dphi_dz;
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B[2][2 * p] = phi / r_q;
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B[3][2 * p] = dphi_dz;
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B[3][2 * p + 1] = dphi_dr;
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}
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const Matrix stiffness_q = B.T().dot(C).dot(B);
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const double weight = kTwoPi * area * qp.weight * r_q;
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for (size_t i = 0; i < 6; ++i)
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for (size_t j = 0; j < 6; ++j)
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Ae[i][j] += weight * stiffness_q[i][j];
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const Point body_force = f(q);
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for (size_t p = 0; p < 3; ++p) {
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const double shape_value = qp.phi[p];
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Fe[2 * p] += weight * shape_value * body_force.x;
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Fe[2 * p + 1] += weight * shape_value * body_force.y;
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}
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}
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auto dof = [T](size_t i) -> size_t {
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if (i == 0) return 2 * T.a;
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if (i == 1) return 2 * T.a + 1;
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if (i == 2) return 2 * T.b;
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if (i == 3) return 2 * T.b + 1;
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if (i == 4) return 2 * T.c;
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return 2 * T.c + 1;
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};
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for (size_t i = 0; i < 6; ++i) {
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F[dof(i)] += Fe[i];
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for (size_t j = 0; j < 6; ++j)
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A[dof(i)][dof(j)] += Ae[i][j];
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}
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}
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}
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inline double boundary_segment_weight(const Point& midpoint, double len) {
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return 0.5 * kTwoPi * midpoint.x * len;
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}
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} // namespace fem_axisymmetric
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