const auto &c = all_vertices[vertex_indices[2]];
const auto &d = all_vertices[vertex_indices[3]];
- return (1.0 / 6.0) * std::abs((a - d) * cross_product_3d(b - d, c - d));
+ return (1.0 / 6.0) * (d - a) * cross_product_3d(b - a, c - a);
}
+ else if (vertex_indices.size() == 5) // pyramid
+ {
+ // This remarkably simple formula comes from Equation 4 of
+ // "Calculation of the volume of a general hexahedron for flow
+ // predictions", Davies and Salmond, AIAA Journal vol. 23 no. 6.
+ const auto &x0 = all_vertices[vertex_indices[0]];
+ const auto &x1 = all_vertices[vertex_indices[1]];
+ const auto &x2 = all_vertices[vertex_indices[2]];
+ const auto &x3 = all_vertices[vertex_indices[3]];
+ const auto &x4 = all_vertices[vertex_indices[4]];
+
+ const auto v01 = x1 - x0;
+ const auto v02 = x2 - x0;
+ const auto v03 = x3 - x0;
+ const auto v04 = x4 - x0;
+ const auto v21 = x2 - x1;
+
+ // doing high - low consistently puts us off by -1 from the original
+ // paper in the first term
+ return -v04 * cross_product_3d(v21, v03) / 6.0 +
+ v03 * cross_product_3d(v01, v02) / 12.0;
+ }
else if (vertex_indices.size() == 6) // wedge
{
- /*
- The following python/sympy script was used:
-
- #!/usr/bin/env python
- # coding: utf-8
- import sympy as sp
- from sympy.simplify.cse_main import cse
- xs = list(sp.symbols(" ".join(["x{}".format(i) for i in range(6)])))
- ys = list(sp.symbols(" ".join(["y{}".format(i) for i in range(6)])))
- zs = list(sp.symbols(" ".join(["z{}".format(i) for i in range(6)])))
- xi, eta, zeta = sp.symbols("xi eta zeta")
- tphi = [(1 - xi - eta)*(1 - zeta),
- (xi)*(1 - zeta),
- (eta)*(1 - zeta),
- (1 - xi - zeta)*(zeta),
- (xi)*(zeta),
- (eta)*(zeta)]
- x_real = sum(xs[i]*tphi[i] for i in range(len(xs)))
- y_real = sum(ys[i]*tphi[i] for i in range(len(xs)))
- z_real = sum(zs[i]*tphi[i] for i in range(len(xs)))
- J = sp.Matrix([[var.diff(v) for v in [xi, eta, zeta]]
- for var in [x_real, y_real, z_real]])
- detJ = J.det()
- detJ2 = detJ.expand().collect(zeta).collect(eta).collect(xi)
- for x in xs:
- detJ2 = detJ2.collect(x)
- for y in ys:
- detJ2 = detJ2.collect(y)
- for z in zs:
- detJ2 = detJ2.collect(z)
- measure = sp.integrate(sp.integrate(sp.integrate(detJ2, (xi, 0, 1)),
- (eta, 0, 1)), (zeta, 0, 1))
- measure2 = measure
- for vs in [xs, ys, zs]:
- for v in vs:
- measure2 = measure2.collect(v)
- pairs, expression = cse(measure2)
- for pair in pairs:
- print("const double " + sp.ccode(pair[0]) + " = "
- + sp.ccode(pair[1]) + ";")
- print("const double result = " + sp.ccode(expression[0]) + ";")
- print("return result;")
+ /* Script used to generate volume code:
+
+ #!/usr/bin/env python
+ # coding: utf-8
+ import sympy as sp
+ from sympy.simplify.cse_main import cse
+ n_vertices = 6
+ xs = list(sp.symbols(" ".join(["x{}".format(i)
+ for i in range(n_vertices)])))
+ ys = list(sp.symbols(" ".join(["y{}".format(i)
+ for i in range(n_vertices)])))
+ zs = list(sp.symbols(" ".join(["z{}".format(i)
+ for i in range(n_vertices)])))
+ xi, eta, zeta = sp.symbols("xi eta zeta")
+ tphi = [(1 - xi - eta)*(1 - zeta),
+ (xi)*(1 - zeta),
+ (eta)*(1 - zeta),
+ (1 - xi - eta)*(zeta),
+ (xi)*(zeta),
+ (eta)*(zeta)]
+ x_real = sum(xs[i]*tphi[i] for i in range(n_vertices))
+ y_real = sum(ys[i]*tphi[i] for i in range(n_vertices))
+ z_real = sum(zs[i]*tphi[i] for i in range(n_vertices))
+ J = sp.Matrix([[var.diff(v) for v in [xi, eta, zeta]]
+ for var in [x_real, y_real, z_real]])
+ detJ = J.det()
+ detJ2 = detJ.expand().collect(zeta).collect(eta).collect(xi)
+ for x in xs:
+ detJ2 = detJ2.collect(x)
+ for y in ys:
+ detJ2 = detJ2.collect(y)
+ for z in zs:
+ detJ2 = detJ2.collect(z)
+ measure = sp.integrate(sp.integrate(
+ sp.integrate(detJ2, (eta, 0, 1 - xi)),
+ (xi, 0, 1)), (zeta, 0, 1))
+ measure2 = measure
+ for vs in [xs, ys, zs]:
+ for v in vs:
+ measure2 = measure2.collect(v)
+
+ pairs, expression = cse(measure2)
+ for vertex_no in range(n_vertices):
+ for (coordinate, index) in [('x', 0), ('y', 1), ('z', 2)]:
+ print(
+ "const double {}{} = all_vertices[vertex_indices[{}]][{}];"
+ .format(coordinate, vertex_no, vertex_no, index))
+
+ for pair in pairs:
+ print("const double " + sp.ccode(pair[0]) + " = "
+ + sp.ccode(pair[1]) + ";")
+ print("const double result = " + sp.ccode(expression[0]) + ";")
+ print("return result;")
*/
- const double x0 = all_vertices[vertex_indices[0]](0);
- const double y0 = all_vertices[vertex_indices[0]](1);
- const double z0 = all_vertices[vertex_indices[0]](2);
- const double x1 = all_vertices[vertex_indices[1]](0);
- const double y1 = all_vertices[vertex_indices[1]](1);
- const double z1 = all_vertices[vertex_indices[1]](2);
- const double x2 = all_vertices[vertex_indices[2]](0);
- const double y2 = all_vertices[vertex_indices[2]](1);
- const double z2 = all_vertices[vertex_indices[2]](2);
- const double x3 = all_vertices[vertex_indices[3]](0);
- const double y3 = all_vertices[vertex_indices[3]](1);
- const double z3 = all_vertices[vertex_indices[3]](2);
- const double x4 = all_vertices[vertex_indices[4]](0);
- const double y4 = all_vertices[vertex_indices[4]](1);
- const double z4 = all_vertices[vertex_indices[4]](2);
- const double x5 = all_vertices[vertex_indices[5]](0);
- const double y5 = all_vertices[vertex_indices[5]](1);
- const double z5 = all_vertices[vertex_indices[5]](2);
-
- const double x6 = (1.0 / 6.0) * z5;
- const double x7 = (1.0 / 12.0) * z1;
- const double x8 = -x7;
- const double x9 = (1.0 / 12.0) * z2;
- const double x10 = -x9;
- const double x11 = (1.0 / 4.0) * z5;
- const double x12 = -x11;
- const double x13 = (1.0 / 12.0) * z0;
- const double x14 = x12 + x13;
- const double x15 = (1.0 / 4.0) * z2;
- const double x16 = (1.0 / 6.0) * z4;
- const double x17 = (1.0 / 4.0) * z1;
- const double x18 = (1.0 / 6.0) * z0;
- const double x19 = x17 - x18;
- const double x20 = -x6;
- const double x21 = (1.0 / 4.0) * z0;
- const double x22 = -x21;
- const double x23 = -x17;
- const double x24 = -x15;
- const double x25 = (1.0 / 6.0) * z3;
- const double x26 = x24 - x25;
- const double x27 = x18 + x23;
- const double x28 = (1.0 / 3.0) * z2;
- const double x29 = (1.0 / 12.0) * z5;
- const double x30 = (1.0 / 12.0) * z3;
- const double x31 = -x30;
- const double x32 = (1.0 / 4.0) * z4;
- const double x33 = x31 + x32;
- const double x34 = (1.0 / 3.0) * z1;
- const double x35 = (1.0 / 12.0) * z4;
- const double x36 = -x16;
- const double x37 = x15 + x25;
- const double x38 = -x13;
- const double x39 = x11 + x38;
- const double x40 = -x32;
- const double x41 = x30 + x40;
- const double x42 = (1.0 / 3.0) * z0;
- const double x43 = (1.0 / 4.0) * z3;
- const double x44 = x32 - x43;
- const double x45 = x40 + x43;
- return x0 * (y1 * (-x28 + x29 + x33) + y2 * (x12 + x31 + x34 - x35) +
- y3 * (x20 + x7 + x9) + y4 * (x23 + x6 + x9) +
- y5 * (x36 + x37 + x8)) +
- x1 * (y0 * (x28 - x29 + x41) + y2 * (x11 + x33 - x42) +
- y3 * (x39 + x9) + y4 * (x12 + x21 + x24) +
- y5 * (x13 + x24 + x44)) +
- x2 * (y0 * (x11 + x30 - x34 + x35) + y1 * (x12 + x41 + x42) +
- y3 * (x39 + x8) + y4 * (x12 + x17 + x38) +
- y5 * (x17 + x22 + x44)) +
- x3 * (-x6 * y4 + y0 * (x10 + x6 + x8) + y1 * (x10 + x14) +
- y2 * (x14 + x7) + y5 * (x15 + x16 + x19)) +
- x4 * (x6 * y3 + y0 * (x10 + x17 + x20) + y1 * (x11 + x15 + x22) +
- y2 * (x11 + x13 + x23) + y5 * (x26 + x27)) +
- x5 * (y0 * (x16 + x26 + x7) + y1 * (x15 + x38 + x45) +
- y2 * (x21 + x23 + x45) + y3 * (x24 + x27 + x36) +
- y4 * (x19 + x37));
+ const double x0 = all_vertices[vertex_indices[0]][0];
+ const double y0 = all_vertices[vertex_indices[0]][1];
+ const double z0 = all_vertices[vertex_indices[0]][2];
+ const double x1 = all_vertices[vertex_indices[1]][0];
+ const double y1 = all_vertices[vertex_indices[1]][1];
+ const double z1 = all_vertices[vertex_indices[1]][2];
+ const double x2 = all_vertices[vertex_indices[2]][0];
+ const double y2 = all_vertices[vertex_indices[2]][1];
+ const double z2 = all_vertices[vertex_indices[2]][2];
+ const double x3 = all_vertices[vertex_indices[3]][0];
+ const double y3 = all_vertices[vertex_indices[3]][1];
+ const double z3 = all_vertices[vertex_indices[3]][2];
+ const double x4 = all_vertices[vertex_indices[4]][0];
+ const double y4 = all_vertices[vertex_indices[4]][1];
+ const double z4 = all_vertices[vertex_indices[4]][2];
+ const double x5 = all_vertices[vertex_indices[5]][0];
+ const double y5 = all_vertices[vertex_indices[5]][1];
+ const double z5 = all_vertices[vertex_indices[5]][2];
+ const double x6 = (1.0 / 12.0) * z1;
+ const double x7 = -x6;
+ const double x8 = (1.0 / 12.0) * z3;
+ const double x9 = x7 + x8;
+ const double x10 = (1.0 / 12.0) * z2;
+ const double x11 = -x8;
+ const double x12 = x10 + x11;
+ const double x13 = (1.0 / 6.0) * z2;
+ const double x14 = (1.0 / 12.0) * z4;
+ const double x15 = (1.0 / 6.0) * z1;
+ const double x16 = (1.0 / 12.0) * z5;
+ const double x17 = -x16;
+ const double x18 = x16 + x7;
+ const double x19 = -x14;
+ const double x20 = x10 + x19;
+ const double x21 = (1.0 / 12.0) * z0;
+ const double x22 = x19 + x21;
+ const double x23 = -x10;
+ const double x24 = x14 + x23;
+ const double x25 = (1.0 / 6.0) * z0;
+ const double x26 = x17 + x21;
+ const double x27 = x23 + x8;
+ const double x28 = -x21;
+ const double x29 = x16 + x28;
+ const double x30 = x17 + x6;
+ const double x31 = x14 + x28;
+ const double x32 = x11 + x6;
+ const double x33 = (1.0 / 6.0) * z5;
+ const double x34 = (1.0 / 6.0) * z4;
+ const double x35 = (1.0 / 6.0) * z3;
+ const double result =
+ x0 * (x12 * y5 + x9 * y4 + y1 * (-x13 + x14 + x8) +
+ y2 * (x11 + x15 + x17) + y3 * (x18 + x20)) +
+ x1 * (x22 * y3 + x24 * y5 + y0 * (x11 + x13 + x19) +
+ y2 * (x14 + x16 - x25) + y4 * (x26 + x27)) +
+ x2 * (x29 * y3 + x30 * y4 + y0 * (-x15 + x16 + x8) +
+ y1 * (x17 + x19 + x25) + y5 * (x31 + x32)) +
+ x3 * (x26 * y2 + x31 * y1 + y0 * (x24 + x30) + y4 * (x28 + x33 + x7) +
+ y5 * (x10 + x21 - x34)) +
+ x4 * (x18 * y2 + x32 * y0 + y1 * (x12 + x29) + y3 * (x21 - x33 + x6) +
+ y5 * (x23 + x35 + x7)) +
+ x5 * (x20 * y1 + x27 * y0 + y2 * (x22 + x9) + y3 * (x23 + x28 + x34) +
+ y4 * (x10 - x35 + x6));
+ return result;
}
AssertDimension(vertex_indices.size(), GeometryInfo<3>::vertices_per_cell);
projects / dealii.git / commitdiff