test_triangle.py 文件源码

def _integrate_exact(f, triangle):
    #
    # Note that
    #
    #     \int_T f(x) dx = \int_T0 |J(xi)| f(P(xi)) dxi
    #
    # with
    #
    #     P(xi) = x0 * (1-xi[0]-xi[1]) + x1 * xi[0] + x2 * xi[1].
    #
    # and T0 being the reference triangle [(0.0, 0.0), (1.0, 0.0), (0.0,
    # 1.0)].
    # The determinant of the transformation matrix J equals twice the volume of
    # the triangle. (See, e.g.,
    # <http://math2.uncc.edu/~shaodeng/TEACHING/math5172/Lectures/Lect_15.PDF>).
    #
    xi = sympy.DeferredVector('xi')
    x_xi = \
        + triangle[0] * (1 - xi[0] - xi[1]) \
        + triangle[1] * xi[0] \
        + triangle[2] * xi[1]
    abs_det_J = 2 * quadpy.triangle.volume(triangle)
    exact = sympy.integrate(
        sympy.integrate(abs_det_J * f(x_xi), (xi[1], 0, 1-xi[0])),
        (xi[0], 0, 1)
        )
    return float(exact)