From: Stefano Dominici Date: Mon, 3 Dec 2018 13:25:35 +0000 (+0100) Subject: Modified implementation in favor of a constant eccentricity manifold X-Git-Tag: v9.1.0-rc1~498^2 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=refs%2Fpull%2F7442%2Fhead;p=dealii.git Modified implementation in favor of a constant eccentricity manifold Unit tests, comments, and pictures were modified accordingly. Minor modifications - Adjusted style. - Used `std_cxx14` instead of `std` for `make_unique()`. - Added guard in `pull_back()` to prevent NaNs in `std::acos()`. This may happen due to numerical errors: when the argument of acos() falls outside the range [-1,1]. Included a special case for pull_back() If the first coordinate is exactly 0, the center of the manifold is returned. --- diff --git a/doc/doxygen/images/elliptical_hyper_shell.png b/doc/doxygen/images/elliptical_hyper_shell.png index c4e70c6278..7da426131d 100644 Binary files a/doc/doxygen/images/elliptical_hyper_shell.png and b/doc/doxygen/images/elliptical_hyper_shell.png differ diff --git a/include/deal.II/grid/manifold_lib.h b/include/deal.II/grid/manifold_lib.h index dfcc581858..6eb5d7e328 100644 --- a/include/deal.II/grid/manifold_lib.h +++ b/include/deal.II/grid/manifold_lib.h @@ -468,11 +468,25 @@ private: * href="https://en.wikipedia.org/wiki/Elliptic_coordinate_system">Wikipedia * . * + * This is based on the definition of elliptic coordinates $(u,v)$ + * @f[ + * \left\lbrace\begin{align*} + * x &= x_0 + c \cosh(u) \cos(v) \\ + * y &= y_0 + c \sinh(u) \sin(v) + * \end{align*}\right. + * @f] + * in which $(x_0,y_0)$ are coordinates of the center of the cartesian system. + * + * The current implementation uses coordinates $(c,v)$, instead of $(u,v)$, and + * fixes $u$ according to a given eccentricity. Therefore, this choice + * of coordinates generates an elliptical manifold characterized by a constant + * eccentricity: $e=\frac{1}{\cosh(u)}$, with $e\in\left]0,1\right[$. + * * The constructor of this class will throw an exception if both dim and * spacedim are different from two. * * This manifold can be used to produce hyper_shells with elliptical curvature. - * As an example, the test elliptical_manifold_01 produces the following + * As an example, the test elliptical_manifold_01 produces the following * triangulation: * @image html elliptical_hyper_shell.png * @@ -486,19 +500,19 @@ class EllipticalManifold : public ChartManifold public: /** * Constructor that takes the center of the manifold system, the direction of - * the major axis, and the eccentricity parameter. + * the major axis, and the manifold eccentricity. * The default major axis is the x-axis. The manifold is rotated in * order to align the major axis to the direction specified in input. * @param center Center of the manifold. * @param major_axis_direction Direction of the major axis of the * manifold. - * @param c_parameter Parameter controlling the eccentricity of the - * manifold. + * @param eccentricity Eccentricity of the + * manifold $e\in\left]0,1\right[$. * */ EllipticalManifold(const Point & center, const Tensor<1, spacedim> &major_axis_direction, - const double c_parameter); + const double eccentricity); virtual std::unique_ptr> clone() const override; @@ -532,9 +546,10 @@ protected: */ const Point center; /** - * The parameter controlling the eccentricity of the manifold. + * Parameters deriving from the eccentricity of the manifold. */ - const double c_parameter; + const double cosh_u; + const double sinh_u; private: /** diff --git a/source/grid/manifold_lib.cc b/source/grid/manifold_lib.cc index e809bcf27e..7387e35f89 100644 --- a/source/grid/manifold_lib.cc +++ b/source/grid/manifold_lib.cc @@ -1156,18 +1156,20 @@ template EllipticalManifold::EllipticalManifold( const Point & center, const Tensor<1, spacedim> &major_axis_direction, - const double c_parameter) + const double eccentricity) : ChartManifold( EllipticalManifold::get_periodicity()) , direction(major_axis_direction) , center(center) - , c_parameter(c_parameter) + , cosh_u(1.0 / eccentricity) + , sinh_u(std::sqrt(cosh_u * cosh_u - 1.0)) { - // throws an exception if dim!=2 || spacedim!=2. + // Throws an exception if dim!=2 || spacedim!=2. Assert(dim == 2 && spacedim == 2, ExcNotImplemented()); - Assert(this->c_parameter > 0.0, + // Throws an exception if eccentricity is not in range. + Assert(std::signbit(cosh_u * cosh_u - 1.0) == false, ExcMessage( - "Invalid ellipsis parameter 'c': It must satisfy c > 0.0.")); + "Invalid eccentricity: It must satisfy 0 < eccentricity < 1.")); const double direction_norm = direction.norm(); Assert(direction_norm != 0.0, ExcMessage( @@ -1181,9 +1183,9 @@ template std::unique_ptr> EllipticalManifold::clone() const { - return std::make_unique>(center, - direction, - c_parameter); + const double eccentricity = 1.0 / cosh_u; + return std_cxx14::make_unique>( + center, direction, eccentricity); } @@ -1215,14 +1217,12 @@ template <> Point<2> EllipticalManifold<2, 2>::push_forward(const Point<2> &chart_point) const { - const double ch = std::cosh(chart_point[0]); - const double sh = std::sinh(chart_point[0]); const double cs = std::cos(chart_point[1]); const double sn = std::sin(chart_point[1]); // Coordinates in the reference frame (i.e. major axis direction is // x-axis) - const double x = c_parameter * ch * cs; - const double y = c_parameter * sh * sn; + const double x = chart_point[0] * cosh_u * cs; + const double y = chart_point[0] * sinh_u * sn; // Rotate them according to the major axis direction const Point<2> p(direction[0] * x - direction[1] * y, direction[1] * x + direction[0] * y); @@ -1246,45 +1246,29 @@ Point<2> EllipticalManifold<2, 2>::pull_back(const Point<2> &space_point) const { // Moving space_point in the reference coordinate system. - const double c2 = c_parameter * c_parameter; const double x0 = space_point[0] - center[0]; const double y0 = space_point[1] - center[1]; const double x = direction[0] * x0 + direction[1] * y0; const double y = -direction[1] * x0 + direction[0] * y0; - // From here we try to find solutions of two equations: - // x^2/(1-q)-y^2/q = c^2 for q, given q = -sinh^2(pt[0]), - // and - // x^2/(1-p)-y^2/p = c^2 for p, given p = sin^2(pt[1]). - // Note that, in the end, p and q are solutions to the same quadratic - // equation. - const double x2 = x * x; - const double y2 = y * y; - // - const double b = x2 + y2 - c2; - const double srdelta = std::sqrt(b * b + 4.0 * c2 * y2); - double p = (-b + srdelta) / (2.0 * c2); - // Since p=sin^2(pt[1]) is bounded in [0,1], - // guard against numerical overflows. - if (p < 0.0) - p = 0.0; - if (p > 1.0) - p = 1.0; - const double q = (-b - srdelta) / (2.0 * c2); - const bool x_is_positive = !std::signbit(x); - const bool y_is_positive = !std::signbit(y); - const double eta0 = std::asin(std::sqrt(p)); - // Given q = -sinh^2(pt[0]), - // pt[0] is calculated by straight inversion since q <= 0 for any - // pt[0]. - Point<2> pt(std::log(std::sqrt(-q) + std::sqrt(1.0 - q)), eta0); - // Unfolding pt[1] according to the quadrant. - if (x_is_positive && !y_is_positive) - pt[1] = 2.0 * numbers::PI - eta0; - else if (!x_is_positive && y_is_positive) - pt[1] = numbers::PI - eta0; - else if (!x_is_positive && !y_is_positive) - pt[1] = numbers::PI + eta0; - return pt; + const double pt0 = + std::sqrt((x * x) / (cosh_u * cosh_u) + (y * y) / (sinh_u * sinh_u)); + // If the radius is exactly zero, the point coincides with the origin. + if (pt0 == 0.0) + { + return center; + } + double cos_eta = x / (pt0 * cosh_u); + if (cos_eta < -1.0) + { + cos_eta = -1.0; + } + if (cos_eta > 1.0) + { + cos_eta = 1.0; + } + const double eta = std::acos(cos_eta); + const double pt1 = (std::signbit(y) ? 2.0 * numbers::PI - eta : eta); + return Point<2>(pt0, pt1); } @@ -1305,15 +1289,13 @@ DerivativeForm<1, 2, 2> EllipticalManifold<2, 2>::push_forward_gradient( const Point<2> &chart_point) const { - const double ch = std::cosh(chart_point[0]); - const double sh = std::sinh(chart_point[0]); const double cs = std::cos(chart_point[1]); const double sn = std::sin(chart_point[1]); DerivativeForm<1, 2, 2> dX; - dX[0][0] = c_parameter * sh * cs; - dX[0][1] = -c_parameter * ch * sn; - dX[1][0] = c_parameter * ch * sn; - dX[1][1] = c_parameter * sh * cs; + dX[0][0] = cosh_u * cs; + dX[0][1] = -chart_point[0] * cosh_u * sn; + dX[1][0] = sinh_u * sn; + dX[1][1] = chart_point[1] * sinh_u * cs; return dX; } diff --git a/tests/manifold/elliptical_manifold_01.cc b/tests/manifold/elliptical_manifold_01.cc index a9a6b0affd..cc1bf38599 100644 --- a/tests/manifold/elliptical_manifold_01.cc +++ b/tests/manifold/elliptical_manifold_01.cc @@ -41,64 +41,27 @@ // hyper_shell made of 8 elements. namespace { - /* The following system of equations - // / - // |x = ellPar*cosh(u)*cos(v) - // |y = ellPar*sinh(u)*sin(v) - // \ - // allows to transform a point (u,v) in the chart space to a - // point (x,y) in the cartesian space. - // - // Setting v=0, the system of equations becomes - // / - // |x = x_0 = ellPar*cosh(u) - // |y = 0 - // \ - // which represents the crossing point of an ellipsis, non-rotated and - // centered at the origin of the cartesian system, with its major axis. - // - // For the sake of completeness, by setting v=pi/2, one finds y_0 = - // ellPar*sinh(u) as the crossing point on the minor axis. - // - // This function takes in input a point pt={x_0,v} - // and transforms it into (x,y) in accordance with the first system of - // equations above. - // - // On a final note, the condition x_0 >= ellPar has to be satisfied. This is - // required because min(cosh(u)) = 1 is obtained for u = 0, and imposing v = 0 - // the minimum value that x can assume is x = ellPar. - // For the sake of completeness, it's worth to notice that there would be no - // restrictions on the input if we passed y_0, instead of x_0, as pt[0]. - */ - Point<2> - chart_to_cartesian(const Point<2> &pt, const double ellPar) - { - Point<2> c; - const double ch = pt[0] / ellPar; - const double sh = std::sqrt(ch * ch - 1.0); - c[0] = ellPar * ch * std::cos(pt[1]); - c[1] = ellPar * sh * std::sin(pt[1]); - return c; - } - - - std::vector> generate_shell_points(const Point<2> ¢er, const double radius0, const double radius1, - const double ellPar) + const double eccentricity) { - const std::array radii{{radius0, radius1}}; + const std::array radii{ + {radius0 * eccentricity, radius1 * eccentricity}}; const std::array angle{ {0, numbers::PI_4, numbers::PI_2, 3.0 * numbers::PI_4, numbers::PI}}; std::vector> points; + // Create an elliptical manifold to use push_forward() + Tensor<1, 2> axis; + axis[0] = 1.0; + axis[1] = 0.0; + EllipticalManifold<2, 2> mani(center, axis, eccentricity); for (auto &i : radii) { for (auto &j : angle) { - points.emplace_back(chart_to_cartesian(Point<2>(i, j), ellPar) + - center); + points.emplace_back(mani.push_forward(Point<2>(i, j)) + center); } } std::array dup{{1, 2, 3, 6, 7, 8}}; @@ -117,19 +80,15 @@ namespace // center, and the major axis oriented in the direction of the x-axis. // // inner_radius and outer_radius parameters correspond to the - // distances from the center of the manifold of the - // crossing points of two concentric ellipsis with their major axis. + // semi-major axis length for the inner and outer ellipsis. // - // Input parameters must respect the following constrains: - // c_parameter < inner_radius < outer_radius. + // Eccentricity must be in range ]0,1[ // - // To understand the constrain on the c_parameter refer to the - // documentation of function chart_to_cartesian() whithin this namespace. void build_simple_hyper_shell(Triangulation<2, 2> &grid, const Point<2> & center, const double inner_radius, const double outer_radius, - const double c_param) + const double eccentricity) { unsigned int cell[][4] = {{5, 6, 0, 1}, {6, 7, 1, 2}, @@ -140,7 +99,7 @@ namespace {15, 14, 12, 11}, {9, 15, 4, 12}}; std::vector> points = - generate_shell_points(center, inner_radius, outer_radius, c_param); + generate_shell_points(center, inner_radius, outer_radius, eccentricity); std::vector> cells(8, CellData<2>()); for (int i = 0; i < 8; ++i) for (int j = 0; j < 4; ++j) @@ -151,7 +110,7 @@ namespace Tensor<1, 2> axis; axis[0] = 1.0; axis[1] = 0.0; - grid.set_manifold(0, EllipticalManifold<2, 2>(center, axis, c_param)); + grid.set_manifold(0, EllipticalManifold<2, 2>(center, axis, eccentricity)); grid.set_all_manifold_ids(0); } @@ -168,13 +127,11 @@ template void test(unsigned int ref = 1) { - // deallog << "Testing dim " << dim << ", spacedim " << spacedim << std::endl; - Triangulation tria; switch (dim) { case 2: - build_simple_hyper_shell(tria, Point(), 1.5, 1.8, 1.4); + build_simple_hyper_shell(tria, Point(), 1.0, 2.0, 0.8); break; default: AssertThrow(0, ExcNotImplemented()); diff --git a/tests/manifold/elliptical_manifold_01.output b/tests/manifold/elliptical_manifold_01.output index d380473c18..edd50e266f 100644 --- a/tests/manifold/elliptical_manifold_01.output +++ b/tests/manifold/elliptical_manifold_01.output @@ -1,582 +1,582 @@ $NOD 576 -1 1.50000 0.00000 0 -2 1.06066 0.380789 0 -3 9.18485e-17 0.538516 0 -4 -1.06066 0.380789 0 -5 -1.50000 6.59492e-17 0 -6 1.80000 0.00000 0 -7 1.27279 0.800000 0 -8 1.10218e-16 1.13137 0 -9 -1.27279 0.800000 0 -10 -1.80000 1.38553e-16 0 -11 1.06066 -0.380789 0 -12 9.18485e-17 -0.538516 0 -13 -1.06066 -0.380789 0 -14 1.27279 -0.800000 0 -15 1.10218e-16 -1.13137 0 -16 -1.27279 -0.800000 0 -17 1.38582 0.206081 0 -18 1.38582 -0.206081 0 -19 0.574025 0.497524 0 -20 -0.574025 0.497524 0 -21 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a/tests/manifold/elliptical_manifold_02.cc +++ b/tests/manifold/elliptical_manifold_02.cc @@ -28,7 +28,7 @@ #include #include -#define TOLERANCE 1e-12 +#define TOLERANCE 1e-14 @@ -36,13 +36,13 @@ template Point test_push(const Tensor<1, spacedim> &axis, - const double c_parameter, + const double eccentricity, const Point & space_point, unsigned int ref = 1) { EllipticalManifold manifold(Point(), axis, - c_parameter); + eccentricity); const Point chart_point(manifold.push_forward(space_point)); deallog << space_point << " -> "; deallog << chart_point << std::endl; @@ -54,13 +54,13 @@ test_push(const Tensor<1, spacedim> &axis, template Point test_pull(const Tensor<1, spacedim> &axis, - const double c_parameter, + const double eccentricity, const Point & chart_point, unsigned int ref = 1) { EllipticalManifold manifold(Point(), axis, - c_parameter); + eccentricity); const Point space_point(manifold.pull_back(chart_point)); deallog << space_point << " <- "; deallog << chart_point << std::endl; @@ -72,11 +72,11 @@ test_pull(const Tensor<1, spacedim> &axis, // Function that tests pull_back() and push_forward(). void local_test(const Tensor<1, 2> &axis, - const double c_parameter, + const double eccentricity, const Point<2> & space_point) { - const Point<2> pt1 = test_push<2, 2>(axis, c_parameter, space_point); - const Point<2> pt2 = test_pull<2, 2>(axis, c_parameter, pt1); + const Point<2> pt1 = test_push<2, 2>(axis, eccentricity, space_point); + const Point<2> pt2 = test_pull<2, 2>(axis, eccentricity, pt1); if ((space_point - pt2).norm() < TOLERANCE) { deallog << "OK" << std::endl; @@ -95,20 +95,20 @@ test() { // test on a default manifold const Tensor<1, 2> axis({1.0, 0.0}); - const double c_parameter(0.1); - local_test(axis, c_parameter, Point<2>(1.1, 0.0)); - local_test(axis, c_parameter, Point<2>(2.0, 0.0)); - local_test(axis, c_parameter, Point<2>(3.0, 0.0)); - local_test(axis, c_parameter, Point<2>(4.0, 2.0)); + const double eccentricity(0.1); + local_test(axis, eccentricity, Point<2>(1.1, 0.0)); + local_test(axis, eccentricity, Point<2>(2.0, 0.0)); + local_test(axis, eccentricity, Point<2>(3.0, 0.0)); + local_test(axis, eccentricity, Point<2>(4.0, 2.0)); } { // test on a rotated manifold const Tensor<1, 2> axis({1.0, -1.0}); - const double c_parameter(0.1); - local_test(axis, c_parameter, Point<2>(1.1, 0.0)); - local_test(axis, c_parameter, Point<2>(2.0, 0.0)); - local_test(axis, c_parameter, Point<2>(3.0, 0.0)); - local_test(axis, c_parameter, Point<2>(4.0, 2.0)); + const double eccentricity(0.1); + local_test(axis, eccentricity, Point<2>(1.1, 0.0)); + local_test(axis, eccentricity, Point<2>(2.0, 0.0)); + local_test(axis, eccentricity, Point<2>(3.0, 0.0)); + local_test(axis, eccentricity, Point<2>(4.0, 2.0)); } } diff --git a/tests/manifold/elliptical_manifold_02.output b/tests/manifold/elliptical_manifold_02.output index 755935b8f6..0cf687080f 100644 --- a/tests/manifold/elliptical_manifold_02.output +++ b/tests/manifold/elliptical_manifold_02.output @@ -1,25 +1,25 @@ -DEAL::1.10000 0.00000 -> 0.166852 0.00000 -DEAL::1.10000 0.00000 <- 0.166852 0.00000 +DEAL::1.10000 0.00000 -> 11.0000 0.00000 +DEAL::1.10000 0.00000 <- 11.0000 0.00000 DEAL::OK -DEAL::2.00000 0.00000 -> 0.376220 0.00000 -DEAL::2.00000 0.00000 <- 0.376220 0.00000 +DEAL::2.00000 0.00000 -> 20.0000 0.00000 +DEAL::2.00000 0.00000 <- 20.0000 0.00000 DEAL::OK -DEAL::3.00000 0.00000 -> 1.00677 0.00000 -DEAL::3.00000 0.00000 <- 1.00677 0.00000 +DEAL::3.00000 0.00000 -> 30.0000 0.00000 +DEAL::3.00000 0.00000 <- 30.0000 0.00000 DEAL::OK -DEAL::4.00000 2.00000 -> -1.13642 2.48147 -DEAL::4.00000 2.00000 <- -1.13642 2.48147 +DEAL::4.00000 2.00000 -> -16.6459 36.1896 +DEAL::4.00000 2.00000 <- -16.6459 36.1896 DEAL::OK -DEAL::1.10000 0.00000 -> 0.117982 -0.117982 -DEAL::1.10000 0.00000 <- 0.117982 -0.117982 +DEAL::1.10000 0.00000 -> 7.77817 -7.77817 +DEAL::1.10000 0.00000 <- 7.77817 -7.77817 DEAL::OK -DEAL::2.00000 0.00000 -> 0.266027 -0.266027 -DEAL::2.00000 0.00000 <- 0.266027 -0.266027 +DEAL::2.00000 0.00000 -> 14.1421 -14.1421 +DEAL::2.00000 0.00000 <- 14.1421 -14.1421 DEAL::OK -DEAL::3.00000 0.00000 -> 0.711891 -0.711891 -DEAL::3.00000 0.00000 <- 0.711891 -0.711891 +DEAL::3.00000 0.00000 -> 21.2132 -21.2132 +DEAL::3.00000 0.00000 <- 21.2132 -21.2132 DEAL::OK -DEAL::4.00000 2.00000 -> 0.951088 2.55823 -DEAL::4.00000 2.00000 <- 0.951088 2.55823 +DEAL::4.00000 2.00000 -> 13.8195 37.3603 +DEAL::4.00000 2.00000 <- 13.8195 37.3603 DEAL::OK