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If your program is a subroutine library, you +may consider it more useful to permit linking proprietary applications with +the library. If this is what you want to do, use the GNU Lesser General +Public License instead of this License. But first, please read +. diff --git a/contrib/overlap/README.md b/contrib/overlap/README.md new file mode 100644 index 00000000..d1cac1cc --- /dev/null +++ b/contrib/overlap/README.md @@ -0,0 +1,81 @@ +[![Build Status](https://travis-ci.org/severinstrobl/overlap.svg?branch=master)](https://travis-ci.org/severinstrobl/overlap) + +# Exact calculation of the overlap volume and area of spheres and mesh elements + +Calculating the intersection or overlapping volume of a sphere and one of the +typically used mesh elements such as a tetrahedron or hexahedron is +surprisingly challenging. This header-only library implements a numerically +robust method to determine this volume. + +The mathematical expressions and algorithms used in this code are described in +S. Strobl et al.: Exact calculation of the overlap volume of spheres and mesh +elements, Journal of Computational Physics, 2016 +(http://dx.doi.org/10.1016/j.jcp.2016.02.003). So if you use the code in +projects resulting in any publications, please cite this paper. + +Employing the concepts and routines used for the calculation of the overlap +volume, the intersection or overlap *area* of a sphere and the facets of a mesh +element can also be calculated with this library. + +## Dependencies + +The compile-time dependencies of this code are: +- Eigen3 (http://eigen.tuxfamily.org) +- C++11 compliant compiler + +The software was successfully compiled and tested using the following +compilers: +- GNU G++ compiler (versions 4.8.4, 4.9.3, and 5.4.0) +- Intel C++ compiler (version 15.0.2) +- Clang C++ compiler (versions 3.6.1, 3.9.1, and 5.0.0) + +## Usage + +The library is implemented as a pure header-only library written in plain +C++11. To use it in your code, simply include the header file *overlap.hpp* and +make sure the Eigen3 headers can be found by your compiler or build system. A +minimal example calculating the overlap of a hexahedron with a side length of 2 +centered at the origin and a sphere with radius 1 centered at a corner of the +hexahedron could look something like this: +``` +vector_t v0{-1, -1, -1}; +vector_t v1{ 1, -1, -1}; +vector_t v2{ 1, 1, -1}; +vector_t v3{-1, 1, -1}; +vector_t v4{-1, -1, 1}; +vector_t v5{ 1, -1, 1}; +vector_t v6{ 1, 1, 1}; +vector_t v7{-1, 1, 1}; + +Hexahedron hex{v0, v1, v2, v3, v4, v5, v6, v7}; +Sphere s{vector_t::Constant(1), 1}; + +scalar_t result = overlap(s, hex); +``` +This code snippet calculates the correct result (pi/6) for this simple +configuration. + +To obtain the overlap area of a sphere and the facets of a tetrahedron, the +function *overlapArea* can be employed as such: +``` +vector_t v0{-std::sqrt(3) / 6.0, -1.0 / 2.0, 0}; +vector_t v1{std::sqrt(3) / 3.0, 0, 0}; +vector_t v2{-std::sqrt(3) / 6.0, +1.0 / 2.0, 0}; +vector_t v3{0, 0, std::sqrt(6) / 3.0}; + +Tetrahedron tet{v0, v1, v2, v3}; +Sphere s{{0, 0, 1.5}, 1.25}; + +auto result = overlapArea(s, tet); + +std::cout << "surface area of sphere intersecting tetrahedron: " << + result[0] << std::endl; + +std::cout << "overlap areas per face:" << std::endl; +// The indices of the faces are NOT zero-based here! +for(size_t f = 1; f < result.size() - 1; ++f) + std::cout << " face #" << (f - 1) << ": " << result[f] << std::endl; + +std::cout << "total surface area of tetrahedron intersecting sphere: " << + result.back() << std::endl; +``` diff --git a/contrib/overlap/overlap.hpp b/contrib/overlap/overlap.hpp new file mode 100644 index 00000000..9dbeb064 --- /dev/null +++ b/contrib/overlap/overlap.hpp @@ -0,0 +1,2204 @@ +/*! + * Exact calculation of the overlap volume of spheres and mesh elements. + * http://dx.doi.org/10.1016/j.jcp.2016.02.003 + * + * Copyright (C) 2015-2018 Severin Strobl + * + * This program is free software: you can redistribute it and/or modify + * it under the terms of the GNU General Public License as published by + * the Free Software Foundation, either version 3 of the License, or + * (at your option) any later version. + * + * This program is distributed in the hope that it will be useful, + * but WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + * GNU General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program. If not, see . + */ + +#ifndef OVERLAP_HPP +#define OVERLAP_HPP + +// Eigen +#include +#include + +// C++ +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +namespace OVERLAP { + +// typedefs +typedef double scalar_t; +typedef Eigen::Matrix vector_t; +typedef Eigen::Matrix vector2_t; + +// Pretty-printing of Eigen matrices. +static const Eigen::IOFormat pretty(Eigen::StreamPrecision, + Eigen::DontAlignCols, " ", ";\n", "", "", "[", "]"); + +// constants +const scalar_t pi = scalar_t(4) * std::atan(scalar_t(1.0)); + +namespace detail { + +static const scalar_t tinyEpsilon(2 * + std::numeric_limits::epsilon()); + +static const scalar_t mediumEpsilon(1e2 * tinyEpsilon); +static const scalar_t largeEpsilon(1e-10); + +// Robust calculation of the normal vector of a polygon using Newell's method +// and a pre-calculated center. +// Ref: Christer Ericson - Real-Time Collision Detection (2005) +template +inline vector_t normalNewell(Iterator begin, Iterator end, const vector_t& + center) { + + const size_t count = end - begin; + vector_t n(vector_t::Zero()); + + for(size_t i = 0; i < count; ++i) + n += (*(begin + i) - center).cross(*(begin + ((i + 1) % count)) - + center); + + scalar_t length = n.stableNorm(); + + if(length) + return n / length; + else + return n; +} + +// This implementation of double_prec is based on: +// T.J. Dekker, A floating-point technique for extending the available +// precision, http://dx.doi.org/10.1007/BF01397083 + +template +struct double_prec_constant; + +template<> +struct double_prec_constant { + // Constant used to split double precision values: + // 2^(24 - 24/2) + 1 = 2^12 + 1 = 4097 + static const uint32_t value = 4097; +}; + +template<> +struct double_prec_constant { + // Constant used to split double precision values: + // 2^(53 - int(53/2)) + 1 = 2^27 + 1 = 134217729 + static const uint32_t value = 134217729; +}; + +// For GCC and Clang an attribute can be used to control the FP precision... +#if defined(__GNUC__) && !defined(__clang__) && !defined(__ICC) && \ + !defined(__INTEL_COMPILER) +#define ENFORCE_EXACT_FPMATH_ATTR __attribute__((__target__("ieee-fp"))) +#else +#define ENFORCE_EXACT_FPMATH_ATTR +#endif + +// ... whereas ICC requires a pragma. +#if defined(__ICC) || defined(__INTEL_COMPILER) +#define ENFORCE_EXACT_FPMATH_ATTR +#define USE_EXACT_FPMATH_PRAGMA 1 +#endif + +template +class double_prec; + +template +inline double_prec operator+(const double_prec& lhs, const + double_prec& rhs) ENFORCE_EXACT_FPMATH_ATTR; + +template +inline double_prec operator-(const double_prec& lhs, const + double_prec& rhs) ENFORCE_EXACT_FPMATH_ATTR; + +template +inline double_prec operator*(const double_prec& lhs, const + double_prec& rhs) ENFORCE_EXACT_FPMATH_ATTR; + +template +class double_prec { + private: + static const uint32_t c = detail::double_prec_constant::value; + + template + friend double_prec operator+(const double_prec&, const + double_prec&); + + template + friend double_prec operator-(const double_prec&, const + double_prec&); + + template + friend double_prec operator*(const double_prec&, const + double_prec&); + + public: + inline double_prec() : h_(0), l_(0) { + } + + // This constructor requires floating point operations in accordance + // with IEEE754 to perform the proper splitting. To allow full + // optimization of all other parts of the code, precise floating point + // ops are only requested here. Unfortunately the way to do this is + // extremely compiler dependent. + inline double_prec(const T& val) ENFORCE_EXACT_FPMATH_ATTR : h_(0), + l_(0) { + +#ifdef USE_EXACT_FPMATH_PRAGMA + #pragma float_control(precise, on) +#endif + + T p = val * T(c); + h_ = (val - p) + p; + l_ = val - h_; + } + + private: + inline explicit double_prec(const T& h, const T& l) : h_(h), l_(l) { + } + + public: + inline const T& high() const { + return h_; + } + + inline const T& low() const { + return l_; + } + + inline T value() const { + return h_ + l_; + } + + template + inline TOther convert() const { + return TOther(h_) + TOther(l_); + } + + private: + T h_; + T l_; +}; + +template +inline double_prec operator+(const double_prec& lhs, const + double_prec& rhs) { + +#ifdef USE_EXACT_FPMATH_PRAGMA + #pragma float_control(precise, on) +#endif + + T h = lhs.h_ + rhs.h_; + T l = std::abs(lhs.h_) >= std::abs(rhs.h_) ? + ((((lhs.h_ - h) + rhs.h_) + lhs.l_) + rhs.l_) : + ((((rhs.h_ - h) + lhs.h_) + rhs.l_) + lhs.l_); + + T c = h + l; + + return double_prec(c, (h - c) + l); +} + +template +inline double_prec operator-(const double_prec& lhs, const + double_prec& rhs) { + +#ifdef USE_EXACT_FPMATH_PRAGMA + #pragma float_control(precise, on) +#endif + + T h = lhs.h_ - rhs.h_; + T l = std::abs(lhs.h_) >= std::abs(rhs.h_) ? + ((((lhs.h_ - h) - rhs.h_) - rhs.l_) + lhs.l_) : + ((((-rhs.h_ - h) + lhs.h_) + lhs.l_) - rhs.l_); + + T c = h + l; + + return double_prec(c, (h - c) + l); +} + +template +inline double_prec operator*(const double_prec& lhs, const + double_prec& rhs) { + +#ifdef USE_EXACT_FPMATH_PRAGMA + #pragma float_control(precise, on) +#endif + + double_prec l(lhs.h_); + double_prec r(rhs.h_); + + T p = l.h_ * r.h_; + T q = l.h_ * r.l_ + l.l_ * r.h_; + T v = p + q; + + double_prec c(v, ((p - v) + q) + l.l_ * r.l_); + c.l_ = ((lhs.h_ + lhs.l_) * rhs.l_ + lhs.l_ * rhs.h_) + c.l_; + T z = c.value(); + + return double_prec(z, (c.h_ - z) + c.l_); +} + +// Ref: J.R. Shewchuk - Lecture Notes on Geometric Robustness +// http://www.cs.berkeley.edu/~jrs/meshpapers/robnotes.pdf +inline scalar_t orient2D(const vector2_t& a, const vector2_t& b, const + vector2_t& c) { + + typedef double_prec real_t; + + real_t a0(a[0]); + real_t a1(a[1]); + real_t b0(b[0]); + real_t b1(b[1]); + real_t c0(c[0]); + real_t c1(c[1]); + + real_t result = (a0 - c0) * (b1 - c1) - (a1 - c1) * (b0 - c0); + + return result.convert(); +} + +// Numerically robust calculation of the normal of the triangle defined by +// the points a, b, and c. +// Ref: J.R. Shewchuk - Lecture Notes on Geometric Robustness +// http://www.cs.berkeley.edu/~jrs/meshpapers/robnotes.pdf +inline vector_t triangleNormal(const vector_t& a, const vector_t& b, const + vector_t& c) { + + scalar_t xy = orient2D(vector2_t(a[0], a[1]), vector2_t(b[0], b[1]), + vector2_t(c[0], c[1])); + + scalar_t yz = orient2D(vector2_t(a[1], a[2]), vector2_t(b[1], b[2]), + vector2_t(c[1], c[2])); + + scalar_t zx = orient2D(vector2_t(a[2], a[0]), vector2_t(b[2], b[0]), + vector2_t(c[2], c[0])); + + return vector_t(yz, zx, xy).normalized(); +} + +// Numerically robust routine to calculate the angle between normalized +// vectors. +// Ref: http://www.plunk.org/~hatch/rightway.php +inline scalar_t angle(const vector_t& v0, const vector_t& v1) { + if(v0.dot(v1) < scalar_t(0)) + return pi - scalar_t(2) * std::asin(scalar_t(0.5) * (v0 + + v1).stableNorm()); + else + return scalar_t(2) * std::asin(scalar_t(0.5) * (v0 - v1).stableNorm()); +} + +template +inline std::array gramSchmidt(const Eigen::MatrixBase& + arg0, const Eigen::MatrixBase& arg1) { + + vector_t v0(arg0.normalized()); + vector_t v1(arg1); + + std::array result; + result[0] = v0; + result[1] = (v1 - v1.dot(v0) * v0).normalized(); + + return result; +} + +inline scalar_t clamp(scalar_t value, scalar_t min, scalar_t max, scalar_t + limit) { + + assert(min <= max && limit >= scalar_t(0)); + + value = (value < min && value > (min - limit)) ? min : value; + value = (value > max && value < (max + limit)) ? max : value; + + return value; +} + +} // namespace detail + +class Transformation { + public: + Transformation(const vector_t& t, const scalar_t& s) : translation(t), + scaling(s) { + } + + vector_t translation; + scalar_t scaling; +}; + +template +class Polygon { + private: + static_assert(VertexCount >= 3 && VertexCount <= 4, + "Only triangles and quadrilateral are supported."); + + public: + static const size_t vertex_count = VertexCount; + + protected: + Polygon() : vertices(), center(), normal(), area() { + } + + template + Polygon(const vector_t& v0, Types... verts) : vertices{{v0, + verts...}}, center(), normal(), area() { + + center = scalar_t(1.0 / vertex_count) * + std::accumulate(vertices.begin(), vertices.end(), + vector_t::Zero().eval()); + + // For a quadrilateral, Newell's method can be simplified + // significantly. + // Ref: Christer Ericson - Real-Time Collision Detection (2005) + if(VertexCount == 4) { + normal = ((vertices[2] - vertices[0]).cross(vertices[3] - + vertices[1])).normalized(); + } else { + normal = detail::normalNewell(vertices.begin(), vertices.end(), + center); + } + } + + void apply(const Transformation& t) { + for(auto& v : vertices) + v = t.scaling * (v + t.translation); + + center = t.scaling * (center + t.translation); + } + + public: + bool isPlanar(const scalar_t epsilon = detail::largeEpsilon) const { + if(VertexCount == 3) + return true; + + for(auto& v : vertices) + if(std::abs(normal.dot(v - center)) > epsilon) + return false; + + return true; + } + + public: + std::array vertices; + vector_t center; + vector_t normal; + scalar_t area; +}; + +class Triangle : public Polygon<3> { + public: + Triangle() : Polygon<3>() { + } + + template + Triangle(const vector_t& v0, Types... verts) : Polygon<3>(v0, + verts...) { + + init(); + } + + void apply(const Transformation& t) { + Polygon<3>::apply(t); + init(); + } + + private: + void init() { + area = scalar_t(0.5) * ((vertices[1] - vertices[0]).cross( + vertices[2] - vertices[0])).stableNorm(); + } +}; + +class Quadrilateral : public Polygon<4> { + public: + Quadrilateral() : Polygon<4>() { + } + + template + Quadrilateral(const vector_t& v0, Types... verts) : Polygon<4>(v0, + verts...) { + + init(); + } + + void apply(const Transformation& t) { + Polygon<4>::apply(t); + init(); + } + + private: + void init() { + area = scalar_t(0.5) * (((vertices[1] - vertices[0]).cross( + vertices[2] - vertices[0])).stableNorm() + + ((vertices[2] - vertices[0]).cross( + vertices[3] - vertices[0])).stableNorm()); + } +}; + +// Forward declarations of the mesh elements. +class Tetrahedron; +class Wedge; +class Hexahedron; + +namespace detail { + +// Some tricks are required to keep this code header-only. +template +struct mappings; + +template +struct mappings { + // Map edges of a tetrahedron to vertices and faces. + static const uint32_t edge_mapping[6][2][2]; + + // Map vertices of a tetrahedron to edges and faces. + // 0: local IDs of the edges intersecting at this vertex + // 1: 0 if the edge is pointing away from the vertex, 1 otherwise + // 2: faces joining at the vertex + static const uint32_t vertex_mapping[4][3][3]; + + // This mapping contains the three sets of the two edges for each of the + // faces joining at a vertex. The indices are mapped to the local edge IDs + // using the first value field of the 'vertex_mapping' table. + static const uint32_t face_mapping[3][2]; +}; + +template +const uint32_t mappings::edge_mapping[6][2][2] = { + { { 0, 1 }, { 0, 1 } }, { { 1, 2 }, { 0, 2 } }, { { 2, 0 }, { 0, 3 } }, + { { 0, 3 }, { 1, 3 } }, { { 1, 3 }, { 1, 2 } }, { { 2, 3 }, { 2, 3 } } +}; + +template +const uint32_t mappings::vertex_mapping[4][3][3] = { + { { 0, 2, 3 }, { 0, 1, 0 }, { 0, 1, 3 } }, + { { 0, 1, 4 }, { 1, 0, 0 }, { 0, 1, 2 } }, + { { 1, 2, 5 }, { 1, 0, 0 }, { 0, 2, 3 } }, + { { 3, 4, 5 }, { 1, 1, 1 }, { 1, 3, 2 } } +}; + +template +const uint32_t mappings::face_mapping[3][2] = { + { 0, 1 }, { 0, 2 }, { 1, 2 } +}; + +typedef mappings tet_mappings; + +} // namespace detail + +class Tetrahedron : public detail::tet_mappings { + public: + template + Tetrahedron(const vector_t& v0, Types... verts) : vertices{{v0, + verts...}}, faces(), center(), volume() { + +#ifndef NDEBUG + // Make sure the ordering of the vertices is correct. + assert((vertices[1] - vertices[0]).cross(vertices[2] - + vertices[0]).dot(vertices[3] - vertices[0]) >= scalar_t(0)); +#endif // NDEBUG + + init(); + } + + Tetrahedron(const std::array& verts) : vertices(verts), + faces(), center(), volume() { + + init(); + } + + Tetrahedron() : vertices{{vector_t::Zero(), vector_t::Zero(), + vector_t::Zero(), vector_t::Zero()}}, faces(), center(), volume() { + } + + void apply(const Transformation& t) { + for(auto& v : vertices) + v = t.scaling * (v + t.translation); + + for(auto& f : faces) + f.apply(t); + + center = scalar_t(0.25) * std::accumulate(vertices.begin(), + vertices.end(), vector_t::Zero().eval()); + + volume = calcVolume(); + } + + scalar_t surfaceArea() const { + scalar_t area(0); + for(const auto& f : faces) + area += f.area; + + return area; + } + + private: + void init() { + // 0: v2, v1, v0 + faces[0] = Triangle(vertices[2], vertices[1], vertices[0]); + + // 1: v0, v1, v3 + faces[1] = Triangle(vertices[0], vertices[1], vertices[3]); + + // 2: v1, v2, v3 + faces[2] = Triangle(vertices[1], vertices[2], vertices[3]); + + // 3: v2, v0, v3 + faces[3] = Triangle(vertices[2], vertices[0], vertices[3]); + + center = scalar_t(0.25) * std::accumulate(vertices.begin(), + vertices.end(), vector_t::Zero().eval()); + + volume = calcVolume(); + } + + scalar_t calcVolume() const { + return scalar_t(1.0 / 6.0) * std::abs((vertices[0] - + vertices[3]).dot((vertices[1] - vertices[3]).cross( + vertices[2] - vertices[3]))); + } + + public: + std::array vertices; + std::array faces; + vector_t center; + scalar_t volume; +}; + +namespace detail { + +template +struct mappings { + // Map edges of a wedge to vertices and faces. + static const uint32_t edge_mapping[9][2][2]; + + // Map vertices of a wedge to edges and faces. + // 0: local IDs of the edges intersecting at this vertex + // 1: 0 if the edge is pointing away from the vertex, 1 otherwise + // 2: faces joining at the vertex + static const uint32_t vertex_mapping[6][3][3]; + + // This mapping contains the three sets of the two edges for each of the + // faces joining at a vertex. The indices are mapped to the local edge IDs + // using the first value field of the 'vertex_mapping' table. + static const uint32_t face_mapping[3][2]; +}; + +template +const uint32_t mappings::edge_mapping[9][2][2] = { + { { 0, 1 }, { 0, 1 } }, { { 1, 2 }, { 0, 2 } }, { { 2, 0 }, { 0, 3 } }, + { { 0, 3 }, { 1, 3 } }, { { 1, 4 }, { 1, 2 } }, { { 2, 5 }, { 2, 3 } }, + { { 3, 4 }, { 1, 4 } }, { { 4, 5 }, { 2, 4 } }, { { 5, 3 }, { 3, 4 } } +}; + +template +const uint32_t mappings::vertex_mapping[6][3][3] = { + { { 0, 2, 3 }, { 0, 1, 0 }, { 0, 1, 3 } }, + { { 0, 1, 4 }, { 1, 0, 0 }, { 0, 1, 2 } }, + { { 1, 2, 5 }, { 1, 0, 0 }, { 0, 2, 3 } }, + + { { 3, 6, 8 }, { 1, 0, 1 }, { 1, 3, 4 } }, + { { 4, 6, 7 }, { 1, 1, 0 }, { 1, 2, 4 } }, + { { 5, 7, 8 }, { 1, 1, 0 }, { 2, 3, 4 } } +}; + +template +const uint32_t mappings::face_mapping[3][2] = { + { 0, 1 }, { 0, 2 }, { 1, 2 } +}; + +typedef mappings wedge_mappings; + +} // namespace detail + +class Wedge : public detail::wedge_mappings { + public: + template + Wedge(const vector_t& v0, Types... verts) : vertices{{v0, + verts...}}, faces(), center(), volume() { + + init(); + } + + Wedge(const std::array& verts) : vertices(verts), + faces(), center(), volume() { + + init(); + } + + Wedge() : vertices{{vector_t::Zero(), vector_t::Zero(), + vector_t::Zero(), vector_t::Zero(), vector_t::Zero(), + vector_t::Zero()}}, faces(), center(), volume() { + } + + void apply(const Transformation& t) { + for(auto& v : vertices) + v = t.scaling * (v + t.translation); + + for(auto& f : faces) + f.apply(t); + + center = scalar_t(1.0 / 6.0) * std::accumulate(vertices.begin(), + vertices.end(), vector_t::Zero().eval()); + + volume = calcVolume(); + } + + scalar_t surfaceArea() const { + scalar_t area(0); + for(const auto& f : faces) + area += f.area; + + return area; + } + + private: + void init() { + // All faces of the wedge are stored as quadrilaterals, so an + // additional point is inserted between v0 and v1. + // 0: v2, v1, v0, v02 + faces[0] = Quadrilateral(vertices[2], vertices[1], vertices[0], + scalar_t(0.5) * (vertices[0] + vertices[2])); + + // 1: v0, v1, v4, v3 + faces[1] = Quadrilateral(vertices[0], vertices[1], vertices[4], + vertices[3]); + + // 2: v1, v2, v5, v4 + faces[2] = Quadrilateral(vertices[1], vertices[2], vertices[5], + vertices[4]); + + // 3: v2, v0, v3, v5 + faces[3] = Quadrilateral(vertices[2], vertices[0], vertices[3], + vertices[5]); + + // All faces of the wedge are stored as quadrilaterals, so an + // additional point is inserted between v3 and v5. + // 4: v3, v4, v5, v53 + faces[4] = Quadrilateral(vertices[3], vertices[4], vertices[5], + scalar_t(0.5) * (vertices[5] + vertices[3])); + + center = scalar_t(1.0 / 6.0) * std::accumulate(vertices.begin(), + vertices.end(), vector_t::Zero().eval()); + + volume = calcVolume(); + } + + scalar_t calcVolume() const { + // The wedge is treated as a degenerate hexahedron here by adding + // two fake vertices v02 and v35. + vector_t diagonal(vertices[5] - vertices[0]); + + return scalar_t(1.0 / 6.0) * (diagonal.dot(((vertices[1] - + vertices[0]).cross(vertices[2] - vertices[4])) + + ((vertices[3] - vertices[0]).cross( + vertices[4] - scalar_t(0.5) * (vertices[3] + vertices[5]))) + + ((scalar_t(0.5) * (vertices[0] + vertices[2]) - + vertices[0]).cross(scalar_t(0.5) * (vertices[3] + + vertices[5]) - vertices[2])))); + } + + public: + std::array vertices; + std::array faces; + vector_t center; + scalar_t volume; +}; + +namespace detail { + +template +struct mappings { + // Map edges of a hexahedron to vertices and faces. + static const uint32_t edge_mapping[12][2][2]; + + // Map vertices of a hexahedron to edges and faces. + // 0: local IDs of the edges intersecting at this vertex + // 1: 0 if the edge is pointing away from the vertex, 1 otherwise + // 2: faces joining at the vertex + static const uint32_t vertex_mapping[8][3][3]; + + // This mapping contains the three sets of the two edges for each of the + // faces joining at a vertex. The indices are mapped to the local edge IDs + // using the first value field of the 'vertex_mapping' table. + static const uint32_t face_mapping[3][2]; +}; + +template +const uint32_t mappings::edge_mapping[12][2][2] = { + { { 0, 1 }, { 0, 1 } }, { { 1, 2 }, { 0, 2 } }, + { { 2, 3 }, { 0, 3 } }, { { 3, 0 }, { 0, 4 } }, + + { { 0, 4 }, { 1, 4 } }, { { 1, 5 }, { 1, 2 } }, + { { 2, 6 }, { 2, 3 } }, { { 3, 7 }, { 3, 4 } }, + + { { 4, 5 }, { 1, 5 } }, { { 5, 6 }, { 2, 5 } }, + { { 6, 7 }, { 3, 5 } }, { { 7, 4 }, { 4, 5 } } +}; + +template +const uint32_t mappings::vertex_mapping[8][3][3] = { + { { 0, 3, 4 }, { 0, 1, 0 }, { 0, 1, 4 } }, + { { 0, 1, 5 }, { 1, 0, 0 }, { 0, 1, 2 } }, + { { 1, 2, 6 }, { 1, 0, 0 }, { 0, 2, 3 } }, + { { 2, 3, 7 }, { 1, 0, 0 }, { 0, 3, 4 } }, + + { { 4, 8, 11 }, { 1, 0, 1 }, { 1, 4, 5 } }, + { { 5, 8, 9 }, { 1, 1, 0 }, { 1, 2, 5 } }, + { { 6, 9, 10 }, { 1, 1, 0 }, { 2, 3, 5 } }, + { { 7, 10, 11 }, { 1, 1, 0 }, { 3, 4, 5 } } +}; + +template +const uint32_t mappings::face_mapping[3][2] = { + { 0, 1 }, { 0, 2 }, { 1, 2 } +}; + +typedef mappings hex_mappings; + +} // namespace detail + +class Hexahedron : public detail::hex_mappings { + public: + template + Hexahedron(const vector_t& v0, Types... verts) : vertices{{v0, + verts...}}, faces(), center(), volume() { + + init(); + } + + Hexahedron(const std::array& verts) : vertices(verts), + faces(), center(), volume() { + + init(); + } + + void apply(const Transformation& t) { + for(auto& v : vertices) + v = t.scaling * (v + t.translation); + + for(auto& f : faces) + f.apply(t); + + center = scalar_t(1.0 / 8.0) * std::accumulate(vertices.begin(), + vertices.end(), vector_t::Zero().eval()); + + volume = calcVolume(); + } + + scalar_t surfaceArea() const { + scalar_t area(0); + for(const auto& f : faces) + area += f.area; + + return area; + } + + private: + void init() { + // 0: v3, v2, v1, v0 + faces[0] = Quadrilateral(vertices[3], vertices[2], vertices[1], + vertices[0]); + + // 1: v0, v1, v5, v4 + faces[1] = Quadrilateral(vertices[0], vertices[1], vertices[5], + vertices[4]); + + // 2: v1, v2, v6, v5 + faces[2] = Quadrilateral(vertices[1], vertices[2], vertices[6], + vertices[5]); + + // 3: v2, v3, v7, v6 + faces[3] = Quadrilateral(vertices[2], vertices[3], vertices[7], + vertices[6]); + + // 4: v3, v0, v4, v7 + faces[4] = Quadrilateral(vertices[3], vertices[0], vertices[4], + vertices[7]); + + // 5: v4, v5, v6, v7 + faces[5] = Quadrilateral(vertices[4], vertices[5], vertices[6], + vertices[7]); + + center = scalar_t(1.0 / 8.0) * std::accumulate(vertices.begin(), + vertices.end(), vector_t::Zero().eval()); + + volume = calcVolume(); + } + + scalar_t calcVolume() const { + vector_t diagonal(vertices[6] - vertices[0]); + + return scalar_t(1.0 / 6.0) * diagonal.dot(((vertices[1] - + vertices[0]).cross(vertices[2] - vertices[5])) + + ((vertices[4] - vertices[0]).cross( + vertices[5] - vertices[7])) + + ((vertices[3] - vertices[0]).cross( + vertices[7] - vertices[2]))); + } + + public: + std::array vertices; + std::array faces; + vector_t center; + scalar_t volume; +}; + +class Sphere { + public: + Sphere(const vector_t& c, scalar_t r) : center(c), radius(r), + volume(scalar_t(4.0 / 3.0 * pi) * r * r * r) { + } + + scalar_t capVolume(scalar_t h) const { + if(h <= scalar_t(0)) + return scalar_t(0); + else if(h >= scalar_t(2) * radius) + return volume; + else + return scalar_t(pi / 3.0) * h * h * (scalar_t(3) * radius - h); + } + + scalar_t capSurfaceArea(scalar_t h) const { + if(h <= scalar_t(0)) + return scalar_t(0); + else if(h >= scalar_t(2) * radius) + return surfaceArea(); + else + return scalar_t(2 * pi) * radius * h; + } + + scalar_t diskArea(scalar_t h) const { + if(h <= scalar_t(0) || h >= scalar_t(2) * radius) + return scalar_t(0); + else + return pi * h * (scalar_t(2) * radius - h); + } + + scalar_t surfaceArea() const { + return (scalar_t(4) * pi) * (radius * radius); + } + + public: + vector_t center; + scalar_t radius; + scalar_t volume; +}; + +class Plane { + public: + Plane(const vector_t& c, const vector_t& n) : center(c), normal(n) { + } + + public: + vector_t center; + vector_t normal; +}; + +class AABB { + public: + AABB(const vector_t& minimum = vector_t::Constant(std::numeric_limits< + scalar_t>::infinity()), const vector_t& maximum = + vector_t::Constant(-std::numeric_limits::infinity())) : + min(minimum), max(maximum) { + } + + bool intersects(const AABB& aabb) const { + if((min.array() > aabb.max.array()).any() || + (max.array() < aabb.min.array()).any()) + return false; + + return true; + } + + AABB overlap(const AABB& aabb) const { + return AABB(min.cwiseMax(aabb.min), max.cwiseMin(aabb.max)); + } + + bool contains(const vector_t& p) const { + if((p.array() < min.array()).any() || + (p.array() > max.array()).any()) + return false; + + return true; + } + + void include(const vector_t& point) { + min = min.cwiseMin(point); + max = max.cwiseMax(point); + } + + template + void include(const std::array& points) { + for(const auto& p : points) + include(p); + } + + scalar_t volume() const { + vector_t size(max - min); + + return size[0] * size[1] * size[2]; + } + + public: + vector_t min, max; +}; + +// Decomposition of a tetrahedron into 4 tetrahedra. +inline void decompose(const Tetrahedron& tet, std::array& + tets) { + + tets[0] = Tetrahedron(tet.vertices[0], tet.vertices[1], tet.vertices[2], + tet.center); + + tets[1] = Tetrahedron(tet.vertices[0], tet.vertices[1], tet.center, + tet.vertices[3]); + + tets[2] = Tetrahedron(tet.vertices[1], tet.vertices[2], tet.center, + tet.vertices[3]); + + tets[3] = Tetrahedron(tet.vertices[2], tet.vertices[0], tet.center, + tet.vertices[3]); +} + +// Decomposition of a hexahedron into 2 wedges. +inline void decompose(const Hexahedron& hex, std::array& wedges) { + wedges[0] = Wedge(hex.vertices[0], hex.vertices[1], hex.vertices[2], + hex.vertices[4], hex.vertices[5], hex.vertices[6]); + + wedges[1] = Wedge(hex.vertices[0], hex.vertices[2], hex.vertices[3], + hex.vertices[4], hex.vertices[6], hex.vertices[7]); +} + +// Decomposition of a hexahedron into 5 tetrahedra. +inline void decompose(const Hexahedron& hex, std::array& + tets) { + + tets[0] = Tetrahedron(hex.vertices[0], hex.vertices[1], hex.vertices[2], + hex.vertices[5]); + + tets[1] = Tetrahedron(hex.vertices[0], hex.vertices[2], hex.vertices[7], + hex.vertices[5]); + + tets[2] = Tetrahedron(hex.vertices[0], hex.vertices[2], hex.vertices[3], + hex.vertices[7]); + + tets[3] = Tetrahedron(hex.vertices[0], hex.vertices[5], hex.vertices[7], + hex.vertices[4]); + + tets[4] = Tetrahedron(hex.vertices[2], hex.vertices[7], hex.vertices[5], + hex.vertices[6]); +} + +// Decomposition of a hexahedron into 6 tetrahedra. +inline void decompose(const Hexahedron& hex, std::array& + tets) { + + tets[0] = Tetrahedron(hex.vertices[0], hex.vertices[5], hex.vertices[7], + hex.vertices[4]); + + tets[1] = Tetrahedron(hex.vertices[0], hex.vertices[1], hex.vertices[7], + hex.vertices[5]); + + tets[2] = Tetrahedron(hex.vertices[1], hex.vertices[6], hex.vertices[7], + hex.vertices[5]); + + tets[3] = Tetrahedron(hex.vertices[0], hex.vertices[7], hex.vertices[2], + hex.vertices[3]); + + tets[4] = Tetrahedron(hex.vertices[0], hex.vertices[7], hex.vertices[1], + hex.vertices[2]); + + tets[5] = Tetrahedron(hex.vertices[1], hex.vertices[7], hex.vertices[6], + hex.vertices[2]); +} + +inline bool contains(const Sphere& s, const vector_t& p) { + return (s.center - p).squaredNorm() <= s.radius * s.radius; +} + +// The (convex!) polygon is assumed to be planar, making this a 2D problem. +// Check the projection of the point onto the plane of the polygon for +// containment within the polygon. +template +bool contains(const Polygon& poly, const vector_t& point) { + const vector_t proj(point - poly.normal.dot(point - poly.center) * + poly.normal); + + for(size_t n = 0; n < poly.vertices.size(); ++n) { + const auto& v0 = poly.vertices[n]; + const auto& v1 = poly.vertices[(n + 1) % poly.vertices.size()]; + vector_t base(scalar_t(0.5) * (v0 + v1)); + vector_t edge(v1 - v0); + + // Note: Only the sign of the projection is of interest, so this vector + // does not have to be normalized. + vector_t dir(edge.cross(poly.normal)); + + // Check whether the projection of the point lies inside of the + // polygon. + if(dir.dot(proj - base) > scalar_t(0)) + return false; + } + + return true; +} + +inline bool contains(const Tetrahedron& tet, const vector_t& p) { + for(const auto& f : tet.faces) + if(f.normal.dot(p - f.center) > scalar_t(0)) + return false; + + return true; +} + +inline bool contains(const Wedge& wedge, const vector_t& p) { + for(const auto& f : wedge.faces) + if(f.normal.dot(p - f.center) > scalar_t(0)) + return false; + + return true; +} + +inline bool contains(const Hexahedron& hex, const vector_t& p) { + for(const auto& f : hex.faces) + if(f.normal.dot(p - f.center) > scalar_t(0)) + return false; + + return true; +} + +inline bool intersect(const Sphere& s, const Plane& p) { + scalar_t proj = p.normal.dot(s.center - p.center); + + return proj * proj - s.radius * s.radius < scalar_t(0); +} + +template +inline bool intersect(const Sphere& s, const Polygon& poly) { + return intersect(s, Plane(poly.center, poly.normal)) && contains(poly, + s.center); +} + +inline std::pair, size_t> lineSphereIntersection(const + vector_t& origin, const vector_t& direction, const Sphere& s) { + + std::array solutions = {{ + std::numeric_limits::infinity(), + std::numeric_limits::infinity() + }}; + + vector_t originRel(origin - s.center); + scalar_t a = direction.squaredNorm(); + + if(a == scalar_t(0)) + return std::make_pair(solutions, 0); + + scalar_t b = scalar_t(2) * direction.dot(originRel); + scalar_t c = originRel.squaredNorm() - s.radius * s.radius; + + scalar_t discriminant = b * b - scalar_t(4) * a * c; + if(discriminant > scalar_t(0)) { + // Two real roots. + scalar_t q = scalar_t(-0.5) * (b + + std::copysign(std::sqrt(discriminant), b)); + + solutions[0] = q / a; + solutions[1] = c / q; + + if(solutions[0] > solutions[1]) + std::swap(solutions[0], solutions[1]); + + return std::make_pair(solutions, 2); + } else if(std::abs(discriminant) == scalar_t(0)) { + // Double real root. + solutions[0] = (scalar_t(-0.5) * b) / a; + solutions[1] = solutions[0]; + + return std::make_pair(solutions, 1); + } else { + // No real roots. + return std::make_pair(solutions, 0); + } +} + +namespace detail { + +// Calculate the volume of a regularized spherical wedge defined by the radius, +// the distance of the intersection point from the center of the sphere and the +// angle. +inline scalar_t regularizedWedge(scalar_t r, scalar_t d, scalar_t alpha) { +#ifndef NDEBUG + // Clamp slight deviations of the angle to valid range. + if(alpha < scalar_t(0) && alpha > -detail::tinyEpsilon) + alpha = scalar_t(0); + + if(alpha > scalar_t(0.5 * pi) && alpha < scalar_t(0.5 * pi) + tinyEpsilon) + alpha = scalar_t(0.5 * pi); +#endif + + // Check the parameters for validity (debug version only). + assert(r > scalar_t(0)); + assert(d >= scalar_t(0) && d <= r); + assert(alpha >= scalar_t(0) && alpha <= scalar_t(0.5 * pi)); + + const scalar_t sinAlpha = std::sin(alpha); + const scalar_t cosAlpha = std::cos(alpha); + + const scalar_t a = d * sinAlpha; + const scalar_t b = std::sqrt(std::abs(r * r - d * d)); + const scalar_t c = d * cosAlpha; + + return scalar_t(1.0 / 3.0) * a * b * c + + a * (scalar_t(1.0 / 3.0) * a * a - r * r) * std::atan2(b, c) + + scalar_t(2.0 / 3.0) * r * r * r * std::atan2(sinAlpha * b, + cosAlpha * r); +} + +// Wrapper around the above function handling correctly handling the case of +// alpha > pi/2 and negative z. +inline scalar_t regularizedWedge(scalar_t r, scalar_t d, scalar_t alpha, + scalar_t z) { + + if(z >= scalar_t(0)) { + if(alpha > scalar_t(0.5 * pi)) { + scalar_t h = r - z; + + return scalar_t(pi / 3.0) * h * h * (scalar_t(3) * r - h) - + regularizedWedge(r, d, pi - alpha); + } else { + return regularizedWedge(r, d, alpha); + } + } else { + scalar_t vHem = scalar_t(2.0 / 3.0 * pi) * r * r * r; + + if(alpha > scalar_t(0.5 * pi)) { + return vHem - regularizedWedge(r, d, pi - alpha); + } else { + scalar_t h = r + z; + scalar_t vCap = scalar_t(pi / 3.0) * h * h * (scalar_t(3) * r - h); + + return vHem - (vCap - regularizedWedge(r, d, alpha)); + } + } +} + +// Calculate the surface area of a regularized spherical wedge defined by the +// radius, the distance of the intersection point from the center of the sphere +// and the angle. +// Ref: Gibson, K. D. & Scheraga, H. A.: Exact calculation of the volume and +// surface area of fused hard-sphere molecules with unequal atomic radii, +// Molecular Physics, 1987, 62, 1247-1265 +inline scalar_t regularizedWedgeArea(scalar_t r, scalar_t z, scalar_t alpha) { +#ifndef NDEBUG + // Clamp slight deviations of the angle to valid range. + if(alpha < scalar_t(0) && alpha > -detail::tinyEpsilon) + alpha = scalar_t(0); + + if(alpha > pi && alpha < pi + tinyEpsilon) + alpha = pi; +#endif + + // Check the parameters for validity (debug version only). + assert(r > scalar_t(0)); + assert(z >= -r && z <= r); + assert(alpha >= scalar_t(0) && alpha <= pi); + + if(alpha < tinyEpsilon || std::abs(r * r - z * z) <= tinyEpsilon) + return scalar_t(0); + + const scalar_t sinAlpha = std::sin(alpha); + const scalar_t cosAlpha = std::cos(alpha); + const scalar_t factor = scalar_t(1) / std::sqrt(std::abs(r * r - z * z)); + + // Clamp slight deviations of the argument to acos() to valid range. + const scalar_t arg0 = clamp(r * cosAlpha * factor, scalar_t(-1), + scalar_t(1), detail::tinyEpsilon); + + const scalar_t arg1 = clamp((z * cosAlpha * factor) / sinAlpha, + scalar_t(-1), scalar_t(1), detail::tinyEpsilon); + + // Check the argument to acos() for validity (debug version only). + assert(scalar_t(-1) <= arg0 && arg0 <= scalar_t(1)); + assert(scalar_t(-1) <= arg1 && arg1 <= scalar_t(1)); + + return scalar_t(2) * r * r * std::acos(arg0) - + scalar_t(2) * r * z * std::acos(arg1); +} + +} // namespace detail + +// Depending on the dimensionality, either the volume or external surface area +// of the general wedge is computed. +template +inline scalar_t generalWedge(const Sphere& s, const Plane& p0, const Plane& p1, + const vector_t& d) { + + static_assert(Dim == 2 || Dim == 3, + "Invalid dimensionality, must be 2 or 3."); + + scalar_t dist(d.stableNorm()); + + if(dist < detail::tinyEpsilon) { + // The wedge (almost) touches the center, the volume depends only on + // the angle. + scalar_t angle = pi - detail::angle(p0.normal, p1.normal); + + if(Dim == 2) { + return scalar_t(2) * s.radius * s.radius * angle; + } else { + return scalar_t(2.0 / 3.0) * s.radius * s.radius * s.radius * + angle; + } + } + + scalar_t s0 = d.dot(p0.normal); + scalar_t s1 = d.dot(p1.normal); + + // Detect degenerated general spherical wedge that can be treated as + // a regularized spherical wedge. + if(std::abs(s0) < detail::tinyEpsilon || + std::abs(s1) < detail::tinyEpsilon) { + + scalar_t angle = pi - detail::angle(p0.normal, p1.normal); + + if(Dim == 2) { + return detail::regularizedWedgeArea(s.radius, + std::abs(s0) > std::abs(s1) ? s0 : s1, angle); + } else { + return detail::regularizedWedge(s.radius, dist, angle, + std::abs(s0) > std::abs(s1) ? s0 : s1); + } + } + + vector_t dUnit(d * (scalar_t(1) / dist)); + if(dist < detail::largeEpsilon) + dUnit = detail::gramSchmidt(p0.normal.cross(p1.normal), dUnit)[1]; + + // Check the planes specify a valid setup (debug version only). + assert(p0.normal.dot(p1.center - p0.center) <= scalar_t(0)); + assert(p1.normal.dot(p0.center - p1.center) <= scalar_t(0)); + + // Calculate the angles between the vector from the sphere center + // to the intersection line and the normal vectors of the two planes. + scalar_t alpha0 = detail::angle(p0.normal, dUnit); + scalar_t alpha1 = detail::angle(p1.normal, dUnit); + + scalar_t dir0 = dUnit.dot((s.center + d) - p0.center); + scalar_t dir1 = dUnit.dot((s.center + d) - p1.center); + + if(s0 >= scalar_t(0) && s1 >= scalar_t(0)) { + alpha0 = scalar_t(0.5 * pi) - std::copysign(alpha0, dir0); + alpha1 = scalar_t(0.5 * pi) - std::copysign(alpha1, dir1); + + if(Dim == 2) { + return detail::regularizedWedgeArea(s.radius, s0, alpha0) + + detail::regularizedWedgeArea(s.radius, s1, alpha1); + } else { + return detail::regularizedWedge(s.radius, dist, alpha0, s0) + + detail::regularizedWedge(s.radius, dist, alpha1, s1); + } + } else if(s0 < scalar_t(0) && s1 < scalar_t(0)) { + alpha0 = scalar_t(0.5 * pi) + std::copysign(scalar_t(1), dir0) * + (alpha0 - pi); + + alpha1 = scalar_t(0.5 * pi) + std::copysign(scalar_t(1), dir1) * + (alpha1 - pi); + + if(Dim == 2) { + return s.surfaceArea() - + (detail::regularizedWedgeArea(s.radius, -s0, alpha0) + + detail::regularizedWedgeArea(s.radius, -s1, alpha1)); + } else { + return s.volume - (detail::regularizedWedge(s.radius, dist, alpha0, + -s0) + detail::regularizedWedge(s.radius, dist, alpha1, -s1)); + } + } else { + alpha0 = scalar_t(0.5 * pi) - std::copysign(scalar_t(1), dir0 * s0) * + (alpha0 - (s0 < scalar_t(0) ? pi : scalar_t(0))); + + alpha1 = scalar_t(0.5 * pi) - std::copysign(scalar_t(1), dir1 * s1) * + (alpha1 - (s1 < scalar_t(0) ? pi : scalar_t(0))); + + if(Dim == 2) { + scalar_t area0 = detail::regularizedWedgeArea(s.radius, + std::abs(s0), alpha0); + + scalar_t area1 = detail::regularizedWedgeArea(s.radius, + std::abs(s1), alpha1); + + return std::max(area0, area1) - std::min(area0, area1); + } else { + scalar_t volume0 = detail::regularizedWedge(s.radius, dist, alpha0, + std::abs(s0)); + + scalar_t volume1 = detail::regularizedWedge(s.radius, dist, alpha1, + std::abs(s1)); + + return std::max(volume0, volume1) - std::min(volume0, volume1); + } + } +} + +template +struct array_size; + +template +struct array_size> { + + static constexpr size_t value() { + return N; + } +}; + +template +constexpr size_t nrEdges() { + return std::is_same::value ? 12 : + (std::is_same::value ? 9 : + (std::is_same::value ? 6 : -1)); +} + +// Workaround for the Intel compiler, as it does not yet support constexpr for +// template arguments. +template +struct element_trait { + static const size_t nrVertices = + array_size::value(); + + static const size_t nrFaces = + array_size::value(); +}; + +// Depending on the dimensionality, either the volume or external surface area +// of the general wedge is computed. +template +scalar_t generalWedge(const Sphere& sphere, const Element& element, size_t + edge, const std::array, nrEdges()>& + intersections) { + + static_assert(Dim == 2 || Dim == 3, + "Invalid dimensionality, must be 2 or 3."); + + const auto& f0 = element.faces[Element::edge_mapping[edge][1][0]]; + const auto& f1 = element.faces[Element::edge_mapping[edge][1][1]]; + + vector_t edgeCenter(scalar_t(0.5) * ((intersections[edge][0] + + element.vertices[Element::edge_mapping[edge][0][0]]) + + (intersections[edge][1] + + element.vertices[Element::edge_mapping[edge][0][1]]))); + + Plane p0(f0.center, f0.normal); + Plane p1(f1.center, f1.normal); + + return generalWedge(sphere, p0, p1, edgeCenter - sphere.center); +} + +template +scalar_t overlap(const Sphere& sOrig, const Element& elementOrig) { + static_assert(std::is_same::value || + std::is_same::value || + std::is_same::value, + "Invalid element type detected."); + + // Construct AABBs and perform a coarse overlap detection. + AABB sAABB(sOrig.center - vector_t::Constant(sOrig.radius), sOrig.center + + vector_t::Constant(sOrig.radius)); + + AABB eAABB; + eAABB.include(elementOrig.vertices); + + if(!sAABB.intersects(eAABB)) + return scalar_t(0); + + // Use scaled and shifted versions of the sphere and the element. + Transformation transformation(-sOrig.center, scalar_t(1) / sOrig.radius); + + Sphere s(vector_t::Zero(), scalar_t(1)); + + Element element(elementOrig); + element.apply(transformation); + + // Constants: Number of vertices and faces. + static const size_t nrVertices = element_trait::nrVertices; + static const size_t nrFaces = element_trait::nrFaces; + + size_t vOverlap = 0; + // Check whether the vertices lie on or outside of the sphere. + for(const auto& vertex : element.vertices) + if((s.center - vertex).squaredNorm() <= s.radius * s.radius) + ++vOverlap; + + // Check for trivial case: All vertices inside of the sphere, resulting in + // a full overlap. + if(vOverlap == nrVertices) + return elementOrig.volume; + + // Sanity check: All faces of the mesh element have to be planar. + for(const auto& face : element.faces) + if(!face.isPlanar()) + throw std::runtime_error("Non-planer face detected in element!"); + + // Sets of overlapping primitives. + std::bitset vMarked; + std::bitset()> eMarked; + std::bitset fMarked; + + // Initial value: Volume of the full sphere. + scalar_t result = s.volume; + + // The intersection points between the single edges and the sphere, this + // is needed later on. + std::array, nrEdges()> eIntersections; + + // Process all edges of the element. + for(size_t n = 0; n < nrEdges(); ++n) { + vector_t start(element.vertices[Element::edge_mapping[n][0][0]]); + vector_t direction(element.vertices[Element::edge_mapping[n][0][1]] - + start); + + auto solutions = lineSphereIntersection(start, direction, s); + + // No intersection between the edge and the sphere, where intersection + // points close to the surface of the sphere are ignored. + // Or: + // The sphere cuts the edge twice, no vertex is inside of the + // sphere, but the case of the edge only touching the sphere has to + // be avoided. + if(!solutions.second || + (solutions.first[0] >= scalar_t(1) - detail::mediumEpsilon) || + solutions.first[1] <= detail::mediumEpsilon || + (solutions.first[0] > scalar_t(0) && + solutions.first[1] < scalar_t(1) && + (solutions.first[1] - solutions.first[0] < + detail::largeEpsilon))) { + + continue; + } else { + vMarked[Element::edge_mapping[n][0][0]] = + solutions.first[0] < scalar_t(0); + + vMarked[Element::edge_mapping[n][0][1]] = + solutions.first[1] > scalar_t(1); + } + + // Store the two intersection points of the edge with the sphere for + // later usage. + eIntersections[n][0] = solutions.first[0] * direction; + eIntersections[n][1] = (solutions.first[1] - scalar_t(1)) * direction; + eMarked[n] = true; + + // If the edge is marked as having an overlap, the two faces forming it + // have to be marked as well. + fMarked[Element::edge_mapping[n][1][0]] = true; + fMarked[Element::edge_mapping[n][1][1]] = true; + } + + // Check whether the dependencies for a vertex intersection are fulfilled. + for(size_t n = 0; n < nrVertices; ++n) { + if(!vMarked[n]) + continue; + + bool edgesValid = true; + for(size_t eN = 0; eN < 3; ++eN) { + size_t edgeId = Element::vertex_mapping[n][0][eN]; + edgesValid &= eMarked[edgeId]; + } + + // If not all three edges intersecting at this vertex where marked, the + // sphere is only touching. + if(!edgesValid) + vMarked[n] = false; + } + + // Process all faces of the element, ignoring the edges as those where + // already checked above. + for(size_t n = 0; n < nrFaces; ++n) + if(intersect(s, element.faces[n])) + fMarked[n] = true; + + // Trivial case: The center of the sphere overlaps the element, but the + // sphere does not intersect any of the faces of the element, meaning the + // sphere is completely contained within the element. + if(!fMarked.count() && contains(element, s.center)) + return sOrig.volume; + + // Spurious intersection: The initial intersection test was positive, but + // the detailed checks revealed no overlap. + if(!vMarked.count() && !eMarked.count() && !fMarked.count()) + return scalar_t(0); + + // Iterate over all the marked faces and subtract the volume of the cap cut + // off by the plane. + for(size_t n = 0; n < nrFaces; ++n) { + if(!fMarked[n]) + continue; + + const auto& f = element.faces[n]; + scalar_t dist = f.normal.dot(s.center - f.center); + scalar_t vCap = s.capVolume(s.radius + dist); + + result -= vCap; + } + + // Handle the edges and add back the volume subtracted twice above in the + // processing of the faces. + for(size_t n = 0; n < nrEdges(); ++n) { + if(!eMarked[n]) + continue; + + scalar_t edgeCorrection = generalWedge<3, Element>(s, element, n, + eIntersections); + + result += edgeCorrection; + } + + // Handle the vertices and subtract the volume added twice above in the + // processing of the edges. + for(size_t n = 0; n < nrVertices; ++n) { + if(!vMarked[n]) + continue; + + // Collect the points where the three edges intersecting at this + // vertex intersect the sphere. + // Both the relative and the absolute positions are required. + std::array intersectionPointsRelative; + std::array intersectionPoints; + for(size_t e = 0; e < 3; ++e) { + auto edgeIdx = Element::vertex_mapping[n][0][e]; + intersectionPointsRelative[e] = + eIntersections[edgeIdx][Element::vertex_mapping[n][1][e]]; + + intersectionPoints[e] = intersectionPointsRelative[e] + + element.vertices[n]; + } + + // This triangle is constructed by hand to have more freedom of how + // the normal vector is calculated. + Triangle coneTria; + coneTria.vertices = {{ intersectionPoints[0], intersectionPoints[1], + intersectionPoints[2] }}; + + coneTria.center = scalar_t(1.0 / 3.0) * + std::accumulate(intersectionPoints.begin(), + intersectionPoints.end(), vector_t::Zero().eval()); + + // Calculate the normal of the triangle defined by the intersection + // points in relative coordinates to improve accuracy. + // Also use double the normal precision to calculate this normal. + coneTria.normal = detail::triangleNormal(intersectionPointsRelative[0], + intersectionPointsRelative[1], intersectionPointsRelative[2]); + + // The area of this triangle is never needed, so it is set to an + // invalid value. + coneTria.area = std::numeric_limits::infinity(); + + std::array, 3> distances; + for(size_t i = 0; i < 3; ++i) + distances[i] = std::make_pair(i, + intersectionPointsRelative[i].squaredNorm()); + + std::sort(distances.begin(), distances.end(), + [](const std::pair& a, + const std::pair& b) -> bool { + return a.second < b.second; + }); + + if(distances[1].second < distances[2].second * detail::largeEpsilon) { + // Use the general spherical wedge defined by the edge with the + // non-degenerated intersection point and the normals of the + // two faces forming it. + scalar_t correction = generalWedge<3, Element>(s, element, + Element::vertex_mapping[n][0][distances[2].first], + eIntersections); + + result -= correction; + + continue; + } + + scalar_t tipTetVolume = scalar_t(1.0 / 6.0) * std::abs( + -intersectionPointsRelative[2].dot((intersectionPointsRelative[0] - + intersectionPointsRelative[2]).cross( + intersectionPointsRelative[1] - intersectionPointsRelative[2]))); + + // Make sure the normal points in the right direction i.e. away from + // the center of the element. + if(coneTria.normal.dot(element.center - coneTria.center) > + scalar_t(0)) { + + coneTria.normal = -coneTria.normal; + } + + Plane plane(coneTria.center, coneTria.normal); + + scalar_t dist = coneTria.normal.dot(s.center - coneTria.center); + scalar_t capVolume = s.capVolume(s.radius + dist); + + // The cap volume is tiny, so the corrections will be even smaller. + // There is no way to actually calculate them with reasonable + // precision, so just the volume of the tetrahedron at the tip is + // used. + if(capVolume < detail::tinyEpsilon) { + result -= tipTetVolume; + continue; + } + + // Calculate the volume of the three spherical segments between + // the faces joining at the vertex and the plane through the + // intersection points. + scalar_t segmentVolume = 0; + + for(size_t e = 0; e < 3; ++e) { + const auto& f = element.faces[Element::vertex_mapping[n][2][e]]; + uint32_t e0 = Element::face_mapping[e][0]; + uint32_t e1 = Element::face_mapping[e][1]; + + vector_t center(scalar_t(0.5) * (intersectionPoints[e0] + + intersectionPoints[e1])); + + scalar_t wedgeVolume = generalWedge<3>(s, plane, Plane(f.center, + -f.normal), center - s.center); + + segmentVolume += wedgeVolume; + } + + // Calculate the volume of the cone and clamp it to zero. + scalar_t coneVolume = std::max(tipTetVolume + capVolume - + segmentVolume, scalar_t(0)); + + // Sanity check: detect negative cone volume. + assert(coneVolume > -std::sqrt(detail::tinyEpsilon)); + + result -= coneVolume; + + // Sanity check: detect negative intermediate result. + assert(result > -std::sqrt(detail::tinyEpsilon)); + } + + // In case of different sized objects the error can become quite large, + // so a relative limit is used. + scalar_t maxOverlap = std::min(s.volume, element.volume); + const scalar_t limit(std::sqrt(std::numeric_limits::epsilon()) * + maxOverlap); + + // Clamp tiny negative volumes to zero. + if(result < scalar_t(0) && result > -limit) + return scalar_t(0); + + // Clamp results slightly too large. + if(result > maxOverlap && result - maxOverlap < limit) + return std::min(sOrig.volume, elementOrig.volume); + + // Perform a sanity check on the final result (debug version only). + assert(result >= scalar_t(0) && result <= maxOverlap); + + // Scale the overlap volume back for the original objects. + result = (result / s.volume) * sOrig.volume; + + return result; +} + +template +scalar_t overlap(const Sphere& s, Iterator eBegin, Iterator eEnd) { + scalar_t sum(0); + + for(Iterator it = eBegin; it != eEnd; ++it) + sum += overlap(s, *it); + + return sum; +} + +// Calculate the surface area of the sphere and the element that are contained +// within the common or intersecting part of the geometries, respectively. +// The returned array of size (N + 2), with N being the number of vertices, +// holds (in this order): +// - surface area of the region of the sphere intersecting the element +// - for each face of the element: area contained within the sphere +// - total surface area of the element intersecting the sphere +template::nrFaces + + 2> +auto overlapArea(const Sphere& sOrig, const Element& elementOrig) -> + std::array { + + static_assert(NrFaces == element_trait::nrFaces + 2, + "Invalid number of faces for the element provided."); + + static_assert(std::is_same::value || + std::is_same::value || + std::is_same::value, + "Invalid element type detected."); + + // Constants: Number of vertices and faces. + static const size_t nrVertices = element_trait::nrVertices; + static const size_t nrFaces = element_trait::nrFaces; + + // Initial value: Zero overlap. + std::array result; + result.fill(scalar_t(0)); + + // Construct AABBs and perform a coarse overlap detection. + AABB sAABB(sOrig.center - vector_t::Constant(sOrig.radius), sOrig.center + + vector_t::Constant(sOrig.radius)); + + AABB eAABB; + eAABB.include(elementOrig.vertices); + + if(!sAABB.intersects(eAABB)) + return result; + + // Use scaled and shifted versions of the sphere and the element. + Transformation transformation(-sOrig.center, scalar_t(1) / sOrig.radius); + + Sphere s(vector_t::Zero(), scalar_t(1)); + + Element element(elementOrig); + element.apply(transformation); + + size_t vOverlap = 0; + // Check whether the vertices lie on or outside of the sphere. + for(const auto& vertex : element.vertices) + if((s.center - vertex).squaredNorm() <= s.radius * s.radius) + ++vOverlap; + + // Check for trivial case: All vertices inside of the sphere, resulting in + // a full coverage of all faces. + if(vOverlap == nrVertices) { + for(size_t n = 0; n < nrFaces; ++n) { + result[n + 1] = elementOrig.faces[n].area; + result[nrFaces + 1] += elementOrig.faces[n].area; + } + + return result; + } + + // Sanity check: All faces of the mesh element have to be planar. + for(const auto& face : element.faces) + if(!face.isPlanar()) + throw std::runtime_error("Non-planer face detected in element!"); + + // Sets of overlapping primitives. + std::bitset vMarked; + std::bitset()> eMarked; + std::bitset fMarked; + + // The intersection points between the single edges and the sphere, this + // is needed later on. + std::array, nrEdges()> eIntersections; + + // Cache the squared radius of the disk formed by the intersection between + // the planes defined by each face and the sphere. + std::array intersectionRadiusSq; + + // Process all edges of the element. + for(size_t n = 0; n < nrEdges(); ++n) { + vector_t start(element.vertices[Element::edge_mapping[n][0][0]]); + vector_t direction(element.vertices[Element::edge_mapping[n][0][1]] - + start); + + auto solutions = lineSphereIntersection(start, direction, s); + + // No intersection between the edge and the sphere, where intersection + // points close to the surface of the sphere are ignored. + // Or: + // The sphere cuts the edge twice, no vertex is inside of the + // sphere, but the case of the edge only touching the sphere has to + // be avoided. + if(!solutions.second || + solutions.first[0] >= scalar_t(1) - detail::mediumEpsilon || + solutions.first[1] <= detail::mediumEpsilon || + (solutions.first[0] > scalar_t(0) && + solutions.first[1] < scalar_t(1) && + solutions.first[1] - solutions.first[0] < + detail::largeEpsilon)) { + + continue; + } else { + vMarked[Element::edge_mapping[n][0][0]] = + solutions.first[0] < scalar_t(0); + + vMarked[Element::edge_mapping[n][0][1]] = + solutions.first[1] > scalar_t(1); + } + + // Store the two intersection points of the edge with the sphere for + // later usage. + eIntersections[n][0] = solutions.first[0] * direction; + eIntersections[n][1] = (solutions.first[1] - scalar_t(1)) * direction; + + eMarked[n] = true; + + // If the edge is marked as having an overlap, the two faces forming it + // have to be marked as well. + fMarked[Element::edge_mapping[n][1][0]] = true; + fMarked[Element::edge_mapping[n][1][1]] = true; + } + + // Check whether the dependencies for a vertex intersection are fulfilled. + for(size_t n = 0; n < nrVertices; ++n) { + if(!vMarked[n]) + continue; + + bool edgesValid = true; + for(size_t eN = 0; eN < 3; ++eN) { + size_t edgeId = Element::vertex_mapping[n][0][eN]; + edgesValid &= eMarked[edgeId]; + } + + // If not all three edges intersecting at this vertex where marked, the + // sphere is only touching. + if(!edgesValid) + vMarked[n] = false; + } + + // Process all faces of the element, ignoring the edges as those where + // already checked above. + for(size_t n = 0; n < nrFaces; ++n) + if(intersect(s, element.faces[n])) + fMarked[n] = true; + + // Trivial case: The center of the sphere overlaps the element, but the + // sphere does not intersect any of the faces of the element, meaning the + // sphere is completely contained within the element. + if(!fMarked.count() && contains(element, s.center)) { + result[0] = sOrig.surfaceArea(); + + return result; + } + + // Spurious intersection: The initial intersection test was positive, but + // the detailed checks revealed no overlap. + if(!vMarked.count() && !eMarked.count() && !fMarked.count()) + return result; + + // Initial value for the surface of the sphere: Surface area of the full + // sphere. + result[0] = s.surfaceArea(); + + // Iterate over all the marked faces and calculate the area of the disk + // defined by the plane as well as the cap surfaces. + for(size_t n = 0; n < nrFaces; ++n) { + if(!fMarked[n]) + continue; + + const auto& f = element.faces[n]; + scalar_t dist = f.normal.dot(s.center - f.center); + result[0] -= s.capSurfaceArea(s.radius + dist); + result[n + 1] = s.diskArea(s.radius + dist); + } + + // Handle the edges and subtract the area of the respective disk cut off by + // the edge and add back the surface area of the spherical wedge defined + // by the edge. + for(size_t n = 0; n < nrEdges(); ++n) { + if(!eMarked[n]) + continue; + + result[0] += generalWedge<2, Element>(s, element, n, eIntersections); + + // The intersection points are relative to the vertices forming the + // edge. + const vector_t chord = + ((element.vertices[Element::edge_mapping[n][0][0]] + + eIntersections[n][0]) - + (element.vertices[Element::edge_mapping[n][0][1]] + + eIntersections[n][1])); + + const scalar_t chordLength = chord.stableNorm(); + + // Each edge belongs to two faces, indexed via + // Element::edge_mapping[n][1][{0,1}]. + for(size_t e = 0; e < 2; ++e) { + const auto faceIdx = Element::edge_mapping[n][1][e]; + const auto& f = element.faces[faceIdx]; + + // Height of the spherical cap cut off by the plane containing the + // face. + const scalar_t dist = f.normal.dot(s.center - f.center) + s.radius; + intersectionRadiusSq[faceIdx] = dist * (scalar_t(2) * s.radius - + dist); + + // Calculate the height of the triangular segment in the plane of + // the base. + const scalar_t factor = std::sqrt(std::max(scalar_t(0), + intersectionRadiusSq[faceIdx] - scalar_t(0.25) * chordLength * + chordLength)); + + const scalar_t theta = scalar_t(2) * std::atan2(chordLength, + scalar_t(2) * factor); + + scalar_t area = scalar_t(0.5) * intersectionRadiusSq[faceIdx] * + (theta - std::sin(theta)); + + // FIXME: Might not be necessary to use the center of the chord. + const vector_t chordCenter = scalar_t(0.5) * + ((element.vertices[Element::edge_mapping[n][0][0]] + + eIntersections[n][0]) + + (element.vertices[Element::edge_mapping[n][0][1]] + + eIntersections[n][1])); + + const vector_t proj(s.center - f.normal.dot(s.center - f.center) * + f.normal); + + // If the projected sphere center and the face center fall on + // opposite sides of the edge, the area has to be inverted. + if(chord.cross(proj - chordCenter).dot( + chord.cross(f.center - chordCenter)) < scalar_t(0)) { + + area = intersectionRadiusSq[faceIdx] * pi - area; + } + + result[faceIdx + 1] -= area; + } + } + + // Handle the vertices and add the area subtracted twice above in the + // processing of the edges. + + // First, handle the spherical surface area of the intersection. + // This is to a large part code duplicated from the volume calculation. + // TODO: Unify the area and volume calculation to remove duplicate code. + for(size_t n = 0; n < nrVertices; ++n) { + if(!vMarked[n]) + continue; + + // Collect the points where the three edges intersecting at this + // vertex intersect the sphere. + // Both the relative and the absolute positions are required. + std::array intersectionPointsRelative; + std::array intersectionPoints; + for(size_t e = 0; e < 3; ++e) { + auto edgeIdx = Element::vertex_mapping[n][0][e]; + intersectionPointsRelative[e] = + eIntersections[edgeIdx][Element::vertex_mapping[n][1][e]]; + + intersectionPoints[e] = intersectionPointsRelative[e] + + element.vertices[n]; + } + + // This triangle is constructed by hand to have more freedom of how + // the normal vector is calculated. + Triangle coneTria; + coneTria.vertices = {{ intersectionPoints[0], intersectionPoints[1], + intersectionPoints[2] }}; + + coneTria.center = scalar_t(1.0 / 3.0) * + std::accumulate(intersectionPoints.begin(), + intersectionPoints.end(), vector_t::Zero().eval()); + + // Calculate the normal of the triangle defined by the intersection + // points in relative coordinates to improve accuracy. + // Also use double the normal precision to calculate this normal. + coneTria.normal = detail::triangleNormal(intersectionPointsRelative[0], + intersectionPointsRelative[1], intersectionPointsRelative[2]); + + // The area of this triangle is never needed, so it is set to an + // invalid value. + coneTria.area = std::numeric_limits::infinity(); + + std::array, 3> distances; + for(size_t i = 0; i < 3; ++i) + distances[i] = std::make_pair(i, + intersectionPointsRelative[i].squaredNorm()); + + std::sort(distances.begin(), distances.end(), + [](const std::pair& a, + const std::pair& b) -> bool { + return a.second < b.second; + }); + + if(distances[1].second < distances[2].second * detail::largeEpsilon) { + // Use the general spherical wedge defined by the edge with the + // non-degenerated intersection point and the normals of the + // two faces forming it. + scalar_t correction = generalWedge<2, Element>(s, element, + Element::vertex_mapping[n][0][distances[2].first], + eIntersections); + + result[0] -= correction; + + continue; + } + + // Make sure the normal points in the right direction, i.e., away from + // the center of the element. + if(coneTria.normal.dot(element.center - coneTria.center) > + scalar_t(0)) { + + coneTria.normal = -coneTria.normal; + } + + Plane plane(coneTria.center, coneTria.normal); + + scalar_t dist = coneTria.normal.dot(s.center - coneTria.center); + scalar_t capSurface = s.capSurfaceArea(s.radius + dist); + + // If cap surface area is small, the corrections will be even smaller. + // There is no way to actually calculate them with reasonable + // precision, so they are just ignored. + if(capSurface < detail::largeEpsilon) + continue; + + // Calculate the surface area of the three spherical segments between + // the faces joining at the vertex and the plane through the + // intersection points. + scalar_t segmentSurface = 0; + for(size_t e = 0; e < 3; ++e) { + const auto& f = element.faces[Element::vertex_mapping[n][2][e]]; + uint32_t e0 = Element::face_mapping[e][0]; + uint32_t e1 = Element::face_mapping[e][1]; + + vector_t center(scalar_t(0.5) * (intersectionPoints[e0] + + intersectionPoints[e1])); + + segmentSurface += generalWedge<2>(s, plane, Plane(f.center, + -f.normal), center - s.center); + } + + // Calculate the surface area of the cone and clamp it to zero. + scalar_t coneSurface = std::max(capSurface - segmentSurface, + scalar_t(0)); + + result[0] -= coneSurface; + + // Sanity checks: detect negative/excessively large intermediate + // result. + assert(result[0] > -std::sqrt(detail::tinyEpsilon)); + assert(result[0] < s.surfaceArea() + detail::tinyEpsilon); + } + + // Second, correct the intersection area of the facets. + for(size_t n = 0; n < nrVertices; ++n) { + if(!vMarked[n]) + continue; + + // Iterate over all the faces joining at this vertex. + for(size_t f = 0; f < 3; ++f) { + // Determine the two edges of this face intersecting at the + // vertex. + uint32_t e0 = Element::face_mapping[f][0]; + uint32_t e1 = Element::face_mapping[f][1]; + std::array edgeIndices = {{ + Element::vertex_mapping[n][0][e0], + Element::vertex_mapping[n][0][e1] + }}; + + // Extract the (relative) intersection points of these edges with + // the sphere furthest from the vertex. + std::array intersectionPoints = {{ + eIntersections[edgeIndices[0]][ + Element::vertex_mapping[n][1][e0]], + + eIntersections[edgeIndices[1]][ + Element::vertex_mapping[n][1][e1]] + }}; + + // Together with the vertex, this determines the triangle + // representing one part of the correction. + const scalar_t triaArea = scalar_t(0.5) * + (intersectionPoints[0].cross( + intersectionPoints[1])).stableNorm(); + + // The second component is the segment defined by the face and the + // intersection points. + const scalar_t chordLength = (intersectionPoints[0] - + intersectionPoints[1]).stableNorm(); + + const auto faceIdx = Element::vertex_mapping[n][2][f]; + + // TODO: Cache theta for each edge. + const scalar_t theta = scalar_t(2) * std::atan2(chordLength, + scalar_t(2) * std::sqrt(std::max(scalar_t(0), + intersectionRadiusSq[faceIdx] - + scalar_t(0.25) * chordLength * chordLength))); + + scalar_t segmentArea = scalar_t(0.5) * + intersectionRadiusSq[faceIdx] * (theta - std::sin(theta)); + + // Determine if the (projected) center of the sphere lies within + // the triangle or not. If not, the segment area has to be + // corrected. + const vector_t d(scalar_t(0.5) * (intersectionPoints[0] + + intersectionPoints[1])); + + const auto& face = element.faces[faceIdx]; + const vector_t proj(s.center - face.normal.dot(s.center - + face.center) * face.normal); + + if(d.dot((proj - element.vertices[n]) - d) > scalar_t(0)) { + segmentArea = intersectionRadiusSq[faceIdx] * pi - segmentArea; + } + + result[faceIdx + 1] += triaArea + segmentArea; + + // Sanity checks: detect excessively large intermediate result. + assert(result[faceIdx + 1] < element.faces[faceIdx].area + + std::sqrt(detail::largeEpsilon)); + } + } + + // Scale the surface areas back for the original objects and clamp + // values within reasonable limits. + const scalar_t scaling = sOrig.radius / s.radius; + const scalar_t sLimit(std::sqrt(std::numeric_limits::epsilon()) * + s.surfaceArea()); + + // As the precision of the area calculation deteriorates quickly with a + // increasing size ratio between the element and the sphere, the precision + // limit applied to the sphere is used as the lower limit for the facets. + const scalar_t fLimit(std::max(sLimit, + std::sqrt(std::numeric_limits::epsilon()) * + element.surfaceArea())); + + // Sanity checks: detect negative/excessively large results for the + // surface area of the facets. +#ifndef NDEBUG + for(size_t n = 0; n < nrFaces; ++n) { + assert(result[n + 1] > -fLimit); + assert(result[n + 1] <= element.faces[n].area + fLimit); + } +#endif // NDEBUG + + // Surface of the sphere. + result[0] = detail::clamp(result[0], scalar_t(0), s.surfaceArea(), sLimit); + result[0] *= (scaling * scaling); + + // Surface of the mesh element. + for(size_t f = 0; f < nrFaces; ++f) { + auto& value = result[f + 1]; + value = detail::clamp(value, scalar_t(0), element.faces[f].area, + fLimit); + + value = value * (scaling * scaling); + } + + result.back() = std::accumulate(result.begin() + 1, result.end() - 1, + scalar_t(0)); + + // Perform some more sanity checks on the final result (debug version + // only). + assert(scalar_t(0) <= result[0] && result[0] <= sOrig.surfaceArea()); + + assert(scalar_t(0) <= result.back() && result.back() <= + elementOrig.surfaceArea()); + + return result; +} + +} +#endif // OVERLAP_HPP + diff --git a/include/auxkernels/MDGranularPorosityAux.h b/include/auxkernels/MDGranularPorosityAux.h new file mode 100644 index 00000000..1e282033 --- /dev/null +++ b/include/auxkernels/MDGranularPorosityAux.h @@ -0,0 +1,39 @@ +/**********************************************************************/ +/* DO NOT MODIFY THIS HEADER */ +/* MAGPIE - Mesoscale Atomistic Glue Program for Integrated Execution */ +/* */ +/* Copyright 2017 Battelle Energy Alliance, LLC */ +/* ALL RIGHTS RESERVED */ +/**********************************************************************/ + +#ifndef MDGRANULARPOROSITYAUX_H +#define MDGRANULARPOROSITYAUX_H + +#include "AuxKernel.h" + +// forward declarations +class MDRunBase; +class MDGranularPorosityAux; + +template <> +InputParameters validParams(); + +class MDGranularPorosityAux : public AuxKernel +{ +public: + MDGranularPorosityAux(const InputParameters & params); + virtual ~MDGranularPorosityAux() {} + + virtual Real computeValue(); + +protected: + /// referene to the MDRunBase user object + const MDRunBase & _md_uo; + + const bool _compute_packing; + + /// property value that is computed only on qp = 0 + Real _packing_fraction; +}; + +#endif // MDGRANULARPOROSITYAUXx_H diff --git a/include/auxkernels/MDGranularPropertyAux.h b/include/auxkernels/MDGranularPropertyAux.h new file mode 100644 index 00000000..e43118c6 --- /dev/null +++ b/include/auxkernels/MDGranularPropertyAux.h @@ -0,0 +1,45 @@ +/**********************************************************************/ +/* DO NOT MODIFY THIS HEADER */ +/* MAGPIE - Mesoscale Atomistic Glue Program for Integrated Execution */ +/* */ +/* Copyright 2017 Battelle Energy Alliance, LLC */ +/* ALL RIGHTS RESERVED */ +/**********************************************************************/ + +#ifndef MDGRANULARPROPERTYAUX_H +#define MDGRANULARPROPERTYAUX_H + +#include "AuxKernel.h" + +// forward declarations +class MDRunBase; +class MDGranularPropertyAux; + +template <> +InputParameters validParams(); + +class MDGranularPropertyAux : public AuxKernel +{ +public: + MDGranularPropertyAux(const InputParameters & params); + virtual ~MDGranularPropertyAux() {} + + virtual Real computeValue(); + + static MooseEnum mdAveragingType(); + +protected: + /// referene to the MDRunBase user object + const MDRunBase & _md_uo; + + /// the type of average to be computed + MooseEnum _average_type; + + /// ID of the desired MD property + unsigned int _property_id; + + /// property value that is computed only on qp = 0 + Real _property_value; +}; + +#endif // MDGRANULARPROPERTYAUX_H diff --git a/include/auxkernels/MDNParticleAux.h b/include/auxkernels/MDNParticleAux.h index 8e7f6b23..5b01c4bc 100644 --- a/include/auxkernels/MDNParticleAux.h +++ b/include/auxkernels/MDNParticleAux.h @@ -28,6 +28,8 @@ class MDNParticleAux : public AuxKernel protected: const MDRunBase & _md_uo; + + std::vector _particles; }; #endif // MDNPARTICLEAUX_H diff --git a/include/userobjects/LAMMPSFileRunner.h b/include/userobjects/LAMMPSFileRunner.h index f02f2a87..f1a24eab 100644 --- a/include/userobjects/LAMMPSFileRunner.h +++ b/include/userobjects/LAMMPSFileRunner.h @@ -55,6 +55,9 @@ class LAMMPSFileRunner : public MDRunBase /// column of x, y, z coordinate in LAMMPS files std::vector _pos_columns; + /// column of properties in LAMMPS files + std::vector _prop_columns; + /// Conversion from FEM time to MD time_stamp Function * _time_conversion; }; diff --git a/include/userobjects/MDRunBase.h b/include/userobjects/MDRunBase.h index aab5a016..e56b4dee 100644 --- a/include/userobjects/MDRunBase.h +++ b/include/userobjects/MDRunBase.h @@ -9,6 +9,9 @@ #ifndef MDRUNBASE_H #define MDRUNBASE_H +#define N_MD_PROPERTIES 8 + +#include "overlap.hpp" #include "GeneralUserObject.h" #include "KDTree.h" #include "libmesh/bounding_box.h" @@ -36,24 +39,37 @@ class MDRunBase : public GeneralUserObject /// particle's position std::vector pos; - /// particle's velocity - std::vector vel; - /// the id of particle in the MD calculation std::vector id; /// the mesh element the particles are in std::vector elem_id; + + /// data attached to each particle + std::vector> properties; }; /// access to the MDParticles const MDParticles & particles() const { return _md_particles; } + // check if the stored particles are granular + bool isGranular() const { return _granular; } + /// access to the element to particle map - const std::vector elemParticles(unique_id_type elem_id) const; + void elemParticles(unique_id_type elem_id, std::vector & elem_particles) const; + + /// access the element to granular map + void granularElementVolumes(unique_id_type elem_id, + std::vector> & gran_vol) const; + + /// access to MD particle's properties + Real particleProperty(unsigned int j, unsigned int prop_id) const; + + /// accessor for md properties that are collected by this UO + MultiMooseEnum properties() const { return _properties; } /// List of quantities to get from MD simulation - MultiMooseEnum getMDQuantities() const; + static MultiMooseEnum mdParticleProperties(); protected: /// call back function to update the particle list @@ -65,6 +81,27 @@ class MDRunBase : public GeneralUserObject /// map MDParticles to elements void mapMDParticles(); + /// update candidates for + void updateElementGranularVolumes(); + + /// helper function to contruct hexahedron + OVERLAP::Hexahedron overlapHex(const Elem * elem) const; + + /// helper function to contruct unit hexahedron + OVERLAP::Hexahedron overlapUnitHex() const; + + /// helper function to contruct tetrahedron + OVERLAP::Tetrahedron overlapTet(const Elem * elem) const; + + /// helper function to construct unit tetrahedron + OVERLAP::Tetrahedron overlapUnitTet() const; + + /// Properties that are requested from MD simulation + MultiMooseEnum _properties; + + /// do the MD particles have extent? + bool _granular; + /// The Mesh we're using MooseMesh & _mesh; @@ -78,19 +115,25 @@ class MDRunBase : public GeneralUserObject std::vector _bbox; /// total number of particles - unsigned int _n_particles; + unsigned int _n_particles = 0; /// number of local particles - unsigned int _n_local_particles; + unsigned int _n_local_particles = 0; /// stores the location of MDParticles _md_particles; - /// a map from elem pointer to particles in this element + /// a map from elem unique id to particles in this element std::map> _elem_particles; + /// a map from element unique id to std::vector of pair(MD particle id, volume of gran. particle in this element) + std::map>> _elem_granular_volumes; + /// A KDTree object to handle md_particles std::unique_ptr _kd_tree; + + /// maximum granular radius + Real _max_granular_radius; }; #endif // MDRUNBASE_H diff --git a/magpie.mk b/magpie.mk index 9f185fd4..c90954ad 100644 --- a/magpie.mk +++ b/magpie.mk @@ -18,3 +18,5 @@ endif include $(MAGPIE_DIR)/contrib/gsl.mk include $(MAGPIE_DIR)/contrib/fftw3.mk + +app_INCLUDES += -I $(MAGPIE_DIR)/contrib/overlap diff --git a/src/auxkernels/MDGranularPorosityAux.C b/src/auxkernels/MDGranularPorosityAux.C new file mode 100644 index 00000000..f866b6cc --- /dev/null +++ b/src/auxkernels/MDGranularPorosityAux.C @@ -0,0 +1,61 @@ +/**********************************************************************/ +/* DO NOT MODIFY THIS HEADER */ +/* MAGPIE - Mesoscale Atomistic Glue Program for Integrated Execution */ +/* */ +/* Copyright 2017 Battelle Energy Alliance, LLC */ +/* ALL RIGHTS RESERVED */ +/**********************************************************************/ + +#include "MDGranularPorosityAux.h" +#include "MDRunBase.h" + +registerMooseObject("MagpieApp", MDGranularPorosityAux); + +template <> +InputParameters +validParams() +{ + InputParameters params = validParams(); + params.addRequiredParam("user_object", "Name of MD runner UserObject"); + params.addParam("compute_packing_fraction", false, "Whether to compute porosity or packing fraction."); + params.addClassDescription( + "Computes porosity or packing fraction (1 - porosity) and injects it into and aux variable."); + return params; +} + +MDGranularPorosityAux::MDGranularPorosityAux(const InputParameters & parameters) + : AuxKernel(parameters), + _md_uo(getUserObject("user_object")), + _compute_packing(getParam("compute_packing_fraction")) +{ + // ensure MD particles are granular + if (!_md_uo.isGranular()) + mooseError("user_object stores non-granular particles."); + + // ensure variable is elemental + if (isNodal()) + mooseError("MDGranularPorosityAux only permits elemental variables."); +} + +Real +MDGranularPorosityAux::computeValue() +{ + if (_qp == 0) + { + _packing_fraction = 0.0; + + // get the overlapping MD particles + std::vector> gran_vol; + _md_uo.granularElementVolumes(_current_elem->unique_id(), gran_vol); + + // add the overlapping volumes + for (auto & p : gran_vol) + _packing_fraction += p.second; + + // divide by element volume + _packing_fraction /= _current_elem->volume(); + } + if (_compute_packing) + return _packing_fraction; + return 1 - _packing_fraction; +} diff --git a/src/auxkernels/MDGranularPropertyAux.C b/src/auxkernels/MDGranularPropertyAux.C new file mode 100644 index 00000000..06e336b0 --- /dev/null +++ b/src/auxkernels/MDGranularPropertyAux.C @@ -0,0 +1,92 @@ +/**********************************************************************/ +/* DO NOT MODIFY THIS HEADER */ +/* MAGPIE - Mesoscale Atomistic Glue Program for Integrated Execution */ +/* */ +/* Copyright 2017 Battelle Energy Alliance, LLC */ +/* ALL RIGHTS RESERVED */ +/**********************************************************************/ + +#include "MDGranularPropertyAux.h" +#include "MDRunBase.h" + +registerMooseObject("MagpieApp", MDGranularPropertyAux); + +template <> +InputParameters +validParams() +{ + InputParameters params = validParams(); + params.addRequiredParam("user_object", "Name of MD runner UserObject"); + params.addParam("md_particle_property", + MDRunBase::mdParticleProperties(), + "Property that is injected into auxiliary variable."); + params.addParam("average_type", + MDGranularPropertyAux::mdAveragingType(), + "The type of average to be taken: " + "granular_sum|granular_densitygranular_interstitial_density."); + params.addClassDescription( + "Injects properties collected for MD particles from MDRunBase object user_object " + "into auxiliary variable."); + return params; +} + +MDGranularPropertyAux::MDGranularPropertyAux(const InputParameters & parameters) + : AuxKernel(parameters), + _md_uo(getUserObject("user_object")), + _average_type(getParam("average_type")) +{ + // check length of MultiMooseEnum parameter, get it and check that UO has it + MultiMooseEnum mme = getParam("md_particle_property"); + if (mme.size() != 1) + mooseError("md_particle_property must contain a single property."); + _property_id = mme.get(0); + if (!_md_uo.properties().contains(mme)) + mooseError("Property ", _property_id, " not available from user_object."); + + // ensure MD particles are granular + if (!_md_uo.isGranular()) + mooseError("user_object stores non-granular particles."); + + // ensure variable is elemental + if (isNodal()) + mooseError("MDGranularPropertyAux only permits elemental variables."); +} + +Real +MDGranularPropertyAux::computeValue() +{ + if (_qp == 0) + { + _property_value = 0.0; + + // get the overlapping MD particles + std::vector> gran_vol; + _md_uo.granularElementVolumes(_current_elem->unique_id(), gran_vol); + + // loop over the overlapping MD particles and add property value + Real denominator = 0; + for (auto & p : gran_vol) + { + _property_value += p.second * _md_uo.particleProperty(p.first, _property_id); + denominator += p.second; + } + + // compute property value depending on what average type is requested + if (_average_type == 1) + _property_value /= _current_elem->volume(); + else if (_average_type == 2) + { + if (denominator == 0.0) + _property_value = 0.0; + else + _property_value /= denominator; + } + } + return _property_value; +} + +MooseEnum +MDGranularPropertyAux::mdAveragingType() +{ + return MooseEnum("granular_sum=0 granular_density=1 granular_interstitial_density=2"); +} diff --git a/src/auxkernels/MDNParticleAux.C b/src/auxkernels/MDNParticleAux.C index 4d446028..c53f7b90 100644 --- a/src/auxkernels/MDNParticleAux.C +++ b/src/auxkernels/MDNParticleAux.C @@ -30,5 +30,6 @@ MDNParticleAux::MDNParticleAux(const InputParameters & parameters) Real MDNParticleAux::computeValue() { - return _md_uo.elemParticles(_current_elem->unique_id()).size(); + _md_uo.elemParticles(_current_elem->unique_id(), _particles); + return _particles.size(); } diff --git a/src/userobjects/LAMMPSFileRunner.C b/src/userobjects/LAMMPSFileRunner.C index 1785f23a..d709eb4c 100644 --- a/src/userobjects/LAMMPSFileRunner.C +++ b/src/userobjects/LAMMPSFileRunner.C @@ -29,7 +29,9 @@ validParams() "a sequence."); params.addRequiredParam>( "xyz_columns", "Column ids of the x, y, and z coordinates of particles."); - + std::vector empty = {}; + params.addParam>( + "property_columns", empty, "Column ids of the properties."); params.addParam("time_conversion", "A conversion from FEM simulation time to MD time stamps."); params.addClassDescription("Allows coupling FEM calculations to LAMMPS dump files."); @@ -41,6 +43,7 @@ LAMMPSFileRunner::LAMMPSFileRunner(const InputParameters & parameters) _time_sequence(getParam("time_sequence")), _lammps_file(getParam("lammps_file")), _pos_columns(getParam>("xyz_columns")), + _prop_columns(getParam>("property_columns")), _time_conversion(nullptr) { ; @@ -50,6 +53,9 @@ LAMMPSFileRunner::LAMMPSFileRunner(const InputParameters & parameters) if (isParamValid("time_conversion")) _time_conversion = &getFunction("time_conversion"); + + if (_properties.size() != _prop_columns.size()) + mooseError("properties array must have same length as property_columns"); } void @@ -86,6 +92,10 @@ LAMMPSFileRunner::updateParticleList() // update the mapping to mesh if mesh has changed or ... mapMDParticles(); + + // update the granular candidates as well if necessary + if (_granular) + updateElementGranularVolumes(); } void @@ -180,10 +190,12 @@ LAMMPSFileRunner::readLAMMPSFile(FileName filename) // zero out the position & id vector position.clear(); _md_particles.id.clear(); + _md_particles.properties.clear(); // size the md particle vector considering not all processors have all particles position.reserve(std::ceil(2.0 * _n_particles / _nproc)); _md_particles.id.reserve(std::ceil(2.0 * _n_particles / _nproc)); + _md_particles.properties.reserve(std::ceil(2.0 * _n_particles / _nproc)); // read particle entries _n_local_particles = 0; @@ -195,6 +207,23 @@ LAMMPSFileRunner::readLAMMPSFile(FileName filename) std::vector elements; MooseUtils::tokenize(line, elements, 1, " "); +#if DEBUG + // check that enough columns exist in debug + // find largest requested column + unsigned int largest_column = 0; + for (unsigned int i = 0; i < _dim; ++i) + if (_pos_columns[i] > largest_column) + largest_column = _pos_columns[i]; + + for (unsigned int i = 0; i < _prop_columns.size(); ++i) + if (_prop_columns[i] > largest_column) + largest_column = _prop_columns[i]; + + if (largest_column >= elements.size()) + mooseError("Error reading ", filename, " on line ", line); +#endif + + // read location Point pos; for (unsigned int i = 0; i < _dim; ++i) { @@ -202,6 +231,15 @@ LAMMPSFileRunner::readLAMMPSFile(FileName filename) sstr >> pos(i); } + // read properties + std::array props; + for (unsigned int i = 0; i < _prop_columns.size(); ++i) + { + unsigned int prop_id = _properties.get(i); + std::stringstream sstr(elements[_prop_columns[i]]); + sstr >> props[prop_id]; + } + // check if this particle is in this processor BBox if (!_bbox[processor_id()].contains_point(pos)) continue; @@ -209,6 +247,7 @@ LAMMPSFileRunner::readLAMMPSFile(FileName filename) ++_n_local_particles; position.push_back(pos); _md_particles.id.push_back(j); + _md_particles.properties.push_back(props); } file.close(); @@ -264,10 +303,12 @@ LAMMPSFileRunner::readLAMMPSFileHistory(std::pair filenames, // zero out the position & id vector position.clear(); _md_particles.id.clear(); + _md_particles.properties.clear(); // size the md particle vector considering not all processors have all particles position.reserve(std::ceil(2.0 * _n_particles / _nproc)); _md_particles.id.reserve(std::ceil(2.0 * _n_particles / _nproc)); + _md_particles.properties.reserve(std::ceil(2.0 * _n_particles / _nproc)); // read particle entries _n_local_particles = 0; @@ -282,6 +323,22 @@ LAMMPSFileRunner::readLAMMPSFileHistory(std::pair filenames, std::getline(file_before, line_before); MooseUtils::tokenize(line_before, elements, 1, " "); +#if DEBUG + // check that enough columns exist in debug + // find largest requested column + unsigned int largest_column = 0; + for (unsigned int i = 0; i < _dim; ++i) + if (_pos_columns[i] > largest_column) + largest_column = _pos_columns[i]; + + for (unsigned int i = 0; i < _prop_columns.size(); ++i) + if (_prop_columns[i] > largest_column) + largest_column = _prop_columns[i]; + + if (largest_column >= elements.size()) + mooseError("Error reading ", filenames.first, " on line ", line_before); +#endif + Point pos_before; for (unsigned int i = 0; i < _dim; ++i) { @@ -289,6 +346,15 @@ LAMMPSFileRunner::readLAMMPSFileHistory(std::pair filenames, sstr >> pos_before(i); } + // read properties from before file + std::array props_before; + for (unsigned int i = 0; i < _prop_columns.size(); ++i) + { + unsigned int prop_id = _properties.get(i); + std::stringstream sstr(elements[_prop_columns[i]]); + sstr >> props_before[prop_id]; + } + // get the MD particle from the after file elements.clear(); std::string line_after; @@ -296,6 +362,10 @@ LAMMPSFileRunner::readLAMMPSFileHistory(std::pair filenames, MooseUtils::tokenize(line_after, elements, 1, " "); + // we have determined largest_column in DEBUG so we can use it in mooseAssert + mooseAssert(largest_column < elements.size(), + "Error reading " << filenames.second << " on line " << line_after); + Point pos_after; for (unsigned int i = 0; i < _dim; ++i) { @@ -303,6 +373,15 @@ LAMMPSFileRunner::readLAMMPSFileHistory(std::pair filenames, sstr >> pos_after(i); } + // read properties from before file + std::array props_after; + for (unsigned int i = 0; i < _prop_columns.size(); ++i) + { + unsigned int prop_id = _properties.get(i); + std::stringstream sstr(elements[_prop_columns[i]]); + sstr >> props_after[prop_id]; + } + // check if this particle is in this processor BBox if (!_bbox[processor_id()].contains_point(pos_after) && !_bbox[processor_id()].contains_point(pos_before)) @@ -311,6 +390,9 @@ LAMMPSFileRunner::readLAMMPSFileHistory(std::pair filenames, ++_n_local_particles; position.push_back((1 - weight) * pos_before + weight * pos_after); _md_particles.id.push_back(j); + _md_particles.properties.push_back(std::array()); + for (unsigned int i = 0; i < N_MD_PROPERTIES; ++i) + _md_particles.properties.back()[i] = (1 - weight) * props_before[i] + weight * props_after[i]; } file_before.close(); file_after.close(); diff --git a/src/userobjects/MDRunBase.C b/src/userobjects/MDRunBase.C index c410f250..2931189e 100644 --- a/src/userobjects/MDRunBase.C +++ b/src/userobjects/MDRunBase.C @@ -17,9 +17,9 @@ inline void dataStore(std::ostream & stream, MDRunBase::MDParticles & pl, void * context) { dataStore(stream, pl.pos, context); - dataStore(stream, pl.vel, context); dataStore(stream, pl.id, context); dataStore(stream, pl.elem_id, context); + dataStore(stream, pl.properties, context); } template <> @@ -27,9 +27,9 @@ inline void dataLoad(std::istream & stream, MDRunBase::MDParticles & pl, void * context) { dataLoad(stream, pl.pos, context); - dataLoad(stream, pl.vel, context); dataLoad(stream, pl.id, context); dataLoad(stream, pl.elem_id, context); + dataLoad(stream, pl.properties, context); } template <> @@ -39,7 +39,9 @@ validParams() InputParameters params = validParams(); params.set("execute_on") = EXEC_TIMESTEP_BEGIN; params.suppressParameter("execute_on"); - + params.addParam("md_particle_properties", + MDRunBase::mdParticleProperties(), + "Properties of MD particles to be obtained and stored."); params.addClassDescription( "Base class for execution of coupled molecular dynamics MOOSE calculations."); return params; @@ -47,16 +49,30 @@ validParams() MDRunBase::MDRunBase(const InputParameters & parameters) : GeneralUserObject(parameters), + _properties(getParam("md_particle_properties")), + _granular(_properties.contains("radius")), _mesh(_subproblem.mesh()), _dim(_mesh.dimension()), _nproc(_app.n_processors()), _bbox(_nproc) { + // check _properties array and MooseEnum for consistency + if (N_MD_PROPERTIES != mdParticleProperties().getNames().size()) + mooseError("Mismatch of MD particle property enum and property array. Report on github."); + + for (auto & id : mdParticleProperties().getIDs()) + if (id < 0 || id >= N_MD_PROPERTIES) + mooseError("Property id ", id, " is either < 0 or larger than array size."); } void MDRunBase::initialSetup() { + // TODO: need to inflate the bounding box for granular particles + if (_granular) + mooseDoOnce(mooseWarning("Granular particles can lead to wrong answers in parallel runs " + "because processor bounding boxes do not account for particle size.")); + for (unsigned int j = 0; j < _nproc; ++j) _bbox[j] = MeshTools::create_processor_bounding_box(_mesh, j); } @@ -78,12 +94,37 @@ MDRunBase::updateKDTree() _kd_tree = libmesh_make_unique(_md_particles.pos, 50); } -const std::vector -MDRunBase::elemParticles(unique_id_type elem_id) const +void +MDRunBase::elemParticles(unique_id_type elem_id, std::vector & elem_particles) const { if (_elem_particles.find(elem_id) != _elem_particles.end()) - return _elem_particles.find(elem_id)->second; - return std::vector(0); + elem_particles = _elem_particles.find(elem_id)->second; + else + elem_particles = {}; +} + +void +MDRunBase::granularElementVolumes(unique_id_type elem_id, + std::vector> & gran_vol) const +{ + mooseAssert(_granular, + "Radius must be provided as MD property to allow granular volume computation."); + if (_elem_granular_volumes.find(elem_id) != _elem_granular_volumes.end()) + gran_vol = _elem_granular_volumes.find(elem_id)->second; + else + gran_vol = {}; +} + +Real +MDRunBase::particleProperty(unsigned int j, unsigned int prop_id) const +{ + // ensure that entry exists + mooseAssert(j < _md_particles.properties.size(), + "Particle index " << j << " not found in _md_particles. properties vector has length " + << _md_particles.properties.size()); + mooseAssert(prop_id < N_MD_PROPERTIES, + "prop_id must be smaller than " << N_MD_PROPERTIES << ". Provided value " << prop_id); + return _md_particles.properties[j][prop_id]; } void @@ -156,8 +197,133 @@ MDRunBase::mapMDParticles() } } +void +MDRunBase::updateElementGranularVolumes() +{ + // clear the granular volume map + _elem_granular_volumes.clear(); + + // _max_granular_radius + _max_granular_radius = 0; + for (auto & p : _md_particles.properties) + if (p[7] > _max_granular_radius) + _max_granular_radius = p[7]; + + /// loop over all local elements + ConstElemRange * active_local_elems = _mesh.getActiveLocalElementRange(); + for (const auto & elem : *active_local_elems) + { + // find all points within an inflated bounding box + std::vector> indices_dist; + BoundingBox bbox = elem->loose_bounding_box(); + Point center = 0.5 * (bbox.min() + bbox.max()); + + // inflate the search sphere by the maximum granular radius + Real radius = (bbox.max() - center).norm() + _max_granular_radius; + _kd_tree->radiusSearch(center, radius, indices_dist); + + // prepare _elem_granular_candidates entry + _elem_granular_volumes[elem->unique_id()] = {}; + + // construct this element's overlap object + ElemType t = elem->type(); + OVERLAP::Hexahedron hex = overlapUnitHex(); + OVERLAP::Tetrahedron tet = overlapUnitTet(); + if (t == HEX8) + hex = overlapHex(elem); + else if (t == TET4) + tet = overlapTet(elem); + else + mooseError("Element type ", t, "not implemented"); + + // loop through all MD particles that the search turned up and test overlap + for (unsigned int j = 0; j < indices_dist.size(); ++j) + { + // construct OVERLAP::sphere object from MD granular particle + unsigned int k = indices_dist[j].first; + OVERLAP::Sphere sph(OVERLAP::vector_t{_md_particles.pos[k](0), + _md_particles.pos[k](1), + _md_particles.pos[k](2)}, + _md_particles.properties[k][7]); + + // compute the overlap + Real ovlp = 0.0; + if (t == HEX8) + ovlp = OVERLAP::overlap(sph, hex); + else if (t == TET4) + ovlp = OVERLAP::overlap(sph, tet); + + // if the overlap is larger than 0, make entry in _elem_granular_volumes + if (ovlp > 0.0) + _elem_granular_volumes[elem->unique_id()].push_back(std::pair(k, ovlp)); + } + } +} + MultiMooseEnum -MDRunBase::getMDQuantities() const +MDRunBase::mdParticleProperties() +{ + return MultiMooseEnum("vel_x=0 vel_y=1 vel_z=2 force_x=3 force_y=4 force_z=5 charge=6 radius=7"); +} + +OVERLAP::Hexahedron +MDRunBase::overlapHex(const Elem * elem) const +{ + Point p; + p = elem->point(0); + OVERLAP::vector_t v0{p(0), p(1), p(2)}; + p = elem->point(1); + OVERLAP::vector_t v1{p(0), p(1), p(2)}; + p = elem->point(2); + OVERLAP::vector_t v2{p(0), p(1), p(2)}; + p = elem->point(3); + OVERLAP::vector_t v3{p(0), p(1), p(2)}; + p = elem->point(4); + OVERLAP::vector_t v4{p(0), p(1), p(2)}; + p = elem->point(5); + OVERLAP::vector_t v5{p(0), p(1), p(2)}; + p = elem->point(6); + OVERLAP::vector_t v6{p(0), p(1), p(2)}; + p = elem->point(7); + OVERLAP::vector_t v7{p(0), p(1), p(2)}; + return OVERLAP::Hexahedron{v0, v1, v2, v3, v4, v5, v6, v7}; +} + +OVERLAP::Hexahedron +MDRunBase::overlapUnitHex() const +{ + OVERLAP::vector_t v0{-1, -1, -1}; + OVERLAP::vector_t v1{1, -1, -1}; + OVERLAP::vector_t v2{1, 1, -1}; + OVERLAP::vector_t v3{-1, 1, -1}; + OVERLAP::vector_t v4{-1, -1, 1}; + OVERLAP::vector_t v5{1, -1, 1}; + OVERLAP::vector_t v6{1, 1, 1}; + OVERLAP::vector_t v7{-1, 1, 1}; + return OVERLAP::Hexahedron{v0, v1, v2, v3, v4, v5, v6, v7}; +} + +OVERLAP::Tetrahedron +MDRunBase::overlapTet(const Elem * elem) const +{ + Point p; + p = elem->point(0); + OVERLAP::vector_t v0{p(0), p(1), p(2)}; + p = elem->point(1); + OVERLAP::vector_t v1{p(0), p(1), p(2)}; + p = elem->point(2); + OVERLAP::vector_t v2{p(0), p(1), p(2)}; + p = elem->point(3); + OVERLAP::vector_t v3{p(0), p(1), p(2)}; + return OVERLAP::Tetrahedron{v0, v1, v2, v3}; +} + +OVERLAP::Tetrahedron +MDRunBase::overlapUnitTet() const { - return MultiMooseEnum("vel_x=0 vel_y=1 vel_z=2 force_x=3 force_y=4 force_z=5 charge=6"); + OVERLAP::vector_t v0{0, 0, 0}; + OVERLAP::vector_t v1{1, 0, 0}; + OVERLAP::vector_t v2{0, 1, 0}; + OVERLAP::vector_t v3{0, 0, 1}; + return OVERLAP::Tetrahedron{v0, v1, v2, v3}; } diff --git a/tests/userobjects/lammps_file/gold/granular_out.e b/tests/userobjects/lammps_file/gold/granular_out.e new file mode 100644 index 00000000..23d57189 Binary files /dev/null and b/tests/userobjects/lammps_file/gold/granular_out.e differ diff --git a/tests/userobjects/lammps_file/gold/granular_sequence_out.e b/tests/userobjects/lammps_file/gold/granular_sequence_out.e new file mode 100644 index 00000000..cf148764 Binary files /dev/null and b/tests/userobjects/lammps_file/gold/granular_sequence_out.e differ diff --git a/tests/userobjects/lammps_file/gold/granular_tet_out.e b/tests/userobjects/lammps_file/gold/granular_tet_out.e new file mode 100644 index 00000000..801972d8 Binary files /dev/null and b/tests/userobjects/lammps_file/gold/granular_tet_out.e differ diff --git a/tests/userobjects/lammps_file/granular.0.xyz b/tests/userobjects/lammps_file/granular.0.xyz new file mode 100644 index 00000000..ca1927fc --- /dev/null +++ b/tests/userobjects/lammps_file/granular.0.xyz @@ -0,0 +1,14 @@ +ITEM: TIMESTEP +0 +ITEM: NUMBER OF ATOMS +5 +ITEM: BOX BOUNDS pp pp pp + 0 1 + 0 1 + 0 1 +ITEM: ATOMS x y z radius charge + 0.25 0.25 0.5 0.1 1 + 0.75 0.25 0.51 0.1 1 + 0.75 0.75 0.525 0.1 1 + 0.25 0.75 0.55 0.1 1 + 0.5 0.5 0.5 0.1 1 diff --git a/tests/userobjects/lammps_file/granular.1.xyz b/tests/userobjects/lammps_file/granular.1.xyz new file mode 100644 index 00000000..61fa77ae --- /dev/null +++ b/tests/userobjects/lammps_file/granular.1.xyz @@ -0,0 +1,14 @@ +ITEM: TIMESTEP +1 +ITEM: NUMBER OF ATOMS +5 +ITEM: BOX BOUNDS pp pp pp + 0 1 + 0 1 + 0 1 +ITEM: ATOMS x y z radius charge + 0.25 0.25 0.5 0.1 1 + 0.75 0.25 0.51 0.1 1 + 0.75 0.75 0.525 0.1 1 + 0.25 0.75 0.55 0.1 1 + 0.5 0.5 0.5 0.1 1 diff --git a/tests/userobjects/lammps_file/granular.i b/tests/userobjects/lammps_file/granular.i new file mode 100644 index 00000000..beebb2f7 --- /dev/null +++ b/tests/userobjects/lammps_file/granular.i @@ -0,0 +1,117 @@ +[Mesh] + type = GeneratedMesh + dim = 3 + xmin = 0 + xmax = 1 + nx = 2 + ymin = 0 + ymax = 1 + ny = 2 + zmin = 0 + zmax = 1 + nz = 2 +[] + +[Problem] + kernel_coverage_check = false +[] + +[Variables] + [./d] + [../] +[] + +[AuxVariables] + [./packing] + family = MONOMIAL + order = CONSTANT + [../] + [./porosity] + family = MONOMIAL + order = CONSTANT + [../] +[] + +[AuxKernels] + [./packing_fraction] + type = MDGranularPropertyAux + variable = packing + average_type = granular_density + md_particle_property = charge + user_object = lammps_runner + [../] + + [./porosity] + type = MDGranularPorosityAux + variable = porosity + user_object = lammps_runner + [../] +[] + +[UserObjects] + [./lammps_runner] + type = LAMMPSFileRunner + lammps_file = 'granular.0.xyz' + xyz_columns = '0 1 2' + md_particle_properties = 'radius charge' + property_columns = '3 4' + [../] +[] + +[Postprocessors] + [./mid1] + type = PointValue + variable = packing + point = '0.25 0.25 0.25' + [../] + [./mid2] + type = PointValue + variable = packing + point = '0.75 0.25 0.25' + [../] + [./mid3] + type = PointValue + variable = packing + point = '0.75 0.75 0.25' + [../] + [./mid4] + type = PointValue + variable = packing + point = '0.25 0.75 0.25' + [../] + [./mid5] + type = PointValue + variable = packing + point = '0.25 0.25 0.75' + [../] + [./mid6] + type = PointValue + variable = packing + point = '0.75 0.25 0.75' + [../] + [./mid7] + type = PointValue + variable = packing + point = '0.75 0.75 0.75' + [../] + [./mid8] + type = PointValue + variable = packing + point = '0.25 0.75 0.75' + [../] + + [./mid8_porosity] + type = PointValue + variable = porosity + point = '0.25 0.75 0.75' + [../] +[] + +[Executioner] + type = Transient + num_steps = 1 +[] + +[Outputs] + exodus = true +[] diff --git a/tests/userobjects/lammps_file/granular_sequence.i b/tests/userobjects/lammps_file/granular_sequence.i new file mode 100644 index 00000000..2ff0eeda --- /dev/null +++ b/tests/userobjects/lammps_file/granular_sequence.i @@ -0,0 +1,103 @@ +[Mesh] + type = GeneratedMesh + dim = 3 + xmin = 0 + xmax = 1 + nx = 2 + ymin = 0 + ymax = 1 + ny = 2 + zmin = 0 + zmax = 1 + nz = 2 +[] + +[Problem] + kernel_coverage_check = false +[] + +[Variables] + [./d] + [../] +[] + +[AuxVariables] + [./packing] + family = MONOMIAL + order = CONSTANT + [../] +[] + +[AuxKernels] + [./packing_fraction] + type = MDGranularPropertyAux + variable = packing + average_type = granular_density + md_particle_property = charge + user_object = lammps_runner + [../] +[] + +[UserObjects] + [./lammps_runner] + type = LAMMPSFileRunner + lammps_file = 'granular' + time_sequence = true + xyz_columns = '0 1 2' + md_particle_properties = 'radius charge' + property_columns = '3 4' + [../] +[] + +[Postprocessors] + [./mid1] + type = PointValue + variable = packing + point = '0.25 0.25 0.25' + [../] + [./mid2] + type = PointValue + variable = packing + point = '0.75 0.25 0.25' + [../] + [./mid3] + type = PointValue + variable = packing + point = '0.75 0.75 0.25' + [../] + [./mid4] + type = PointValue + variable = packing + point = '0.25 0.75 0.25' + [../] + [./mid5] + type = PointValue + variable = packing + point = '0.25 0.25 0.75' + [../] + [./mid6] + type = PointValue + variable = packing + point = '0.75 0.25 0.75' + [../] + [./mid7] + type = PointValue + variable = packing + point = '0.75 0.75 0.75' + [../] + [./mid8] + type = PointValue + variable = packing + point = '0.25 0.75 0.75' + [../] +[] + +[Executioner] + type = Transient + num_steps = 2 + dt = 0.5 +[] + +[Outputs] + exodus = true +[] diff --git a/tests/userobjects/lammps_file/granular_tet.i b/tests/userobjects/lammps_file/granular_tet.i new file mode 100644 index 00000000..e471de6f --- /dev/null +++ b/tests/userobjects/lammps_file/granular_tet.i @@ -0,0 +1,66 @@ +[Mesh] + type = GeneratedMesh + dim = 3 + xmin = 0 + xmax = 1 + nx = 1 + ymin = 0 + ymax = 1 + ny = 1 + zmin = 0 + zmax = 1 + nz = 1 + elem_type = TET4 +[] + +[Problem] + kernel_coverage_check = false +[] + +[Variables] + [./d] + [../] +[] + +[AuxVariables] + [./packing] + family = MONOMIAL + order = CONSTANT + [../] +[] + +[AuxKernels] + [./packing_fraction] + type = MDGranularPorosityAux + variable = packing + compute_packing_fraction = true + user_object = lammps_runner + [../] +[] + +[UserObjects] + [./lammps_runner] + type = LAMMPSFileRunner + lammps_file = 'single.0.xyz' + xyz_columns = '0 1 2' + md_particle_properties = 'radius' + property_columns = '3' + [../] +[] + +[Postprocessors] + [./sphere_volume] + type = ElementIntegralVariablePostprocessor + variable = packing + [../] +[] + +[Executioner] + type = Transient + num_steps = 1 + dt = 1 +[] + +[Outputs] + exodus = true +[] diff --git a/tests/userobjects/lammps_file/single.0.xyz b/tests/userobjects/lammps_file/single.0.xyz new file mode 100644 index 00000000..e71ae71b --- /dev/null +++ b/tests/userobjects/lammps_file/single.0.xyz @@ -0,0 +1,10 @@ +ITEM: TIMESTEP +0 +ITEM: NUMBER OF ATOMS +1 +ITEM: BOX BOUNDS pp pp pp +0 1 +0 1 +0 1 +ITEM: ATOMS x y z r +0 0 0 1 diff --git a/tests/userobjects/lammps_file/tests b/tests/userobjects/lammps_file/tests index 6fd55bd8..555b28dd 100644 --- a/tests/userobjects/lammps_file/tests +++ b/tests/userobjects/lammps_file/tests @@ -18,4 +18,22 @@ cli_args = 'UserObjects/lammps_runner/lammps_file="sequence_reverse/simple" Outputs/file_base=lammps_file_runner_sequence_reverse' [../] + + [./granular] + type = 'Exodiff' + input = 'granular.i' + exodiff = 'granular_out.e' + [../] + + [./granular_sequence] + type = 'Exodiff' + input = 'granular_sequence.i' + exodiff = 'granular_sequence_out.e' + [../] + + [./granular_tet] + type = 'Exodiff' + input = 'granular_tet.i' + exodiff = 'granular_tet_out.e' + [../] [] diff --git a/unit/src/OverlapTest.C b/unit/src/OverlapTest.C new file mode 100644 index 00000000..c37ea294 --- /dev/null +++ b/unit/src/OverlapTest.C @@ -0,0 +1,33 @@ +/****************************************************************/ +/* DO NOT MODIFY THIS HEADER */ +/* MOOSE - Multiphysics Object Oriented Simulation Environment */ +/* */ +/* (c) 2010 Battelle Energy Alliance, LLC */ +/* ALL RIGHTS RESERVED */ +/* */ +/* Prepared by Battelle Energy Alliance, LLC */ +/* Under Contract No. DE-AC07-05ID14517 */ +/* With the U. S. Department of Energy */ +/* */ +/* See COPYRIGHT for full restrictions */ +/****************************************************************/ + +//Magpie includes +#include "overlap.hpp" +#include + +TEST(OverlapTest, tet) +{ + // construct tet + OVERLAP::vector_t v0{0, 0, 0}; + OVERLAP::vector_t v1{1.2, 0, 0}; + OVERLAP::vector_t v2{0, 1.2, 0}; + OVERLAP::vector_t v3{0, 0, 1.2}; + OVERLAP::Tetrahedron tet = OVERLAP::Tetrahedron{v0, v1, v2, v3}; + + // construct sphere + OVERLAP::Sphere sph(OVERLAP::vector_t{0.0, 0.0, 0.0}, 1.0); + + // answer of 0.283529 is confirmed by Monte-Carlo + EXPECT_NEAR(OVERLAP::overlap(sph, tet), 0.283529, 1e-5) << "Tet overlap is wrong"; +}