overlap 0.1.1

Exact calculation of the overlap volume and area of spheres and mesh elements

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. 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.

Usage

Supported primitives

The overlap calculation directly supports these element types:

• tetrahedra (4 nodes/vertices, data type Tetrahedron)
• pentahedra/wedges/triangular prisms (5 nodes/vertices, data type Wedge)
• hexahedra (6 nodes/vertices, data type Hexahedron)

The elements must be convex and have to be specified as a list of three-dimensional nodes/vertices, while the sphere (data type Sphere) requires a center point and the radius.

Node ordering

The element types of the overlap library follow the node numbering conventions of the CFD General Notation System (CGNS) project. Please refer to the CGNS documentation for the order of the nodes of hexahedral, tetrahedral, and pentahedral/wedge-shaped elements of linear order, respectively. Also the ordering of the faces uses the conventions of CGNS. This should make interfacing this library with existing codes rather easy, often even without the need to reorder nodes.

Dependencies

The compile-time dependencies of this code are:

• Eigen3, tested with versions 3.3.4 and above
• compiler supporting C++11

The software is currently continuously compiled and tested with the following compilers:

Compiler	Versions
GNU G++	10.3.0, 9.3.0, 8.4.0, 7.5.0
Clang/LLVM	12.0.0, 11.0.0, 10.0.0, 9.0.1, 8.0.1

Additionally, the Intel C++ compiler starting with version 15.0 should work, albeit this configuration is not part of the CI process.

C++

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. The library creates two relevant type aliases, namely scalar_t for double and vector_t for Eigen::Matrix<scalar_t, 3, 1, Eigen::DontAlign>, which are used in the public interface for scalar and vectorial quantities, respectively. In principle, these types can be adjusted to specific needs, yet reducing the numerical precision of the scalar floating point type will have a significant impact on the precision and stability of the calculations.

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 (π/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;

Python

The Python version of the overlap library is available via the Python Package Index (PyPI), so for most environments installation should be possible simply via pip install overlap.

In case no pre-built package or wheel is available for your system, compilation of the wrapper code is required which in turn requires the requirements listed above for the C++ version to be fulfilled.

The interface of Python version closely resembles the interface of the C++ version:

import numpy as np
import overlap

vertices = np.array((
    (-1, -np.sqrt(1./3.), -np.sqrt(1./6.)),
    (1, -np.sqrt(1./3.), -np.sqrt(1./6.)),
    (0, np.sqrt(4./3.), -np.sqrt(1./6.)),
    (0, 0, np.sqrt(3./2.))
))

tet = overlap.Tetrahedron(vertices)
sphere = overlap.Sphere((0, 0, 0.5), 1)

result = overlap.overlap(sphere, tet)

Calculation of the overlap area instead of the overlap volume is possible via the function overlap_area() of the package.

License

The overlap library is distributed under the GNU General Public License v3, please refer to the LICENSE file for the full license text.

This distribution bundles external third-party software covered by separate license terms. For details please consult the corresponding license terms included with each package in the respective subdirectory.

Package	License
Eigen	MPL2
Google Test	3-clause BSD
pybind11	3-clause BSD