Package: overlap
Exact calculation of the overlap volume and area of spheres and mesh elementsCalculating 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 inS. 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 overlapareaof a sphere and the facets of a mesh
element can also be calculated with this library.UsageSupported primitivesThe overlap calculation directly supports these element types:tetrahedra (4 nodes/vertices, data typeTetrahedron)pentahedra/wedges/triangular prisms (5 nodes/vertices, data typeWedge)hexahedra (6 nodes/vertices, data typeHexahedron)The elements must be convex and have to be specified as a list of three-dimensional nodes/vertices,
while the sphere (data typeSphere) requires a center point and the radius.Node orderingThe element types of the overlap library follow the node numbering conventions
of theCFD General Notation System (CGNS)project.
Please refer to the CGNS documentation for the order of the nodes ofhexahedral,tetrahedral, andpentahedral/wedge-shapedelements 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.DependenciesThe compile-time dependencies of this code are:Eigen3, tested with versions 3.3.4 and abovecompiler supporting C++11The software is currently continuously compiled and tested with the following
compilers:CompilerVersionsGNU G++10.3.0, 9.3.0, 8.4.0, 7.5.0Clang/LLVM12.0.0, 11.0.0, 10.0.0, 9.0.1, 8.0.1Additionally, 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 fileoverlap.hppand
make sure theEigen3headers can be found by your compiler or build system.
The library creates two relevant type aliases, namelyscalar_tfordoubleandvector_tforEigen::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_tv0{-1,-1,-1};vector_tv1{1,-1,-1};vector_tv2{1,1,-1};vector_tv3{-1,1,-1};vector_tv4{-1,-1,1};vector_tv5{1,-1,1};vector_tv6{1,1,1};vector_tv7{-1,1,1};Hexahedronhex{v0,v1,v2,v3,v4,v5,v6,v7};Spheres{vector_t::Constant(1),1};scalar_tresult=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
functionoverlapArea()can be employed as such:vector_tv0{-std::sqrt(3)/6.0,-1.0/2.0,0};vector_tv1{std::sqrt(3)/3.0,0,0};vector_tv2{-std::sqrt(3)/6.0,+1.0/2.0,0};vector_tv3{0,0,std::sqrt(6)/3.0};Tetrahedrontet{v0,v1,v2,v3};Spheres{{0,0,1.5},1.25};autoresult=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_tf=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;PythonThe Python version of theoverlaplibrary is available via thePython
Package Index (PyPI), so for most
environments installation should be possible simply viapip install overlap.In case no pre-built package orwheelis 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:importnumpyasnpimportoverlapvertices=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 functionoverlap_area()of the package.LicenseTheoverlaplibrary is distributed under the GNU General Public
License v3, please refer to theLICENSEfile 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.PackageLicenseEigenMPL2Google Test3-clause BSDpybind113-clause BSD