polyhedral-gravity-model
Table of Contents
References
This code is a validated implementation in C++17 of the Polyhedral Gravity Model by Tsoulis et al.. Additionally, the model provides a Python binding. It was initially created in a collaborative project between TU Munich and ESA's Advanced Concepts Team.
If this implementation proves useful to you, please consider citing the accompanying paper published in the Journal of Open Source Software.
The implementation is based on the paper Tsoulis, D., 2012. Analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals. Geophysics, 77(2), pp.F1-F11. and its corresponding implementation in FORTRAN.
Supplementary details can be found in the more recent paper TSOULIS, Dimitrios; GAVRIILIDOU, Georgia. A computational review of the line integral analytical formulation of the polyhedral gravity signal. Geophysical Prospecting, 2021, 69. Jg., Nr. 8-9, S. 1745-1760. and its corresponding implementation in MATLAB, which is strongly based on the former implementation in FORTRAN.
Documentation & Examples
[!NOTE] The GitHub Pages of this project contain the full extensive documentation of the C++ Library and Python Interface as well as background on the gravity model and advanced settings not detailed here.
Input & Output (C++ and Python)
Input
The evaluation of the polyhedral gravity model requires the following parameters:
| Name |
|---|
| Polyhedral Mesh (either as vertices & faces or as polyhedral source files) |
| Constant Density $\rho$ |
The mesh and the constant density's unit must match. Have a look at the documentation to view the supported mesh files.
Output
The calculation outputs the following parameters for every Computation Point P. The units of the respective output depend on the units of the input parameters (mesh and density)! Hence, if e.g., your mesh is in $km$, the density must match. Further, output units will be different accordingly.
| Name | Unit (if mesh in $[m]$ and $\rho$ in $[kg/m^3]$) | Comment |
|---|---|---|
| $V$ | $\frac{m^2}{s^2}$ or $\frac{J}{kg}$ | The potential or also called specific energy |
| $V_x$, $V_y$, $V_z$ | $\frac{m}{s^2}$ | The gravitational acceleration in the three cartesian directions |
| $V_{xx}$, $V_{yy}$, $V_{zz}$, $V_{xy}$, $V_{xz}$, $V_{yz}$ | $\frac{1}{s^2}$ | The spatial rate of change of the gravitational acceleration |
[!NOTE] This gravity model's output obeys to the geodesy and geophysics sign conventions. Hence, the potential $V$ for a polyhedron with a mass $m > 0$ is defined as positive. Accordingly, the accelerations are defined as $\textbf{g} = + \nabla V$.
Minimal Python Example
The following example shows how to use the python interface to compute the gravity around a cube:
import numpy as np
from polyhedral_gravity import Polyhedron, GravityEvaluable, evaluate, PolyhedronIntegrity, NormalOrientation, MetricUnit
# We define the cube as a polyhedron with 8 vertices and 12 triangular faces
# The polyhedron's normals point outwards (see below for checking this)
# The density is set to 1.0
cube_vertices = np.array(
[[-1, -1, -1], [1, -1, -1], [1, 1, -1], [-1, 1, -1],
[-1, -1, 1], [1, -1, 1], [1, 1, 1], [-1, 1, 1]]
)
cube_faces = np.array(
[[1, 3, 2], [0, 3, 1], [0, 1, 5], [0, 5, 4], [0, 7, 3], [0, 4, 7],
[1, 2, 6], [1, 6, 5], [2, 3, 6], [3, 7, 6], [4, 5, 6], [4, 6, 7]]
)
cube_density = 1.0
computation_point = np.array([0, 0, 0])
We first define a constant density Polyhedron from vertices and faces
cube_polyhedron = Polyhedron(
polyhedral_source=(cube_vertices, cube_faces),
density=cube_density,
)
In case you want to hand over the polyhedron via a supported file format,
just replace the polyhedral_source argument with a list of strings,
where each string is the path to a supported file format, e.g. polyhedral_source=["eros.node","eros.face"] or polyhedral_source=["eros.mesh"].
Continuing, the simplest way to compute the gravity is to use the evaluate function:
potential, acceleration, tensor = evaluate(
polyhedron=cube_polyhedron,
computation_points=computation_point,
parallel=True,
)
The more advanced way is to use the GravityEvaluable class. It caches the
internal data structure and properties which can be reused for multiple
evaluations. This is especially useful if you want to compute the gravity
for multiple computation points but don't know the "future points" in advance.
evaluable = GravityEvaluable(polyhedron=cube_polyhedron) # stores intermediate computation steps
potential, acceleration, tensor = evaluable(
computation_points=computation_point,
parallel=True,
)
# Any future evaluable call after this one will be faster
Note that the computation_point could also be (N, 3)-shaped array to compute multiple points at once.
In this case, the return value of evaluate(..) or an GravityEvaluable will
be a list of triplets comprising potential, acceleration, and tensor.
The gravity model requires that all the polyhedron's plane unit normals consistently
point outwards or inwards the polyhedron. You can specify this via the normal_orientation.
This property is - by default - checked when constructing the Polyhedron! So, don't worry, it
is impossible if not explicitly disabled to create an invalid Polyhedron.
You can disable/ enable this setting via the optional integrity_check flag and can even
automatically repair the ordering via HEAL.
If you are confident that your mesh is defined correctly (e.g., checked once with the integrity check)
you can disable this check (via DISABLE) to avoid the additional runtime overhead of the check.
Also, you can set the metric unit of the mesh and the density.
This also influences the output unit. E.g., Density in $kg/m^3$, Mesh in $m$, then the potential is given in $m^2/s^2$.
cube_polyhedron = Polyhedron(
polyhedral_source=(cube_vertices, cube_faces),# coordinates in m (default), km, or unitless
density=cube_density, # kg/m^3 (default) or kg/km^3 or unitless
normal_orientation=NormalOrientation.INWARDS, # OUTWARDS (default) or INWARDS
integrity_check=PolyhedronIntegrity.VERIFY, # VERIFY (default), DISABLE or HEAL
metric_unit=MetricUnit.METER, # METER (default), KILOMETER, UNITLESS
)
[!TIP] More examples and plots are depicted in the jupyter notebook and the second jupyter notebook
Minimal C++ Example
The following example shows how to use the C++ library to compute the gravity. It works analogously to the Python example above.
// Defining the input like above in the Python example
std::vector<std::array<double, 3>> vertices = ...
std::vector<std::array<size_t, 3>> faces = ...
double density = 1.0;
// The constant density polyhedron is defined by its vertices & faces
// It also supports the hand-over of NormalOrientation and PolyhedronIntegrity as optional arguments
// as above described for the Python Interface
Polyhedron polyhedron{vertices, faces, density};
std::vector<std::array<double, 3>> points = ...
std::array<double, 3> point = points[0];
bool parallel = true;
The C++ library provides also two ways to compute the gravity. Via
the free function evaluate...
const auto[pot, acc, tensor] = GravityModel::evaluate(polyhedron, point, parallel);
... or via the GravityEvaluable class.
// Instantiation of the GravityEvaluable object
GravityEvaluable evaluable{polyhedron};
// From now, we can evaluate the gravity model for any point with
const auto[potential, acceleration, tensor] = evaluable(point, parallel);
// or for multiple points with
const auto results = evaluable(points, parallel);
Similarly to Python, the C++ implementation also provides mesh checking capabilities.
[!TIP] For reference, have a look at the main method of the C++ executable.
Installation
With conda
The python interface can be easily installed with conda:
conda install -c conda-forge polyhedral-gravity-model
With pip
As a second option, you can also install the python interface with pip from PyPi.
pip install polyhedral-gravity
Binaries for the most common platforms are available on PyPI, including
Windows, Linux, and macOS. For macOS and Linux, binaries for
x86_64 and aarch64 are provided.
In case pip uses the source distribution, please make sure that
you have a C++17 capable compiler and CMake installed.
From source
The project uses the following dependencies, all of them are automatically set up via CMake:
- GoogleTest (1.15.2 or compatible), only required for testing
- spdlog (1.13.0 or compatible), required for logging
- tetgen (1.6 or compatible), required for I/O
- yaml-cpp (0.8.0 or compatible), required for I/O
- thrust (2.1.0 or compatible), required for parallelization and utility
- xsimd (11.1.0 or compatible), required for vectorization of the
atan(..) - pybind11 (2.12.0 or compatible), required for the Python interface, but not the C++ standalone
The module will be built using a C++17 capable compiler, CMake. Just execute the following command in the repository root folder:
pip install .
To modify the build options (like parallelization) have a look
at the next paragraph. The options
are modified by setting the environment variables before executing
the pip install . command, e.g.:
export POLYHEDRAL_GRAVITY_PARALLELIZATION="TBB"
pip install .
(Optional: For a faster build, you can install all dependencies available for your system in your local python environment. That way, they won't be fetched from GitHub.)
C++ Library & Executable
Building the C++ Library & Executable
The program is built by using CMake. So first make sure that you installed CMake and then follow these steps:
mkdir build
cd build
cmake .. <options>
cmake --build .
The following options are available:
| Name (Default) | Options |
|---|---|
POLYHEDRAL_GRAVITY_PARALLELIZATION (CPP) |
CPP = Serial Execution / OMP or TBB = Parallel Execution with OpenMP or Intel's TBB |
POLYHEDRAL_GRAVITY_LOGGING_LEVEL (INFO) |
TRACE, DEBUG, INFO, WARN, ERROR, CRITICAL, OFF |
BUILD_POLYHEDRAL_GRAVITY_DOCS (OFF) |
Build this documentation |
BUILD_POLYHEDRAL_GRAVITY_TESTS (ON) |
Build the Tests |
BUILD_POLYHEDRAL_GRAVITY_PYTHON_INTERFACE (ON) |
Build the Python interface |
During testing POLYHEDRAL_GRAVITY_PARALLELIZATION=TBB has been the most performant.
It is further not recommended to change the POLYHEDRAL_GRAVITY_LOGGING_LEVEL to something else than INFO=2.
The recommended CMake settings using the TBB backend would look like this:
cmake .. -POLYHEDRAL_GRAVITY_PARALLELIZATION="TBB"
Running the C++ Executable
After the build, the gravity model can be run by executing:
./polyhedralGravity <YAML-Configuration-File>
where the YAML-Configuration-File contains the required parameters.
Examples for Configuration Files and Polyhedral Source Files can be
found in this repository in the folder /example-config/.
Input Configuration File
The configuration should look similar to the given example below. It is required to specify the source-files of the polyhedron's mesh (more info about the supported file in the documentation), the density of the polyhedron, and the wished computation points where the gravity tensor shall be computed. Further, one must specify the name of the .csv output file.
---
gravityModel:
input:
polyhedron: # polyhedron source-file(s)
- "../example-config/data/tsoulis.node" # .node contains the vertices
- "../example-config/data/tsoulis.face" # .face contains the triangular faces
density: 2670.0 # constant density, units must match with the mesh (see a section below)
# Depends on metric_unit: 'km' -> kg/km^3, 'm' -> kg/m^3, 'unitless' -> 'unitless'
points: # Location of the computation point(s) P
- [ 0, 0, 0 ] # Here it is situated at the origin
check_mesh: true # Fully optional, enables mesh autodetect+repair of
# the polyhedron's vertex ordering (not given: true)
metric_unit: m # Unit of mesh: One of 'm', 'km' or 'unitless' (not given: 'm')
output:
filename: "gravity_result.csv" # The name of the output file
Output
The executable produces a CSV file containing $V$, $V_x$, $V_y$, $V_z$, $V_{xx}$, $V_{yy}$, $V_{zz}$, $V_{xy}$, $V_{xz}$, $V_{yz}$ for every computation point P.
Testing
The project uses GoogleTest for testing. In order to execute those tests, just execute the following command in the build directory:
ctest
For the Python test suite, please execute the following command in the repository root folder:
pytest
Contributing
We are happy to accept contributions to the project in the form of suggestions, bug reports, and pull requests. Please have a look at the contributing guidelines for more information.
1 maintainer
Metadata
1 maintainer
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