polyhedral-gravity 3.0

Package to compute full gravity tensor of a given constant density polyhedron for arbitrary points according to the geodetic convention

polyhedral-gravity-model

Table of Contents

• References
• Documentation & Examples
  • Input & Output (C++ and Python)
  • Minimal Python Example
  • Minimal C++ Example
• Installation
  • With conda
  • With pip
  • From source
• C++ Library & Executable
  • Building the C++ Library & Executable
  • Running the C++ Executable
• Testing
• Contributing

References

This code is a validated implementation in C++17 of the Polyhedral Gravity Model by Tsoulis et al.. It was created in a collaborative project between TU Munich and ESA's Advanced Concepts Team. Please refer to the project report for extensive information about the theoretical background, related work, implementation & design decisions, application, verification, and runtime measurements of the presented code.

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 constants density's unit must match. Have a look 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 accerleration 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 accleration

[!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

# 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.

cube_polyhedron = Polyhedron(
  polyhedral_source=(cube_vertices, cube_faces),
  density=cube_density,
  normal_orientation=NormalOrientation.INWARDS, # OUTWARDS (default) or INWARDS
  integrity_check=PolyhedronIntegrity.VERIFY,   # VERIFY (default), DISABLE or HEAL
)

[!TIP] More examples and plots are depicted in the 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 (1.13.0 or compatible), only required for testing
• spdlog (1.11.0 or compatible), required for logging
• tetgen (1.6 or compatible), required for I/O
• yaml-cpp (0.7.0 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 (2.10.4 or compatible), required for the Python interface, but not the C++ standalone

The module will be build 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 build 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
LOGGING_LEVEL (2)	0 = TRACE/ 1 = DEBUG/ 2 = INFO / 3 = WARN/ 4 = ERROR/ 5 = CRITICAL/ 6 = OFF
USE_LOCAL_TBB (OFF)	Use a local installation of TBB instead of setting it up via CMake
BUILD_POLYHEDRAL_GRAVITY_DOCS (OFF)	Build this documentation
BUILD_POLYHEDRAL_GRAVITY_TESTS (ON)	Build the Tests
BUILD_POLYHEDRAL_PYTHON_INTERFACE (ON)	Build the Python interface

During testing POLYHEDRAL_GRAVITY_PARALLELIZATION=TBB has been the most performant. It is further not recommend to change the 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 section below)
    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)
  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 oder 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.