Python API - polyhedral_gravity#

Module Overview#

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 \(\rho\)

In the Python Interface, you define these parameters as polyhedral_gravity.Polyhedron

from polyhedral_gravity import Polyhedron, GravityEvaluable, evaluate, PolyhedronIntegrity, NormalOrientation

polyhedron = Polyhedron(
    polyhedral_source=(vertices, faces),           # (N,3) and (M,3) array-like
    density=density,                               # Density of the Polyhedron, Unit must match to the mesh's scale
    normal_orientation=NormalOrientation.OUTWARDS, # Possible values OUTWARDS (default) or INWARDS
    integrity_check=PolyhedronIntegrity.VERIFY,    # Possible values AUTOMATIC (default), VERIFY, HEAL, DISABLE
)

Note

Tsoulis et al.’s formulation requires that the normals point OUTWARDS. The implementation can handle both cases and also can automatically determine the property if initiall set wrong. Using AUTOMATIC (default for first-time-user) or VERIFY raises a ValueError if the polyhedral_gravity.NormalOrientation is wrong. Using HEAL will re-order the vertex sorting to fix errors. Using DISABLE will turn this check off and avoid \(O(n^2)\) runtime complexcity of this check! Highly recommened, when you “know your mesh”!

The polyhedron’s mesh’s units must match with the constant density! For example, if the mesh is in \([m]\), then the constant density should be in \([\frac{kg}{m^3}]\).

Afterwards one can use polyhedral_gravity.evaluate() the gravity at a single point P via:

potential, acceleration, tensor = evaluate(
    polyhedron=polyhedron,
    computation_points=P,
    parallel=True,
)

or via use the cached approach polyhedral_gravity.GravityEvaluable (desriable for subsequent evaluations using the same polyhedral_gravity.Polyhedron)

evaluable = GravityEvaluable(polyhedron=polyhedron)
potential, acceleration, tensor = evaluable(
    computation_points=P,
    parallel=True,
)

Note

If P would be an array of points, the return value would be a List[Tuple[potential, acceleration, tensor]]!

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, the output units will match the input units.

Name	If mesh \([m]\) and density \([\frac{kg}{m^3}]\)	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

This 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.

The accelerations \(V_x\), \(V_y\), \(V_z\) are defined as

\[\textbf{g} = + \nabla V = \left( \frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z} \right)\]

Accordingly, the second derivative tensor is defined as the derivative of \(\textbf{g}\).

Polyhedron#

class polyhedral_gravity.Polyhedron#

A constant density Polyhedron stores the mesh data consisting of vertices and triangular faces. The density and the coordinate system in which vertices and faces are defined need to have the same scale/ units. Tsoulis et al.’s polyhedral gravity model requires that the plane unit normals of every face are pointing outwards of the polyhedron. Otherwise the results are negated. The class by default enforces this constraints and offers utility to (automatically) make the input data obey to this constraint.

__getitem__(self: polyhedral_gravity.Polyhedron, index: int) → Annotated[list[Annotated[list[float], FixedSize(3)]], FixedSize(3)]#

Returns the the three cooridnates of the vertices making the face at the requested index. This does not return the face as list of vertex indices, but resolved with the actual coordinates.

Parameters:: index – The index of the face
Returns:: The resolved face
Return type:: \((3, 3)\)-array-like
Raises:: IndexError if face index is out-of-bounds –

__init__(self: polyhedral_gravity.Polyhedron, polyhedral_source: Union[tuple[list[Annotated[list[float], FixedSize(3)]], list[Annotated[list[int], FixedSize(3)]]], list[str]], density: float, normal_orientation: polyhedral_gravity.NormalOrientation = <NormalOrientation.OUTWARDS: 0>, integrity_check: polyhedral_gravity.PolyhedronIntegrity = <PolyhedronIntegrity.AUTOMATIC: 2>) → None#

Creates a new Polyhedron from vertices and faces and a constant density. If the integrity_check is not set to DISABLE, the mesh integrity is checked (so that it fits the specification of the polyhedral model by Tsoulis et al.)

Parameters:

polyhedral_source – The vertices (\((N, 3)\)-array-like) and faces (\((M, 3)\)-array-like) of the polyhedron as pair or The filenames of the files containing the vertices & faces as list of strings
density – The constant density of the polyhedron, it must match the mesh’s units, e.g. mesh in \([m]\) then density in \([kg/m^3]\)
normal_orientation – The pointing direction of the mesh’s plane unit normals, i.e., either OUTWARDS or INWARDS of the polyhedron. One of polyhedral_gravity.NormalOrientation. (default: OUTWARDS)
integrity_check –
Conducts an Integrity Check (degenerated faces/ vertex ordering) depending on the values. One of polyhedral_gravity.PolyhedronIntegrity:
- AUTOMATIC (Default): Prints to stdout and throws ValueError if normal_orientation is wrong/ inconsisten
- VERIFY: Like AUTOMATIC, but does not print to stdout
- DISABLE: Recommened, when you are familiar with the mesh to avoid \(O(n^2)\) runtime cost. Disables ALL checks
- HEAL: Automatically fixes the normal_orientation and vertex ordering to the correct values

Raises:

ValueError – If the faces array does not contain a reference to vertex 0 indicating an index start at 1
ValueError – If integrity_check is set to AUTOMATIC or VERIFY and the mesh is inconsistent

Note

The integrity_check is automatically enabled to avoid wrong results due to the wrong vertex ordering. The check requires \(O(n^2)\) operations. You want to turn this off, when you know you mesh!

__repr__(self: polyhedral_gravity.Polyhedron) → str#: str: A string representation of this polyhedron

check_normal_orientation(self: polyhedral_gravity.Polyhedron) → tuple[polyhedral_gravity.NormalOrientation, set[int]]#

Returns a tuple consisting of majority plane unit normal orientation, i.e. the direction in which at least more than half of the plane unit normals point, and the indices of the faces violating this orientation, i.e. the faces whose plane unit normals point in the other direction. The set of incides vioalting the property is empty if the mesh has a clear ordering. The set contains values if the mesh is incosistent.

Returns:: Tuple consisting consisting of majority plane unit normal orientation and the indices of the faces violating this orientation.

Note

This utility is mainly for diagnostics and debugging purposes. If the polyhedron is constrcuted with integrity_check set to AUTOMATIC or VERIFY, the construction fails anyways. If set to HEAL, this method should return an empty set (but maybe a different ordering than initially specified) Only if set to DISABLE, then this method might actually return a set with faulty indices. Hence, if you want to know your mesh error. Construct the polyhedron with integrity_check=DISABLE and call this method.

property density#

The density of the polyhedron (Read/ Write)

Type:: float

property faces#

The faces of the polyhedron (Read-Only)

Type:: (M, 3)-array-like of int

property normal_orientation#

The orientation of the plane unit normals (Read-Only)

Type:: polyhedral_gravity.NormalOrientation

property vertices#

The vertices of the polyhedron (Read-Only)

Type:: (N, 3)-array-like of float

Enums to specify MeshChecks#

class polyhedral_gravity.NormalOrientation#: The orientation of the plane unit normals of the polyhedron. Tsoulis et al. equations require the normals to point outwards of the polyhedron. If the opposite hold, the result is negated. The implementation can handle both cases.

Members:

OUTWARDS : Outwards pointing plane unit normals

INWARDS : Inwards pointing plane unit normals

class polyhedral_gravity.PolyhedronIntegrity#: The pointing direction of the normals of a Polyhedron. They can either point outwards or inwards the polyhedron.

Members:

DISABLE : All activities regarding MeshChecking are disabled. No runtime overhead!

VERIFY : Only verification of the NormalOrientation. A misalignment (e.g. specified OUTWARDS, but is not) leads to a runtime_error. Runtime Cost \(O(n^2)\)

AUTOMATIC : Like VERIFY, but also informs the user about the option in any case on the runtime costs. This is the implicit default option. Runtime Cost: \(O(n^2)\) and output to stdout in every case!

HEAL : Verification and Autmatioc Healing of the NormalOrientation. A misalignemt does not lead to a runtime_error, but to an internal correction of vertices ordering. Runtime Cost: \(O(n^2)\)

GravityModel#

Single Function#

polyhedral_gravity.evaluate(polyhedron: polyhedral_gravity.Polyhedron, computation_points: Union[Annotated[list[float], FixedSize(3)], list[Annotated[list[float], FixedSize(3)]]], parallel: bool = True) → Union[tuple[float, Annotated[list[float], FixedSize(3)], Annotated[list[float], FixedSize(6)]], list[tuple[float, Annotated[list[float], FixedSize(3)], Annotated[list[float], FixedSize(6)]]]]#

Evaluates the polyhedral gravity model for a given constant density polyhedron at a given computation point.

Parameters:

polyhedron – The polyhedron for which to evaluate the gravity model
computation_points – The computation points as tuple or list of points
parallel – If True, the computation is done in parallel (default: True)

Returns:

Either a triplet of potential \(V\), acceleration \([V_x, V_y, V_z]\) and second derivatives \([V_{xx}, V_{yy}, V_{zz}, V_{xy},V_{xz}, V_{yz}]\) at the computation points or if multiple computation points are given a list of these triplets

Cached Evaluation#

class polyhedral_gravity.GravityEvaluable#

A class to evaluate the polyhedral gravity model for a given constant density polyhedron at a given computation point. It provides a poylhedral_gravity.GravityEvaluable.__call__() method to evaluate the polyhedral gravity model for computation points while also caching the polyhedron & intermediate results over the lifetime of the object.

__call__(self: polyhedral_gravity.GravityEvaluable, computation_points: Union[Annotated[list[float], FixedSize(3)], list[Annotated[list[float], FixedSize(3)]]], parallel: bool = True) → Union[tuple[float, Annotated[list[float], FixedSize(3)], Annotated[list[float], FixedSize(6)]], list[tuple[float, Annotated[list[float], FixedSize(3)], Annotated[list[float], FixedSize(6)]]]]#

Evaluates the polyhedral gravity model for a given constant density polyhedron at a given computation point.

Parameters:

computation_points – The computation points as tuple or list of points
parallel – If True, the computation is done in parallel (default: True)

Returns:

__init__(self: polyhedral_gravity.GravityEvaluable, polyhedron: polyhedral_gravity.Polyhedron) → None#

Creates a new GravityEvaluable for a given constant density polyhedron. It provides a poylhedral_gravity.GravityEvaluable.__call__() method to evaluate the polyhedral gravity model for computation points while also caching the polyhedron & intermediate results over the lifetime of the object.

Parameters:: polyhedron – The polyhedron for which to evaluate the gravity model

__repr__(self: polyhedral_gravity.GravityEvaluable) → str#: str: A string representation of this GravityEvaluable

Python API - polyhedral_gravity

Contents

Python API - polyhedral_gravity#

Module Overview#

Polyhedron#

Enums to specify MeshChecks#

GravityModel#

Single Function#

Cached Evaluation#