
Quick search LESARD: Tools¶ The local library `LocalLib` contains pythonic hacks that are gathered as simple functions externally accessible. Page navigation Simple pythonic hacks Tools handling quadrature points (external library) Tools apparented to Barycentric coordinates Tools apparented to Barycentric basis functions Simple pythonic hacks ¶ `3_LESARD.LocalLib.ReversedFunctools.partial`(func, args)¶ Partial function applying the last arguments Parameters func* (function callback) – the function of interest args (arguments list) – the arguments to parse to the function from the end. Returns the lambda function defining the partial callback from the last arguments Return type Lmb (function callback) Tools handling quadrature points (external library)¶ `3_LESARD.LocalLib.LobattoQuadratureNodes.GaussLobatto`(n, eps)¶ Computes the Legendre-Gauss-Lobatto nodes, weights and the LGL Vandermonde matrix. The LGL nodes are the zeros of (1-x^2)P’_N(x). Useful for numerical integration and spectral methods. Initial content from Greg von Winckel - 04/17/2004, Python translation of lglnodes.m, translated and modified into Python by Jacob Schroder - 9/15/2018 Parameters n (integer) – requesting an nth-order Gauss-quadrature rule on [-1, 1] epse (float) – safety net for division Returns representing the quadrature nodes and weights. Return type (nodes, weights) (tuple) Example >>> from lglnodes import * >>> (nodes, weights) = lglnodes(3) >>> print(str(nodes) + " " + str(weights)) [-1. -0.4472136 0.4472136 1. ] [0.16666667 0.83333333 0.83333333 0.16666667] Note (n+1) nodes and weights are returned Reference on LGL nodes and weights: C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Tang, “Spectral Methods in Fluid Dynamics,” Section 2.3. Springer-Verlag 1987 Tools apparented to Barycentric coordinates ¶ `3_LESARD.LocalLib.BarycentricTools.GetBarycentricCoordinates`(points, vertices)¶ Barycentric coordinates from the given vertices (forming a triangles) Parameters Points (2D numpy array) – Cartesian coordinates of the points to convert (NbPoints x 2) vertices (2D numpy array) – Cartesian coordinates of the points taken as reference, ordered. (3 x 2) Returns Barycentric coordinates of points with respect to the vertices (NbPoints x 3) Return type bcoors (2D numpy array) `3_LESARD.LocalLib.BarycentricTools.GetCartesianCoordinates`(bcoords, vertices)¶ Barycentric coordinates from the given vertices (forming a triangles) Parameters bcoors (2D numpy array) – Barycentric coordinates of points with respect to the vertices (NbPoints x 3) vertices (2D numpy array) – Cartesian coordinates of the points taken as reference, ordered. (3 x 2) Returns Cartesian coordinates of the points to convert (NbPoints x 2) Return type Points (2D numpy array) Tools apparented to Barycentric basis functions ¶ `3_LESARD.LocalLib.BarycentricBasisFunctions.BarycentricBasis`(Nele, Type, Order, points)¶ Simple mapping to the implementation giving out the barycentic basis functions according to the elment’s type (simplicial, quad, …) Parameters Nele (integer) – the number of edges of the considered element to compute the basis functions for Type (string) – type of wished basis functions Order (integer) – order of the wished basis functions points (2D numpy array) – the points at which to evaluate the basis functons (NbPoints, 3) Returns the value of all the basis functions at each given point (NbBasisFunc x NbPoints) through the call of the relevant function. Return type EvaluatedBase (2D numpy array) `3_LESARD.LocalLib.BarycentricBasisFunctions.BarycentricGradients`(Nele, Type, Order, points, n, vol)¶ Simple mapping to the implementation giving out the barycentic basis functions gradients according to the elment’s type (simplicial, quad, …) Parameters Nele (integer) – the number of edges of the considered element to compute the basis functions for Type (string) – type of wished basis functions Order (integer) – order of the wished basis functions points (2D numpy array) – the points at which to evaluate the basis functons (NbPoints, 3) n (2D numpy array) – the normal vector to each face, in the order corresponding to the vertices furnishing the barycentric coordinates referential vol (float) – the volume of the element furnishing the barycentric coordinates referential Returns the value of all the x and y gradient’s coordinates at each given point, ordered accordingly to the vertices from which the barycentic coordinates have been generated (NbBasisFunc x NbPoints x 2), through the call of the relevant function. Return type EvaluatedGradient (3D numpy array)

LESARD: Tools¶

The local library LocalLib contains pythonic hacks that are gathered as simple functions externally accessible.

Page navigation

Simple pythonic hacks
Tools handling quadrature points (external library)
Tools apparented to Barycentric coordinates
Tools apparented to Barycentric basis functions

Simple pythonic hacks ¶

3_LESARD.LocalLib.ReversedFunctools.partial(func, *args)¶

Partial function applying the last arguments

Parameters

func (function callback) – the function of interest
*args (arguments list) – the arguments to parse to the function from the end.

Returns

the lambda function defining the partial callback from the last arguments

Return type

Lmb (function callback)

Tools handling quadrature points (external library)¶

3_LESARD.LocalLib.LobattoQuadratureNodes.GaussLobatto(n, eps)¶

Computes the Legendre-Gauss-Lobatto nodes, weights and the LGL Vandermonde matrix. The LGL nodes are the zeros of (1-x^2)*P’_N(x). Useful for numerical integration and spectral methods. Initial content from Greg von Winckel - 04/17/2004, Python translation of lglnodes.m, translated and modified into Python by Jacob Schroder - 9/15/2018

Parameters

n (integer) – requesting an nth-order Gauss-quadrature rule on [-1, 1]
epse (float) – safety net for division

Returns

representing the quadrature nodes and weights.

Return type

(nodes, weights) (tuple)

Example

>>> from lglnodes import *
>>> (nodes, weights) = lglnodes(3)
>>> print(str(nodes) + "   " + str(weights))
[-1.        -0.4472136  0.4472136  1.       ]   [0.16666667 0.83333333 0.83333333 0.16666667]

Note

(n+1) nodes and weights are returned
Reference on LGL nodes and weights: C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Tang, “Spectral Methods in Fluid Dynamics,” Section 2.3. Springer-Verlag 1987

Tools apparented to Barycentric coordinates ¶

3_LESARD.LocalLib.BarycentricTools.GetBarycentricCoordinates(points, vertices)¶

Barycentric coordinates from the given vertices (forming a triangles)

Parameters

Points (2D numpy array) – Cartesian coordinates of the points to convert (NbPoints x 2)
vertices (2D numpy array) – Cartesian coordinates of the points taken as reference, ordered. (3 x 2)

Returns

Barycentric coordinates of points with respect to the vertices (NbPoints x 3)

Return type

bcoors (2D numpy array)

3_LESARD.LocalLib.BarycentricTools.GetCartesianCoordinates(bcoords, vertices)¶

Barycentric coordinates from the given vertices (forming a triangles)

Parameters

bcoors (2D numpy array) – Barycentric coordinates of points with respect to the vertices (NbPoints x 3)
vertices (2D numpy array) – Cartesian coordinates of the points taken as reference, ordered. (3 x 2)

Returns

Cartesian coordinates of the points to convert (NbPoints x 2)

Return type

Points (2D numpy array)

Tools apparented to Barycentric basis functions ¶

3_LESARD.LocalLib.BarycentricBasisFunctions.BarycentricBasis(Nele, Type, Order, points)¶

Simple mapping to the implementation giving out the barycentic basis functions according to the elment’s type (simplicial, quad, …)

Parameters

Nele (integer) – the number of edges of the considered element to compute the basis functions for
Type (string) – type of wished basis functions
Order (integer) – order of the wished basis functions
points (2D numpy array) – the points at which to evaluate the basis functons (NbPoints, 3)

Returns

the value of all the basis functions at each given point (NbBasisFunc x NbPoints): through the call of the relevant function.

Return type

EvaluatedBase (2D numpy array)

3_LESARD.LocalLib.BarycentricBasisFunctions.BarycentricGradients(Nele, Type, Order, points, n, vol)¶

Simple mapping to the implementation giving out the barycentic basis functions gradients according to the elment’s type (simplicial, quad, …)

Parameters

Nele (integer) – the number of edges of the considered element to compute the basis functions for
Type (string) – type of wished basis functions
Order (integer) – order of the wished basis functions
points (2D numpy array) – the points at which to evaluate the basis functons (NbPoints, 3)
n (2D numpy array) – the normal vector to each face, in the order corresponding to the vertices furnishing the barycentric coordinates referential
vol (float) – the volume of the element furnishing the barycentric coordinates referential

Returns

the value of all the x and y gradient’s coordinates at each given point, ordered accordingly to the: vertices from which the barycentic coordinates have been generated (NbBasisFunc x NbPoints x 2), through the call of the relevant function.

Return type

EvaluatedGradient (3D numpy array)

LESARD: Tools¶

Simple pythonic hacks¶

Tools handling quadrature points (external library)¶

Tools apparented to Barycentric coordinates¶

Tools apparented to Barycentric basis functions¶

Simple pythonic hacks ¶

Tools apparented to Barycentric coordinates ¶

Tools apparented to Barycentric basis functions ¶