quad2D module#
Created on Sun Sep 3 10:38:57 2023
@author: Pavan Pranjivan Mehta
- class quad2D.quad_pts_weights#
Bases:
objectComputes 2D Quadrature Points and weights
- legendre()#
2D Qudrature points and weights of Legendre Polynomials in [-1,1] x [-1,1]
Parameters#
- degreeint
degree of the polynomial
Returns#
2D Qudrature points and weights of Legendre polynomials within the interval [-1,1] x [-1,1]
- qx: float, 1D array
quadrature points in x direction
- qy: float, 1D array
quadrature points in y direction
- w: float, 2D array
weights
TODO#
Pending implementation for m and n, where m != n are the max. degree in x and y directions repectively
- shift_legendre()#
Shifted 2D Qudrature points and weights of Legendre Polynomials in [0,1] x [0,1]
- degreeint
degree of the polynomial
2D Qudrature points and weights of Legendre polynomials within the interval [0,1] x [0,1]
- qx: float, 1D array
quadrature points in x direction
- qy: float, 1D array
quadrature points in y direction
- w: float, 2D array
weights
Pending implementation for m and n, where m != n are the max. degree in x and y directions repectively
check the weight and quand computation
- homogenous_legendre()#
2D Qudrature points and weights of Homogenous basis using Legendre Polynomials in [-1,1] x [-1,1]
phi_i = l_{i+2} - l_{i}, where l is legendre polynomial in [-1,1] of degree i <= degree phi_j = l_{j+2} - l_{j}, where l is legendre polynomial in [-1,1] of degree j <= degree
phi_{ij} = phi_i o phi_j , o is composition operator
Parameters#
- degreeint
degree of the polynomial
Returns#
2D Qudrature points and weights of Homogenous basis using Legendre polynomials within the interval [-1,1] x[-1,1]
- q2x: float, 1D array
In x direction, associted with degree+2 Legendre Polynomial
- q2y: float, 1D array
In y direction, quadrature points associted with degree+2 Legendre Polynomial
- w2: float, 2D array
2D weights associted with degree+2 Legendre Polynomial
- q0x: float, 1D array
In x direction, quadrature points associted with degree Legendre Polynomial
- q0y: float, 1D array
In y direction, quadrature points associted with degree Legendre Polynomial
- w0: float, 1D array
2D weights associted with degree Legendre Polynomial
TODO#
Pending implementation for m and n, where m != n are the max. degree in x and y directions repectively
- homogenous_shift_legendre()#
2D Qudrature points and weights of Homogenous basis using Shited Legendre Polynomials in [0,1]
phi_i = l_{i+2} - l_{i}, where l is shifted egendre polynomial in [0,1] of degree i <= degree phi_j = l_{j+2} - l_{j}, where l is shifted legendre polynomial in [0,1] of degree j <= degree
phi_{ij} = phi_i o phi_j , o is composition operator
Parameters#
- degreeint
degree of the polynomial
Returns#
2d Qudrature points and weights of Homogenous basis using shifted Legendre polynomials within the interval [0,1]
- q2x: float, 1D array
In x direction, associted with degree+2 shifted Legendre Polynomial
- q2y: float, 1D array shifted
In y direction, quadrature points associted with degree+2 shifted Legendre Polynomial
- w2: float, 2D array
2D weights associted with degree+2 Legendre Polynomial
- q0x: float, 1D array
In x direction, quadrature points associted with degree shifted Legendre Polynomial
- q0y: float, 1D array
In y direction, quadrature points associted with degree shifted Legendre Polynomial
- w0: float, 1D array
2D weights associted with degree shifted Legendre Polynomial
TODO#
Pending implementation for m and n, where m != n are the max. degree in x and y directions repectively
- class quad2D.gauss_quad#
Bases:
objectComputes Gauss qundrature in two dimensions (2D)
- legendre(degree)#
2D Gauss Legendre Quadrature in [-1,1] x [-1,1]
Parameters#
- f: float, 2D array
f(x) at 2D quadrature points
- degreeint
degree of the polynomial
Returns#
Value of 2D Gauss Legendre Quadrature within the interval [-1,1] x [-1,1]
TODO#
Pending implementation for m and n, where m != n are the max. degree in x and y directions repectively
- shift_legendre(degree)#
2D Gauss Legendre Quadrature (shifted) in [0,1] x [0,1]
Parameters#
- f: float, 2D array
f(x) at 2D quadrature points
- degreeint
degree of the polynomial
Returns#
Value of Shifted 2D Gauss Legendre Quadrature within the interval [0,1] x [0,1]
TODO#
Pending implementation for m and n, where m != n are the max. degree in x and y directions repectively
- homogenous_legendre(f0x, f2y, f0y, degree_i, degree_j)#
2D Gauss Legendre Quadrature for homogenous in [-1,1] x [-1,1]
phi_i = l_{i+2} - l_{i}, where l is legendre polynomial in [-1,1] of degree i <= degree phi_j = l_{j+2} - l_{j}, where l is legendre polynomial in [-1,1] of degree j <= degree
phi_{ij} = phi_i o phi_j , o is composition operator
Parameters#
- f2x: float array
f(x, y) in “x” direction at quadrature points with associted degree “k+2” legendre polynomials
- f0x: float array
f(x, y) in “x” direction at quadrature point with associted degree “k” legendre polynomials
- f2y: float array
f(x, y) in “y” direction at quadrature points with associted degree “k+2” legendre polynomials
- f0y: float array
f(x, y) in “y” direction at quadrature point with associted degree “k” legendre polynomials
- degree_iint
degree of the polynomial in “x” direction
- degree_jint
degree of the polynomial in “y” direction
Returns#
Value of 2D Gauss Legendre Quadrature for homogenous within the interval [-1,1] x [-1,1]
TODO#
Pending implementation for m and n, where m != n are the max. degree in x and y directions repectively
- homogenous_shift_legendre(f0x, f2y, f0y, degree_i, degree_j)#
2D Gauss Legendre Quadrature for homogenous in [0,1] x [0,1]
phi_i = l_{i+2} - l_{i}, where l is shifted legendre polynomial in [0,1] of degree i <= degree phi_j = l_{j+2} - l_{j}, where l is shifted legendre polynomial in [0,1] of degree j <= degree
phi_{ij} = phi_i o phi_j , o is composition operator
Parameters
- f2x: float array
f(x, y) in “x” direction at quadrature points with associted degree “k+2” shifted legendre polynomials
- f0x: float array
f(x, y) in “x” direction at quadrature point with associted degree “k” shifted legendre polynomials
- f2y: float array
f(x, y) in “y” direction at quadrature points with associted degree “k+2” shifted legendre polynomials
- f0y: float array
f(x, y) in “y” direction at quadrature point with associted degree “k” shifted legendre polynomials
- degree_iint
degree of the polynomial in “x” direction
- degree_jint
degree of the polynomial in “y” direction
Returns#
Value of 2D Gauss Legendre Quadrature for homogenous within the interval [0,1] x [0,1]
TODO#
Pending implementation for m and n, where m != n are the max. degree in x and y directions repectively