stiffness_matrix2D module#

Created on Sun Sep 3 16:57:29 2023

@author: Pavan Pranjivan Mehta

class stiffness_matrix2D.stiffness_matrix2D#

Bases: object

Computes stiffness matrix in 2 dimensions

Uses the trick,

K2 = K1 M1 + M1 K1, wherw K1 and M1 are 1D stiffness and mass matrix Credits for the formula : Prof. Luca Heltai, SISSA MathLab, Trieste, Italy

legendre(min_degree=1, h=1e-06)#

Computes stiffness matrix Legendre Polynomials in [-1,1] of dim = (k^2 x k^2), where k = (max_degree - min_degree)

Parameters#

min_degreeint: min degree of the polynomial, default = 1
max_degreeint: max degree of the polynomial

Returns#

2D Stiffness martix Legendre polynomials within the interval [-1,1] x [-1,1]

M: numpy array: 2D Matrix of dim = (k^2 x k^2), where k = (max_degree - min_degree) of Legengre Polynomials

homogenous_legendre(min_degree=0, h=1e-06)#

Computes 2D Stiffness matrix for Homogenous basis function using Legendre Polynomials in [-1,1] x [-1,1] of dim = (k^2 x k^2), where k = (max_degree - min_degree)

Parameters#

min_degreeint: min degree of the polynomial, default = 0
max_degreeint: max degree of the polynomial

Returns#

2D Stiffness martix of homogenous basis using Legendre polynomials within the interval [-1,1]

M: numpy array: 2D Matrix of dim = (k^2 x k^2), where k = (max_degree - min_degree) of Polynomials

homogenous_shift_legendre(min_degree=0, h=1e-06)#

Computes 2D Stiffness matrix Homogenous basis function using Shifted Legendre Polynomials in [0,1] x [0,1] of dim = (k^2 x k^2), where k = (max_degree - min_degree)

Parameters#

min_degreeint: min degree of the polynomial, default = 0
max_degreeint: max max_degree of the polynomial

Returns#

2D Stiffness martix of homogenous basis using shifted Legendre polynomials within the interval [0,1]

M: numpy array: 2D Matrix of dim = (k^2 x k^2), where k = (max_degree - min_degree) of shifted Legengre Polynomials

class stiffness_matrix2D.test#

Bases: object

homogenous_legendre(min_degree=0)#

runge()#

funx()#

funy()#

func(y)#

der2funcX()#

der2funcY()#

laplacian_homogenous_legendre(min_degree=0)#

Solves for Laplacian using Homogenous basis function using Legendre Polynomials in [-1,1]

phi_{k} = L_{k+2} - L_{k}, where L_{k} is k-th order Legendre Polynomial

homogenous_legendre_proj_err()#

laplacian_homogenous_shift_legendre(min_degree=0)#

Solves for Laplacian using Homogenous basis function using Shifted Legendre Polynomials in [0,1]

phi_{k} = L_{k+2} - L_{k}, where L_{k} is k-th order Legendre Polynomial

homogenous_shift_legendre_proj_err()#

proj_err()#