stiffness_matrix module#

Created on Fri Sep 1 16:26:35 2023

@author: Pavan Pranjivan Mehta

class stiffness_matrix.stiffness_matrix#

Bases: object

Computes stiffness matrix in 1 dimension

constrect the space V = span {Pn}, where Pn is Poly of degree n, such that : min_degree <= n <= mas_degree; deafult, min_degree = 1.

Note#

Note that, n = 0 is a singular matrix

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

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

Parameters#

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

Returns#

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

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

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

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

Parameters#

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

Returns#

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

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

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

Computes Stiffness matrix Homogenous basis function using Shifted Legendre Polynomials in [0,1] of dim = (k x k), 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#

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

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

class stiffness_matrix.test#

Bases: object

legendre(min_degree=1)#

homogenous_legendre(min_degree=0)#

runge()#

func()#

der2func()#

laplacian_legendre(min_degree=1)#

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

TODO#

bcboundary conditions: Implement bc, since Legendre Poly does not satisfy homogenous bc

legendre_proj_err()#

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()#