linear_mappint.sagews

%typeset_mode True

var('xi, eta')
# x_nodes = [-0.9, -1, 1, 1.1]
## y_nodes = [1.1, -1.0, -1.2, 1.4]
x_nodes = [-1, -1, -0.8, -0.8]
y_nodes = [1, 0.8, 0.8, 1]

(xi, eta)

#N_0 = 0.25 * (1 - xi) * (1 - eta) #Aman
#N_1 = 0.25 * (1 + xi) * (1 - eta)
#N_2 = 0.25 * (1 + xi) * (1 + eta)
#N_3 = 0.25 * (1 - xi) * (1 + eta)

N_0 = +(1/4) * (1-xi) * (1+eta) # Balavarun
N_1 = +(1/4) * (1-eta) * (1-xi)
N_2 = +(1/4) * (1+xi) * (1-eta)
N_3 = +(1/4) * (1+xi) * (1+eta)

x = N_0 * x_nodes[0] + N_1 * x_nodes[1] + N_2 * x_nodes[2] + N_3 * x_nodes[3]
y = N_0 * y_nodes[0] + N_1 * y_nodes[1] + N_2 * y_nodes[2] + N_3 * y_nodes[3]

x.simplify_full()
y.simplify_full()

0.1*xi - 0.8999999999999999 0.1*eta + 0.9

dx_dxi  = diff(x, xi)
dx_deta = diff(x, eta)
dy_dxi  = diff(y, xi)
dy_deta = diff(y, eta)

dx_dxi
dx_deta
dy_dxi 
dy_deta

0.100000000000000 0.000000000000000 0.000000000000000 0.100000000000000

J_xy_xiEta = matrix(2, 2, [dx_dxi, dx_deta, dy_dxi, dy_deta])

J_xiEta_xy = (~J_xy_xiEta)

J_xiEta_xy[0][0].simplify_full()
J_xiEta_xy[1][1].simplify_full()

10.0 10.0