83 lines
No EOL
1.5 KiB
Python
83 lines
No EOL
1.5 KiB
Python
import numpy as np
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import matplotlib.pyplot as plt
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np.set_printoptions(precision=2)
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N = 2 # number of elements
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split = np.sqrt(2)/2
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#split = 0.5
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N_lower = round(N*split) # number of elements in (0, split)
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print(N_lower, "elements below sqrt(2)/2")
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N_upper = N - N_lower # number of elements in (split, 1)
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print(N_upper, "elements above sqrt(2)/2")
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assert(N == N_lower + N_upper)
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x_lower = np.linspace(0, split, N_lower + 1)
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x_upper = np.linspace(split, 1, N_upper + 1)
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x = np.concatenate([x_lower, x_upper[1:]])
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print(x)
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A = np.zeros((N + 1, N + 1))
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for i in range(1, N + 1):
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h = x[i] - x[i - 1]
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lam = 1
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if(x[i] > split):
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lam = 10
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a_11 = lam/h
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a_12 = -lam/h
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a_21 = -lam/h
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a_22 = lam/h
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A[i - 1, i - 1] += a_11
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A[i - 1, i] += a_12
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A[i, i - 1] += a_21
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A[i, i] += a_22
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print("A =\n", A)
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# take dirichlet data into account
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u_g = np.zeros(N + 1)
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u_g[0] = 0
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u_g[N] = 1
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print("u_g =\n", u_g)
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# remove first and last row of A
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A_g = A[1:N, :]
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#print("A_g =\n", A_g)
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# assemble RHS with dirichlet data
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f = -A_g.dot(u_g)
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#print(f)
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# matrix for the inner nodes (excluding nodes with dirichlet bcs)
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A_0 = A[1:N, 1:N]
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#print(A_0)
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# solve for u_0 (free dofs)
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u_0 = np.linalg.solve(A_0, f)
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# assemble "u = u_0 + u_g"
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u = np.concatenate([[0], u_0, [1]])
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print("u =\n", u)
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plt.plot(x, u, '-')
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plt.xlabel('x')
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plt.ylabel('u_h(x)')
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plt.grid()
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plt.show() |