115 lines
2.9 KiB
Python
115 lines
2.9 KiB
Python
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import matplotlib.pyplot as plt
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import numpy as np
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import numpy.linalg as lin
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import scipy.sparse as sp
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from scipy.sparse.linalg import spsolve
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def kappa_integral(x,y):
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# todo 3 a)
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return 0.5 * (y**2 - x**2) + y -x
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def build_massMatrix(N):
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# todo 3 b)
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M = np.identity(N)
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return M
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def build_rigidityMatrix(N):
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# todo 3 b)
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# Be careful with the indices!
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# kappa_integral could be helpful here
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M = np.zeros((N,N))
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h = 1/(N+1)
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# The case of i=j
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for i in range(N):
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M[i,i] = kappa_integral((h-1)*i, h*i) \
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+ kappa_integral(h*i, (h+1)*i)
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# to be filled
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# The case of i-j=1
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for i in range(1,N):
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M[i, i-1] = - kappa_integral(h*i, h*(i+1))# to be filled
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# The case of j-i=1
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for i in range(N-1):
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M[i, i+1] = - kappa_integral(h*(i-1),h*i)# to be filled
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return M
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def f(t,x):
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# todo 3 c)
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return ((x + 1) * np.pi **2 - 1) * np.exp(-t) * np.sin(np.pi * x) - np.pi * np.exp(-t) * np.cos(np.pi * x)
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def initial_value(x):
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# todo 3 c)
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return np.sin(np.pi * x)
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def exact_solution_at_1(x):
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# todo 3 c)
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return np.exp(-1) * np.sin(np.pi * x)
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def build_F(t,N):
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# todo 3 d)
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h = 1/(N+1)
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x = np.linspace(0,1,N+1)
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return h/3 * (f(t, x - h/2) + f(t,x) + f(t, x + h/2))
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def FEM_theta(N,M,theta):
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# todo 3 e)
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h = 1/(N+1)
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k = 1/(M+1)
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u = initial_value(np.linspace(0,1,N+1))
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M_matrix = build_massMatrix(N)
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B = theta * M_matrix
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C = (1 - theta) * M_matrix - k * build_rigidityMatrix(N)
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F = build_F(0,N)
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for m in range(M):
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if (theta != 0):
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u = np.linalg.solve(B, C @ u + F)
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elif (theta == 0):
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u = C @ u + F
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F = build_F(m*k,N)
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return u
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#### error analysis ####
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nb_samples = 5
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exponents = np.arange(2,7)
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N = np.pow(2,exponents) - 1 # fill in this line for f)-g)
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M = np.pow(2, exponents) # fill in this line for f)-g)
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theta= .3# fill in this line for f)-g)
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#### Do not change any code below! ####
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l2error = np.zeros(nb_samples)
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k = 1 / M
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try:
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for i in range(nb_samples):
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l2error[i] = (1 / (N[i]+1)) ** (1 / 2) * lin.norm(exact_solution_at_1((1/(N[i]+1))*(np.arange(N[i])+1)) - FEM_theta(N[i], M[i],theta), ord=2)
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if np.isnan(l2error[i])==True:
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raise Exception("Error unbounded. Plots not shown.")
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conv_rate = np.polyfit(np.log(k), np.log(l2error), deg=1)
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if conv_rate[0]<0:
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raise Exception("Error unbounded. Plots not shown.")
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print(f"FEM method with theta={theta} converges: Convergence rate in discrete $L^2$ norm with respect to time step $k$: {conv_rate[0]}")
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plt.figure(figsize=[10, 6])
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plt.loglog(k, l2error, '-x', label='error')
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plt.loglog(k, k, '--', label='$O(k)$')
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plt.loglog(k, k**2, '--', label='$O(k^2)$')
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plt.title('$L^2$ convergence rate', fontsize=13)
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plt.xlabel('$k$', fontsize=13)
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plt.ylabel('error', fontsize=13)
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plt.legend()
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plt.plot()
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plt.show()
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except Exception as e:
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print(e)
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