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83
ex6/adaptivity_schemes.py
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83
ex6/adaptivity_schemes.py
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import numpy as np
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def flux_jumps(mesh, u):
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N = len(mesh) - 1 # number of elements
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jumps = np.zeros(N + 1) # N-1 edges
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for i in range(1, N):
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upper = (u[i + 1] - u[i])/(mesh[i + 1] - mesh[i])
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lower = (u[i] - u[i - 1])/(mesh[i] - mesh[i - 1])
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jumps[i] = upper - lower
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return jumps
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def residual_errors(mesh, u):
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N = len(mesh) - 1 # number of elements
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errors = np.zeros(N)
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jumps = flux_jumps(mesh, u)
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for i in range(N):
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errors[i] = np.sqrt((jumps[i]**2 + jumps[i + 1]**2)/2) # Braess (8.10)
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#print("errors:\n", errors)
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return errors
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def adapt_h(mesh, u, alpha):
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N = len(mesh) - 1 # number of elements
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errors = residual_errors(mesh, u)
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threshhold = alpha * abs(max(errors))
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# refine mesh
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refined_mesh = [mesh[0]]
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for i in range(N):
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if abs(errors[i]) <= threshhold:
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refined_mesh.append(mesh[i + 1])
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else:
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refined_mesh.append(mesh[i] + (mesh[i + 1] - mesh[i])/2)
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refined_mesh.append(mesh[i + 1])
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#print("refined mesh:\n", refined_mesh)
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return refined_mesh
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def adapt_r(mesh, u):
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N = len(mesh) - 1 # number of elements
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rho = np.abs(flux_jumps(mesh, u)) # rho ... mesh density function
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p = np.zeros(N) # piecewise constant function on the mesh elements
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for i in range(N):
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p[i] = (rho[i] + rho[i + 1])/2
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P = np.zeros(N + 1) # \int_0^{x_j} p(x) dx for j = 1,...,N
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for j in range(1, N + 1):
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h_j = mesh[j] - mesh[j - 1]
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P[j] = P[j - 1] + h_j*p[j - 1] # add integral over j-th interval
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moved_mesh = np.zeros(N + 1)
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moved_mesh[0] = mesh[0]
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moved_mesh[-1] = mesh[-1]
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for j in range(1, N): # calculate the new nodes with De Boor's algorithm
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xi_j = j/N
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k = np.searchsorted(P, xi_j*P[-1], side="left") # searches for index k, such that xi_j*P[-1] > P[k]
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assert(P[k - 1] < xi_j*P[-1])
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assert(xi_j*P[-1] <= P[k])
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moved_mesh[j] = mesh[k - 1] + (xi_j*P[-1] - P[k - 1])/p[k - 1]
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print("orign_mesh[j] =", mesh[j])
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print("moved_mesh[j] =", moved_mesh[j])
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print("done\n")
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#print("moved mesh:\n", moved_mesh)
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return moved_mesh
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139
ex6/ex_6A.py
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139
ex6/ex_6A.py
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import numpy as np
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import scipy.integrate as integrate
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import matplotlib.pyplot as plt
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import adaptivity_schemes
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np.set_printoptions(precision=2)
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def Solve_6A(mesh, p):
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N = len(mesh) - 1 # number of elements
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f = lambda x : 2*p**3*x/((p**2*x**2 + 1)**2)
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g_b = p/(p**2 + 1)
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A = np.zeros((N + 1, N + 1))
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f_vec = np.zeros(N + 1)
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for i in range(1, N + 1):
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h = mesh[i] - mesh[i - 1]
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a_11 = 1./h
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a_12 = -1./h
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a_21 = -1./h
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a_22 = 1./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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phi_lower = lambda x : (mesh[i] - x)/h
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f_vec[i-1] += integrate.quad(lambda x : f(x)*phi_lower(x), mesh[i - 1], mesh[i])[0]
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phi_upper = lambda x : (x - mesh[i - 1])/h
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f_vec[i] += integrate.quad(lambda x : f(x)*phi_upper(x), mesh[i - 1], mesh[i])[0]
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# take Neumann data into account
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A[N, N] += 0
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f_vec[N] += g_b
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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] = -np.arctan(p)
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#print("u_g =\n", u_g)
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# remove first row of A
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A_g = A[1:N+1, :]
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#print("A_g =\n", A_g)
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# remove first row of f_vec
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f_g = f_vec[1:N+1]
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# assemble RHS with dirichlet data
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f_g -= A_g.dot(u_g)
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#print("f_g =\n", f_g)
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# matrix for the inner nodes (excluding nodes with dirichlet bcs)
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A_0 = A[1:N+1, 1:N+1]
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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_g)
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# assemble "u = u_0 + u_g"
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u = np.concatenate([[u_g[0]], u_0])
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return u
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p = 100
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########## h-adaptivity ##########
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N = 5 # number of elements
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mesh = np.linspace(-1, 1, N + 1)
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u = Solve_6A(mesh, p)
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plt.plot(mesh, u, '-o')
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plt.grid()
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plt.xlabel('x')
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plt.ylabel('u_h(x)')
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plt.title("h-adaptivity")
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N_vec = ["0 refinements, " + str(N) + " elements"]
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refinements = 4 # number of refinements
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for i in range(refinements):
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mesh = adaptivity_schemes.adapt_h(mesh, u, 0.7)
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u = Solve_6A(mesh, p)
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plt.plot(mesh, u, '-o')
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N_vec.append(str(i + 1) + " refinements, " + str(len(mesh) - 1) + " elements")
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# plot exact solution
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x = np.linspace(-1, 1, 50)
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plt.plot(x, np.arctan(p*x))
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N_vec.append("exact")
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plt.legend(N_vec)
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plt.show()
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# ########## r-adaptivity ##########
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N = 5
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mesh = np.linspace(-1, 1, N + 1)
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u = Solve_6A(mesh, p)
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plt.plot(mesh, u, '-o')
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title = "r-adaptivity with " + str(N) + " elements"
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plt.title(title)
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adaptations_vec = ["0 adaptations"]
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adaptations = 5 # number of iterations
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for i in range(adaptations):
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mesh = adaptivity_schemes.adapt_r(mesh, u)
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u = Solve_6A(mesh, p)
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plt.plot(mesh, u, '-o')
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adaptations_vec.append(str(i + 1) + " adaptations")
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# plot exact solution
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x = np.linspace(-1, 1, 50)
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plt.plot(x, np.arctan(p*x))
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adaptations_vec.append("exact")
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plt.legend(adaptations_vec)
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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()
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120
ex6/ex_6B.py
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120
ex6/ex_6B.py
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import numpy as np
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import matplotlib.pyplot as plt
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import adaptivity_schemes
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np.set_printoptions(precision=2)
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def lam_func(x):
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n = len(x)
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lam_vec = np.zeros(n)
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for i in range(n):
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if (x[i] > 1/np.sqrt(2)):
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lam_vec[i] = 10
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else:
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lam_vec[i] = 1
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return lam_vec
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def Solve_6B(mesh):
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N = len(mesh) - 1 # number of elements
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A = np.zeros((N + 1, N + 1))
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lam_vec = lam_func(mesh)
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for i in range(1, N + 1):
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h = mesh[i] - mesh[i - 1]
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a_11 = lam_vec[i]/h
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a_12 = -lam_vec[i]/h
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a_21 = -lam_vec[i]/h
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a_22 = lam_vec[i]/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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return u
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########## h-adaptivity ##########
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N = 2 # number of elements
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mesh = np.linspace(0, 1, N + 1)
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u = Solve_6B(mesh)
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plt.plot(mesh, u, '-o')
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plt.grid()
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plt.xlabel('x')
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plt.ylabel('u_h(x)')
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plt.title("h-adaptivity")
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N_vec = ["0 refinements, " + str(N) + " elements"]
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refinements = 5 # number of refinements
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for i in range(refinements):
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mesh = adaptivity_schemes.adapt_h(mesh, lam_func(mesh)*u, 0.9)
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u = Solve_6B(mesh)
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plt.plot(mesh, u, '-o')
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N_vec.append(str(i + 1) + " refinements, " + str(len(mesh) - 1) + " elements")
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plt.legend(N_vec)
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plt.show()
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########## r-adaptivity ##########
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N = 5
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mesh = np.linspace(0, 1, N + 1)
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u = Solve_6B(mesh)
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plt.plot(mesh, u, '-o')
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title = "r-adaptivity with " + str(N) + " elements"
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plt.title(title)
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adaptations_vec = ["0 adaptations"]
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adaptations = 4 # number of iterations
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for i in range(adaptations):
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mesh = adaptivity_schemes.adapt_r(mesh, lam_func(mesh)*u)
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u = Solve_6B(mesh)
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plt.plot(mesh, u, '-o')
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adaptations_vec.append(str(i + 1) + " adaptations")
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plt.legend(adaptations_vec)
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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()
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127
ex6/ex_6C.py
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127
ex6/ex_6C.py
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import numpy as np
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import matplotlib.pyplot as plt
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import adaptivity_schemes
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np.set_printoptions(precision=2)
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def Solve_6C(mesh, p):
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N = len(mesh) - 1 # number of elements
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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 = mesh[i] - mesh[i - 1]
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a_11 = 1./h - p/2.
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a_12 = -1./h + p/2.
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a_21 = -1./h - p/2.
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a_22 = 1./h + p/2.
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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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return u
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p = 70
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######### h-adaptivity ##########
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N = 5 # number of elements
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mesh = np.linspace(0, 1, N + 1)
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u = Solve_6C(mesh, p)
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plt.plot(mesh, u, '-o')
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plt.grid()
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plt.xlabel('x')
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plt.ylabel('u_h(x)')
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plt.title("h-adaptivity")
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N_vec = ["0 refinements, " + str(N) + " elements"]
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refinements = 4 # number of refinements
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for i in range(refinements):
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mesh = adaptivity_schemes.adapt_h(mesh, u, 0.7)
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u = Solve_6C(mesh, p)
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plt.plot(mesh, u, '-o')
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N_vec.append(str(i + 1) + " refinements, " + str(len(mesh) - 1) + " elements")
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# plot exact solution
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x = np.linspace(0, 1, 50)
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plt.plot(x, (np.exp(p*x) - 1.)/(np.exp(p) - 1.))
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N_vec.append("exact")
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plt.legend(N_vec)
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plt.show()
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########## r-adaptivity ##########
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N = 10
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mesh = np.linspace(0, 1, N + 1)
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u = Solve_6C(mesh, p)
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plt.plot(mesh, u, '-o')
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title = "r-adaptivity with " + str(N) + " elements"
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plt.title(title)
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adaptations_vec = ["0 adaptations"]
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adaptations = 4 # number of iterations
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for i in range(adaptations):
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mesh = adaptivity_schemes.adapt_r(mesh, u)
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u = Solve_6C(mesh, p)
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plt.plot(mesh, u, '-o')
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adaptations_vec.append(str(i + 1) + " adaptations")
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# plot exact solution
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x = np.linspace(0, 1, 50)
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plt.plot(x, (np.exp(p*x) - 1.)/(np.exp(p) - 1.))
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adaptations_vec.append("exact")
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plt.legend(adaptations_vec)
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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()
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