100 lines
3 KiB
Python
100 lines
3 KiB
Python
import numpy as np
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from adapt import *
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# PDE:
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# -(lambda(x)u'(x))' = 0 x in (0,1)
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# u(0) = 0
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# u(1) = 1
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# lambda(x) = | 1 x in (0,1/sqrt(2))
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# | 10 x in (1/sqrt(2),1)
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# rhs
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def f(x):
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return 0
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def lam(x):
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if x >= 0 and x <= 1/np.sqrt(2):
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return 1
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elif x <= 1 and x > 1/np.sqrt(2):
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return 10
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else:
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return print("lambda undefined")
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# Stiffness and Load
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def K_loc(k, mesh):
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K_loc = np.zeros((2,2))
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K_loc[0,0] = integrate.quad(lambda x: lam(x)*phi_prime(k-1, mesh, x)**2, mesh[k-1], mesh[k])[0]
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K_loc[1,0] = integrate.quad(lambda x: lam(x)*phi_prime(k-1, mesh, x)*phi_prime(k, mesh, x), mesh[k-1], mesh[k])[0]
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K_loc[0,1] = integrate.quad(lambda x: lam(x)*phi_prime(k, mesh, x)*phi_prime(k-1, mesh, x), mesh[k-1], mesh[k])[0]
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K_loc[1,1] = integrate.quad(lambda x: lam(x)*phi_prime(k, mesh, x)**2, mesh[k-1], mesh[k])[0]
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return K_loc
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def F_loc(k, mesh):
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F_loc = np.zeros(2)
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F_loc[0] = integrate.quad(lambda x: f(x) * phi(k-1, mesh, x), mesh[k-1], mesh[k])[0]
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F_loc[1] = integrate.quad(lambda x: f(x) * phi(k, mesh, x), mesh[k-1], mesh[k])[0]
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return F_loc
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# Assembling
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def Assemble(mesh):
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m = len(mesh)
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n = m-1
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K = np.zeros((m,m))
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F = np.zeros(m)
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for k in range(1,m):
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K[k-1:k+1,k-1:k+1] += K_loc(k, mesh)
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F[k-1:k+1] += F_loc(k, mesh)
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# Boundary conditions
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# Dirichlet: u(0) = 0
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K[0,:] = 0
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K[0,0] = 1
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F[0] = 0
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# Dirichlet: u(1) = 1
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K[-1,:] = 0
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K[-1,-1] = 1
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F[-1] = 1
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return K,F
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def plotting(mesh, u, comment):
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exact_x = [0, 1/np.sqrt(2), 1]
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exact = [0, 10/(np.sqrt(2)+9), 1]
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plt.plot(exact_x, exact, "--", linewidth=1, color="red", label="exact")
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plt.title(f"n = {n} | {comment}")
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plt.xlabel("x")
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plt.ylabel("u(x)")
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plt.plot(mesh, u, "-o", label="u_h")
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plt.xticks(mesh, labels=[])
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plt.legend()
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plt.grid(True)
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plt.tight_layout()
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plt.savefig("task_b.png", dpi=300)
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plt.show()
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return 0
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##############################################################################
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mesh = np.linspace(0, 1, 10)
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m = len(mesh)
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n = m-1
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K, F = Assemble(mesh) # assemble
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u = np.linalg.solve(K, F) # solve
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jumps = flux_jumps(mesh, u) # flux jumps
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plotting(mesh, u, "before adapting") # plotting
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iterations = 3
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for it in range(iterations):
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# mesh = adapt_h(mesh, jumps) # h-adaptivity
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mesh = adapt_r(mesh, np.abs(jumps)) # r-adaptivity (positive density (jumps)!)
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m = len(mesh)
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n = m-1
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K, F = Assemble(mesh) # assemble
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u = np.linalg.solve(K, F) # solve
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jumps = flux_jumps(mesh, u) # flux jumps
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# plotting(mesh, u, f"iteration {it+1}") # plotting each iteration
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# print(jumps)
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plotting(mesh, u, f"after {iterations} iterations") # plotting
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