From 89326dd329856c6b35561175857a0ee249315472 Mon Sep 17 00:00:00 2001 From: "dino.celebic" Date: Tue, 23 Dec 2025 22:34:09 +0100 Subject: [PATCH] ex6 --- .gitignore | 1 + ex6/code/adapt.py | 83 ++++++++++++++++++++++++++++++++ ex6/code/task_a.py | 104 +++++++++++++++++++++++++++++++++++++++++ ex6/code/task_b.py | 100 +++++++++++++++++++++++++++++++++++++++ ex6/code/task_c.py | 100 +++++++++++++++++++++++++++++++++++++++ ex6/ex6_sheet.pdf | Bin 0 -> 151801 bytes ex6/task_a_adapt_h.png | Bin 0 -> 77451 bytes ex6/task_a_adapt_r.png | Bin 0 -> 77238 bytes ex6/task_b_adapt_h.png | Bin 0 -> 84529 bytes ex6/task_b_adapt_r.png | Bin 0 -> 80857 bytes ex6/task_c_adapt_h.png | Bin 0 -> 70996 bytes ex6/task_c_adapt_r.png | Bin 0 -> 72799 bytes ex7/ex7_sheet.pdf | Bin 0 -> 296740 bytes 13 files changed, 388 insertions(+) create mode 100644 ex6/code/adapt.py create mode 100644 ex6/code/task_a.py create mode 100644 ex6/code/task_b.py create mode 100644 ex6/code/task_c.py create mode 100644 ex6/ex6_sheet.pdf create mode 100644 ex6/task_a_adapt_h.png create mode 100644 ex6/task_a_adapt_r.png create mode 100644 ex6/task_b_adapt_h.png create mode 100644 ex6/task_b_adapt_r.png create mode 100644 ex6/task_c_adapt_h.png create mode 100644 ex6/task_c_adapt_r.png create mode 100644 ex7/ex7_sheet.pdf diff --git a/.gitignore b/.gitignore index 9682f6d..d8e9a70 100644 --- a/.gitignore +++ b/.gitignore @@ -3,4 +3,5 @@ main *.o .vscode/ +__pycache__/ ex1/code/out_1.txt \ No newline at end of file diff --git a/ex6/code/adapt.py b/ex6/code/adapt.py new file mode 100644 index 0000000..edc6ff5 --- /dev/null +++ b/ex6/code/adapt.py @@ -0,0 +1,83 @@ +import numpy as np +import matplotlib.pyplot as plt +import scipy.integrate as integrate + +# Basis functions +# ---- TODO: order p? ---- +def phi(k, mesh, x): + if x > mesh[k-1] and x < mesh[k]: + return (x-mesh[k-1]) / (mesh[k] - mesh[k-1]) + elif x > mesh[k] and x < mesh[k+1]: + return (mesh[k+1]-x) / (mesh[k+1] - mesh[k]) + elif x == mesh[k]: + return 1 + else: + return 0 +def phi_prime(k, mesh, x): + if x > mesh[k-1] and x < mesh[k]: + return 1 / (mesh[k] - mesh[k-1]) + elif x > mesh[k] and x < mesh[k+1]: + return -1 / (mesh[k+1] - mesh[k]) + elif x == mesh[k]: + return print("oh no") + else: + return 0 + +# vertex flux jump between elements +def flux_jumps(mesh, u): + m = len(mesh) + + r = (u[2:]-u[1:-1]) / (mesh[2:]-mesh[1:-1]) + l = (u[1:-1]-u[:-2]) / (mesh[1:-1]-mesh[:-2]) + jumps = np.zeros(m) # 0 jump at bnd? + jumps[1:-1] = r - l + + jumps[0] = jumps[1] # or same jump at bnd? + jumps[-1] = jumps[-2] # or same jump at bnd? + return jumps + +# h-adaptivity (refine neighboring elments, if flux jump over certain threshold) +def adapt_h(mesh, jumps): + new_mesh = [mesh[0]] + jumps = jumps[1:-1] # only interior jumps (excluding bnd nodes needed for De Boor) + + # define threshold + threshold = 0.5 * np.max(np.abs(jumps)) + # mark elements + marked_nodes = np.abs(jumps) > threshold + marked_el = np.zeros(len(mesh)-1, dtype=bool) + for i, refine in enumerate(marked_nodes): + if refine: + marked_el[i] = True + marked_el[i+1] = True + + # make new mesh + for k, refine in enumerate(marked_el): + l = mesh[k] + r = mesh[k+1] + if refine: + new_mesh.extend(np.linspace(l, r, 3)[1:]) + else: + new_mesh.append(r) + + return np.asarray(new_mesh, dtype=float) + +# r-adaptivity (one iteration of De Boor's algorithm, moving mesh nodes for equidistributing mesh) +def adapt_r(mesh, rho): + m = len(mesh) + + p = 0.5 * (rho[:-1] + rho[1:]) # piecewise constant function on elements + + P = np.zeros(m) + for i in range(1,m): + P[i] = P[i-1] + (mesh[i]-mesh[i-1])*p[i-1] # approx integral of p, from node 0 to node i + Pb = P[-1] # integral of p, over whole mesh + + new_mesh = mesh.copy() + for j in range(1, m-1): + xi_j = (j)/(m-1) + k = np.searchsorted(P, xi_j*Pb) # search k s.t.: P[node k-1] < xi_j*Pb < P[node k] + k = max(k,1) # if k=0 + new_mesh[j] = mesh[k-1] + 2*(xi_j*Pb - P[k-1]) / (rho[k-1]+rho[k]) # new node j + return new_mesh + diff --git a/ex6/code/task_a.py b/ex6/code/task_a.py new file mode 100644 index 0000000..18ae409 --- /dev/null +++ b/ex6/code/task_a.py @@ -0,0 +1,104 @@ +import numpy as np +from adapt import * + +# PDE: +# -u''(x) = f(x) x in (-1,1) +# u(-1) = -arctan(p) +# u'(1) = p / (p^2 + 1) +# +# weak form +# int u'v' dx = int f(x) * v(x) dx + p/(p^2+1) * v(1) +# + +# rhs +def f(x): + return 2 * p**3 * x / (p**2 * x**2 + 1)**2 + +# Stiffness and Load +def K_loc(k, mesh): + K_loc = np.zeros((2,2)) + K_loc[0,0] = integrate.quad(lambda x: phi_prime(k-1, mesh, x)**2, mesh[k-1], mesh[k])[0] + K_loc[1,0] = integrate.quad(lambda x: phi_prime(k-1, mesh, x)*phi_prime(k, mesh, x), mesh[k-1], mesh[k])[0] + K_loc[0,1] = integrate.quad(lambda x: phi_prime(k, mesh, x)*phi_prime(k-1, mesh, x), mesh[k-1], mesh[k])[0] + K_loc[1,1] = integrate.quad(lambda x: phi_prime(k, mesh, x)**2, mesh[k-1], mesh[k])[0] + return K_loc +def F_loc(k, mesh): + F_loc = np.zeros(2) + F_loc[0] = integrate.quad(lambda x: f(x) * phi(k-1, mesh, x), mesh[k-1], mesh[k])[0] + F_loc[1] = integrate.quad(lambda x: f(x) * phi(k, mesh, x), mesh[k-1], mesh[k])[0] + return F_loc + +# Assembling +def Assemble(mesh): + m = len(mesh) + n = m-1 + + K = np.zeros((m,m)) + F = np.zeros(m) + for k in range(1,m): + K[k-1:k+1,k-1:k+1] += K_loc(k, mesh) + F[k-1:k+1] += F_loc(k, mesh) + + # Boundary conditions + # Dirichlet: u(-1) = -arctan(p) + K[0,:] = 0 + K[0,0] = 1 + F[0] = -np.arctan(p) + # Neumann: u'(1) = p / (p^2 + 1) + F[-1] += p/(p**2+1) * phi(n, mesh, 1) + return K,F + +def plotting(mesh, u, comment): + exact_x = np.linspace(-1,1,1000) + exact = np.arctan(p*exact_x) + plt.plot(exact_x, exact, "--", linewidth=1, color="red", label="exact") + plt.title(f"p = {p} | n = {n} | {comment}") + plt.xlabel("x") + plt.ylabel("u(x)") + plt.plot(mesh, u, "-o", label="u_h") + plt.xticks(mesh, labels=[]) + plt.legend() + plt.grid(True) + plt.tight_layout() + plt.savefig("task_a.png", dpi=300) + plt.show() + return 0 + +############################################################################## + +# parameters +p_list = [5,10,20,100] +for p in p_list: + + # h-adaptivity + mesh = np.array([-1.0, -0.2, 0, 0.7, 1.0]) # with 0 as node + # mesh = np.array([-1.0,-0.131,0.372,1.0]) # without 0 as node + + # r-adaptivity + # mesh = np.linspace(-1, 1, 11) # with 0 as node + # mesh = np.linspace(-1, 1, 10) # without 0 as node + + m = len(mesh) + n = m-1 + + K, F = Assemble(mesh) # assemble + u = np.linalg.solve(K, F) # solve + jumps = flux_jumps(mesh, u) # flux jumps + plotting(mesh, u, "before adapting") # plotting + + iterations = 6 + for it in range(iterations): + mesh = adapt_h(mesh, jumps) # h-adaptivity + # mesh = adapt_r(mesh, np.abs(jumps)) # r-adaptivity (positive density (jumps)!) + m = len(mesh) + n = m-1 + + K, F = Assemble(mesh) # assemble + u = np.linalg.solve(K, F) # solve + jumps = flux_jumps(mesh, u) # flux jumps + # plotting(mesh, u, f"iteration {it+1}") # plotting each iteration + + # print(jumps) + plotting(mesh, u, f"after {iterations} iterations") # plotting + + diff --git a/ex6/code/task_b.py b/ex6/code/task_b.py new file mode 100644 index 0000000..69a4d5f --- /dev/null +++ b/ex6/code/task_b.py @@ -0,0 +1,100 @@ +import numpy as np +from adapt import * + +# PDE: +# -(lambda(x)u'(x))' = 0 x in (0,1) +# u(0) = 0 +# u(1) = 1 +# lambda(x) = | 1 x in (0,1/sqrt(2)) +# | 10 x in (1/sqrt(2),1) + +# rhs +def f(x): + return 0 + +def lam(x): + if x >= 0 and x <= 1/np.sqrt(2): + return 1 + elif x <= 1 and x > 1/np.sqrt(2): + return 10 + else: + return print("lambda undefined") + +# Stiffness and Load +def K_loc(k, mesh): + K_loc = np.zeros((2,2)) + K_loc[0,0] = integrate.quad(lambda x: lam(x)*phi_prime(k-1, mesh, x)**2, mesh[k-1], mesh[k])[0] + 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] + 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] + K_loc[1,1] = integrate.quad(lambda x: lam(x)*phi_prime(k, mesh, x)**2, mesh[k-1], mesh[k])[0] + return K_loc +def F_loc(k, mesh): + F_loc = np.zeros(2) + F_loc[0] = integrate.quad(lambda x: f(x) * phi(k-1, mesh, x), mesh[k-1], mesh[k])[0] + F_loc[1] = integrate.quad(lambda x: f(x) * phi(k, mesh, x), mesh[k-1], mesh[k])[0] + return F_loc + +# Assembling +def Assemble(mesh): + m = len(mesh) + n = m-1 + + K = np.zeros((m,m)) + F = np.zeros(m) + for k in range(1,m): + K[k-1:k+1,k-1:k+1] += K_loc(k, mesh) + F[k-1:k+1] += F_loc(k, mesh) + + # Boundary conditions + # Dirichlet: u(0) = 0 + K[0,:] = 0 + K[0,0] = 1 + F[0] = 0 + # Dirichlet: u(1) = 1 + K[-1,:] = 0 + K[-1,-1] = 1 + F[-1] = 1 + return K,F + +def plotting(mesh, u, comment): + exact_x = [0, 1/np.sqrt(2), 1] + exact = [0, 10/(np.sqrt(2)+9), 1] + plt.plot(exact_x, exact, "--", linewidth=1, color="red", label="exact") + plt.title(f"n = {n} | {comment}") + plt.xlabel("x") + plt.ylabel("u(x)") + plt.plot(mesh, u, "-o", label="u_h") + plt.xticks(mesh, labels=[]) + plt.legend() + plt.grid(True) + plt.tight_layout() + plt.savefig("task_b.png", dpi=300) + plt.show() + return 0 + +############################################################################## + +mesh = np.linspace(0, 1, 10) +m = len(mesh) +n = m-1 + +K, F = Assemble(mesh) # assemble +u = np.linalg.solve(K, F) # solve +jumps = flux_jumps(mesh, u) # flux jumps +plotting(mesh, u, "before adapting") # plotting + +iterations = 3 +for it in range(iterations): + # mesh = adapt_h(mesh, jumps) # h-adaptivity + mesh = adapt_r(mesh, np.abs(jumps)) # r-adaptivity (positive density (jumps)!) + m = len(mesh) + n = m-1 + + K, F = Assemble(mesh) # assemble + u = np.linalg.solve(K, F) # solve + jumps = flux_jumps(mesh, u) # flux jumps + # plotting(mesh, u, f"iteration {it+1}") # plotting each iteration + +# print(jumps) +plotting(mesh, u, f"after {iterations} iterations") # plotting + diff --git a/ex6/code/task_c.py b/ex6/code/task_c.py new file mode 100644 index 0000000..da469ec --- /dev/null +++ b/ex6/code/task_c.py @@ -0,0 +1,100 @@ +import numpy as np +from adapt import * + +# Peclet problem: +# -u''(x) + pu'(x) = 0 x in (0,1) +# u(0) = 0 +# u(1) = 1 + +# rhs +def f(x): + return 0 + +# Stiffness and Load +def K_loc(k, mesh): + K_loc = np.zeros((2,2)) + K_loc[0,0] = integrate.quad(lambda x: phi_prime(k-1, mesh, x)**2, mesh[k-1], mesh[k])[0] + K_loc[1,0] = integrate.quad(lambda x: phi_prime(k-1, mesh, x)*phi_prime(k, mesh, x), mesh[k-1], mesh[k])[0] + K_loc[0,1] = integrate.quad(lambda x: phi_prime(k, mesh, x)*phi_prime(k-1, mesh, x), mesh[k-1], mesh[k])[0] + K_loc[1,1] = integrate.quad(lambda x: phi_prime(k, mesh, x)**2, mesh[k-1], mesh[k])[0] + + K_loc[0,0] += p*integrate.quad(lambda x: phi_prime(k-1, mesh, x) * phi(k-1, mesh, x), mesh[k-1], mesh[k])[0] + K_loc[1,0] += p*integrate.quad(lambda x: phi_prime(k-1, mesh, x) * phi(k, mesh, x), mesh[k-1], mesh[k])[0] + K_loc[0,1] += p*integrate.quad(lambda x: phi_prime(k, mesh, x) * phi(k-1, mesh, x), mesh[k-1], mesh[k])[0] + K_loc[1,1] += p*integrate.quad(lambda x: phi_prime(k, mesh, x) * phi(k, mesh, x), mesh[k-1], mesh[k])[0] + return K_loc +def F_loc(k, mesh): + F_loc = np.zeros(2) + F_loc[0] = integrate.quad(lambda x: f(x) * phi(k-1, mesh, x), mesh[k-1], mesh[k])[0] + F_loc[1] = integrate.quad(lambda x: f(x) * phi(k, mesh, x), mesh[k-1], mesh[k])[0] + return F_loc + +# Assembling +def Assemble(mesh): + m = len(mesh) + n = m-1 + + K = np.zeros((m,m)) + F = np.zeros(m) + for k in range(1,m): + K[k-1:k+1,k-1:k+1] += K_loc(k, mesh) + F[k-1:k+1] += F_loc(k, mesh) + + # Boundary conditions + # Dirichlet: u(0) = 0 + K[0,:] = 0 + K[0,0] = 1 + F[0] = 0 + # Dirichlet: u(1) = 1 + K[-1,:] = 0 + K[-1,-1] = 1 + F[-1] = 1 + return K,F + +def plotting(mesh, u, comment): + exact_x = np.linspace(0,1,1000) + exact = (np.exp(p*exact_x)-1)/(np.exp(p)-1) + plt.plot(exact_x, exact, "--", linewidth=1, color="red", label="exact") + plt.title(f"p = {p} | n = {n} | {comment}") + plt.xlabel("x") + plt.ylabel("u(x)") + plt.plot(mesh, u, "-o", label="u_h") + plt.xticks(mesh, labels=[]) + plt.legend() + plt.grid(True) + plt.tight_layout() + plt.savefig("task_c.png", dpi=300) + plt.show() + return 0 + +############################################################################## + +# parameters +p_list = [-70, 70] +for p in p_list: + mesh = np.linspace(0, 1, 30) + m = len(mesh) + n = m-1 + + K, F = Assemble(mesh) # assemble + u = np.linalg.solve(K, F) # solve + jumps = flux_jumps(mesh, u) # flux jumps + plotting(mesh, u, "before adapting") # plotting + + # NOTE: mesh very different every iteration for adapt_r() + iterations = 21 + for it in range(iterations): + # mesh = adapt_h(mesh, jumps) # h-adaptivity + mesh = adapt_r(mesh, np.abs(jumps)) # r-adaptivity (positive density (jumps)!) + m = len(mesh) + n = m-1 + + K, F = Assemble(mesh) # assemble + u = np.linalg.solve(K, F) # solve + jumps = flux_jumps(mesh, u) # flux 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