From 36a12731efe70693b8e9d4ed578ab005a1556591 Mon Sep 17 00:00:00 2001 From: "dino.celebic" Date: Tue, 25 Nov 2025 19:21:38 +0100 Subject: [PATCH] ex4 --- ex3/code/main.cpp | 12 ++++----- ex4/code/task_a.py | 57 ++++++++++++++++++++++++++++++++++++++++++ ex4/code/task_b.py | 61 +++++++++++++++++++++++++++++++++++++++++++++ ex4/code/task_c.py | 61 +++++++++++++++++++++++++++++++++++++++++++++ ex4/ex4_sheet.pdf | Bin 0 -> 144136 bytes ex4/task_a.png | Bin 0 -> 95695 bytes ex4/task_b.png | Bin 0 -> 83422 bytes ex4/task_c.png | Bin 0 -> 133969 bytes 8 files changed, 185 insertions(+), 6 deletions(-) create mode 100644 ex4/code/task_a.py create mode 100644 ex4/code/task_b.py create mode 100644 ex4/code/task_c.py create mode 100644 ex4/ex4_sheet.pdf create mode 100644 ex4/task_a.png create mode 100644 ex4/task_b.png create mode 100644 ex4/task_c.png diff --git a/ex3/code/main.cpp b/ex3/code/main.cpp index dc96dfa..244f247 100644 --- a/ex3/code/main.cpp +++ b/ex3/code/main.cpp @@ -255,12 +255,12 @@ void task_7() { } int main() { - // task_1(); - // task_2(); - // task_3(); - // task_4(); - // task_5(); - // task_6(); + task_1(); + task_2(); + task_3(); + task_4(); + task_5(); + task_6(); task_7(); printf("\n\n"); diff --git a/ex4/code/task_a.py b/ex4/code/task_a.py new file mode 100644 index 0000000..6028299 --- /dev/null +++ b/ex4/code/task_a.py @@ -0,0 +1,57 @@ +import numpy as np +import matplotlib.pyplot as plt + +# PDE: +# -u''(x) + a*u(x) = f(x) x in (0,1) +# u(0) = 0 +# u'(1) = \alpha*(g_b - u(1)) + +# parameters +a = 1 +f = 3 +alpha = 1 +gb = 1 +n = 10 # elements + +# mesh +m = n+1 # nodes +h = 1.0/n +x = np.linspace(0,1,m) + +# local stiffness matrix +K_loc = np.zeros((2,2)) +A = (1.0/h) * np.array([[ 1,-1], [-1, 1]]) +B = (a*h/6) * np.array([[ 2, 1], [ 1, 2]]) +K_loc = A+B + +# Assembling +K = np.zeros((m,m)) +F = np.zeros(m) +for i in range(n): + K[i:i+2,i:i+2] += K_loc + F[i:i+2] += np.full(2, f*h/2) + +# Boundary conditions +# Dirichlet: u(0) = 0 +K[0,:] = 0 +K[0,0] = 1 +F[0] = 0 +# Neumann: u'(1) = \alpha*(g_b - u(1)) +A[-1,-1] += alpha +F[-1] += alpha*gb + +u = np.linalg.solve(K, F) +plt.plot(x, u, "-o", label="u_h") +plt.title(f"n = {n} | h = {h} | a = {a} | f(x) = {f} | alpha = {alpha} | gb = {gb}") +plt.xlabel("x") +plt.ylabel("u(x)") +plt.legend() +plt.grid(True) + +print("K = ", K) +print("f = ", F) +print("u = ", u) + +plt.tight_layout() +plt.savefig("../task_a.png", dpi=300) +plt.show() \ No newline at end of file diff --git a/ex4/code/task_b.py b/ex4/code/task_b.py new file mode 100644 index 0000000..f642051 --- /dev/null +++ b/ex4/code/task_b.py @@ -0,0 +1,61 @@ +import numpy as np +import matplotlib.pyplot as plt + +# 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) + +# parameters +n = 2 # elements (must be even) + +# mesh +m = n+1 # nodes +nh = int(n/2) # elements per subdomain +mh = nh+1 # nodes per subdomain +jump = 1/np.sqrt(2) +x1 = np.linspace(0,jump, mh) +x2 = np.linspace(jump, 1, mh) +x = np.concatenate((x1[:-1],x2)) +h1 = jump/nh +h2 = (1-jump)/nh + +# local stiffness matrix +K_loc1 = ( 1.0/h1) * np.array([[ 1,-1], [-1, 1]]) +K_loc2 = (10.0/h2) * np.array([[ 1,-1], [-1, 1]]) + +# Assembling +K = np.zeros((m,m)) +F = np.zeros(m) +for i in range(nh): + K[i:i+2,i:i+2] += K_loc1 +for i in range(nh,n): + K[i:i+2,i:i+2] += K_loc2 + +# 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 + +u = np.linalg.solve(K, F) +plt.plot(x, u, "-o", label="u_h") +plt.title(f"n = {n} (already exact)") +plt.xlabel("x") +plt.ylabel("u(x)") +plt.legend() +plt.grid(True) + +print("K = ", K) +print("f = ", F) +print("u = ", u) + +plt.tight_layout() +plt.savefig("../task_b.png", dpi=300) +plt.show() \ No newline at end of file diff --git a/ex4/code/task_c.py b/ex4/code/task_c.py new file mode 100644 index 0000000..31102b9 --- /dev/null +++ b/ex4/code/task_c.py @@ -0,0 +1,61 @@ +import numpy as np +import matplotlib.pyplot as plt + +# Peclet problem: +# -u''(x) + pu'(x) = 0 x in (0,1) +# u(0) = 0 +# u(1) = 1 + +# parameters +p = 70 +n_values = [10,20,30,40,70] # elements + +# exact solution +x_exact = np.linspace(0,1,1000) +u_exact = (np.exp(p*x_exact)-1)/(np.exp(p)-1) + +for n in n_values: + # mesh + m = n+1 # nodes + h = 1.0/n + x = np.linspace(0,1,m) + + # local stiffness matrix + K_loc = np.zeros((2,2)) + A = (1.0/h) * np.array([[ 1,-1], [-1, 1]]) + B = (p/2) * np.array([[ -1, 1], [ -1, 1]]) + K_loc = A+B + + # Assembling + K = np.zeros((m,m)) + F = np.zeros(m) + for i in range(n): + K[i:i+2,i:i+2] += K_loc + + # 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 + + u = np.linalg.solve(K, F) + plt.plot(x, u, "-o", markersize=2, label=f"n = {n}") + +plt.plot(x_exact, u_exact, "black", label="exact") +plt.title(f"p = {p}") 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