535 lines
21 KiB
Python
535 lines
21 KiB
Python
# main.py
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import numpy as np
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import pandas as pd
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import matplotlib.pyplot as plt
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import case1
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import case2
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import common
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data_dict = {}
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data_dict["a_values"] = np.array([0.29, 0.5, 1.35, 2.75, 4.75, 9.15])
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def prob1():
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'''Analytical solutions'''
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print("Problem 1")
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# Problem 1 - Analytical solution
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data_dict["prob1"] = {}
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data_dict["prob1"]["x_values"] = np.array([0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1.0])
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data_dict["prob1"]["x_values_fine"] = np.linspace(0, 1)
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# Case 1 temperature calculations
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results = []
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results_fine = []
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for a_value in data_dict["a_values"]:
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result = case1.analytical(a=a_value, x=data_dict["prob1"]["x_values"])
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result_fine = case1.analytical(a=a_value, x=data_dict["prob1"]["x_values_fine"])
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results.append(result)
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results_fine.append(result_fine)
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data_dict["prob1"]["case1_temp_results"] = np.array(results_fine)
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df_case1 = pd.DataFrame(results, index=[f'a = {a}' for a in data_dict["a_values"]], columns=[f'x = {x:.3f}' for x in data_dict["prob1"]["x_values"]])
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# Pretty print DataFrame for Case 1 temperature
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print("Analytical Case 1 Temperature Results:")
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print(df_case1)
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print("\n" * 2)
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# Plotting Case 1 temperature
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plt.figure(figsize=(10, 6))
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for idx, a_value in enumerate(data_dict["a_values"]):
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plt.plot(data_dict["prob1"]["x_values_fine"], data_dict["prob1"]["case1_temp_results"][idx], label=f'α = {a_value}')
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plt.xlabel('Position (cm)')
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plt.ylabel('Temperature (°C)')
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plt.legend()
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plt.grid(True)
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plt.savefig('prob1_case1_analytical_temperature_vs_position.png', dpi=300)
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#plt.show()
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plt.close()
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# Case 2 temperature calculations
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results = []
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results_fine = []
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for a_value in data_dict["a_values"]:
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result = case2.analytical(a=a_value, x=data_dict["prob1"]["x_values"])
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result_fine = case2.analytical(a=a_value, x=data_dict["prob1"]["x_values_fine"])
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results.append(result)
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results_fine.append(result_fine)
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data_dict["prob1"]["case2_temp_results"] = np.array(results_fine)
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df_case2 = pd.DataFrame(results, index=[f'a = {a}' for a in data_dict["a_values"]], columns=[f'x = {x:.3f}' for x in data_dict["prob1"]["x_values"]])
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# Pretty print DataFrame for Case 2
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print("Analytical Case 2 Temperature Results:")
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print(df_case2)
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print("\n" * 2)
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# Plotting Case 2
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plt.figure(figsize=(10, 6))
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for idx, a_value in enumerate(data_dict["a_values"]):
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plt.plot(data_dict["prob1"]["x_values_fine"], data_dict["prob1"]["case2_temp_results"][idx], label=f'α = {a_value}')
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plt.xlabel('Position (cm)')
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plt.ylabel('Temperature (°C)')
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plt.legend()
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plt.grid(True)
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plt.savefig('prob1_case2_analytical_temperature_vs_position.png', dpi=300)
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#plt.show()
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plt.close()
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# Comparison plots between cases for temperature distribution
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fig, axes = plt.subplots(3, 2, figsize=(12, 12), sharex=True, sharey=True)
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for idx, a_value in enumerate(data_dict["a_values"]):
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ax = axes[idx // 2, idx % 2]
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ax.plot(data_dict["prob1"]["x_values_fine"], data_dict["prob1"]["case1_temp_results"][idx], label='Case 1')
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ax.plot(data_dict["prob1"]["x_values_fine"], data_dict["prob1"]["case2_temp_results"][idx], label='Case 2')
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ax.set_title(f'α = {a_value}')
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ax.set_xlabel('Position (cm)')
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ax.set_ylabel('Temperature (°C)')
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ax.legend()
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ax.grid(True)
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plt.tight_layout(rect=[0, 0, 1, 0.95]) # Adjust layout to make room for the main title
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plt.savefig('prob1_comparison_cases.png', dpi=300)
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#plt.show()
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plt.close()
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# Case 1 heat flux calculations
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results = []
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for a_value in data_dict["a_values"]:
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result = case1.analytical_heat_loss(a=a_value)
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results.append(result)
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data_dict["prob1"]["case1_heat_flux_results"] = np.array(results)
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# Create a DataFrame with alpha as the index and heat flux as the data
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df_case1_heat_flux = pd.DataFrame(results, index=[f'a = {a}' for a in data_dict["a_values"]], columns=['Heat Flux'])
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# Pretty print DataFrame for Case 1 heat flux
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print("Analytical Case 1 Heat Flux Results:")
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print(df_case1_heat_flux)
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print("\n" * 2)
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# Case 2 heat flux calculations
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results = []
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for a_value in data_dict["a_values"]:
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result = case2.analytical_heat_loss(a=a_value)
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results.append(result)
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data_dict["prob1"]["case2_heat_flux_results"] = np.array(results)
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# Create a DataFrame with alpha as the index and heat flux as the data
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df_case2_heat_flux = pd.DataFrame(results, index=[f'a = {a}' for a in data_dict["a_values"]], columns=['Heat Flux'])
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# Pretty print DataFrame for Case 2 heat flux
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print("Analytical Case 2 Heat Flux Results:")
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print(df_case2_heat_flux)
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print("\n" * 2)
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# Plotting heat flux vs alpha for both cases
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plt.figure(figsize=(10, 6))
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plt.plot(data_dict["a_values"], data_dict["prob1"]["case1_heat_flux_results"], label='Case 1')
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plt.plot(data_dict["a_values"], data_dict["prob1"]["case2_heat_flux_results"], label='Case 2')
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plt.xlabel('Alpha ($cm^{-1}$)')
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plt.ylabel('Heat Flux (W)')
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plt.legend()
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plt.grid(True)
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plt.savefig('prob1_heat_flux_comparison.png', dpi=300)
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#plt.show()
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plt.close()
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def prob2():
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'''FDM with dx = 0.125 and alpha varying'''
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print("Problem 2")
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data_dict["prob2"] = {}
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data_dict["prob2"]["num_point"] = 8 # This does not consider the final T(L) point. So actually we have 9 points
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data_dict["prob2"]["dx"] = 0.125
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# Case 1
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results = []
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# num_point does not consider T(L). So "4" points is T0, T1, T2, T3, and then we add back T(L)
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for a_value in data_dict["a_values"]:
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fdm_array = case1.fdm_array(data_dict["prob2"]["num_point"], a = a_value)
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fdm_array_inverse = np.linalg.inv(fdm_array)
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right_array = case1.right_array(data_dict["prob2"]["num_point"])
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unkown_temps_array = fdm_array_inverse @ right_array
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solution = [0] # First point is 0 from BC
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solution[1:] = unkown_temps_array
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solution.append(100) # T(L) BC
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results.append(solution)
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data_dict["prob2"]["case1"] = {}
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data_dict["prob2"]["case1"]["temp_results"] = np.array(results)
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data_dict["prob2"]["x_values"] = np.array([0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1.0])
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df_case1 = pd.DataFrame(results, index=[f'a = {a}' for a in data_dict["a_values"]], columns=[f'x = {x:.3f}' for x in data_dict["prob2"]["x_values"]])
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# Pretty print DataFrame for Case 1 temperature
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print("FDM Case 1 Temperature Results:")
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print(df_case1)
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print("\n" * 2)
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# Plotting Case 1 temperature
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plt.figure(figsize=(10, 6))
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for idx, a_value in enumerate(data_dict["a_values"]):
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plt.plot(data_dict["prob2"]["x_values"], data_dict["prob2"]["case1"]["temp_results"][idx], label=f'α = {a_value}')
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plt.xlabel('Position (cm)')
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plt.ylabel('Temperature (°C)')
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plt.legend()
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plt.grid(True)
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plt.savefig('prob2_case1_fdm_temperature_vs_position.png', dpi=300)
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#plt.show()
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plt.close()
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# Case 2
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results = []
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# num_point does not consider T(L). So "4" points is T0, T1, T2, T3, and then we add back T(L)
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for a_value in data_dict["a_values"]:
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fdm_array = case2.fdm_array(data_dict["prob2"]["num_point"], a = a_value)
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fdm_array_inverse = np.linalg.inv(fdm_array)
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right_array = case2.right_array(data_dict["prob2"]["num_point"])
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unkown_temps_array = fdm_array_inverse @ right_array
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solution = unkown_temps_array.tolist()
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solution.append(100) # T(L) BC
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results.append(solution)
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data_dict["prob2"]["case2"] = {}
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data_dict["prob2"]["case2"]["temp_results"] = np.array(results)
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data_dict["prob2"]["x_values"] = np.array([0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1.0])
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df_case2 = pd.DataFrame(results, index=[f'a = {a}' for a in data_dict["a_values"]], columns=[f'x = {x:.3f}' for x in data_dict["prob2"]["x_values"]])
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# Pretty print DataFrame for Case 2 temperature
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print("FDM Case 2 Temperature Results:")
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print(df_case2)
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print("\n" * 2)
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# Plotting Case 2 temperature
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plt.figure(figsize=(10, 6))
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for idx, a_value in enumerate(data_dict["a_values"]):
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plt.plot(data_dict["prob2"]["x_values"], data_dict["prob2"]["case2"]["temp_results"][idx], label=f'α = {a_value}')
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plt.xlabel('Position (cm)')
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plt.ylabel('Temperature (°C)')
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plt.legend()
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plt.grid(True)
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plt.savefig('prob2_case2_fdm_temperature_vs_position.png', dpi=300)
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#plt.show()
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plt.close()
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# Plot the comparisons similar to problem 1
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# Comparison plots between cases for temperature distribution
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fig, axes = plt.subplots(3, 2, figsize=(12, 12), sharex=True, sharey=True)
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for idx, a_value in enumerate(data_dict["a_values"]):
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ax = axes[idx // 2, idx % 2]
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ax.plot(data_dict["prob2"]["x_values"], data_dict["prob2"]["case1"]["temp_results"][idx], label='Case 1')
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ax.plot(data_dict["prob2"]["x_values"], data_dict["prob2"]["case2"]["temp_results"][idx], label='Case 2')
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ax.set_title(f'α = {a_value}')
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ax.set_xlabel('Position (cm)')
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ax.set_ylabel('Temperature (°C)')
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ax.legend()
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ax.grid(True)
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plt.tight_layout(rect=[0, 0, 1, 0.95]) # Adjust layout to make room for the main title
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plt.savefig('prob2_comparison_cases.png', dpi=300)
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#plt.show()
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plt.close()
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# Heat flux extraction
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# Case 1
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results = []
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for i in range(len(data_dict["a_values"])):
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a = data_dict["a_values"][i]
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(t_0, t_1) = data_dict["prob2"]["case1"]["temp_results"][i][-2:]
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dx = data_dict["prob2"]["dx"]
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first_order = common.taylor_extraction(t_0, t_1, dx, a, 1)
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second_order = common.taylor_extraction(t_0, t_1, dx, a, 2)
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results.append([first_order, second_order])
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data_dict["prob2"]["case1"]["heat_results"] = np.array(results)
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# Create a DataFrame with alpha as the index and heat flux as the data
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df_case1_heat_flux = pd.DataFrame(results, index=[f'a = {a}' for a in data_dict["a_values"]], columns=['1st Order', '2nd Order'])
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# Pretty print DataFrame for Case 1 heat flux
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print("FDM Case 1 Heat Flux Results:")
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print(df_case1_heat_flux)
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print("\n" * 2)
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# Plotting heat flux vs alpha for both extractions
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plt.figure(figsize=(10, 6))
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plt.plot(data_dict["a_values"], [i[0] for i in data_dict["prob2"]["case1"]["heat_results"]], label='1st Order Extraction')
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plt.plot(data_dict["a_values"], [i[1] for i in data_dict["prob2"]["case1"]["heat_results"]], label='2nd Order Extraction')
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plt.xlabel('Alpha ($cm^{-1}$)')
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plt.ylabel('Heat Flux (W)')
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plt.legend()
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plt.grid(True)
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plt.savefig('prob2_case1_heat_flux.png', dpi=300)
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#plt.show()
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plt.close()
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# Case 2
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results = []
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for i in range(len(data_dict["a_values"])):
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a = data_dict["a_values"][i]
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(t_0, t_1) = data_dict["prob2"]["case2"]["temp_results"][i][-2:]
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dx = data_dict["prob2"]["dx"]
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first_order = common.taylor_extraction(t_0, t_1, dx, a, 1)
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second_order = common.taylor_extraction(t_0, t_1, dx, a, 2)
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results.append([first_order, second_order])
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data_dict["prob2"]["case2"]["heat_results"] = np.array(results)
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# Create a DataFrame with alpha as the index and heat flux as the data
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df_case2_heat_flux = pd.DataFrame(results, index=[f'a = {a}' for a in data_dict["a_values"]], columns=['1st Order', '2nd Order'])
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# Pretty print DataFrame for Case 1 heat flux
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print("FDM Case 2 Heat Flux Results:")
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print(df_case2_heat_flux)
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print("\n" * 2)
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# Plotting heat flux vs alpha for both extractions
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plt.figure(figsize=(10, 6))
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plt.plot(data_dict["a_values"], [i[0] for i in data_dict["prob2"]["case2"]["heat_results"]], label='1st Order Extraction')
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plt.plot(data_dict["a_values"], [i[1] for i in data_dict["prob2"]["case2"]["heat_results"]], label='2nd Order Extraction')
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plt.xlabel('Alpha ($cm^{-1}$)')
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plt.ylabel('Heat Flux (W)')
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plt.legend()
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plt.grid(True)
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plt.savefig('prob2_case2_heat_flux.png', dpi=300)
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#plt.show()
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plt.close()
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def prob3():
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'''Convergence analysis of the cases'''
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print("Problem 3")
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data_dict["prob3"] = {}
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data_dict["prob3"]["num_points"] = [2**i for i in range(2, 8)] # This does not consider the final T(L) point. So actually we have n + 1 points
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data_dict["prob3"]["dx_values"] = [1 / i for i in data_dict["prob3"]["num_points"]]
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# Case 1
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data_dict["prob3"]["case1"] = {}
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data_dict["prob3"]["case1"]["temp_results"] = []
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# num_point does not consider T(L). So "4" points is T0, T1, T2, T3, and then we add back T(L)
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for num_point in data_dict["prob3"]["num_points"]:
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fdm_array = case1.fdm_array(num_point)
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fdm_array_inverse = np.linalg.inv(fdm_array)
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right_array = case1.right_array(num_point)
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unkown_temps_array = fdm_array_inverse @ right_array
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solution = [0] # First point is 0 from BC
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solution[1:] = unkown_temps_array
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solution.append(100) # T(L) BC
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data_dict["prob3"]["case1"]["temp_results"].append(solution)
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#data_dict["prob3"]["x_values_fine"] = np.linspace(0, 1)
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#data_dict["prob3"]["case1"]["true_temp_results"] = case1.analytical(a=2.75, x=data_dict["prob3"]["x_values_fine"])
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# We have all of the temperature results for varying dx
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# For each dx, calculate the 1st and 2nd order heat flux
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results = []
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for i in range(len(data_dict["prob3"]["dx_values"])):
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a = 2.75
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dx = data_dict["prob3"]["dx_values"][i]
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(t_0, t_1) = data_dict["prob3"]["case1"]["temp_results"][i][-2:]
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first_order = common.taylor_extraction(t_0, t_1, dx, a, 1)
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second_order = common.taylor_extraction(t_0, t_1, dx, a, 2)
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results.append([first_order, second_order])
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data_dict["prob3"]["case1"]["heat_results"] = np.array(results)
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data_dict["prob3"]["case1"]["heat_results_true"] = case1.analytical_heat_loss(a=2.75)
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# Calculate % error and beta values
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errors = []
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betas = []
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q_exact = data_dict["prob3"]["case1"]["heat_results_true"]
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for i in range(0, len(results)):
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# % Error calculation
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q_1_curr, q_2_curr = results[i]
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first_order_error = common.calc_error(q_exact, q_1_curr)
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second_order_error = common.calc_error(q_exact, q_2_curr)
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# Beta (convergence rate) calculation
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if i != 0:
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q_1_prev, q_2_prev = results[i-1]
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dx_1 = data_dict["prob3"]["dx_values"][i-1]
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dx_2 = data_dict["prob3"]["dx_values"][i]
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first_order_beta = common.calc_beta(q_exact, q_1_curr, q_1_prev, dx_2, dx_1)
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second_order_beta = common.calc_beta(q_exact, q_2_curr, q_2_prev, dx_2, dx_1)
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else:
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first_order_beta = 0
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second_order_beta = 0
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errors.append([first_order_error, second_order_error])
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betas.append([first_order_beta, second_order_beta])
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data_dict["prob3"]["case1"]["errors"] = errors
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data_dict["prob3"]["case1"]["betas"] = betas
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# Combine dx values, results, errors, and betas into a DataFrame for display
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table_data = []
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for i in range(len(data_dict["prob3"]["dx_values"])):
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table_data.append([
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results[i][0], # 1st order result
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results[i][1], # 2nd order result
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data_dict["prob3"]["case1"]["heat_results_true"], # True Q
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errors[i][0], # 1st order % error
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errors[i][1], # 2nd order % error
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betas[i][0], # 1st order beta
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||
betas[i][1] # 2nd order beta
|
||
])
|
||
|
||
columns = ['1st Q', '2nd Q', 'True Q', '1st % Error', '2nd % Error', '1st β', '2nd β']
|
||
df_case1 = pd.DataFrame(table_data, index=[f'dx = {i}' for i in data_dict["prob3"]["dx_values"]], columns=columns)
|
||
|
||
print("FDM Case 1 Heat Flux Convergence Results:")
|
||
print(df_case1)
|
||
print("\n" * 2)
|
||
|
||
# Plotting
|
||
plt.figure(figsize=(8, 6))
|
||
plt.plot(data_dict["prob3"]["dx_values"], [err[0] for err in data_dict["prob3"]["case1"]["errors"]], label='1st Order Error')
|
||
plt.plot(data_dict["prob3"]["dx_values"], [err[1] for err in data_dict["prob3"]["case1"]["errors"]], label='2nd Order Error')
|
||
plt.xscale('log')
|
||
plt.yscale('log')
|
||
plt.xlabel('Δx')
|
||
plt.ylabel('% Error')
|
||
plt.legend()
|
||
plt.grid(True)
|
||
plt.savefig("prob3_case1_convergence.png", dpi=300)
|
||
#plt.show()
|
||
plt.close()
|
||
|
||
|
||
|
||
|
||
# Case 2
|
||
data_dict["prob3"]["case2"] = {}
|
||
data_dict["prob3"]["case2"]["temp_results"] = []
|
||
# num_point does not consider T(L). So "4" points is T0, T1, T2, T3, and then we add back T(L)
|
||
for num_point in data_dict["prob3"]["num_points"]:
|
||
fdm_array = case2.fdm_array(num_point)
|
||
fdm_array_inverse = np.linalg.inv(fdm_array)
|
||
|
||
right_array = case2.right_array(num_point)
|
||
|
||
unkown_temps_array = fdm_array_inverse @ right_array
|
||
solution = []
|
||
solution[0:] = unkown_temps_array
|
||
solution.append(100) # T(L) BC
|
||
|
||
data_dict["prob3"]["case2"]["temp_results"].append(solution)
|
||
|
||
# We have all of the temperature results for varying dx
|
||
# For each dx, calculate the 1st and 2nd order heat flux
|
||
results = []
|
||
for i in range(len(data_dict["prob3"]["dx_values"])):
|
||
a = 2.75
|
||
dx = data_dict["prob3"]["dx_values"][i]
|
||
(t_0, t_1) = data_dict["prob3"]["case2"]["temp_results"][i][-2:]
|
||
|
||
first_order = common.taylor_extraction(t_0, t_1, dx, a, 1)
|
||
second_order = common.taylor_extraction(t_0, t_1, dx, a, 2)
|
||
results.append([first_order, second_order])
|
||
|
||
data_dict["prob3"]["case2"]["heat_results"] = np.array(results)
|
||
|
||
data_dict["prob3"]["case2"]["heat_results_true"] = case2.analytical_heat_loss(a=2.75)
|
||
|
||
# Calculate % error and beta values
|
||
errors = []
|
||
betas = []
|
||
q_exact = data_dict["prob3"]["case2"]["heat_results_true"]
|
||
for i in range(0, len(results)):
|
||
|
||
# % Error calculation
|
||
q_1_curr, q_2_curr = results[i]
|
||
first_order_error = common.calc_error(q_exact, q_1_curr)
|
||
second_order_error = common.calc_error(q_exact, q_2_curr)
|
||
|
||
# Beta (convergence rate) calculation
|
||
if i != 0:
|
||
q_1_prev, q_2_prev = results[i-1]
|
||
dx_1 = data_dict["prob3"]["dx_values"][i-1]
|
||
dx_2 = data_dict["prob3"]["dx_values"][i]
|
||
first_order_beta = common.calc_beta(q_exact, q_1_curr, q_1_prev, dx_2, dx_1)
|
||
second_order_beta = common.calc_beta(q_exact, q_2_curr, q_2_prev, dx_2, dx_1)
|
||
else:
|
||
first_order_beta = 0
|
||
second_order_beta = 0
|
||
|
||
errors.append([first_order_error, second_order_error])
|
||
betas.append([first_order_beta, second_order_beta])
|
||
|
||
data_dict["prob3"]["case2"]["errors"] = errors
|
||
data_dict["prob3"]["case2"]["betas"] = betas
|
||
|
||
# Combine dx values, results, errors, and betas into a DataFrame for display
|
||
table_data = []
|
||
|
||
for i in range(len(data_dict["prob3"]["dx_values"])):
|
||
table_data.append([
|
||
results[i][0], # 1st order result
|
||
results[i][1], # 2nd order result
|
||
data_dict["prob3"]["case2"]["heat_results_true"], # True Q
|
||
errors[i][0], # 1st order % error
|
||
errors[i][1], # 2nd order % error
|
||
betas[i][0], # 1st order beta
|
||
betas[i][1] # 2nd order beta
|
||
])
|
||
|
||
columns = ['1st Q', '2nd Q', 'True Q', '1st % Error', '2nd % Error', '1st β', '2nd β']
|
||
df_case2 = pd.DataFrame(table_data, index=[f'dx = {i}' for i in data_dict["prob3"]["dx_values"]], columns=columns)
|
||
|
||
print("FDM Case 2 Heat Flux Convergence Results:")
|
||
print(df_case2)
|
||
print("\n" * 2)
|
||
|
||
# Plotting
|
||
plt.figure(figsize=(8, 6))
|
||
plt.plot(data_dict["prob3"]["dx_values"], [err[0] for err in data_dict["prob3"]["case2"]["errors"]], label='1st Order Error')
|
||
plt.plot(data_dict["prob3"]["dx_values"], [err[1] for err in data_dict["prob3"]["case2"]["errors"]], label='2nd Order Error')
|
||
plt.xscale('log')
|
||
plt.yscale('log')
|
||
plt.xlabel('Δx')
|
||
plt.ylabel('% Error')
|
||
plt.legend()
|
||
plt.grid(True)
|
||
plt.savefig("prob3_case2_convergence.png", dpi=300)
|
||
#plt.show()
|
||
plt.close()
|
||
|
||
|
||
|
||
|
||
if __name__ == "__main__":
|
||
print("Homework 1 by Judson Upchurch")
|
||
print("Please note that some of the tables are transposed compared to the LaTeX equivalent")
|
||
print("")
|
||
|
||
prob1() # Analyitical temperature distribution
|
||
|
||
prob2() # FDM with dx = 0.125 and varying alpha
|
||
|
||
prob3() # Convergence analysis |