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Functions

1. laplacian2

Apply a 2D Laplacian filter to a given 2D array.

Parameters

  • state (np.ndarray): A 2D NumPy array representing the input data or image on which the Laplacian filter will be applied.

Returns

  • np.ndarray: A 2D NumPy array representing the filtered result after applying the Laplacian filter. The output has the same shape as the input.

Notes

The Laplacian filter is applied using a 3x3 kernel defined as follows:

[[ 0,  1,  0],
 [ 1, -4,  1],
 [ 0,  1,  0]]

The 'same' mode is used for convolution, ensuring that the output has the same dimensions as the input. The 'symm' boundary condition is used for handling boundary pixels.

2. Cahn_Hilliard

Calculate the time derivative of concentration field u at a given timestep t according to Cahn-Hilliard equation for phase separation in a 2D system.

Parameters

  • u (np.ndarray): A 2D NumPy array representing the concentration field at the current time.
  • t (float): The current time.
  • dx (float): Spatial step size (grid spacing) in the x and y directions.
  • D (float): Diffusion coefficient controlling the rate of phase separation.
  • a (float): Parameter affecting the interfacial energy between phases.

Returns

  • np.ndarray: A 2D NumPy array representing the time derivative concentration field u at the current time step t.

Notes

The Cahn-Hilliard equation is defined as follows:

du/dt = (D / dx^2) * laplacian2(u^3 - u - (a / dx^2) * laplacian2(u))

Where laplacian2(u) represents the Laplacian of u calculated using the laplacian2 function.

The function calculates the right-hand side of the equation for the given concentration field u, time t, spatial step size dx, diffusion coefficient D, and interfacial energy parameter a.

Example

>>> initial_concentration = np.random.rand(100, 100)
>>> t = 0.0
>>> dx = 0.1
>>> D = 0.1
>>> a = 1.0
>>> dudt = Cahn_Hilliard(initial_concentration, t, dx, D, a)

3. integrate

Numerically integrate a given differential equation over time using the Euler method.

Parameters

  • func (callable): The function representing the differential equation to be integrated. It should accept the following arguments:
    • u (np.ndarray): The current state of the system.
    • t (float): The current time.
    • *args (tuple): Additional arguments to be passed to func. The function should return the rate of change of u at the given time t.
  • u0 (np.ndarray): A 2D NumPy array representing the initial state of the system.
  • t (np.ndarray): A 1D NumPy array containing the discrete time steps at which to compute the solution.
  • *args (tuple, optional): Additional arguments to be passed to func.

Returns

  • np.ndarray: A 3D NumPy array representing the solution of the differential equation over time. The first dimension corresponds to time steps, and the shape of each step matches the shape of u0.

4. Sol_CahnHilliard

A class for simulating and analyzing solutions of the Cahn-Hilliard equation.

Parameters

  • L (float): Length of the spatial domain.
  • N (int): Number of spatial grid points.
  • D (float): Diffusion coefficient controlling the rate of phase separation.
  • a (float): Parameter affecting the interfacial energy between phases.

Attributes

  • D (float): Diffusion coefficient.
  • a (float): Interfacial energy parameter.
  • x (np.ndarray): 1D NumPy array representing the spatial grid.
  • sol (np.ndarray): 3D NumPy array representing the concentration field over time.
  • t (np.ndarray): 1D NumPy array representing time steps.
  • step (int): Step size for data analysis and visualization. Default value is 10.
  • ft_sol (np.ndarray): 2D NumPy array representing the Fourier transformed concentration profiles.
  • ft_t (np.ndarray): 1D NumPy array representing time steps for Fourier analysis.
  • histo (list): List of histograms representing concentration distributions over time.

Methods

compute_sol(self, c0, t)

Simulate the evolution of the concentration field.

  • c0 (np.ndarray): Initial concentration field.
  • t (np.ndarray): 1D NumPy array representing time steps.
set_step(self, step)

Set the step size for data analysis and visualization.

  • step (int): New step size.
ComputeFT(self)

Compute the average Fourier transform of concentration profiles along x and y at times specified by self.step.

Compute_histo(self)

Compute histograms of concentration at times specified by self.step.

MakeGif_sol(self, file_name='cahn_hilliard.gif')

Create a GIF animation of concentration field evolution.

  • file_name (str, optional): Name of the output GIF file.
MakeGif_FT(self, file_name='ft_vs_time.gif')

Create a GIF animation of Fourier components over time.

  • file_name (str, optional): Name of the output GIF file.
MakeGif_tot(self, file_name='cahn_hilliard_tot.gif')

Create a GIF animation of concentration and Fourier analysis over time.

  • file_name (str, optional): Name of the output GIF file.

This class allows you to simulate and analyze solutions of the Cahn-Hilliard equation, providing methods for data analysis and visualization.