The accompanying Jupyter notebook can be obtained here 3_transient_analysis

Heat equation

The heat equation is a fundamental partial differential equation that describes the evolution of temperature distribution over time in a given domain. In this tutorial, we will perform a 1D transient analysis of the heat equation.

The 1D heat equation is given by:

\[ \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} + f(x, t), \]

where \(u(x, t)\) is the temperature distribution at position \(x\) and time \(t\), \(k\) is the thermal conductivity, and \(f(x, t)\) represents any source term in the equation. For this analysis, we consider a 1D domain \([0, L]\).

https://hplgit.github.io/fenics-tutorial/pub/html/._ftut1006.html

from fenics import *
from matplotlib import pyplot as plt

# Create a mesh
L = 1.0  # Length of the domain
nx = 100  # Number of spatial nodes
mesh = IntervalMesh(nx, 0, L)

# Define the function space
degree = 1  # Linear elements
S = FunctionSpace(mesh, 'CG', degree)

Initial Condition and Boundary Conditions

We specify an initial temperature distribution \(u_{\text{initial}}(x)\) at \(t = 0\). For this analysis, we use the expression \(u_{\text{initial}}(x) = \exp(-100(x - 0.5)^2)\). Additionally, we need to impose boundary conditions to complete the problem formulation.

# Define initial condition and boundary conditions
u_initial = Expression('20*sin(pi*x[0])', degree=2, domain=mesh)
u_n = interpolate(u_initial, S)
u_n_minus_1 = Function(S)

Temporal Discretization

We define the total simulation time \(T\) and the number of time steps \(num\_steps\). The time step size \(dt\) is calculated as \(dt = \frac{T}{num\_steps}\).

# Define time discretization parameters
T = 1.0   # Total simulation time
num_steps = 5  # Number of time steps
dt = T / num_steps  # Time step size

# Define the heat equation
u = TrialFunction(S)
v = TestFunction(S)
k = Constant(1.0e-1)  # Thermal conductivity
f = Constant(0.0)  # Source term (zero for this example)

support = CompiledSubDomain("on_boundary")
# bc = DirichletBC(S, Constant(0), support)
bc = []

a = u*v*dx + dt*k*inner(grad(u), grad(v))*dx
L = (u_n*v*dx + dt*f*v*dx)

Time-stepping Loop

We use a time-stepping loop to iteratively solve the heat equation at each time step. At every time step, we update the temperature distribution based on the discretized equation.

u = Function(S)
t = 0
# Create a figure with the specified size
fig, ax = plt.subplots(figsize=(13, 8))
label = 'time: {0:3.1f}'.format(t)  # Label for each curve
ax.plot(mesh.coordinates(), u_n.vector()[:], label=label, linewidth=3)

for n in range(num_steps):
    t += dt
    solve(a == L, u, bc)

    # Update solution for the next time step
    u_n_minus_1.assign(u_n)
    u_n.assign(u)

    label = 'time: {0:3.1f}'.format(t)  # Label for each curve
    ax.plot(mesh.coordinates(), u.vector()[:], label=label, linewidth=3)

# Add labels and legend
ax.set_xlabel('x')
ax.set_ylabel('Temperature')
ax.legend()

# Show the plot
ax.grid(True)
plt.show()

png