Tutorial 5-3 Transient analysis

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 $u_{initial} (x)$ at $t = 0$ . For this analysis, we use the expression $u_{initial} (x) = \exp (- 100 (x - 0.5)^{2})$ . 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 $n u m_s t e p s$ . The time step size $d t$ is calculated as $d t = \frac{T}{n u m_s t e p s}$ .

# 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()

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