Implementing Mixed Finite Element Methods for Stokes Equations with Nutils

Implementing mixed finite element methods for the Stokes equations using Nutils involves several steps including defining the problem, setting up the finite element spaces, assembling the system of equations, and solving them. Below is a concise guide to help you get started.

1. Installation

Ensure you have Nutils installed in your Python environment. You can install it via pip:

pip install nutils

2. Define the Problem

The Stokes equations consist of the momentum equation and the continuity equation. In a mixed formulation, you typically solve for both the velocity field mathbf{u} and the pressure p.

3. Set Up the Domain and Finite Element Spaces

Define your computational domain and the finite element spaces for velocity and pressure. Here’s an example using a 2D unit square domain:

import nutils Define the mesh domain ([0, 1], [0, 1], elements(20, 20)) Define finite element spaces velocity_space _(domain, order2, dim2, family'Lagrange') pressure_space _(domain, order1, family'Lagrange')

4. Define the Stokes Equations

The Stokes equations in mixed form can be expressed as:

( - abla cdot sigma abla p f quad text{in } Omega )

( abla cdot mathbf{u} 0 quad text{in } Omega )

Where ( sigma 2muvarepsilon(mathbf{u}) ) is the stress tensor, ( mu ) is the viscosity, and ( varepsilon(mathbf{u}) ) is the strain rate tensor.

5. Weak Formulation

You need to derive the weak form of the equations. For the mixed formulation, it typically involves integrating by parts and applying the divergence theorem.

6. Assemble the System

Using Nutils, you can define the bilinear forms and linear forms to assemble the system of equations.

Define parameters mu 1.0 # viscosity f [0, -1] # body force Define the weak forms def weak_formu(p): return 2 * mu * u @ v - p * div(v) - f @ v, div(u) * q Assemble the system problem ([velocity_space, pressure_space], weak_formweak_formu)

7. Apply Boundary Conditions

You need to apply appropriate boundary conditions for your problem, which could be Dirichlet or Neumann conditions.

Apply boundary conditions problem.bc[u] [0, 0] # No-slip condition

8. Solve the System

Finally, solve the assembled system of equations.

solution ()

9. Post-Processing

You can visualize the results or perform further analysis on the solution.

import as plt Extract velocity and pressure u, p solution Plotting plt.quiver(u[0], u[1], p) plt.title('Velocity field')

Conclusion

This is a basic outline for implementing a mixed finite element method for the Stokes equations in Nutils. Depending on your specific problem geometry, boundary conditions, etc., you may need to adjust the implementation accordingly. You can refer to the Nutils documentation for more detailed examples and functions available for your specific needs.

By following these steps, you can effectively implement and solve the Stokes equations using mixed finite element methods with Nutils.