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# Elasticity using algebraic multigrid

Copyright © 2020-2022 Garth N. Wells and Michal Habera

This demo ({download}`demo_elasticity.py`) solves the equations of
static linear elasticity using a smoothed aggregation algebraic
multigrid solver. It illustrates how to:

- Use a smoothed aggregation algebraic multigrid solver
- Use {py:class}`Expression <dolfinx.fem.Expression>` to compute
  derived quantities of a solution

The required modules are first imported:

```python
import numpy as np

import dolfinx
import ufl
from dolfinx import la
from dolfinx.fem import (Expression, Function, FunctionSpaceBase, dirichletbc,
                         form, functionspace, locate_dofs_topological)
from dolfinx.fem.petsc import (apply_lifting, assemble_matrix, assemble_vector,
                               set_bc)
from dolfinx.io import XDMFFile
from dolfinx.mesh import (CellType, GhostMode, create_box,
                          locate_entities_boundary)
from ufl import dx, grad, inner

from mpi4py import MPI
from petsc4py import PETSc

dtype = PETSc.ScalarType  # type: ignore
```

## Create the operator near-nullspace

Smooth aggregation algebraic multigrid solvers require the so-called
'near-nullspace', which is the nullspace of the operator in the
absence of boundary conditions. The below function builds a
`PETSc.NullSpace` object for a 3D elasticity problem. The nullspace is
spanned by six vectors -- three translation modes and three rotation
modes.

```python
def build_nullspace(V: FunctionSpaceBase):
    """Build PETSc nullspace for 3D elasticity"""

    # Create vectors that will span the nullspace
    bs = V.dofmap.index_map_bs
    length0 = V.dofmap.index_map.size_local
    basis = [la.vector(V.dofmap.index_map, bs=bs, dtype=dtype) for i in range(6)]
    b = [b.array for b in basis]

    # Get dof indices for each subspace (x, y and z dofs)
    dofs = [V.sub(i).dofmap.list.flatten() for i in range(3)]

    # Set the three translational rigid body modes
    for i in range(3):
        b[i][dofs[i]] = 1.0

    # Set the three rotational rigid body modes
    x = V.tabulate_dof_coordinates()
    dofs_block = V.dofmap.list.flatten()
    x0, x1, x2 = x[dofs_block, 0], x[dofs_block, 1], x[dofs_block, 2]
    b[3][dofs[0]] = -x1
    b[3][dofs[1]] = x0
    b[4][dofs[0]] = x2
    b[4][dofs[2]] = -x0
    b[5][dofs[2]] = x1
    b[5][dofs[1]] = -x2

    _basis = [x._cpp_object for x in basis]
    dolfinx.cpp.la.orthonormalize(_basis)
    assert dolfinx.cpp.la.is_orthonormal(_basis)

    basis_petsc = [PETSc.Vec().createWithArray(x[:bs * length0], bsize=3, comm=V.mesh.comm) for x in b]  # type: ignore
    return PETSc.NullSpace().create(vectors=basis_petsc)  # type: ignore
```

## Problem definition

+++

Create a box Mesh:

```python
msh = create_box(MPI.COMM_WORLD, [np.array([0.0, 0.0, 0.0]),
                                  np.array([2.0, 1.0, 1.0])], [16, 16, 16],
                 CellType.tetrahedron, ghost_mode=GhostMode.shared_facet)
```

Create a centripetal source term $f = \rho \omega^2 [x_0, \, x_1]$:

```python
ω, ρ = 300.0, 10.0
x = ufl.SpatialCoordinate(msh)
f = ufl.as_vector((ρ * ω**2 * x[0], ρ * ω**2 * x[1], 0.0))
```

Define the elasticity parameters and create a function that computes
an expression for the stress given a displacement field.

```python
E = 1.0e9
ν = 0.3
μ = E / (2.0 * (1.0 + ν))
λ = E * ν / ((1.0 + ν) * (1.0 - 2.0 * ν))


def σ(v):
    """Return an expression for the stress σ given a displacement field"""
    return 2.0 * μ * ufl.sym(grad(v)) + λ * ufl.tr(ufl.sym(grad(v))) * ufl.Identity(len(v))
```

A function space space is created and the elasticity variational
problem defined:

```python
V = functionspace(msh, ("Lagrange", 1, (msh.geometry.dim,)))
u, v = ufl.TrialFunction(V), ufl.TestFunction(V)
a = form(inner(σ(u), grad(v)) * dx)
L = form(inner(f, v) * dx)
```

A homogeneous (zero) boundary condition is created on $x_0 = 0$ and
$x_1 = 1$ by finding all facets on these boundaries, and then creating
a Dirichlet boundary condition object.

```python
facets = locate_entities_boundary(msh, dim=2,
                                  marker=lambda x: np.logical_or(np.isclose(x[0], 0.0),
                                                                 np.isclose(x[1], 1.0)))
bc = dirichletbc(np.zeros(3, dtype=dtype),
                 locate_dofs_topological(V, entity_dim=2, entities=facets), V=V)
```

## Assemble and solve

The bilinear form `a` is assembled into a matrix `A`, with
modifications for the Dirichlet boundary conditions. The call
`A.assemble()` completes any parallel communication required to
compute the matrix.

```python
A = assemble_matrix(a, bcs=[bc])
A.assemble()
```

The linear form `L` is assembled into a vector `b`, and then modified
by {py:func}`apply_lifting <dolfinx.fem.petsc.apply_lifting>` to
account for the Dirichlet boundary conditions. After calling
{py:func}`apply_lifting <dolfinx.fem.petsc.apply_lifting>`, the method
`ghostUpdate` accumulates entries on the owning rank, and this is
followed by setting the boundary values in `b`.

```python
b = assemble_vector(L)
apply_lifting(b, [a], bcs=[[bc]])
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)  # type: ignore
set_bc(b, [bc])
```

Create the near-nullspace and attach it to the PETSc matrix:

```python
ns = build_nullspace(V)
A.setNearNullSpace(ns)
A.setOption(PETSc.Mat.Option.SPD, True)  # type: ignore
```

Set PETSc solver options, create a PETSc Krylov solver, and attach the
matrix `A` to the solver:

```python
# Set solver options
opts = PETSc.Options()  # type: ignore
opts["ksp_type"] = "cg"
opts["ksp_rtol"] = 1.0e-8
opts["pc_type"] = "gamg"

# Use Chebyshev smoothing for multigrid
opts["mg_levels_ksp_type"] = "chebyshev"
opts["mg_levels_pc_type"] = "jacobi"

# Improve estimate of eigenvalues for Chebyshev smoothing
opts["mg_levels_ksp_chebyshev_esteig_steps"] = 10

# Create PETSc Krylov solver and turn convergence monitoring on
solver = PETSc.KSP().create(msh.comm)  # type: ignore
solver.setFromOptions()

# Set matrix operator
solver.setOperators(A)
```

Create a solution {py:class}`Function<dolfinx.fem.Function>` `uh` and
solve:

```python
uh = Function(V)

# Set a monitor, solve linear system, and display the solver
# configuration
solver.setMonitor(lambda _, its, rnorm: print(f"Iteration: {its}, rel. residual: {rnorm}"))
solver.solve(b, uh.vector)
solver.view()

# Scatter forward the solution vector to update ghost values
uh.x.scatter_forward()
```

## Post-processing

The computed solution is now post-processed. Expressions for the
deviatoric and Von Mises stress are defined:

```python
sigma_dev = σ(uh) - (1 / 3) * ufl.tr(σ(uh)) * ufl.Identity(len(uh))
sigma_vm = ufl.sqrt((3 / 2) * inner(sigma_dev, sigma_dev))
```

Next, the Von Mises stress is interpolated in a piecewise-constant
space by creating an {py:class}`Expression<dolfinx.fem.Expression>`
that is interpolated into the
{py:class}`Function<dolfinx.fem.Function>` `sigma_vm_h`.

```python
W = functionspace(msh, ("Discontinuous Lagrange", 0))
sigma_vm_expr = Expression(sigma_vm, W.element.interpolation_points())
sigma_vm_h = Function(W)
sigma_vm_h.interpolate(sigma_vm_expr)
```

Save displacement field `uh` and the Von Mises stress `sigma_vm_h` in
XDMF format files.

```python
with XDMFFile(msh.comm, "out_elasticity/displacements.xdmf", "w") as file:
    file.write_mesh(msh)
    file.write_function(uh)

# Save solution to XDMF format
with XDMFFile(msh.comm, "out_elasticity/von_mises_stress.xdmf", "w") as file:
    file.write_mesh(msh)
    file.write_function(sigma_vm_h)
```

Finally, we compute the $L^2$ norm of the displacement solution
vector. This is a collective operation (i.e., the method `norm` must
be called from all MPI ranks), but we print the norm only on rank 0.

```python
unorm = uh.x.norm()
if msh.comm.rank == 0:
    print("Solution vector norm:", unorm)
```

The solution vector norm can be a useful check that the solver is
computing the same result when running in serial and in parallel.
