Writing Fluid Simulations in ΦFlow¶

Google Collab Book

There are two main viewpoints for simulating fluids:

  • Eulerian simulations use grids, tracking fluid distribution and fluid velocity at fixed sample points
  • Lagrangian simulations track particles that move with the fluid.

ΦFlow supports both methods to some extent but mainly focuses on Eulerian simulations.

Before we discuss the various operations required for fluid simulations, let's define our variables and initial state. In this case, we will create a 64×96 grid, sampling velocity vectors in staggered form and marker values at the centroids.

In [1]:
from tqdm.notebook import trange
from phi.jax.flow import *  # imports sub-modules + core classes

velocity = StaggeredGrid(Noise(), 'periodic', x=64, y=96)
plot({"velocity": velocity, "vorticity": field.curl(velocity)})

/opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phi/vis/_matplotlib/_matplotlib_plots.py:167: UserWarning: This figure includes Axes that are not compatible with tight_layout, so results might be incorrect.
  plt.tight_layout()  # because subplot titles can be added after figure creation
Out[1]:

Operator Splitting¶

The Navier-Stokes equations for fluids, $\frac{\partial u}{\partial t} = - (u \cdot \nabla) u - \nu \nabla^2 u - \frac 1 \rho \nabla p + g$, comprise multiple terms.

Operator splitting enables writing fast and stable fluid simulations by sequentially evaluating the different terms. For each of the terms, ΦFlow provides functions to compute them:

  • Advection: advect.semi_lagrangian [Stam 1999], advect.mac_cormack [MacCormack 2002]
  • Diffusion: diffuse.explicit, diffuse.implicit
  • Pressure projection: fluid.make_incompressible [Chorin and Temam 1968]

All of these functions take in a state variable and return the new state after a certain time dt has passed. In the following example, the velocity is self-advected and made incompressible, while the marker is passively advected.

In [2]:
@jit_compile
def operator_split_step(v, p, dt, viscosity=0.1):
    v = advect.semi_lagrangian(v, v, dt)  # velocity self-advection
    v = diffuse.explicit(v, viscosity, dt)
    v, p = fluid.make_incompressible(v, (), Solve(x0=p, rank_deficiency=0))
    return v, p

velocity0, pressure0 = fluid.make_incompressible(velocity)
velocity1, pressure1 = operator_split_step(velocity0, None, dt=1.)
plot({'initial vorticity': field.curl(velocity0), 'after step': field.curl(velocity1)})

/opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/backend/_linalg.py:337: SparseEfficiencyWarning: spsolve requires A be CSC or CSR matrix format
  x = spsolve(lin[batch], y[batch])  # returns nan when diverges

---------------------------------------------------------------------------
Diverged                                  Traceback (most recent call last)
Cell In[2], line 8
      5     v, p = fluid.make_incompressible(v, (), Solve(x0=p, rank_deficiency=0))
      6     return v, p
----> 8 velocity0, pressure0 = fluid.make_incompressible(velocity)
      9 velocity1, pressure1 = operator_split_step(velocity0, None, dt=1.)
     10 plot({'initial vorticity': field.curl(velocity0), 'after step': field.curl(velocity1)})

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phi/physics/fluid.py:156, in make_incompressible(velocity, obstacles, solve, active, order, correct_skew, wide_stencil)
    154 if wide_stencil is None:
    155     wide_stencil = not velocity.is_staggered
--> 156 pressure = math.solve_linear(masked_laplace, div, solve, velocity.boundary, hard_bcs, active, wide_stencil=wide_stencil, order=order, implicit=None, upwind=None, correct_skew=correct_skew)
    157 # --- Subtract grad p ---
    158 grad_pressure = field.spatial_gradient(pressure, input_velocity.extrapolation, at=velocity.sampled_at, order=order, scheme='green-gauss')

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:666, in solve_linear(f, y, solve, grad_for_f, f_kwargs, *f_args, **f_kwargs_)
    663         return result  # must return exactly `x` so gradient isn't computed w.r.t. other quantities
    665     _matrix_solve = attach_gradient_solve(_matrix_solve_forward, auxiliary_args=f'is_backprop,solve{",matrix" if matrix.backend == NUMPY else ""}', matrix_adjoint=grad_for_f)
--> 666     return _matrix_solve(y - bias, solve, matrix)
    667 else:  # Matrix-free solve
    668     f_args = cached(f_args)

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_functional.py:963, in CustomGradientFunction.__call__(self, *args, **kwargs)
    958             if len(self.traces) >= 8:
    959                 warnings.warn(f"""{self.__name__} has been traced {len(self.traces)} times.
    960 To avoid memory leaks, call {f_name(self.f)}.traces.clear(), {f_name(self.f)}.recorded_mappings.clear().
    961 Traces can be avoided by jit-compiling the code that calls custom gradient functions.
    962 """, RuntimeWarning, stacklevel=2)
--> 963         native_result = self.traces[key](*natives)  # With PyTorch + jit, this does not call forward_native every time
    964         output_key = match_output_signature(key, self.recorded_mappings, self)
    965         output_tensors = assemble_tensors(native_result, output_key.specs)

    [... skipping hidden 9 frame]

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_functional.py:919, in CustomGradientFunction._trace.<locals>.forward_native(*natives)
    917 kwargs = assemble_tree(in_key.tree, in_tensors, attr_type=variable_attributes)
    918 ML_LOGGER.debug(f"Running forward pass of custom op {forward_native.__name__} given args {tuple(kwargs.keys())} containing {len(natives)} native tensors")
--> 919 result = self.f(**kwargs, **in_key.auxiliary_kwargs)  # Tensor or tuple/list of Tensors
    920 nest, out_tensors = disassemble_tree(result, cache=True, attr_type=variable_attributes)
    921 result_natives, result_shapes, specs = disassemble_tensors(out_tensors, expand=True)

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:662, in solve_linear.<locals>._matrix_solve_forward(y, solve, matrix, is_backprop)
    660     idx = b.concat([idx, new_col, new_row], 0)
    661     nat_matrix = b.sparse_coo_tensor(idx, data, (N+1, N+1))
--> 662 result = _linear_solve_forward(y, solve, nat_matrix, pattern_dims_in, pattern_dims_out, preconditioner, backend, is_backprop)
    663 return result

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:775, in _linear_solve_forward(y, solve, native_lin_op, pattern_dims_in, pattern_dims_out, preconditioner, backend, is_backprop)
    773 for tape in _SOLVE_TAPES:
    774     tape._add(solve, trj, result)
--> 775 result.convergence_check(is_backprop and 'TensorFlow' in backend.name)  # raises ConvergenceException
    776 return final_x

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:200, in SolveInfo.convergence_check(self, only_warn)
    198             warnings.warn(self.msg, ConvergenceWarning)
    199         else:
--> 200             raise Diverged(self)
    201 if not self.converged.trajectory[-1].all:
    202     if NotConverged not in self.solve.suppress:

Diverged: Solve diverged within -1 iterations using scipy.sparse.linalg.spsolve.
In [3]:
@jit_compile
def operator_split_step(v, p, dt):
    v = advect.semi_lagrangian(v, v, dt)  # velocity self-advection
    v = diffuse.explicit(v, 0.1, dt)
    v, p = fluid.make_incompressible(v, (), Solve(x0=p, rank_deficiency=0))
    return v, p

velocity0, pressure0 = fluid.make_incompressible(velocity)
velocity1, pressure1 = operator_split_step(velocity0, None, dt=1.)
plot({'initial vorticity': field.curl(velocity0), 'after step': field.curl(velocity1)})

---------------------------------------------------------------------------
Diverged                                  Traceback (most recent call last)
Cell In[3], line 8
      5     v, p = fluid.make_incompressible(v, (), Solve(x0=p, rank_deficiency=0))
      6     return v, p
----> 8 velocity0, pressure0 = fluid.make_incompressible(velocity)
      9 velocity1, pressure1 = operator_split_step(velocity0, None, dt=1.)
     10 plot({'initial vorticity': field.curl(velocity0), 'after step': field.curl(velocity1)})

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phi/physics/fluid.py:156, in make_incompressible(velocity, obstacles, solve, active, order, correct_skew, wide_stencil)
    154 if wide_stencil is None:
    155     wide_stencil = not velocity.is_staggered
--> 156 pressure = math.solve_linear(masked_laplace, div, solve, velocity.boundary, hard_bcs, active, wide_stencil=wide_stencil, order=order, implicit=None, upwind=None, correct_skew=correct_skew)
    157 # --- Subtract grad p ---
    158 grad_pressure = field.spatial_gradient(pressure, input_velocity.extrapolation, at=velocity.sampled_at, order=order, scheme='green-gauss')

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:666, in solve_linear(f, y, solve, grad_for_f, f_kwargs, *f_args, **f_kwargs_)
    663         return result  # must return exactly `x` so gradient isn't computed w.r.t. other quantities
    665     _matrix_solve = attach_gradient_solve(_matrix_solve_forward, auxiliary_args=f'is_backprop,solve{",matrix" if matrix.backend == NUMPY else ""}', matrix_adjoint=grad_for_f)
--> 666     return _matrix_solve(y - bias, solve, matrix)
    667 else:  # Matrix-free solve
    668     f_args = cached(f_args)

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_functional.py:963, in CustomGradientFunction.__call__(self, *args, **kwargs)
    958             if len(self.traces) >= 8:
    959                 warnings.warn(f"""{self.__name__} has been traced {len(self.traces)} times.
    960 To avoid memory leaks, call {f_name(self.f)}.traces.clear(), {f_name(self.f)}.recorded_mappings.clear().
    961 Traces can be avoided by jit-compiling the code that calls custom gradient functions.
    962 """, RuntimeWarning, stacklevel=2)
--> 963         native_result = self.traces[key](*natives)  # With PyTorch + jit, this does not call forward_native every time
    964         output_key = match_output_signature(key, self.recorded_mappings, self)
    965         output_tensors = assemble_tensors(native_result, output_key.specs)

    [... skipping hidden 9 frame]

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_functional.py:919, in CustomGradientFunction._trace.<locals>.forward_native(*natives)
    917 kwargs = assemble_tree(in_key.tree, in_tensors, attr_type=variable_attributes)
    918 ML_LOGGER.debug(f"Running forward pass of custom op {forward_native.__name__} given args {tuple(kwargs.keys())} containing {len(natives)} native tensors")
--> 919 result = self.f(**kwargs, **in_key.auxiliary_kwargs)  # Tensor or tuple/list of Tensors
    920 nest, out_tensors = disassemble_tree(result, cache=True, attr_type=variable_attributes)
    921 result_natives, result_shapes, specs = disassemble_tensors(out_tensors, expand=True)

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:662, in solve_linear.<locals>._matrix_solve_forward(y, solve, matrix, is_backprop)
    660     idx = b.concat([idx, new_col, new_row], 0)
    661     nat_matrix = b.sparse_coo_tensor(idx, data, (N+1, N+1))
--> 662 result = _linear_solve_forward(y, solve, nat_matrix, pattern_dims_in, pattern_dims_out, preconditioner, backend, is_backprop)
    663 return result

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:775, in _linear_solve_forward(y, solve, native_lin_op, pattern_dims_in, pattern_dims_out, preconditioner, backend, is_backprop)
    773 for tape in _SOLVE_TAPES:
    774     tape._add(solve, trj, result)
--> 775 result.convergence_check(is_backprop and 'TensorFlow' in backend.name)  # raises ConvergenceException
    776 return final_x

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:200, in SolveInfo.convergence_check(self, only_warn)
    198             warnings.warn(self.msg, ConvergenceWarning)
    199         else:
--> 200             raise Diverged(self)
    201 if not self.converged.trajectory[-1].all:
    202     if NotConverged not in self.solve.suppress:

Diverged: Solve diverged within -1 iterations using scipy.sparse.linalg.spsolve.

We can use iterate to compute a trajectory by repeatedly calling operator_split_step. All intermediate states are stacked along the specified dimension which we call time.

In [4]:
velocity_trj, pressure_trj = iterate(operator_split_step, batch(time=100), velocity0, pressure0, dt=1., range=trange)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[4], line 1
----> 1 velocity_trj, pressure_trj = iterate(operator_split_step, batch(time=100), velocity0, pressure0, dt=1., range=trange)

NameError: name 'velocity0' is not defined

Alternatively, we could have written a for loop, added all intermediate states to a list, and stacked the results afterward. Now, let's plot this trajectory by animating the time dimension.

In [5]:
plot(field.curl(velocity_trj), animate='time', same_scale=False)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Cell In[5], line 1
----> 1 plot(field.curl(velocity_trj), animate='time', same_scale=False)

NameError: name 'velocity_trj' is not defined

Higher-Order Simulations¶

The operator splitting approach is not compatible with more accurate numerical schemes. For more accurate simulations, we can use higher-order spatial schemes as well as time integration. In that case, we define a momentum equation which computes the PDE terms directly, without integrating them in time. The following example computes explicit fourth-order accurate advection and diffusion.

In [6]:
def momentum_equation(v, viscosity=0.1):
    advection = advect.finite_difference(v, v, order=4, implicit=None)
    diffusion = diffuse.finite_difference(v, viscosity, order=4, implicit=None)
    return advection + diffusion

Next, we perform time integration with the incompressibility constraint. This is considerably more expensive than the previous approach but yields much more accurate results.

In [7]:
@jit_compile
def rk4_step(v, p, dt):
    return fluid.incompressible_rk4(momentum_equation, v, p, dt, pressure_order=4)

velocity = CenteredGrid(Noise(vector='x,y'), 'periodic', x=64, y=96)
velocity0, pressure0 = fluid.make_incompressible(velocity, order=4)
velocity_trj, pressure_trj = iterate(rk4_step, batch(time=100), velocity0, pressure0, dt=.5, substeps=2, range=trange)
plot(field.curl(velocity_trj), animate='time', same_scale=False)

---------------------------------------------------------------------------
Diverged                                  Traceback (most recent call last)
Cell In[7], line 6
      3     return fluid.incompressible_rk4(momentum_equation, v, p, dt, pressure_order=4)
      5 velocity = CenteredGrid(Noise(vector='x,y'), 'periodic', x=64, y=96)
----> 6 velocity0, pressure0 = fluid.make_incompressible(velocity, order=4)
      7 velocity_trj, pressure_trj = iterate(rk4_step, batch(time=100), velocity0, pressure0, dt=.5, substeps=2, range=trange)
      8 plot(field.curl(velocity_trj), animate='time', same_scale=False)

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phi/physics/fluid.py:156, in make_incompressible(velocity, obstacles, solve, active, order, correct_skew, wide_stencil)
    154 if wide_stencil is None:
    155     wide_stencil = not velocity.is_staggered
--> 156 pressure = math.solve_linear(masked_laplace, div, solve, velocity.boundary, hard_bcs, active, wide_stencil=wide_stencil, order=order, implicit=None, upwind=None, correct_skew=correct_skew)
    157 # --- Subtract grad p ---
    158 grad_pressure = field.spatial_gradient(pressure, input_velocity.extrapolation, at=velocity.sampled_at, order=order, scheme='green-gauss')

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:666, in solve_linear(f, y, solve, grad_for_f, f_kwargs, *f_args, **f_kwargs_)
    663         return result  # must return exactly `x` so gradient isn't computed w.r.t. other quantities
    665     _matrix_solve = attach_gradient_solve(_matrix_solve_forward, auxiliary_args=f'is_backprop,solve{",matrix" if matrix.backend == NUMPY else ""}', matrix_adjoint=grad_for_f)
--> 666     return _matrix_solve(y - bias, solve, matrix)
    667 else:  # Matrix-free solve
    668     f_args = cached(f_args)

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_functional.py:963, in CustomGradientFunction.__call__(self, *args, **kwargs)
    958             if len(self.traces) >= 8:
    959                 warnings.warn(f"""{self.__name__} has been traced {len(self.traces)} times.
    960 To avoid memory leaks, call {f_name(self.f)}.traces.clear(), {f_name(self.f)}.recorded_mappings.clear().
    961 Traces can be avoided by jit-compiling the code that calls custom gradient functions.
    962 """, RuntimeWarning, stacklevel=2)
--> 963         native_result = self.traces[key](*natives)  # With PyTorch + jit, this does not call forward_native every time
    964         output_key = match_output_signature(key, self.recorded_mappings, self)
    965         output_tensors = assemble_tensors(native_result, output_key.specs)

    [... skipping hidden 9 frame]

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_functional.py:919, in CustomGradientFunction._trace.<locals>.forward_native(*natives)
    917 kwargs = assemble_tree(in_key.tree, in_tensors, attr_type=variable_attributes)
    918 ML_LOGGER.debug(f"Running forward pass of custom op {forward_native.__name__} given args {tuple(kwargs.keys())} containing {len(natives)} native tensors")
--> 919 result = self.f(**kwargs, **in_key.auxiliary_kwargs)  # Tensor or tuple/list of Tensors
    920 nest, out_tensors = disassemble_tree(result, cache=True, attr_type=variable_attributes)
    921 result_natives, result_shapes, specs = disassemble_tensors(out_tensors, expand=True)

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:662, in solve_linear.<locals>._matrix_solve_forward(y, solve, matrix, is_backprop)
    660     idx = b.concat([idx, new_col, new_row], 0)
    661     nat_matrix = b.sparse_coo_tensor(idx, data, (N+1, N+1))
--> 662 result = _linear_solve_forward(y, solve, nat_matrix, pattern_dims_in, pattern_dims_out, preconditioner, backend, is_backprop)
    663 return result

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:775, in _linear_solve_forward(y, solve, native_lin_op, pattern_dims_in, pattern_dims_out, preconditioner, backend, is_backprop)
    773 for tape in _SOLVE_TAPES:
    774     tape._add(solve, trj, result)
--> 775 result.convergence_check(is_backprop and 'TensorFlow' in backend.name)  # raises ConvergenceException
    776 return final_x

File /opt/hostedtoolcache/Python/3.12.9/x64/lib/python3.12/site-packages/phiml/math/_optimize.py:200, in SolveInfo.convergence_check(self, only_warn)
    198             warnings.warn(self.msg, ConvergenceWarning)
    199         else:
--> 200             raise Diverged(self)
    201 if not self.converged.trajectory[-1].all:
    202     if NotConverged not in self.solve.suppress:

Diverged: Solve diverged within -1 iterations using scipy.sparse.linalg.spsolve.

Further Reading¶

The Kolmogorov flow notebebook shows higher-order fluid flow with forcing.

For a comparison of various schemes in both accuracy and performance is given here.

Coupling between centered and staggered fields can be seen in the smoke plume notebook and Python script.