FEM Toolkit#

The warp.fem module is designed to facilitate solving physical systems described as differential equations. For example, it can solve PDEs for diffusion, convection, fluid flow, and elasticity problems using finite-element-based (FEM) Galerkin methods and allows users to quickly experiment with various FEM formulations and discretization schemes.

Integrands#

The core functionality of the FEM toolkit is the ability to integrate constant, linear, and bilinear forms over various domains and using arbitrary interpolation basis.

The main mechanism is the integrand() decorator, for instance:

@integrand
def linear_form(
    s: Sample,
    domain: Domain,
    v: Field,
):
    x = domain(s)
    return v(s) * wp.max(0.0, 1.0 - wp.length(x))


@integrand
def diffusion_form(s: Sample, u: Field, v: Field, nu: float):
    return nu * wp.dot(
        grad(u, s),
        grad(v, s),
    )

Integrands are normal Warp kernels, meaning that they may contain arbitrary Warp functions. However, they accept a few special parameters:

  • Sample contains information about the current integration sample point, such as the element index and coordinates in element.

  • Field designates an abstract field, which will be replaced at call time by the actual field type such as a discrete field, field.TestField or field.TrialField defined over some FunctionSpace, an ImplicitField wrapping an arbitrary function, or any other of the available Fields. A field u can then be evaluated at a given sample s using the usual call operator as u(s). Several other operators are available, such as the gradient grad; see the Operators section.

  • Domain designates an abstract integration domain. Evaluating a domain at a sample s as domain(s) yields the corresponding world position, and several operators are also provided domains, for example evaluating the normal at a given sample:

    @integrand
def boundary_form(
    s: Sample,
    domain: Domain,
    u: Field,
):
    nor = normal(domain, s)
    return wp.dot(u(s), nor)

Integrands cannot be used directly with warp.launch(), but must be called through integrate() or interpolate() instead. The Sample and Domain arguments of the root integrand (integrand parameter passed to integrate() or interpolate() call) will get automatically populated. Field arguments must be passed as a dictionary in the fields parameter of the launcher function, and all other standard Warp types arguments must be passed as a dictionary in the values parameter of the launcher function, for instance:

integrate(diffusion_form, fields={"u": trial, "v": test}, values={"nu": viscosity})

Basic Workflow#

The typical steps for solving a linearized PDE with warp.fem are as follow:

  • Define a Geometry (grid, mesh, etc). At the moment, 2D and 3D regular grids, NanoVDB volumes, and triangle, quadrilateral, tetrahedron and hexahedron unstructured meshes are supported.

  • Define one or more FunctionSpace, by equipping the geometry elements with shape functions. See make_polynomial_space(). At the moment, continuous/discontinuous Lagrange (\(P_{k[d]}, Q_{k[d]}\)) and Serendipity (\(S_k\)) shape functions of order \(k \leq 3\) are supported, as well as linear Nédélec (first kind) and Raviart-Thomas vector-valued shape functions.

  • Define an integration domain, for instance the geometry’s cells (Cells) or boundary sides (BoundarySides).

  • Integrate linear forms to build the system’s right-hand-side. Define a test function over the function space using make_test(), a Quadrature formula (or let the module choose one based on the function space degree), and call integrate() with the linear form integrand. The result is a warp.array containing the integration result for each of the function space degrees of freedom.

  • Integrate bilinear forms to build the system’s left-hand-side. Define a trial function over the function space using make_trial(), then call integrate() with the bilinear form integrand. The result is a warp.sparse.BsrMatrix containing the integration result for each pair of test and trial function space degrees of freedom. Note that the trial and test functions do not have to be defined over the same function space, so that Mixed FEM is supported.

  • Solve the resulting linear system using the solver of your choice, for instance one of the built-in Iterative Linear Solvers.

The following excerpt from the introductory example warp/examples/fem/example_diffusion.py outlines this procedure:

# Grid geometry
geo = Grid2D(n=50, cell_size=2)

# Domain and function spaces
domain = Cells(geometry=geo)
scalar_space = make_polynomial_space(geo, degree=3)

# Right-hand-side (forcing term)
test = make_test(space=scalar_space, domain=domain)
rhs = integrate(linear_form, fields={"v": test})

# Weakly-imposed boundary conditions on Y sides
boundary = BoundarySides(geo)
bd_test = make_test(space=scalar_space, domain=boundary)
bd_trial = make_trial(space=scalar_space, domain=boundary)
bd_matrix = integrate(y_mass_form, fields={"u": bd_trial, "v": bd_test})

# Diffusion form
trial = make_trial(space=scalar_space, domain=domain)
matrix = integrate(diffusion_form, fields={"u": trial, "v": test}, values={"nu": viscosity})

# Assemble linear system (add diffusion and boundary condition matrices)
matrix += bd_matrix * boundary_strength

# Solve linear system using Conjugate Gradient
x = wp.zeros_like(rhs)
bsr_cg(matrix, b=rhs, x=x)

Note

The integrate() function does not check that the passed integrands are actually linear or bilinear forms; it is up to the user to ensure that they are. To solve non-linear PDEs, one can use an iterative procedure and pass the current value of the studied function DiscreteField argument to the integrand, in which arbitrary operations are permitted. However, the result of the form must remain linear in the test and trial fields. This strategy is demonstrated in the example_mixed_elasticity.py example.

Introductory Examples#

warp.fem ships with a list of examples in the warp/examples/fem directory demonstrating how to solve classical model problems.

  • example_diffusion.py: 2D diffusion with homogeneous Neumann and Dirichlet boundary conditions

    • example_diffusion_3d.py: 3D variant of the diffusion problem

  • example_convection_diffusion.py: 2D convection-diffusion using semi-Lagrangian advection

    • example_convection_diffusion_dg.py: 2D convection-diffusion using Discontinuous Galerkin with upwind transport and Symmetric Interior Penalty

  • example_burgers.py: 2D inviscid Burgers using Discontinuous Galerkin with upwind transport and slope limiter

  • example_stokes.py: 2D incompressible Stokes flow using mixed \(P_k/P_{k-1}\) or \(Q_k/P_{(k-1)d}\) elements

  • example_navier_stokes.py: 2D Navier-Stokes flow using mixed \(P_k/P_{k-1}\) elements

  • example_mixed_elasticity.py: 2D nonlinear elasticity using mixed continuous/discontinuous \(S_k/P_{(k-1)d}\) elements

  • example_distortion_energy.py: Parameterization of a 3D surface minimizing a 2D nonlinear distortion energy

  • example_magnetostatics.py: 2D magnetostatics using a curl-curl formulation

  • example_elastic_shape_optimization.py: Shape optimization of a 2D elastic cantilever beam

  • example_darcy_ls_optimization.py: Level-set-based optimization of a 2D Darcy flow

Advanced Usages#

High-order (curved) geometries#

It is possible to convert any Geometry (grids and explicit meshes) into a curved, high-order variant by deforming them with an arbitrary-order displacement field using the make_deformed_geometry() method. The process looks as follows:

# Define a base geometry
base_geo = fem.Grid3D(res=resolution)

# Define a displacement field on the base geometry
deformation_space = fem.make_polynomial_space(base_geo, degree=deformation_degree, dtype=wp.vec3)
deformation_field = deformation_space.make_field()

# Populate the field value by interpolating an expression
fem.interpolate(deformation_field_expr, dest=deformation_field)

# Construct the deformed geometry from the displacement field
deform_geo = deformation_field.make_deformed_geometry()

# Define new function spaces on the deformed geometry
scalar_space = fem.make_polynomial_space(deformed_geo, degree=scalar_space_degree)

See example_deformed_geometry.py for a complete example. It is also possible to define the deformation field from an ImplicitField, as done in example_magnetostatics.py.

Particle-based quadrature and position lookups#

The global lookup and partition_lookup operators allow generating a Sample from an arbitrary position; this is illustrated in the example_streamlines.py example for generating 3D streamlines by tracing through a velocity field.

Note

Non-grid-based geometry types require building a Bounding Volume Hierarchy (BVH) acceleration structure for lookup and similar operators to be functional. This can be done by calling Geometry.build_bvh() or passing build_bvh=True to the geometry constructor. In case the geometry vertex positions are later modified, the BVH can be refit using Geometry.update_bvh().

This operator is also leveraged by the PicQuadrature to provide a way to define Particle-In-Cell quadratures from a set of arbitrary particles, making it possible to implement MPM-type methods. The particles are automatically bucketed to the geometry cells when the quadrature is initialized. This is illustrated by the example_stokes_transfer.py and example_apic_fluid.py examples.

Nonconforming fields#

Fields defined on a given Geometry cannot be directly used for integrating over a distinct geometry; however, they may be wrapped in a NonconformingField for this purpose. This is leveraged by the example_nonconforming_contact.py to simulate contacting bodies that are discretized separately.

Note

Currently NonconformingField does not support wrapping a trial field, so it is not yet possible to define bilinear forms over different geometries.

Note

The mapping between the different geometries is position based, so a NonconformingField is not able to accurately capture discontinuous function spaces. Moreover, the integration domain must support the lookup operator (see Particle-based quadrature and position lookups).

Partitioning#

The FEM toolkit makes it possible to perform integration on a subset of the domain elements, possibly re-indexing degrees of freedom so that the linear system contains the local ones only. This is useful for distributed computation (see warp/examples/fem/example_diffusion_mgpu.py), or simply to limit the simulation domain to a subset of active cells (see warp/examples/fem/example_stokes_transfer.py).

A partition of the simulation geometry can be defined using subclasses of GeometryPartition such as LinearGeometryPartition or ExplicitGeometryPartition.

Function spaces can then be partitioned according to the geometry partition using make_space_partition(). The resulting SpacePartition object allows translating between space-wide and partition-wide node indices, and differentiating interior, frontier and exterior nodes.

The Subdomain class can be used to integrate over a subset of elements while keeping the full set of degrees of freedom, i.e, without reindexing; this is illustrated in the example_streamlines.py example to define inflow and outflow boundaries.

Adaptivity#

While unstructured mesh refinement is currently out of scope, warp.fem provides an adaptive version of the sparse grid geometry, AdaptiveNanogrid, with power-of-two voxel scales. Helpers for building such geometries from hierarchy of grids or a refinement oracle are also provided, see adaptive_nanogrid_from_field() and adaptive_nanogrid_from_hierarchy(). An example is provided in warp/examples/fem/example_adaptive_grid.py.

Note

The existence of “T-junctions” at resolution boundaries mean that usual tri-polynomial shape functions will no longer be globally continuous. Discontinuous–Galerkin or similar techniques may be used to take into account the “jump” at multi-resolution faces.

Memory management#

Several warp.fem functions require allocating temporary buffers to perform their computations. If such functions are called many times in a tight loop, those many allocations and de-allocations may degrade performance, though this is a lot less significant when Stream-Ordered Memory Pool Allocators are in use. To overcome this issue, a TemporaryStore object may be created to persist and reuse temporary allocations across calls, either globally using set_default_temporary_store() or at a per-function granularity using the corresponding argument.

Fields#

Fields represent functions defined over a geometry. The following field types are available:

  • DiscreteField: A field defined by interpolating values at the nodes of a function space.

  • ImplicitField: A field wrapping an arbitrary function.

  • UniformField: A constant field with the same value everywhere.

  • NonconformingField: A wrapper for evaluating fields defined on a different geometry.

  • GeometryField: A field representing the deformation of a geometry.

Fields can be evaluated at a Sample using the call operator, e.g., u(s) evaluates field u at sample s.

Additionally, test and trial fields (field.TestField and field.TrialField) are created using make_test() and make_trial() for building linear and bilinear forms.

Operators#

The following operators are available for use within integrands:

Field operators:

  • grad: Gradient of a field.

  • div: Divergence of a vector field.

  • curl: Curl of a vector field.

  • D: Generic derivative operator.

Domain operators:

  • position: World position at a sample (same as calling domain(s)).

  • normal: Normal vector at a sample on a boundary.

  • measure: Integration measure (area or volume element).

  • measure_ratio: Ratio of measures between a domain and its reference element.

  • deformation_gradient: Deformation gradient of the domain.

Discontinuous Galerkin operators (for use on interior sides):

  • inner: Value on the inner side of an interface.

  • outer: Value on the outer side of an interface.

  • average: Average of values across an interface.

  • jump: Jump of values across an interface.

  • grad_average: Average of gradients across an interface.

  • grad_jump: Jump of gradients across an interface.

  • grad_outer: Gradient on the outer side of an interface.

  • div_outer: Divergence on the outer side of an interface.

Visualization#

Most function spaces define a vtk_cells method that returns a list of VTK-compatible cell types and node indices. This can be used to visualize discrete fields in VTK-aware viewers such as pyvista, for instance:

import numpy as np
import pyvista

import warp as wp
import warp.fem as fem


@fem.integrand
def ackley(s: fem.Sample, domain: fem.Domain):
   x = domain(s)
   return (
      -20.0 * wp.exp(-0.2 * wp.sqrt(0.5 * wp.length_sq(x)))
      - wp.exp(0.5 * (wp.cos(2.0 * wp.pi * x[0]) + wp.cos(2.0 * wp.pi * x[1])))
      + wp.e
      + 20.0
   )


# Define field
geo = fem.Grid2D(res=wp.vec2i(64, 64), bounds_lo=wp.vec2(-4.0, -4.0), bounds_hi=wp.vec2(4.0, 4.0))
space = fem.make_polynomial_space(geo, degree=3)
field = space.make_field()
fem.interpolate(ackley, dest=field)

# Extract cells, nodes and values
cells, types = field.space.vtk_cells()
nodes = field.space.node_positions().numpy()
values = field.dof_values.numpy()
positions = np.hstack((nodes, values[:, np.newaxis]))

# Visualize with pyvista
grid = pyvista.UnstructuredGrid(cells, types, positions)
grid.point_data["scalars"] = values
plotter = pyvista.Plotter()
plotter.add_mesh(grid)
plotter.show()