warp.fem

Finite Element Method (FEM) toolkit for solving differential equations.

This module provides tools for solving physical systems described as partial differential equations (PDEs) using finite-element-based Galerkin methods. It supports diffusion, convection, fluid flow, and elasticity problems with various FEM formulations and discretization schemes.

The core workflow involves defining geometries (grids, meshes, NanoVDB volumes), function spaces with shape functions, integration domains, and using the integrate() function with integrand()-decorated kernels to build linear and bilinear forms for solving linear systems.

Usage:

This module must be explicitly imported:

import warp.fem

See also

FEM Toolkit for comprehensive documentation and examples.

Submodules

These modules are automatically available when you import warp.fem.

• warp.fem.cache

• warp.fem.field

• warp.fem.geometry

• warp.fem.linalg

• warp.fem.polynomial

• warp.fem.space

• warp.fem.utils

API

AdaptiveNanogrid	Adaptive sparse grid
BasisSpace	Interface class for defining shape functions over a geometry.
BoundarySides	A Domain containing boundary sides of the geometry or geometry partition
CellBasedGeometryPartition	Geometry partition based on a subset of cells.
Cells	A Domain containing all cells of the geometry or geometry partition
Coords	alias of vec3f
DiscreteField	Explicitly-valued field defined over a partition of a discrete function space
DofMapper	Base class from mapping node degrees of freedom to function values
Domain	Tag for domain-like integrand arguments
Element	Enumeration of reference element types
ElementBasis	Choice of basis function to equip individual elements
ElementIndex	alias of int
ElementKind	Type of geometry elements
ExplicitGeometryPartition
ExplicitQuadrature	Quadrature using explicit per-cell points and weights.
Field	Tag for field-like integrand arguments
FieldLike	Base class for integrable fields
FrontierSides	A Domain containing frontier sides of the geometry partition (sides shared with at least another partition)
FunctionSpace	Interface class for function spaces, i.e. geometry + interpolation basis.
Geometry	Interface class for discrete geometries
GeometryDomain	Interface class for domains, i.e. (partial) views of elements in a Geometry.
GeometryField	Base class for fields defined over a geometry
GeometryPartition	Base class for geometry partitions, i.e. subset of cells and sides.
Grid2D	Two-dimensional regular grid geometry
Grid3D	Three-dimensional regular grid geometry
Hexmesh	Hexahedral mesh geometry
ImplicitField	Field defined from an arbitrary function over a domain.
Integrand	An integrand is a device function containing arbitrary expressions over Field and Domain variables.
LinearGeometryPartition
Nanogrid	Sparse grid geometry
NodalQuadrature	Quadrature using space node points as quadrature points
NodeIndex	alias of int
NonconformingField	Field defined as the map of a DiscreteField over a non-conforming geometry.
Operator	Operators provide syntactic sugar over Field and Domain evaluation functions and arguments
PicQuadrature	Particle-based quadrature formula, using a global set of points unevenly spread out over geometry elements.
PointBasisSpace
Polynomial	Polynomial family defining interpolation nodes over an interval
Quadmesh2D	Two-dimensional quadrilateral mesh
Quadmesh3D	Three-dimensional quadrilateral mesh
Quadrature	Interface class for quadrature rules
QuadraturePointIndex	alias of int
RegularQuadrature	Regular quadrature formula, using a constant set of quadrature points per element
Sample	Per-sample point context for evaluating fields and related operators in integrands
ShapeBasisSpace	Base class for defining shape-function-based basis spaces.
ShapeFunction	Interface class for defining scalar-valued shape functions over a single element
Sides	A Domain containing all (interior and boundary) sides of the geometry or geometry partition
SkewSymmetricTensorMapper	Orthonormal isomorphism from R^{n (n-1)} to nxn skew-symmetric tensors, using usual L2 norm for vectors and half Frobenius norm, (tau : tau)/2 for tensors.
SpacePartition
SpaceRestriction	Restriction of a space partition to a given GeometryDomain
SpaceTopology	Interface class for defining the topology of a function space.
Subdomain	Subdomain -- restriction of domain to a subset of its elements
SymmetricTensorMapper	Orthonormal isomorphism from R^{n (n+1)} to nxn symmetric tensors, using usual L2 norm for vectors and half Frobenius norm, (tau : tau)/2 for tensors.
Temporary	alias of array
TemporaryStore	Shared pool of temporary arrays that will be persisted and reused across invocations of warp.fem functions.
Tetmesh	Tetrahedral mesh geometry
Trimesh2D	2D Triangular mesh geometry
Trimesh3D	3D Triangular mesh geometry
UniformField	Field defined as a constant value over a domain.
D	Symmetric part of the (inner) gradient of the field at s
adaptive_nanogrid_from_field	Constructs a warp.fem.AdaptiveNanogrid from a coarse grid and a refinement field.
adaptive_nanogrid_from_hierarchy	Constructs a warp.fem.AdaptiveNanogrid from a non-overlapping grid hierarchy.
at_node	Returns a copy of the Sample s moved to the coordinates of a local node of the field f.
average	Average between inner and outer element values
borrow_temporary	Borrows and returns a temporary array with specified attributes from a shared pool.
borrow_temporary_like	Borrows and returns a temporary array with the same attributes as another array or temporary.
cells	Converts a domain defined on geometry sides to a domain defined of cells.
curl	Skew part of the (inner) gradient of the field at s, as a vector such that wp.cross(curl(u), v) = skew(grad(u)) v
deformation_gradient	Evaluates the gradient of the domain position with respect to the element reference space at the sample point s
degree	Polynomial degree of a field
div	Evaluates the field divergence at a sample point s.
div_outer	Evaluates the field divergence at a sample point s.
element_closest_point	Computes the coordinates of the closest point to a world position within a given element.
element_coordinates	Returns the coordinates in an element reference system corresponding to a work position.
element_index	Returns the index in the geometry of the domain_element_index'th domain element.
element_partition_index	Returns the index of the passed cell in the domain's geometry partition, or NULL_ELEMENT_INDEX if not part of the partition.
grad	Evaluates the field gradient at a sample point s.
grad_average	Average between inner and outer element gradients
grad_jump	Jump between inner and outer element gradients on an interior side.
grad_outer	Evaluates the field gradient at a sample point s.
inner	Evaluates the field at a sample point s.
integrand	Decorator for functions to be integrated (or interpolated) using warp.fem
integrate	Integrates a constant, linear or bilinear form, and returns a scalar, array, or sparse matrix, respectively.
interpolate	Interpolates a function at a finite set of sample points and optionally assigns the result to a discrete field, raw warp array, or sparse matrix.
jump	Jump between inner and outer element values on an interior side.
lookup	Looks-up the sample point corresponding to a world position x, projecting to the closest point on the geometry.
make_collocated_function_space	Constructs a function space from a scalar-valued basis space and a value type, such that all degrees of freedom of the value type are stored at each of the basis nodes.
make_contravariant_function_space	Constructs a contravariant function space from a vector-valued basis space
make_covariant_function_space	Constructs a covariant function space from a vector-valued basis space
make_discrete_field	Constructs a zero-initialized discrete field over a function space or partition
make_element_based_space_topology	Makes a space topology from a geometry and an element-based shape function.
make_element_shape_function	Equips a reference element with a shape function basis.
make_free_sample	Returns a Sample that is not associated to any quadrature point or dof
make_polynomial_basis_space	Equips a geometry with a polynomial basis.
make_polynomial_space	Equips a geometry with a collocated, polynomial function space.
make_restriction	Restricts a discrete field to a subset of elements.
make_space_partition	Computes the subset of nodes from a function space topology that touch a geometry partition
make_space_restriction	Restricts a function space partition to a Domain, i.e. a subset of its elements.
make_test	Constructs a test field over a function space or its restriction
make_trial	Constructs a trial field over a function space or partition
measure	Returns the measure (volume, area, or length) determinant of an element at a sample point s
measure_ratio	Returns the maximum ratio between the measure of this element and that of higher-dimensional neighbors.
node_count	Returns the number of nodes associated to the field f in the element containing the sample s
node_index	Returns the index in the function space of a local node of the field f.
node_inner_weight	Returns the inner element weight associated to a local node of the field f at the sample point s.
node_inner_weight_gradient	Returns the gradient (w.r.t world coordinates) of the inner element weight associated to a local node of the field f at the sample point s.
node_outer_weight	Returns the outer element weight associated to a local node of the field f at the sample point s.
node_outer_weight_gradient	Returns the gradient (w.r.t world coordinates) of the outer element weight associated to a local node of the field f at the sample point s.
node_partition_index	For a NodalField f, returns the index of a given node in the fields's space partition, or NULL_NODE_INDEX if it does not exists
normal	Evaluates the element normal at the sample point s.
normalize_dirichlet_projector	Scale projector so that it becomes idempotent, and apply the same scaling to fixed_value if provided
outer	Evaluates the field at a sample point s.
partition_lookup	Looks-up the sample point corresponding to a world position x, projecting to the closest point on the geometry partition.
position	Evaluates the world position of the sample point s
project_linear_system	Projects both the left-hand-side and right-hand-side of a linear system to enforce Dirichlet boundary conditions
project_system_matrix	Projects the right-hand-side of a linear system to enforce Dirichlet boundary conditions
project_system_rhs	Projects the right-hand-side of a linear system to enforce Dirichlet boundary conditions
set_default_temporary_store	Globally sets the default TemporaryStore instance to use for temporary allocations in warp.fem functions.
to_cell_side	Converts a Sample defined on a cell to a sample defined on one of its side.
to_inner_cell	Converts a Sample defined on a side to a sample defined on the side's inner cell
to_outer_cell	Converts a Sample defined on a side to a sample defined on the side's outer cell
NULL_ELEMENT_INDEX
NULL_NODE_INDEX
NULL_QP_INDEX
OUTSIDE