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 for 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	Geometry partition defined by an explicit cell selection mask.
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	Geometry partition that uniformly divides cells by index range.
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	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	Unstructured basis space with radial basis kernel functions.
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	Partition of a function space over a geometry partition.
SpaceRestriction	Restriction of a space partition to a given GeometryDomain.
SpaceTopology	Interface class for defining the topology of a function space.
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	Return 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	Borrow and return a temporary array with specified attributes from a shared pool.
borrow_temporary_like	Borrow and return a temporary array with the same attributes as another array or temporary.
cells	Convert 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	Evaluate 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	Evaluate the field divergence at a sample point s.
div_outer	Evaluate the field divergence at a sample point s.
element_closest_point	Compute the coordinates of the closest point to a world position within a given element.
element_coordinates	Return the coordinates in an element reference system corresponding to a work position.
element_index	Return the index in the geometry of the domain_element_index'th domain element.
element_partition_index	Return the index of the passed cell in the domain's geometry partition, or NULL_ELEMENT_INDEX if not part of the partition.
grad	Evaluate 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	Evaluate the field gradient at a sample point s.
inner	Evaluate 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	Look 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	Equip a reference element with a shape function basis.
make_free_sample	Return 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	Compute 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	Return the measure (volume, area, or length) determinant of an element at a sample point s.
measure_ratio	Return the maximum ratio between the measure of this element and that of higher-dimensional neighbors.
node_count	Return the number of nodes associated to the field f in the element containing the sample s.
node_index	Return the index in the function space of a local node of the field f.
node_inner_weight	Return the inner element weight associated to a local node of the field f at the sample point s.
node_inner_weight_gradient	Return 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	Return the outer element weight associated to a local node of the field f at the sample point s.
node_outer_weight_gradient	Return 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	Evaluate 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	Evaluate the field at a sample point s.
partition_lookup	Look up the sample point corresponding to a world position x, projecting to the closest point on the geometry partition.
position	Evaluate 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	Convert a Sample defined on a cell to a sample defined on one of its side.
to_inner_cell	Convert a Sample defined on a side to a sample defined on the side's inner cell.
to_outer_cell	Convert a Sample defined on a side to a sample defined on the side's outer cell.
NULL_ELEMENT_INDEX	Constant indicating an invalid element index.
NULL_NODE_INDEX	Constant indicating an invalid node index.
NULL_QP_INDEX	Constant indicating an invalid quadrature point index.
OUTSIDE	Constant indicating an invalid element coordinate.