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Discrete Exterior Calculus
More Complete Introduction
• See Chapter 7 “Discrete Differential Forms
for Computational Modeling” in the
SIGGRAPH 2006 Discrete Differential
Geometry Course Notes
• Notes are available from
http://ddg.cs.columbia.edu, or Google
“discrete differential geometry”
• Topology, interpolation, more differential
operators.
Motivation
• We often work on problems where physical or
geometric quantities are defined throughout
space, on a surface, or on a curve.
• When we discretize the geometry, where do we
put those quantities? What do the numbers
mean? How do we integrate and differentiate
them?
• DEC provides a scheme that preserves
important properties and structures from
continuous calculus.
Nutshell: Integration
• We discretize geometry as a simplicial complex
(triangle mesh, tet mesh) with oriented faces,
edges, and vertices.
• Where we store quantities corresponds to how
we integrate them, i.e. we integrate divergence
over volumes so it goes on tets.
• Quanities are “pre-integrated.” The stored
divergence value on a tet is actually an integral
of divergence over the tet.
• Integration over a domain is an oriented sum of
the numbers on the simplices composing the
domain.
Nutshell: Differentiation
• The discrete exterior derivative d maps the oriented sum
of values on the boundary of a simplex to the simplex.
• Think first fundamental theorem of calculus, Gauss’
theorem, Stokes’ theorem.
 f x dx  F b  F a 
b
a
• d is implemented as a matrix of 0, 1, and -1 representing
the incidence of k and k+1 simplices.
• For instance, it might be a matrix with |E| columns and
|F| rows applied to a column vector of |E| values
Nutshell: Dual
• Sometimes we have a quantity defined on simplices of
one dimension that we would like to integrate on
simplices of a different dimension.
• The Hodge star transfers a value from a k-simplex to a
dual n-k simplex, i.e. from an edge to a dual face in 3D.
• The value is transformed based on the difference
between the primal and dual geometry.
• In the notes, the transformation is a scaling based on the
ratio of primal and dual element sizes.
• This can be implemented as a diagonal matrix.
Nutshell: Recap
• Integration is a weighted sum, think dot
product.
• Differentiation goes from boundaries to
simplices, think incidence matrix.
• Hodge star goes from primal to dual
elements with a scaling factor, think
diagonal matrix.
Simplifcial Complexes
Dual Complex
Boundary Operator
What is a form?
• A form is something ready to be integrated,
f(x)dx is a form.
• (Intuitively) A form is an association of a number
with an oriented piece of geometry.
• The dimensionality of the differential or domain
of integration determines the dimensionality of
the form.
• 1-forms for curves, 2-forms for surfaces, etc.
• Discretely, 1-form is values stored on edges, etc.
Examples
d
• Flux lives on faces while divergence lives on tets.
• The sum of the flux over the boundary of a volume
equals the integral of the divergence over the volume.
• Divergence is the discrete exterior derivative of flux.
• d takes the sum of values defined on the boundary of a
simplex and puts it on the simplex
• d is an incidence matrix, the transpose of the boundary
operator
d Does Everything
• There is one d for each dimensionality of
form.
• d for 0-forms is gradient, d for 1-forms is
curl, d for 2-forms is divergence
Structure Preservation

d   

 f x dx  F b  F a 
b
a
   F  d   F  dr

   F dV   F  ndS
V
V
Hodge Star in Action
• A typical 1-form might be based on the dot
product of a vector field with the tangent to a
curve.
• A typical 2-form might be based on the dot
product of a vector field with the normal to a
surface.
• The tangent to an edge is normal to a dual face.
• The Hodge star extracts the dot product from the
integral over the primal edge and reintegrates it
over the dual face.
Circulation and Vorticity
• The flux on a face is the integral over the face of
a dot product of a velocity vector with the normal
to the face.
• The circulation on an edge is the integral over
the edge of the dot product of the velocity vector
with the tangent to the edge.
• The Hodge star of flux is circulation.
• In the paper, take the discrete derivative of
circulation to get vorticity, which is a 2-form.