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.

• 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

aFbFdxxfb

a

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

b

aaFbFdxxf

drFdF

VVndSFdVF

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.