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 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

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

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.