Discrete Exterior Calculus. More Complete Introduction See Chapter 7 “Discrete Differential Forms...
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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.
• 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.
• 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.
• 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
• 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.
• 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.
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.
• 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
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
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.