# Discrete exterior calculus

## API docs

CombinatorialSpaces.DiscreteExteriorCalculusModule

The discrete exterior calculus (DEC) for simplicial sets.

This module provides the dual complex associated with a delta set (the primal complex), which is a discrete incarnation of Hodge duality, as well as the many operators of the DEC that depend on it, such as the Hodge star, codifferential, wedge product, interior product, and Lie derivative. The main reference for this module is Hirani's 2003 PhD thesis.

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CombinatorialSpaces.DiscreteExteriorCalculus.DualChainType

Wrapper for chain of dual cells of dimension n.

In an $N$-dimensional complex, the elementary dual simplices of each $n$-simplex together comprise the dual $(N-n)$-cell of the simplex. Using this correspondence, a basis for primal $n$-chains defines the basis for dual $(N-n)$-chains.

Note

In (Hirani 2003, Definition 3.4.1), the duality operator assigns a certain sign to each elementary dual simplex. For us, all of these signs should be regarded as positive because we have already incorporated them into the orientation of the dual simplices.

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Catlab.Theories.ΔMethod

Laplace-de Rham operator on discrete forms.

This linear operator on primal $n$-forms is defined by $Δ := δ d + d δ$. Restricted to 0-forms, it reduces to the negative of the Laplace-Beltrami operator ∇²: $Δ f = -∇² f$.

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CombinatorialSpaces.DiscreteExteriorCalculus.:∧Method

Wedge product of discrete forms.

The wedge product of a $k$-form and an $l$-form is a $(k+l)$-form.

The DEC and related systems have several flavors of wedge product. This one is the discrete primal-primal wedge product introduced in (Hirani, 2003, Chapter 7) and (Desbrun et al 2005, Section 8). It depends on the geometric embedding and requires the dual complex.

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CombinatorialSpaces.DiscreteExteriorCalculus.:⋆Method

Hodge star operator from primal 1-forms to dual 1-forms.

This specific hodge star implementation is based on the hodge star presented in (Ayoub et al 2020), which generalizes the operator presented in (Hirani 2003). This reproduces the diagonal hodge for a dual mesh generated under circumcentric subdivision and provides off-diagonal correction factors for meshes generated under other subdivision schemes (e.g. barycentric).

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CombinatorialSpaces.DiscreteExteriorCalculus.:⋆Method

Hodge star operator from primal $n$-forms to dual $N-n$-forms.

Note

Some authors, such as (Hirani 2003) and (Desbrun 2005), use the symbol $⋆$ for the duality operator on chains and the symbol $*$ for the Hodge star operator on cochains. We do not explicitly define the duality operator and we use the symbol $⋆$ for the Hodge star.

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CombinatorialSpaces.DiscreteExteriorCalculus.elementary_dualsMethod

List of elementary dual simplices corresponding to primal simplex.

In general, in an $n$-dimensional complex, the elementary duals of primal $k$-simplices are dual $(n-k)$-simplices. Thus, in 1D dual complexes, the elementary duals of...

• primal vertices are dual edges
• primal edges are (single) dual vertices

In 2D dual complexes, the elementary duals of...

• primal vertices are dual triangles
• primal edges are dual edges
• primal triangles are (single) dual triangles
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CombinatorialSpaces.DiscreteExteriorCalculus.interior_productMethod

Interior product of a vector field (or 1-form) and a $n$-form.

Specifically, this operation is the primal-dual interior product defined in (Hirani 2003, Section 8.2) and (Desbrun et al 2005, Section 10). Thus it takes a primal vector field (or primal 1-form) and a dual $n$-forms and then returns a dual $(n-1)$-form.

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CombinatorialSpaces.DiscreteExteriorCalculus.∇²Method

Laplace-Beltrami operator on discrete forms.

This linear operator on primal $n$-forms defined by $∇² α := -δ d α$, where δ is the codifferential and d is the exterior derivative.

Note

For following texts such as Abraham-Marsden-Ratiu, we take the sign convention that makes the Laplace-Beltrami operator consistent with the Euclidean Laplace operator (the divergence of the gradient). Other authors, such as (Hirani 2003), take the opposite convention, which has the advantage of being consistent with the Laplace-de Rham operator Δ.

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CombinatorialSpaces.DiscreteExteriorCalculus.♯Method

Sharp operator for converting 1-forms to vector fields.

A generic function for discrete sharp operators. Currently only the primal-primal flat from (Hirani 2003, Definition 5.8.1 and Remark 2.7.2) is implemented.

Note

A PP-flat is also defined in (Desbrun et al 2005, Definition 7.4) but differs in two ways: Desbrun et al's notation suggests a unit normal vector, whereas the gradient of Hirani's primal-primal interpolation function is not necessarily a unit vector. More importantly, Hirani's vector is a normal to a different face than Desbrun et al's, with further confusion created by the fact that Hirani's Figure 5.7 agrees with Desbrun et al's description rather than his own. That being said, to the best of our knowledge, our implementation is the correct one and agrees with Hirani's description, if not his figure.

See also: the flat operator ♭.

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