Neighborhoods

The Neighborhood concept represents the differential geometric notion of the derivative of a chart in a point of its parameter domain. Not only can it be queried for the parametric and Cartesian coordinates of the input and output point respectively, it also contains information on tangents and Jacobian determinants. For the special case where the dimension of the range manifold is one less than that of the surrounding space, access is provided to the unit normal.

This concept is an enriched point concept and should allow the construction of the majority of kernels encountered in finite and boundary element methods.

CompScienceMeshes.parametric — Function

parametric(neighborhood) -> point

Return the parameters where neighborhood is located.

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CompScienceMeshes.cartesian — Function

cartesian(neighborhood) -> point

Return the cartesian coordinates of the point where neighborhood is located.

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CompScienceMeshes.jacobian — Function

A number defines a neighborhood in euclidian space

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CompScienceMeshes.tangents — Function

tangents(neighborhod, i) -> tangent_i

Return the i-th tangent vector at the neighborhood.

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CompScienceMeshes.normal — Function

normal(neighborhood)

Return the normal at a neighborhood on a surface.

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In addition to this interface, a neighborhood has appropriate methods for getindex defined, so that it can be used with any function that expects a tuple of coordinates. In other words, a neighborhood is a model for Point.

Numerical Quadrature

This package provides a number of routines that aim to facilitate numerical quadrature over charts. In addition, it includes a default set of quadrature rules for segments (1D) and triangles (2D).

CompScienceMeshes.quadpoints — Function

quadpoints(refspace, charts, rules)

Computed a matrix of vectors containing (weight, point, value) triples that can be used in numerical integration over the elements described by the charts. Internally, this method used quadpoints(chart, rule) to retrieve the points and weights for a certain quadrature rule over chart.

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