Revisiting numerical data and visualization requirements

 

Recent numerical methods deliver a large amount of data on a lot of different mesh types. The following list shows some of these types.

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Structured or unstructured meshes (from Finite Difference or Finite Element / Volume methods) with a variety of ansatz functions on them.

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Adaptive meshes consisting of a single or of mixed element types, e. g. simplicial, prismatic, rectangular or cuboidal structure or of more general kind (CSG modeling of surfaces with smooth corners naturally leads to triangles or pentagons, e. g.).

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Conforming or non-conforming meshes, where the neighbourhood of elements across an element side is not one-to-one (for example, locally refined rectangular meshes lead naturally to non-conforming meshes).

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Meshes with parametric (curved) elements of any of the above mentioned types, with globally constant or locally changing polynomial order of parameterization.

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Hierarchically structured meshes suitable for efficient numerical solvers (which give the possibility to choose a coarse hierarchy level for interactivity of time-consuming operations).

Finite Element / Finite Volume data is usually given by coefficients to ansatz functions with local support and can easily be evaluated in local coordinates. Therefore a function f on a specific element e of such a mesh can easily be evaluated by a call like f(e,c) where c is the local coordinate vector. For example, for piecewise linear data on triangles, the barycentric coordinates of a point in the triangle are equal to the values of the nodal basis functions. On the other hand, a piecewise polynomial interpolation of Finite Difference data, which are given only at the nodal points of a regular grid, can easily be evaluated using local coordinates as parameters.

As we want to handle `general' elements, we use the following notion of local coordinates that belong to an element: For each element, we specify the dimension of the local coordinate system and the coordinates of all vertices of the element in that local system.

Usually, an n-dimensional element can be parameterized by an n-dimensional local coordinate system, but there are situations where it is more intuitive or appropriate to use a higher dimensional local system. Examples of this are:

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Usage of the full (n+1)-dimensional barycentric coordinate system on an n-dimensional simplex. If are the vertices of the simplex, there is a unique representation of every point P of the simplex as with and .

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Smooth corners in a surface built out of rectangular patches may be naturally described as triangles with 3 local coordinates or pentagons with 5 local coordinates.

As a result, we have to permit local coordinate systems of arbitrary dimension. It is natural to assume that the underlying coordinate space is a standard Euclidian vector space. The following figure shows some examples of possible local coordinate systems for distinct elements. These elements are given as the following range of local coordinates: