naginterfaces.library.mesh.dim2_​renumber

naginterfaces.library.mesh.dim2_renumber(nnzmax, coor, edge, conn, itrace, io_manager=None)

dim2_renumber renumbers the vertices of a given mesh using a Gibbs method, in order the reduce the bandwidth of Finite Element matrices associated with that mesh.

For full information please refer to the NAG Library document for d06cc

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/d06/d06ccf.html

Parameters

nnzmaxint

The maximum number of nonzero entries in the matrix based on the input mesh. It is the dimension of the arrays $irow$ and $icol$ as declared in the function from which dim2_renumber is called.

coorfloat, array-like, shape $(2,\text{nv})$

$coor[0,\text{i}-1]$ contains the $x$ coordinate of the $\text{i}$th input mesh vertex, for $\text{i}=1,\dots ,\text{nv}$; while $coor[1,\text{i}-1]$ contains the corresponding $y$ coordinate.

edgeint, array-like, shape $(3,\text{nedge})$

The specification of the boundary or interface edges. $edge[0,j-1]$ and $edge[1,j-1]$ contain the vertex numbers of the two end points of the $j$th boundary edge. $edge[2,j-1]$ is a user-supplied tag for the $j$th boundary or interface edge: $edge[2,j-1]=0$ for an interior edge and has a nonzero tag otherwise.

connint, array-like, shape $(3,\text{nelt})$

The connectivity of the mesh between triangles and vertices. For each triangle $\text{j}$, $conn[\text{i}-1,\text{j}-1]$ gives the indices of its three vertices (in anticlockwise order), for $\text{j}=1,2,\dots ,\text{nelt}$, for $\text{i}=1,2,\dots ,3$.

itraceint

The level of trace information required from dim2_renumber.

$itrace\le 0$

No output is generated.

$itrace=1$

Information about the effect of the renumbering on the finite element matrix are output. This information includes the half bandwidth and the sparsity structure of this matrix before and after renumbering.

$itrace>1$

The output is similar to that produced when $itrace=1$ but the sparsities (for each row of the matrix, indices of nonzero entries) of the matrix before and after renumbering are also output.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

nnzint

The number of nonzero entries in the matrix based on the input mesh.

coorfloat, ndarray, shape $(2,\text{nv})$

$coor[0,\text{i}-1]$ will contain the $x$ coordinate of the $\text{i}$th renumbered mesh vertex, for $\text{i}=1,\dots ,\text{nv}$; while $coor[1,\text{i}-1]$ will contain the corresponding $y$ coordinate.

edgeint, ndarray, shape $(3,\text{nedge})$

The renumbered specification of the boundary or interface edges.

connint, ndarray, shape $(3,\text{nelt})$

The renumbered connectivity of the mesh between triangles and vertices.

irowint, ndarray, shape $\left(nnzmax\right)$

The first $nnz$ elements contain the row and column indices of the nonzero elements supplied in the finite element matrix $A$.

icolint, ndarray, shape $\left(nnzmax\right)$

The first $nnz$ elements contain the row and column indices of the nonzero elements supplied in the finite element matrix $A$.

Raises

NagValueError

(errno $1$)

On entry, $\text{nv}=⟨value⟩$.

Constraint: $\text{nv}\ge 3$.

(errno $1$)

On entry, $\text{nedge}=⟨value⟩$.

Constraint: $\text{nedge}\ge 1$.

(errno $1$)

On entry, $edge(\text{I},\text{J})=⟨value⟩$, $\text{I}=⟨value⟩$, $\text{J}=⟨value⟩$ and $\text{nv}=⟨value⟩$.

Constraint: $edge(\text{I},\text{J})\ge 1$ and $edge(\text{I},\text{J})\le \text{nv}$.

(errno $1$)

On entry, $\text{nelt}=⟨value⟩$ and $\text{nv}=⟨value⟩$.

Constraint: $\text{nelt}\le 2\times \text{nv}-1$.

(errno $1$)

On entry, $conn(\text{I},\text{J})=⟨value⟩$, $\text{I}=⟨value⟩$, $\text{J}=⟨value⟩$ and $\text{nv}=⟨value⟩$.

Constraint: $conn(\text{I},\text{J})\ge 1$ and $conn(\text{I},\text{J})\le \text{nv}$.

(errno $1$)

On entry, $nnzmax=⟨value⟩$, $\text{nelt}=⟨value⟩$ and $\text{nv}=⟨value⟩$.

Constraint: $nnzmax\ge (4\times \text{nelt}+\text{nv})$ and $nnzmax\le {\text{nv}}^2$.

(errno $1$)

On entry, the end points of the edge $\text{J}$ have the same index $\text{I}$: $\text{J}=⟨value⟩$ and $\text{I}=⟨value⟩$.

(errno $1$)

On entry, vertices $2$ and $3$ of the triangle $\text{K}$ have the same index $\text{I}$: $\text{K}=⟨value⟩$ and $\text{I}=⟨value⟩$.

(errno $1$)

On entry, vertices $1$ and $3$ of the triangle $\text{K}$ have the same index $\text{I}$: $\text{K}=⟨value⟩$ and $\text{I}=⟨value⟩$.

(errno $1$)

On entry, vertices $1$ and $2$ of the triangle $\text{K}$ have the same index $\text{I}$: $\text{K}=⟨value⟩$ and $\text{I}=⟨value⟩$.

(errno $2$)

An error has occurred during the computation of the compact sparsity of the finite element matrix. Check the Triangle/Vertices connectivity.

(errno $3$)

A serious error has occurred in an internal call to the renumbering function. Check the input mesh especially the connectivity. Seek expert help.

Notes

dim2_renumber uses a Gibbs method to renumber the vertices of a given mesh in order to reduce the bandwidth of the associated finite element matrix $A$. This matrix has elements $a_{ij}$ such that:

$$a_{ij}\ne 0\Rightarrow i\text{and}j\text{are vertices belonging to the same triangle.}$$

This function reduces the bandwidth $m$, which is the smallest integer such that $a_{ij}\ne 0$ whenever $|i-j|>m$ (see Gibbs et al. (1976) for details about that method). dim2_renumber also returns the sparsity structure of the matrix associated with the renumbered mesh.

This function is derived from material in the MODULEF package from INRIA (Institut National de Recherche en Informatique et Automatique).

References

Gibbs, N E, Poole, W G Jr and Stockmeyer, P K, 1976, An algorithm for reducing the bandwidth and profile of a sparse matrix, SIAM J. Numer. Anal. (13), 236–250