Geometric element

A mesh consists mainly of a list of nodes and a list of geometric elements. XLiFE++ makes a distinction between two types of geometric element:

• the complete element with all geometric information (MeshElement class)

• the side element that refers to a side of an other geometric element (GeomElement class)

Actually, complete elements are GeomElement objects that handle a MeshElement allocated pointer whereas side elements are GeomElement objects that handle a list of parent elements (a pair of GeomElement* and side number). By default, MeshElement pointer is not allocated for side element, but it can be allocated using the GeomElement::buildSideMeshElement function.

In advanced use, users may have to manipulate GeomElement/MeshElement objects.

Table 3 Main data handled by GeomElement

number_	unique id number of geometric element
meshElement_p	MeshElement pointer (may be null when a side element)
parentSides_	list of parents (vector<GeoNumPair>) if a side element, empty when a mesh element
materialId	material id (Number, default= 0)
domainId	domain id (Number, default= 0)
color	useful to mark an element (Real, default= 0)
theta, phi	local curvatures angles (x-y plane, x-z plane); (Real, default= 0)
twin_p	twin GeomElement in case of GeomElement on an interface
userData_p	a free pointer (void*) allowing users to attach data of any type to the Element
userData_ps	a free void pointers vector (vector<void*>) to attach data of any type to the Element (useful when mutiltithreading)

Note

userData_p and userData_ps are free pointers dedicated to users to do specific calculations that are not managed by XLiFE++; the casting and memory management associated with these pointers are the responsibility of users.

Table 4 Main data handled by MeshElement

nodes	a subset of Mesh::nodes (direct access to nodes, vector<Point*>)
nodeNumbers	a subset of mesh node numbers (direct access to node numbers, vector<Number*>)
vertexNumbers	a subset of mesh node numbers (vector<Number*>, may be equal to nodeNumbers, index starts from 1)
measures	length or area or volume of element and of its sides (vector<Real*>)
size	characteristic size of element (Real, set by computeMeasures)
centroid	centroid of the element (Point, set by computeMeasures)
orientation	element orientation (sign of jacobian determinant, 0 means unset orientation)
refElt_p	pointer to the reference element (RefElement*) attached to the mesh element
geomMapData_p	pointer to a geometric calculation structure (GeomMapData*), handling map from reference element, jacobian,…
linearMap	true if the geometric map \(\varphi_\ell\) from reference element is linear (for instance first order simplicial element)

The GeomElement class collects all the member functions related to geometric element being either a mesh element or a side element.

Table 5 Useful functions attached to GeomElement

number	returns element number as Number
elementDim, spaceDim	return the dimension of the element and the nodes dimansion as Dimen
shapeType(s)	return the shape type of element (s=0) and its sides s>0 (_segment, _triangle, …)
meshElement, isSideElement	return MeshElement* pointer (null if not allocated) and true if it is a side element
buildSideMeshElement	create a MeshElement object when element is a side element (not automatic)
numberOfParents	return number of parents if a side element
parentSide(i), parentSides	return return i-th parent and its side (no check) or access to parentSides_ as a vector<GeoNumPair>
numberOfNodes, numberOfVertices, numberOfSides, numberOfSideOfSides	returns number of nodes, of vertices, of sides, of sides of side of element
vertexNumber(i), nodeNumber(i)	return the global number of vertex/node i (i=1,…)
vertexNumbers(s), nodeNumber(s)	return the global numbers of vertices/nodes if s=0, of side s else (as a vector<Number>)
vertex(i, s), sideOfSideVertex(i, s), edgeVertex(i, e)	return as Point the i-th vertex of element or side, side of side or edge s>=1 (i=1,…)
elementSharingSide(s), elementsSharingVertex(i)	return the adjacent element by side s or vertex i as a GeoNumPair
center, size, measure(s),	return the center of element (as a Point), a characteristic size of element (diameter of the circumscribed ball or max of length of edges), the measure of element if s = 0 or the measure of side s > 0
normalVector(s), tangentVector(s)	return outward normal/tangent vector on side s>0, if s=0 normal/tangent vector to element
contains(pt)	return true if mesh element contains a point pt given as a Point
projection(pt, dist)	projection of a point onto mesh element (return a Point and the distance to element)
print, printVertices or operator <<	display information related to current mesh element

Most of member functions of GeomElement call member functions of MeshElement. But there are specific functions attached to MeshElement:

Table 6 Additional functions attached to MeshElement

refElement(s)	return reference element ( as a RefElement*) related to element if s =0 or related to side element s >0
geomMapData(pt)	build GeomMapData structure at point pt and return a reference to
clearGeomMapDa	clear GeomMapData object
computeMeasures	compute measure of element and its side
computeOrientation	compute orientation of element: sign(det(jacobian))

MeshElement manages a pointer to a GeomMapData object that handles data and tools attached to the map \(\varphi_\ell\) from reference element to geometric element:

Table 7 Main data related to GeomMapData class

geomElement_p	pointer to GeomElement object
currentPoint	point in reference element used for calculation
jacobianMatrix, inverseJacobianMatrix	jacobian matrix of mapping from reference element to current one, inverse jacobian matrix (when needed)
jacobianDeterminant, differentialElement	jacobian determinant and differential element for elementary integrals
normalVector, tangentVector, bitangentVector,	unit outward normal/tangent/bi-tangent vector at a given boundary point
metricTensor, metricTensorDeterminant	symetric metric tensor (t_i|t_j) and metric_tensor determinant

Table 8 Main function related to GeomMapData class

geomMap(pt,*s*)	mapping from ref. elt onto geom. elt (return a Point).
geomMapInverse(pt)	inverse mapping from elt onto ref. elt.
piolaMap	Piola mapping from ref. elt onto geom. elt. Element (shape functions known)
covariantPiolaMap(pt), contravariantPiolaMap(pt)	return the covariant/contraariant Piola map matrix at current point (Matrix<real_t>)
computeJacobianMatrix(pt)	compute jacobian matrix at a point (return a Matrix<real_t>))
computeJacobianDeterminant	compute jacobian determinant (return a real_t))
invertJacobianMatrix	compute inverse of jacobian matrix (jacobian already computed)
computeDifferentialElement	compute differential element assuming jacobian matrix is update
computeNormalVector, computeOrientedNormal	computes (oriented) normal vector at a point of element, assuming jacobian matrix is update
computeTangentVector	compute tangent (and bitangent) vector at currernr point
omputeMetricTensor	compute metric tensor for surface differential geometry
computeSurfaceGradient	compute surface gradient gradS from 2D reference gradient (grad0, grad1)
print and operator <<	display information