The union of all elements defines the domain of the PDE.
Each element is defined by the nodes used to describe its shape. In Figure 7.1 the element,
which has type Tri3,
with element reference number is defined by the nodes
with reference numbers
,
and 0 . Notice that the order is counterclockwise.
The coefficients of the PDE are evaluated at integration nodes with each individual element.
For quadrilateral elements a Gauss quadrature scheme is used. In the case of triangular elements a
modified form is applied. The boundary of the domain is also subdivided into elements. In Figure 7.1
line elements with two nodes are used. The elements are also defined by their describing nodes, e.g.
the face element reference number
which has type Line2 is defined by the nodes
with the reference numbers
and 0. Again the order is crucial, if moving from the first
to second node the domain has to lie on the left hand side (in the case of a two dimension surface element
the domain has to lie on the left hand side when moving counterclockwise). If the gradient on the
surface of the domain is to be calculated rich face elements face to be used. Rich elements on a face
are identical to interior elements but with a modified order of nodes such that the 'first' face of the element aligns
with the surface of the domain. In Figure 7.1
elements of the type Tri3Face are used.
The face element reference number
as a rich face element is defined by the nodes
with reference numbers
, 0 and
. Notice that the face element
is identical to the
interior element
except that, in this case, the order of the node is different to align the first
edge of the triangle (which is the edge starting with the first node) with the boundary of the domain.
Be aware that face elements and elements in the interior of the domain must match, i.e. a face element must be the face of an interior element or, in case of a rich face element, it must be identical to an interior element. If no face elements are specified esys.finley implicitly assumes homogeneous natural boundary conditions , i.e. d=0 and y=0, on the entire boundary of the domain. For inhomogeneous natural boundary conditions , the boundary must be described by face elements.
If discontinuities of the PDE solution are considered contact elements
are introduced to describe the contact region
even if
and
are zero. Figure 7.2 shows a simple example of a mesh
of rectangular elements around a contact region
.
The contact region is described by the
elements
,
and
. Their element type is Line2_Contact.
The nodes
,
,
,
define contact element
, where the coordinates of nodes
and
and
nodes
and
are identical with the idea that nodes
and
are located above and
nodes
and
below the contact region.
Again, the order of the nodes within an element is crucial. There is also the option of using rich elements
if the gradient is to be calculated on the contact region. Similarly to the rich face elements
these are constructed from two interior elements by reordering the nodes such that
the 'first' face of the element above and the 'first' face of the element below the
contact regions line up. The rich version of element
is of type Rec4Face_Contact and is defined by the nodes
,
,
,
,
,
, 0 and
.
Table 7.1 shows the interior element types and the corresponding element types to be used on the face and contacts. Figure 7.3, Figure 7.4 and Figure 7.5 show the ordering of the nodes within an element.
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The native esys.finley file format is defined as follows.
Each node i has dim spatial coordinates Node[i], a reference number
Node_ref[i], a degree of freedom Node_DOF[i] and tag Node_tag[i].
In most cases Node_DOF[i]=Node_ref[i] however, for periodic boundary conditions,
Node_DOF[i] is chosen differently, see example below. The tag can be used to mark nodes sharing
the same properties. Element i is defined by the Element_numNodes nodes Element_Nodes[i]
which is a list of node reference numbers. The order is crucial.
It has a reference number Element_ref[i] and a tag Element_tag[i]. The tag
can be used to mark elements sharing the same properties. For instance elements above
a contact region are marked with and elements below a contact region are marked with
.
Element_Type and Element_Num give the element type and the number of elements in the mesh.
Analogue notations are used for face and contact elements. The following Python script
prints the mesh definition in the esys.finley file format:
The following example of a mesh file defines the mesh shown in Figure 7.2:
Example 1 2D Nodes 16 0 0 0 0. 0. 2 2 0 0.33 0. 3 3 0 0.66 0. 7 4 0 1. 0. 5 5 0 0. 0.5 6 6 0 0.33 0.5 8 8 0 0.66 0.5 10 10 0 1.0 0.5 12 12 0 0. 0.5 9 9 0 0.33 0.5 13 13 0 0.66 0.5 15 15 0 1.0 0.5 16 16 0 0. 1.0 18 18 0 0.33 1.0 19 19 0 0.66 1.0 20 20 0 1.0 1.0 Rec4 6 0 1 0 2 6 5 1 1 2 3 8 6 2 1 3 7 10 8 5 2 12 9 18 16 7 2 13 19 18 9 10 2 20 19 13 15 Line2 0 Line2_Contact 3 4 0 9 12 6 5 3 0 13 9 8 6 6 0 15 13 10 8 Point1 0Notice that the order in which the nodes and elements are given is arbitrary. In the case that rich contact elements are used the contact element section gets the form
Rec4Face_Contact 3 4 0 9 12 16 18 6 5 0 2 3 0 13 9 18 19 8 6 2 3 6 0 15 13 19 20 10 8 3 7Periodic boundary condition can be introduced by altering Node_DOF. It allows identification of nodes even if they have different physical locations. For instance, to enforce periodic boundary conditions at the face
2D Nodes 16 0 0 0 0. 0. 2 2 0 0.33 0. 3 3 0 0.66 0. 7 0 0 1. 0. 5 5 0 0. 0.5 6 6 0 0.33 0.5 8 8 0 0.66 0.5 10 5 0 1.0 0.5 12 12 0 0. 0.5 9 9 0 0.33 0.5 13 13 0 0.66 0.5 15 12 0 1.0 0.5 16 16 0 0. 1.0 18 18 0 0.33 1.0 19 19 0 0.66 1.0 20 16 0 1.0 1.0
(0,0) (0,0)2 (1,0)(1,0)28 (30,0)2
(0,0) (0,0)2 (1,0)(1,0)28 (30,0)2
(0,30) (0,0)2 (0,1)(0,1)28 (0,30)2
(0,30) (0,0)2 (0,1)(0,1)28 (0,30)2
(0,30) (0,0)(-1,1)30 (0,0)2 (-30,30)2
(0,15) (0,0)(-4,3)20 (0,0)2 (-20,15)2
(0,-15) (0,0)(-4,-3)20 (0,0)2 (-20,-15)2
(0,-15) (-0.7,-0.7)(-4,-3)18.7 (0,0)2 (-20,-15)2
(0,15) (0,0)(-2,3)10 (0,0)2 (-10,15)2
(0,15) (0,0)(-2,3)9.4 (0,0)2 (-10,15)2
(0,0) (0,0)2 (10,0)2 (1,0)(1,0)28 (20,0)2 (30,0)2
(0,0) (0,0)2 (10,0)2 (1,0)(10,0)3(1,0)8 (20,0)2 (30,0)2
(0,30) (0,0)2 (0,10)2 (0,1)(0,1)28 (0,20)2 (0,30)2
(0,30) (0,0)2 (0,10)2 (0,1)(0,10)3(0,1)8 (0,20)2 (0,30)2
(0,30) (0,0)(-1,1)30 (0,0)2 (-10,10)2 (-20,20)2 (-30,30)2
(0,15) (0,0)(-4,3)20 (0,0)2 (-6.66,5)2 (-13.33,10)2 (-20,15)2
(0,-15) (0,0)(-4,-3)20 (0,0)2 (-6.66,-5)2 (-13.33,-10)2 (-20,-15)2
(0,-15) (-0.7,-0.7)(-6.66,-5)3(-4,-3)5.1 (0,0)2 (-6.66,-5)2 (-13.33,-10)2 (-20,-15)2
(0,15) (0,0)(-2,3)10 (0,0)2 (-3.33,5)2 (-6.66,10)2 (-10,15)2
(0,15) (-0.6,0.8)(-3.33,5)3(-2,3)2.35 (0,0)2 (-3.33,5)2 (-6.66,10)2 (-10,15)2
(0,0) (0,0)2 (15,0)2 (1,0)(1,0)28 (30,0)2
(0,0) (0,0)2 (15,0)2 (1,0)(15,0)2(1,0)13 (30,0)2
(0,30) (0,0)2 (0,15)2 (0,1)(0,1)28 (0,30)2
(0,30) (0,0)2 (0,15)2 (0,1)(0,15)2(0,1)13 (0,30)2
(0,30) (0,0)(-1,1)30 (0,0)2 (-15,15)2 (-30,30)2
(0,15) (0,0)(-4,3)20 (0,0)2 (-10,7.5)2 (-20,15)2
(0,-15) (0,0)(-4,-3)20 (0,0)2 (-10,-7.5)2 (-20,-15)2
(0,-15) (-0.7,-0.7)(-10,-7.5)2(-4,-3)8.4 (0,0)2 (-10,-7.5)2 (-20,-15)2
(0,15) (0,0)(-2,3)10 (0,0)2 (-5,7.5)2 (-10,15)2
(0,15) (-0.6,0.8)(-5,7.5)2(-2,3)3.9 (0,0)2 (-5,7.5)2 (-10,15)2
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