3D Continuum Elements

2.8 3D Continuum Elements

This section contains description of continuum elements.

2.8.1 LSpace element

Implementation of Linear 3d eight - node finite element. Each node has 3 degrees of freedom. The element features are summarized in Table 38.

Figure 16: LSpace element (Node numbers in black, side numbers in blue, and surface numbers in red).


Keyword	lspace
Description	Linear isoparametric brick element
Specific parameters	[NIP #(in)]

Parameters	NIP: allows to set the number of integration points (possible completions are 1, 8 (default), or 27).
Unknowns	Three dofs (u-displacement, v-displacement, w-displacement) are required in each node.
Approximation	Linear approximation of displacement and geometry.
Integration	Full integration of all strain components.
Features	Supports adaptivity, geometric nonlinearity, and layered cross section support
CS properties	-
Loads	-
Nlgeo	0,1,2.
Status	Reliable
Tests/Examples	tests/sm/patch302.in, tests/sm/compoDamMat.in, tests/sm/deadweight02.in

Table 38: lspace element summary

2.8.2 LSpaceBB element

Implementation of 3d brick eight - node linear approximation element with selective integration of deviatoric and volumetric strain contributions (B-bar formulation) for incompressible problems. Features and description identical to conventional lspace element, see section 2.8.1.

2.8.3 QSpace element

Implementation of quadratic 3d 20-node finite element. Each node has 3 degrees of freedom. The element features are summarized in Table 39.

Figure 17: QSpace element.


Keyword	qspace
Description	Quadratic isoparametric brick element
Specific parameters	[NIP #(in)]

Parameters	NIP: allows to set the number of integration points (possible completions are 1, 8 (default), or 27).
Unknowns	Three dofs (u-displacement, v-displacement, w-displacement) are required in each node.
Approximation	Quadratic approximation of displacement and geometry.
Integration	Full integration of all strain components.
Features	Layered cross section support.
CS properties	-
Loads	-
Nlgeo	0,1,2.
Status	Reliable
Tests/Examples	tests/sm/cantilever_Qspace.in, tests/sm/layered_cube.in

Table 39: qspace element summary

2.8.4 LTRSpace element

Implementation of tetrahedra four-node finite element. Each node has 3 degrees of freedom. The element features are summarized in Table 40. Following node numbering convention is adopted (see also Fig. 18):

Select a face that will contain the first three corners. The excluded corner will be the last one.
Number these three corners in a counterclockwise sense when looking at the face from the excluded corner.

Figure 18: LTRSpace element. Definition and node numbering convention.


Keyword	LTRSpace
Description	Linear tetrahedra element
Specific parameters	-

Unknowns	Three dofs (u-displacement, v-displacement, w-displacement) are required in each node.
Approximation	Linear approximation of displacements and geometry using linear volume coordinates.
Integration	Full integration of all strain components using four point Gauss integration formula.
Features	Adaptivity support, Geometric nonlinearity support.
CS properties	-
Loads	Surface and Edge loadings supported.
Nlgeo	0,1,2.
Status	Reliable
Tests/Examples	tests/sm/patch300.in, tests/sm/patch301.in

Table 40: LTRSpace element summary

2.8.5 QTRSpace element

Implementation of tetrahedra ten-node finite element. Each node has 3 degrees of freedom. The element features are summarized in Table 41. Following node numbering convention is adopted (see also Fig. 19):

Figure 19: QTRSpace element. Definition and node numbering convention.


Keyword	QTRSpace
Description	3D tetrahedra element with quadratic interpolation
Specific parameters	[NIP #(in)]

Parameters	NIP: allows to alter the default integration formula (possible completions are 1, 4 (default), 5, 11, 15, 24, and 45 point intergartion formulas).
Unknowns	Three dofs (u-displacement, v-displacement, w-displacement) are required in each node.
Approximation	Quadratic approximation of displacements and geometry using linear volume coordinates.
Integration	Full integration of all strain components using four point Gauss integration formula.
Features	-
CS properties	-
Loads	-
Nlgeo	0,1,2.
Status	Reliable
Tests/Examples	-

Table 41: QTRSpace element summary

2.8.6 LWedge element

Implementation of wedge six-node finite element. Each node has 3 degrees of freedom. The element features are summarized in Table 42. Following node numbering convention is adopted (see also Fig. 20):

Figure 20: LWedge element. Node numbering convention in black, edge numbering in blue and face numbering in red.


Keyword	LWedge
Description	3D wedge six-node finite element with linear interpolation
Specific parameters	[NIP #(in)]

Parameters	NIP: allows to alter the default integration formula (possible completions are 2 (default) and 9 point integration formulas).
Unknowns	Three dofs (u-displacement, v-displacement, w-displacement) are required in each node.
Approximation	Linear approximation of displacements and geometry.
Integration	Full integration of all strain components using four point Gauss integration formula.
Features	Layered cross section support.
CS properties	-
Loads	-
Nlgeo	0,1,2.
Status	Reliable
Tests/Examples	-

Table 42: LWedge element summary

2.8.7 QWedge element

Implementation of wedge fifteen-node finite element. Each node has 3 degrees of freedom. The element features are summarized in Table 43. Following node numbering convention is adopted (see also Fig. 21):

Figure 21: QWedge element. Node numbering convention in black, edge numbering in blue and face numbering in red.


Keyword	QWedge
Description	3D wedge six-node finite element with quadratic interpolation
Specific parameters	[NIP #(in)]

Parameters	NIP: allows to alter the default integration formula (possible completions are 2 (default) and 9 point integration formulas).
Unknowns	Three dofs (u-displacement, v-displacement, w-displacement) are required in each node.
Approximation	Quadratic approximation of displacements and geometry.
Integration	Full integration of all strain components using four point Gauss integration formula.
Features	Layered cross section support.
CS properties	-
Loads	-
Nlgeo	0,1,2.
Status	Reliable

Table 43: QWedge element summary

2.8.8 Layer stacking sequence definition for 3D elements

Selected 3D elements (bricks and wedge geometries) support using the LayeredCrossSection model to define layer stack as a sequence of individual layers. Individual layers are assumed to lie in element parametric ξ -η plane and are stacked along parametric ζ coordinate of the element. The direction of parametric coordinates is determined by element node numbering convention, see figures with element geometries above. Note, that the stacking direction is in general the function of element geometry.

It is important to understand concept of element and material coordinate systems.

The element coordinate system (elemCS) coincides, by default, with the global coordinate system. The user-defined element coordinate system can be defined using lcs parameter. The lcs parameter defines an array of size 6, where the first 3 components define direction of local element-axis and remaining 3 components define direction of local element y-axis. The local element z axis is computed using vector product e_z = e_x × e_y.

The material properties of each layer are defined in material coordinate system (matCS). Also the solver output for individual layers is done in matCS. By default, the material coordinate system coincides with global coordinate system. Additionally, the material coordinate system for individual layer can be rotated around material CS z-axis by angle, defined by layered cross section rotations keyword. This array parameter allows to define rotation angle for individual layers and should be defined in degrees not radians. If matcs element keyword is present, but no lcs element record is defined, then the following definition of elemCS is assumed: e_x = { dx(ξ,η,ζ)-
dξ , dy(ξ,η,ζ)
dξ , dz(ξ,η,ζ)-
dξ },h = { dx(ξ,η,ζ)
dη , dy(ξ,η,ζ)-
dη , dz(ξ,η,ζ)
dη },e_z = e_x × h,e_y = e_z × e_x, where ξ,η,ζ are parametric element coordinates. The strains and stresses in individual layers are always reported in material coordinate system. LayeredCS integration The layered cross section integration can be set up using number of integrations points in layer plane (nintegrationpoints parameter ) and using number of integration points per layer thickness (layerintegrationpoints parameter).

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