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Subsections

7.2.4 Element Types

Regular three-dimensional structural element types for bodies in DIANA can be tetrahedrons, pentahedrons, or hexahedrons. The elements may have linear, quadratic, or cubic interpolation functions for the displacement field. For elements with linear interpolation functions there are two nodes at each edge, whereas for elements with quadratic interpolation functions there are three nodes at each edge, and for elements with cubic interpolation functions the number of nodes along an edge is equal to four. The next paragraphs [§7.2.4.1 to §7.2.4.11] give an overview of the different element types where the numbers display the relative node numbers for the element connectivity. In DIANA the relative node numbers are always defined counterclockwise. The isoparametric coordinates $ \xi$ , $ \eta$ , and $ \zeta$ are also displayed for the different element shapes.


7.2.4.1 TE12L - tetrahedron, 3 sides, 4 nodes

Figure 7.38: TE12L
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The TE12L element [Fig.7.38] is a four-node, three-side isoparametric solid tetrahedron element. It is based on linear interpolation and numerical integration. The polynomials for the translations uxyz can be expressed as

ui($\displaystyle \xi$,$\displaystyle \eta$,$\displaystyle \zeta$) = a0 + a1$\displaystyle \xi$ + a2$\displaystyle \eta$ + a3$\displaystyle \zeta$ (7.38)

These polynomials yield a constant strain and stress distribution over the element volume. By default DIANA applies a 1-point [ nlc = 1 ] integration scheme over the volume, 4- and 5-point are suitable options. Schemes higher than 5-point are not available.

7.2.4.2 PY15L - pyramid, 5 nodes

Figure 7.39: PY15L
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The PY15L element [Fig.7.39] is a five-node isoparametric solid pyramid element as described by the University of Colorado [79]. It is based on linear interpolation and numerical integration. The polynomials for the translations uxyz are based on a isoparametric eight-node brick element were nodes 5 to 8 of the brick element are collapsed to apex 5 of the pyramid element. The polynomials can be expressed as

\begin{equation*}\begin{aligned}u_{1} ( \xi,\eta,\zeta) &= \frac{1}{8}\left(1-\x...
... \xi,\eta,\zeta) &= \frac{1}{2}\left(1+\zeta\right) \end{aligned}\end{equation*}

Typically, a pyramid element approximates the following strain and stress distribution over the element volume. The strain $ \varepsilon_{{xx}}^{}$ and stress $ \sigma_{{xx}}^{}$ are constant in x direction and vary linearly in y and z direction. The strain $ \varepsilon_{{yy}}^{}$ and stress $ \sigma_{{yy}}^{}$ are constant in y direction and vary linearly in x and z direction. The strain $ \varepsilon_{{zz}}^{}$ and stress $ \sigma_{{zz}}^{}$ are constant in z direction and vary linearly in x and y direction. By default DIANA applies a 5-point [ nlc = 5 ] integration scheme over the volume, a 1-point integration scheme is a suitable option.


7.2.4.3 TP18L - wedge, 6 nodes

Figure 7.40: TP18L
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The TP18L element [Fig.7.40] is a six-node isoparametric solid wedge element. It is based on linear area interpolation in the triangular domain and a linear isoparametric interpolation in the $ \zeta$ direction. The polynomials for the translations uxyz can be expressed as

ui($\displaystyle \xi$,$\displaystyle \eta$,$\displaystyle \zeta$) = a0 + a1$\displaystyle \xi$ + a2$\displaystyle \eta$ + a3$\displaystyle \zeta$ + a4$\displaystyle \xi$$\displaystyle \zeta$ + a5$\displaystyle \eta$$\displaystyle \zeta$ (7.40)

These polynomials yield a constant strain and stress distribution over the element volume. By default DIANA applies a 1-point integration [ nlc = 1 ] scheme in the triangular domain and 2-point [ n$\scriptstyle \zeta$ = 2 ] in the $ \zeta$ direction. Schemes higher than 1 x 2 are unsuitable.


7.2.4.4 HX24L - brick, 8 nodes

Figure 7.41: HX24L
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The HX24L element [Fig.7.41] is an eight-node isoparametric solid brick element. It is based on linear interpolation and Gauss integration. The polynomials for the translations uxyz can be expressed as

ui($\displaystyle \xi$,$\displaystyle \eta$,$\displaystyle \zeta$) = a0 + a1$\displaystyle \xi$ + a2$\displaystyle \eta$ + a3$\displaystyle \zeta$ + a4$\displaystyle \xi$$\displaystyle \eta$ + a5$\displaystyle \eta$$\displaystyle \zeta$ + a6$\displaystyle \zeta$$\displaystyle \xi$ + a7$\displaystyle \xi$$\displaystyle \eta$$\displaystyle \zeta$ (7.41)

Typically, a rectangular brick element approximates the following strain and stress distribution over the element volume. The strain $ \varepsilon_{{xx}}^{}$ and stress $ \sigma_{{xx}}^{}$ are constant in x direction and vary linearly in y and z direction. The strain $ \varepsilon_{{yy}}^{}$ and stress $ \sigma_{{yy}}^{}$ are constant in y direction and vary linearly in x and z direction. The strain $ \varepsilon_{{zz}}^{}$ and stress $ \sigma_{{zz}}^{}$ are constant in z direction and vary linearly in x and y direction. If bending behaviour is dominant the use of incompatible bubble displacement modes is advised. By default DIANA applies a 2 x 2 x 2 [n$\scriptstyle \xi$ = 2 , n$\scriptstyle \eta$ = 2 , n$\scriptstyle \zeta$ = 2 ] integration scheme, 1 x 1 x 1 is a suitable option if all assumed strain options are suppressed. The 1 x 1 x 1 integration scheme is based on uniform strain assumption with orthogonal hourglass control according to Flanagan and Belytschko [26]. Schemes higher than 2 x 2 x 2 are unsuitable.


7.2.4.5 CTE30 - tetrahedron, 3 sides, 10 nodes

Figure 7.42: CTE30
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The CTE30 element [Fig.7.42] is a ten-node, three-side isoparametric solid tetrahedron element. It is based on quadratic interpolation and numerical integration. The polynomials for the translations uxyz can be expressed as

\begin{displaymath}\begin{split}u_{i} ( \xi,\eta,\zeta) = &a_{0} + a_{1} \xi + a...
... + a_{7} \xi^{2} + a_{8} \eta^{2} + a_{9} \zeta^{2} \end{split}\end{displaymath} (7.42)

Typically, these polynomials yield a linearly varying strain and stress distribution over the element volume. By default DIANA applies a 4-point [ nlc = 4 ] integration scheme over the volume, 1- and 5-point are suitable options. Schemes higher than 5-point are not available.

7.2.4.6 CPY39 - pyramid, 13 nodes

Figure 7.43: CPY39
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The CPY39 element [Fig.7.43] is a thirteen-node isoparametric solid pyramid element as described by the University of Colorado [79]. It is based on quadratic interpolation and integration. The polynomials for the translations uxyz are based on a isoparametric twnety-node brick element were nodes 13 to 20 of the brick element are collapsed to apex 13 of the pyramid element. Face bubble functions are added to assure compatiblity over the apex faces when attached to a ten-node tetrahedron element. The polynomials can be expressed as

\begin{equation*}\begin{aligned}u_{1} ( \xi,\eta,\zeta) &= -\frac{1}{16}\left(1-...
...\eta,\zeta) &= \frac{1}{2}\zeta\left(1+\zeta\right) \end{aligned}\end{equation*}

Typically, a thirteen-node pyramid element approximates the following strain and stress distribution over the element volume. The strain $ \varepsilon_{{xx}}^{}$ and stress $ \sigma_{{xx}}^{}$ vary linearly in x direction and quadratically in y and z direction. The strain $ \varepsilon_{{yy}}^{}$ and stress $ \sigma_{{yy}}^{}$ vary linearly in y direction and quadratically in x and z direction. The strain $ \varepsilon_{{zz}}^{}$ and stress $ \sigma_{{zz}}^{}$ vary linearly in z direction and quadratically in x and y direction. DIANA always applies a 13-point [ nlc = 13 ] integration scheme over the volume.


7.2.4.7 CTP45 - wedge, 15 nodes

Figure 7.44: CTP45
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The CTP45 element [Fig.7.44] is a fifteen-node isoparametric solid wedge element. It is based on quadratic interpolation and numerical integration. The polynomials for the translations uxyz can be expressed as

\begin{displaymath}\begin{split}u_{i} ( \xi,\eta,\zeta) = &a_{0} + a_{1} \xi + a...
...zeta + a_{13} \xi \zeta^{2} + a_{14} \eta \zeta^{2} \end{split}\end{displaymath} (7.44)

These polynomials yield a strain and stress distribution which varies approximately linearly over the element volume. By default DIANA applies a 4-point integration [ nlc = 4 ] scheme in the triangular domain and a 2-point scheme in the $ \zeta$ direction. [ n$\scriptstyle \zeta$ = 2 ] In a small element patch ($ \pm$ two elements), the 3 x 2 scheme may lead to zero-energy modes and a too soft behaviour. It is sufficient to apply a 4-point scheme in the triangular domain and a 3-point scheme in $ \zeta$ direction, schemes of higher order are unsuitable.


7.2.4.8 CHX60 - brick, 20 nodes

Figure 7.45: CHX60
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The CHX60 element [Fig.7.45] is a twenty-node isoparametric solid brick element. It is based on quadratic interpolation and Gauss integration. The polynomials for the translations uxyz can be expressed as

\begin{displaymath}\begin{split}u_{i} ( \xi,\eta,\zeta) = &a_{0} + a_{1} \xi + a...
...{18} \xi \eta^{2} \zeta + a_{19} \xi \eta \zeta^{2} \end{split}\end{displaymath} (7.45)

Typically, a rectangular brick element approximates the following strain and stress distribution over the element volume. The strain $ \varepsilon_{{xx}}^{}$ and stress $ \sigma_{{xx}}^{}$ vary linearly in x direction and quadratically in y and z direction. The strain $ \varepsilon_{{yy}}^{}$ and stress $ \sigma_{{yy}}^{}$ vary linearly in y direction and quadratically in x and z direction. The strain $ \varepsilon_{{zz}}^{}$ and stress $ \sigma_{{zz}}^{}$ vary linearly in z direction and quadratically in x and y direction. By default DIANA applies a 3 x 3 x 3 integration scheme. [n$\scriptstyle \xi$ = 3 , n$\scriptstyle \eta$ = 3 , n$\scriptstyle \zeta$ = 3 ] A suitable option in a patch of more than one element is 2 x 2 x 2 which yields optimal stress points. Schemes lower than 2 x 2 x 2 or higher than 3 x 3 x 3 are unsuitable.


7.2.4.9 CTE48 - tetrahedron, 3 sides, 16 nodes

Figure 7.46: CTE48
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The CTE48 element [Fig.7.46] is a sixteen-node, three-side isoparametric solid tetrahedron element. It is based on third-order interpolation and numerical integration. The polynomials for the translations uxyz can be expressed as

\begin{displaymath}\begin{split}u_{i} ( \xi,\eta,\zeta) = &a_{0} + a_{1} \xi + a...
...{2} + a_{14} \eta^{2} \zeta + a_{15} \eta \zeta^{2} \end{split}\end{displaymath} (7.46)

Typically, these polynomials approximate the following strain and stress distribution. The strain $ \varepsilon_{{xx}}^{}$ and stress $ \sigma_{{xx}}^{}$ vary linearly in x direction and quadratically in y and z direction. The strain $ \varepsilon_{{yy}}^{}$ and stress $ \sigma_{{yy}}^{}$ vary linearly in y direction and quadratically in x and z direction. The strain $ \varepsilon_{{zz}}^{}$ and stress $ \sigma_{{zz}}^{}$ vary linearly in z direction and quadratically in x and y direction. By default DIANA applies a 64-point integration [ nlc = 64 ] scheme over the volume. Other schemes are unsuitable.


7.2.4.10 CTP72 - wedge, 24 nodes

Figure 7.47: CTP72
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The CTP72 element [Fig.7.47] is a twenty-four-node isoparametric solid wedge element. It is based on third-order interpolation and numerical integration. The polynomials for the translations uxyz can be expressed as

\begin{displaymath}\begin{split}u_{i} ( \xi,\eta,\zeta) = &a_{0} + a_{1} \xi + a...
...zeta^{3} + a_{22} \eta \zeta^{3} + a_{23} \zeta^{3} \end{split}\end{displaymath} (7.47)

Typically, these polynomials approximate the following strain and stress distribution. The strain $ \varepsilon_{{xx}}^{}$ and stress $ \sigma_{{xx}}^{}$ vary linearly in x direction and quadratically in y and z direction. The strain $ \varepsilon_{{yy}}^{}$ and stress $ \sigma_{{yy}}^{}$ vary linearly in y direction and quadratically in x and z direction. The strain $ \varepsilon_{{zz}}^{}$ and stress $ \sigma_{{zz}}^{}$ vary linearly in z direction and quadratically in x and y direction. By default DIANA applies a 9-point integration [ nlc = 9 ] scheme in the triangular domain and a 4-point scheme [ n$\scriptstyle \zeta$ = 4 ] in the $ \zeta$ direction. Options are 6- and 7-point in the triangular domain, and 3-point in the $ \zeta$ direction. Other schemes are unsuitable.


7.2.4.11 CHX96 - brick, 32 nodes

Figure 7.48: CHX96
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The CHX96 element [Fig.7.48] is a thirty-two-node isoparametric solid brick element. It is based on third-order interpolation and Gauss integration. The polynomials for the translations uxyz can be expressed as

\begin{displaymath}\begin{split}u_{i} ( \xi,\eta,\zeta) = &a_{0} + a_{1} \xi + a...
... \xi \eta^{3} \zeta + \\ &a_{31} \xi \eta \zeta^{3} \end{split}\end{displaymath} (7.48)

Typically, a rectangular brick element approximates the following strain and stress distribution over the element volume. The strain $ \varepsilon_{{xx}}^{}$ and stress $ \sigma_{{xx}}^{}$ vary linearly in x direction and quadratically in y and z direction. The strain $ \varepsilon_{{yy}}^{}$ and stress $ \sigma_{{yy}}^{}$ vary linearly in y direction and quadratically in x and z direction. The strain $ \varepsilon_{{zz}}^{}$ and stress $ \sigma_{{zz}}^{}$ vary linearly in z direction and quadratically in x and y direction. By default DIANA applies a 4 x 4 x 4 integration [n$\scriptstyle \xi$ = 4 , n$\scriptstyle \eta$ = 4 , n$\scriptstyle \zeta$ = 4 ] scheme. A suitable option is 3 x 3 x 3 which yields optimal stress points and is sufficient. Schemes lower than 3 x 3 x 3 or higher than 4 x 4 x 4 are unsuitable.


next up previous contents index
Next: 7.2.5 Integration Schemes and Up: 7.2 Three-dimensional Bodies or Previous: 7.2.3 Strains and Stresses   Contents   Index
DIANA-10.0 User's Manual - Theory
First ed.

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