# Hourglass Modes

Hourglass modes are element distortions that have zero strain energy. Thus, no stresses are created within the element. There are 12 hourglass modes for a brick element, 4 modes for each of the 3 coordinate directions. $\Gamma$ represents the hourglass mode vector, as defined by Flanagan-Belytschko. 1 They produce linear strain modes, which cannot be accounted for using a standard one point integration scheme.
${\Gamma }^{1}=\left(+1,-1,+1,-1,+1,-1,+1,-1\right)$
${\Gamma }^{2}=\left(+1,+1,-1,-1,-1,-1,+1,+1\right)$
${\Gamma }^{3}=\left(+1,-1,-1,+1,-1,+1,+1,-1\right)$

${\Gamma }^{4}=\left(+1,-1,+1,-1,-1,+1,-1,+1\right)$

To correct this phenomenon, it is necessary to introduce anti-hourglass forces and moments. Two possible formulations are presented hereafter.

## Kosloff and Frasier Formulation

The Kosloff-Frasier hourglass formulation 2 uses a simplified hourglass vector. The hourglass velocity rates are defined as:(1)
$\frac{\partial {q}_{i}^{\alpha }}{\partial t}=\sum _{I=1}^{8}{\Gamma }_{I}^{\alpha }\cdot {v}_{iI}$
Where,
$\Gamma$
Non-orthogonal hourglass mode shape vector
$\nu$
Node velocity vector
$i$
Direction index, running from 1 to 3
$I$
Node index, from 1 to 8
$\alpha$
Hourglass mode index, from 1 to 4

This vector is not perfectly orthogonal to the rigid body and deformation modes.

All hourglass formulations except the physical stabilization formulation for solid elements in Radioss use a viscous damping technique. This allows the hourglass resisting forces to be given by:(2)
${f}_{iI}^{hgr}=\frac{1}{4}\rho ch{\left(\sqrt[3]{\Omega }\right)}^{2}\sum _{\alpha }\frac{\partial {q}_{i}^{\alpha }}{\partial t}\cdot {\Gamma }_{I}^{\alpha }$
Where,
$\rho$
Material density
$c$
Sound speed
$h$
Dimensional scaling coefficient defined in the input
$\Omega$
Volume

## Flanagan-Belytschko Formulation

In the Kosloff-Frasier formulation seen in Kosloff and Frasier Formulation, the hourglass base vector ${\Gamma }_{I}^{\alpha }$ is not perfectly orthogonal to the rigid body and deformation modes that are taken into account by the one point integration scheme. The mean stress/strain formulation of a one point integration scheme only considers a fully linear velocity field, so that the physical element modes generally contribute to the hourglass energy. To avoid this, the idea in the Flanagan-Belytschko formulation is to build an hourglass velocity field which always remains orthogonal to the physical element modes. This can be written as:(3)
${v}_{iI}^{Hour}={v}_{iI}-{v}_{iI}^{Lin}$
The linear portion of the velocity field can be expanded to give:(4)
${v}_{iI}^{Hour}={v}_{iI}-\left(\overline{{v}_{iI}}+\frac{\partial {v}_{iI}}{\partial {x}_{j}}\cdot \left({x}_{j}-\overline{{x}_{j}}\right)\right)$
Decomposition on the hourglass vectors base gives 1:(5)
$\frac{\partial {q}_{i}^{\alpha }}{\partial t}={\Gamma }_{I}^{\alpha }\cdot {v}_{iI}^{Hour}=\left({v}_{iI}-\frac{\partial {v}_{il}}{\partial {x}_{j}}\cdot {x}_{j}\right)\cdot {\Gamma }_{I}^{\alpha }$
Where,
$\frac{\partial {q}_{i}^{\alpha }}{\partial t}$
Hourglass modal velocities
${\Gamma }_{I}^{\alpha }$
Hourglass vectors base
Remembering that $\frac{\partial {v}_{i}}{\partial {x}_{j}}=\frac{\partial {\Phi }_{j}}{\partial {x}_{j}}\cdot {v}_{iJ}$ and factorizing Equation 5 gives:(6)
$\frac{\partial {q}_{i}^{\alpha }}{\partial t}={v}_{iI}\cdot \left({\Gamma }_{I}^{\alpha }-\frac{\partial {\Phi }_{j}}{\partial {x}_{j}}{x}_{j}{\Gamma }_{I}^{\alpha }\right)$
(7)
${\gamma }_{I}^{\alpha }={\Gamma }_{I}^{\alpha }-\frac{\partial {\Phi }_{j}}{\partial {x}_{j}}{x}_{j}{\Gamma }_{J}^{\alpha }$

is the hourglass shape vector used in place of ${\Gamma }_{I}^{\alpha }$ in Equation 2.

## Physical Hourglass Formulation

You also try to decompose the internal force vector as:(8)
$\left\{{f}_{I}^{\mathrm{int}}\right\}=\left\{{\left({f}_{I}^{\mathrm{int}}\right)}^{0}\right\}+\left\{{\left({f}_{I}^{\mathrm{int}}\right)}^{H}\right\}$
In elastic case, you have:(9)
$\begin{array}{l}\left\{{f}_{I}^{\mathrm{int}}\right\}={{\int }_{\Omega }\left[{B}_{I}\right]}^{t}\left[C\right]\sum _{j=1}^{8}\left[{B}_{J}\right]\left\{{v}^{J}\right\}d\Omega \\ =\underset{\Omega }{\int }{\left({\left[{B}_{I}\right]}^{0}+{\left[{\overline{B}}_{I}\right]}^{H}\right)}^{t}\left[C\right]\sum _{j=1}^{8}\left({\left[{B}_{J}\right]}^{0}+{\left[{\overline{B}}_{J}\right]}^{H}\right)\left\{{v}^{J}\right\}d\Omega \end{array}$

The constant part $\left\{{\left({f}_{I}^{\mathrm{int}}\right)}^{0}\right\}=\underset{\Omega }{\int }{\left({\left[{B}_{I}\right]}^{0}\right)}^{t}\left[C\right]\sum _{j=1}^{8}{\left[{B}_{J}\right]}^{0}\left\{{v}^{J}\right\}d\Omega$ is evaluated at the quadrature point just like other one-point integration formulations mentioned before, and the non-constant part (Hourglass) will be calculated as:

Taking the simplification of $\frac{\partial {x}_{i}}{\partial {\xi }_{j}}=0;\left(i\ne j\right)$ (that is the Jacobian matrix of Strain Rate, Equation 1 is diagonal), you have:(10)
${\left({f}_{iI}^{\mathrm{int}}\right)}^{H}=\sum _{\alpha =1}^{4}{Q}_{i\alpha }{\gamma }_{I}^{\alpha }$
with 12 generalized hourglass stress rates ${\stackrel{.}{Q}}_{i\alpha }$ calculated by:(11)
$\begin{array}{l}{\stackrel{.}{Q}}_{ii}=\mu \left[\left({H}_{jj}+{H}_{kk}\right){\stackrel{˙}{q}}_{i}^{i}+{H}_{ij}{\stackrel{˙}{q}}_{j}^{j}+{H}_{ik}{\stackrel{˙}{q}}_{k}^{k}\right]\\ {\stackrel{.}{Q}}_{jj}=\mu \left[\frac{1}{1-\nu }{H}_{ii}{\stackrel{˙}{q}}_{i}^{j}+\overline{\nu }{H}_{ij}{\stackrel{˙}{q}}_{j}^{i}\right]\\ {\stackrel{.}{Q}}_{i4}=2\mu \frac{1+\nu }{3}{H}_{ii}{\stackrel{˙}{q}}_{j}^{4}\end{array}$
and(12)
$\begin{array}{l}{H}_{ii}=\underset{\Omega }{\int }{\left(\frac{\partial {\varphi }_{j}}{\partial {x}_{i}}\right)}^{2}d\Omega =\underset{\Omega }{\int }{\left(\frac{\partial {\varphi }_{k}}{\partial {x}_{i}}\right)}^{2}d\Omega =3\underset{\Omega }{\int }{\left(\frac{\partial {\varphi }_{4}}{\partial {x}_{i}}\right)}^{2}d\Omega \\ {H}_{ij}=\underset{\Omega }{\int }\frac{\partial {\varphi }_{i}}{\partial {x}_{j}}\frac{\partial {\varphi }_{j}}{\partial {x}_{i}}d\Omega \end{array}$

Where, $i$ , $j$ , $k$ are permuted between 1 to 3 and ${\stackrel{˙}{q}}_{i}^{\alpha }$ has the same definition than in Equation 6.

Extension to nonlinear materials has been done simply by replacing shear modulus $\mu$ by its effective tangent values which is evaluated at the quadrature point. For the usual elastoplastic materials, use a more sophistic procedure which is described in Advanced Elasto-plastic Hourglass Control.

## Advanced Elasto-plastic Hourglass Control

With one-point integration formulation, if the non-constant part follows exactly the state of constant part for the case of elasto-plastic calculation, the plasticity will be under-estimated due to the fact that the constant equivalent stress is often the smallest one in the element and element will be stiffer. Therefore, defining a yield criterion for the non-constant part seems to be a good idea to overcome this drawback.
Plastic yield criterion
The von Mises type of criterion is written by:(13)
$f={\sigma }_{eq}^{2}\left(\xi ,\eta ,\zeta \right)-{\sigma }_{y}^{2}=0$
for any point in the solid element, where ${\sigma }_{y}^{}$ is evaluated at the quadrature point.
As only one criterion is used for the non-constant part, two choices are possible:
1. taking the mean value, i.e.: $f=f\left({\overline{\sigma }}_{eq}^{}\right);{\overline{\sigma }}_{eq}^{}=\frac{1}{\Omega }\underset{\Omega }{\int }{\sigma }_{eq}^{}d\Omega$
2. taking the value by some representative points, for example: eight Gauss points
The second choice has been used in this element.
Elastro-plastic hourglass stress calculation
The incremental hourglass stress is computed by:
• Elastic increment

${\left({\sigma }_{i}\right)}_{n+1}^{trH}={\left({\sigma }_{i}\right)}_{n}^{H}+\left[C\right]{\left\{\stackrel{˙}{\epsilon }\right\}}^{H}\text{Δ}t$

• Check the yield criterion
• If $f\ge 0$ , the hourglass stress correction will be done by un radial return

${\left({\sigma }_{i}\right)}_{n+1}^{H}=P\left({\left({\sigma }_{i}\right)}_{n+1}^{trH},f\right)$

1 Flanagan D. and Belytschko T., “A Uniform Strain Hexahedron and Quadrilateral with Orthogonal Hourglass Control”, Int. Journal Num. Methods in Engineering, 17 679-706, 1981.
2 Kosloff D. and Frazier G., “Treatment of hourglass pattern in low order finite element code”, International Journal for Numerical and Analytical Methods in Geomechanics, 1978.