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### Eight-node brick element with reduced integration (C3D8R)

This is a general purpose linear brick element, with reduced integration (1 integration point). The shape functions are the same as for the C3D8 element and can be found in [11]. The node numbering follows the convention of Figure 8 and the integration point is shown in Fig 10.

Figure 10: 1x1x1 integration point scheme in hexahedral elements

Due to the reduced integation, the locking phenomena observed in the C3D8 element do not show. However, the element exhibits other shortcomings:

- The element tends to be not stiff enough in bending.
- Stresses, strains.. are most accurate in the integation points. The integration point of the C3D8R element is located in the middle of the element. Thus, small elements are required to capture a stress concentration at the boundary of a structure.
- There are 12 spurious zero energy modes leading to massive hourglassing: this means that the correct solution is superposed by arbitrarily large displacements corresponding to the zero energy modes. Thus, the displacements are completely wrong. Since the zero energy modes do no lead to any stresses, the stress field is still correct. In practice, the C3D8R element is not very useful without hourglass control. So far, no hourglass control has been implemented in CalculiX.

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