# Simple Averaging

Simple averaging means that tensor and vector components are extracted and the invariants are computed prior to averaging.

For components, the corresponding components from each element corner are extracted and then they are averaged. For invariants, the corresponding invariants are calculated from each tensor at the element corners and then averaged.

For example, as shown in the Nodal Averaging of Elemental Results topic, there are four tensors, [A2], [B1], [C3], and [D4] at four corners at Node 400.

The average of the xx component at Node 400 is:
For the average of an invariant such as von Mises, the von Mises value for all tensor [A2], [B1], [C3], and [D4] are computed, then they are averaged as follows:
The averaging methods for solid elements and shell elements are different.
Element Type Reference System Projection Rule Description
Solid Global N/A All corresponding tensors and vectors are transformed to the global coordinate system and then a component or invariant is averaged as explained above.
Analysis N/A All corresponding tensors and vectors are left in their original coordinate system (no transformation occurs) and then a component or invariant is averaged as explained above. In this case, each participating element result can be in a different system. Simple averaging for components ignores variations in systems. Since invariance does not depend on the coordinate system, all reference systems will produce the same results for invariants.
Elemental N/A All corresponding tensors and vectors are transformed to the elemental coordinate system and then a component or invariant is averaged as explained above. In this case, each participating element result can also be in a different system. Simple averaging for components ignores variations in systems. Since invariance does not depend on the coordinate system, all reference systems will produce the same results for invariants.
User-defined N/A All corresponding tensors and vectors are transformed to the user-defined coordinate system and then a component or invariant is averaged as explained above.
Shell Global ON All corresponding tensors and vectors are transformed to the projected system (following the projection rule) of the participating elements and then the component or invariants are averaged as explained above.

As a result, even if the reference system is the same, the projected systems in participating elements can be different. Simple averaging for components ignores variations in systems.

OFF All corresponding tensors and vectors are transformed to the global coordinate system and a component or invariant is then averaged as explained above.
Analysis N/A All corresponding tensors and vectors are left in their original coordinate system (no transformation occurs) and then a component or invariant is averaged as explained above. In this case, each participating element result could be in a different system. Simple averaging for components ignores variations in systems. Since invariance does not depend on the coordinate system, all reference systems will produce the same results for invariants.
Elemental N/A All corresponding tensors and vectors are transformed to the elemental coordinate system and then a component or invariant is averaged as explained above. In this case, each participating element result could also be in a different system. Simple averaging for components ignores variations in systems. Since invariance does not depend on the coordinate system, all reference systems will produce the same results for invariants.
User-defined ON All corresponding tensors and vectors are transformed to the projected system (following the projection rule) of the participating elements and then the component or invariants are averaged as explained above.

As a result, even if the reference system is the same the projected systems in participating elements can be different. Simple averaging for components ignores variations in systems.

OFF All corresponding tensors and vectors are transformed to the User-defined coordinate system and a component or invariant is then averaged as explained above.

Valid reference coordinate system: All (Global, Analysis, Elemental, and User-defined)