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# Regarding Stiffness Matrix & Element Nodal Forces

## Regarding Stiffness Matrix & Element Nodal Forces

(OP)
Hello Everyone, I am calculating stiffness matrix for a hexahedron element using C code and I am using gauss points. However if I compare my output with the stiffness matrix output by Abaqus, I see some differences ? Is it normal ?

Also I am trying to transfer element nodal forces from one mesh to another. The transfer has to be conservative, so what is the best way to acheive this ?

### RE: Regarding Stiffness Matrix & Element Nodal Forces

If your element has the same formulation as the Abaqus element (or is intended to have the same formulation), then it should give almost exactly the same results. If it doesn't, it won't.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

### RE: Regarding Stiffness Matrix & Element Nodal Forces

8-node hex elements often have all sorts of different formulations because the basic textbook one is quite terrible. Several programs seem to use identical formulations for 10-node tets though. 20-node hex is also more likely to be the same.

Other things to check:
• Make sure you have the same number of Gauss points. It might be 1 or 8.
• Make sure the row order is the same. It might be ordered by the element's node numbering or by the global node numbers or arbitrarily mixed up in some other way.
• See if the non-zero structure is the same.
• Check for correctness by summing the rows. Each row should sum to zero.
Not sure how you'd transfer node forces to a different mesh. I guess the most correct way would be to first find a continuous (ie. not discrete) force field that the node forces define, and then apply that distributed load to each element in the new mesh by integrating it over the element's face. For that second stage, you could use Gauss points to sample the field, but there's no guarantee they'll capture all the details if it's very non-uniform. You could use a much larger number of integration points to sample it more accurately. If you already know the field that the node forces represent then that will take care of the first stage which I have no idea how to do otherwise.

That's a very complex job. If you a dirty shortcut, maybe just sum them to find the net force then apply a uniform force that's the same? That might cause the wrong moment though.

### RE: Regarding Stiffness Matrix & Element Nodal Forces

(OP)
Thanks Doug and Whitwas for your suggestions.

@whitwas, The row sum came to be zero. I will try your suggestions and update. Thanks a lot.

### RE: Regarding Stiffness Matrix & Element Nodal Forces

(OP)
Hello Everyone, I found why there was a difference in results when cmpared to Abaqus. For Hex8, Abaqus does both Full Integration using 8 Gauss Points plus Reduced Integration using 1 Gauss point. The stiffness term is split into two parts - dilational and deviatoric (dont know the exact meaning).

### RE: Regarding Stiffness Matrix & Element Nodal Forces

"dilational" ... relating to "dilation" (oh, great !) ... "or compression" ... ok, that's more helpful.

### RE: Regarding Stiffness Matrix & Element Nodal Forces

Dilational typically refers to stresses that change the volume, though I usually hear it referred to as hydrostatic, not dilational. However, dilational is probably the better term since hydrostatic indicates water is involved which doesn't have to be the case.

For dilational stresses, think of a balloon floating in a chamber of gas. As the pressure in the surrounding gas goes up, the balloon shrinks in size. If you slowly increase the pressure the balloon will keep shrinking in size, but is unlikely to pop. Dilational stresses usually aren't what cause material failure, though they still cause strain. Deviatoric is more like shear stresses. They cause materials to fail. If you tried to cut that balloon with scissors after it shrank it would definitely pop. The scissors are introducing mostly deviatoric stresses.

In reality, both of these are derived from the typical stress tensor. When you add the two together it should give you back your typical stress tensor.

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