How to calculate volume of a hex ( brick ) element?

[OnlineFEASolver]

Anyone know offhand the formula for calculating the volume of an 8-noded brick element. I have got all the nodal coordinates but not sure what the formula is? I realize it may not be a simple answer, but will likely involve some vector math..

Principal -

http://www.OnlineFEASolver.com

General FEA Consulting Services

 

[johnhors]

You can only do this correctly using the element shape functions, as found in any book on the theory of finite elements.

 

[prost]

I really like the idea of using the shape functions. It will take some algebra, but you should be able to do it easily by hand with the element shape functions. For instance, the 8 shape functions are N(1), N(2)...N(8).

x=sum(X(i)*N(i))=X(1)*N(1)+....X(8)*N(8),..X(i) are the
x coordinates of the nodes.
y=sum(Y(i)*N(i)), z=sum(Z(i)*N(i)). The shape functions N(i) are a function of the parent element coordinates, xi, eta, zeta.
Volume=integral(1*dx*dy*dz)=integral(J*dxi*deta*dzeta), where J is the Jacobian, the determinant of a 3 x 3 matrix, first row are the derivatives dx/dxi, dy/dxi, and dz/dxi, second row are the derivates with respect to eta, third row zeta. The derivatives are easy to compute:
dx/dxi=sum(X(i)*dN(i)/dxi)=X(1)*dN(1)/dxi+X(2)*dN(2)/dxi+...X(8)*dN(8)/dxi, same for the other 8 derivatives.

Once you compute those derivatives, compute the Jacobian, the integral(J*dxi*deta*dzeta) you can do by hand I think. Even if not, you may be able to write a small VBA in Excel to compute the integral numerically.

 

[rb1957]

or you could make a full-size model of it, put it in a bath-tub, and see how much water it displaces

 

[johnhors]

Eureka !

 

[rb1957]

sensibly tho', I'd've thought that codes would have "spat" volume of elements back as part of the data check (just like they've tell the area of a 2D shell element); probably by the calculation suggested by prost, above.

If the solid has planar sides then it'll reduce to a prizmatic portion and some (maximum of 6?) pyramids ??

 

[GregLocock]

Would it not show up in the mass of the elemet?

Cheers

Greg Locock

Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.