# Stiffness Matrix for Linear Tetrahedral Element

1. Jun 16, 2014

### Hassan2

Dear all,

I am writing my finite element code for linear elasticity analysis with linear tetrahedral elements. I am not new to the finite element method and my code already works well with element types of triangles, quadrangles in 2D and hexahedrons in 3D but I don't get the correct result with tetrahedrons. I doubt the calculation of the element stiffness matrix and need a valid stiffness matrix of a single tetrahedron for comparison. Does anyone has hands on such software ?

Attachment contains the stiffness matrix and its eigenvalues from my code for a unit tetrahedron whose base are at (1,0,0), (0,1,0), and (0,0,1) and whose apex at (0,0,0), with Young's modulus of 200 MPa and Poisson's ratio of 0.3.

Hassan

#### Attached Files:

• ###### tetrahedron.png
File size:
46 KB
Views:
640
2. Jun 16, 2014

### AlephZero

3. Jun 17, 2014

### Hassan2

Thank you very much AlephZero.

My code calculates the matrix correctly. When I run the code for a beam whose mesh is fine enough, the displacement vectors look fine but their magnitudes are scaled down with a factor usually between 1.0 and 2.0 and the factor depends on the mesh and loading. I don't think the error is due to element type.

Last edited: Jun 17, 2014
4. Jun 17, 2014

### AlephZero

A linear tetrahedron with exact integration is not a good element for continuum mechanics. You can probably find examples where the errors are much worse than 50%, if you search the literature. It is a better element for scalar field problems like heat transfer, especially for nonlinear problems where the gradient discontinuities between the elements may be physically realistic (e.g. they correspond to changes of state in the material).

On the other hand the quadratic (10 node) tetrahedron is very well behaved for structural analysis, if you want to use an automatic tetrahedral mesh generator.

5. Jun 17, 2014

### Hassan2

That could be the reason. I have been spending several days on this problem. Before trying higher order tetrahedrons, I should explain that in 2D, I didn't find much difference between triangular and quad elements so I didn't expect much differences between tetrahedrons and beam elements too.

My test model is a cantilever beam with rectangular cross-section. The original mesh is constructed with beam elements and I believe the mesh is fine enough. By dividing each element into 5 tetrahedrons, I expect the results to be nearly the same but the difference is more than 50 %.

6. Jun 17, 2014