Monday, May 12, 2008

What Is The Difference Between H-Element And P-Element - Explanation With Example Problem

3D Elements - Hexahedrons & Tetrahedrons.

3d elements take the form of cubes called hexahedrons (hexes) or 3d triangles called tetrahedrons (tets). In the early days there were few choices when creating models As technology has advanced, engineers have been forced to make more decisions. Many of these decisions hinge on understanding the role of the element shape and 'the order of interpolation' in the element creation. The order of interpolation refers to the degree of the complete polynomial appearing in the element shape functions.

Every element has a size h and a order p. There are three basic approaches to FEA: the h, p, and the h-p methods.

  • With the h method, the element order, p is kept constant, but the mesh is refined by making the size h smaller.
  • With the p method, the element size h is kept constant and the element order p is increased.
  • With the h-p method, the h is made smaller as the p is increased to create higher order h elements.

Either reducing the element size or increasing the element order reduces the error in the FEA approximations.

Element types include eight node brick elements, four node tets, and 10 node tets. There are many reasons the eight node hex element produces more accurate results than other elements in the finite element analysis of real world models.

The eight node hexahedral element is linear (p=1) with linear strain variation displacement mode. Tetrahedral elements are also linear but have more function error because the have a constant strain. The tets requires many more elements to produce a converged solution than does a hex. It takes five tets to fill the volume of one hex. These five tets together will have more function error than a single hex because they cannot assume all the displacement fields handled by the brick.

Test have shown that eight node hexes produce more accurate and faster FEA results than a number of other models, including 4 node tets, 10 node tets, and hybrids. In addition, meshes comprised of hexes are easier to visualize than meshes comprised of tets. Plus, the reaction of hex elements to applications of body loads more precisely corresponds to loads under real world conditions.

So then, are 8 node 'bricks' best for building a solid mesh model?

Proponents of higher order elements which require more nodes per element, claim that using a smaller number of larger sized elements results in less computational time and achieves the same accuracy as lower order h-elements. The basis for this claim of less computational time is that the higher order elements have less function error, even for the course mesh. But there is a major flaw in this logic" Most parts and products have complex geometry's which require a fine mesh to accurately resolve the geometry as a solid mesh. The mesh size is so small the function error does not exceed what is required for engineering accuracy. Use of p elements and higher order h elements with mid sized nodes, therefore, offers no benefit over the use of eight - node hexes.

The p method suffers from its own accuracy problems, related to the fact that the larger the elements, the greater the effect of each elements has on the entire FEA result. The error in an element typically stems from a geometric or load singularity present in the solution over that element.

This error can permeate adjacent elements and seriously effect the accuracy of results because it affects stresses and fluxes. Since the geometry and load singularity are common in most designs parts or products, p elements and higher order h elements have to be refined in size to cope with large gradients and discontinuities in the solution near the points of singularity. Refinement of the elements defeats the very purpose of using p elements or higher order h elements for FEA because the refinement takes time to make.

The result from the h method quadrilaterals is the most accurate, and with the automatic mesher, overall solution time is slashed. Thus, the h method accompanied by a robust hex generator is simply the best solution for 99 out of 100 design situations.

Letters to the editor:

More on hexes vs. Tets

Victor Wiengartens raised some interesting points in his response to the above article about the benefits of using eight node hexahedral elements.

We do not feel eight node hexahedral elements for all applications. As a matter of fact, we feel that 20 node hexahedral elements are essential for accurate analysis for specific engineering applications which involve conductivity such as fluid flow, heat transfer and electrostatics. We also agree with Dr. Wiengartens opinion that the mid nodes of 20-node hexahedrons enable users to capture the internal geometry of a model with curves.

While Dr. Wiengartens points out that users can greatly increase accuracy by replacing eight node hex elements with 20 node hex elements, we don not believe the increase in accuracy is great enough to warrant the use of 20 node hex elements in all cases. The time to run the same number of 20 node elements is significantly slower, compared to eight node hex elements and may not be worth the slight increase in accuracy. The best software products will enable users to change some elements into 20 node elements to accommodate various features on a model such as curves or fillet openings.

To state that ALGOR does not advocate the use of the higher order elements is untrue since our Houdini product, the only automatic interface between virtually all major CAD programs and all major FEA programs, can automatically generate 20 node hexahedrons as well as eight node hexahedrons.

On a final note, most problems encountered with analysis and meshing software products are usually the results of the user not being educated about the types of elements that are best for each different engineering application. As a developer of these products, we p[lace a strong emphasis on educating our users about how to maximize the use of the products to ensure that they are used correctly for the fastest and most accurate results.

Michael Bussler and Thomas Koehler

Algor Inc, Pittsburgh, PA


If you would like to test the difference between using H And P Elements, Try this problem..