For given solid model geometry the finite element analyst has relied on building the mesh by hand using 8-node hexahedral elements. However, for complex solid models the analyst must rely on an automatic and semi-automatic mesh generators.

A 3-D domain cannot always be meshed into hexahedral elements. However, it can be decomposed into tetrahedral elements more easily when compared to hexahedral elements. Thus automatic mesh generators for tetrahedral elements have become more popular. This is leading to increasing popularity of tetrahedral elements in finite element analysis. The requires that element performance tests and guidelines for element acceptance should be developed for the previously less popular tetrahedron elements. As a first step, the relative merits of Tetrahedron and Hexahedron elements for modeling well understood problems needs to be studied. In the existing literature of papers that address the issues of performance of Hexahedron and Tetrahedron elements performance is given is scant. It is proposed here to investigate the performance of Tetrahedron and Hexahedron elements. Some comments of automatic mesh generators for arbitrary solid models are made.

A.O. Cifuentes and A. Kalbag

IBM Research Division

P.O. Box 218

Yorktown Heights, NY 10598 USA

Finite Elements in Analysis and Design 12 (1992) 313-318

Elsevier

In this paper four problems are considered, each problem dominated by a different deformation pattern.

Problem 1 Bending. L =8, b =1, h =1, load per unit length = 0.01, E=1000, Poisson’s ratio=0.15

Problem 2 Shear. L =1, b =0.6, h =1, load per unit length =1,E =1000, Poisson’s ratio=0.15

Problem 3 Torsion. L =16, b =1.h =1, torsional moment =0.1146, E =1000, Poisson’s ratio=0.15

Problem 4 Axial behavior. L =4, b =1, h=1, mass density per unit of volume = 1, E =1000, Poisson’s ratio=0.0

For analyses each problem was solved using four different models(four different meshes):

 Mesh 1. This mesh is made of linear tetrahedral elements.

 Mesh 2. This mesh is made of quadratic tetrahedral elements.

 Mesh 3. This mesh is made of linear tetrahedral elements with element size smaller then element

size in mesh 1.

 Mesh 4. This mesh is of hexahedral elements with element sizing same as mesh 2.

RESULTS:

The analysis of above 4 problems where performed on an IBM RISC/600 workstation using Abaqus. The table 1 to 4 summarizes the results. The nomenclature is as follows:

N : # of nodes.

E : # of elements.

x,y,z : the node spacing in x-, y- and z- direction.

T : the total CPU time in seconds.

Table 1

Result for problem 1(bending)

R : represents the vertical displacement at the end of the beam center line.

Correct R = 6.254x10 ^–2.

Table 2

Result for problem 2 (shear)

R : represents the vertical displacement at the end of the beam center line.

Correct R = 5.375x10 ^–3.

Table 3

Result for problem 3 (torsion)

R : represents the maximum shear stress on the cross section of the beam.

Correct R = 0.5511.

Table 4

Result for problem 4 (axial vibration)

R : represents the fundamental frequency for axial vibration.

Correct R = 3.953

This paper has two main conclusions:

1.The results obtained with Hex and Tet elements in terms of accuracy and CPU time are roughly equivalent. Therefore analysts who rely on automatic mesh generators do not have a disadvantage compared to those analysts who use brick elements.

2. The best approach to take if a model made of linear Tet dose not give satisfactory results is to increase the order of element rather than refining the mesh with smaller liner elements as it takes more time.

The paper does not consider incompressibility(that is for different Poisson’s ratio).

The paper concludes that quadratic tets and hexes were equivalent. That seems correct, but it also seems to me that there is no clear choice across these test problems. Only mesh 3 didn't win on some problem 

1. Melon and P. Ward

Structural Dynamics Research Corporation

Milford, Ohio, USA and Hitchin, Great Britain

SDRC technical report.

In this paper three problems are considered:

Length = 10, width = 2, depth = 1.5, E = 1.5E+6, poisson’s ratio = .29, density = 0.0617981

Problem 1 bending.

Problem 2 shear.

Problem 3 first bending mode.

For analysis a total of 8 different models were used. 3 models were made from parabolic hexahedron. Each of these was used to create parabolic tetrahedron models, where each hexahedron element was broken into 5 tetrahedron element.

Model 1 four Hexahedron elements.

Model 2 Tetrahedra from model 1.

Model 3 distorted Hexahedra.

Model 4 Tetrahedra from model 3

Model 5 Eight Hexahedra (elements are flat)

Model 6 Tetrahedra from model 5 (elements are more flat)

Model 7 distorted Tetrahedra

Model 8 Free mesh generated in I-DEAS

RESULTS:

The analysis of the above problems were performed with the SDRC I-DEAS Model Solution Finite Element solver. The results are presented in the following tables

Table 1

Problem 1(bending)

Correct R1(displacement) = 1.000

Correct R2(Strain energy) = 3000

Table 2

Problem 2(shear)

Correct R1(Displacement) = 1.019

Correct R2(Strain Energy) = 2339.6

Table 3

Problem 3(shear)

Correct R(frequency) = 15.56 Hz

The efficiency benefit in using the parabolic Tet element for modeling solid structures warrants the use of the element. I-DEAS Model Solution was used to produce the results presented in these tests. Model Solution has had this element in the element library for 5 years. MSC/NASTRAN and ANSYS have adopted this element into their element library and others are sure to come. This proves the increasing popularity of the element, which means element performance tests and guidelines for element acceptance tests should be developed.

The Tetrahedron elements behave well compared to hex element by a narrow margin.

Looking at model 5 and 6. On denser mesh hex element win in problem 2 and 3 while in problem 1 there is a tie.

For bending problem the Tet element are not affected very much if the elements become near flat but in shear load problem Tet elements show large deviation from theoretical values.

Steven E. Benzley, Ernest Perry,Karl Merkley, Brett Clark

Brigham Young University

Provo, Utah

Greg Sjaardama

Sandia National Laboratories

Albuquerque, NM.

In this paper a cantilever beam with the following dimensions is considered.

Length = 10, width = 1, height = 1, E = 10000, Poissons ratio = 0.3 and 0.49

Density = 0.1

Static linear analysis:

Two structural cases are used to compare the results of statically loaded all Hex and all Tet meshes.

Problem 1 (Bending) A force of 0.25 applied at free end and the reference point is 5 units from fixed end. Analytical magnitude of normal displacement and bending stress at the reference point,using classical beam theory are 0.000125 and 30.0 respectively.(independent of poissons ratio)

Problem 2 ( Torsion) A shear force of 0.25 is applied at free end and the reference point is 5 units from fixed end. Analytical magnitude of shear stress was 0.68(independent of poisson’s ratio) The rotational displacement is 0.000003269 for poissons ratio 0.3 and 0.000003747 for poissons ratio of 0.49.

Results

1. linear Tet elements produce the maximum errors.
2. Hex elements perform better in bending.
3. Hex elements perform better in 3 of the torsion cases.
4. Quadratic elements do better than linear elements with quadratic Hex showing slightly better performance over the quadratic Tet.
5. Degradation in all solutions as poisson's ratio approaches 0.5

Dynamic model analysis:

Natural modes of vibration are compared in this section. solution for bending mode is 317.5 cycles/sec and torsional vibration mode is 2614 cycles/sec.

Results:

The linear Tet performs poorly in all cases. The quadratic elements are adequate with a slight advantage given to the Hex mesh.

Static nonlinear elasto-plastic analysis:

In this section a comparison of Hex and Tet mesh is made for static nonlinear, elasto-plastic calculations. Bending and torsion cases as before are evaluated with Yield Stress of 10,000.

Load which is applied at free end is increased incrementally. The displacement of the loaded tip of the beam and the distance to the incipient yield front on surface of the beam from fixed end are quantities used for comparison between Tet and Hex meshes.

The comparison of linear static bending situation indicated that LT models produced errors between 10 to 70 % in both displacement and stress calculation. Such errors are obviously unacceptable for stress analysis work. However LH, QH, and QT models all provided acceptable results, even with relatively coarse meshes. In all cases, the error was significantly grater with a nearly incompressible material model(i.e. poisson's ratio =0.49)

The linear static torsion problem again showed that the LT element produced errors of an unacceptable magnitude. This problem also demonstrated that, because selective integration is only effective on the bending problem, the LH element, without a significant number of degrees of freedom, produce poor results. Here, as in the previous problem, the QH element is superior.

Dynamic model analysis comparisons reflected the results of the previous 2 comparisons. That is. LT models produce more error that is acceptable and QH models are generally preferred to insure accuracy with both torsion and nearly incompressible materials.

The analytical solution for Elasto_plastic analysis is not given. Figures 8 and 9 are not clear to make comparison; therefore I think this data is not reliable.

I think stiffness matrix eigenvalues for quadratic tetrahedrons needs to be calculated to do comparison.

This is the most thorough paper, but even still there are only a few data points for each problem. 

 4. The controversy over Hex or Tet Meshing

Victor I. Weingarten

Presidnt & CEO

Structural Research and Analysis Corp.

Santa Monica Calif.

In this paper victor says the real problem is not the type of element (Hex or Tet) which is most suitable for parts build with solid modeling programs, but the real problem lies in finding an efficient way to perform FEA for complex solid models without straning computing time.

Using 4 node Tet give less accurate solution than using 20 node Hex unless a highly refined Tet mesh is used . some engineers who use 10-node tet elements say that they have more confidence in fully automatic Tet than in Hex meshes to describe the geometry of complex designs.this is because automatic hex meshers contains so many limitation they are really semi-outomatic at the best. Using hex elements may be more time consuming than using 10-node tet elements, which take advantage of the speed of fully automatic meshing.

Structural Reasearch & Analysis Corp. conducted several tests and concluded that for like number of same number of elements the 20-node hex proved more accurate than 8-node tet and 10-node tet more accurate than the 4-node tet. In case of same number of DOF the 20-node hex proved as accurate as or more accurate than the 10-node tet.

Finally Victor says to use p-method to increase accuracy and the DOF of the 10-node tet so that it matches or exceeds, the 20-node hex in accuracy.

This is not a serious technical paper, and really more about p-version element than Hexahedron and Tetrahedron element performance investigation. With this paper the writer talks about another efficient way to perform finite element analysis for complex solid models.

In the numerical study performed here two problems with analytical solutions are considered. These two problems are selected to investigate the modeling capability of these elements in "shear locking", and "volumetric locking" situations. The first problem is cantilever beam bending problem that studies the shear locking problem. The second is a circular cylindrical pipe with internal pressure. This problem is modeling with varying poisson’s ratio to study volumetric locking. Both the problems are modeled with tetrahedron and hexahedron elements. The finite element analysis was done using ANSYS finite element program. The meshes for both the elements were generated using the programs automatic mesh generators.

Load = 400

Length (L) = 10

Width (b) = 1

Height (h) = 1

Young’s modulus (E) = 10⁷

Poisson’s ratio = 0.3

Analytical value for vertical displacement at the free end of the beam centerline is 0.16

Two different meshes for each of the element types are considered.

The hexahedron finite element meshes for this problem are shown in figures 1 and 2. The tetrahedron element meshes are shown in figures 3 and 4.

Results:

Solid45: 8 node Hex

Solid92: 10 node Tet

NES: No extra shapes functions

ES: Extra shape functions

Solid 45 with extra shapes gives good results. This mesh also solves less number of degrees of freedom and as a result using less computer memory.

A thick walled cylinder of inner radius Ri = 3 and outer radius Ro = 9 is subjected to an internal pressute p = 1. Thickness of the cylinder t = 1.Young’s modulus = 1000. Poisson’s ratio varying from 0.0 to 0.49999999. This problem is chosen to investigate the issue of volumetric locking in both the element types which occurs in incompressible deformations. Incompressible deformations occur when Poisson’s ratio is 0.5 in elastic analysis. The analytical result of radial displacements for varying Poissons’s ratio is given in "A.results" column of the table.

Results:

DOF for solid92 = 267

DOF for solid45 = 48

Solid45: 8 node Hex with reduced integration

Solid92: 10 node Tet

Hex elements give better results than Tet elements for incompressibility.

• Linear hexahedron elements with special formulation like the use of extra shape functions and reduced integration result in solutions that are very accurate.

• Quadratic tetrahedron elements perform well for modeling bending problems. However they do not perform well when modeling incompressible situations where Poisson’s ratio is close to 0.5 or equal to 0.5

• Linear Hexahedron elements result in fewer degrees of freedom when comapred to the Quadratic Tetrahedron elements. Thus they require less CPU and disk storage

• There is need to develop automatic hexahedron mesh generators for use in analysis

• There are good Tetrahedron element automatic mesh generation algorithm like

1. Advancing front method
2. Voronoi method

• The CPU time for creation of Hex mesh is more Than Tet mesh for same number of nodes.

• Good Tetrahedron elements can be fitted at the pointed corner of a solid model by using methods like advance front method.

• Tetrahedron mesh refinement is easier than Hexahedron mesh refinement as algorithm for Tetrahedron refinement is clearly defined when compared to Hexahedron elements. In local and global refinement Hexahedrons can be split into good 5 or 6 Tetrahedrons while splitting of Tetrahedrons into Hexahedrons can be defined, but, generally this type of operation leads to bad results for elements shapes.

• Always the face of a Tetrahedron element lies in a plane but a face of a Hex element may not lie on the plane surface and large amount of this warping means not a good element for finite element analysis.

"Automatic Mesh Generation", P.L. George, John Wiley & Sons.