The Eight-Node Hexahedral "Brick" Element in Finite Element Analysis
ALGOR Development Staff
ALGOR, Inc., Pittsburgh, PA U.S.A
Once a part or product has been designed, it must be tested to ensure it will perform according to specifications in the real world. Rather than go through the expensive and time-consuming process of building prototypes and subjecting them to destructive testing, the standard practice for the past decade has been to test a model of the part or product on the computer using finite element analysis (FEA).
The objective of finite element analysis of real world models is to simulate destructive testing using a minimum amount of computer memory, computation time and modeling time. The concept of FEA is simple and well-understood: The design is turned into a mesh of finite elements. FEA software then tests each finite element for how it responds to such phenomena as stress, heat, fluid flow or electrostatics. FEA has been key in transferring design and analysis from drafting boards and slide rules to computer screens.
Figure 1: The diameter of the circle (h) defines the size of this 2-D triangular element.
A mechanical engineer simulating a design can select from a variety of element types when building an FEA model. The principal issue in selecting a finite element type is accuracy. Until recently, the engineer would build the solid mesh manually, attempting to make an accurate representation of the part design.
The physical system describing the design of a typical part or die often has a complex geometry, and building the software model is therefore an intricate process. A number of software programs now exist which automatically or semi-automatically build the mesh, in some cases, directly from the CAD design. Because the engineer typically goes through many design and analysis cycles before determining the optimal design, automatic mesh generators such as Algors HypergenTM and HexagenTM have become popular. All other variables being equal, an automatic mesh generator is by definition more accurate, since it minimizes the element of human error in the transformation of a design to a solid finite element mesh.
When determining which mesh generation software to use, the engineer must evaluate the type of finite element that will be the basis of the FEA model. Elements differ in many ways, but for analysis, the most significant items are the shape of the element and its order of interpolation, which refers to the degree of the complete polynomial appearing in the element shape functions. There will be an order of polynomial for the element, termed the p. There is also a size for the element, termed the h. Size h is usually the diameter of the smallest circle (smallest sphere for a three-dimensional element) that encloses the element (see Figure 1). Every element has a size h and an order p.
FEA is a discretization technique that provides approximate answers to a physical system described in terms of a mathematical model, which is usually a system of partial differential equations (PDEs). Discretization is a method that approximates a physical system that has an infinite number of degrees of freedom, using a model that allows only finite degrees of freedom.
For example, consider a simple beam. It can bend an infinite number of ways, depending on the load and restraints. A discretization technique may assume that the beam can only bend such that its deformed shape is a cubic spline. Clearly, the physical system has many more degrees than the model created by the discretization technique. It also follows that there is more than one way of discretizing a physical system, with an infinite number of degrees of freedom. Incidentally, this example would not have sufficient discretization for accurate FEA.
FEA, therefore, provides approximate answers to a physical system. If u is the exact solution for the PDE, FEA will produce an approximation uh. The approximation uh will converge to the exact solution u of the mathematical model under certain conditions: when the mesh size (h) decreases to zero or when the element order (p) is increased to infinity.
There are three basic approaches to FEA: the h, p and h-p methods. With the h method, the element order (p) is kept constant, but the mesh is refined infinitely by making the element 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 will reduce the error in the FEA approximation. FEA software exists for all three methods. Before examining which may be superior, one must first determine which element type results in greater model, and therefore analysis, accuracy.
Element types include eight-node hexahedrons, four-node tetrahedrons and ten-node tetrahedrons, but eight-node hexahedrons, which part and die designers call bricks, lead to more reliable FEA solutions. There are many reasons why the eight-node hexahedral 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 a linear strain variation displacement mode. Tetrahedral elements are also linear, but can have more discretization error because they have a constant strain.
One cannot really compare the discretization error of a single eight-node hexahedral element and a single four-node tetrahedral element, since the solution cost is directly proportional to the number of nodes. A more appropriate comparison is between an eight-node hexahedron comprised of five tetrahedrons and a single eight-node hexahedron (see Figure 2., which was generated using HypergenTM, Algors automatic tetrahedral mesh generator). The five tetrahedrons will together have more discretization error than the eight-node "brick" because the five tetrahedrons cannot assume all the displacement fields handled by the eight-node element.
Figure 2: Five (5) 4-node tetrahedral elements comprise a single 8-node brick element.
Besides being more accurate, the hexahedral element presents other advantages in FEA model building. Meshes comprised of hexahedrons are easier to visualize than meshes comprised of tetrahedrons. In addition, the reaction of hexahedral elements to the application of body loads more precisely corresponds to loads under real world conditions. The eight-node hexahedral elements are therefore superior to tetrahedral elements for finite element analysis.
The question remains as to whether eight-node brick linear hexahedrons are superior to higher-order elements (p > 1), be they p elements (p method) or higher-order h elements (h-p method; see Figure 3) for building the solid mesh model of the part or die. Proponents of higher order elements (which require more nodes per element) claim that using a smaller number of larger-size 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 higher order elements have less discretization error, even for a coarse mesh.
There is a major logical flaw in this claim: Most parts and products have complex geometries which require fine meshing to accurately resolve the geometry as a solid mesh. The mesh size is so small that the discretization error does not exceed what is required for engineering accuracy. Use of p elements and higher order h elements with mid-side nodes therefore offers no practical engineering benefit over use of eight-node hexahedrons.
The p method suffers from its own accuracy problems, related to the fact that the larger the elements, the greater the effect of each element 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 pollute, that is, permeate adjacent elements. The pollution problem can seriously impact the accuracy of results because it affects stresses and fluxes. Since geometric and load singularities are common in most designed 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 singularities. Refinement of these elements defeats the very purpose of using p or higher order h elements for FEA because the refinements take time to make.
Figure 3: This "high-order" brick element includes mid-side nodes.
Eight-node hexahedrons capture the singularities of the model at much less cost because they consume much less computer time and memory than the processing of p and higher order h elements. For a mesh of p or higher order h elements, the bandwidth minimizer consumes more disk space and CPU time, and also produces much wider bandwidths. A larger bandwidth increases solution time, since the solution time is proportional to the square of the bandwidth. Finally, eight-node brick hexahedral elements can be easily degenerated to lower order elements (transition and degenerative elements) maintaining spatial isotropy; the same cannot be said for higher order elements.
In conclusion, while there may be perceived theoretical advantages to the p or h-p methods, the eight-node hexahedral "brick" element using the h method is superior to other element formulations for the practical purpose of accurate and fast finite element analysis of real world part and products. A more accurate FEA model leads to more accurate analysis, which in turn results in manufactured products that perform to specification.