Extended modules > Creo Elements/Direct Finite Element Analysis > Basic finite element theory
  
Basic finite element theory
One of the important concepts needed to understand stress analysis (Linear Static Analysis) using the finite element method is Hooke's Law. Robert Hooke determined what is now the basis for modern finite element linear static stress analysis in 1678. Hooke's Law states, "elastic bodies stretch (strain) in proportion to the forces (stresses) on them." The formula can be expressed as:
F=kx
F= force
k= proportional constant
x = strain distance
Many other types of physical phenomena can be handled using formulae derived from Hooke's Law.
Imagine a glass placed on a table. It is then divided mathematically into 5000 little "pyramid" elements. Each element has 5 corners, or nodes. All nodes on the bottom of the glass are fixed, or "constrained", so they cannot move (in technical terms, we impose zero displacement on these nodes). Now, if we press down on one node near the top of the glass, we apply a load.
Since all materials have some elasticity, the node will move slightly. That movement would be according to the formula, F=kx, for that individual element but other elements are in the way or impeding its movement. Nevertheless, the force transmitted through the first element has an effect on the other nodes and they will effect others throughout the glass.
In the finite element method, an essential calculation is made called "element stiffness formulation." In this step a proportional constant, "k", is created for the relationship between all the nodes of each element. Therefore, we can imagine every node as being connected to the other nodes on an individual element by a spring. And the spring will respond to stresses according to Hooke's equation, F=kx.
Now we have reduced the glass to a collection of interrelated springs. When we carry out the analysis a value for each "x" and "F" is determined at every node using the formula F=kx. Note: "F" and "x" are vectors as each has a value and a direction.
In the last phase of analysis the stresses are determined by knowing the value of "F" at each node and the geometry of each element.
H-element adaptivity and p-element adaptivity
This section provides a brief description of:
h-adaptivity and h-elements
p-adaptivity and p-elements
In traditional finite element analysis as the number of elements increases, the accuracy of the solution improves. The accuracy of the problem can be measured quantitatively with various entities, such as strain energies, displacements, and stresses, as well as in various error estimation methods, such as simple mathematical norm or root-mean-square methods. The goal is to perform an accurate prediction on the behavior of your actual model by using these error analysis methods.
You can modify a series of finite element analyses either manually or automatically by reducing the size and increasing the number of elements. This is the standard h-adaptivity method. Each element is formulated mathematically with a certain predetermined order of shape functions. This polynomial order does not change in the h-adaptivity method. The elements associated with this type of capability are called the h-elements.
A different method to modify the subsequent finite element analyses on the same problem is to increase the polynomial order in each element while maintaining the original finite element size and mesh. The increase of the interpolation order is internal, and the solution stops automatically once a specified error tolerance is satisfied. This is known as the p-adaptivity method. The elements associated with this capability are called the p-elements.
P-elements have higher-order polynomials, which provide better representation of complex stress fields.
Geometry and loads can be represented more accurately
Accuracy of the analysis is controlled primarily by the polynomial level, not by the element size.
As with the loads, it is important to use distributed boundary conditions for p-version elements in detailed stress analysis. Concentrated boundary conditions also cause singularities in the stress field and should be avoided.
In summary:
H-Adaptivity:
The size of elements is reduced and their number is increased.
P-Adaptivity:
The polynomial order is increased in each element while maintaining the original finite element size and mesh. The increase of the interpolation order is internal, and the solution stops automatically once a specified error tolerance is satisfied.