Overview:
There
are two common methods of solving problems within finite element programs. These methods are the h-method and the
p-method. With each method the geometry
to be analyzed is broken into finite elements.
The difference between the two methods lies in how these elements are
treated. The h-method uses many simple
elements, whereas the p-method uses few complex elements.
H-Method:
The
simplest type of element has a linear shape function. This means that the function for displacement
across the element is linear. With the
h-method, the shape function of the element will usually be linear. In an actual part, it is quite uncommon for
the displacement to vary linearly. The
h-method accounts for this by increasing the number of elements. More accurate information is obtained by
increasing the number of elements.
The
name for the h-method is borrowed from mathematics. The finite element method was originally
developed by the work of mathematicians, particularly those who worked in the
area of numeric integration. The
variable h is used to specify the step size in numeric integration. This variable name carried over into finite
element analysis.
The
upside to only using linear shape functions is that it easy to solve the element
equations. The downside is that the
strain across the element must be constant.
Strain is defined as the change in displacement divided by the original
length. Since the displacement function
is linear, strain must be constant throughout the element. Stress is derived from strain by using the
modulus of elasticity, which is a constant.
Therefore, stress in an element with a linear shape function must be
constant.
Suppose that the actual stress across a part varied by the function represented by the curve in Figure 1. If the problem was analyzed using linear shape functions, then the results for a course mesh would be represented by the bars in Figure 1
Figure 1: H-Method with Course Mesh
If a part is modeled with a very course mesh, then the stress distribution across the part will be very inaccurate. In order to more accurately find the stress distribution across the part, we will need to increase the number of elements. If the numbers of elements are doubled, then the stress distribution would be represented by the bars in Figure 2.
Figure 2: H-Method with Fine Mesh
To save computing time, it is most beneficial to increase the number of nodes only in the areas where more nodes are necessary. If a large section of the part is under a constant stress, then only a few elements will be required. This will save a lot of computing time.
The number of elements must only be increased in areas where the stress is changes quickly over a small distance. This could be the area where a load is applied, around a hole, or where geometry is changing. In these areas the stress can change dramatically over a very small distance. It is up to the user to determine where more elements will be required to obtain an accurate solution.
P-Method:
With
p-elements, once a mesh is created, it does not need to be changed. Rather than changing the number of elements,
the shape function of the element will be changed to handle non-linear
displacement functions. In areas where
the stress is changing quickly, the complexity of the shape function is changed
rather than changing the size of the elements.
More accurate information is obtained by increasing the complexity of
the shape function.
The
p in p-method stands for polynomial.
Increasing the polynomial order of the shape function changes the
accuracy of the p-method. This allows a
very complex displacement function to be approximated across a large element.
If
the shape function is second order, then the strain across the element will be
linear. Using the same example stress
distribution as before, the results would be represented by the bars in Figure 3.
Figure 3:
P-Method with 2nd Order Polynomial
If
this does not accurately reflect the strain in the element, then the order of
the shape function can be increased to third order. This will allow strain over the element to be
a second order function. The results
would be represented by the bars in Figure
4.
Figure 4: P-Method with 3rd Order Polynomial
Often
with p-method programs, the polynomial order can be increased as high as
nine. This allows for an eighth order
strain function over the element
The
upside to the p-method is that high ordered shape function can approximate the
strain distribution in an element very closely.
The downside is that it requires a lot of computing time to solve a high
order shape function. In order to save
computing time, it is beneficial to only increase the polynomial order in the
elements where more complex shape functions are needed.
Summary:
In
terms of computing time, the h and p methods will have about the same
requirements. In each case you are
solving for about the same number of nodes.
The h-method increases the number of nodes by adding more elements. The p-method increases the number of nodes by
increasing the order of the shape function.
