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.

h-method采用简单的形状函数，要增加节点通过增加单元数量而各个单元形状函数不变

p-method采用比较复杂单元形状函数，要增加节点通过增加单元阶数，而数量不变。

h-method适用于任何类型分析，但是p-method只适用于线性结构静力分析，根据所要解决的问题，h-method需要的网格比p-method更细。p-method用较粗糙的网格也能达到较高的精度。

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 number 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.

h-method采用简单的形状函数，要增加节点通过增加单元数量而各个单元形状函数不变

p-method采用比较复杂单元形状函数，要增加节点通过增加单元阶数，而数量不变。