Using P-Elements

Introduction

This tutorial was completed using ANSYS 7.0. This tutorial outlines the steps necessary for solving a model meshed with p-elements. The p-method manipulates the polynomial level (p-level) of the finite element shape functions which are used to approximate the real solution. Thus, rather than increasing mesh density, the p-level can be increased to give a similar result. By keeping mesh density rather coarse, computational time can be kept to a minimum. This is the greatest advantage of using p-elements over h-elements.

A uniform load will be applied to the right hand side of the geometry shown below. The specimen was modeled as steel with a modulus of elasticity of 200 GPa.

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Preprocessing: Defining the Problem

1. Give example a TitleUtility Menu > File > Change Title ...
   /title, P-Method Meshing
2. Activate the p-Method Solution OptionsANSYS Main Menu > Preferences
   /PMETH,ONSelect p-Method Struct. as shown below
   
3. Open preprocessor menuANSYS Main Menu > Preprocessor
   /PREP7
4. Define KeypointsPreprocessor > Modeling > Create > Keypoints > In Active CS...
   K,#,x,y,zWe are going to define 12 keypoints for this geometry as given in the following table:
   
   

   Keypoint	Coordinates (x,y,z)
   1	(0,0)
   2	(0,100)
   3	(20,100)
   4	(45,52)
   5	(55,52)
   6	(80,100)
   7	(100,100)
   8	(100,0)
   9	(80,0)
   10	(55,48)
   11	(45,48)
   12	(20,0)

5. Create AreaPreprocessor > Modeling > Create > Areas > Arbitrary > Through KPs
   A,1,2,3,4,5,6,7,8,9,10,11,12Click each of the keypoints in numerical order to create the area shown below.
   
6. Define the Type of Element
Preprocessor > Element Type > Add/Edit/Delete...For this problem we will use the PLANE145 (p-Elements 2D Quad) element. This element has eight nodes with 2 degrees of freedom each (translation along the X and Y axes). It can support a polynomial with maximum order of eight.
After clicking OK to select the element, click Options... to open the keyoptions window, shown below. Choose Plane stress + TK for Analysis Type.

Keyopts 1 and 2 can be used to set the starting and maximum p-level for this element type. For now we will leave them as default.
Other types of p-elements exist in the ANSYS library. These include Solid127 and Solid128 which have electrostatic DOF's, and Plane145, Plane146, Solid147, Solid148 and Shell150 which have structural DOF's. For more information on these elements, go to the Element Library in the help file.

7. Define Real Constants
Preprocessor > Real Constants... > Add...In the 'Real Constants for PLANE145' window, enter the following geometric properties:

1. Thickness THK: 10
This defines an element with a thickness of 10 mm.

8. Define Element Material PropertiesPreprocessor > Material Props > Material Models > Structural > Linear > Elastic > IsotropicIn the window that appears, enter the following geometric properties for steel:
   
   1. Young's modulus EX: 200000
   2. Poisson's Ratio PRXY: 0.3
9. Define Mesh SizePreprocessor > Meshing > Size Cntrls > ManualSize > Areas > All Areas...For this example we will use an element edge length of 5mm.
10. Mesh the framePreprocessor > Meshing > Mesh > Areas > Free > click 'Pick All'

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Solution Phase: Assigning Loads and Solving

1. Define Analysis Type
Solution > Analysis Type > New Analysis > Static
ANTYPE,0

2. Set Solution Controls
Solution > Analysis Type > Sol'n ControlsThe following window will pop up.

A) Set Time at end of loadstep to 1 and Automatic time stepping to ON
B) Set Number of substeps to 20, Max no. of substeps to 100, Min no. of substeps to 20.
C) Set the Frequency to Write every substep

3. Apply Constraints
Solution > Define Loads > Apply > Structural > Displacement > On LinesFix the left side of the area (ie all DOF constrained)

4. Apply Loads
Solution > Define Loads > Apply > Pressure > On LinesApply a pressure of -100 N/mm^2

The applied loads and constraints should now appear as shown in the figure below.


5. Solve the System
Solution > Solve > Current LS
SOLVE

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Postprocessing: Viewing the Results

1. Read in the Last Data SetGeneral Postproc > Read Results > Last Set
2. Plot Equivalent Stress
General Postproc > Plot Results > Contour Plot > Element SoluIn the window that pops up, select Stress > von Mises SEQV

The following stress distribution should appear.


3. Plot p-Levels
General Postproc > Plot Results > p-Method > p-LevelsThe following distribution should appear.

Note how the order of the polynomial increased in the area with the greatest range in stress. This allowed the elements to more accurately model the stress distribution through that area. For more complex geometries, these orders may go as high as 8. As a comparison, a plot of the stress distribution for a normal h-element (PLANE2) model using the same mesh, and one with a mesh 5 times finer are shown below.


As one can see from the two plots, the mesh density had to be increased by 5 times to get the accuracy that the p-elements delivered. This is the benefit of using p-elements. You can use a mesh that is relatively coarse, thus computational time will be low, and still get reasonable results. However, care should be taken using p-elements as they can sometimes give poor results or take a long time to converge.

ANSYS Command Listing

finish
/clear

/title, P-Method Meshing
/pmeth,on   ! Initialize p-method in ANSYS

/prep7    ! Enter preprocessor

k,1,0,0    ! Keypoints defining geometry
k,2,0,100
k,3,20,100
k,4,45,52
k,5,55,52
k,6,80,100
k,7,100,100
k,8,100,0
k,9,80,0
k,10,55,48
k,11,45,48
k,12,20,0

a,1,2,3,4,5,6,7,8,9,10,11,12 ! Create area from keypoints

et,1,plane145   ! Element type
keyopt,1,3,3   ! Plane stress with thickness option
r,1,10    ! Real constant - thickness
mp,ex,1,200000   ! Young's modulus
mp,prxy,1,0.3   ! Poisson's ratio

esize,5    ! Element size
amesh,all   ! Mesh area

finish
/solu    ! Enter solution phase

antype,0   ! Static analysis
nsubst,20,100,20  ! Number of substeps  
outres,all,all   ! Output data for all substeps
time,1    ! Time at end = 1

lsel,s,loc,x,0   ! Line select at x=0
dl,all,,all   ! Constrain the line, all DOF's
lsel,all   ! Re-select all lines

lsel,s,loc,x,100  ! Line select at x=100
sfl,all,pres,-100  ! Apply a pressure
lsel,all   ! Re-select all lines

solve
finish

/post1    ! Enter postprocessor
set,last   ! Select last set of data
plesol,s,eqv   ! Plot the equivalent stress