Instruction for Implementing Tetrahedron Shape Functions to Finite Element Formulator in CCISE Software Programs (Sequential Testbed I)

Prepared exclusively for Physics 681, Spring 1999 Materials Simulations by David Chen (04/18/99)

• Our objectives are to:
  1. implement the shape functions (or the basic trial functions) for three kinds of tetrahedron elements: linear T4 tetrahedron (T4), quadratic isoparametric T10 tetrahedron (T10), and hierarchical quadratic TetP2 tetrahedron (TP2)
  2. study the solution accuracy and computational time with these element trial functions.

     We will work on the same examples to verify our implementations and fill up the following tables for comparisons when we proceed:

     Table: CubeSimple

     	SigmaYY	Elapsed Time	Iterative Steps
     T4
     T10
     TP2
     REF	1.0

     Table: CubeCrack

     	KI (x=0.5)	Elapsed Time	Iterative Steps
     T4
     T10
     TP2
     REF	5.23 (Single Edge Notch, 2D FRANC2D)

• Open two X terminals. We will use one for FEM analysis and the other for FEM programming implementation.
• Copy the examples into your directory
  cp ~sethna/FEM/examples.tar .
  tar xvf examples.tar
  You will find two examples named CubeSimple and CubeCrack in "examples" directory
• Copy the source codes into your directory
  cp ~sethna/FEM/FEM_Src.tar .
  tar xvf FEM_Src.tar
  You will find the C++ source codes for the FEM formulator in "FEM_Src" directory
• Take a look at the source codes
  [make sure you have "set DISPLAY" to your localhost and have /usr/local/bin in your PATH]
  You can edit the source codes using "nc", "emacs" or "vi". I recommend "nc" if you are new to the UNIX world.
  Type "make" to rebuild all the source codes in your directory. Once the codes are compiled, you should have two executable files "formulator" and "ndstress".
• Let's first implement the shape function for the linear T4 tetrahedron.
  [make sure you are in FEM_Src directory]
  nc ElemShape.cc implement the shape functions and shape function derivatives for T4 in functions "FemVector *T4Shape::shape_functions" and "FemMatrix *T4Shape::shape_derivatives"
  Type "make" to compile the programs
• Let's run FEM analyses with the T4 shape functions [make sure you are in examples directory]
  cp CubeSimple.geo CubeSimple_T4.geo
  cp CubeSimple.msh CubeSimple_T4.msh
  ../FEM_Src/formulate -i4 CubeSimple_T4 [note: the i4 argument for T4]
  ebesolve CubeSimple_T4 CubeSimple_T4 CubeSimple_T4 1.0e-15 2000
  [write down the final elapsed time and iteration steps in the above table]
  fulldisp CubeSimple_T4.neq CubeSimple_T4.dsp
  ../FEM_Src/ndstress -i4 CubeSimple_T4.geo CubeSimple_T4.msh CubeSimple_T4.disp tmp > CubeSimple_T4.str
  [note: the i4 argument for T4]
  franc3d
  "Read FRANC3D File" "CubeSimple.fys" "accept"
  "Read/Write Analysis File", "3D FEM I/O", "Read GeoFmt Results Files" "Visualize/Analyze Results", "Deformation & Contour"
  Show SigmaYY contour plot and write down the value in the table. Do you see a uniform stress distribution for the cube? Do you think the T4 element is adequate to approximate the unknown displacement and stress fields for this problem? Why?
  "end"

  Perform the same procedures for CubeCrack_T4.
  cp CubeCrack.geo CubeCrack_T4.geo
  cp CubeCrack.msh CubeCrack_T4.msh
  ....
  Also, in franc3d do:
  "Visualize/Analyze Results", "Fracture Analysis", "Stress Intensity Factor", accept defaults. It should show three plots with the stress intensity factors along the crack front. Write down KI value at x=0.5.

  Now, do you see a uniform stress distribution for the cube? Do you think the T4 element is adequate to approximate the unknown displacement and stress fields for this problem? Why?

• Let's do the shape function implementation for the quadratic isoparametric T10 tetrahedron. [make sure you are in FEM_Src directory]
  nc ElemShape.cc
  implement the shape functions and shape function derivatives for T10 in functions "FemVector *T10Shape::shape_functions" and "FemMatrix *T10Shape::shape_derivatives"
  Type "make" to compile the programs
• Perform FEM analyses with the T10 shape functions
  cp CubeSimple.geo CubeSimple_T10.geo
  cp CubeSimple.msh CubeSimple_T10.msh
  ../FEM_Src/formulate CubeSimple_T10
  [note: T10 uses the default argument]
  ebesolve CubeSimple_T10 CubeSimple_T10 CubeSimple_T10 1.0e-15 2000
  fulldisp CubeSimple_T10.neq CubeSimple_T10.dsp
  ../FEM_Src/ndstress CubeSimple_T10.geo CubeSimple_T10.msh CubeSimple_T10.disp tmp > CubeSimple_T10.str
  [note: T10 uses the default argument]
  ...
  cp CubeCrack.geo CubeCrack_T10.geo
  cp CubeCrack.msh CubeCrack_T10.msh
  ...

  Now, what do you think? Does the quadratic shape function (T10) give you a better approximation of the solutions than the linear shape function (T4)? Does it take longer computational time and/or iterative steps to solve the problems?

• Finally let's do the hierarchical shape function implementation for the tetrahedron. We will implement a particular case for p=2 (quadratic). [make sure you are in FEM_Src directory]
  nc ElemShape.cc
  implement the shape functions and shape function derivatives for T10 in functions "FemVector *TetP2Shape::shape_functions" and "FemMatrix *TetP2Shape::shape_derivatives"
  Type "make" to compile the programs
• Perform FEM analyses with the TP2 shape functions
  cp CubeSimple.geo CubeSimple_TP2.geo
  cp CubeSimple.msh CubeSimple_TP2.msh
  ../FEM_Src/formulate -h CubeSimple_TP2 [note: the h argument for TP2]
  ebesolve CubeSimple_TP2 CubeSimple_TP2 CubeSimple_TP2 1.0e-15 2000
  fulldisp CubeSimple_TP2.neq CubeSimple_TP2.dsp
  ../FEM_Src/ndstress -h CubeSimple_TP2.geo CubeSimple_TP2.msh CubeSimple_TP2.disp tmp > CubeSimple_TP2.str
  [note: the h argument for TP2]
  ...
  cp CubeCrack.geo CubeCrack_TP2.geo
  cp CubeCrack.msh CubeCrack_TP2.msh
  ...
  
  Now, what do you think? Does the TP2 give you a better approximation of the solutions than T10? Why? How about the computational time and iterative steps? Why?
• You can run the analyses of your own models with T4, T10, and TP2 shape functions as you see fit. Do you reach the same conclusions with other problems? Can you propose a general guideline for using the FEM shape functions?

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).