Linear Structural Analysis for a 3D Body.

The Problem.
 This case study solves a linear structural analysis problem for a 3-D object tracted by nodal loads. The components of the solution U = (U1,U2,U3) are the displacements in the x-, y-, and z-directions. This example was provided with VECFEM version 1.2 and we use it here with permission. For a theoretical discussion of the VECFEM implementation of linear structural analaysis, click here.

Use PDELab to ...

Why use PDELab?
 PDELab's VECFEM stress template is used to input the required parameters (elasticity and Passion's number) which descibe the isotropic material from which the 3D object is constructed. A PATRAN neutral file describing the nodes, elements, and the nodal forces must be supplied by the user. PDELab generates the necessary VECFEM problem definition functions and parameters automatically from within the Framework Editor. The output PATRAN components file contains the x-, y-, and z- displacements.

The original VECFEM Package (version 1.2) requires users to write fortran functions to define the system of PDEs using the variational form. Users must also write a main driver program which allocates the necessary memory and calls the appropriate solver routines.
PDELab simplifies the process of specifying a problem to VECFEM by providing

Define the Problem
Step 1:When PDELab top level window appears, select 3-D and FEM and click on New File. The PDELab 3-D Finite Element Method Session appears. The Session toolkit is to the right of the Session window.

Step 2: Click on the Framework Editor in the Session toolkit.

Step 3: Choose VECFEM from the Framework menu and Stress Template from the Equation Form menu. The Framework Editor allows users to select the framework and form of the equation for defining the problem. In this case, selecting the VECFEM stress template tells the symbolic processor that a linear structural analysis problem will be solved, and the user should provide only the values for elasticity and Poisson's number which describe the material. The framework selection also determines the code that will be generated automatically by GenCray to define the problem for the PDELab-VECFEM interface.
Click on PDE System and enter the two parameter values in the PDE System editor as shown here. Click on Done when finished.

Step 4: The template values have been specified, so click on Generate Framework. The Maxima symbolic processor is accessed to check for errors, and the GenCray code generator generates the appropriate PDELab problem specification. Any input errors are printed out in the Maxima window of the Framework Editor.

Step 5: When the input has been successfully processed, select Quit from the File menu. The PDELab language program appears in the Session window. The generated program is shown below. The boundary segment lists an external specification for the domain definition. The PDELab triple lists the vecfem solver with default parameters.

PDELab language `.e' file 

OPTIONS.
         vecfem
         space dimension of system = 3
         pdes in system = 3
         number of right-hand-sides = 1
EQUATION.
         vecfem
BOUNDARY.
         external
TRIPLE.
         vecfem &
       (displacement = .true., nodalforce = .true., &
        dirichlet = .true., outflag = .true., order=3, &
        method=123, precond=1, symflag = .true., &
        optflag=.true., tol =  1.D-7, &
        pellpackoutflag=.true., pellpackoutfile='stress.out')

SUBPROGRAM.
      real function vfl3d (t, group, comp1, comp2, x1, x2, x, y, z,
     .                l, lt, ele, nk, dim, u, dudx, ut, dutdx)
      integer group, comp1, comp2, x1, x2, l, lt, ele, nk, dim
      real t, x, y, z, u(l,nk), dudx(l, nk, dim), ut(lt, nk),
     .      dutdx(lt, nk, dim)
      real data(9,9)
      real E, NY
      E = 1.93d4
      NY = 0.3

      c11=E*(1.-NY)/(1+NY)/(1-2*NY)
      c22=E*(1.-NY)/(1+NY)/(1-2*NY)
      c33=E*(1.-NY)/(1+NY)/(1-2*NY)
      c44=E/2./(1+NY)
      c55=E/2./(1+NY)
      c66=E/2./(1+NY)
      c12=E*NY/(1+NY)/(1-2*NY)
      c13=E*NY/(1+NY)/(1-2*NY)
      c23=E*NY/(1+NY)/(1-2*NY)

c     initialize the 9x9 matrix
      do 10 i1 = 1, 9
         do 20 i2 = 1, 9
            data(i1,i2) = 0.0
 20      continue
 10   continue
      data(1,1) = c11
      data(1,5) = c12
      data(1,9) = c13
      data(2,2) = c66
      data(2,4) = c66
      data(3,3) = c55
      data(3,7) = c55
      data(4,2) = c66
      data(4,4) = c66
      data(5,1) = c12
      data(5,5) = c22
      data(5,9) = c23
      data(6,6) = c44
      data(6,8) = c44
      data(7,3) = c55
      data(7,7) = c55
      data(8,8) = c44
      data(9,1) = c13
      data(9,5) = c23
      data(9,9) = c33

      if ((x1 .eq. 0) .or. (x2 .eq. 0)) then
        vfl3d = 0.0
      else
        vfl3d = data ((comp1-1)*3+x1, (comp2-1)*3+x2)
      endif

      return
      end
END.


Specify the Solution
Step 6: We will input the neutral format mesh file generated by the external mesh generator (PATRAN). Click on the MeshGenerator and identify the mesh to use for the problem.

Step 7: Click on Save and the following information is placed in the PDELab session: 

 
mesh.
   read neutral from file  &
      (filename='elbow.ntl', &
      i1ndim=3, i3nint=5, i3ntyp=2, &
      i1mnpt=352, i1melm=1134, &
      edge=2.365844, npatches=2)
You can now close the Mesh Generator by clicking on Quit.

Note that PDELab provides other 3D mesh generators, such as GEOMPACK and QMG, which generate meshes with tetrahedral and other element types. These mesh generators require a poly-file description of the surface of the object. Surface patches can also be established during the domain building process which will be maintained during mesh generation. The boundary conditions are then applied to the surface patches or to specific surface nodal points identified by the user. PDELab's 3D Boundary Conditions Tool supportes the application of boundary conditions to a 3D mesh stored in neutral format or PDELab format.

Step 8: You can save the PDELab language program to a `.e' file by clicking on the SaveAs button in the Session toolkit. A Filename dialog requests a name for your .e file.

A portion of the completed '.e' file in the PDELab Session is shown here, with the 3D FEM toolkit to the right of the Session window.

Execute the Problem
Step 9: We are now ready to enter the ExecuteTool environment, where the PDELab language program will be processed. A fortran main program will be generated from the .e file. It will be compiled and linked with the PDELab module libraries and the standard PDELab I/O libraries. Click on the ExecuteTool button in the Session toolkit.

Step 10: The PDELab .e file is loaded automatically. Click on the Run button The trace of the process for compiling and linking the PDELab executable appears in the trace window. When the compiling and linking process is complete, the executable program is run and the solution and trance files are generated and placed in your user directory. Click on Quit to exit the ExecuteTool.

Visualize the Solution
Step 11: The solution output has been generated during execution. You should find the output file containing the PDELab format data in the location specified by the save segment. If no path was specified, this file is located in the execution directory. The file contains the solutions U1, U2, and U3. Click on the OutputTool in the Session toolkit. Select PDELab Solution in the File menu. Select the output file that was generated by your program when the OutputTool requests the file name. The Select Solution Box is displayed after the two files are loaded. The selection box contains the solutions U1,U2 and U3.

 
Deformation of the body
 X Displacement
 
Y Displacement
  
Z Displacement
/TD>