5.8 MOTORCYCLE CRANKSHAFT, TETRAHEDRON NO. 16

Copy the sample file B11_G.COS to the Z88 input file Z88G.COS.

We want to compute a crankshaft for a monocylinder motorcycle engine and put a force of -5,000 N onto the piston. The meshing will do Pro/ENGINEER.

The boundary conditions are a bit tricky for this example: Put a reference (or datum) point to the center of the face of the crankshaft. We'll need this point to fix the crankshaft in Z direction, i.e. lengthwise.

The ball bearings, which allow always some angular movement, and, thus, should be regarded as moment- free supports, are fastened to the larger shaft axises. The flange facings of the shaft axises are to be fixed in X and Y direction. Because whole surfaces are fixed, don't allow one ore more of these surfaces to be fixed in Z direction, too. This would result in blocking the angular movement - try it, if you won't believe it.

A total force of -5,000 N will be put onto the peripheral surface of the crankshaft journal.

The mesh is automatically generated by Pro/MECHANICA featuring parabolic tetrahedrons. After storing the COSMOS file, a Z88 session may start:

Copy B11_G.COS to Z88G.COS, the COSMOS file for the converter Z88G

Start converting Z88G.COS with Z88G

Windows: COSMOS converter Z88G. Looks quite similar on UNIX machines.

and proceed with the Cuthill- McKee algorithm Z88H, because we'll expect a very bad node-numbering for the parabolic tetrahedrons.

Windows: Cuthill-McKee program Z88H. Looks quite similar on UNIX machines.

The first line of Z88I1.TXT tells you the following values:

- 6,826 nodes
- 3,941 elemente
- 20,478 dof

MAXKOI must have as a minimum 3,941 elements * 10 nodes per element = 39,410.

Thus, Z88.DYN should look as follows:

MAXGS when starting, any value
MAXKOI minimum 39410
MAXK minimum 6826
MAXE minimum 3941
MAXNFG minimum 20478
MAXNEG minimum 1

Proceed with a look at the structure with Z88O .

The computing time with Z88F is about 16 sec. on a PC (AMD Athlon 64 X2 3800+ processor, 4 GByte memory, Windows XP). Enter a value of about 11,400,000 for MAXGS..

See the deflected structure with Z88O. The angular deflection of the axises is quite amazing. Now you would read off the deflections of distinguished nodes, multiply with the appropriate lever arms and check with the bearing catalogue if your ball bearings will allow this angular movement without problems.

Windows: Computing deflections with Z88F. Looks quite similar on UNIX machines.

Windows: Plot programm Z88O, undeflected structure.

Windows: Plot programm Z88O, deflected structure.

Now we'll launch the sparse matrix iteration solver Z88I1 and Z88I2. To begin with, we'll try 1,000,000 for MAXIEZ in Z88.DYN :

COMMON START

MAXGS 11500000 > has for Z88I1 no meaning !

MAXKOI 40000 > must always be large enough !

MAXK 7000 > read off from Z88I1.TXT

MAXE 4000 > read off from Z88I1.TXT

MAXNFG 21000 > read off from Z88I1.TXT

MAXNEG 1 > read off from Z88I1.TXT

MAXPR 1 > has no meaning for this example

MAXRB 903 > read off from Z88I2.TXT

MAXIEZ 1000000 > important for Z88I1

MAXGP 500000 > used by Z88O for the Gauss points

COMMON END

``

Windows: Part 1 of the Sparse Matrix Solver

Our entries did work properly (otherwise, you would have to increase MAXIEZ) and the sorting times was just a breeze.

Read off for MAXGS: 768,687, rounded up 770,000. This looks fairly better than the direct Cholesky solver Z88F with its need of 11,381,064 8-Byte elements = 87 Mbyte. The second part of the iteration solver, i.e. Z88I2, will only need 768,687 8-Bytes elements = 6 MByte.

Thus, we would adjust the memory in Z88.DYN as follows (feel free to enter even bigger values):

COMMON START

MAXGS 770000 > important !

MAXKOI 40000 > must always be large enough !

MAXK 7000 > read off from Z88I1.TXT

MAXE 4000 > read off from Z88I1.TXT

MAXNFG 21000 > read off from Z88I1.TXT

MAXNEG 1 > read off from Z88I1.TXT

MAXPR 1 > has no meaning for this example

MAXRB 903 > read off from Z88I2.TXT

MAXIEZ 1000000 > not used by Z88I2

MAXGP 500000 > used by Z88O for the Gauss points

COMMON END

``

If you adjust the iteration parameters in Z88I4.TXT (chapter 3.6) as follows:

`10000 1e-7 0.0001 1. 1`

i.e. a maximum of 10,000 iterations, EPS with 1E-7 and ALPHA with 0.0001, then this results in a computing time of about 12 sec. on a PC (AMD Athlon 64 X2 3800+ processor, 4 GByte memory, Windows XP).

In this case, both the iteration solver and the direct Cholesky solver need about the same time, but the iteration solver needs fewer than one tenth of memory. For large structures, things get even worse for the Cholesky solver! But pay attention to the fact, that you can't really compare the computing times. Try other entries for EPS, for example 1E-5 (resulting in 296 iterations and 11 seconds) or 1E-10 (resulting in 329 iterations and 13 sec.), and see the different computing times.

Windows: The sparse matrix iteration solver Part 2, i.e. Z88I2.

However, a very nice experiment is this:

Start from the very beginning, run Z88G, but not the Cuthill- McKee algorithm Z88H. Launch directly after Z88G a test run with Z88F :

Windows: The direct Cholesky solver in test mode.

Gee, see the faces falling: now we would need 184,122,663 8- Byte elements = 1,4 GByte. Absolutely no need for this!

However, run again the iteration solver part 1, i.e. Z88I1. This will again result in only 768,687 elements for the total stiffness matrix. Calculate, please:

184,122,663 : 768,687 = 240 : 1

The second part of the iteration solver, i.e. Z88I2, needs now some more iterations (350 in contrary to 415 with an equal EPS of 1E-7), because the matrix features the same number of non-zero elements, though, but the condition is worse because of the very bad node-numbering of Pro/MECHANICA. That means: When using the iteration solver you don't need to run the Cuthill-McKee algorithm Z88H for reducing the storage needs of the iteration solver (in contrary to the direct Cholesky solver Z88F, which may depend heavily on Z88H for larger structures!). However, Z88H may improve the matrix condition anyway.

Try now the direct Sparse Matrix Solver with Fill-In:

Much faster runs the solver pair Z88I1 and Z88PAR. Adjust the 5th entry in Z88I4.TXT to the number of CPUs. Keep in mind that Z88PAR deals heavily with dynamic memory when running. This may cause serious trouble when computing very large structures. For this example the elapsed time was ~ 4 sec. with two CPUs.

Windows: The direct Sparse Matrix Solver Z88PAR.