RE: Solve of inverse quadratic finite element problem - resend

• To: mathgroup at smc.vnet.net
• Subject: [mg36482] RE: [mg36453] Solve of inverse quadratic finite element problem - resend
• From: "DrBob" <drbob at bigfoot.com>
• Date: Sun, 8 Sep 2002 03:31:26 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```If we do think of x1...x20 as constants, it's still fourth-order, with
three simultaneous equations.  Evaluate this:

Solve[a x^4 + b x^3 + c x^2 + d x + e == 0, x]

and look at the four solutions found -- for a single variable and one
equation!

For your problem, I'm thinking there'd be at least 4^3 solutions, each
of them even more complicated than these, and an unknown number of
contingencies to deal with, depending on values of the xi.  Check each
radical for a negative argument, and the special cases multiply quickly.
For some values of the xi, there'd be no solutions; for other values,
many solutions.

If you do manage to solve this, I'd love to see the method!

Bobby

-----Original Message-----
From: DrBob [mailto:drbob at bigfoot.com]
To: mathgroup at smc.vnet.net
Subject: [mg36482] RE: [mg36453] Solve of inverse quadratic finite element problem
- resend

If x1 through x20 aren't unknowns, you should have told me their values.
If you can't, they are UNKNOWN.

True, you're not solving for them... but they affect the dimensionality
of the problem just as if you WERE solving for them.  You have a
20-dimensional space full of contingencies -- solution forms that depend
on the values of x1 through x20.  The Solve function won't deal with
contingencies even if it could, and if it tried, there would be too
many.

Bobby

-----Original Message-----
From: purcell [mailto:chris.purcell at drdc-rddc.gc.ca]
To: mathgroup at smc.vnet.net
Subject: [mg36482] RE: [mg36453] Solve of inverse quadratic finite element problem
- resend

The constants x1 through x20 are constants.
That leaves 3 equations in 3 unknowns.
Thanks for looking at this. Chris.

At 06:40 PM 9/7/2002 -0500, you wrote:
>You have three nonlinear (fourth-order) equations and 23 unknowns.  A
>faster computer won't help any.
>
>Bobby
>
>-----Original Message-----
>From: purcell [mailto:chris.purcell at drdc-rddc.gc.ca]
To: mathgroup at smc.vnet.net
>Sent: Saturday, September 07, 2002 1:54 AM
>Subject: [mg36482] [mg36453] Solve of inverse quadratic finite element problem -
>resend
>
>
>Would someone with a very fast machine and lots of memory be willing to
>try
>this Solve for me?
>It is the inverse of the 20 node quadratic hexahedral mapping used in
>finite element analysis.
>None of my computers can handle this - they run out of memory (using
>Version 4.2).
>
>(*Hex20 Node definition in global coordinates *)
>Clear[
>x1, y1, z1,
>x2, y2, z2,
>x3, y3, z3,
>x4, y4, z4,
>x5, y5, z5,
>x6, y6, z6,
>x7, y7, z7,
>x8, y8, z8,
>x9, y9, z9,
>x10, y10, z10,
>x11, y11, z11,
>x12, y12, z12,
>x13, y13, z13,
>x14, y14, z14,
>x15, y15, z15,
>x16, y16, z16,
>x17, y17, z17,
>x18, y18, z18,
>x19, y19, z19,
>x20, y20, z20];
>
>(* local coordinates *)
>Clear[u, v, w];
>
>(* Global co-ordinates *)
>Clear[x, y, z];
>
>(* corner nodes *)
>N1= (1-u)*(1-v)*(1-w)*(-2-u-v-w)/8;
>N3= (1+u)*(1-v)*(1-w)*(-2+u-v-w)/8;
>N5= (1+u)*(1+v)*(1-w)*(-2+u+v-w)/8;
>N7= (1-u)*(1+v)*(1-w)*(-2-u+v-w)/8;
>N13=(1-u)*(1-v)*(1+w)*(-2-u-v+w)/8;
>N15=(1+u)*(1-v)*(1+w)*(-2+u-v+w)/8;
>N17=(1+u)*(1+v)*(1+w)*(-2+u+v+w)/8;
>N19=(1-u)*(1+v)*(1+w)*(-2-u+v+w)/8;
>(*  to u nodes *)
>N2= (1-u^2)*(1-v)*(1-w)/4;
>N6= (1-u^2)*(1+v)*(1-w)/4;
>N14=(1-u^2)*(1-v)*(1+w)/4;
>N18=(1-u^2)*(1+v)*(1+w)/4;
>(*  to v nodes *)
>N4= (1+u)*(1-v^2)*(1-w)/4;
>N8= (1-u)*(1-v^2)*(1-w)/4;
>N16=(1+u)*(1-v^2)*(1+w)/4;
>N20=(1-u)*(1-v^2)*(1+w)/4;
>(*  to w nodes *)
>N9= (1-u)*(1-v)*(1-w^2)/4;
>N10=(1+u)*(1-v)*(1-w^2)/4;
>N11=(1+u)*(1+v)*(1-w^2)/4;
>N12=(1-u)*(1-v)*(1-w^2)/4;
>
>(* solve the inverse transform *)
>Solve[{
>x1*N1+x2*N2+x3*N3+x4*N4+x5*N5+x6*N6+x7*N7+x8*N8+x9*N9+x10*N10+
>x11*N11+x12*N12+x13*N13+x14*N14+x15*N15+x16*N16+x17*N17+x18*N18+x19*N19
+
>x20*N20-x==0,
>y1*N1+y2*N2+y3*N3+y4*N4+y5*N5+y6*N6+y7*N7+y8*N8+y9*N9+y10*N10+
>y11*N11+y12*N12+y13*N13+y14*N14+y15*N15+y16*N16+y17*N17+y18*N18+y19*N19
+
>y20*N20-y==0,
>z1*N1+z2*N2+z3*N3+z4*N4+z5*N5+z6*N6+z7*N7+z8*N8+z9*N9+z10*N10+
>z11*N11+z12*N12+z13*N13+z14*N14+z15*N15+z16*N16+z17*N17+z18*N18+z19*N19
+
>z20*N20-z==0},
>{u,v,w}]
>Christopher J. Purcell
>9 Grove St., PO Box 1012
>Tel 902-426-3100 x389, Fax 902-426-9654
>E-mail:    chris.purcell at drdc-rddc.gc.ca

Christopher J. Purcell
9 Grove St., PO Box 1012