Solve of inverse quadratic finite element problem - resend

• To: mathgroup at smc.vnet.net
• Subject: [mg36453] Solve of inverse quadratic finite element problem - resend
• From: purcell <chris.purcell at drdc-rddc.gc.ca>
• Date: Sat, 7 Sep 2002 02:54:18 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```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