General Mechanical

pure Neumann problem(heat conduction 2d plate)

• vkr535
Subscriber

Hello Team,

I was trying to model a 2d heat conduction of a rectangular plate with all edge neumann boundary conditions as given in the example 4.9 in the attached image

This problem is ill posed and may be solved using the additional constraint as suggested above. I was trying to reproduce this problem in ANSYS and see if i could solve, but i couldn't do it as the solver catches the pivot error. I attached the snapshot of the model, which is a rectangular plate with length, L and width W. The bottom edge is subjected to non-zero flux condition, while all other edges are insulated(zero-flux).

Approach:

1. The basic idea was to augment the exiting stiffness matrix with extra dof's and then use the additional constraint. So, firstly I meshed the plate with plane55 elements. Then I tried to increase the dof of the model, by adding the extra nodes at the bottom edge and create the combin14(thermal dof) elements with zero stiffness connecting the domain nodes at the bottom edge to the newly created extra nodes.

2. I then applied the constraint equation(CE) to the newly created nodes as per the eqn(4.129).

I end up with the pivot error issue and couldnt make it work. I wrote the APDL input below wherein i tried to use the above approach.

Could you take a look ino this issue and let me know if its possible to solve this

APDL input:

FINI
/CLEAR,NOSTART
!/FILNAME,dpl_str
/CONFIG,NRES,100000
/PREP7
scopt,no
seltol,1e-15
pi = ACOS(-1.0)
dx=2e-06
area=0.1
MP,KXX,1,1

length_1=100e-06
length_2=2*length_1
HEIGHT_1=100e-06
HEIGHT_3=HEIGHT_1+100e-06

!! Modelling
R,1,area
ET,1,plane55
RECTNG,0,length_1,HEIGHT_1,HEIGHT_3

! Meshing
TYPE,1
MAT,1
REAL,1
esiz,dx
allsel,all,all
AMESH,all

nsel,s,loc,y,HEIGHT_1
nsel,r,loc,x,0,0.5*length_1
cm,edg1,nodes
allsel,all,all

nsel,s,loc,y,HEIGHT_1
nsel,r,loc,x,0.5*length_1,length_1
cm,edg3,nodes
allsel,all,all

*del,sft
nsel,s,,,edg1
nsel,a,,,edg3
*VGET,sft,NODE, ,NSEL,, , ,2
allsel,all,all
*voper,sft,sft,eq,1
ndmax=ndinqr(0,14)
node_begin=ndmax+1
*del,ndx
*del,ttmp
*dim,ndx,,ndmax,2
*dim,ttmp,,ndmax
*vfill,ttmp,ramp,1,1
*VFILL,ndx(1,1),RAMP,node_begin,1
*vfun,ndx(1,2),copy,ttmp
N,ndx(1:ndmax,1),NX(1:ndmax),NY(1:ndmax),NZ(1:ndmax)
etyp_combn14=etyiqr(0,14)+1
ET,etyp_combn14,COMBIN14
keyop,etyp_combn14,2,8
r,2
TYPE,etyp_combn14
real,2
EN,ndx(1:ndmax),(1:ndmax),ndx(1:ndmax)

ce,1,0,ndx(1),temp,1,ndx(2),temp,1,ndx(3),temp,1
*do,ii,3,48
ce,high,,ndx(ii+1),temp,1,ndx(ii+2),temp,1,ndx(ii+3),temp,1
*enddo

/solu
nsel,s,,,edg1
sf,all,hflux,0.2
allsel,all
nsel,a,,,edg3
sf,all,hflux,0.5
allsle
solve
fini

• vkr535
Subscriber

attached images of the example problem