# Contact — Welcome to LS-DYNA Examples

This problem includes three tools a punch, a pressure pad, a die and a workpiece. A workpiece is deep drawn by the hemispherical punch while the pressure pad and die prevents wrinkling. The load on the pressure pad is ramped, then the punch displaces in the y direction.

## Introduction

```*CONTROL_CONTACT
Hemispherical Punch
LS-DYNA Manual Section: *CONTROL_CONTACT

*MAT_POWER_LAW_PLASTICITY
*RIGIDWALL_PLANAR

Example: Hemispherical Punch

Filename: control_contact.hemi-draw.k

Description:
This problem includes three tools a punch, a pressure pad, a die and a
workpiece. A workpiece is deep drawn by the hemispherical punch while the
then the punch displaces in the y direction.

Model:
The workpiece measures 80 mm in radius and 1 mm in thickness. The punch radius
is 50.0 mm and the die torus radius is 6.35 mm. The workpiece contains 528
Belytschko Tsay shell elements with 5 integration points through the thickness.
The tools are rigid members. Only 1/4 of the system is modeled because of
symmetry.

Input:
The number of integration points is 5 for the workpiece.(*SECTION_SHELL) This
model contains two options to consider shell thickness. The first option is
the contact surfaces are projected to the true surface of shell
(*CONTRO L_CONTACT). The second option is membrane straining results in
thickness changes (*CONTROL_CONTACT). The motion of the punch follows a sine
function represented by load curve number 2 (Section 22).

Reference:
Honecker, A. and Mattiason, K.```

## Keywords

```*BOUNDARY_PRESCRIBED_MOTION_RIGID
*CONTACT_SURFACE_TO_SURFACE
*CONTROL_CONTACT
*CONTROL_ENERGY
*CONTROL_OUTPUT
*CONTROL_SHELL
*CONTROL_TERMINATION
*DATABASE_BINARY_D3PLOT
*DATABASE_BINARY_D3THDT
*DATABASE_EXTENT_BINARY
*DATABASE_GLSTAT
*DATABASE_HISTORY_NODE
*DATABASE_MATSUM
*DATABASE_NODOUT
*DATABASE_RCFORC
*DEFINE_CURVE
*ELEMENT_SHELL
*END
*KEYWORD
*MAT_POWER_LAW_PLASTICITY
*MAT_RIGID
*NODE
*PART
*RIGIDWALL_PLANAR
*SECTION_SHELL
*SET_NODE_LIST
*TITLE```

## Reduced Input

```  *KEYWORD
*TITLE
Hemispherical Deep Draw
\$
\$  LSTC Example
\$
\$
\$  Units: kg, mm, ms, kN, GPa, kN-mm
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$\$\$\$  Control Ouput
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$...>....1....>....2....>....3....>....4....>....5....>....6....>....7....>....8
\$
*CONTROL_TERMINATION
\$   endtim    endcyc     dtmin    endneg    endmas
6.0
\$
\$\$\$\$   shell thickness is considered during contact: shlthk = 1
\$
*CONTROL_CONTACT
\$   slsfac    rwpnal    islchk    shlthk    penopt    thkchg     orien
1.0                   1
\$   usrstr    usrfac     nsbcs    interm    xpenen

\$
*CONTROL_ENERGY
\$     hgen      rwen    slnten     rylen
2         2         2
\$
*CONTROL_OUTPUT
\$    npopt    neecho    nrefup    iaccop     opifs    ipnint    ikedit
1         3         0         0                   2      1000
\$
\$\$\$\$   membrane straining causes thickness change: istupd = 1
\$
*CONTROL_SHELL
\$   wrpang    itrist     irnxx    istupd    theory       bwc     miter
1
\$
\$
*DATABASE_BINARY_D3PLOT
\$       dt      lcdt
0.20
\$
*DATABASE_EXTENT_BINARY
\$    neiph     neips    maxint    strflg    sigflg    epsflg    rltflg    engflg
1
\$   cmpflg    ieverp    beamip
1
\$
*DATABASE_BINARY_D3THDT
\$       dt      lcdt
12.00E+00
\$
*DATABASE_GLSTAT
\$       dt
0.05
\$
*DATABASE_MATSUM
\$       dt
0.05
\$
*DATABASE_NODOUT
\$       dt
0.05
\$
*DATABASE_HISTORY_NODE
\$      id1       id2       id3       id4       id5       id6       id7       id8
1333
\$
*DATABASE_RCFORC
\$       dt
0.05
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$\$\$\$  Define Contacts
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$...>....1....>....2....>....3....>....4....>....5....>....6....>....7....>....8
\$
\$\$\$\$\$\$\$\$\$\$  contact between workpiece and punch
\$
*CONTACT_SURFACE_TO_SURFACE
\$     ssid      msid     sstyp     mstyp    sboxid    mboxid       spr       mpr
1         2         3         3
\$       fs        fd        dc        vc       vdc    penchk        bt        dt
0.15      0.15
\$      sfs       sfm       sst       mst      sfst      sfmt       fsf       vsf

\$
\$\$\$\$\$\$\$\$\$\$  contact between workpiece and holder
\$
*CONTACT_SURFACE_TO_SURFACE
\$     ssid      msid     sstyp     mstyp    sboxid    mboxid       spr       mpr
1         3         3         3
\$       fs        fd        dc        vc       vdc    penchk        bt        dt
0.15      0.15
\$      sfs       sfm       sst       mst      sfst      sfmt       fsf       vsf

\$
\$\$\$\$\$\$\$\$\$\$  contact between workpiece and die
\$
*CONTACT_SURFACE_TO_SURFACE
\$     ssid      msid     sstyp     mstyp    sboxid    mboxid       spr       mpr
1         4         3         3
\$       fs        fd        dc        vc       vdc    penchk        bt        dt
0.15      0.15
\$      sfs       sfm       sst       mst      sfst      sfmt       fsf       vsf

\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$\$\$\$  Define Parts and Materials
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$...>....1....>....2....>....3....>....4....>....5....>....6....>....7....>....8
\$
*PART
\$      pid       sid       mid     eosid      hgid    adpopt
Workpiece
1         1         1
Punch
2         1         2
3         1         2
Die
4         1         2
\$
\$
*MAT_POWER_LAW_PLASTICITY
\$      mid        ro         e        pr         k         n       src       srp
1  7.83e-06      69.0     0.300     0.598     0.216       0.0       0.0
\$
*MAT_RIGID
\$      mid        ro         e        pr         n    couple         m     alias
2  7.83e-06      69.0     0.300
\$      cmo     con1       con2

\$   lco/a1        a2        a3        v1        v2        3

\$
\$\$\$\$\$  All parts use this section, thus all shells have 1 mm thicknesses.
\$\$\$\$\$  Those parts that aren't rigid, use B-T shell formulation with
\$\$\$\$\$  five through the thickness integration points.
\$
*SECTION_SHELL
\$      sid    elform      shrf       nip     propt   qr/irid     icomp
1         2                   5
\$       t1        t2        t3        t4      nloc
1.0       1.0       1.0       1.0       0.0
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$...>....1....>....2....>....3....>....4....>....5....>....6....>....7....>....8
\$
\$\$\$\$\$  Define motion of the punch.
\$
*BOUNDARY_PRESCRIBED_MOTION_RIGID
\$      pid       dof       vad      lcid        sf       vid
2         2         0         2      -1.0         0
\$
\$\$\$\$\$  Pressure load on the holder.
\$
\$     lcid        sf        at        n1        n2        n3        n4
1 1.000E+00 0.000E+00       907       900       901       908
1 1.000E+00 0.000E+00       914       907       908       915
1 1.000E+00 0.000E+00       921       914       915       922
1 1.000E+00 0.000E+00       928       921       922       929
1 1.000E+00 0.000E+00       935       928       929       936
1 1.000E+00 0.000E+00       942       935       936       943
1 1.000E+00 0.000E+00       949       942       943       950
1 1.000E+00 0.000E+00       956       949       950       957
1 1.000E+00 0.000E+00       963       956       957       964
1 1.000E+00 0.000E+00       970       963       964       971
1 1.000E+00 0.000E+00       977       970       971       978
1 1.000E+00 0.000E+00       984       977       978       985
1 1.000E+00 0.000E+00       991       984       985       992
1 1.000E+00 0.000E+00       998       991       992       999
1 1.000E+00 0.000E+00      1005       998       999      1006
1 1.000E+00 0.000E+00      1012      1005      1006      1013
1 1.000E+00 0.000E+00      1019      1012      1013      1020
1 1.000E+00 0.000E+00      1026      1019      1020      1027
1 1.000E+00 0.000E+00      1033      1026      1027      1034
1 1.000E+00 0.000E+00      1040      1033      1034      1041
1 1.000E+00 0.000E+00      1047      1040      1041      1048
1 1.000E+00 0.000E+00      1054      1047      1048      1055
1 1.000E+00 0.000E+00      1061      1054      1055      1062
1 1.000E+00 0.000E+00      1068      1061      1062      1069
1 1.000E+00 0.000E+00       908       901       902       909
1 1.000E+00 0.000E+00       915       908       909       916
1 1.000E+00 0.000E+00       922       915       916       923
1 1.000E+00 0.000E+00       929       922       923       930
1 1.000E+00 0.000E+00       936       929       930       937
1 1.000E+00 0.000E+00       943       936       937       944
1 1.000E+00 0.000E+00       950       943       944       951
1 1.000E+00 0.000E+00       957       950       951       958
1 1.000E+00 0.000E+00       964       957       958       965
1 1.000E+00 0.000E+00       971       964       965       972
1 1.000E+00 0.000E+00       978       971       972       979
1 1.000E+00 0.000E+00       985       978       979       986
1 1.000E+00 0.000E+00       992       985       986       993
1 1.000E+00 0.000E+00       999       992       993      1000
1 1.000E+00 0.000E+00      1006       999      1000      1007
1 1.000E+00 0.000E+00      1013      1006      1007      1014
1 1.000E+00 0.000E+00      1020      1013      1014      1021
1 1.000E+00 0.000E+00      1027      1020      1021      1028
1 1.000E+00 0.000E+00      1034      1027      1028      1035
1 1.000E+00 0.000E+00      1041      1034      1035      1042
1 1.000E+00 0.000E+00      1048      1041      1042      1049
1 1.000E+00 0.000E+00      1055      1048      1049      1056
1 1.000E+00 0.000E+00      1062      1055      1056      1063
1 1.000E+00 0.000E+00      1069      1062      1063      1070
1 1.000E+00 0.000E+00       909       902       903       910
1 1.000E+00 0.000E+00       916       909       910       917
1 1.000E+00 0.000E+00       923       916       917       924
1 1.000E+00 0.000E+00       930       923       924       931
1 1.000E+00 0.000E+00       937       930       931       938
1 1.000E+00 0.000E+00       944       937       938       945
1 1.000E+00 0.000E+00       951       944       945       952
1 1.000E+00 0.000E+00       958       951       952       959
1 1.000E+00 0.000E+00       965       958       959       966
1 1.000E+00 0.000E+00       972       965       966       973
1 1.000E+00 0.000E+00       979       972       973       980
1 1.000E+00 0.000E+00       986       979       980       987
1 1.000E+00 0.000E+00       993       986       987       994
1 1.000E+00 0.000E+00      1000       993       994      1001
1 1.000E+00 0.000E+00      1007      1000      1001      1008
1 1.000E+00 0.000E+00      1014      1007      1008      1015
1 1.000E+00 0.000E+00      1021      1014      1015      1022
1 1.000E+00 0.000E+00      1028      1021      1022      1029
1 1.000E+00 0.000E+00      1035      1028      1029      1036
1 1.000E+00 0.000E+00      1042      1035      1036      1043
1 1.000E+00 0.000E+00      1049      1042      1043      1050
1 1.000E+00 0.000E+00      1056      1049      1050      1057
1 1.000E+00 0.000E+00      1063      1056      1057      1064
1 1.000E+00 0.000E+00      1070      1063      1064      1071
1 1.000E+00 0.000E+00       910       903       904       911
1 1.000E+00 0.000E+00       917       910       911       918
1 1.000E+00 0.000E+00       924       917       918       925
1 1.000E+00 0.000E+00       931       924       925       932
1 1.000E+00 0.000E+00       938       931       932       939
1 1.000E+00 0.000E+00       945       938       939       946
1 1.000E+00 0.000E+00       952       945       946       953
1 1.000E+00 0.000E+00       959       952       953       960
1 1.000E+00 0.000E+00       966       959       960       967
1 1.000E+00 0.000E+00       973       966       967       974
1 1.000E+00 0.000E+00       980       973       974       981
1 1.000E+00 0.000E+00       987       980       981       988
1 1.000E+00 0.000E+00       994       987       988       995
1 1.000E+00 0.000E+00      1001       994       995      1002
1 1.000E+00 0.000E+00      1008      1001      1002      1009
1 1.000E+00 0.000E+00      1015      1008      1009      1016
1 1.000E+00 0.000E+00      1022      1015      1016      1023
1 1.000E+00 0.000E+00      1029      1022      1023      1030
1 1.000E+00 0.000E+00      1036      1029      1030      1037
1 1.000E+00 0.000E+00      1043      1036      1037      1044
1 1.000E+00 0.000E+00      1050      1043      1044      1051
1 1.000E+00 0.000E+00      1057      1050      1051      1058
1 1.000E+00 0.000E+00      1064      1057      1058      1065
1 1.000E+00 0.000E+00      1071      1064      1065      1072
1 1.000E+00 0.000E+00       911       904       905       912
1 1.000E+00 0.000E+00       918       911       912       919
1 1.000E+00 0.000E+00       925       918       919       926
1 1.000E+00 0.000E+00       932       925       926       933
1 1.000E+00 0.000E+00       939       932       933       940
1 1.000E+00 0.000E+00       946       939       940       947
1 1.000E+00 0.000E+00       953       946       947       954
1 1.000E+00 0.000E+00       960       953       954       961
1 1.000E+00 0.000E+00       967       960       961       968
1 1.000E+00 0.000E+00       974       967       968       975
1 1.000E+00 0.000E+00       981       974       975       982
1 1.000E+00 0.000E+00       988       981       982       989
1 1.000E+00 0.000E+00       995       988       989       996
1 1.000E+00 0.000E+00      1002       995       996      1003
1 1.000E+00 0.000E+00      1009      1002      1003      1010
1 1.000E+00 0.000E+00      1016      1009      1010      1017
1 1.000E+00 0.000E+00      1023      1016      1017      1024
1 1.000E+00 0.000E+00      1030      1023      1024      1031
1 1.000E+00 0.000E+00      1037      1030      1031      1038
1 1.000E+00 0.000E+00      1044      1037      1038      1045
1 1.000E+00 0.000E+00      1051      1044      1045      1052
1 1.000E+00 0.000E+00      1058      1051      1052      1059
1 1.000E+00 0.000E+00      1065      1058      1059      1066
1 1.000E+00 0.000E+00      1072      1065      1066      1073
1 1.000E+00 0.000E+00       912       905       906       913
1 1.000E+00 0.000E+00       919       912       913       920
1 1.000E+00 0.000E+00       926       919       920       927
1 1.000E+00 0.000E+00       933       926       927       934
1 1.000E+00 0.000E+00       940       933       934       941
1 1.000E+00 0.000E+00       947       940       941       948
1 1.000E+00 0.000E+00       954       947       948       955
1 1.000E+00 0.000E+00       961       954       955       962
1 1.000E+00 0.000E+00       968       961       962       969
1 1.000E+00 0.000E+00       975       968       969       976
1 1.000E+00 0.000E+00       982       975       976       983
1 1.000E+00 0.000E+00       989       982       983       990
1 1.000E+00 0.000E+00       996       989       990       997
1 1.000E+00 0.000E+00      1003       996       997      1004
1 1.000E+00 0.000E+00      1010      1003      1004      1011
1 1.000E+00 0.000E+00      1017      1010      1011      1018
1 1.000E+00 0.000E+00      1024      1017      1018      1025
1 1.000E+00 0.000E+00      1031      1024      1025      1032
1 1.000E+00 0.000E+00      1038      1031      1032      1039
1 1.000E+00 0.000E+00      1045      1038      1039      1046
1 1.000E+00 0.000E+00      1052      1045      1046      1053
1 1.000E+00 0.000E+00      1059      1052      1053      1060
1 1.000E+00 0.000E+00      1066      1059      1060      1067
1 1.000E+00 0.000E+00      1073      1066      1067      1074
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$\$\$\$  Rigidwalls
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$\$\$\$\$  Prevent nodes on the holder from moving in the positive y-direction.
\$
*RIGIDWALL_PLANAR
\$     nsid    nsidex     boxid
1
\$       xt        yt        zt        xh        yh        zh      fric
0.000     0.000     0.000   0.00000   1.00000   0.00000     0.000
\$
*SET_NODE_LIST
\$      sid
1
\$     nid1      nid2      nid3      nid4      nid5      nid6      nid7      nid8
900       901       902       903       904       905       906       907
908       909       910       911       912       913       914       915
916       917       918       919       920       921       922       923
924       925       926       927       928       929       930       931
932       933       934       935       936       937       938       939
940       941       942       943       944       945       946       947
948       949       950       951       952       953       954       955
956       957       958       959       960       961       962       963
964       965       966       967       968       969       970       971
972       973       974       975       976       977       978       979
980       981       982       983       984       985       986       987
988       989       990       991       992       993       994       995
996       997       998       999      1000      1001      1002      1003
1004      1005      1006      1007      1008      1009      1010      1011
1012      1013      1014      1015      1016      1017      1018      1019
1020      1021      1022      1023      1024      1025      1026      1027
1028      1029      1030      1031      1032      1033      1034      1035
1036      1037      1038      1039      1040      1041      1042      1043
1044      1045      1046      1047      1048      1049      1050      1051
1052      1053      1054      1055      1056      1057      1058      1059
1060      1061      1062      1063      1064      1065      1066      1067
1068      1069      1070      1071      1072      1073      1074
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$\$\$\$  Define Curves
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$...>....1....>....2....>....3....>....4....>....5....>....6....>....7....>....8
\$
*DEFINE_CURVE
\$     lcid      sidr      scla      sclo      offa      offo
1
\$                  a                   o
0.000E+00           0.000E+00
1.000E+00           1.000E-03
8.000E+00           1.000E-03
\$
*DEFINE_CURVE
\$     lcid      sidr      scla      sclo      offa      offo
2
\$                  a                   o
0.000E+00           0.000E+00
1.000E+00           0.000E+00
1.125E+00           1.479E+00
1.250E+00           2.949E+00
1.375E+00           4.400E+00
1.500E+00           5.825E+00
1.625E+00           7.213E+00
1.750E+00           8.558E+00
1.875E+00           9.849E+00
2.000E+00           1.108E+01
2.125E+00           1.224E+01
2.250E+00           1.333E+01
2.375E+00           1.433E+01
2.500E+00           1.525E+01
2.625E+00           1.607E+01
2.750E+00           1.680E+01
2.875E+00           1.741E+01
3.000E+00           1.793E+01
3.125E+00           1.833E+01
3.250E+00           1.862E+01
3.375E+00           1.879E+01
3.500E+00           1.885E+01
3.625E+00           1.879E+01
3.750E+00           1.862E+01
3.875E+00           1.833E+01
4.000E+00           1.793E+01
4.125E+00           1.741E+01
4.250E+00           1.680E+01
4.375E+00           1.607E+01
4.500E+00           1.525E+01
4.625E+00           1.433E+01
4.750E+00           1.333E+01
4.875E+00           1.224E+01
5.000E+00           1.108E+01
5.125E+00           9.849E+00
5.250E+00           8.558E+00
5.375E+00           7.213E+00
5.500E+00           5.825E+00
5.625E+00           4.400E+00
5.750E+00           2.949E+00
5.875E+00           1.479E+00
6.000E+00           0.000E+00
8.000E+00           0.000E+00
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$\$\$\$  Define Nodes and Elements
\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$
\$
\$\$\$\$\$  Note: Boundary conditions on many of the nodes are defined here.
\$
*END```