# 3.1 Test 4: Thermal Problem

## 3.1.1 Description

Calculate the temperature distribution in a two-dimensional domain.

Figure 3.1: Two-dimensional heat problem

## 3.1.2 The boundary value problem

Find u(x) which satisfies:

where

This problem allows one exact solution:

The type of finite element utilized in order to solve this problem: TRIA 2P3D (straight P3 Hermitian triangle)

Figure 3.2: Suggested mesh ( TRNOXX)

## 3.1.3 Execution of the test

Step 1: Mesh

 Execute program: D4NOXX Modules used: APNOPO ... 2D mesh generator [MODULEF User Guide - 3] TRIHER ... construction of mesh using Voronoi's polyhedra theory REGMA2 ... regularization of mesh GIBBS ... renumbering Libraries: NOP2, NOPO, UTSD , UTII Input data file: D4NOPO.D Output data structure: T4NOPO

Step2: Interpolation [13]

 Execute program: D4COXX Module used: COMACO Libraries: THER, COSD, UTSD, UTII Element type: TRIA 2P3D Output data structures: T4MAIL T4COOR

Step 3: Description of the data [14]

 Call preprocessor: FOMIXX Data File: D4FOMI.D ... in library TESD Output data structures: T4FORC T4MILI

Step 4: Calculation, assembly of the element arrays, description of boundary conditions and solution

 Execute program: D4ARXX Modules used: CONDL1 ... construction of pointers on the degrees of freedom associated with each node [20] COBDC1 ... description of boundary conditions PREPAC ... allocation of pointers THEASS ... calculates element arrays and assembly CLIMPC ... introduction of boundary conditions CHOLPC ... factorization by Cholesky DRCHPC ... forward and backward substitution Libraries: COSD, THER, RESB, RESO, RESR, UTSD, UTII Output data structure: T4B

Remark: The finite element TRIA 2P3D utilizes the direct access file POBA. In this test the name of the file is POBA. In order to change the name of this file it is necessary to modify the variable NOMPOB in program D4ARXX. The file POBA is created, once the tape is loaded, by program POBAXX in PPAL (see Part iii).

The user function VALCLR allows for non-constant boundary conditions. This function is found at the end of program D4ARXX in file D4ARXX.F in library TEST.

The number of words necessary to store the matrix : 5903

The number of words necessary to store the corresponding right-hand-side (RHS) vector : 365

## 3.1.4 Memory usage and Execution time

The super array M is dimensioned equal to 45 000 words in all the programs.

 Step Dimension of Execution time the super array LM (DPS8 Multics) Mesh 6 000 Interpolation 4 600 3.38 Preparation 5 000 52.18 Solution 10 000 8.54

## 3.1.5 Results

The complete output of the D.S. B, containing the temperature and the temperature derivative in x and y at each node in the mesh, is found in the file S4B.D in library TESD.

The remainder of this section contains a partial or total printout of the various output D.S. of the different programs. These data structures can be viewed by calling preprocessor IMAGXX and specifying the names of the files containing the desired data structures.

``` &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

PRINT OUT OF D.S. NOPO OF LEVEL  1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
TITLE                           : BENCHMARK MODULEF NUMBER 4 :THERMAL TEST
DATE AND USER'S NAME            : 19/11/90  dutoit
DATA STRUCTURE TYPE             : NOPO
LEVEL AND STATE NUMBER          :      1     0
NUMBER OF ASSOCIATED TABLES     :      1

TABLE  1 : CF01 OF TYPE CHARACTER    , HAS     10 WORDS
CONTENTS OF THIS TABLE :   CURVE ASSOCIATED :      1
NAME :   CF01 TYPE :  4 VALUES  :
1   8COURBE01(X,Y)=X**2+(Y-1)**2-.16;

TABLE  N O P 2
--------------
CHARACTERISTICS OF THE MESH :

SPACE DIMENSION                        (NDIM ) :      2
MAXIMUM NUMBER OF REFERENCES           (NDSR ) :      2
MAXIMUM NUMBER OF SUB-DOMAINS          (NDSD ) :      1
NODES AND POINTS COINCIDE             (NCOPNP) :      1
NUMBER OF ELEMENTS                     (NE   ) :    129
NUMBER OF TRIANGLES                    (NTRI ) :    129
NUMBER OF BOUNDARY ELEMENTS            (NEF  ) :     57
NUMBER OF NODES                        (NOE  ) :     81
NUMBER OF NODES BY SEGMENT (NO EXTREMITIES)    :      0
TYPE OF COORDINATE VALUES             (NTYCOO) : REEL1MOT
MAX DIFFERENCE +1 BETWEEN 2 NODES OF AN ELEMENT :    13
NUMBER OF COARSE ELEMENTS             (NBEGM ) :      0
NUMBER OF WORDS FOR TABLE NOP5        (LNOP5 ) :   1224
REFERENCE AXIS    X,Y,Z               (NTACOO) :      1

TABLE  N O P 4
----------------
COORDINATES OF POINTS

-------------------------------------------------------------------------------
| POINT |     X        |     Y       |   | POINT |     X        |     Y       |
-------------------------------------------------------------------------------
|     1 |  -1.00000    |  1.00000    |   |     2 |  -.880000    |  1.00000    |
|     3 |  -.760000    |  1.00000    |   |     4 |  -.927111    | 0.920056    |
|     5 |  -1.00000    | 0.875000    |   |     6 |  -.701447    | 0.913023    |
|     7 |  -.640000    |  1.00000    |   |     8 |  -1.00000    | 0.750000    |
|     9 |  -.808490    | 0.872889    |   |    10 |  -.906893    | 0.796814    |
--------------------------------------- ---------------------------------------

TABLE  N O P 5
----------------
DESCRIPTION OF ELEMENTS

| ELEMENT :      1 | TRIANGLE     | S-D :   1 | NUMBER OF ASSOCIATED WORDS :  0

NUMBER OF  NODES  :  3 :       79       76       75

| ELEMENT :      2 | TRIANGLE     | S-D :   1 | NUMBER OF ASSOCIATED WORDS :  4

NUMBER OF  NODES :  3 :       79       75       78
VERTEX REFERENCE :        0        0        2

| ELEMENT :      3 | TRIANGLE     | S-D :   1 | NUMBER OF ASSOCIATED WORDS :  0

NUMBER OF  NODES :  3 :       11       12        6

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

PRINT OUT OF D.S. MAIL OF LEVEL  1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
TITLE                           :
DATE AND USER'S NAME            : 19/09/90  dutoit
DATA STRUCTURE TYPE             : MAIL
LEVEL AND STATE NUMBER          :      1     0
NUMBER OF ASSOCIATED TABLES     :      0

TABLE  M A I 2
--------------

SPACE DIMENSION                           (NDIM) :      2
NUMBER OF UNKNOWNS OF THE VARIATIONAL FORMULATION (NINCFV) :      1
NUMBER OF TYPES OF ELEMENTS             (NTYELM) :      1
NUMBER OF REFERENCES                       (NNR) :      2
NUMBER OF TYPES OF NODES                (NTYNOE) :      1
NUMBER OF TYPES OF POINTS               (NTYPOI) :      1
NUMBER OF WORDS OF TABLE MAI3            (LMAI3) :     36
POINTS ARE ONLY DEFINED BY THEIR COORDINATES
NODES AND POINTS ACOINCIDE EVERYWHERE   (NCOPNP) :      1
MAX NUMBER OF WORDS FOR AN ELEMENT      (NMMAEL) :     18
NUMBER OF WORDS OF THE TABLE MAIL        (LMAIL) :   1482
STORAGE OPTION                           (NOPFI) :      0

TABLE  M A I 3
--------------

NUMBER OF ELEMENTS                          (NE) :    129
NUMBER OF NODES                            (NOE) :     81
NUMBER OF POINTS                            (NP) :     81
NUMBER OF SUB-DOMAINS                     (NDSD) :      1

*** TABLE OF THE TYPES OF ELEMENTS ***

--------------------------------------------------------------------------------
|  S-D |      NODE       |    SEGMENT      |     TRIANGLE    |    QUADRANGLE   |
--------------------------------------------------------------------------------
|      | STRAIGHT CURVED | STRAIGHT CURVED | STRAIGHT CURVED | STRAIGHT CURVED |
--------------------------------------------------------------------------------
|  1   |        |        |        |        |TRIA2P3D|        |        |        |
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
|  S-D |   TETRAHEDRON   |   PENTAHEDRON   |   HEXAHEDRON    |   SUPER-ELEMENT |
--------------------------------------------------------------------------------
|      | STRAIGHT CURVED | STRAIGHT CURVED | STRAIGHT CURVED | STRAIGHT CURVED |
--------------------------------------------------------------------------------
|  1   |        |        |        |        |        |        |        |        |
--------------------------------------------------------------------------------

TABLES   M A I 5   TO   M A I 8
-------------------------------
DESCRIPTION OF NODE TYPES

NUMBER OF VARIATIONAL UNKNOWNS FOR EACH NODE TYPE
-----------------------------------------------------

TYPE  1 :   1

NODE TYPE         1 :
--------------------
TEMPERATURE      (UNKNOWN  :  1) : VN
TEMPERATURE      (UNKNOWN  :  1) : DX
TEMPERATURE      (UNKNOWN  :  1) : DY

TABLES    M A I B  TO   M A I K
-------------------------------
DESCRIPTION OF ELEMENT TYPES

ELEMENT TYPE      1     :
-------------------
ELEMENT NAME                 : THERTRIA2P3D
ELEMENT CODE                 : 100004
THE ELEMENT IS A             : TRIANGLE
NUMBER OF NODES              :      3
NUMBER OF POINTS             :      3
NUMBER OF ASSOCIATED TABLES  :      1
-------------------------------------------------------------------------------
| NODE  | TYPE  |    COORDINATES ON THE UNIT ELEMENT                          |
-------------------------------------------------------------------------------
|     1 |     1 |   0.0000000      0.0000000
|     2 |     1 |    1.000000      0.0000000
|     3 |     1 |   0.0000000       1.000000
-------------------------------------------------------------------------------
-------------------------------------------------------------------------------
| POINT | TYPE  |    COORDINATES ON THE UNIT ELEMENT                          |
-------------------------------------------------------------------------------
|     1 |     1 |   0.0000000      0.0000000
|     2 |     1 |    1.000000      0.0000000
|     3 |     1 |   0.0000000       1.000000
-------------------------------------------------------------------------------
TABLE        1 ( TYPE ENTIER   ) :   1  MAX VALUE OF INDEX(ES) :     7

TABLE  M A I L
--------------
DESCRIPTION OF ELEMENTS IF NOPTFI = 0

-------------------------------------------------------------------
| ELEMENT | TYPE | NBER OF WORDS| S-D | TABLES   | NBER OF NODES  |
-------------------------------------------------------------------
|     1   |  1   |     0        |   1 |     0    |     3          |
NODE NUMBERS   :    79    76    75

-------------------------------------------------------------------
|     2   |  1   |     4        |   1 |     1    |     3          |
NODE NUMBERS   :    79    75    78
REFERENCE (VERTICES) :      0     0     2
-------------------------------------------------------------------
|     3   |  1   |     0        |   1 |     0    |     3          |
NODE NUMBERS   :    11    12     6
-------------------------------------------------------------------

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

PRINT OUT OF D.S. COOR OF LEVEL  1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
TITLE                           :
DATE AND USER'S NAME            : 19/09/90  dutoit
DATA STRUCTURE TYPE             : COOR
LEVEL AND STATE NUMBER          :      1     0
NUMBER OF ASSOCIATED TABLES     :      0

TABLE C O O 2
-------------
TYPE OF THE TABLE COO4               (NTYT) :     2
NUMBER OF ITS INDEX                 (NINDI) :     2
SPACE DIMENSION                      (NDIM) :     2
MAX VALUE OF THE SECOND INDEX          (M2) :    81
SEGMENTATION CODE                   (NCODS) :     1
NUMBER OF BLOCS                     (NBLOC) :     1
TYPE OF THE COORDINATE AXIS        (NTACOO) :     1

TABLE C O O 3
-------------
TABLE OF THE LAST COLUMN NUMBER OF EACH PAGE

----------------------------------------------
PAGE     1 :     81   PAGE

----------------------------------------------

TABLE C O O 4
-------------

-------------------------------------------------------------------------------
| POINT |     X        |     Y       |   | POINT |     X        |     Y       |
-------------------------------------------------------------------------------
|     1 |  -1.00000    |  1.00000    |   |     2 |  -.880000    |  1.00000    |
|     3 |  -.760000    |  1.00000    |   |     4 |  -.927111    | 0.920056    |
|     5 |  -1.00000    | 0.875000    |   |     6 |  -.701447    | 0.913023    |
|     7 |  -.640000    |  1.00000    |   |     8 |  -1.00000    | 0.750000    |
|     9 |  -.808490    | 0.872889    |   |    10 |  -.906893    | 0.796814    |
--------------------------------------- ---------------------------------------

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

PRINT OUT OF D.S. FORC OF LEVEL  1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
TITLE                           :
DATE AND USER'S NAME            : 31/01/94  dutoit
DATA STRUCTURE TYPE             : FORC
LEVEL AND STATE NUMBER          :      1     0
NUMBER OF ASSOCIATED TABLES     :      1

TABLE  1 : FOO1 OF TYPE REEL1MOT    , IT HAS     10 WORDS
CONTENTS OF THIS TABLE :    F OMEGA AND F GAMMA
NAME :   FOO1  TYPE :  2 VALUES  :
-0.4000000E+01 -0.4000000E+01 -0.4000000E+01 -0.4000000E+01 -0.4000000E+01
-0.4000000E+01 -0.4000000E+01 -0.3200000E+00 -0.3200000E+00 -0.3200000E+00

TABLE  F O R 2
--------------
INFORMATION ( NUMBER OF SETS ) OR NOT ( 0 )

--------------------------------------------------
|            |                 |                 |
|            |   SUB-DOMAINS   |     ELEMENTS    |
|            |                 |                 |
--------------------------------------------------
|  VOLUMES   |          0      |          0      |
|  SURFACES  |          1      |          0      |
|  LINES     |          1      |          0      |
|  POINTS    |          0      |          0      |
--------------------------------------------------

TABLE  F O R 4
--------------
DESCRIPTION OF REFERENCES FOR SURFACES

REFERENCE  ( OR SUB-DOMAIN ) NUMBER :      1
TYPE OF THE ELEMENT ( > 0 : ISOTROPIC, < 0 : ANISOTROPIC ) :      1
USE OF A  TABLE
NAME OF THE TABLE : FOO1

TABLE  F O R 5
--------------
DESCRIPTION OF REFERENCES FOR LINES

REFERENCE  ( OR SUB-DOMAIN ) NUMBER :      1
TYPE OF THE ELEMENT ( > 0 : ISOTROPIC, < 0 : ANISOTROPIC ) :      1
USE OF A  TABLE
NAME OF THE TABLE : FOO1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

PRINT OUT OF D.S. MILI OF LEVEL  1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
TITLE                           :
DATE AND USER'S NAME            : 31/01/94  dutoit
DATA STRUCTURE TYPE             : MILI
LEVEL AND STATE NUMBER          :      1     0
NUMBER OF ASSOCIATED TABLES     :      1

TABLE  1 : MIL1 OF TYPE REEL1MOT    , IT HAS      2 WORDS
CONTENTS OF THIS TABLE :     YOUNG_POISSON_EPAISSEUR
NAME :   MIL  TYPE :  5 VALUES  :
0.1000000E+01  0.3000000E+01

TABLE  M I L 2
--------------
INFORMATION ( NUMBER OF SETS ) OR NOT ( 0 )

--------------------------------------------------
|            |                 |                 |
|            |   SUB-DOMAINS   |     ELEMENTS    |
|            |                 |                 |
--------------------------------------------------
|  VOLUMES   |          0      |          0      |
|  SURFACES  |          1      |          0      |
|  LINES     |          1      |          0      |
|  POINTS    |          0      |          0      |
--------------------------------------------------

TABLE  M I L 4
--------------
DESCRIPTION OF REFERENCES FOR SURFACES

REFERENCE  ( OR SUB-DOMAIN ) NUMBER :      1
TYPE OF THE ELEMENT ( > 0 : ISOTROPIC, < 0 : ANISOTROPIC ) :      1
USE OF A  TABLE
NAME OF THE TABLE : MIL1

TABLE  M I L 5
--------------
DESCRIPTION OF REFERENCES FOR LINES

REFERENCE  ( OR SUB-DOMAIN ) NUMBER :      1
TYPE OF THE ELEMENT ( > 0 : ISOTROPIC, < 0 : ANISOTROPIC ) :      1
USE OF A  TABLE
NAME OF THE TABLE : MIL1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

PRINT OUT OF D.S. B    OF LEVEL  1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
TITLE                           :
DATE AND USER'S NAME            : 12/09/90  dutoit
DATA STRUCTURE TYPE             : B
LEVEL AND STATE NUMBER          :      1     2
NUMBER OF ASSOCIATED TABLES     :      0

TABLE  B 2
----------
TYPE OF THE TABLE                           (NTYT) :      2
NUMBER OF INDICES AND THEIR MAX VALUES    (NIND..) :      2     1   243
OPTION (1: PAGES OF SAME SIZE , 0: IF NOT)  (NCOD) :      1
NUMBER OF PAGES OF THE TABLE B4            (NBLOC) :      1
NUMBER OF NODES                              (NOE) :     81
NUMBER OF TABLES B4 IN THIS D.S.          (NBBLOC) :      1
CONSTANT NUMBER OF D.O.F. AT EACH NODE OR 0   (ND) :      3
NUMBER OF D.O.F. OR LENGHT OF A B4 PAGE     (NTDL) :    243
STORAGE OPTION FOR B4                     (NCODSB) :     -1

TABLE  B 3
----------
POINTER TO THE LAST ROW OF EACH PAGE OF B

1           0     2         243

TABLE B4(NDSM,NTDL) NUMBER :      1
-----------------------------------

TABLE B4(NDSM,NTDL) PAGINATED IN 1 OR MORE PAGES
------------------------------------------------
PAGE     1
----------
------------------------------------------------------
| NODE   | DOF | GLOBAL DOF|  LOAD CASE(S)           |
------------------------------------------------------
|      1 |   1 |        1  |     1.00000
|   2 |        2  |    -2.00000
|   3 |        3  |    0.000000
|      2 |   1 |        4  |    0.774400
|   2 |        5  |    -1.76000
|   3 |        6  |    0.000000
|      3 |   1 |        7  |    0.577600
|   2 |        8  |    -1.52000
|   3 |        9  |    0.000000
|      4 |   1 |       10  |    0.865922
```

## 3.1.6 Post-processing

A plot of the isotherms is generated by the preprocessor TRMCXX as follows:

 Call preprocessor: TRMCXX Specify input data structures: T4MAIL T4COOR T4B

Figure 3.3: Isotherms ( TRMCXX)

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