# 2.2 Test 1: Thermal Problem

This test has been proposed by NAFEMS (National Agency for Finite Element Methods & Standards) [Cameron et al.].

## 2.2.1 Description

Consider the thermal problem described below:

Figure 2.5: Two-dimensional thermal test with heat transfer

The internal heat production is zero.

 Boundary conditions: on boundary AB T = 100°C . on boundary DA Zero flux. on boundaries BC, CD Heat transfer. Surrounding temperature 0°C

Material properties:

Isotropic material with thermal conductivity:

Coefficient of heat transfer on boundaries BC, CD:

## 2.2.2 The boundary value problem

The type of finite element utilized in order to solve this problem: TRIA 2P2D.

Figure 2.6: Suggested mesh

Result: The known temperature at point E, or at vertex 23 with the coordinates (0.6 , 0.2) is:

## 2.2.3 Execution of the test

As shown in the previous section, we execute the test by following various steps, indicated below. The data files have already been created for your convenience, i.e., all that remains to be done is to call the preprocessor and to choose the EXECUTE option from the main menu.

Step 1: Mesh

 Call preprocessor: APNOXX Libraries used: CONV, NOP2, NOPO, UTII, UTSD Modules used: APNOPO ... 2D mesh generator [MODULEF User Guide - 3] QUACOO ... generation of quadrangles MODNOP ... refinement transformation ADPNOP ... addition of nodes GIBBS ... renumbering Specify input data file: D1NOPO.D Output data structure: T1NOPO

Step 2: Interpolation [13]

 Call preprocessor: COMAXX Libraries used: CONV, COSD, THER, UTII Module used: COMACO Element type: TRIA 2P2D Specify input data file: D1COMA.D Output data structures: T1MAIL T1COOR

Step 3: Calculation of the element arrays (constant case) [14]

 Call preprocessor: THECXX Libraries used: CONV, COSD, THER, UTII Module used: THERCT Specify input data file: D1TAE.D Output data structure: T1TAE

Step 4: Description of boundary conditions [18]

 Call preprocessor: COBDXX Libraries used: CONV, COSD, UTII Modules used: COBDC1 COTYNO CALMAI Specify input data file: D1BDCL.D Output data structure: T1BDCL

Step 5: Assembly and solution by a direct method (Cholesky)
[4] [MODULEF User Guide - 5]

 Call preprocessor: CHOLXX Libraries used: CONV, COSD, RESO, RESB, UTII, UTSD Modules used: PREPAC ... allocation of pointers ASSMUA ... assembly of matrix in main memory (m.m.) ASEMBV ... assembly of RHS vector in main memory CLIMPC ... factorization by Cholesky DRCHPC ... forward- and backward substitution Specify input data file: D1B.D Output data structure: T1B

The number of words necessary to store the matrix : 922

The number of words necessary to store the corresponding right-hand-side (RHS) vector : 217

## 2.2.4 Memory usage and execution time

The table below only represents the maximum values.

 Step Dimension of Execution time the super array (DPS8 Multics) Mesh 6 000 9.5 Interpolation 3 000 4.9 Element matrices 4 000 5.0 Boundary conditions 2 000 3.8 Solution 3 000 6.4

## 2.2.5 Results

The complete output of the D.S. B, containing the value the temperature at different nodes in the mesh, is found in S1B.D in library TESD.

The temperature calculated at point E (node 70) equals:

The relative error is given by:

The remainder of this section contains a partial printout of the output data structures (O.D.S.) resulting from the different steps. These data structures can be viewed by calling preprocessor IMAGXX and specifying the names of the files containing the desired data structures, as follows:

```B\$ imagxx
******************
EXECUTING IM'D.S.'
******************

-- NAME OF THE FILE STORING THE D.S. ?
T1NOPO
-- DIRECT ACCESS FILE (YES-NO) ?
N
-- PRINT PARAMETER ?
5
-- SCREEN OUTPUT (YES-NO) ?
Y
```

The output corresponding to the data structures generated in steps 1 to 5 of test 1 are given below:

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

PRINT OUT OF D.S. NOPO OF LEVEL  5

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
TITLE                           : BENCHMARK MODULEF NUMBER 1 :THERMAL PROBLEM
DATE AND USER'S NAME            : 16/11/90  dutoit
DATA STRUCTURE TYPE             : NOPO
LEVEL AND STATE NUMBER          :      3     0
NUMBER OF ASSOCIATED TABLES     :      0

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

SPACE DIMENSION                        (NDIM ) :      2
MAXIMUM NUMBER OF REFERENCES           (NDSR ) :      3
MAXIMUM NUMBER OF SUB-DOMAINS          (NDSD ) :      1
NODES AND POINTS DO NOT COINCIDE      (NCOPNP) :      0
NUMBER OF ELEMENTS                     (NE   ) :     30
NUMBER OF TRIANGLES                    (NTRI ) :     30
NUMBER OF BOUNDARY ELEMENTS            (NEF  ) :     24
NUMBER OF NODES                        (NOE  ) :     77
NUMBER OF NODES BY SEGMENT (NO EXTREMITIES)    :      1
NUMBER OF POINTS                        (NP  ) :     24
TYPE OF COORDINATE VALUES             (NTYCOO) : REEL1MOT
MAX DIFFERENCE +1 BETWEEN 2 NODES OF AN ELEMENT :    17
NUMBER OF COARSE ELEMENTS             (NBEGM ) :      0
NUMBER OF WORDS FOR TABLE NOP5        (LNOP5 ) :    558
REFERENCE AXIS    X,Y,Z               (NTACOO) :      1

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

-------------------------------------------------------------------------------
| POINT |     X        |     Y       |   | POINT |     X        |     Y       |
-------------------------------------------------------------------------------
|     1 |  0.000000    |  1.00000    |   |     2 |  0.000000    | 0.800000    |
|     3 |  0.000000    | 0.600000    |   |     4 |  0.000000    | 0.400000    |
|     5 |  0.000000    | 0.200000    |   |     6 |  0.000000    | 0.000000    |
|     7 |  0.200000    |  1.00000    |   |     8 |  0.200000    | 0.800000    |
|     9 |  0.200000    | 0.600000    |   |    10 |  0.200000    | 0.400000    |
|    11 |  0.200000    | 0.200000    |   |    12 |  0.200000    | 0.000000    |
|    13 |  0.400000    |  1.00000    |   |    14 |  0.400000    | 0.800000    |
|    15 |  0.400000    | 0.600000    |   |    16 |  0.400000    | 0.400000    |
|    17 |  0.400000    | 0.200000    |   |    18 |  0.400000    | 0.000000    |
|    19 |  0.600000    |  1.00000    |   |    20 |  0.600000    | 0.800000    |
|    21 |  0.600000    | 0.600000    |   |    22 |  0.600000    | 0.400000    |
|    23 |  0.600000    | 0.200000    |   |    24 |  0.600000    | 0.000000    |
--------------------------------------- ---------------------------------------

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

| ELEMENT :      1 | TRIANGLE     | S-D :   1 | NUMBER OF STORED WORDS  :  7

NUMBER OF  NODES  :  6 :        1        5        6       2       4       3

NUMBER OF  POINTS :  3 :        1        2        7
EDGE REFERENCE        :        3        0        2
VERTEX REFERENCE      :        2        3        2

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

NUMBER OF  NODES  :  6 :        6        5        11       4       7       8

NUMBER OF  POINTS :  3 :        7        2        8
VERTEX REFERENCE      :        2        3        0

| ELEMENT :      3 | TRIANGLE     | S-D :   1 | NUMBER OF STORED WORDS  :  7

NUMBER OF  NODES  :  6 :        5        13       11       9       12       7

NUMBER OF  POINTS :  3 :        2        3        8
EDGE REFERENCE        :        3        0        0
VERTEX REFERENCE      :        3        3        0

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

PRINT OUT OF D.S. MAIL OF LEVEL  1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
TITLE                           :
DATE AND USER'S NAME            : 16/11/90  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) :      3
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 COORDINATES
NODES AND POINTS ARE NOT IDENTICAL      (NCOPNP) :      0
MAX NUMBER OF WORDS FOR AN ELEMENT      (NMMAEL) :     25
NUMBER OF WORDS OF THE TABLE MAIL        (LMAIL) :    618
STORAGE OPTION                           (NOPFI) :      0

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

NUMBER OF ELEMENTS                          (NE) :     30
NUMBER OF NODES                            (NOE) :     77
NUMBER OF POINTS                            (NP) :     24
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   |        |        |        |        |TRIA2P2D|        |        |        |
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
|  S-D |    TETRAEDRON   |    PENTAEDRON   |    HEXAEDRON    |   SUPER-ELEMENT |
--------------------------------------------------------------------------------
|      | STRAIGHT CURVED | STRAIGHT CURVED | STRAIGHT CURVED | STRAIGHT CURVED |
--------------------------------------------------------------------------------
|  1   |        |        |        |        |        |        |        |        |
--------------------------------------------------------------------------------

TABLES   M A I 5   TO   M A I 8
-------------------------------
DESCRIPTOR OF THE TYPES OF NODES

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

TYPE  1 :   1

TYPE OF NODE      1 :
--------------------
TEMPERATURE      (UNKNOWN  :  1) : VN

TABLES    M A I B  TO   M A I K
-------------------------------
DESCRIPTOR OF THE TYPES OF ELEMENTS

ELEMENT OF TYPE   1   :
-------------------
NAME OF THE ELEMENT          : THERTRIA2P2D
CODE OF THE ELEMENT          : 100002
THE ELEMENT IS A             : TRIANGLE
NUMBER OF NODES              :      6
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
|     4 |     1 |   0.5000000      0.0000000
|     5 |     1 |   0.5000000      0.5000000
|     6 |     1 |   0.0000000      0.5000000
-------------------------------------------------------------------------------
-------------------------------------------------------------------------------
| 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
--------------
DESCRIPTOR OF ELEMENTS IF NOPTFI = 0

-------------------------------------------------------------------
| ELEMENT | TYPE | NBER OF WORDS| S-D | TABLES   | NBER OF NODES  |
-------------------------------------------------------------------
|     1   |  1   |     7        |   1 |     1    |     6          |
LIST OF NODES   :     1     5     6     2     4     3
NUMBER OF POINTS   3 :      1     2     7
REFERENCE (EDGES VERTICES) :      3     0     2     2     3     2

-------------------------------------------------------------------
|     2   |  1   |     4        |   1 |     1    |     6          |
LIST OF NODES   :     6     5    11     4     7     8
NUMBER OF POINTS   3 :      7     2     8
REFERENCE (VERTICES) :      2     3     0
-------------------------------------------------------------------
|     3   |  1   |     7        |   1 |     1    |     6          |
LIST OF NODES   :     5    13    11     9    12     7
NUMBER OF POINTS   3 :      2     3     8
REFERENCE (EDGES VERTICES) :      3     0     0     3     3     0

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

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

PRINT OUT OF D.S. COOR OF LEVEL  1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
TITLE                           :
DATE AND USER'S NAME            : 16/11/90  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) :    24
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 :     24   PAGE

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

TABLE C O O 4
-------------
-------------------------------------------------------------------------------
| POINT |     X        |     Y       |   | POINT |     X        |     Y       |
-------------------------------------------------------------------------------
|     1 |  0.000000    |  1.00000    |   |     2 |  0.000000    | 0.800000    |
|     3 |  0.000000    | 0.600000    |   |     4 |  0.000000    | 0.400000    |
|     5 |  0.000000    | 0.200000    |   |     6 |  0.000000    | 0.000000    |
|     7 |  0.200000    |  1.00000    |   |     8 |  0.200000    | 0.800000    |
|     9 |  0.200000    | 0.600000    |   |    10 |  0.200000    | 0.400000    |
|    11 |  0.200000    | 0.200000    |   |    12 |  0.200000    | 0.000000    |
|    13 |  0.400000    |  1.00000    |   |    14 |  0.400000    | 0.800000    |
|    15 |  0.400000    | 0.600000    |   |    16 |  0.400000    | 0.400000    |
|    17 |  0.400000    | 0.200000    |   |    18 |  0.400000    | 0.000000    |
|    19 |  0.600000    |  1.00000    |   |    20 |  0.600000    | 0.800000    |
|    21 |  0.600000    | 0.600000    |   |    22 |  0.600000    | 0.400000    |
|    23 |  0.600000    | 0.200000    |   |    24 |  0.600000    | 0.000000    |
--------------------------------------- ---------------------------------------

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

PRINT OUT OF D.S. BDCL OF LEVEL  1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
TITLE                           :
DATE AND USER'S NAME            : 16/11/90  dutoit
DATA STRUCTURE TYPE             : BDCL
LEVEL AND STATE NUMBER          :      1     0
NUMBER OF ASSOCIATED TABLES     :      0

TABLE  B D C 2
--------------
TYPE OF CONSTRAINED VALUES                (NTYB) :      2
NUMBER OF CARDS DESCRIBING CONSTRAINTS   (NCART) :      7
NUMBER OF LINEAR RELATIONS BETWEEN CONSTRAINTS  (NCLRL) :      0
STORAGE OPTION                          (NOPTFI) :      0
NUMBER OF WORDS OF TABLE BDC3           (NMMAT3) :     21
NUMBER OF WORDS OF TABLE BDC4           (NMMAT4) :      7
NUMBER OF WORDS OF TABLE BDC5           (NMMAT5) :      0
NUMBER OF WORDS OF TABLE BDC6           (NMMAT6) :      0

TABLE       B D C 3  B D C 4
-----------------------------
CLAMPING TYPE :  1 : NODE :    55 , D.O.F. :  1
AT VALUE    :   0.1000000E+03
CLAMPING TYPE :  1 : NODE :    64 , D.O.F. :  1
AT VALUE    :   0.1000000E+03
CLAMPING TYPE :  1 : NODE :    67 , D.O.F. :  1
AT VALUE    :   0.1000000E+03
CLAMPING TYPE :  1 : NODE :    71 , D.O.F. :  1
AT VALUE    :   0.1000000E+03
CLAMPING TYPE :  1 : NODE :    74 , D.O.F. :  1
AT VALUE    :   0.1000000E+03
CLAMPING TYPE :  1 : NODE :    75 , D.O.F. :  1
AT VALUE    :   0.1000000E+03
CLAMPING TYPE :  1 : NODE :    76 , D.O.F. :  1
AT VALUE    :   0.1000000E+03

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

PRINT OUT OF D.S. B    OF LEVEL  1

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
TITLE                           :
DATE AND USER'S NAME           : 16/11/90  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    77
OPTION (1: PAGES OF SAME SIZE , 0: IF NOT)  (NCOD) :      1
NUMBER OF PAGES OF THE TABLE B4            (NBLOC) :      1
NUMBER OF NODES                              (NOE) :     77
NUMBER OF TABLES B4 IN THIS D.S.          (NBBLOC) :      1
CONSTANT NUMBER OF D.O.F. AT EACH NODE OR 0   (ND) :      1
NUMBER OF D.O.F. OR LENGHT OF A B4 PAGE     (NTDL) :     77
STORAGE OPTION FOR B4                     (NCODSB) :     -1

TABLE  B 3
----------
POINTER ON THE LAST ROW OF EACH PAGE OF B4

1           0     2          77

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  |     3.33833
|      2 |   1 |        2  |     8.34988
|      3 |   1 |        3  |     3.29528
|      4 |   1 |        4  |     8.14101
|      5 |   1 |        5  |     13.7792
|      6 |   1 |        6  |     2.99108
|      7 |   1 |        7  |     13.4293
|      8 |   1 |        8  |     7.48959
|      9 |   1 |        9  |     19.9092
|     10 |   1 |       10  |     2.60714
```

## 2.2.6 Post-processing

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

 Call preprocessor: TRMCXX Specify input data structures: T1MAIL T1COOR T1B

Figure 2.7: Isotherms ( TRMCXX)

Next: 2.3 Test 2: 2D Elasticity Problem Up: 2 Conversational tests Prev: 2.1 Introductory Example Contents