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Contents
Calculate the temperature distribution in a two-dimensional domain.
Figure 3.1: Two-dimensional heat 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)
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
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 |
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 ADRESS IN IT : 1 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 ADRESS IN IT : 8 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& 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 ADRESS IN IT : 1 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 ADRESS IN IT : 2 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& 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
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)