Modeling a Composite Slot Cross-Section for Torsional Analysis

Modeling a Composite Slot Cross-Section for Torsional Analysis The cross-section in question is shown below (see also p. 432 in the textbook). Due to double symmetry, only one-quarter of the cross-section will be modeled as shown below. 1 V. N. Kaliakin

To creat this model using PATRAN requires the following actions. Start the PATRAN application. Create a suitable new database (journal) file (e.g. filled_slot.db) in which to record all commands and actions. Click on the Geometry icon. Using Create -> Points -> XYZ, define the coordinates of the nine (9) points shown above. These are used to describe the four (4) recatngular regions that the domain is divided into. Using Create -> Curve -> 2 Point, define the following straight lines:1-2, 2-5,5-4,4-1, 2-3,3-6, 6-5, 5-8, 8-7, 7-4, 6-9, and 9-8. Using Create -> Surface -> Edge -> 4 Edge, create the following four (4) surfaces: surface 1 (curve 1, curve 2, curve 3, curve 4), surface 2 (curve 5, curve 6, curve 7, curve 2), surface 3 (curve 7, curve 11, curve 12, curve 8), surface 4 (curve 3, curve 8, curve 9, curve 10). Note that the curves are selected sequentially (in this case counterclockwise). Click on the Elements icon. Using Create -> Mesh Seed -> Uniform, proceed to define the number of elements desired along given curves (lines) in the various surfaces. In prticular, 3 elements along curve 1, 3 elements along curve 9, 4 elements along curve 10, 6 elements along curve 4, and 8 elements along curves 5 and 12. Using Create -> Mesh -> Surface, create the meshes in surface 1, surface 2, surface 3, and finally in surface 4. In this case quad4 (linear quadrilateral elements have been selected). Since four separate rectangular surfaces have been meshed, duplicate nodes have been created at the same location along common surface boundaries. To condense these common nodes (i.e., do delete unnecessary duplicate nodes) select the Equivalence -> All -> Tolerance Cube command with the default tolerance and press Apply. To next optimize the nodal numbering, select the Optimize -> Nodes -> Cuthill- McKee command with the Profile radio button selected (UD_scalar emplys a profile solver) and press Apply. A window will pop up with the old and new bandwidth, profile, etc. displayed. Close this window by presssing OK. Click on the Loads/BCs icon. Using Create -> Temperature -> Nodal, decsribe the homogeneous essential specificatiosn aloing the external boundary. In particular, choose a New Set Name (e.g., external_boundary). Next press the Input Data button and enter a value of 0.0 for Temperature and press OK. Press the Select Application Region button and select curves 6, 9, 11 and 12. Press the OK button. Once back in the main menu, press Apply. Some red markers will next appear at vertex nodes along the chosen boundary. 2 V. N. Kaliakin

Click on the Properties icon. Create -> 2D -> Shell, decsribe and apply the two material types to the appropriate surfaces. In particular, enter a Proprty Set Name (e.g., material_1). Then press the Input Proerties button and fill in the material name and thickness (equal to 1.0) and press OK. Then proceed to Select Members and choose the appropriate surfaces (surfaces 2, 3, 4 for material 1). Press Apply. Repeat this procedure for the second material (applicable to surface 1). Next write the Patran neutral file by going to the File menu and scrolling down to Export. Make sure that the format selected is indeed a neutral file and then name the file (e.g., filled_slot.out) and save it to an appropriate directory. The neutral file must next be translated to UD_scalar format. To do this run the pat2ud_scalar program. The input file asked for should be the Patran neutral file (e.g., filled_slot.out). The output file name supplied must be something different from the filename used for the neutral file (e.g., filled_slot.dat). The resulting data file (e.g., filled_slot.dat) requires a bit of editing before it can be analyzed using UD_scalar. In particular, Check that the proper element type has been specified in the ELEMENT commands appearing in the input file filled_slot.dat The number of elements needs to be properly specified in the DIMENSION command block. Although the number of elements is given in the translated file, the proper DIMENSION command must reflect the proper type of element used (in this case four-node quadrilateral elements). As such, the following command must be supplied: DIMENSION MAX QS4 110, where the total number of elements (110) value was provided by Patran. The number of material descriptions must also be specified in the DIMENSION command block. Since two different materials are used in the model, it follows that the following command must be supplied: DIMENSION MAX scalar_1 2 The heat source and material descriptions must next follow the FINISHED SETTINGS command. The former is equal to the product 2GQ, while the latter must describe an identity coefficient matrix. Since two different materials are used in the model, it follows that the following command must be supplied: scalar source number 1 source_1 9600.0 scalar source number 2 source_1 4800.0 3 V. N. Kaliakin

! scalar conduc constant number 1 & desc "this is material #1" k11 1.0 k22 1.0 scalar conduc constant number 2 & desc "this is material #2" k11 1.0 k22 1.0 For brevity, the input file associated with a 20-element mesh is shown on the next page (NOTE: some lines of input have been wrapped by the word processing software). 4 V. N. Kaliakin

ana title "torsion of a 'slot' cross-section" ana title "with TWO different materials" ana title "20 element mesh of QS4 elements"! ANALYSIS ACTION ANALYZE ANALYSIS IDEALIZATION PLANE_STRAIN ANALYSIS TEMPORAL STATIC! ECHO WARNINGS OFF! dim max scalar_1 2 dim max nodes = 30 dim max qs4 = 20! FINISHED SETTINGS! scalar source number 1 source_1 9600.0! scalar source number 2 source_1 4800.0! scalar conduc constant number 1 & desc "this is material #1" k11 1.0 k22 1.0! scalar conduc constant number 2 & desc "this is material #2" k11 1.0 k22 1.0! NODES LINE NUMBER 1 x1 x2 4.000000000E+00 x3 NODES LINE NUMBER 2 x1 5.000000000E-01 x2 4.000000000E+00 x3 NODES LINE NUMBER 3 x1 1.000000000E+00 x2 4.000000000E+00 x3 NODES LINE NUMBER 4 x1 2.000000000E+00 x2 4.000000000E+00 x3 NODES LINE NUMBER 5 x1 3.000000000E+00 x2 4.000000000E+00 x3 NODES LINE NUMBER 6 x1 x2 3.500000000E+00 x3 NODES LINE NUMBER 7 x1 5.000000000E-01 x2 3.500000000E+00 x3 NODES LINE NUMBER 8 x1 1.000000000E+00 x2 3.500000000E+00 x3 NODES LINE NUMBER 9 x1 2.000000000E+00 x2 3.500000000E+00 x3 NODES LINE NUMBER 10 x1 3.000000000E+00 x2 3.500000000E+00 x3 NODES LINE NUMBER 11 x1 x2 3.000000000E+00 x3 NODES LINE NUMBER 12 x1 x2 2.000000000E+00 x3 NODES LINE NUMBER 13 x1 x2 1.000000119E+00 x3 NODES LINE NUMBER 14 x1 x2 x3 NODES LINE NUMBER 15 x1 5.000000000E-01 x2 3.000000000E+00 x3 5 V. N. Kaliakin

NODES LINE NUMBER 16 x1 1.000000000E+00 x2 3.000000000E+00 x3 NODES LINE NUMBER 17 x1 2.000000000E+00 x2 3.000000000E+00 x3 NODES LINE NUMBER 18 x1 3.000000000E+00 x2 3.000000000E+00 x3 NODES LINE NUMBER 19 x1 5.000000000E-01 x2 2.000000000E+00 x3 NODES LINE NUMBER 20 x1 5.000000000E-01 x2 1.000000119E+00 x3 NODES LINE NUMBER 21 x1 5.000000000E-01 x2 x3 NODES LINE NUMBER 22 x1 1.000000000E+00 x2 2.000000000E+00 x3 NODES LINE NUMBER 23 x1 2.000000000E+00 x2 2.000000238E+00 x3 NODES LINE NUMBER 24 x1 3.000000000E+00 x2 2.000000000E+00 x3 NODES LINE NUMBER 25 x1 1.000000000E+00 x2 1.000000000E+00 x3 NODES LINE NUMBER 26 x1 1.000000000E+00 x2 x3 NODES LINE NUMBER 27 x1 2.000000000E+00 x2 1.000000119E+00 x3 NODES LINE NUMBER 28 x1 3.000000000E+00 x2 1.000000000E+00 x3 NODES LINE NUMBER 29 x1 2.000000000E+00 x2 x3 NODES LINE NUMBER 30 x1 3.000000000E+00 x2 x3! ELEM SCALAR TYPE QS4 SCALAR 2 NODES 14 13 20 21 ELEM SCALAR TYPE QS4 SCALAR 2 NODES 13 12 19 20 ELEM SCALAR TYPE QS4 SCALAR 2 NODES 12 11 15 19 ELEM SCALAR TYPE QS4 SCALAR 2 NODES 21 20 25 26 ELEM SCALAR TYPE QS4 SCALAR 2 NODES 20 19 22 25 ELEM SCALAR TYPE QS4 SCALAR 2 NODES 19 15 16 22 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 26 25 27 29 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 25 22 23 27 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 22 16 17 23 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 29 27 28 30 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 27 23 24 28 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 23 17 18 24 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 18 17 9 10 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 17 16 8 9 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 10 9 4 5 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 9 8 3 4 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 16 15 7 8 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 15 11 6 7 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 8 7 2 3 ELEM SCALAR TYPE QS4 SCALAR 1 NODES 7 6 1 2! SPEC CONC SCALAR NODE 1 PHI VALUE SPEC CONC SCALAR NODE 2 PHI VALUE SPEC CONC SCALAR NODE 3 PHI VALUE SPEC CONC SCALAR NODE 4 PHI VALUE SPEC CONC SCALAR NODE 5 PHI VALUE 6 V. N. Kaliakin

SPEC CONC SCALAR NODE 10 PHI VALUE SPEC CONC SCALAR NODE 18 PHI VALUE SPEC CONC SCALAR NODE 24 PHI VALUE SPEC CONC SCALAR NODE 28 PHI VALUE SPEC CONC SCALAR NODE 30 PHI VALUE! FINISHED DATA! SOLUTION TIME FINAL 1.0 INCR 1 OUTPUT 1:10:1! FINISHED LOADING! A plot of the mesh, obtained using Tecplot, is shown below. The results obtained analyzing the above data by the ud_scalar computer program are given on the following page (NOTE: some lines of input have been wrapped by the word processing software). 7 V. N. Kaliakin

_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/ _/ _/ _/ ud_scalar: version 1.00: 23.04.08 _/ _/ _/ _/ _/ _/ DATE OF ANALYSIS :: day:16 month:03 year:10 _/ _/ _/ _/ ANALYSIS INITIATED AT TIME :: 15:11:38 _/ _/ _/ _/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/ data file translated from PATRAN neutral file using PAT2UD_SCALAR version 1.0 P3/PATRAN Neutral File from: C:\filled_slot.db neutral file creation date: 10-Mar-08 neutral file creation time: 14:13:19 PATRAN release number: 3.0 torsion of a 'slot' cross-section with TWO different materials 80 element mesh of QS4 elements ====================================================================== D Y N A M I C S T O R A G E A L L O C A T I O N ====================================================================== Largest NODE number which can used in the mesh = 99 Max. no. of CONSTANT scalar conductivity idealizations = 2 Max. no. of 4-node quad. "scalar" (QS4) elements = 80 ====================================================================== = G E N E R A L A N A L Y S I S I N F O R M A T I O N = ====================================================================== --> analysis with SCALAR primary dependent variables shall be performed --> TWO-DIMENSIONAL solution domain assumed (PLANE STRAIN idealization) --> solver type used : SKYLINE --> storage type : SYMMETRIC --> "Isoparametric" mesh generation scheme used --> LINEAR analysis ====================================================================== = S C A L A R M A T E R I A L P A R A M E T E R S = 8 V. N. Kaliakin

====================================================================== --> idealization no.: 1 ~~~~~~~~~~~~~~~ type : constant scalar conductivity coefficients info. : this is material #1 "Conductivities" (material parameters) : k_11 = 1.000E+00 k_12 = 0.000E+00 k_13 = 0.000E+00 k_22 = 1.000E+00 k_23 = 0.000E+00 k_33 = 1.000E+00 source term S_1 = 9.600E+03 source term S_2 = 0.000E+00 --> idealization no.: 2 ~~~~~~~~~~~~~~~ type : constant scalar conductivity coefficients info. : this is material #2 "Conductivities" (material parameters) : k_11 = 1.000E+00 k_12 = 0.000E+00 k_13 = 0.000E+00 k_22 = 1.000E+00 k_23 = 0.000E+00 k_33 = 1.000E+00 source term S_1 = 4.800E+03 source term S_2 = 0.000E+00 ====================================================================== = N O D A L C O O R D I N A T E S = ====================================================================== 1 x1 = 3.000E+00 x2 = 4.000E+00 2 x1 = 2.500E+00 x2 = 4.000E+00 3 x1 = 2.000E+00 x2 = 4.000E+00 4 x1 = 1.500E+00 x2 = 4.000E+00 5 x1 = 1.000E+00 x2 = 4.000E+00 6 x1 = 7.500E-01 x2 = 4.000E+00 7 x1 = 5.000E-01 x2 = 4.000E+00 8 x1 = 2.500E-01 x2 = 4.000E+00 9 x1 = 0.000E+00 x2 = 4.000E+00 10 x1 = 3.000E+00 x2 = 3.750E+00 11 x1 = 2.500E+00 x2 = 3.750E+00 12 x1 = 2.000E+00 x2 = 3.750E+00 13 x1 = 1.500E+00 x2 = 3.750E+00 14 x1 = 1.000E+00 x2 = 3.750E+00 15 x1 = 7.500E-01 x2 = 3.750E+00 16 x1 = 5.000E-01 x2 = 3.750E+00 17 x1 = 2.500E-01 x2 = 3.750E+00 18 x1 = 0.000E+00 x2 = 3.750E+00 19 x1 = 3.000E+00 x2 = 3.500E+00 20 x1 = 2.500E+00 x2 = 3.500E+00 21 x1 = 2.000E+00 x2 = 3.500E+00 22 x1 = 1.500E+00 x2 = 3.500E+00 23 x1 = 1.000E+00 x2 = 3.500E+00 24 x1 = 7.500E-01 x2 = 3.500E+00 25 x1 = 5.000E-01 x2 = 3.500E+00 9 V. N. Kaliakin

26 x1 = 2.500E-01 x2 = 3.500E+00 27 x1 = 0.000E+00 x2 = 3.500E+00 28 x1 = 3.000E+00 x2 = 3.250E+00 29 x1 = 2.500E+00 x2 = 3.250E+00 30 x1 = 2.000E+00 x2 = 3.250E+00 31 x1 = 1.500E+00 x2 = 3.250E+00 32 x1 = 1.000E+00 x2 = 3.250E+00 33 x1 = 7.500E-01 x2 = 3.250E+00 34 x1 = 5.000E-01 x2 = 3.250E+00 35 x1 = 2.500E-01 x2 = 3.250E+00 36 x1 = 0.000E+00 x2 = 3.250E+00 37 x1 = 3.000E+00 x2 = 3.000E+00 38 x1 = 2.500E+00 x2 = 3.000E+00 39 x1 = 2.000E+00 x2 = 3.000E+00 40 x1 = 1.500E+00 x2 = 3.000E+00 41 x1 = 1.000E+00 x2 = 3.000E+00 42 x1 = 7.500E-01 x2 = 3.000E+00 43 x1 = 5.000E-01 x2 = 3.000E+00 44 x1 = 2.500E-01 x2 = 3.000E+00 45 x1 = 0.000E+00 x2 = 3.000E+00 46 x1 = 3.000E+00 x2 = 2.500E+00 47 x1 = 2.500E+00 x2 = 2.500E+00 48 x1 = 2.000E+00 x2 = 2.500E+00 49 x1 = 1.500E+00 x2 = 2.500E+00 50 x1 = 1.000E+00 x2 = 2.500E+00 51 x1 = 7.500E-01 x2 = 2.500E+00 52 x1 = 5.000E-01 x2 = 2.500E+00 53 x1 = 2.500E-01 x2 = 2.500E+00 54 x1 = 0.000E+00 x2 = 2.500E+00 55 x1 = 3.000E+00 x2 = 2.000E+00 56 x1 = 3.000E+00 x2 = 1.500E+00 57 x1 = 3.000E+00 x2 = 1.000E+00 58 x1 = 3.000E+00 x2 = 5.000E-01 59 x1 = 3.000E+00 x2 = 0.000E+00 60 x1 = 2.500E+00 x2 = 2.000E+00 61 x1 = 2.000E+00 x2 = 2.000E+00 62 x1 = 1.500E+00 x2 = 2.000E+00 63 x1 = 1.000E+00 x2 = 2.000E+00 64 x1 = 7.500E-01 x2 = 2.000E+00 65 x1 = 5.000E-01 x2 = 2.000E+00 66 x1 = 2.500E-01 x2 = 2.000E+00 67 x1 = 0.000E+00 x2 = 2.000E+00 68 x1 = 2.500E+00 x2 = 1.500E+00 69 x1 = 2.500E+00 x2 = 1.000E+00 70 x1 = 2.500E+00 x2 = 5.000E-01 71 x1 = 2.500E+00 x2 = 0.000E+00 72 x1 = 2.000E+00 x2 = 1.500E+00 73 x1 = 1.500E+00 x2 = 1.500E+00 74 x1 = 1.000E+00 x2 = 1.500E+00 75 x1 = 7.500E-01 x2 = 1.500E+00 76 x1 = 5.000E-01 x2 = 1.500E+00 77 x1 = 2.500E-01 x2 = 1.500E+00 78 x1 = 0.000E+00 x2 = 1.500E+00 79 x1 = 2.000E+00 x2 = 1.000E+00 80 x1 = 2.000E+00 x2 = 5.000E-01 81 x1 = 2.000E+00 x2 = 0.000E+00 82 x1 = 0.000E+00 x2 = 1.000E+00 83 x1 = 0.000E+00 x2 = 5.000E-01 84 x1 = 0.000E+00 x2 = 0.000E+00 85 x1 = 1.500E+00 x2 = 1.000E+00 86 x1 = 1.000E+00 x2 = 1.000E+00 87 x1 = 7.500E-01 x2 = 1.000E+00 88 x1 = 5.000E-01 x2 = 1.000E+00 10 V. N. Kaliakin

89 x1 = 2.500E-01 x2 = 1.000E+00 90 x1 = 1.500E+00 x2 = 5.000E-01 91 x1 = 1.500E+00 x2 = 0.000E+00 92 x1 = 2.500E-01 x2 = 5.000E-01 93 x1 = 2.500E-01 x2 = 0.000E+00 94 x1 = 1.000E+00 x2 = 5.000E-01 95 x1 = 7.500E-01 x2 = 5.000E-01 96 x1 = 5.000E-01 x2 = 5.000E-01 97 x1 = 1.000E+00 x2 = 0.000E+00 98 x1 = 5.000E-01 x2 = 0.000E+00 99 x1 = 7.500E-01 x2 = 0.000E+00 ====================================================================== = E L E M E N T I N F O R M A T I O N = ====================================================================== --> number : 1 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 84 83 92 93 --> number : 2 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 83 82 89 92 --> number : 3 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 82 78 77 89 --> number : 4 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 78 67 66 77 --> number : 5 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 67 54 53 66 --> number : 6 (type : QS4 ) (kind : SCALAR ) 11 V. N. Kaliakin

~~~~~~ nodes : 54 45 44 53 --> number : 7 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 93 92 96 98 --> number : 8 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 92 89 88 96 --> number : 9 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 89 77 76 88 --> number : 10 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 77 66 65 76 --> number : 11 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 66 53 52 65 --> number : 12 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 53 44 43 52 --> number : 13 (type : QS4 ) (kind : SCALAR ) 12 V. N. Kaliakin

~~~~~~ nodes : 98 96 95 99 --> number : 14 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 96 88 87 95 --> number : 15 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 88 76 75 87 --> number : 16 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 76 65 64 75 --> number : 17 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 65 52 51 64 --> number : 18 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 52 43 42 51 --> number : 19 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 99 95 94 97 --> number : 20 (type : QS4 ) (kind : SCALAR ) 13 V. N. Kaliakin

~~~~~~ nodes : 95 87 86 94 --> number : 21 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 87 75 74 86 --> number : 22 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 75 64 63 74 --> number : 23 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 64 51 50 63 --> number : 24 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 51 42 41 50 --> number : 25 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 97 94 90 91 --> number : 26 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 94 86 85 90 --> number : 27 (type : QS4 ) (kind : SCALAR ) 14 V. N. Kaliakin

~~~~~~ nodes : 86 74 73 85 --> number : 28 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 74 63 62 73 --> number : 29 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 63 50 49 62 --> number : 30 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 50 41 40 49 --> number : 31 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 91 90 80 81 --> number : 32 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 90 85 79 80 --> number : 33 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 85 73 72 79 --> number : 34 (type : QS4 ) (kind : SCALAR ) 15 V. N. Kaliakin

~~~~~~ nodes : 73 62 61 72 --> number : 35 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 62 49 48 61 --> number : 36 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 49 40 39 48 --> number : 37 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 81 80 70 71 --> number : 38 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 80 79 69 70 --> number : 39 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 79 72 68 69 --> number : 40 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 72 61 60 68 --> number : 41 (type : QS4 ) (kind : SCALAR ) 16 V. N. Kaliakin

~~~~~~ nodes : 61 48 47 60 --> number : 42 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 48 39 38 47 --> number : 43 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 71 70 58 59 --> number : 44 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 70 69 57 58 --> number : 45 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 69 68 56 57 --> number : 46 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 68 60 55 56 --> number : 47 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 60 47 46 55 --> number : 48 (type : QS4 ) (kind : SCALAR ) 17 V. N. Kaliakin

~~~~~~ nodes : 47 38 37 46 --> number : 49 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 37 28 29 38 --> number : 50 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 28 19 20 29 --> number : 51 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 19 10 11 20 --> number : 52 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 10 1 2 11 --> number : 53 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 38 29 30 39 --> number : 54 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 29 20 21 30 --> number : 55 (type : QS4 ) (kind : SCALAR ) 18 V. N. Kaliakin

~~~~~~ nodes : 20 11 12 21 --> number : 56 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 11 2 3 12 --> number : 57 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 39 30 31 40 --> number : 58 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 30 21 22 31 --> number : 59 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 21 12 13 22 --> number : 60 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 12 3 4 13 --> number : 61 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 40 31 32 41 --> number : 62 (type : QS4 ) (kind : SCALAR ) 19 V. N. Kaliakin

~~~~~~ nodes : 31 22 23 32 --> number : 63 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 22 13 14 23 --> number : 64 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 13 4 5 14 --> number : 65 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 41 32 33 42 --> number : 66 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 32 23 24 33 --> number : 67 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 23 14 15 24 --> number : 68 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 14 5 6 15 --> number : 69 (type : QS4 ) (kind : SCALAR ) 20 V. N. Kaliakin

~~~~~~ nodes : 42 33 34 43 --> number : 70 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 33 24 25 34 --> number : 71 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 24 15 16 25 --> number : 72 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 15 6 7 16 --> number : 73 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 43 34 35 44 --> number : 74 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 34 25 26 35 --> number : 75 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 25 16 17 26 --> number : 76 (type : QS4 ) (kind : SCALAR ) 21 V. N. Kaliakin

~~~~~~ nodes : 16 7 8 17 --> number : 77 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 44 35 36 45 --> number : 78 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 35 26 27 36 --> number : 79 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 26 17 18 27 --> number : 80 (type : QS4 ) (kind : SCALAR ) ~~~~~~ nodes : 17 8 9 18 ====================================================================== = N O D E P O I N T S P E C I F I C A T I O N S = ====================================================================== Node ( c o o r d i n a t e s ) Number s p e c i f i c a t i o n: ~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ 1 : ( x1 = 3.000E+00, x2 = 4.000E+00 ) 2 : ( x1 = 2.500E+00, x2 = 4.000E+00 ) 3 : ( x1 = 2.000E+00, x2 = 4.000E+00 ) 22 V. N. Kaliakin

4 : ( x1 = 1.500E+00, x2 = 4.000E+00 ) 5 : ( x1 = 1.000E+00, x2 = 4.000E+00 ) 6 : ( x1 = 7.500E-01, x2 = 4.000E+00 ) 7 : ( x1 = 5.000E-01, x2 = 4.000E+00 ) 8 : ( x1 = 2.500E-01, x2 = 4.000E+00 ) 9 : ( x1 = 0.000E+00, x2 = 4.000E+00 ) 10 : ( x1 = 3.000E+00, x2 = 3.750E+00 ) 19 : ( x1 = 3.000E+00, x2 = 3.500E+00 ) 28 : ( x1 = 3.000E+00, x2 = 3.250E+00 ) 37 : ( x1 = 3.000E+00, x2 = 3.000E+00 ) 46 : ( x1 = 3.000E+00, x2 = 2.500E+00 ) 55 : ( x1 = 3.000E+00, x2 = 2.000E+00 ) 56 : ( x1 = 3.000E+00, x2 = 1.500E+00 ) 57 : ( x1 = 3.000E+00, x2 = 1.000E+00 ) 58 : ( x1 = 3.000E+00, x2 = 5.000E-01 ) 59 : ( x1 = 3.000E+00, x2 = 0.000E+00 ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ end of mathematical model data ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ====================================================================== = E L E M E N T F L U X E S = ====================================================================== --> element 1 ( type = QS4 ) : [ 23 V. N. Kaliakin

@(x1 = 1.250E-01, x2 = 2.500E-01 ) : flux_1 = 3.369E+02 ; flux_2 = 5.229E+02 --> element 2 ( type = QS4 ) : [ @(x1 = 1.250E-01, x2 = 7.500E-01 ) : flux_1 = 3.212E+02 ; flux_2 = 1.600E+03 --> element 3 ( type = QS4 ) : [ @(x1 = 1.250E-01, x2 = 1.250E+00 ) : flux_1 = 2.904E+02 ; flux_2 = 2.771E+03 --> element 4 ( type = QS4 ) : [ @(x1 = 1.250E-01, x2 = 1.750E+00 ) : flux_1 = 2.476E+02 ; flux_2 = 4.091E+03 --> element 5 ( type = QS4 ) : [ @(x1 = 1.250E-01, x2 = 2.250E+00 ) : flux_1 = 2.008E+02 ; flux_2 = 5.595E+03 --> element 6 ( type = QS4 ) : [ @(x1 = 1.250E-01, x2 = 2.750E+00 ) : flux_1 = 1.572E+02 ; flux_2 = 7.281E+03 --> element 7 ( type = QS4 ) : [ @(x1 = 3.750E-01, x2 = 2.500E-01 ) : flux_1 = 1.015E+03 ; flux_2 = 5.149E+02 --> element 8 ( type = QS4 ) : [ @(x1 = 3.750E-01, x2 = 7.500E-01 ) : flux_1 = 9.673E+02 ; flux_2 = 1.577E+03 --> element 9 ( type = QS4 ) : [ @(x1 = 3.750E-01, x2 = 1.250E+00 ) : flux_1 = 8.744E+02 ; flux_2 = 2.733E+03 --> element 10 ( type = QS4 ) : [ @(x1 = 3.750E-01, x2 = 1.750E+00 ) : flux_1 = 7.441E+02 ; flux_2 = 4.043E+03 --> element 11 ( type = QS4 ) : [ @(x1 = 3.750E-01, x2 = 2.250E+00 ) : flux_1 = 6.006E+02 ; flux_2 = 5.548E+03 24 V. N. Kaliakin

--> element 12 ( type = QS4 ) : [ @(x1 = 3.750E-01, x2 = 2.750E+00 ) : flux_1 = 4.719E+02 ; flux_2 = 7.241E+03 --> element 13 ( type = QS4 ) : [ @(x1 = 6.250E-01, x2 = 2.500E-01 ) : flux_1 = 1.704E+03 ; flux_2 = 4.990E+02 --> element 14 ( type = QS4 ) : [ @(x1 = 6.250E-01, x2 = 7.500E-01 ) : flux_1 = 1.625E+03 ; flux_2 = 1.529E+03 --> element 15 ( type = QS4 ) : [ @(x1 = 6.250E-01, x2 = 1.250E+00 ) : flux_1 = 1.469E+03 ; flux_2 = 2.656E+03 --> element 16 ( type = QS4 ) : [ @(x1 = 6.250E-01, x2 = 1.750E+00 ) : flux_1 = 1.245E+03 ; flux_2 = 3.943E+03 --> element 17 ( type = QS4 ) : [ @(x1 = 6.250E-01, x2 = 2.250E+00 ) : flux_1 = 9.930E+02 ; flux_2 = 5.450E+03 --> element 18 ( type = QS4 ) : [ @(x1 = 6.250E-01, x2 = 2.750E+00 ) : flux_1 = 7.848E+02 ; flux_2 = 7.171E+03 --> element 19 ( type = QS4 ) : [ @(x1 = 8.750E-01, x2 = 2.500E-01 ) : flux_1 = 2.414E+03 ; flux_2 = 4.751E+02 --> element 20 ( type = QS4 ) : [ @(x1 = 8.750E-01, x2 = 7.500E-01 ) : flux_1 = 2.302E+03 ; flux_2 = 1.458E+03 --> element 21 ( type = QS4 ) : [ @(x1 = 8.750E-01, x2 = 1.250E+00 ) : flux_1 = 2.081E+03 ; flux_2 = 2.538E+03 --> element 22 ( type = QS4 ) : [ 25 V. N. Kaliakin

@(x1 = 8.750E-01, x2 = 1.750E+00 ) : flux_1 = 1.758E+03 ; flux_2 = 3.788E+03 --> element 23 ( type = QS4 ) : [ @(x1 = 8.750E-01, x2 = 2.250E+00 ) : flux_1 = 1.369E+03 ; flux_2 = 5.284E+03 --> element 24 ( type = QS4 ) : [ @(x1 = 8.750E-01, x2 = 2.750E+00 ) : flux_1 = 1.078E+03 ; flux_2 = 7.087E+03 --> element 25 ( type = QS4 ) : [ @(x1 = 1.250E+00, x2 = 2.500E-01 ) : flux_1 = 4.740E+03 ; flux_2 = 4.215E+02 --> element 26 ( type = QS4 ) : [ @(x1 = 1.250E+00, x2 = 7.500E-01 ) : flux_1 = 4.581E+03 ; flux_2 = 1.296E+03 --> element 27 ( type = QS4 ) : [ @(x1 = 1.250E+00, x2 = 1.250E+00 ) : flux_1 = 4.259E+03 ; flux_2 = 2.267E+03 --> element 28 ( type = QS4 ) : [ @(x1 = 1.250E+00, x2 = 1.750E+00 ) : flux_1 = 3.774E+03 ; flux_2 = 3.414E+03 --> element 29 ( type = QS4 ) : [ @(x1 = 1.250E+00, x2 = 2.250E+00 ) : flux_1 = 3.149E+03 ; flux_2 = 4.838E+03 --> element 30 ( type = QS4 ) : [ @(x1 = 1.250E+00, x2 = 2.750E+00 ) : flux_1 = 2.479E+03 ; flux_2 = 6.717E+03 --> element 31 ( type = QS4 ) : [ @(x1 = 1.750E+00, x2 = 2.500E-01 ) : flux_1 = 8.781E+03 ; flux_2 = 3.278E+02 --> element 32 ( type = QS4 ) : [ @(x1 = 1.750E+00, x2 = 7.500E-01 ) : flux_1 = 8.562E+03 ; flux_2 = 1.011E+03 26 V. N. Kaliakin

--> element 33 ( type = QS4 ) : [ @(x1 = 1.750E+00, x2 = 1.250E+00 ) : flux_1 = 8.112E+03 ; flux_2 = 1.780E+03 --> element 34 ( type = QS4 ) : [ @(x1 = 1.750E+00, x2 = 1.750E+00 ) : flux_1 = 7.407E+03 ; flux_2 = 2.712E+03 --> element 35 ( type = QS4 ) : [ @(x1 = 1.750E+00, x2 = 2.250E+00 ) : flux_1 = 6.418E+03 ; flux_2 = 3.925E+03 --> element 36 ( type = QS4 ) : [ @(x1 = 1.750E+00, x2 = 2.750E+00 ) : flux_1 = 5.088E+03 ; flux_2 = 5.630E+03 --> element 37 ( type = QS4 ) : [ @(x1 = 2.250E+00, x2 = 2.500E-01 ) : flux_1 = 1.304E+04 ; flux_2 = 2.085E+02 --> element 38 ( type = QS4 ) : [ @(x1 = 2.250E+00, x2 = 7.500E-01 ) : flux_1 = 1.277E+04 ; flux_2 = 6.442E+02 --> element 39 ( type = QS4 ) : [ @(x1 = 2.250E+00, x2 = 1.250E+00 ) : flux_1 = 1.222E+04 ; flux_2 = 1.140E+03 --> element 40 ( type = QS4 ) : [ @(x1 = 2.250E+00, x2 = 1.750E+00 ) : flux_1 = 1.132E+04 ; flux_2 = 1.754E+03 --> element 41 ( type = QS4 ) : [ @(x1 = 2.250E+00, x2 = 2.250E+00 ) : flux_1 = 1.001E+04 ; flux_2 = 2.577E+03 --> element 42 ( type = QS4 ) : [ @(x1 = 2.250E+00, x2 = 2.750E+00 ) : flux_1 = 8.118E+03 ; flux_2 = 3.761E+03 --> element 43 ( type = QS4 ) : [ 27 V. N. Kaliakin

@(x1 = 2.750E+00, x2 = 2.500E-01 ) : flux_1 = 1.755E+04 ; flux_2 = 7.157E+01 --> element 44 ( type = QS4 ) : [ @(x1 = 2.750E+00, x2 = 7.500E-01 ) : flux_1 = 1.726E+04 ; flux_2 = 2.215E+02 --> element 45 ( type = QS4 ) : [ @(x1 = 2.750E+00, x2 = 1.250E+00 ) : flux_1 = 1.665E+04 ; flux_2 = 3.932E+02 --> element 46 ( type = QS4 ) : [ @(x1 = 2.750E+00, x2 = 1.750E+00 ) : flux_1 = 1.564E+04 ; flux_2 = 6.080E+02 --> element 47 ( type = QS4 ) : [ @(x1 = 2.750E+00, x2 = 2.250E+00 ) : flux_1 = 1.414E+04 ; flux_2 = 8.991E+02 --> element 48 ( type = QS4 ) : [ @(x1 = 2.750E+00, x2 = 2.750E+00 ) : flux_1 = 1.191E+04 ; flux_2 = 1.326E+03 --> element 49 ( type = QS4 ) : [ @(x1 = 2.750E+00, x2 = 3.125E+00 ) : flux_1 = 9.694E+03 ; flux_2 = 1.786E+03 --> element 50 ( type = QS4 ) : [ @(x1 = 2.750E+00, x2 = 3.375E+00 ) : flux_1 = 7.693E+03 ; flux_2 = 2.218E+03 --> element 51 ( type = QS4 ) : [ @(x1 = 2.750E+00, x2 = 3.625E+00 ) : flux_1 = 5.172E+03 ; flux_2 = 2.824E+03 --> element 52 ( type = QS4 ) : [ @(x1 = 2.750E+00, x2 = 3.875E+00 ) : flux_1 = 1.880E+03 ; flux_2 = 3.760E+03 --> element 53 ( type = QS4 ) : [ @(x1 = 2.250E+00, x2 = 3.125E+00 ) : flux_1 = 6.293E+03 ; flux_2 = 5.001E+03 28 V. N. Kaliakin

--> element 54 ( type = QS4 ) : [ @(x1 = 2.250E+00, x2 = 3.375E+00 ) : flux_1 = 4.738E+03 ; flux_2 = 6.115E+03 --> element 55 ( type = QS4 ) : [ @(x1 = 2.250E+00, x2 = 3.625E+00 ) : flux_1 = 2.941E+03 ; flux_2 = 7.562E+03 --> element 56 ( type = QS4 ) : [ @(x1 = 2.250E+00, x2 = 3.875E+00 ) : flux_1 = 9.919E+02 ; flux_2 = 9.504E+03 --> element 57 ( type = QS4 ) : [ @(x1 = 1.750E+00, x2 = 3.125E+00 ) : flux_1 = 3.858E+03 ; flux_2 = 7.370E+03 --> element 58 ( type = QS4 ) : [ @(x1 = 1.750E+00, x2 = 3.375E+00 ) : flux_1 = 2.865E+03 ; flux_2 = 8.843E+03 --> element 59 ( type = QS4 ) : [ @(x1 = 1.750E+00, x2 = 3.625E+00 ) : flux_1 = 1.771E+03 ; flux_2 = 1.062E+04 --> element 60 ( type = QS4 ) : [ @(x1 = 1.750E+00, x2 = 3.875E+00 ) : flux_1 = 6.006E+02 ; flux_2 = 1.269E+04 --> element 61 ( type = QS4 ) : [ @(x1 = 1.250E+00, x2 = 3.125E+00 ) : flux_1 = 1.970E+03 ; flux_2 = 8.673E+03 --> element 62 ( type = QS4 ) : [ @(x1 = 1.250E+00, x2 = 3.375E+00 ) : flux_1 = 1.531E+03 ; flux_2 = 1.041E+04 --> element 63 ( type = QS4 ) : [ @(x1 = 1.250E+00, x2 = 3.625E+00 ) : flux_1 = 9.665E+02 ; flux_2 = 1.237E+04 --> element 64 ( type = QS4 ) : [ 29 V. N. Kaliakin

@(x1 = 1.250E+00, x2 = 3.875E+00 ) : flux_1 = 3.298E+02 ; flux_2 = 1.455E+04 --> element 65 ( type = QS4 ) : [ @(x1 = 8.750E-01, x2 = 3.125E+00 ) : flux_1 = 9.761E+02 ; flux_2 = 9.054E+03 --> element 66 ( type = QS4 ) : [ @(x1 = 8.750E-01, x2 = 3.375E+00 ) : flux_1 = 8.358E+02 ; flux_2 = 1.104E+04 --> element 67 ( type = QS4 ) : [ @(x1 = 8.750E-01, x2 = 3.625E+00 ) : flux_1 = 5.443E+02 ; flux_2 = 1.315E+04 --> element 68 ( type = QS4 ) : [ @(x1 = 8.750E-01, x2 = 3.875E+00 ) : flux_1 = 1.882E+02 ; flux_2 = 1.540E+04 --> element 69 ( type = QS4 ) : [ @(x1 = 6.250E-01, x2 = 3.125E+00 ) : flux_1 = 6.530E+02 ; flux_2 = 9.120E+03 --> element 70 ( type = QS4 ) : [ @(x1 = 6.250E-01, x2 = 3.375E+00 ) : flux_1 = 5.282E+02 ; flux_2 = 1.124E+04 --> element 71 ( type = QS4 ) : [ @(x1 = 6.250E-01, x2 = 3.625E+00 ) : flux_1 = 3.467E+02 ; flux_2 = 1.342E+04 --> element 72 ( type = QS4 ) : [ @(x1 = 6.250E-01, x2 = 3.875E+00 ) : flux_1 = 1.204E+02 ; flux_2 = 1.571E+04 --> element 73 ( type = QS4 ) : [ @(x1 = 3.750E-01, x2 = 3.125E+00 ) : flux_1 = 3.790E+02 ; flux_2 = 9.204E+03 --> element 74 ( type = QS4 ) : [ @(x1 = 3.750E-01, x2 = 3.375E+00 ) : flux_1 = 2.978E+02 ; flux_2 = 1.137E+04 30 V. N. Kaliakin

--> element 75 ( type = QS4 ) : [ @(x1 = 3.750E-01, x2 = 3.625E+00 ) : flux_1 = 1.925E+02 ; flux_2 = 1.359E+04 --> element 76 ( type = QS4 ) : [ @(x1 = 3.750E-01, x2 = 3.875E+00 ) : flux_1 = 6.679E+01 ; flux_2 = 1.589E+04 --> element 77 ( type = QS4 ) : [ @(x1 = 1.250E-01, x2 = 3.125E+00 ) : flux_1 = 1.244E+02 ; flux_2 = 9.251E+03 --> element 78 ( type = QS4 ) : [ @(x1 = 1.250E-01, x2 = 3.375E+00 ) : flux_1 = 9.659E+01 ; flux_2 = 1.143E+04 --> element 79 ( type = QS4 ) : [ @(x1 = 1.250E-01, x2 = 3.625E+00 ) : flux_1 = 6.189E+01 ; flux_2 = 1.367E+04 --> element 80 ( type = QS4 ) : [ @(x1 = 1.250E-01, x2 = 3.875E+00 ) : flux_1 = 2.139E+01 ; flux_2 = 1.598E+04 maximum values of element variables : ----------------------------------- max flux_1 = 1.755E+04 @ x1 = 2.750E+00, x2 = 2.500E-01 max flux_2 = 1.598E+04 @ x1 = 1.250E-01, x2 = 3.875E+00 ====================================================================== = N O D A L Q U A N T I T I E S = ====================================================================== 1 ( x1 = 3.000E+00, x2 = 4.000E+00 ), phi = 9.401E-18 2 ( x1 = 2.500E+00, x2 = 4.000E+00 ), phi = 4.068E-17 3 ( x1 = 2.000E+00, x2 = 4.000E+00 ), phi = 6.697E-17 4 ( x1 = 1.500E+00, x2 = 4.000E+00 ), phi = 8.199E-17 5 ( x1 = 1.000E+00, x2 = 4.000E+00 ), phi = 6.717E-17 6 ( x1 = 7.500E-01, x2 = 4.000E+00 ), phi = 5.828E-17 7 ( x1 = 5.000E-01, x2 = 4.000E+00 ), phi = 5.922E-17 8 ( x1 = 2.500E-01, x2 = 4.000E+00 ), phi = 5.974E-17 9 ( x1 = 0.000E+00, x2 = 4.000E+00 ), phi = 2.995E-17 10 ( x1 = 3.000E+00, x2 = 3.750E+00 ), phi = 8.939E-18 11 ( x1 = 2.500E+00, x2 = 3.750E+00 ), phi = 1.880E+03 12 ( x1 = 2.000E+00, x2 = 3.750E+00 ), phi = 2.872E+03 13 ( x1 = 1.500E+00, x2 = 3.750E+00 ), phi = 3.473E+03 31 V. N. Kaliakin

14 ( x1 = 1.000E+00, x2 = 3.750E+00 ), phi = 3.802E+03 15 ( x1 = 7.500E-01, x2 = 3.750E+00 ), phi = 3.897E+03 16 ( x1 = 5.000E-01, x2 = 3.750E+00 ), phi = 3.957E+03 17 ( x1 = 2.500E-01, x2 = 3.750E+00 ), phi = 3.990E+03 18 ( x1 = 0.000E+00, x2 = 3.750E+00 ), phi = 4.001E+03 19 ( x1 = 3.000E+00, x2 = 3.500E+00 ), phi = 1.824E-17 20 ( x1 = 2.500E+00, x2 = 3.500E+00 ), phi = 3.292E+03 21 ( x1 = 2.000E+00, x2 = 3.500E+00 ), phi = 5.241E+03 22 ( x1 = 1.500E+00, x2 = 3.500E+00 ), phi = 6.411E+03 23 ( x1 = 1.000E+00, x2 = 3.500E+00 ), phi = 7.048E+03 24 ( x1 = 7.500E-01, x2 = 3.500E+00 ), phi = 7.226E+03 25 ( x1 = 5.000E-01, x2 = 3.500E+00 ), phi = 7.339E+03 26 ( x1 = 2.500E-01, x2 = 3.500E+00 ), phi = 7.402E+03 27 ( x1 = 0.000E+00, x2 = 3.500E+00 ), phi = 7.422E+03 28 ( x1 = 3.000E+00, x2 = 3.250E+00 ), phi = 2.532E-17 29 ( x1 = 2.500E+00, x2 = 3.250E+00 ), phi = 4.401E+03 30 ( x1 = 2.000E+00, x2 = 3.250E+00 ), phi = 7.190E+03 31 ( x1 = 1.500E+00, x2 = 3.250E+00 ), phi = 8.884E+03 32 ( x1 = 1.000E+00, x2 = 3.250E+00 ), phi = 9.778E+03 33 ( x1 = 7.500E-01, x2 = 3.250E+00 ), phi = 1.002E+04 34 ( x1 = 5.000E-01, x2 = 3.250E+00 ), phi = 1.017E+04 35 ( x1 = 2.500E-01, x2 = 3.250E+00 ), phi = 1.026E+04 36 ( x1 = 0.000E+00, x2 = 3.250E+00 ), phi = 1.028E+04 37 ( x1 = 3.000E+00, x2 = 3.000E+00 ), phi = 6.060E-17 38 ( x1 = 2.500E+00, x2 = 3.000E+00 ), phi = 5.294E+03 39 ( x1 = 2.000E+00, x2 = 3.000E+00 ), phi = 8.798E+03 40 ( x1 = 1.500E+00, x2 = 3.000E+00 ), phi = 1.096E+04 41 ( x1 = 1.000E+00, x2 = 3.000E+00 ), phi = 1.204E+04 42 ( x1 = 7.500E-01, x2 = 3.000E+00 ), phi = 1.229E+04 43 ( x1 = 5.000E-01, x2 = 3.000E+00 ), phi = 1.246E+04 44 ( x1 = 2.500E-01, x2 = 3.000E+00 ), phi = 1.256E+04 45 ( x1 = 0.000E+00, x2 = 3.000E+00 ), phi = 1.260E+04 46 ( x1 = 3.000E+00, x2 = 2.500E+00 ), phi = 9.715E-17 47 ( x1 = 2.500E+00, x2 = 2.500E+00 ), phi = 6.619E+03 48 ( x1 = 2.000E+00, x2 = 2.500E+00 ), phi = 1.123E+04 49 ( x1 = 1.500E+00, x2 = 2.500E+00 ), phi = 1.416E+04 50 ( x1 = 1.000E+00, x2 = 2.500E+00 ), phi = 1.556E+04 51 ( x1 = 7.500E-01, x2 = 2.500E+00 ), phi = 1.585E+04 52 ( x1 = 5.000E-01, x2 = 2.500E+00 ), phi = 1.607E+04 53 ( x1 = 2.500E-01, x2 = 2.500E+00 ), phi = 1.620E+04 54 ( x1 = 0.000E+00, x2 = 2.500E+00 ), phi = 1.624E+04 55 ( x1 = 3.000E+00, x2 = 2.000E+00 ), phi = 1.113E-16 56 ( x1 = 3.000E+00, x2 = 1.500E+00 ), phi = 1.208E-16 57 ( x1 = 3.000E+00, x2 = 1.000E+00 ), phi = 1.269E-16 58 ( x1 = 3.000E+00, x2 = 5.000E-01 ), phi = 1.304E-16 59 ( x1 = 3.000E+00, x2 = 0.000E+00 ), phi = 6.574E-17 60 ( x1 = 2.500E+00, x2 = 2.000E+00 ), phi = 7.518E+03 61 ( x1 = 2.000E+00, x2 = 2.000E+00 ), phi = 1.291E+04 62 ( x1 = 1.500E+00, x2 = 2.000E+00 ), phi = 1.640E+04 63 ( x1 = 1.000E+00, x2 = 2.000E+00 ), phi = 1.815E+04 64 ( x1 = 7.500E-01, x2 = 2.000E+00 ), phi = 1.854E+04 65 ( x1 = 5.000E-01, x2 = 2.000E+00 ), phi = 1.882E+04 66 ( x1 = 2.500E-01, x2 = 2.000E+00 ), phi = 1.899E+04 67 ( x1 = 0.000E+00, x2 = 2.000E+00 ), phi = 1.905E+04 68 ( x1 = 2.500E+00, x2 = 1.500E+00 ), phi = 8.126E+03 69 ( x1 = 2.500E+00, x2 = 1.000E+00 ), phi = 8.519E+03 70 ( x1 = 2.500E+00, x2 = 5.000E-01 ), phi = 8.741E+03 71 ( x1 = 2.500E+00, x2 = 0.000E+00 ), phi = 8.812E+03 72 ( x1 = 2.000E+00, x2 = 1.500E+00 ), phi = 1.406E+04 73 ( x1 = 1.500E+00, x2 = 1.500E+00 ), phi = 1.797E+04 74 ( x1 = 1.000E+00, x2 = 1.500E+00 ), phi = 2.000E+04 75 ( x1 = 7.500E-01, x2 = 1.500E+00 ), phi = 2.048E+04 76 ( x1 = 5.000E-01, x2 = 1.500E+00 ), phi = 2.083E+04 32 V. N. Kaliakin

77 ( x1 = 2.500E-01, x2 = 1.500E+00 ), phi = 2.103E+04 78 ( x1 = 0.000E+00, x2 = 1.500E+00 ), phi = 2.110E+04 79 ( x1 = 2.000E+00, x2 = 1.000E+00 ), phi = 1.480E+04 80 ( x1 = 2.000E+00, x2 = 5.000E-01 ), phi = 1.523E+04 81 ( x1 = 2.000E+00, x2 = 0.000E+00 ), phi = 1.536E+04 82 ( x1 = 0.000E+00, x2 = 1.000E+00 ), phi = 2.249E+04 83 ( x1 = 0.000E+00, x2 = 5.000E-01 ), phi = 2.329E+04 84 ( x1 = 0.000E+00, x2 = 0.000E+00 ), phi = 2.355E+04 85 ( x1 = 1.500E+00, x2 = 1.000E+00 ), phi = 1.900E+04 86 ( x1 = 1.000E+00, x2 = 1.000E+00 ), phi = 2.123E+04 87 ( x1 = 7.500E-01, x2 = 1.000E+00 ), phi = 2.179E+04 88 ( x1 = 5.000E-01, x2 = 1.000E+00 ), phi = 2.218E+04 89 ( x1 = 2.500E-01, x2 = 1.000E+00 ), phi = 2.241E+04 90 ( x1 = 1.500E+00, x2 = 5.000E-01 ), phi = 1.959E+04 91 ( x1 = 1.500E+00, x2 = 0.000E+00 ), phi = 1.978E+04 92 ( x1 = 2.500E-01, x2 = 5.000E-01 ), phi = 2.321E+04 93 ( x1 = 2.500E-01, x2 = 0.000E+00 ), phi = 2.347E+04 94 ( x1 = 1.000E+00, x2 = 5.000E-01 ), phi = 2.194E+04 95 ( x1 = 7.500E-01, x2 = 5.000E-01 ), phi = 2.254E+04 96 ( x1 = 5.000E-01, x2 = 5.000E-01 ), phi = 2.296E+04 97 ( x1 = 1.000E+00, x2 = 0.000E+00 ), phi = 2.217E+04 98 ( x1 = 5.000E-01, x2 = 0.000E+00 ), phi = 2.321E+04 99 ( x1 = 7.500E-01, x2 = 0.000E+00 ), phi = 2.278E+04 max phi = 2.355E+04 @ node 84 ( 0.000E+00, 0.000E+00) ud_scalar -> end of analysis........ A contour plot of the primary dependent variable, drawn using Tecplot, is provided on the following page. 33 V. N. Kaliakin

34 V. N. Kaliakin