A Verification Study of ABAQUS AC3D8R Elements for Acoustic Wave Propagation

Transcription

1 A Verification Study of ABAQUS AC3D8R Elements for Acoustic Wave Propagation by Michael Robert Hubenthal A Project Submitted to the Graduate Faculty of Rensselaer Polytechnic Institute in Partial Fulfillment of the Requirements for the degree of Master of Mechanical Engineering Approved: Dr. Ernesto Gutierrez-Miravete, Project Adviser Rensselaer Polytechnic Institute Hartford, Connecticut December, 2010 (For Graduation June 2011) i

2 Copyright 2010 by Michael Hubenthal All Rights Reserved ii

3 CONTENTS Verification of ABAQUS AC3D8R Elements for Acoustic Wave Propagation... i LIST OF TABLES... iv LIST OF FIGURES... v NOMENCLATURE... vi ACKNOWLEDGMENTS... vii ABSTRACT... viii 1. INTRODUCTION METHODOLOGY FINITE ELEMENT MODEL DESCRIPTION General Description Element Size Mesh Density Aspect Ratio Skew Orientation Propagation Length RESULTS AND DISCUSSION Element Size Mesh Density Aspect Ratio Skew Orientation Propagation Length CONCLUSIONS REFERENCES APPENDIX A: Input Deck Example iii

4 LIST OF TABLES Table 1 Properties of Water... 2 Table 2 Index of Analysis Cases Table 3 Element Size Accuracy Comparison Table 4 Density Test vs. Element Size Test Accuracy Comparison Table 5 Aspect Ratio Accuracy Comparison Table 6 Aspect Ratio Test vs. Element Size Test Accuracy Comparison Table 7 Element Skew Accuracy Comparison Table 8 Element Orientation Accuracy Comparison Table 9 Propagation Length Accuracy Comparison iv

5 LIST OF FIGURES Figure 1 Pressure Wave Profile... 3 Figure 2 Aspect Ratio Variation... 4 Figure 3 Element Skew... 5 Figure 4 Element Orientation... 5 Figure 5 Baseline Model... 6 Figure 6 Element Size Study Models... 7 Figure 7 Mesh Density Study Models... 8 Figure 8 Aspect Ratio Study Models... 8 Figure 9 Skewed Element Study Models... 9 Figure 10 Element Orientation Study Model... 9 Figure 11 Propagation Length Figure 12 Element Size Test Results Figure 13 Mesh Density (2x2) Accuracy Comparison Figure 14 Mesh Density (4x4) Accuracy Comparison Figure 15 Aspect Ratio Test Results Figure 16 1:4 Aspect Ratio vs. Element Size Test Figure 17 4:1 Aspect Ratio vs. Element Size Test Figure 18 Element Skew Results Figure 19 Orientation Test Results Figure 20 Propagation Length Test Results v

6 NOMENCLATURE Symbol K c ρ t L m s sec N kg Description Bulk modulus Speed of sound Density Time Length Meter Second Second Newton kilogram vi

7 ACKNOWLEDGMENTS Soli Deo Gloria Proverbs 2:6 I would also like to thank my wife for graciously allowing me the time to work on this project during the new and especially busy time in our life, parenthood. vii

8 ABSTRACT The AC3D8R acoustic element in the ABAQUS finite element code is used for acoustic wave propagation. The element can be a very powerful tool for solving fluidstructure interaction problems in ABAQUS Explicit, but the pressure wave propagation must be correct before a correct structural response can be obtained. This project seeks to determine which geometric characteristics such as element size, aspect ratio, skew, element orientation, and propagation distance most severely affect wave propagation accuracy of the AC3D8R element. To test the quality of various geometric characteristics of the AC3D8R element, finite element fluid models that each exhibit one geometric characteristic of interest are created and exercised. A known wave is propagated through the fluid mesh, and the wave response in the fluid mesh is compared to the theoretical response. Results from the tests indicate that the element length in the direction of wave propagation is the geometric element characteristic that affects results the most. Neither the number of elements in the cross-sectional area normal to the wave nor the element frontal area affected the solution accuracy. Skew had also little effect on the pressure response, but element orientation slightly reduced the speed of wave propagation. Element size in the direction of wave propagation is the most important element geometric characteristic to control when creating AC3D8R fluid mesh. The solution accuracy also decreases with increasing distance through which the fluid must propagate. Therefore, it is important to limit the extent of the modeled fluid in the direction of propagation as much as possible to prevent accuracy losses as the wave travels through the fluid. viii

9 1. INTRODUCTION The ABAQUS finite element code has an eight-noded three-dimensional acoustic continuum element with reduced integration and hourglass control (Reference 1), the AC3D8R element, used for acoustic wave propagation. This element is often coupled with a structural finite element model to study the structure s response to pressure loading from a wave propagating through the fluid. This element can be a very powerful tool for solving fluid-structure interaction problems in ABAQUS Explicit. In order to solve the structural response, the fluid pressure loading profile must first propagate through the fluid. When the wave accurately propagates through the fluid, the correct structural response can be obtained, but if the pressure wave does not correctly propagate through the fluid, it is impossible to get the correct structural response regardless of the structural mesh quality. When using AC3D8R elements, it has been observed that for a pressure wave loading, the wave propagation through the fluid can vary with the fluid discretization. The Abaqus Verification Manual (Reference 2) runs test cases to show that wave propagation initializes properly in the fluid, but the manual does not provide documentation on the accuracy of the AC3DR element wave propagation solution as the wave progresses through the fluid. There are several geometric characteristics that can potentially affect the quality of the wave propagation through the fluid mesh such as element size, aspect ratio, skew, and element orientation relative to the wave. This project seeks to determine which characteristics most severely affect the accuracy of the fluid propagation. 1

10 2. METHODOLOGY To test the quality of various geometric characteristics of the AC3D8R element, finite element fluid models are exercised that each exhibit one geometric characteristic of interest. A wave is propagated through the fluid mesh, and the wave response in the fluid mesh is compared to the theoretical response. In order to determine the quality of a fluid element like the AC3D8R, the correct wave propagation solution must first be determined so that numerical solutions can be compared to predicted values from physics. The theoretical speed of sound, c, of a wave traveling through a fluid is calculated using a fluid s bulk modulus, K, and the fluid density, ρ, as shown below in Equation 1. This equation is solved for the speed of sound in water. Table 1 shows the properties of water. c= c= K ρ x10 N / m 3 999kg / m 2 = m / s (1) Table 1 Properties of Water Bulk Modulus, K 2.15x10 9 N/m 2 Density, ρ 999kg/m 3 Speed of sound, c m/s The theoretical time, t, for a wave to propagate through one meter of water can be determined using Equation 1, above, and a known fluid length, L. This is shown in Equation 2. L t = c m t = = x m / s 1 4 sec (2) Using the time required for a wave to travel through one meter of water, a general pressure-time history wave is developed that is then propagated through the fluid. A 2

11 simple triangular wave profile is selected as the general wave to propagate through the fluid for tests being performed in this analysis. The triangular wave is assigned an arbitrarily selected amplitude of 500 N/m 2. The wave is assigned a period of x10-4 sec, which is approximately half the time required for a wave to propagate through one meter of water. A picture of the pressure wave is shown below in Figure 1. Pressure vs. Time Pressure N/m^ E E E E E E E E E-04 time (sec) Figure 1 Pressure Wave Profile A baseline finite element fluid model is developed that uses the same mesh size and discretization as the ABAQUS verification manual s incident wave test of an AC3D8R element subjected to a pressure amplitude defined planar incident wave in ABAQUS Explicit (Reference 2). The triangular wave is propagated through the fluid, and the wave s pressure profile in the discretized fluid is compared to the expected profile from the theoretical wave traveling at the speed of sound in water. The baseline model will be the model that the geometry variations will be compared to. 3

12 To study element size effects, fluid volumes are created that have the same distance to propagate as the baseline mesh, but the element size is varied from ¼ of the baseline element length to elements with edges 10 times the length of the baseline elements. For each of these volumes, the cross sectional area is composed of a single cube shaped element normal to the incident wave and a constant 1m length in the direction of propagation. The triangular wave is propagated through the various sized meshes and compared to the speed and amplitude of the baseline wave. Cases with elements that have the same dimensions as the baseline model but smaller element sizes are also tested in order to discover whether results are affected by having multiple elements across the frontal the cross sectional area. To study aspect ratio effects, a fluid volume is created that has the same dimensions as the baseline test. The volume is meshed with elements that have a constant area normal to the pressure wave, but the element length aspect ratio varies from 1:4 to 4:1. The triangular wave is propagated through the mesh and compared to the speed and amplitude of the baseline wave. Figure 2 shows an example of aspect ratio variation. Figure 2 Aspect Ratio Variation To study element skew effects, a fluid volume is created that has the same number of elements as the baseline test, but the top surface of the elements is skewed. The skew is varied from 5 degrees to 45 degrees. The same plane, longitudinal wave is propagated through the mesh and compared to the speed and amplitude of the baseline wave. Figure 3 gives an example of element skew. 4

13 Figure 3 Element Skew To study the effects of element orientation relative to the incident wave, a mesh with cube elements will be built with the elements rotated 45 degrees from the baseline orientation. Geometry cannot be duplicated as in other cases without adding wedge elements, but the length of the mesh in the direction of the wave will be maintained. The same wave is propagated through the mesh and compared to the speed and amplitude of the baseline wave. Figure 4 displays fluid mesh with an element oriented at 45 degrees from the baseline orientation. Figure 4 Element Orientation The results of each of these cases will be compared relative to one another to reveal which geometric considerations most greatly affect the solution accuracy. In addition to the element geometry tests, an additional case will be considered to determine to what degree the fluid propagation results are affected by the overall distance the wave must travel as it propagates through the fluid. To study this, the baseline, ½ baseline, and ¼ baseline element size models will be extended to 10x s the original model length, and the wave will be propagated through the extended fluid. 5

14 3. FINITE ELEMENT MODEL DESCRIPTION 3.1 General Description The baseline finite element model consists of a 100 element AC3D8R acoustic fluid volume. Each element is a cube with 0.01m long edges. The model is loaded with a planar incident wave originating 10 meters from the end of the fluid at the wave source point indicated below. The wave s position at the beginning of the analysis is determined by the wave initialization point. The wave initialization point, or standoff, locates the wave at time = 0.0 sec as if the wave had already propagated from the source. This saves analysis time and disc space by allowing the analysis to start with the wave at the location of interest instead of explicitly stepping through the time necessary for the wave to propagate from the source point to the modeled fluid. A non-reflecting surface boundary condition is applied at the far end of the fluid volume. This boundary effect is equivalent to having additional fluid beyond the modeled region as opposed to a solid wall at the end that could reflect the wave s energy. The model s wave formulation uses total wave formulation, which accounts for incident and scattered waves. References 3 and 4 provide more thorough explanations of source point, wave initialization, total wave formulation, and acoustic boundary conditions. Wave source Wave initialization point Incident wave loaded surface Non-reflecting end surface Figure 5 Baseline Model 6

15 3.2 Element Size The first geometric characteristic being considered was element size. For this study, the length of fluid was maintained while the element size varied. The finite element models used are shown below in Figure 6 along with isometric views of the different elements sizes. The models each utilize the same loads and boundary conditions as the baseline model from Figure 5. ¼ baseline element size ½ baseline element size baseline element size 2x baseline element size 3x baseline element size 4x baseline element size 10x baseline element size Figure 6 Element Size Study Models 7

16 3.3 Mesh Density The models in the mesh density study have the volume as the baseline model as well as identical loads and boundary conditions. However, the models in the mesh density study have more elements in the same volume than the baseline model. Figure 7 shows the mesh density study fluid models. baseline element size density: 2 x 2 (element size ½ baseline) density: 4 x 4 (element size ¼ baseline) 3.4 Aspect Ratio Figure 7 Mesh Density Study Models Models for the aspect ratio study have the same volume, loads, and boundary conditions as the baseline model, but the elements vary in aspect ratio from 1:4 to 4:1. The models for the aspect ratio study are shown below in Figure 8. 1:4 1:3 1:2 baseline 2:1 3:1 4:1 Figure 8 Aspect Ratio Study Models 8

17 3.5 Skew The models used to evaluate element skew have the same number of elements and the same loads and boundary conditions as the baseline case. However, the top nodes in the skewed models are shifted to produce non-orthogonal elements. Elements with skew angle from 5 to 45 are investigated in this test. Figure 9, below, displays the skewed models. 5 skew 15 skew 30 skew 45 skew Figure 9 Skewed Element Study Models 3.6 Orientation The finite element model used to study the effects of element orientation is shown below in Figure 10. Figure 10 Element Orientation Study Model This model has the same overall dimensions, loads, and boundary conditions as the baseline model, but the AC3D8R elements are oriented at 45 from horizontal. This model also includes AC3D6 wedge elements that are necessary to maintain the overall geometry of the fluid volume. 9

18 3.7 Propagation Length The models for the propagation length study match the loads, element sizes, and boundary conditions from the element size test models (baseline, ½ baseline, and ¼ baseline). The only difference is that the models in this study have overall fluid lengths increased to 10x s the original one meter distance. Figure 11 shows an example of the increased propagation length. Figure 11 Propagation Length 10

19 Table 2. An overview of all the analysis cases run for all the tests is included below in Test Element Size Mesh Density Aspect Ratio Table 2 Index of Analysis Cases ¼ baseline ½ baseline Baseline 2x baseline 3x baseline 4x baseline 10x baseline Density 2x2 Density 4x4 Aspect 4:1 Aspect 2:1 Description m x m x0.0025m cube elements, 1 meter long fluid, 1 element cross section 0.005m x 0.005m x0.005m cube elements, 1 meter long fluid, 1 element cross section 0.01m x 0.01m x0.01m cube elements, 1 meter long fluid, 1 element cross section 0.02m x 0.02m x0.02m cube elements, 1 meter long fluid, 1 element cross section 0.033m x 0.033m x0.033m cube elements, 1 meter long fluid, 1 element cross section 0.04m x 0.04m x0.04m cube elements, 1 meter long fluid, 1 element cross section 0.1m x 0.1m x0.1m cube elements, 1 meter long fluid, 1 element cross section 0.005m x 0.005m x0.005m cube elements, 1 meter long fluid, 2 elements by 2 elements cross section m x m x0.0025m cube elements, 1 meter long fluid, 4 elements by 4 elements cross section 0.01m x 0.01m x0.0025m elements, 1 meter long fluid, 1 element cross section 0.01m x 0.01m x0.005m elements, 1 meter long fluid, 1 element cross section 11

20 Aspect 1:2 Aspect 1:3 Aspect 1:4 Skew 5 degree skew 15 degree skew 30 degree skew 45 degree skew Orientation Orientation model Propagation Length Baseline long ½ baseline element size long ¼ baseline element size long 0.01m x 0.01m x0.02m elements, 1 meter long fluid, 1 element cross section 0.01m x 0.01m x0.033m elements, 1 meter long fluid, 1 element cross section 0.01m x 0.01m x0.04m elements, 1 meter long fluid, 1 element cross section 0.01m length elements, 5 degree skew, 1 meter long fluid, 1 element cross section 0.01m length elements, 15 degree skew, 1 meter long fluid, 1 element cross section 0.01m length elements, 30 degree skew, 1 meter long fluid, 1 element cross section 0.01m length elements, 45 degree skew, 1 meter long fluid, 1 element cross section m x m x m cube elements, cubes rotated 45 degrees, 1 meter long fluid, 1 element cross section 0.01m x 0.01m x0.01m cube elements, 10 meter long fluid, 1 element cross section 0.005m x 0.005m x0.005m cube elements, 10 meter long fluid, 1 element cross section m x m x0.0025m cube elements, 10 meter long fluid, 1 element cross section 12

21 4. RESULTS AND DISCUSSION 4.1 Element Size The results from the element size test are shown below in Figure 12 at a time of seconds. Table 3 shows the time of peak pressure, peak pressure magnitude, peak pressure accuracy as a percentage compared to the theoretical peak pressure of 500 N/m 2, and the norm (a measure of overall error) from each model. The equation for the norm is given below in Equation 3. 1 L 2 Norm = ( ptheoetical x p el x )dx L ( ) mod ( ) (3) Element Size (t= sec) Pressure (N/m^2) theoretical element size 1/4 of baseline element size 1/2 of baseline baseline element size 2x baseline element size 3x baseline element size 4x baseline element size 10x baseline distance (m) Figure 12 Element Size Test Results Table 3 Element Size Accuracy Comparison 13

22 Model Peak Pressure Maximum Time Pressure (N/m 2 ) Accuracy Norm Theoretical 0.75 sec ¼ baseline sec % 1.82 ½ baseline sec % 7.73 Baseline sec % x baseline sec % x baseline sec % x baseline sec % x baseline sec % From the results above, it is apparent that using smaller sized elements increases the solution accuracy. The peak pressure time, peak accuracy, and norm all trend toward the theoretical as the element size gets smaller. The 10x baseline model results do show a peak value that is higher than the 4x baseline model, but the 10x baseline solution is more severely out of phase from the theoretical solution than the 4x baseline model and has a norm over eight times as large as the 4x baseline model. From Figure 12, it is also apparent that the results behind the wave are sensitive to element size. Models with larger elements have increasingly large error after the pressure wave has passed. The models with smaller element size have less oscillation after the wave has passed. 4.2 Mesh Density The element size test demonstrates that using smaller elements produces a peak pressure magnitude and time history results that more closely match the theoretical response than the larger elements, but do the results vary when there are additional similar sized elements connected in the direction normal to the wave propagation direction? To determine this, the results of the mesh density studies are shown compared with the element size tests having elements of like sizes. In Figure 13, the 2x2 density model results are compared to the results from the ½ baseline element size model. Both of these models have cube elements with edge lengths of 0.005in. Figure 14 shows the 4x4 density model results compared with the ¼ baseline element size model. The models in Figure 14 each have elements with edge lengths in long on each side. 14

23 Figure 13 Mesh Density (2x2) Accuracy Comparison Figure 14 Mesh Density (4x4) Accuracy Comparison 15

24 As is seen from the results in Table 4, both the 2x2 and 4x4 density results lay very nearly on top of the results of the element size tests. This indicates that the additional elements across the fluid volume s cross-sectional area do not affect the solution accuracy. Table 4 Density Test vs. Element Size Test Accuracy Comparison Model Peak Pressure Maximum Time Pressure (N/m 2 ) Accuracy Norm Density 2x sec % 7.73 ½ baseline sec % 7.73 Density 4x sec % 1.94 ¼ baseline sec % Aspect Ratio The results from the aspect ratio test are shown below in Figure 15 and Table Aspect Ratio (t= sec) Pressure (N/m^2) Theoretical aspect 4 to 1 aspect 2 to 1 baseline aspect 1 to 2 aspect 1 to 3 aspect 1 to distance (m) Figure 15 Aspect Ratio Test Results 16

25 Table 5 Aspect Ratio Accuracy Comparison Model Peak Pressure Maximum Time Pressure (N/m 2 ) Accuracy Norm Theoretical 0.75 sec Aspect 4: sec % 2.08 Aspect 2: sec % 8.48 Baseline sec % Aspect 1: sec % Aspect 1: sec % Aspect 1: % Consistent with the element size test results, the results from the aspect ratio test show that the solution accuracy improves as the number of elements in the direction of wave propagation increases. The 4:1 aspect ratio element model has a roughly 10% better accuracy compared with the 1:4 aspect ratio model results. From the results above, it is not immediately apparent if the frontal area of the elements affects the solution accuracy or only the element length in the direction of propagation matters. To determine whether or not the cross sectional area of the element affects solution results, the results of the aspect ratio test are compared to the corresponding element size test results. The results from models with large elements, the 1:4 aspect ratio model and 4x element size model, are compared in Figure 16. Both of these models have poor results compared to the theoretical response, but the results from the aspect ratio test match those of the corresponding element size test. Figure 17 shows a results comparison for the more accurate 4:1 aspect ratio model and the ¼ element size model. Both figures show that the aspect ratio results land almost exactly on top of the results from the element size tests. This indicates that the frontal area of the element does not affect the solution results, but rather the element length in the direction of wave propagation is the critical parameter. Table 6 provides a tabular overview of the results from these analyses. 17

26 Frontal Area 0.04m x0.04m (t= sec) 500 aspect 1 to element size 4x baseline Pressure (N/m^2) distance (m) Figure 16 1:4 Aspect Ratio vs. Element Size Test Frontal Area m x m (t= sec) aspect 4 to 1 element size 1/4 of baseline Pressure (N/m^2) distance (m) Figure 17 4:1 Aspect Ratio vs. Element Size Test 18

27 Table 6 Aspect Ratio Test vs. Element Size Test Accuracy Comparison Model Peak Pressure Maximum Time Pressure (N/m 2 ) Accuracy Norm Aspect 1: sec % x baseline sec % Aspect 4: sec % 2.08 ¼ baseline sec % Skew The results from the skew test are shown below in Figure 18. These results show all the skewed cases closely matching the baseline analysis. This indicates that the element skew plays a minor role in solution quality. 500 Element Skew (t= sec) Pressure (N/m^2) deg. Skew 15 deg. Skew 30 deg. Skew 45 deg. Skew Theoretical baseline distance (in) Figure 18 Element Skew Results 19

28 Table 7 shows the maximum pressure for each of the skewed cases as well as the baseline case. Table 7 Element Skew Accuracy Comparison Model Peak Pressure Maximum Time Pressure (N/m 2 ) Accuracy Norm Theoretical 0.75 sec Baseline sec % degree skew sec % degree skew sec % degree skew sec % degree skew sec %

29 4.5 Orientation The results from the element orientation test are shown below in Figure 19. These results show that the elements oriented at 45 degrees from the baseline orientation produce results similar in magnitude and shape to the baseline analysis, but the orientation model results lag behind the baseline results in time. Table 8 shows the results comparison between the baseline model and orientation model Element Orientation (t= sec) theoretical baseline orientation Pressure (N/m^2) distance (m) Figure 19 Orientation Test Results Table 8 Element Orientation Accuracy Comparison Model Peak Pressure Maximum Time Pressure (N/m 2 ) Accuracy Norm Theoretical 0.75 sec Baseline sec % Orientation sec %

30 4.6 Propagation Length The results from the propagation length test are shown below in Figure 20. In Table 9, the results are compared to the element size tests. In the propagation length tests, the smaller size elements are again more accurate than the larger elements. However, the overall accuracy decreases for each of the models compared to the results from the one meter length models. The extended baseline model had the most significant drop in peak pressure, but the extended ¼ size baseline model developed a small bias error of about 3 N/m 2 in the areas behind the wave. This bias contributed to the noticeable increase in the norm for the extended ¼ size baseline model from the one meter length analysis Extended Length Propagation (t= sec) Theoretical 1/2 size baseline-long 1/4 size baseline-long baseline_long Pressure (N/m^2) Distance (m) Figure 20 Propagation Length Test Results 22

31 Model Table 9 Propagation Length Accuracy Comparison Peak Pressure Time (sec) 1 meter length 10 meter length Peak Maximum Maximum Pressure Pressure Accuracy Norm Pressure (N/m 2 Time ) (N/m 2 ) (sec) Accuracy Norm Theoretical Baseline % % ½ baseline element size ¼ baseline element size % % % %

32 5. CONCLUSIONS From the tests performed, it is concluded that the element length in the direction of wave propagation is the geometric element characteristic that affects results the most. Neither the number of elements in the cross section nor the cross-sectional area affected the solution accuracy. Skew had little effect on the pressure magnitude. Element orientation did affect the solution accuracy, but the effects of element size were much more severe than the lag caused by element orientation. Element size in the direction of wave propagation is the most important element geometric characteristic to control when creating AC3D8R fluid mesh. From the propagation length test, it is also apparent that the solution accuracy decreases with increasing length through which the fluid must propagate. Therefore, it is important to limit the extent of the modeled fluid in the direction of propagation to prevent accuracy losses as the wave travels through the fluid. 24

33 6. REFERENCES 1. Abaqus Analysis User s Manual Section , Three-dimensional solid element library, Abaqus Version Abaqus Verification Manual Section , Incident Wave Loading, Abaqus Version Abaqus Analysis User s Manual Section 6.9.1, Acoustic, shock and coupled acoustic-structural analysis, Abaqus Version Abaqus Analysis User s Manual Section , Acoustic and shock loads, Abaqus Version

34 7. APPENDIX A: Input Deck Example The input deck for the baseline case is shown below. *Heading ** Job name: baseline_cae Model name: baseline_wave_analysis ** Generated by: Abaqus/CAE *Preprint, echo=no, model=no, history=no, contact=no ** ** PARTS ** *Part, name=part-1 *Node 1, 0., 0., 0. 2, , 0., 0. 3, , 0., 0. 4, , 0., 0. 5, , 0., 0. 6, , 0., 0. 7, , 0., 0. 8, , 0., 0. 9, , 0., 0. 10, , 0., 0. 11, , 0., 0. 12, , 0., 0. 13, , 0., 0. 14, , 0., 0. 15, , 0., 0. 16, , 0., 0. 17, , 0., 0. 18, , 0., 0. 19, , 0., 0. 20, , 0., 0. 21, , 0., 0. 22, , 0., 0. 23, , 0., 0. 24, , 0., 0. 25, , 0., 0. 26, 0.25, 0., 0. 27, , 0., 0. 28, , 0., 0. 29, , 0., 0. 30, , 0., 0. 31, , 0., 0. 32, , 0., 0. 33, , 0., 0. 34, , 0., 0. 35, , 0., 0. 36, , 0., 0. 37, , 0., 0. 38, , 0., 0. 39, , 0., 0. 40, , 0., 0. 41, , 0., 0. 42, , 0., 0. 43, , 0., 0. 44, , 0., 0. 45, , 0., 0. 46, , 0., 0. 47, , 0., 0. 48, , 0., 0. 49, , 0., 0. 50, , 0., 0. 51, 0.5, 0., 0. 52, , 0., 0. 53, , 0., 0. 54, , 0., 0. 26

35 55, , 0., 0. 56, , 0., 0. 57, , 0., 0. 58, , 0., 0. 59, , 0., 0. 60, , 0., 0. 61, , 0., 0. 62, , 0., 0. 63, , 0., 0. 64, , 0., 0. 65, , 0., 0. 66, , 0., 0. 67, , 0., 0. 68, , 0., 0. 69, , 0., 0. 70, , 0., 0. 71, , 0., 0. 72, , 0., 0. 73, , 0., 0. 74, , 0., 0. 75, , 0., 0. 76, 0.75, 0., 0. 77, , 0., 0. 78, , 0., 0. 79, , 0., 0. 80, , 0., 0. 81, , 0., 0. 82, , 0., 0. 83, , 0., 0. 84, , 0., 0. 85, , 0., 0. 86, , 0., 0. 87, , 0., 0. 88, , 0., 0. 89, , 0., 0. 90, , 0., 0. 91, , 0., 0. 92, , 0., 0. 93, , 0., 0. 94, , 0., 0. 95, , 0., 0. 96, , 0., 0. 97, , 0., 0. 98, , 0., 0. 99, , 0., , , 0., , 1., 0., , 0., , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0. 27

36 1025, , , , 0.25, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0.5, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0.75, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0. 28

37 1096, , , , , , , , , , , , , , , , 1., , , 0., , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0.25, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0.5, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

38 1566, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0.75, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 1., , , 0., 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , 0.25, 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0.,

39 2536, , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , 0.5, 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , 0.75, 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , , 0., , 1., 0., *Element, type=ac3d8r 1, 1, 2, 2502, 2501, 1001, 1002, 1502, , 2, 3, 2503, 2502, 1002, 1003, 1503, , 3, 4, 2504, 2503, 1003, 1004, 1504, , 4, 5, 2505, 2504, 1004, 1005, 1505,

40 5, 5, 6, 2506, 2505, 1005, 1006, 1506, , 6, 7, 2507, 2506, 1006, 1007, 1507, , 7, 8, 2508, 2507, 1007, 1008, 1508, , 8, 9, 2509, 2508, 1008, 1009, 1509, , 9, 10, 2510, 2509, 1009, 1010, 1510, , 10, 11, 2511, 2510, 1010, 1011, 1511, , 11, 12, 2512, 2511, 1011, 1012, 1512, , 12, 13, 2513, 2512, 1012, 1013, 1513, , 13, 14, 2514, 2513, 1013, 1014, 1514, , 14, 15, 2515, 2514, 1014, 1015, 1515, , 15, 16, 2516, 2515, 1015, 1016, 1516, , 16, 17, 2517, 2516, 1016, 1017, 1517, , 17, 18, 2518, 2517, 1017, 1018, 1518, , 18, 19, 2519, 2518, 1018, 1019, 1519, , 19, 20, 2520, 2519, 1019, 1020, 1520, , 20, 21, 2521, 2520, 1020, 1021, 1521, , 21, 22, 2522, 2521, 1021, 1022, 1522, , 22, 23, 2523, 2522, 1022, 1023, 1523, , 23, 24, 2524, 2523, 1023, 1024, 1524, , 24, 25, 2525, 2524, 1024, 1025, 1525, , 25, 26, 2526, 2525, 1025, 1026, 1526, , 26, 27, 2527, 2526, 1026, 1027, 1527, , 27, 28, 2528, 2527, 1027, 1028, 1528, , 28, 29, 2529, 2528, 1028, 1029, 1529, , 29, 30, 2530, 2529, 1029, 1030, 1530, , 30, 31, 2531, 2530, 1030, 1031, 1531, , 31, 32, 2532, 2531, 1031, 1032, 1532, , 32, 33, 2533, 2532, 1032, 1033, 1533, , 33, 34, 2534, 2533, 1033, 1034, 1534, , 34, 35, 2535, 2534, 1034, 1035, 1535, , 35, 36, 2536, 2535, 1035, 1036, 1536, , 36, 37, 2537, 2536, 1036, 1037, 1537, , 37, 38, 2538, 2537, 1037, 1038, 1538, , 38, 39, 2539, 2538, 1038, 1039, 1539, , 39, 40, 2540, 2539, 1039, 1040, 1540, , 40, 41, 2541, 2540, 1040, 1041, 1541, , 41, 42, 2542, 2541, 1041, 1042, 1542, , 42, 43, 2543, 2542, 1042, 1043, 1543, , 43, 44, 2544, 2543, 1043, 1044, 1544, , 44, 45, 2545, 2544, 1044, 1045, 1545, , 45, 46, 2546, 2545, 1045, 1046, 1546, , 46, 47, 2547, 2546, 1046, 1047, 1547, , 47, 48, 2548, 2547, 1047, 1048, 1548, , 48, 49, 2549, 2548, 1048, 1049, 1549, , 49, 50, 2550, 2549, 1049, 1050, 1550, , 50, 51, 2551, 2550, 1050, 1051, 1551, , 51, 52, 2552, 2551, 1051, 1052, 1552, , 52, 53, 2553, 2552, 1052, 1053, 1553, , 53, 54, 2554, 2553, 1053, 1054, 1554, , 54, 55, 2555, 2554, 1054, 1055, 1555, , 55, 56, 2556, 2555, 1055, 1056, 1556, , 56, 57, 2557, 2556, 1056, 1057, 1557, , 57, 58, 2558, 2557, 1057, 1058, 1558, , 58, 59, 2559, 2558, 1058, 1059, 1559, , 59, 60, 2560, 2559, 1059, 1060, 1560, , 60, 61, 2561, 2560, 1060, 1061, 1561, , 61, 62, 2562, 2561, 1061, 1062, 1562, , 62, 63, 2563, 2562, 1062, 1063, 1563, , 63, 64, 2564, 2563, 1063, 1064, 1564, , 64, 65, 2565, 2564, 1064, 1065, 1565, , 65, 66, 2566, 2565, 1065, 1066, 1566, , 66, 67, 2567, 2566, 1066, 1067, 1567, , 67, 68, 2568, 2567, 1067, 1068, 1568, , 68, 69, 2569, 2568, 1068, 1069, 1569, , 69, 70, 2570, 2569, 1069, 1070, 1570, , 70, 71, 2571, 2570, 1070, 1071, 1571, , 71, 72, 2572, 2571, 1071, 1072, 1572, , 72, 73, 2573, 2572, 1072, 1073, 1573, , 73, 74, 2574, 2573, 1073, 1074, 1574, , 74, 75, 2575, 2574, 1074, 1075, 1575, , 75, 76, 2576, 2575, 1075, 1076, 1576,

41 76, 76, 77, 2577, 2576, 1076, 1077, 1577, , 77, 78, 2578, 2577, 1077, 1078, 1578, , 78, 79, 2579, 2578, 1078, 1079, 1579, , 79, 80, 2580, 2579, 1079, 1080, 1580, , 80, 81, 2581, 2580, 1080, 1081, 1581, , 81, 82, 2582, 2581, 1081, 1082, 1582, , 82, 83, 2583, 2582, 1082, 1083, 1583, , 83, 84, 2584, 2583, 1083, 1084, 1584, , 84, 85, 2585, 2584, 1084, 1085, 1585, , 85, 86, 2586, 2585, 1085, 1086, 1586, , 86, 87, 2587, 2586, 1086, 1087, 1587, , 87, 88, 2588, 2587, 1087, 1088, 1588, , 88, 89, 2589, 2588, 1088, 1089, 1589, , 89, 90, 2590, 2589, 1089, 1090, 1590, , 90, 91, 2591, 2590, 1090, 1091, 1591, , 91, 92, 2592, 2591, 1091, 1092, 1592, , 92, 93, 2593, 2592, 1092, 1093, 1593, , 93, 94, 2594, 2593, 1093, 1094, 1594, , 94, 95, 2595, 2594, 1094, 1095, 1595, , 95, 96, 2596, 2595, 1095, 1096, 1596, , 96, 97, 2597, 2596, 1096, 1097, 1597, , 97, 98, 2598, 2597, 1097, 1098, 1598, , 98, 99, 2599, 2598, 1098, 1099, 1599, , 99, 100, 2600, 2599, 1099, 1100, 1600, , 100, 101, 2601, 2600, 1100, 1101, 1601, 1600 *Elset, elset=channel, generate 1, 100, 1 ** Section: Section-1-CHANNEL *Solid Section, elset=channel, material=water, *End Part ** ** ** ASSEMBLY ** *Assembly, name=assembly ** *Instance, name=part-1-1, part=part-1 *End Instance ** *Node 1, 0., 0., 0. *Node 2, -10., 0., 0. *Nset, nset=_pickedset13, internal 2, *Nset, nset=_pickedset14, internal 1, *Nset, nset=channel, instance=part-1-1 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96 97, 98, 99, 100, 101, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, , 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, , 1029, 1030, 1031, 1032, 1033, 1034, 1035, 1036, 1037, 1038, 1039, 1040, 1041, 1042, , 1045, 1046, 1047, 1048, 1049, 1050, 1051, 1052, 1053, 1054, 1055, 1056, 1057, 1058, , 1061, 1062, 1063, 1064, 1065, 1066, 1067, 1068, 1069, 1070, 1071, 1072, 1073, 1074,