Advantages, Limitations and Error Estimation of Mixed Solid Axisymmetric Modeling Sudeep Bosu TATA Consultancy Services Baskaran Sundaram TATA Consultancy Services, 185 Lloyds Road,Chennai-600086,INDIA Abstract The key to successful analysis of large structures often depends upon the analyst's ability to minimise the computational effort. An efficient method to reduce the computational effort is the mixed modeling of axisymmetric and 3D solid elements in ANSYS. This approach can be adopted for geometries possessing significant axisymmetric regions like turbine and compressor disks. The model size in these cases is large and consequently a large memory and disk space are needed to solve these models. Also, a variety of analyses are required to be executed on these models, like, thermal, structural, modal, etc. The finite element model building exercise is operationally more intensive than conventional method, which involves full 3D sectorial modeling, but it is more than offset by the time saved while running model for different analyses. However one needs to fully comprehend the limitations and errors that might be introduced and equate them with the advantages, while adopting this technique. The scope of this study is to investigate the advantages and limitations of this approach for different analyses. A study is conducted to establish the validity of the interface conditions, material and load scaling by comparing with a sectored 3D model for a simple model where results are easy to comprehend. The model was also analysed for determining the various limitations and drawbacks the methodology can impose for different analyses and the conclusions in this regard are made. In this approach, the section of the model which is completely axisymmetric is modeled using axisymmetric elements, sections which are cyclic symmetric are modeled with 3D elements as one sector with symmetry boundary conditions and sections which are not cyclically symmetric are modeled in full. Introduction The continuing challenge for any CAE analyst is to provide reliable solution in the shortest possible time. The time constraint of the situation and computational resources available, result in the analyst's assumptions towards simplification of the simulation. These assumptions could be the root cause for errors and limitations (apart from those which are always present in any numerical simulation). It is therefore required to get a clear idea as to the limitations, capabilities and errors that are likely to be present whenever a simulation methodology is adopted. This work is aimed at providing an insight into the methodology, advantages and limitations and errors that are likely to be present while using the mixed 3Daxisymmetric modeling approach. Methodology This section describes the steps to be followed while creating a model using the axisymmetric-solid approach. It also provides the justification for the application of scaled boundary conditions and material properties. The crucial concept to keep in mind when using axisymmetric and solid elements in the same model is that the axisymmetric elements represent the full 360 behavior of the geometry and the solid elements should be modified to represent the physical characteristics of the entire 360.
The steps involved in modeling are: 1. Identify axisymmetric and non-axisymmetric sections of the model 2. Generate 2-D and 3-D geometry. Add a small part of the axisymmetric section in the 3-D model to have sufficient margin for the transition to take place 3. Generate mapped mesh of linear MESH200 elements on the surface of 3-D model that connects to the axisymmetric section. Mesh 3-D section with either hexahedral or tetrahedral elements 4. Generate 2-D mesh of the axisymmetric section with linear axisymmetric elements along the edge that connects to the 3-D geometry. Make sure that the axisymmetric elements follow the requirements of ANSYS for example, global Y axis being the axial direction and the elements lying on a XY plane with Z coordinate being zero. 5. Rotate the solid section nodes about their Y-axis so their X-axis projects radially away from the global Y-axis (Figure 1) Figure 1. Solid element nodes rotated in axisymmetric coordinate System 6. Couple nodes on the interface that are located at a common radius in the radial and axial directions (Figure 2) Figure 2. Compatible boundary conditions at the interface
7. From this nodal set, constrain all the nodes (except nodes shared by axisymmetric elements) in the circumferential direction (Figure 3) Figure 3. Boundary conditions for the solid region 8. Apply appropriate symmetry boundary conditions on the cut surfaces of 3-D model (Figure 3) 9. Scale up the following material properties of the solid section, by N (where N is the total no. of solid sectors in the model) Elastic modulus, Density, Thermal conductivity, Specific heat (in case of transient analysis involving mass, you need to scale up only density and not specific heat). The material properties are not to be scaled for the axisymmetric section. 10. Scale up the following loads by N, Force, Pressure, Moment and Convection coefficient on the solid section. The loads on the axisymmetric section are not to be scaled. Normal Finite Element Analysis is degree of freedom (Dof) based and this requires us to maintain Dof compatibility between the solid and axisymmetric zones. ANSYS assembles the stiffness matrix for axisymmetric elements considering a 360 perspective as, K = 2π 1 1 0 1 1 B T EBr J dξdηdθ (eqn. 1) for the solid elements it is, K = 1 1 1 1 1 1 B T EB J dξdηdζ (eqn. 2)
where, B is the element strain displacement matrix, E is the material property matrix, J is the jacobian matrix and ξ,ζ,η are the natural coordinates. Since the solid model is sectored, the stiffness calculated for the same, will be the result of integration over the respective sector angle and not 360 or 2π radians as shown in equation 2. This implies the K matrix will be under valued by a factor of N. To over come this numerical issue the E matrix (in turn the material properties) is scaled up by the same factor. During postprocessing, the Dof's (displacements for structural and temperatures for thermal) are calculated correctly where as the integrated quantities like stresses and heat flux in the solid zone are scaled up by N. To postprocess derived terms, 1. First select the axisymmetric elements and retrieve results only for the selected elements 2. Select only the 3-D elements, scale these results by 1/N and append them to the results already stored in step 1. 3. Plot stresses for the entire model. Model Description A solid model featuring symmetric and axisymmetric regions is shown in Figure 4 and solid-axisymmetric model in Figure 5. Figure 4. Sector Model
Figure 5. Solid-Axisymmetric Model The model is built with SOLID92/87, SOLID95/90 and/or PLANE42/55 elements. The model details are: Table 1 Finite Element Model Details No of Elements No of Nodes Active Dof's (Thermal) Active Dof's (Structural) Sector Model 26082 98192 289953 98192 Solid Axisymmetric Model 12457 30361 89354 30361 The model is subjected to different types of analysis, namely, thermal, structural and modal. All the analyses mentioned in this paper are performed using ANSYS v7.0, on Dell Precision workstation 340, Windows 2000, P4, 2 GHz, 2 GB RAM machine. The analyses were carried out with default memory model. Analysis Results & Discussion Thermal Table 2 and Figure 6 below illustrate the time taken for the 45 sector model and the mixed model, to solve a steady state thermal problem using three different solvers available in ANSYS.
Table 2 Disk requirements and Thermal runtimes for the Sector and Mixed models PCG Sparse Frontal Disk space(mb) CP Time (sec) Disk space(mb) CP Time (sec) Disk space(mb) CP Time (sec) 45 Sector 34.88 68.39 66.19 76.78 1363.38 621.27 Solid-axi 15.06 40.55 23.38 30.50 251.19 69.25 % Savings 56.81 40.71 64.68 60.28 81.58 88.85 Percentage reduction 100 90 80 70 60 50 40 30 20 10 0 Percentage savings between sector and mixed model for Thermal Analysis 40.7 56.8 60.3 64.7 88.9 PCG Sparse Frontal Solver Used 81.6 CPU Time Disk Space Figure 6. Percentage savings for thermal ANSYS It may be observed from the above table and graph that there is a consistent amount of saving in both solver time and disk space. The reduction in solution time is more pronounced in case of direct solvers (in this case, sparse and frontal). There is also a reduction in the requirement for space as the assembled matrices are much smaller. The thermal analysis has the advantage of carrying only one DOF per node. On the whole, it was observed that the method reduces computational effort to a large extent without compromising on the quality of solution. Structural Structural analysis is performed successively after the thermal analysis, with structural loads and temperature distribution from the thermal analysis. Table 3 and Figure 7 below show comparison between the mixed and sector models. Table 3 Disk requirements and structural runtimes for the Sector and Mixed models PCG Sparse Frontal Disk space(mb) CP Time (sec) Disk space(mb) CP Time (sec) Disk space(mb) CP Time (sec) 45 Sector 51.31 213.92 296.37 299.34 13956.58 12524.74 Solid-axi 28.38 72.78 90.50 73.36 1655.06 773.20 % Savings 44.70 65.98 69.46 75.49 88.14 93.83
Percentage savings between sector and mixed model for Structural Analysis Percentage reduction 110 100 90 80 70 60 50 40 30 20 10 0 93.8 88.1 75.5 66.0 69.5 44.7 PCG Sparse Frontal Solver Used CPU Time Disk Space Figure 7. Percentage savings for structural analysis There is a marked reduction in the solution time between the sectored model and the mixed model. This may be attributed to the difference in the active DoF's. The benefit increases with the adoption of direct solvers for solution. Above results indicate quite substantial savings both in solution time and disk space, on the adoption of the mixed modelling technique. All this is possible with much less memory requirement as can be seen in the case of PCG solver which requires almost 65% less memory to solve the mixed model as compared to the sector model. This sometimes can be the dividing line between a feasible simulation and an impractical one such as the structural analysis with frontal solver as mentioned in Table 3 and Graph 2. This results in huge disk space savings to the tune of Giga bytes for large models. The sector model required a space of 13.6 GB for the formulation of the triangulated matrix while the mixed model required only 1.6 GB for the same analysis. The major concern in performing structural analysis with mixed model is the axisymmetric loading and the response of the structure and the incursions of the non-axisymmetric stresses and deflections into the axisymmetric zone. This puts extra responsibility on the analyst to select the zones appropriately using proper engineering judgment. Modal Modal analysis is computationally intensive as it involves the assembly of multiple matrices and their solution. The solution process is also more complicated as the iterative solver operates enclosed by the eigenvalue solver. The extension of the mixed model approach to the modal regime needs some modification to the boundary conditions mentioned in the methodology section. Modal analysis is best performed with the whole model as even though most modes may repeat in a symmetrical model but the full model is required to capture certain modes undetectable in sector models. As the simulation of the whole model requires large computational resources, analysts often resort to application of different boundary conditions on a cyclic symmetric sector model to capture all the modes. These often include application of symmetric, antisymmetric and coupled end boundaries. ANSYS provides the cyclic symmetry module to overcome these issues.
In order to understand the modal characteristics of the mixed approach, three simple shaft models were created as listed below 1) With the solid and axisymmetric interface being along an axial plane (hereafter referred as Axial Zoning)(figure 8) Figure 8 Shaft model with axial zoning 2) With the solid and axisymmetric interface being along a radial plane (hereafter referred as Radial Zoning)(figure 9) Figure 9 Shaft model with radial zoning 3) The full sectoral 3D solid shaft model (hereafter referred as Solid Model) The boundary condition mentioned previously permits the eigenvalue solution of the model assuming symmetric BC's which results in the incapability of the model to detect certain frequencies topped by the torsional ones. Table 4 below shows the first ten frequencies obtained from a shaft model,
Table 4 Frequencies from analysis with symmetric boundary conditions Mode # Mixed model (Axial -Zoning) Mixed Model (Radial-Zoning) Solid model 1 127.98 128.62 128.68 2 183.79 184.29 184.40 3 329.35 335.00 331.39 4 349.31 356.17 350.69 5 386.42 391.58 388.18 6 396.34 406.42 400.19 7 449.03 454.84 451.20 8 532.96 549.90 538.71 9 534.44 550.62 539.90 10 559.16 568.83 562.80 From the above, it may be possible to conclude mathematically, that there is no appreciable difference between the frequencies with the radial model being stiffer and axial softer than the sector model. However the errors are within a range of 2%. Another set of eigenvalues can be obtained by the assumption of antisymmetric boundary conditions at the sector faces. Table 5 Frequencies from analysis with antisymmetric boundary conditions Mode # Mixed model (Axial -Zoning) Mixed Model (Radial-Zoning) Solid model 1 227.15 121.76 242.48 2 238.91 285.85 323.76 3 347.40 321.28 404.50 4 376.82 403.99 490.95 5 396.92 411.77 552.63 6 449.51 465.02 587.06 7 536.33 478.79 588.10 8 572.42 521.84 603.49 9 585.07 651.85 606.74 10 589.46 714.64 649.42 The frequency values of the mixed model in Table 5 are not consistent with the sector model as the stiffness in this case in not able to duplicate the solid model due to the inherent symmetricity present in the axisymmetric elements. Inorder to overcome this problem of inbuilt symmetricity the simulation was performed by replacing the axisymmetric PLANE42 elements with harmonic PLANE25 elements.
Table 6 Frequencies from analysis incorporating harmonic elements with antisymmetric boundary conditions Mode # Mixed model with Harmonic Elements (Axial -Zoning) Mixed model with Harmonic Elements (Radial -Zoning) 1 170.28 121.76 2 235.85 149.00 3 238.91 285.85 4 347.40 286.12 5 376.82 321.28 6 377.82 403.99 7 399.88 412.22 8 449.51 465.02 9 536.33 479.17 10 572.42 521.84 The major improvement was observed in the axial zoning case, where the first torsional frequency was not only captured and also expanded. The use of ANSYS cyclic symmetric option overcomes these limitations to some extent for the mixed modeling approach. However, it was observed that the full model frequencies predicted did not match the mixed model cyclic symmetry analysis. This could be due to the inherent limitations in ANSYS cyclic symmetry procedure which forces us to redefine interface conditions. A more detailed study, in order to obtain boundary conditions compatible with the ANSYS cyclic module is required. Conclusion With judicious selection of the solid and axisymmetric zones, the methodology of mixed simulation using solid and axisymmetric elements is quite advantageous for thermal and structural analysis. The extension of the method to modal analysis though may be of interest but will lead to errors in prediction of eigen values which is more pronounced for torsional or non symmetric modes. Additionally mode shape visualization is not reliable as the even though the eigenvalues may be mathematically predicted by the solver the expansion pass is incapable of simulating the same. Improvisations can be made with the substitution of harmonic elements in place of axisymmetric elements particularly in order to predict the torsional frequencies. However the authors would recommend a far cautious approach while resorting to modal analysis using this method. References 1. Cook, Robert D. Finite Element Modeling for Stress Analysis, John Wiley & Sons Inc.:1995 2. Adams Vince, Akenazi Abraham, "Building Better Products with Finite Element Analysis", Onward press :1999 3. ANSYS ANSYS Theory Reference, Release 7.0, ANSYS, Inc.:2002 4. ANSYS ANSYS Modeling and Meshing Guide, Release 7.0, ANSYS, Inc.:2002 5. Crawford, John, Combining Axisymmetric and 3-D Solid Elements, ANSYS Solutions, Volume 3, Number 4, pp. 28-29