- Email: [email protected]
Current researches by FEM
Journal of Wind Engineering and Industrial Aerodynamics, 46 & 47 (1993) 897-898 Elsevier
897
Current researches by F E M T. NOMURA Associate Professor, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113, Japan
1. I N T R O D U C T I O N There are three approaches currently in the finite element analysis of vortex-induced vibration of bluff cylinders. In one approach, fixed finite element meshes are employed in association with velocity boundary conditions on the cylinder surface [e.g., Li, et al., JFM, 1992]. M o v i n g meshes are employed in the others. One approach is to translate variables to updated mesh [e.g., Anagnqstopoulos, JCP, 1989]. More sophisticated method i~a this approach is recently proposed by Tezduyar, et al. [CMAME, 1992], who are developing d e f o r m a b l e s p a c e - t i m e e l e m e n t . The last approach is to employ t h e a r b i t r a r y L a g r a n g i a n - E u l e r i a n ( A L E ) c o n c e p t , which was introduced by Hirt, et al. [JCP, 1974] and has been polished up by Hughes, et al. [CMAME, 1981] and others. Applications of the ALE concept to the finite element analysis of fluid-structure interaction problems can be found in early '80s though its application to vortex-induced vibration of cylinders is reported only recently by Nomura &: Iijima [JSCE, 1989], Shimura &: Zienkiewicz [1991], Nomura & Hughes [CMAME, 1992]. In the remainder of this article, we will briefly introduce some numerical examples of ours regarding the ALE finite element analysis of vortex-induced vibration of a bluff cylinder as well as computational techniques supporting such analysis.
2. N U M E R I C A L
EXAMPLES
Flows around a vibrating thin-H plate are computed by using the mesh deformation pattern as shown in Fig.1. Figure 2 shows the comparison of computed timelines around the transversely vibrating plate with a sinusoidally prescribed displacement history at Re = 1200 with the flow visualization experiments by Nakamura, et al. [JFM, 1984]. The same finite element mesh is used for the coupling analysis of the transverse vibration of an elastically mounted plate in the uniform water current. The mass of the plate is 27.2 g, the "viscous damping is 1.513 g/s and the spring stiffness is 332.0 g/s 2. Figure 3(a) shows the response amplitudes and the lift coefficients over some Reynolds number range. There are two sudden increases of the amplitude around Re = 1000 and Re = 2400. Figure 3(b) shows the phase lags of the lift and the web pressure to the displacement history. This result suggests that, up to Re = 2400, the eddies along the web have negative contribution to the plate vibration, in other words, the vortices Elsevier SciencePublishers B.V.
898 behind the plate excite the vibration. Then, beyond Re = 24{}0, the eddies along t,he web take the role of exciting the vibration. This fe~ture rn~.y b(e support¢:d by the flow patterns in Fig. 4.
3. C O M P U T A T I O N A L
TECHNIQUES
We have introduced some computational techniques to decrease computational burden. One is to reduce the ALE region leaving the remainder of the analysis domain as the Eulerian, fixed mesh region. The other is to provide internal arrays to store ele m e n t c o n n e c t i v i t y to avoid un-vectorizable s e a r c h e s in re-assembly of the global coefficient matrix. Using these strategies, we can estimate that the CPU time ratio of the ALE computations to ordinary fixed mesh computations per each iteration is not more than four, This suggests that ALE computations are not surprisingly large in comparison with the fixed mesh analysis. In addition, we are developing a method to generate mesh deformation patterns, in which local concentration of element distortion can be avoided [Fig.5]. This method can be applied to complicated cylinder cross-sections, or multi-body problems [Nomura &: Nishimura, JSCE, to appear].
,:0 {1.2
_ l Rcy~ids number
Fig.1 Mesh deformation pattern
1
o-o Lif~ Web Pier,re
1O00 2000 R~ynold~n~mbct
j 3000
Fig.3 (a, left) Response amplitude and llft coefficients, (b,right) phase lags to the displacement history
Flg.4 Streamlines at the maximum displacement; (left} Re=f200, {right}Re=-2400.
1)
]7)i/"
( I"YI-;/77"i"/.i, Fig.2 Time lines around transversely Fig.5 Generated meshdeformation pattern; (left) before vibrating thin-H plate, adjusting element distortion, (right) after adjusting,
897
Current researches by F E M T. NOMURA Associate Professor, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113, Japan
1. I N T R O D U C T I O N There are three approaches currently in the finite element analysis of vortex-induced vibration of bluff cylinders. In one approach, fixed finite element meshes are employed in association with velocity boundary conditions on the cylinder surface [e.g., Li, et al., JFM, 1992]. M o v i n g meshes are employed in the others. One approach is to translate variables to updated mesh [e.g., Anagnqstopoulos, JCP, 1989]. More sophisticated method i~a this approach is recently proposed by Tezduyar, et al. [CMAME, 1992], who are developing d e f o r m a b l e s p a c e - t i m e e l e m e n t . The last approach is to employ t h e a r b i t r a r y L a g r a n g i a n - E u l e r i a n ( A L E ) c o n c e p t , which was introduced by Hirt, et al. [JCP, 1974] and has been polished up by Hughes, et al. [CMAME, 1981] and others. Applications of the ALE concept to the finite element analysis of fluid-structure interaction problems can be found in early '80s though its application to vortex-induced vibration of cylinders is reported only recently by Nomura &: Iijima [JSCE, 1989], Shimura &: Zienkiewicz [1991], Nomura & Hughes [CMAME, 1992]. In the remainder of this article, we will briefly introduce some numerical examples of ours regarding the ALE finite element analysis of vortex-induced vibration of a bluff cylinder as well as computational techniques supporting such analysis.
2. N U M E R I C A L
EXAMPLES
Flows around a vibrating thin-H plate are computed by using the mesh deformation pattern as shown in Fig.1. Figure 2 shows the comparison of computed timelines around the transversely vibrating plate with a sinusoidally prescribed displacement history at Re = 1200 with the flow visualization experiments by Nakamura, et al. [JFM, 1984]. The same finite element mesh is used for the coupling analysis of the transverse vibration of an elastically mounted plate in the uniform water current. The mass of the plate is 27.2 g, the "viscous damping is 1.513 g/s and the spring stiffness is 332.0 g/s 2. Figure 3(a) shows the response amplitudes and the lift coefficients over some Reynolds number range. There are two sudden increases of the amplitude around Re = 1000 and Re = 2400. Figure 3(b) shows the phase lags of the lift and the web pressure to the displacement history. This result suggests that, up to Re = 2400, the eddies along the web have negative contribution to the plate vibration, in other words, the vortices Elsevier SciencePublishers B.V.
898 behind the plate excite the vibration. Then, beyond Re = 24{}0, the eddies along t,he web take the role of exciting the vibration. This fe~ture rn~.y b(e support¢:d by the flow patterns in Fig. 4.
3. C O M P U T A T I O N A L
TECHNIQUES
We have introduced some computational techniques to decrease computational burden. One is to reduce the ALE region leaving the remainder of the analysis domain as the Eulerian, fixed mesh region. The other is to provide internal arrays to store ele m e n t c o n n e c t i v i t y to avoid un-vectorizable s e a r c h e s in re-assembly of the global coefficient matrix. Using these strategies, we can estimate that the CPU time ratio of the ALE computations to ordinary fixed mesh computations per each iteration is not more than four, This suggests that ALE computations are not surprisingly large in comparison with the fixed mesh analysis. In addition, we are developing a method to generate mesh deformation patterns, in which local concentration of element distortion can be avoided [Fig.5]. This method can be applied to complicated cylinder cross-sections, or multi-body problems [Nomura &: Nishimura, JSCE, to appear].
,:0 {1.2
_ l Rcy~ids number
Fig.1 Mesh deformation pattern
1
o-o Lif~ Web Pier,re
1O00 2000 R~ynold~n~mbct
j 3000
Fig.3 (a, left) Response amplitude and llft coefficients, (b,right) phase lags to the displacement history
Flg.4 Streamlines at the maximum displacement; (left} Re=f200, {right}Re=-2400.
1)
]7)i/"
( I"YI-;/77"i"/.i, Fig.2 Time lines around transversely Fig.5 Generated meshdeformation pattern; (left) before vibrating thin-H plate, adjusting element distortion, (right) after adjusting,