The application of staggered solution methods in coupled problems representing different phenomena is more obvious, though, as it turns out, more difficult.
For instance, let us consider the linear discrete fluid-structure equations with damping omitted, written as [see Eqs (19.26) and (19.28)]
(19.1 12) where we have omitted the tilde superscript for simplicity.
variables and write using Eq. (19.82)
For illustration purposes we shall use the GN22 type of approximation for both
(19.1 13)
which together with Eq. (19.112) written at t = completes the system of equations requiring simultaneous solution for A&+ and Ap, +
Now a staggered solution of a fairly obvious kind would be to write the first set of equations (19.112) corresponding to the structural behaviour with a predicted (approximate) value of P , , + ~ = p:+ 1 , as this would allow an independent solution for Aii, + writing
MU,+1 + K u , + l = -f+Qp:+l (19.114) This would then be followed by the solution of the fluid problem for Apn+ I writing Spn+l + H u , + ~ = - q - Q T U , + l (19.1 15) This scheme turns out, however, to be only conditionally stable,47 even if pi and
Pi are chosen so that unconditional stability of a simultaneous solution is achieved.
(The stability limit is indeed the same as if a fully explicit scheme were chosen for the fluid phase.)
Various stabilization schemes can be used here.25i47 One of these is given below. In this Eq. (19.114) is augmented to
1 T P (19.1 16)
- 1 T
MUn+l+ ( K + Q S Q )un+1 = - f + Q P f : + l +QS- Q u n + l
before solving for Ai$,+ 1. It turns out that this scheme is now unconditionally stable provided the usual conditions
P2 2 P1 P1 2 ;
are satisfied.
Such stabilization involves the inverse of S but again it should be noted that this needs to be obtained only for the coupling nodes on the interface. Another stable scheme involves a similar inversion of H and is useful as incompressible behaviour is automatically given.
Similar stabilization processes have been applied with success to the soil-fluid system.60'61
References
1. O.C. Zienkiewicz. Coupled problems and their numerical solution. In Numerical Methods in Coupled Systems (Eds R.W. Lewis, P. Bettis and E. Hinton), pp. 65-68, John Wiley and Sons, Chichester, 1984.
2. O.C. Zienkiewicz, E. Oiiate, and J.C. Heinrich. A general formulation for coupled thermal flow of metals using finite elements. Internat. J. Num. Meth. Eng., 17, 1497-514, 1980.
3. B.A. Boley and J.H. Weiner. Theory of Thermal Stresses. John Wiley & Sons, New York, 1960. Reprinted by R.E. Krieger Publishing Co., Malabar Florida, 1985.
4. R.W. Lewis, P. Bettess, and E. Hinton, editors. Numerical Methods in Coupled Systems, Chichester, 1984. John Wiley & Sons.
5. R.W. Lewis, E. Hinton, P. Bettess, and B.A. Schrefler, editors. Numerical Methods in Coupled Systems, Chichester, 1987. John Wiley & Sons.
6. J.C. Simo and T.J.R. Hughes. Computational Inelasticity, volume 7 of Interdisciplinary Applied Mathematics. Springer-Verlag, Berlin, 1998.
7. O.C. Zienkiewicz and R.E. Newton. Coupled vibration of a structure submerged in a com- pressible fluid. In Proc. Znt. symp. on Finite Element Techniques, pp. 1-15, Stuttgart, 1969.
8. P. Bettess and O.C. Zienkiewicz. Diffraction and refraction of surface waves using finite and infinite elements. Internat. J. Num. Meth. Eng., 11, 1271-90, 1977.
9. O.C. Zienkiewicz, D.W. Kelly, and P. Bettess. The Sommerfield (radiation) condition on infinite domains and its modelling in numerical procedures. In Proc. IRIA 3rd Znt. Symp.
on Computing Methods in Applied Science and Engineering, Versailles, December 1977.
10. O.C. Zienkiewicz, P. Bettess, and D.W. Kelly. The finite element method for determining fluid loadings on rigid structures. Two- and three-dimensional formulations. In O.C.
Zienkiewicz, R.W. Lewis, and K.G. Stagg, editors, Numerical Methods in Offshore Engineering, pp. 141-183. John Wiley & Sons, Chichester, 1978.
11. O.C. Zienkiewicz and P. Bettess. Dynamic fluid-structure interaction. Numerical modelling of the coupled problem. In O.C. Zienkiewicz, R.W. Lewis, and K.G. Stagg, editors, Numerical Methods in Offshore Engineering, pp. 185-193. John Wiley & Sons, Chichester, 1978.
12. O.C. Zienkiewicz and P. Bettess. Fluid-structure dynamic interaction and wave forces. An introduction to numerical treatment. Znternat. J. Num. Meth. Eng., 13, 1-16, 1978.
13. O.C. Zienkiewicz and P. Bettess. Fluid-structure dynamic interaction and some 'unified' approximation processes. In Proc. 5th Znt. Symp. on Unification of Finite Elements, Finite Diyerences and Calculus of Variations, University of Connecticut, May 1980.
14. R. Ohayon. Symmetric variational formulations for harmonic vibration problems coupling primal and dual variables - applications to fluid-structure coupled systems. La Rechereche Aerospatiale, 3, 69-77, 1979.
15. R. Ohayon. True symmetric formulation of free vibrations for fluid-structure interaction in bounded media. In Numerical Methods in Coupled Systems, Chichester, 1984. John Wiley &
Sons.
16. R. Ohayon. Fluid-structure interaction. Proc. of the ECCM’99 Conference, IACMI ECCM’99, 3 1 August-3 September 1999, Munich, Germany.
17. H. Morand and R. Ohayon. Fluid-Structure Interaction, Wiley, 1995.
18. M.P. Paidoussis and P.P. Friedmann (eds). 4th International Symposium on Fluid-Struc- ture Interactions, Aeroelasticity, Flow-Induced Vibration and Noise, vol. 1, 2, 3, ASME/
Winter Annual Meeting, 16-21 November 1997, Dallas, Texas, AD-vol. 52-3.
19. T. Kvamsdal et al. (eds). Computational Methods for Fluid-Structure Interaction, Tapir Publishers, Trondheim, 1999.
20. R. Ohayon and C.A. Felippa (eds). Computational Methods for Fluid-Structure Inter- action and Coupled Problems. Comp. Meth. in Appl. Mech. Eng., special issue, to appear 2000.
21. M. Geradin, G. Roberts, and J. Huck. Eigenvalue analysis and transient response of fluid structure interaction problems. Eng. Comp., 1, 152-60, 1984.
22. G. Sandberg and P. Gorensson. A symmetric finite element formation of acoustic fluid- structure interaction analysis. J. Sound Vib., 123, 507-15, 1988.
23. K.K. Gupta. On a numerical solution of the supersonic panel flutter eigenproblem.
Internat. J. Num. Meth. Eng., 10, 637-45, 1976.
24. B.M. Irons. The role of part inversion in fluid-structure problems with mixed variables. J AIAA, 7 , 568, 1970.
25. W.J.T. Daniel. Modal methods in finite element fluid-structure eigenvalue problems.
Internat. J . Num. Meth. Eng., 15, 1161-75, 1980.
26. C.A. Felippa. Symmetrization of coupled eigenproblems by eigenvector augmentation.
Commun. Appl. Num. Meth., 4, 561-63, 1988.
27. J. Holbeche. Ph. D . Thesis. PhD thesis, University of Wales, Swansea, 1971.
28. A.K. Chopra and S. Gupta. Hydrodynamic and foundation interaction effects in earthquake response of a concrete gravity dam. J. Struct. Div. Am. SOC. Civ. Eng., 578, 1399-412, 1981.
29. J.F. Hall and A.K. Chopra. Hydrodynamic effects in the dynamic response of concrete gravity dams. Earthquake Eng. Struct. Dyn., 10, 333-95, 1982.
30. O.C. Zienkiewicz and R.L. Taylor. Coupled problems - a simple time-stepping procedure.
Comm. Appl. Num. Meth., 1, 233-39, 1985.
31. O.C. Zienkiewicz, R.W. Clough, and H.B. Seed. Earthquake analysis procedures for concrete and earth dams - state of the art. Technical Report Bulletin 32, Int. Commission on Large Dams, Paris, 1986.
32. R.E. Newton. Finite element study of shock induced cavitation. In ASCE Spring Conven- tion, Portland, Oregon, 1980.
33. O.C. Zienkiewicz, D.K. Paul, and E. Hinton. Cavitation in fluid-structure response (with particular reference to dams under earthquake loading). Earthquake Eng. Struct. Dyn., 11,
34. M.A. Biot. Theory of propagation of elastic waves in a fluid saturated porous medium, Part I: low frequency range; Part 11: high frequency range. J. Acoust. SOC. Am., 28, 168-91, 1956.
35. O.C. Zienkiewicz, C.T. Chang, and E. Hinton. Non-linear seismic responses and liquefac- tion. Internat. J. Num. Anal. Meth. Geomech., 2, 381-404, 1978.
36. O.C. Zienkiewicz and T. Shiomi. Dynamic behaviour of saturated porous media, the generalized Biot formulation and its numerical solution. Internat. J. Num. Anal. Meth.
Geomech., 8, 71-96, 1984.
37. O.C. Zienkiewicz, K.H. Leung, and M. Pastor. Simple model for transient soil loading in earthquake analysis: Part I - basic model and its application. Internat. J. Num. Anal. Meth.
Geomech., 9,453-76, 1985.
463-81, 1983.
38. O.C. Zienkiewicz, K.H. Leung, and M. Pastor. Simple model for transient soil loading in earthquake analysis: Part I1 - non-associative models for sands. Internat. J. Num. Anal.
Meth. Geomech., 9, 477-98, 1985.
39. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, and T. Shiomi. Computational approach to soil dynamics. In A.S. Czamak, editor, Soil Dynamics and Liquefaction, volume Develop- ments in Geotechnial Engineering 42. Elsevier, Amsterdam, 1987.
40. O.C. Zienkiewicz, A.H.C. Chan, M. Pastor, D.K. Paul, and T. Shiomi. Static and dynamic behaviour of soils: A rational approach to quantitative solutions, I. Proc. Roy. SOC.
London, 429,285-309, 1990.
41. O.C. Zienkiewicz, Y.M. Xie, B.A. Schrefler, A. Ledesma, and N. Bicanic. Static and dynamic behaviour of soils: A rational approach to quantitative solutions, 11. Proc. Roy.
SOC. London, 429, 311-21, 1990.
42. O.C. Zienkiewicz, A.H. Chan, M. Pastor, B.A. Schrefler and T. Shiomi. Computational Geomechanics with Special Reference to Earthquake Engineering, John Wiley and Sons, Chichester, 1999.
43. B.R. Simon, J. S-S. Wu, M.W. Carlton, L.E. Kazarian, E/P. France, J.H. Evans, and O.C.
Zienkiewicz. Poroelastic dynamic structural models of rhesus spinal motion segments.
Spine, 10(6), 494-507, 1985.
44. B.R. Simon, J. S-S. Wu, and O.C. Zienkiewicz. Higher order mixed and Hermitian finite element procedures for dynamic analysis of saturated porous media. Internat. J. Num.
Meth. Eng., 10, 483-99, 1986.
45. R.W. Lewis and B.A. Schrefler. The Finite Element method in the Deformation and Consolidation of Porous Media. John Wiley & Sons, Chichester, 1987.
46. X.K. Li, O.C. Zienkiewicz, and Y.M. Xie. A numerical model for immiscible two-phase fluid flow in porous media and its time domain solution. Internat. J. Num. Meth. Eng.,
47. O.C. Zienkiewicz and A.H.C. Chan. Coupled problems and their numerical solution. In Advanced in Computational Non-linear Mechanics, chapter 3, pp. 109- 176. Springer- Verlag, Berlin, 1988.
48. T. Belytschko and R. Mullen. Stability of explicit-implicit time domain solution. Internat.
J. Num. Meth. Eng., 12, 1575-86, 1978.
49. T.J.R. Hughes and W.K. Liu. Implicit-explicit finite elements in transient analyses. Part I and Part 11. J. Appl. Mech., 45, 371-78, 1978.
50. T. Belytschko and T.J.R. Hughes, editors. Computational Methods for Transient Analysis.
North-Holland, Amsterdam, 1983.
51. C.A. Felippa and K.C. Park. Staggered transient analysis procedures for coupled mechan- ical systems: formulation. Comp. Meth. Appl. Mech. Eng., 24, 61-111, 1980.
52. K.C. Park. Partitioned transient analysis procedures for coupled field problems: stability analysis. J . Appl. Mech., 47, 370-76, 1980.
53. K.C. Park and C.A. Felippa. Partitioned transient analysis procedures for coupled field problems: accuracy analysis. J. Appl. Mech., 47, 919-26, 1980.
54. O.C. Zienkiewicz, E. Hinton, K.H. Leung, and R.L. Taylor. Staggered time marching schemes in dynamic soil analysis and selective explicit extrapolation algorithms. In R.
Shaw et al., editors, Proc. Con$ on Innovative Numerical Analysis for the Engineering Sciences, University of Virginia Press, 1980.
55. O.C. Zienkiewicz, C.T. Chang, and P. Bettess. Drained, undrained, consolidating dynamic behaviour assumptions in soils. Geotechnique, 30, 385-95, 1980.
56. D.M. Trujillo. An unconditionally stable explicit scheme of structural dynamics. Internat.
J. Num. Meth. Eng., 11, 1579-92, 1977.
57. L.J. Hayes. Implementation of finite element alternating-direction methods on non- rectangular regions. Internat. J. Num. Meth. Eng., 16, 35-49, 1980.
30, 1195-212, 1990.