TRUE AND EFFICIENT SOLUTION OF BEM-FEM ACOUSTIC-STRUCTURAL COUPLING FOR UNIFIED SPACECRAFT FLUID-STRUCTURE COUPLING USING CHIEF REGULARIZATION AND FAST MULTIPOLE EXPANSION
- Paper number
IAC-08.C2.3.8
- Author
Prof. Harijono Djojodihardjo, Universitas Al Azhar Indonesia, Indonesia
- Year
2008
- Abstract
Structural-acoustic interaction, which is a significant issue found in many applications, including modern new and relatively lighter aircrafts operating at very high speed and altitudes, vibration problems can severely and adversely affect spacecraft structures and their payloads, and other engineering applications [1-7]. By modeling structural-acoustic interaction using boundary and finite element coupling, it is possible to couple the boundary element method and the finite element method to solve the structure-acoustic interaction [8-10]. In earlier series of work carried out by the author and colleagues addressing the vibration of structures due to sound wave [11-15], structural-acoustic interaction is modeled and analyzed using boundary and finite element coupling. The computational scheme developed for the calculation of the acoustic radiation as well as the structural dynamic response of the structure using coupled BEM/FEM has given satisfactory results for acoustic disturbance in the low frequency range, which was the range of particular interest in many practical applications. However, for larger frequency range, it is well known that while the solution to the original boundary value problem in the exterior domain to the boundary is perfectly unique for all wave numbers, this is not the case for the numerical treatment of integral equation formulation, which breaks down at certain frequencies known as irregular frequencies or fictitious frequencies. Although such phenomenon is completely nonphysical since there are no discrete eigenvalues for the exterior problems, a method known as CHIEF (Combined Helmholtz Interior integral Equation Formulation) can be utilized to overcome such problem. Test-case applications of CHIEF method to a spherical shell geometry has given excellent results. The work carried out is focused on further treatment in the formulation and solution of the basic problem of acoustic radiation and scattering problem, and the excitation and vibration of elastic structure in a coupled fluid-elastic-structure interaction. In the first part of the work, an in-depth analysis is carried out to review the critical frequencies and other singularities. Since the governing boundary integral equation of equation fails at frequencies coincident with the interior cavity frequencies of homogeneous Dirichlet boundary conditions, the discretized equation of the associated [H] matrix representing the kernel in the equation becomes ill-conditioned when the exciting frequency is close to the interior frequencies, thus providing an erroneous acoustic loading matrix. Several methods suggested in the literature are reviewed and worked out for the specific application problems considered. Many solutions have been addressed to solve this problem, such as those offered by Schenck [16-17], Seybert and other researchers [18-19], Chen [20], Burton and Miller [21], and Chen [22]. In fact, as will be discussed in this paper, the direct computation of various singular integrals can be avoided altogether in the BIE/BEM in most cases, if the weakly singular or non-singular forms of the BIEs are employed, with greater efficiency and without sacrificing any accuracy. This can be achieved in the BEM because the BIEs for most problems do not contain singular integrals at all if they are formulated properly, even if the fundamental solutions employed in BIEs are in general singular[23-25]. In this conjunction, Schenck CHIEF method that was earlier investigated and applied and gives satisfactory results in the generic case of a sphere is further elaborated in view of other variants for efficient BE computation. A potential problem with this approach is the choice of interior points for the supplemental equations. To be effective at one of the critical frequencies the interior point must not be a nodal point of the corresponding interior eigenmode. However, it is generally not difficult to avoid nodal points, and as was shown by Seybert and Rengarajan[18], regardless of the number of interior points chosen, at least one of the points should not lie on or near a nodal surface. The method developed by Wu and Seybert[19] in which each interior constraint equation and its first-order derivatives are enforced in a weighted residual sense over a small volume contained within S is also applied. This method adds more equations for each interior point, but make the proper selection of interior points less critical. Burton and Miller[21] produced an integral equation that was valid for all wave numbers by forming a linear combination of the Helmholtz integral relation used by CHIEF and its normal derivative. The application of this approach involves the evaluation of the hypersingular integrals involving a double normal derivative of the free-space Green’s function, taking advantage of the suggested approach for regularizing these integrals. Another procedure to be investigated in the recently proposed method by Chen [22] to solve the interior eigenproblems and exterior acoustic radiation and scattering problems using a Combined Helmholtz Exterior integral Equation Formulation (CHEEF) method . Analogous to the CHIEF method, in the CHEEF method, the spurious solutions are filtered out by using additional constraints from the exterior points which are chosen carefully. Using generic example for a sphere like before, the optimum numbers and proper positions for selecting the points in the exterior domain are studied. Other approach originated from the suggestion of Cruse [21] and Liu [22] in elastostatics and elastodynamics is also assessed. Their method involves an alternative and indirect way to evaluate the free term Cij. , to arrive at a singular integral in the sense of Cauchy principal value (CPV). Liu substituted and rearranged relevant expression in this approach back into the Boundary Integral equation to arrive at explicit weakly singular form of the BIE for elastostatic problems. The rearrangement of the terms made the two CPV integrals, one in the Cij expression and one in the original form of the BIE to be cancelled out naturally and completely. Therefore, no more strongly singular integrals in BIE appears, and thus significantly reduce the burden on the BEM implementations. These approached originally developed by Cruse, Liu and Rudolphi in elastostatics will also be reviewed and applied for the present acoustic radiation and scattering problem. The results of the above study are again utilized in the composite scheme developed earlier by the author to refine the work for acoustic-structural coupling. Along this line, the method of Mathews[26] to couple the Burton-Miller formulation with the finite element method in order to solve fluid structure interaction problems will be applied as comparison. An example of the result of applying CHIEF technique to avoid fictitious frequencies is illustrated in the following figure. Fig. 1 Surface pressure distribution on pulsating sphere for analytical, BEM, and BEM-CHIEF solution for one and two CHIEF point References: [1] Edson, J.R.,”Review of Testing and Information on Sonic Fatigue,”Doc. No. D-17130, Boeing Co., 1957. [2] Powell, C.A. and Parrott, T.L., “A summary of Acoustic Loads and Response Studies, “Tenth National Aero-Space Plane Technology Symposium, Paper No. 106, April, 1991. [3] Eaton, D.C.G., An Overview of Structural Acoustics and Related High-Frequency-Vibration Activities. http://esapub.esrin.esa.it /bulletin/bullet92/b92eaton.htm. [4] Renshaw, A. A., D’Angelo, C., and Mote, C. D., Jr., 1994, ‘‘Aeroelastically Excited Vibration of a Rotating Disk,’’ J. Sound Vib., 177, pp. 577–590. [5] Kang, Namcheol and Raman, Arvind, Aeroelastic Flutter Mechanisms of a Flexible Disk Rotating in an Enclosed Compressible Fluid, Transactions of the ASME, Vol. 71, January 2004, pp 120-130. [6] Tandon, N. , Rao, V.V.P. and Agrawal, V.P. , Vibration and noise analysis of computer hard disk drives, Measurement 39 (2006) 16–25, www.elsevier.com/locate/measurement [7] J. F. Luo, Y. J. Liu, and E. J. Berger, "Analysis of two-dimensional thin structures (from micro- to nanoscales) using the boundary element method," Computational Mechanics, 22, 404-412 (1998). [8] Chuh Mei, and Carl S. Pates, III, Analysis Of Random Structure-Acoustic Interaction Problems Using Coupled Boundary Element And Finite Element Methods, NASA-CR-195931, 1994. [9] Holström, F., Structure-Acoustic Analysis Using BEM/FEM; Implementation In MATLAB®, Master’s Thesis, Copyright © 2001 by Structural Mechanics & Engineering Acoustics, LTH, Sweden, Printed by KFS i Lund AB, Lund, Sweden, May 2001, Homepage: http://www.akustik.lth [10] Meddahi, S., Marquez, A. and Selgas, V. , “A new BEM–FEM coupling strategy for two-dimensional fluid–solid interaction problems”, Journal of Computational Physics 199 pp. 205–220, 2004. [11] Djojodihardjo, H. and Tendean,E., A Computational Technique for the Dynamics of Structure Subject To Acoustic Excitation, ICAS 2004, Vancouver, October 2004. [12] Djojodihardjo, H. and Safari, I., Further Development of the Computational Technique for Dynamics of Structure Subjected to Acoustic Excitation, paper IAC-05-C2.2.08, presented at the 56th International Astronautical Congress / The World Space Congress-2005, 17-21 October 2005/Fukuoka Japan. [13] Djojodihardjo, H and Safari, I., Unified Computational Scheme For Acoustic Aeroelastomechanic Interaction, paper IAC-06-C2.3.09, presented at the 57th International Astronautical Congress / The World Space Congress-2006, 1-6 October 2006/Valencia Spain. [14] Harijono Djojodihardjo and Irtan Safari, BEM-FEM COUPLING FOR ACOUSTIC EFFECTS ON AEROELASTIC STABILITY OF STRUCTURES, keynote paper, ICCES 07, paper ICCES0720060930168, Miami Beach, Florida, 3-6 January 2007 [15] Harijono Djojodihardjo, BEM-FEM Acoustic-Structural Coupling For Spacecraft Structure Incorporating Treatment Of Irregular Frequencies, paper IAC-07-C2.3.4, 58 th International Astronautical Congress - The World Space Congress-2006, 23-28 September 2007/Hyderabad, India [16] H. A. Schenck, Improved integral formulation for acoustic radiation problems. J. Acoust. Sot. Am. 44, 41-58 (1968). [17] W. Benthien and A. Schenck, Nonexistence And Nonuniqueness Problems Associated With Integral Equation Methods In Acoustics, Computers & Structures Vol. 65, No. 3, pp. 295-305, 1997 [18] F. Seybert and T. K. Rengarajan, The use of CHIEF to obtain unique solutions for acoustic radiation using boundary integral equations. J. Acoust. Sot. Am. 81, 1299-1306 (1987) [19] T. W. Wu and A. F. Seybert, A weighted residual formulation for the CHIEF method in acoustics. J.Acoust. Sot. Am. 90, 1608-1614 (1991). [20] J.T. Chen, I. L. Chen and M. T. Liang, 2001, “Analytical Study and Numerical Experiments for Radiation and Scattering Problems using the CHIEF Method”, Journal of Sound and Vibration, 248(5), pp.809-828. [21] J. Burton and G. F. Miller, The application of the integral equation method to the numerical solution of some exterior boundary value problems. Proc. R. Soc.Lond. A 323, 201-210 (1971). [22] J.T. Chen, I. L. Chen, S. R. Kuo and M. T. Liang, 2001, “A New Method for True and Spurious Eigensolutions of arbitrary cavities using the CHEEF Method”, J. Acoustic Society of America, 109(3), [DOI:10.1121/1.1349187]. [23] Cruse TA. A direct formulation and numerical solution of the general transient elastodynamic problem - II. J Math Anal Appl 1968;22(2):341-355. [24] Y.J. Liu, On the simple-solution method and non-singular nature of the BIE/BEM - a review and some new results, Engineering Analysis with Boundary Elements 24 (2000) 789-795 [25] Y.J. Liu, On the simple-solution method and non-singular nature of the BIE/BEM - a review and some new results, Engineering Analysis with Boundary Elements 24 (2000) 789-795 [26] C. Mathews, Acoustic radiation for three dimensional elastic structures. In: Innovative Numerical Analysis in Applied Engineering Science. Montreal, (1980).
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