Transactions on Modelling and Simulation vol 2, 1993 WIT Press, ISSN X

Transcription

1 Transient fluid-structure interaction using retarded potential and nonlinear finite element M.A. Tamm", C.T. Dyka& "Naval Research Laboratory, Research Computation Division, Washington, USA ^Geo-Centers, Inc., Indian Head Highway, Forf M/aa/tm^om, MD ^07^, [/^A ABSTRACT The transient analysis of coupled fluid-structure systems is an important area of particular concern to the naval community. However, the general state of affairs in this class of problems is far from fully satisfactory. In this paper, the nonlinear finite element method (FEM) program ABAQUS is used to model the structure and coupled to the fluid realm, which is discretized with the boundary element method (BEM) employing the retarded potential (RP) formulation. The RP method has not been extensively applied in the past to such classes of problems, due primarily to storage and stability requirements necessitated by its temporal and spacially non-local nature. However, the RP approach does offer important advantages over both the use of FEM for the fluid, or the widely used Doubly Asymptotic Approximation (DAA), which is an approximation of the retarded potential. Particular attention is paid in this work to trying to increase the stability of the explicit RP method and the coupled system, and decreasing the storage requirements. One approach tried is simply to limit how far back in time and the maximum distance information is allowed to effect current calculations. Initial results are not conclusive. Overall, it is concluded that the RP method may be quite viable for thefluid.however, additional work is still required to better stabilize the RP method - such as a sliding criteria to "shut off information, scaling the pressures, and/or a semi-implicit approach. This work represents a continuing effort to develop a production capability for the transient analysis offluid-structureproblems.

2 568 Boundary Elements INTRODUCTION Transient analysis of real world coupled systems is a difficult problem even with the current levels of supercomputers. In this paper, we will be concerned with coupled, fluid-structure interaction, in particular, with submerged structures subjected to weak shock waves. The dynamic analysis of linear and nonlinear structures using finite element has developed to a very sophisticated level, see for instance [1-3]. Finite difference operators are typcially employed for the time domain and can be grouped into explicit or implicit methods. Explicit operators do not require the forming and factoring of a global stiffness matrix but do have stability concerns which severly restrict the size of the time steps. They are primarily applicable to wave propagation problems. Unconditionally stable implicit operators have been widely applied for linear and nonlinear analyses. Implicit methods can employ much larger time steps but require the formulation and factoring of a global stiffness matrix, which is very costly and even prohibitive for large problems. Overall, the FEM analysis of especially large, nonlinear transient structures remains a difficult and expensive process. Explicit and implicit methods both have advantages and disadvantages which can greatly restrict their efficiency - even on the largest supercomputers. For transientfluid-structureinteraction of submerged structures, FEM is the obvious choice for discretization of the structure which may respond in a linear or nonlinear manner. In the fluid, which is typically approximated as an acoustic media, several choices exist for discretization [4]. Due to efficiency considerations, the coupled system is usually solved in a staggered fashion. The treatment of the fluid media is motivated by the fact that the analyst is concerned primarily with the response of the structure and what effect the fluid has upon it. Thus additional approximations can be introduced. One approach is domain discretization with techniques such as FEM, finite difference or finite volume. Of these, FEM seems to be the most popular due to its ability to model arbitrary geometries. The use of FEM for the fluid domain, however, represents a costly approach. Typically a large number of degrees of freedom (dof) are required and implicit methods are used. Also as with other domain methods, the modeling of infinite boundary conditions raises additional difficulties. The second approach for the fluid media is the use of boundary element methods (BEM) [5]. The introduction of BEM, while eliminating the need to discretize the domain of the fluid, carries with it an assumption of homogeneity. Thus thermal variations, cavitation, etc. can not be easily dealt with if at all. The retarded potential (RP) represents an exact formulation of the transient BEM problem [5-9]. The RP equations are, however, non-local in both space and time. Because of this, difficulties especially with storage and stability have greatly limited the more

3 Boundary Elements 569 widespread application of the retarded potential. The doubly asympotic approximation (DAA) is an approximation to the RP method and has been widely employed [10-13]. In general, DAA removes the non-local nature in time of the boundary element equations. However, there is significant concern over DAA's accuracy especially for highly transient, nonlinear problems of fairly long durations. DAA correctly calculates the fluid loading at early and late time. In the intermediate time region or for the radiation by the structure vibrations in the intermediate frequency range, DAA's accuracy is very suspect. DAA2 [13] represents a generalization of DAA with some improved accuracy in the intermediate frequency range. Overall, the solution of transient fluid-structure interaction problems is certainly not at a fully satisfactory level. In what follows, we discuss the retarded potential formulation and its coupling to the nonlinear finite element code ABAQUS (to form ABAQUS-RP) for the solution of submerged structures subjected to weak shock waves. Discussion is directed toward increasing the stability of the explicit RP method. FORMULATION Consider a submerged structure subjected to an incident pressure wave. We assume that the fluid can be approximated as an ideal compressible media in linear wave motion (acousticfluid).the incident wave p* impinges on the structure to create p*. In addition, the structure's response initiates a radiation pressure p *in the surrounding fluid. The total pressure p is then represented as: _ inc i ^ i ^ P = P + P + P (1) The boundary condition on the "wetted" surface of the structure can be written as: where n is the unit normal into the structure; p is the fluid density; and a is the normal acceleration of the structure. The retarded potential integral is the solution of the linear wave equation: # subject to the boundary conditions of eq. (2). In eq. (3), c is the sonic velocity in the fluid. A discussion of the retarded potential and its discretized approximation, as applied to submerged structures, can be found in [7-9]. The form for the calculation of the total pressure p on the "wetted" surface of the structure subjected to a continuous incident pressure field p"** is as follows:

4 570 Boundary Elements p (x, t) = 2p"" (,, 0 - () -S' + (4) in which r' = t-r/c and /? = j-^ (5) In eqs. (4) and (5), f' is the retarded time; & is the position of the observation point on the surface; a' is the location of an integration point on the surface; R is the distance between the observation point and the integration point; and #' represents the surface normal direction at the integration point orientated away from the fluid. IMPLEMENTATION Equation (4) is discretized with boundary element and linked to the FEM structural analysis code ABAQUS to form the program AB AQUS-RP. The structure will be allowed to respond in a linear or nonlinear fashion. The surface pressure field is approximated by subdividing the surface into K zones or elements of constant pressure. These constant boundary elements coincide with the "wetted" surface of the structure, which is modeled with ABAQUS 8 node quadratic shell elements (S8R). Thus the boundary elements although constant in the pressure are quadratic with respect to the geometry [14]. Figure 1 indicates a typical 8 node shell element and the corresponding boundary element. Each zone or boundary element k is subdivided into a mesh of subzones /. For purposes of measuring the distance between the integration point / and the observation point ; (thereby the time retardation), point ; is located at the center of each boundary element. The integration point / is located at the center of subzone / within the boundary element k. When the observation point ; is within the zone k being integrated, R can go to zero, and singular integration techniques must be employed [8,9]. In this work, the time step dt is assumed to be constant, and the current time can be expressed as: / = m - dt (6) The retardation time between points ; and / (within zone k) is expressed as ffri and follows from eq. (5). In general, where n is an integer and 0 < y< 1. In eq. (4), p, ^- and a can now be interpolated as follows:

5 Boundary Elements 571 tyk The term y is typically expressed by a three point backward difference formula and we have employed this form [7-9]. Groenenboom [15] indicates that this represents an explicit assumption and presents a semiimplicit formulation which requires a matrix inversion. However, even this approach requires the following for stability: dt>max(r/c) (9) Equation (9) and the work in [15] put serious restrictions on the time step and raises questions as to the suitability of of the retarded potential (indeed even DAA since it is an approximation of the RP) for transient fluidstructure interaction. Is the RP method, even employing a semi-implicit form, capable of allowing the linear acoustic fluid to respond correctly to highly transient boundary terms from events such as underwater explosions? The source of the instability in both the explicit and semiimplicit RP approaches is due to small round off errors that tend to increase over time and destabilize the solution. Numerical damping in the structural equations tends to partially alleviate this difficulty and allow time steps smaller than eq. (9). However even with significant numerical damping, unstability seems to eventually enter the response of the coupled system - see the discussion in the APPLICATION section. One approach to this dilemma is to set limits on how far back in time and the maximum distance pressures and accelerations are allowed to influence the observation point. Essentially this amounts to "shutting off information too far in the past and far away in space. In the next section, we will explore the validity of this approach on the explicit RP formulation, and note the effect on both stability and accuracy. Returning to our discussion of the details of the computations, the retarded potential integral, eq. (4), can be discretized and arranged into the form [7- where maxr;,, 1= - r^ + 1, c-dl /=- r + 3 c-dt (11) The terms A*, and B*. depend only on geometry and are computed and stored prior to calculating the pressure. Equation (10) represents the response of the fluid media on the "wetted" surface of the structure of the structure. Both the current pressures pf and the current accelerations a are unknown, and must be determined. The fluid pressure represents a loading term on the governing FEM equations of the structure. The coupled equations for the fluid-structure interaction are solved in a staggered fashion [16]. With a staggered approach, the current pressures p? in eq. (10) are solved using matrix multiplications of the past pressures and accelerations, and the current acceleration estimates obtained

6 572 Boundary Elements from the structural equations. The solution of the pressures pj* thus can require storage of a large number of pressures and accelerations at previous time steps. This along with stability represents a prime difficulty which has seriously handicapped the application of the retarded potential approach in the past. As discussed above, we will be experimenting with increasing the stability by "shutting off or "clipping"information, by limiting the storage of past histories. APPLICATION A retarded potential (RP) capability has been coupled to the ABAQUS program, through the DLOAD user written subroutine, to form ABAQUS- RP. The initial implementation of this capability was in a MICRO-VAX environment. Recently, we have developed a CRAY version of the user written subroutine, which is coupled to ABAQUS version 4.9 and executed on NRL's CRAY-EL computer. However, up to this point in time we have limited ourselves to investigating fluid-structure interaction problems which are relatively small in size and coarse in the time stepping since we are still trying to validate the overall capability. Also, the availability of accurate fluid-structure interaction problems with which to compare to is very limited. An important feature in ABAQUS is the Hilber-Hughes implicit time step operator with controllable numerical damping. Previous experience [7,9,17] has indicated that numerical damping has greatly helped to stabilize the fluid response. However, with an explicit retarded potential approach, even numerical damping has not been sufficient to completely stabilize the response of the fluid-structure system to allow time steps more than approximately one or two orders of magnitude below the restriction imposed by equation (9). Instability seems to eventually creep into the system. As discussed early, in this paper we are proposing to place limits on both how far back in time and/or the maximum distances pressures are allowed to influence observation points. In what follows in this section, the validity of thefirstapproach will be investigated, both in terms of providing additional stability and of course with respect to accuracy. In addition, a variable time stepping capability has been included in the RP program. However, it has been proven to be very unstable with the adaptive time stepping algorithm employed in version 4.9 of ABAQUS. The current version of ABAQUS 5.2 [18] has a more advanced adaptive time stepping procedure that changes time steps less often. This adaptive algorithm along with the "shutting off of information far away in distance and in past time for the RP, is a subject for future work. Thus in what follows, we shall limit ourselves to constant time time stepping. To demonstrate the ABAQUS-RP program, consider Figure 2 in which an elastic sphere is subjected to an incident step plane wave along the x axis. An exact solution to this problem is presented in [19]. Figure 3 indicates the

7 Boundary Elements 573 symmetric ABAQUS model which consists of 54 S8R quadratic shell elements [1] overlaid with constant pressure boundary elements, which are also quadratic in shape. Due to symmetry, only 1/4 of the sphere is modeled. The radius of the sphere r is 1 m, while the thichness 2h is 0.02 m. The material properties of the steel are: Young's modulus E = E11 Pa, Poisson's ratio v =.3, and the mass density p, = kg/ m?. The properties of the surrounding water are: the speed of sound c = m/sec and the mass density p^ = kg/m\ The magnitude of the incident plane wave is assumed to be 14.0 E6 Pa, and the time step employed is dt = 0.10 r/c or 6.84 E-5 seconds. Figure 4 compares the ABAQUS-RP predictions with the closed form solution [19] for the dimensionless displacement history at 9 = 45 and 90 degrees. In general with a relatively coarse grid and time step, the agreement initially appears to be fairly good. Differences in the peaks responds are due primarily to the relatively high numerical damping of the Hilber-Hughes operator used to integrate the structural finite element model. For this problem in an effort to maintain stability, a = 0.3 (see [1]) was used As evidenced in Figure 4 and especially for 0 = 45 degrees, instability begins to be evident at approximately i = 8, even with the significant numerical damping used for the structuralfiniteelement model (a = 0.3). In an effort to apply ABAQUS-RP to a nonlinear structural problem and to assess the effects of plasticity on the stability, the sphere is allowed to yield (a. u = 345. E6 Pa and perfect plasticity is assumed) and large displacements are included. The magnitude of the plane wave is reduced to 8.5 E6 Pa (otherwise the sphere would collapse). The displacement history at 9 = 90 degrees is plotted in Figure 5. There are no closed form solution to this nonlinear problem of course, and comparisons to DAA predictions are not available with which to assess the accuracy of the ABAQUS-RP solution. In general theresultslook reasonable until approximately t = 8 at which point instability again becomes apparent in the response of the sphere. The results thus far indicate the lack of stability for the explicit form of the retarded potential, even with significant numerical damping in the structural temporal operator and also the presence of plasticity. Based upon [15,17], this is not an unexpected result. In an attempt to stabilize the RP, we next will limit how many previous time steps will be allowed to be stored at the boundary nodes. This approach also effectively places limits upon the maximum distance integration points are allowed to influence calculations at an observation point. For an integration point to influence a calculation, the distance, d, to the center of the element that it is contained has to satisfy: d<nxcxdt (12) where n is the number of previous time steps stored. If eq. (12) is not satisfied, the A and B matrices will be multiplied by zeros. Examining

8 574 Boundary Elements equations (10) and (11), it is seen that for the sphere with dt = 0.10 r/c, 7 = 1 1 and P = 13. Because symmetry is present, the values of maxr-^ is effectively doubled, thus making 7 = 21 and / = 23. Placing moderate limits on 7 and F of 15 and 17 did help to extend the range of stability. However, the results at this time are still inconclusive, and further computer runs are necessary to be more definitive concerning this approach of "shutting off or "clipping" past histories. SUMMARY AND FUTURE DIRECTIONS In this work, a brief discussion concerning transient analysis of fluidstructure interaction systems has been presented. Motivated by the present state-of-the-art, an advanced retarded potential capability has been coupled to the ABAQUS nonlinearfiniteelement program to produce ABAQUS-RP and applied. In general, the RP method coupled with finite element offers advantages over current approaches to solving fluid-structure interaction problems. It has not been extensively investigated in the past due primarily to storage requirements. In addition however, the RP method also contains stability requirements for both explicit and implicit formations which limit the minimum allowable time step to levels that are too coarse for general application to the above class of problems. In this paper, we have begun to explore approaches for overcoming these inherent difficulties with the explicit retarded potential formulation while maintaining sufficient accuracy. One approach looked at has been limiting the non-local dependencies in time and space by simply "shutting off or "clipping" information too distant in the past or too far away. This method also alleviates storage problems, which will be more significant in problems where the structural dimensions are much larger, such as in the case of submersibles. The initial results of "shutting off " information, while limiting storage requirements, are not conclusive. In the future, we will more thoroughly investigate this approach and its effect on both stability and accuracy, as well as the use of a sliding criteria with respect to time. However at this point in time, this approach by itself does not appear to be fully adequate to completely stabilize the RP method and maintain sufficient accuracy. Re-examining eq. (4) and in the light of [15], it becomes apparent that the stabiltiy difficulties for the explicit (and implicit as well) originate in the term -J- (or p), and with machine precision. Essentially in eq. (4) p is added to p. The time derivative of the pressure can be expressed as: P = = xa, (13)

9 Boundary Elements 575 Finite difference operators are then introduced into eq. (13). The term A/7 is then expressed as the difference in pressures - which can be very small compared to the actual pressures. Early in the analysis, Ap can be accurately calculated, even if close to zero. Later, however, errors accumulate and precision is lost. The term R/ (caf) now multiplying A/? is much greater than one because the time step is much smaller than dt given by eq. (9). Thus instability eventually begins and grows as the analysis progresses. One possible solution to this problem, which will be pursued in the near future, is to scale the pressure p. Essentially this involves solving eq. (4) for p rather than p where p = factor xp (14) in which factor > 1 Finally one other avenue to be explored in the future is the development and implementation of a semi-implicit formulation for the retarded potential similar to [15]. This in combination with "clipping" information and scaling the pressure may produce a more robust algorithm that is stable and in which storage is greatly reduced. Overall, the retarded potential method offers significant advantages for the analysis of the fluid in transient fluid-structure interaction problems. It has not been extensively investigated in the past due primarily to storage problems, and also because of stability limitations. In this paper, we have begun to address both of these concerns. Initial results with just "clipping" information are not conclusive. Additional work is required to better stabilize the RP method - such as scaling the pressures, a sliding "clipping" algorithm, and/or a semi-implicit approach. Overall our goal is to develop the RP method and ABAQUS-RP into a production capability for the solution of transientfluid-structureinteraction problems.

10 576 Boundary Elements REFERENCES 1. ABAQUS User's and Theoretical Manuals, Hibbitt, Karlsson & Sorenson, Inc., Provdence, R. L, Bathe, K. J., Finite Element Methods in Engineering, Prentice-Hall, Belytschko, T. and Hughes, T. J. R. (Editors), Computational Methods for Transient Analysis, North-Holland, Zienkiewicz, O. C. and Bettess, P., 'Fluid-Structure Dynamic Interaction and Some Unified Approximation Processes,' (Ed. Kardestuncer, H.), Fifth Symposium on the Unification of Finite Element, Finite Differences and Calculus of Variations, University of Connecticut, May Brebbia, C. A., Telles, J. C. F., and Wrobel, L. C, Boundary Element Techniques Theory and Applications in Engineering, Springer-Verlag, Dohner, J. L., Shoureshi, R. and Bernard, R. J., Transient Analysis of Three-Dimensional Wave Propagation Using the Boundary Element Method,' Int. Jour. Num. Meth. Engin., Vol. 24, pp , Huang, H., Everstine, G. C. and Wang, Y. F., 'Retarded Potential Techniques for the Analysis of Submerged Structures Impinged by Weak Shock Waves,' (Ed. Belytschko, T. and Geers, T. L.), Computational Methods for Fluid-Structure Interaction Problems, ASME, AMD Vol. 26, Mitzner, K. M., 'Numerical Solution for Transient Scattering from a Hard Surface of Arbitrary Shape - Retarded Potential Technique,' J. Acoust. Soc. Amer., Vol. 42, No. 2, Tamm, M. A. and Webbon, W. W., 'Submerged Structural Response to Weak Shock by Coupled Three-Dimensional Retarded Potential Fluid- Finite Element Structural Analysis, NRL Memorandum, Report 5903, May Deruntz, J. A., The Underwater Shock Analysis Code and Its Applications,' Lockheed Palo Alto Research Lab Report, Felippa, C. A., Top-Down Derivation of Doubly Asymptotic Approximations for Structure-Fluid Interaction Analysis,' (Ed. Shaw, R. P.), Innovative Numerical Analysis for the Applied Engineering Sciences, University Press of Virginia, Geers, T. L., 'Residual Potential and Approximation Methods for Three- Dimensional Fluid-Structure Interaction Problems,' /. Acout. Soc. Amer., Vol. 49, No. 2, pp , Geers, T. L., 'Doubly Asymptotic Approximations for Transient Motions of Submerged Structures,' /. Acoust. Soc. Amer., Vol. 64, No. 5, pp , Tamm, M. A., 'A Parametric Patch Surface Geometry Definition for the Three Dimensional Retarded Potential Technique,' NRL Memorandum

11 Boundary Elements 577 Report 5904, June Groenenboom, P. H. L., * Solution of the Retarded Potential Problem and Numerical Stability,' (Ed. Brebbia, C. A.), Boundary Element 6, Springer-Verlag, Park, K. C., Felippa, C. A. and Deruntz, J. A., 'Stabilization of Staggered Solution Procedures for Fluid-Structure Interaction Analysis,' (Ed. Belytschko, T. and Geers, T. L.), Computational Methods for Fluid-Structure Interaction Problems, ASME, AMD Vol. 26, Dyka, C. T. and Tamm, M. A., 'Fluid-Structure Interaction Using Retarded Potential and ABAQUS,' ABAQUS User's Conference, published by HKS, Pawtucket, RI, May ABAQUS 5.2 User's and Theoretical Manuals, Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, RI, Huang, H. Transient Interaction of Plane Acoustic Waves with a Spherical Elastic Shell,' J. Acoust. Soc. Amer., Vol. 45, No. 3, pp , 1969.

12 578 Boundary Elements Subzooe S8R Shell Element Structure Fluid RP Constant Boundary Element Figure 1. Quadratic ABAQUS shell element overlaid by a constant RP Boundary Element. INCIDENT WAVE SPHERICAL SHELL Figure 2. Elastic sphere subject to an incident step, plane wave.

13 Boundary Elements 579 Figure 3. Sphere modeled with S8R ABAQUS shell elements. 0.25' = 45* PC = 90* / = c- Figure 4. Shell deflection 6 in the x direction, classical, ABAQUS - RP (elastic sphere).

14 580 Boundary Elements 9 = 90" PC Figure 5. Shell deflection 5 in the x direction, ABAQUS - RP results (elastic-plastic sphere).