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Transient response analysis of multiple submerged structures
Finite Elements in Analysis and Design 6 (1989) 153-172 Elsevier
153
TRANSIENT RESPONSE ANALYSIS OF MULTIPLE SUBMERGED STRUCTURES Gene C. RUZICKA Army Helicopter Research Office, Ames Research Center, Moffett Field, CA 94035, U.S.A.
Thomas L. GEERS Department of Mechanical Engineering~ University of Colorado, Boulder, CO 80309, U.S.A. Received August, 1987 Revised July, 1989 Abstract. This paper describes transient-response analyses of submerged multiple structures with two different computational methods. Both methods approximate the presence of the surrounding infinite fluid with a transmitting boundary based on the first-order doubly asymptotic approximation (DAA1) , but differ in their treatment of multiple scattering by the structures. With the first method, the DAA boundary is placed directly on the structures' wet surfaces, which is strictly valid only for low-frequency components of multiply-scattered waves. The second, more costly, method permits a more valid analysis of multiple scattering through finite-element discretization of the local fluid region. Computational results are presented for simple two-dimensional problems involving two circular cylindrical shells with internal masses. The results produced by the two methods are often in close agreement, with the greatest discrepancies occurring when high-frequency multiple scattering is important.
Introduction
Analysis of the response of underwater structures to transient excitation has long posed a formidable challenge to engineers. The major difficulty in solving these problems is the need to couple structural response to the motion of the surrounding infinite fluid, which makes conventinal finite-element techniques prohibitively expensive.
Background Early efforts concentrated on developing analytical solutions for single, infinite-cylindrical or spherical shells, see, e.g., [13]. Although exact solutions were obtained, the techniques used were applicable only to a narrow class of problems. The limitations of analytical approaches prompted the search for general-purpose numerical tools. A useful step in this direction was the development of Doubly Asymptotic Approximations (DAA's) for treating the interaction of the structure with the surrounding fluid [10,14,6]. In such a treatment, the infinite fluid is represented by a boundary on the structure's wet surface that yields correct fluid behavior at high and low frequencies and effects a smooth transition between. A related advance was the development of staggered solution procedures [8,18], which provide efficient and stable means for integrating separately the structure and DAA equations in time. The DAA method was combined with a staggered solution algorithm to produce the USA (Underwater Shock Analysis) Code [5], which has been found to be an efficient and versatile tool.
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G. C Ruzicka, T.L Geers / Transient response analysis of multiple submerged structures
Unfortunately, the shock analysis of multiple structures constitutes an application where a standard DAA approach is inadequate, for the DAA does not properly account for the high-frequency components of multiply scattered waves, although the low-frequency components are treated correctly. This limitation has been overcome, however, by development of the USA--STAGS--CFA Code [7], which allows a contained-fluid region to be placed between the structural assembly and the DAA boundary. This region is merely a local portion of the surrounding fluid, and is modelled with acoustic finite elements. Because the contained fluid extends into the gaps between the submerged structures, it is possible to obtain with USA--STAGS-CFA a refined analysis of wave scattering in these regions. However, a price must be paid for this refinement. In addition to the burden of constructing a contained-fluid mesh and integrating the contained-fluid equations, the time integration is only conditionally stable, which limits the size of the integration increment.
This study In view of the increased modelling and computational demands of the USA--STAGS--CFAcode, it is natural to inquire if adequate solutions for multiple-structure problems can be obtained using the simpler USA--STAGSCode [3], for which the DAA boundary is placed directly on the structures' wet surfaces. The investigation of this question is the purpose of the present study, in which computational results for two-dimensional shock response analysis of simple multiple structures produced by the USA--STAGSand USA-STAGS-CFA Codes are compared. The complexities of these analyses are such that, to our knowledge, no analytical solutions for them have been obtained. Consequently, the more refined USA--STAGS-CFA results are used as benchmarks against which the USA-STAGS solutions are judged. The next section describes the governing equations for the finite-element/boundary-element methods studied, and also discusses the computational techniques used to solve the equations. The third section describes the two-dimensional analyses used to evaluate USA--STAGStreatment of multiple-structure problems, presents the results obtained, and states some conclusions drawn from the evaluation.
Governing equations The solution of underwater shock problems for multiple structures involves as many as three coupled fields: a structural field, a contained-fluid field of finite extent, and a DAA surface field that approximates the behavior of the surrounding infinite fluid. This section describes the governing field equations and solution methods that are embodied in the USA-STAGS and USA--STAGS--CFACodes. The intent is to provide a summary of the relevant theory; for a more extensive discussion, the reader is referred to the cited references.
Equations for the structure The structural field is treated with the STAGS Code [1], which employs the familiar set of finite-element equations (see, e.g., [19]).
Ms s + Cs s + Ks s
(1)
where x~ is the vector for the structural degrees-of-freedom (DOF's) and ~ is a load vector arising from external applied forces and from pressure exerted by the contained fluid. The matrix M s is a diagonal lumped mass matrix. The stiffness matrix K s is given by
Ks = ( n'on dV, ~K
(2)
G.C. Ruzicka, T.L. Geers/ Transientresponseanalysisof multiplesubmergedstructures
155
where V~ is the structural volume, the superscript t denotes matrix transposition, and D is a constitutive matrix relating stress and strain; the matrix B relates strain at an interior point to nodal displacements and is given by B = L X t, where L is the strain-displacement operator and X is a finite-element shape-function matrix that approximates the physical displacement xs(~, t) at spatial point ~ as
xs(~, t) ~ X t ( ~ ) X s ( t ) .
(3)
The damping matrix C~ in (1) is based on the Rayleigh damping relation C~ = aM~ + flK s (a and fl are constants), which ensures the existence of classical normal modes in the dry structure.
Equations for the contained fluid The contained fluid is an acoustic medium of finite extent that lies between the structure and the DAA boundary. The treatment of this medium described below follows that of [17] and [7]. For inviscid, irrotational motion, the fluid displacement xf can be derived from a displacement potential ~k as -
pxf = V~k,
(4)
where p is the fluid density. The pressure p is given by p =~,
(5)
and the governing equation is the wave equation = c2X72~k,
(6)
The constant c is the speed of sound in the fluid, obtained from c 2 = x/p, where x is the bulk modulus of the fluid. Equation (6) may be discretized by application of the finite-element approximation ap(~', t) --- Nt(~)~b(t)
(7)
where N is a shape-function vector and ~k is a vector of nodal values of ~. An equation for ~p may be derived with the Bubnov-Galerkin method, which requires that the weighted average of the residual error in (6) should vanish, i.e., f v N ( ~ - c:V 2~) d V = 0.
(8)
Application of the divergence theorem to (8), followed by insertion of the finite-element approximation (7), leads to
Q~ + c2ng/= c2b,
(9)
where Q and H are symmetric matrices given by
Q = f N N t dV, v,
H= fv,(VN)(vN)'
(10)
dV,
(11)
and the vector b is a boundary forcing vector to be described momentarily. The matrices Q and H are analogs of the mass and stiffness matrices, respectively, that appear in the structural field equations. As in common structural analysis practice, the response calculations are expedited through replacement of the consistent Q matrix defined in (10) with
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G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
a lumped diagonal matrix Q. The lumping is accomplished by placing the row sums of Q on the diagonal of Q. The boundary forcing vector in (9) is given by
b=fBN~ndW,
(12)
where Bf is the contained fluid boundary and n is the unit normal to the boundary, taken positive outward. The integral in (12) is readily simphfied to a more convenient matrix expression. First, observe that, from (4),
3~k
px~ . n,
3n
(13)
where x~ is the fluid displacement on the boundary. In a manner analogous to the discretization of the governing equation, the boundary displacements may be interpolated from nodal values as
xb(~, t) -- N d ( ~ ) x b ( t ) ,
(14)
where N b is a shape-function vector and x b is a vector of boundary displacements. The use of (14) and (13) in (12) leads to
b = pLfx b,
(15)
where Lf = -
~ NN~,Fb dB
(16)
Br
in which Fb is a diagonal matrix of direction cosines. It is interesting to note that, for nodes on a symmetry plane, b = 0. To see this, observe that the symmetry boundary condition is simply xfb. n = 0. Hence, from (13) and (12), b = 0 on that boundary.
Equations for the DAA boundary If a structure is immersed in a fluid of finite extent, the contained-fluid field just described, in conjunction with appropriate boundary conditions, suffices to define fluid behavior completely. On the other hand, if the fluid domain is infinite, the presence of a contained-fluid field alone is adequate only if the domain extends out far enough to avoid interference from waves reflected at the outer boundary. This so-called island approach has the advantage of being "exact", but is prohibitively expensive for all but short-time calculations. An alternative method is to surround an abbreviated contained-fluid mesh with a transmitting boundary, which allows scattered-wave energy to pass out of the contained-fluid field. The error incurred through use of a transmitting boundary is generally small if the boundary is well formulated.
Asymptotic behavior The transmitting boundary used in this study is based on the first-order Doubly Asymptotic which is exact in the limits of both low-frequency and high-frequency motions. It is appropriate to begin the derivation of the DAA 1 equations with descriptions of fluid behavior at these extremes. In the development that follows, the DAA boundary is subdivided into a mesh of boundary elements, whose behavior is referred to nodes located at the centroids of the elements.
Approximation (DAm1) [10,14,6],
G.C. Ruzicka, 7".L Geers / Transient response analysis of multiple submerged structures
15 7
The fluid motion may always be represented as the sum of an incident wave and a scattered wave. This may be expressed in computational vector form for pressures and normal fluid-particle displacements at the DAA control points as Pd =Pd
i
s +Pd,
i X d = X d -t- X ~ ,
(17)
where the superscripts i and s denote incident and scattered waves, respectively. As the incident wave is specified ab initio, it is only the effects of the scattered wave that must be approximated. At high frequencies, boundary pressures and displacements are related by the plane wave approximation (PWA) [16] P ds- -_
pcica.
(18)
The physical basis for this relation is that, in the high-frequency limit, acoustic wavelengths in the fluid are much shorter than the surface wavelengths characterizing the scattered wave, so each element of the boundary may be thought of as an infinite flat plate radiating plane waves outward. At low frequencies, the virtual mass approximation (VMA) applies [2] which is given by A d p ~ = MdJt a,
(19)
where A d is a diagonal matrix containing the areas of the DAA boundary elements. The symmetric matrix M d is the fluid mass matrix for computing the kinetic energy for irrotational flow of an incompressible fluid that is excited by motions normal to the DAA boundary [4]. The physical rationale for (19) is that, in the low-frequency limit, acoustic wavelengths in the fluid are much longer than the surface wavelengths characterizing the scattered wave, so the fluid behaves as an incompressible medium. DAA equation The complete first-order DAA equation may be written: • S
S
--
..S
n d p d + p c A d p d -- p C M d X d.
(20)
It is readily seen through frequency-domain analysis that this equation embodies the proper asymptotic behavior. Taking the Laplace transform of (20), one obtains sMap~ + pcAdp~ = s2pcnax~,
(21)
where the bar denotes transformed quantities and s is the transform variable. At high frequencies, or large s, (21) becomes p~ = soci~,
(22)
which yields (18) upon back-transformation. At low frequencies, (21) reduces to Ad~a = s2Md2a,
(23)
which becomes (19) upon back-transformation. Applicability of the DAA boundary It was mentioned earlier that use of the DAA boundary is questionable when it is placed directly on the wet surfaces of multiple structures. At low frequencies, the VMA can be validly applied to multiple structures simply by calculating a fluid mass matrix that couples all the structures through the fluid; this extension has been implemented in the USA--STAGS code. Unfortunately, the high-frequency PWA is not applicable to multiple structures because the PWA assumes that scattered waves emanating from the DAA boundary radiate out to infinity. This assumption is clearly violated for multiple structures because waves scattered by one
15 8
G.C. Ruzicka, T. L Geers / Transient response analysis of multiple submerged structures
structure can impinge on another structure and then return. One can argue heuristically, however, that USA--STAGS should give reasonable results if the high-frequency components of multiply scattered waves do not significantly affect response. In the USA-STAGS--erA approach, multiple scattering is accommodated within the contained fluid. This is clearly a more rigorous treatment than that of USA-STAGS, although the overall analysis is still approximate because the interaction with the infinite fluid is modelled with a DAA boundary. In this instance, however, all conditions for applicability of the DAA may be satisfied by fashioning the fluid mesh so that the DAA boundary is everywhere non-concave. Boundary
conditions
A complete description of the component fields requires specification of the boundary conditions that couple the fields. These boundary conditions ensure that force and displacement compatibility is satisfied at interfaces between the domains. method In this method, the DAA boundary is placed directly on the wet surface of the structure and only one interface is present. Force compatibility requires that the structural force vector be equivalent to the pressure-force exerted by the contained fluid, i.e., USA--STAGS
= - GsdAoPd,
(24)
where ~ is the vector of nodal forces exerted on the structure's wet surface and Gsd is a transformation matrix relating structural and DAA-boundary forces. Application of the contragradience principle [15] then yields the displacement compatibility equation
(25)
X d = GstdXs .
USA-STAGS-CFA
method
In this method, the DAA boundary is placed on the exterior surface of the contained fluid so that two interfaces are present, the structure-fluid interface and the f l u i d - D A A interface. At the former, force compatibility requires that the structural force vector be equivalent to the pressure-force exerted by the fluid, i.e., Is = - G s f a f P f ,
(26)
where Gsf is a transformation matrix relating structural and fluid surface forces, A f gives the tributary area on the structural wet surface for each fluid node, and pf is a vector of fluid surface pressures. Displacement compatibility on this interface may be enforced by entering the structural displacements at the appropriate locations in the vector x b appearing in (15). An alternative procedure may be used that eliminates the need to calculate and store the matrix Lf defined in (16). This merely involves application of the contragradience principle to obtain [15] bs = oA,as'fx
,
(27)
where bs is the contribution of the structure-fluid interface to the boundary forcing vector. On the fluid-DAA interface, the force compatibility relation is
Pd= GdfPf,
(28)
where Gaf is a transformation matrix relating DAA control-point pressures to fluid surface pressures. An analysis similar to that used in the derivation of (27) leads to bd = OadG~fxa,
(29)
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
159
where bd is the contribution of the DAA interface to the boundary forcing vector and A d is a diagonal matrix giving the tributary area on the DAA boundary of each fluid node. Assembled response equations
Based on the preceding development, dynamic fluid-structure response may be calculated by step-by-step numerical integration of ordinary differential equations in time. The form of these ordinary differential equations is determined by the computational approach selected. In the following, a USA-STAGS form and a USA--STAGS--CFAform are presented that, in the opinion of the authors, provide the greatest insight into the solution processes. However, a variety of conditions have dictated that neither of these forms be used in the USA-STAGS and USA--STAGS--CFA Codes. The computational approaches actually employed in the codes are described in [3] and [7]. USA-STAGS form
Here, two coupled sets of ordinary differential equations are obtained as follows. The first set is assembled by introducing (24) and the first of (17) into (1) to obtain i
s
Ms)2 S + CsYC~ + K~x~ = - GsdAd( Pd +Pd)"
(30)
The second set is obtained by introducing (25) and the second of (17) into (20) to produce M d p ~ -1- p c A a p ~ = p c M d (Gstd2s -- 2 1 ) .
(31)
These two sets are solved simultaneously by step-by-step numerical integration in time for the response vectors x~ and p,]. The data transfer between STAGS and USA in solving the two sets is illustrated in Fig. 1. USA--STAGS--CFA form
Here, three coupled sets of ordinary differential equations are obtained as follows. The first set is assembled by introducing (26) and (5) into (1) to obtain Ms.~ s + Cs.ic~ + K s x s = - GseAy~s,
(32)
where ~k~ denotes that part of ~ pertaining to nodes on the structure-fluid interface. The second set is obtained by partitioning ~k as ~kt = (~k t
q'ot
~k~),
(33)
where ~ko and ~ka denote the parts of ~k that pertain to interior nodes and to nodes on the fluid-DAA interface, respectively. The introduction of (27), (29) and the second of (17) into (9) then yields
(
AfC Xs
Q ~ + c2H~k = pc2 ~
0
A dG ,
~.
+
(34)
))
Finally, the third set is obtained by introducing the first of (17), (28), and (5) into (20) and
displacements Submerged Structure
DA~. Boundary
<-
forces
Fig. 1. Data transfer in the USA--STAGS Code.
160
G.C. Ruzicka, T L. Geers / Transient response analysis of multiple submerged structures
forces
displacements ">
;>
Fluid Volume
Submerged Structure
forces
DAA
<"
Boundary
displacements
Fig. 2. Data Transfer in the USA-STAGS-CFA Code.
integrating the resulting equation twice (with quiescent initial conditions) to produce p c M d x ~ = n d (Gdf~d -- p d ) "[- pCAd(Gdf l~d --/~d ),
(35)
where the numbers 1 and 2 over p~ denote single and double integration in time, respectively. The equations sets (32), (34) and (35) are solved simultaneously by step-by-step numerical integration in time for the response vectors x s, ff and x~. The data transfer among STAGS, CFA and USA in solving the sets is illustrated in Fig. 2.
Computational procedures The USA-STAGS and USA-STAGS-CFA codes employ staggered solution procedures. Staggered schemes have the advantage of dealing individually with the coefficient matrices that pertain to the individual component fields. These matrices tend to be much more manageable than the motley matrices usually generated when coupled-field equations are merged. In addition, staggering permits the optimum assignment of a time-integration algorithm to each equation set. Yet another advantage of staggering is that it allows the individual field processors to be isolated in separate software modules. The computational procedures used by USA-STAGS and USA--STAGS--CFA are extensively discussed in [3] and [7]. We are content here to state that the USA-STAGS time-integration procedure is unconditionally stable with respect to time increment for linear problems, but that the USA--STAGS--CFA procedure is only conditionally stable. The stability limit for the latter is roughly given by
At < O( l/c),
(36)
where l is the shortest distance between contained fluid nodes. As (36) is generally more stringent than the limit imposed by integration accuracy, USA--STAGSwill run successfully with fewer time steps than the number required by USA--STAGS-CFA for the same shock-response problem.
Two-dimensional analyses The USA--STAGS and USA-STAGS--CFA analysis methods are compared here on the basis of two-dimensional analyses of two identical cylindrical-shell units separated by a distance of one-half radius (Fig. 3). Each shell unit consists of an internal mass connected to a sandwich shell by many uniform and uniformly spaced springs. Parameters varied in the analyses are the circumferential bending stiffness of the shell and the fixed shell natural frequency of the mass-spring oscillator. The excitation consists of an incident step-wave oriented either side-on or end-on to the shell-unit pair. The multiple-shell-unit analyses are supplemented by analyses of corresponding single units. In addition to the USA--STAGSand USA-STAGS--CFA analyses, the single-shell studies incorporate an analytical form of the DAA approach based on the decomposition of shell response into Fourier harmonics [12]. The single-shell studies serve two purposes: they gauge the extent of fluid coupling in the multiple-shell problems, and they indicate the level of error incurred through discretization.
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submergedstructures
161
INCIDENT STEP-WAVE
INCIDENT STEP-WAVE=.
END-ON WAVE
SIDE-ON WAVE
Fig. 3. ConfigurationsUsed in the Multiple-StructureStudies. The earlier discussion under Applicability of the D A A Boundary suggests that, as oscillator natural frequency rises, agreement between USA-STAGS and USA-STAGS--CFA results should deteriorate. Also, it is reasonable to expect greater disagreement between the results for excitation by an end-on wave than that by a side-on wave because of shadowing effects present in the former case.
Structure and fluid models
The shell units are most conveniently described in terms of the dimensionless parameters shown in Fig. 4. The parameter I2 is the natural frequency of the spring-mass oscillator inside a fixed shell. The parameter 7 is the square root of the ratio of the bending stiffness of the sandwich shell to the stiffness of a uniform shell constructed by removing the core and fusing the inner and outer surface layers of combined thickness d. This parameter is needed in order to model the ring stiffeners that characterize actual pressure hulls [9,12]. The material of the inner and outer shell layers is characterized by its density Ps, Poisson's ratio v, and plate velocity c s, the last given by c s2 = E J p s ( 1 - ~,2), where Es is Young's modulus. The values given in Fig. 4 pertain to steel.
Structure models Each shell is modelled with a single ring of STAGS 410 shell elements, with 72 elements to a full ring. Symmetry constraints are applied as required to the shell edges, including axial constraints, to simulate plane-strain conditions. Problem symmetry is exploited by using half-ring models in the single-shell and end-on-excitation studies and by using a single full-ring model in the side-on-excitation studies. The identical oscillator springs are modelled with STAGS200 beam elements that extend from every node on the shell model to an oscillator-mass node at the centroid of the model. The axial stiffness of the beams is adjusted to produce the desired oscillator frequency, while the bending rigidity is zero. The axial stiffness of the beams on the symmetry diameter of the half-ring model is, of course, half the nominal value.
162
G.C. Ruzicka, T.L. Geers / Transientresponse analysis of multiple submerged structures Ps, E s '
v
d/2
7d (APPROX.)
4-3
M
-
2Xa------~B = 0.2,
all c
= 0,
d a
= 0.01
7
=
P.js P
-
v
= 0.3
7.65
1
5.3694
~_s = 3 . 5 3 c 1,
10
Fig. 4. Model Parameters.
Contained-fluid models
Different finite-element grids for the contained-fluid field are required for the single-shell, end-on-wave, and side-on-wave studies (Figs. 5-7). The dement used is an eight-node isoparametric brick with a trilinear interpolation scheme; at present, this is the only fluid element available in CFA. The grid geometry for the single-shell problems is generated automatically. The grids for the two-shdl problems were first sketched by hand and then entered into the database with a digitizer.
I N C I D E N T STEP WAVE
MASS
BOUNDARY
Fig. 5. Single-Shell Configuration: Fluid and StructureMeshes.
G.C. Ruzicka, T . L Geers ~/ Transient response analysis of multiple submerged structures
INCIDENT .
~
S T E P - W ~ l
i i lllll X
163
~A~
MASS SHELL
SHELL
BOUNDARY Fig. 6. Two-Shell Configuration: Fluid and Structure Meshes for the End-on Incident Wave.
DAA models Each D A A mesh matched the corresponding structural or contained-fluid mesh at the interface. This is shown in Figs. 5-7.
Analysis procedures USA-STAGS analysis
The first step here is the execution of the STAGSC-1 Code, which generates the structural mass and stiffness matrices, as well as geometry data and other information needed by the USA Code. The remaining steps involve modules of the USA Code. The FLUMAS module accepts the
IN~ DAA
BOUNDARY
Fig. 7. Two-Shell Configuration: Fluid and Structure Meshes for the Side-on Incident Wave.
164
G. C Ruzicka, T.L Geers / Transient response analysis of multiple submerged structures
geometry of the DAA boundary and generates the fluid mass matrix M d and the DAA-structure transformation matrix Gsd. This is followed by execution of the AUGMAT module, which generates several auxiliary matrices used in solving the response equations. Finally, the TIMINT module performs the time integration. USA-STAGS--CFA
analysis
The first step here is identical to that for a USA-STAGS analysis. The USA-CFA versions of the FLUMAS, AUGMAT, and TIMINT modules are then executed in sequence. These modules are similar in function and input to their USA counterparts, the major difference being that geometry data describing the contained-fluid mesh, rather than the structural mesh, must be prepared and placed in a file to be passed to the VLUMASmodule as input.
Time integration The lengthiest calculations are the time-integration runs. The costs of these are inversely proportional to the time increment, which should therefore be set as large as possible within the constraints imposed by accuracy and stability. For the USA--STAGS analyses, where only accuracy considerations apply, an increment of 0.045a/c was selected on the basis of previous experience. For the USA-STAGS--CFAcalculations, the highest accuracy is achieved by setting the increment as close as possible to the Courant stability limit, which is about O.08a/c for the fluid grids used. However, coupled-system stability requirements reduced the increment to 0.045a/c for the single-shell studies and 0.025a/c for the two-shell studies. Another integration parameter in USA--STAGS--CFAcalculations is the numerical damping parameter for CFA, which is here taken as unity [7]. The response selected for evaluation and display are radial velocities at the front and back of each shell. All calculations were done on a VAX 11/780 computer.
Response results Unstiffened shells, no oscillator The first shell unit to be considered is an unstiffened shell (~, = 1) with the internal oscillator absent. The incident wave is a step wave, with only end-on excitation considered in the two-shell analysis. The single-shell results are shown in Fig. 8. In addition to the discrete-element and analytical DAA results, the figure shows exact solutions obtained using the residual potential method [9-12]. The exact and approximate results are in close agreement. It is appropriate to discuss two features of the velocity histories in Fig. 8. First, the asymptotic translational velocity of the shell considerably exceeds the fluid-particle velocity of the incident wave, which is given by ocV/P = 1, where P is the magnitude of the incident-wave pressure. This occurs because, with the internal oscillator removed, the shell is very positively buoyant, see, e.g., [9]. Second, the small "blips" in the histories are produced by extensional waves propagating around the shell that gradually lose energy via "creeping waves" in the fluid, (see, e.g., [11]). The two-shell results for end-on excitation, along with their single-shell counterparts, are shown in Figs. 9 and 10. With regard to the forward shell (Fig. 9), the presence of the rear shell has little effect on response at the front, but significantly affects response at the back. The nature of the latter effect is as follows. When the incident wave and the scattered wave generated by the forward shell reach the front region of the rear shell, that area moves rapidly inward, generating a rarefactive scattered wave that propagates back toward the forward shell. When this rarefactive wave reaches the back of the forward shell, it pulls that region radially outward, which increases velocity response markedly above that occurring at the back of the
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Fig. 8. Single-Shell Velocity Responses: Unstiffened Shell.
single shell. USA-STAGS satisfactorily captures this effect as it p r o d u c e s results in c l o s e a g r e e m e n t w i t h t h o s e of USA--STAGS-CFA. Figure 10 s h o w s calculated r e s p o n s e histories at the front and b a c k of the rear shell, a l o n g w i t h c o r r e s p o n d i n g single-shell results. In c o n s o n a n c e w i t h the results o f F i g u r e 9, the f o r w a r d shell has little effect o n the r e s p o n s e at the b a c k of the rear shell, but it significantly affects the r e s p o n s e at the front. Here, USA--STAGS d o e s n o t d o so well, p r o d u c i n g a v e l o c i t y h i s t o r y at the C~
front
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Fig. 9. Two-Shell Velocity Responses: Forward Unstiffened Shell, End-on Incident Wave.
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G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
166
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f r o n t that is m a r k e d l y m o r e a b r u p t t h a n its USA--STAGS--CFA c o u n t e r p a r t mating peak response.
and slightly overesti-
Stiffened shell, low-frequency oscillator This shell unit consists o f a stiffened shell ( y = 10) c o n t a i n i n g a l o w - f r e q u e n c y w i t h f i x e d - b a s e n a t u r a l f r e q u e n c y ~2a/c = 0.2. S i d e - o n as w e l l as e n d - o n s t e p - w a v e s two-shell configuration.
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50.0
oscillator e x c i t e the
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
167
to
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ct/a Fig. 12. Two-Shell
Responses:
Velocity
Forward
Stiffened Shell, Low-Frequency
Oscillator, End-on
Incident Wave.
The single-shell results, shown in Fig. 11, show that here, as with the unstiffened shell, there is close agreement between the USA--STAGS and USA--STAGS--CFA results. The slow oscillation in Fig. 11 reflects the interaction, through the springs, of the oscillator mass and the combined mass of the shell and entrained external fluid. The frequency of this motion is somewhat higher than the oscillator fixed-base natural frequency, as befits reduced-mass oscillation. Because the shell unit considered here is neutrally buoyant, the oscillation takes place about the fluid-particle velocity of the incident wave. It is interesting to note that, early in the motion, the in ¢~"
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25.0
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50.0
ct/a Fig. 13. Two-Shell
Velocity Responses:
Rear Stiffened Shell, Low-Frequency
Oscillator, End-on
Incident Wave.
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
168
tO
e4 i
.~°
_
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45.0
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ctla F i g . 14. T w o - S h e l l V e l o c i t y R e s p o n s e s : S t i f f e n e d Shell, L o w - F r e q u e n c y
Oscillator, Side-on Incident Wave.
response histories in Fig. 11 agree closely with their counterparts in Fig. 8. This results from the rather high flexibility of the shell wails (be they stiffened or unstiffened) and the softness of the oscillator springs, which means that early response is dominated by inertial and m e m b r a n e effects, which are identical in the stiffened and unstiffened shells. Figures 12 and 13 show response histories for the two-shell configuration excited by an end-on step-wave. In general, the earlier comments on Figs. 9 and 10 apply equally well here. Once again, the highest level of shell interaction through the fluid, and the greatest discrepancies between the USA-STAGS and USA-STAGS-CFA results occur where shell regions are in close proximity. Velocity histories for excitation by the side-on wave are shown in Fig. 14. As expected, the results are generally intermediate between those for the single-sheU and end-on-wave cases with regard to the extent of shell interaction and agreement between the USA-STAGS and USA-STAGS--CFA results. A somewhat disturbing feature of the stiffened-shell response histories is the presence of small-scale, high-frequency oscillations that become noticeable at ct/a -- 20. These oscillations suggest slowly growing numerical instability, which did not yield to treatment during the present effort; hence it constitutes an item for future work.
Stiffened shell, high frequency oscillator This shell unit consists of a stiffened shell (y = 10) containing a high-frequency oscillator with fixed-base natural frequency 12a/c = 1.0. Here too, side-on as well as end-on step-waves are applied. The single-shell results (Fig. 15) reveal a slight deterioration in agreement, relative to that exhibited in Fig. 11, between USA--STAGS and M o d a l - D A A results on the one hand, and USA-STAGS--CFA results on the other. This is consistent with an earlier study [12] comparing exact and D A A solutions, where it was found that DAA-based methods tend to exaggerate radiation damping. Here, the USA-STAGS--CFA results benefit from the two layers of containedfluid elements (Fig. 5), which reduce excessive damping by moving the D A A boundary away from the shell's surface.
G.C. Ruzicka, T1- Geers / Transient response analysis of multiple submerged structures
169
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4.0
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to.o
t2.o
t4.o
ct/a
Fig. 15. Single-Shell Velocity Responses: Stiffened Shell, High-Frequency Oscillator.
T h e results for e x c i t a t i o n b y the e n d - o n w a v e (Figs. 16 a n d 17) reveal s i m i l a r d e t e r i o r a t i o n in a g r e e m e n t b e t w e e n the USA-STAGS a n d USA--STAGS-CFA results. A s in p r e v i o u s c o m p a r i s o n s , the best a g r e e m e n t occurs w h e r e shell i n t e r a c t i o n is least i m p o r t a n t , viz., at the front o f the f o r w a r d shell a n d at the b a c k o f the r e a r shell. A t the o t h e r two locations, d i s c r e p a n c i e s are p r o n o u n c e d , especially at the front of the r e a r shell (Fig. 17), where USA--STAGS fails to a c c o u n t for e a r l y - t i m e (high-frequency) s h a d o w i n g b y the f o r w a r d shell.
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I I i i. :i
"e.o
to.0
t2.0
z4.0
ctla
Fig. 16. Two-Shell Velocity Responses: Forward Stiffened Shell, High-Frequency Oscillator, End-on Incident Wave.
170
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures O
N"
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8.0
t2.0
14.o
ct/a
Fig. 17. Two-Shell Velocity Responses: Rear Stiffened Shell, High-Frequency Oscillator, End-on Incident Wave.
D i s c r e p a n c i e s b e t w e e n USA-STAGS a n d USA--STAGS--CFA results are also a p p a r e n t in the s i d e - o n w a v e results (Fig. 18), a l t h o u g h , in c o m p a r i s o n w i t h the e n d - o n - w a v e results, t h e y are m o d e s t . A s expected, a g r e e m e n t is best at the front o f each shell, a n d poorer at the b a c k of each shell, w h e r e s o m e m u l t i p l e - s c a t t e r i n g effects m a n i f e s t t h e m s e l v e s . A n interesting feature of the " b a c k " v e l o c i t y histories in Figs. 1 5 - 1 8 is a h i g h - f r e q u e n c y oscillation appearing in the USA--STAGS a n d M o d a l - D A A results d u r i n g the interval I < c t / a < 3 or 3 < c t / a < 5. T h e fact that it takes 20 M o d a I - D A A c i r c u m f e r e n t i a l h a r m o n i c s to capture this
O
;
I
i I
1
,
'
,~.-1. ~-~
I . . . . . .
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......
' ~ I~.
_____~
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r .......... i
p,-....................
;
~'&
• ---
2.0
'
Singlc-Shell
4,0
front
, .-.
i
_ . . . . . . . . . . . . . .
'. . . . . . . . . . .
USA-STAGS USA-STAGS-CFA
Two-Shell
USA-STAGS
Two-shell
USA-STAGS-CFA
I
i,.
i.~"~'~
f
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, ~
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;
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,
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18.
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Velocity
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Stiffened
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High-Frequency
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Side-on
Incident
Wave.
G.C. Ruzicka, T L. Geers / Transient response analysis of multiple submerged structures
171
oscillation in Figure 15 is evidence of the short-structural-wavelength nature of the phenomenon. The absence of this oscillation from the USA-STAGS-CFA results reveals an important drawback in the use of finite-element grids to propagate transient waves. The USA--STAGS method, on the other hand, does not suffer from this drawback, because the interaction of the structure with the infinite fluid is handled entirely with boundary elements.
Conclusion The USA--STAGS and USA--STAGS-CFA results for transiently excited, multiple, submerged structures are likely to agree satisfactorily when response is dominated by low-frequency components. Agreement is less likely, however, when intermediate- and high-frequency components are significant. In these situations, the use of USA--STAGS--CFAappears to be mandatory, although the presence of important high-frequency components requires refined meshing, which in turn incurs high computational costs. In most cases, it should be possible to assess the applicability of the simpler USA--STAGSassembly to a three-dimensional problem by comparing results obtained from the application of both USA--STAGS and USA-STAGS--CFA to related, two-dimensional evaluation problems. Such a procedure is strongly advised in production analyses.
Acknowledgements This study was performed under Contract DNA001-83-C-0244 with the Defense Nuclear Agency when both authors were at the Lockheed Palo Alto Research Laboratory. The authors express their appreciation for valuable advice and assistance to Drs. John A. DeRuntz and Carlos A. Felippa, who developed the software utilized in the study.
References [1] ALMROTH, B.O., et al., Structural Analysis of General Shells: User Instructions for STAGSC, LMSC-D633873, Lockheed Palo Alto Research Laboratory, Palo Alto, CA, 1980. [2] CrlERXOCK, G., "Transient flexural vibrations of ship-like structures exposed to underwater explosions", J. Acoust. Soc. Am. 48, pp. 170-180, 1970. [3] DI~RUNTZ,J.A. and F.A. BROGAN, Underwater Shock Analysis of Nonlinear Structures, A Reference Manual for the USA-STAGS Code (Version 3), DNA 5545F, Defense Nuclear Agency, Washington, DC, 1980. [4] DERuNTZ, J.A. and T.L GEERS, "Added mass computation by the boundary integral method", Int. J. Num. Methods Eng. 12, pp. 531-550, 1978. [5] DI~RUNTZ, J.A., et al., The Underwater Shock Analysis Code ( U S A - Version 3), DNA 5615F, Defense Nuclear Agency, Washington, DC 1980. [6] FELIPPA, C.A., "Top-down derivation of double asymptotic approximations for structure-fluid interaction analysis,', in Innovative Numerical Analysis for the Applied Engineering Sciences, edited by R.P. SHAW et al., University Press of Virginia, Charlottesville, VA, pp. 79-88, 1980. [7] FELIPPA, C.A. and J.A. DERur~TZ, "Finite element analysis of shock-induced hull cavitation", Comput. Methods Appl. Mech. Eng. 44, pp. 297-337, 1984. [8] FEUPPA, C.A., and K.C. PARK, "Staggered solution transient analysis procedures for coupled mechanical systems: formulation", Comput. Methods Appl. Mech. Eng. 24, pp. 61-111, 1980. [9] GEERS, T.L., "Excitation of an elastic cylindrical shell by a transient acoustic wave", J. Appl. Mech. 36, pp. 459-469, 1969. [10] GEERS, T.L., "Residual potential and approximate methods for three-dimensional fluid-structure interaction problems", J. Acoust. Soe. Am. 49, pp. 1505-1510, 1971. [11] GEERS, T.L., "Scattering of a transient acoustic wave by an elastic cylindrical shell", J. Acoust. Soc. Am. 37, pp. 1091-1106, 1972.
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G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
[12] GEERS,T.L., "Shock response analysis of submerged structures", Shock Vib. Bull. 44 (Supp. 3), pp. 17-32, 1974. [13] GEERS, T.L., "Transient response analysis of submerged structures", Finite Element Analysis of Transient Nonlinear Structural Behavior, edited by T. BELVTSCHKOet al., AMD-Vol. 14, American Society of Mechanical Engineers, New York, pp. 59-84, 1975. [14] GEERS,T.L., "Doubly asymptotic approximations for transient motions of submerged structures", J. Acoust. Soc. Am. 64, pp. 1500-1508, 1978. [15] GEERS, T.L. and G.C. RUZICKA, "Finite-element/boundary-element analysis of multiple structures excited by transient acoustic waves", Numerical Methods for Transient and Coupled Problems, edited by R.W. LEWIS et al., Pineridge Press, Swansea, UK, pp. 150-157, 1984. [16] MINDUN, R.D., and H.H. BLEICH, "Response of an elastic cylindrical shell to a transverse step shock wave", J. Appl. Mech. 20, pp. 189-195, 1953. [17] NEWTON, R.E., "Finite element analysis of shock-induced cavitation", Preprint 80-110, ASCE Spring Convention, Portland, Oregon, 1980. [18] PARK, K.C., et al., "Stabilization of staggered solution procedures for fluid-structure interaction analysis", Computational Methods for Fluid-Structure Interaction Problems, edited by T. BELYTSCHKOand T.L. GEERS, AMD-Vol. 26, American Society of Mechanical Engineers, New York, pp. 95-124, 1977. [19] ZIENKIEWICZ,O.C., The Finite Element Metho~ McGraw-Hill, London, 1977.
153
TRANSIENT RESPONSE ANALYSIS OF MULTIPLE SUBMERGED STRUCTURES Gene C. RUZICKA Army Helicopter Research Office, Ames Research Center, Moffett Field, CA 94035, U.S.A.
Thomas L. GEERS Department of Mechanical Engineering~ University of Colorado, Boulder, CO 80309, U.S.A. Received August, 1987 Revised July, 1989 Abstract. This paper describes transient-response analyses of submerged multiple structures with two different computational methods. Both methods approximate the presence of the surrounding infinite fluid with a transmitting boundary based on the first-order doubly asymptotic approximation (DAA1) , but differ in their treatment of multiple scattering by the structures. With the first method, the DAA boundary is placed directly on the structures' wet surfaces, which is strictly valid only for low-frequency components of multiply-scattered waves. The second, more costly, method permits a more valid analysis of multiple scattering through finite-element discretization of the local fluid region. Computational results are presented for simple two-dimensional problems involving two circular cylindrical shells with internal masses. The results produced by the two methods are often in close agreement, with the greatest discrepancies occurring when high-frequency multiple scattering is important.
Introduction
Analysis of the response of underwater structures to transient excitation has long posed a formidable challenge to engineers. The major difficulty in solving these problems is the need to couple structural response to the motion of the surrounding infinite fluid, which makes conventinal finite-element techniques prohibitively expensive.
Background Early efforts concentrated on developing analytical solutions for single, infinite-cylindrical or spherical shells, see, e.g., [13]. Although exact solutions were obtained, the techniques used were applicable only to a narrow class of problems. The limitations of analytical approaches prompted the search for general-purpose numerical tools. A useful step in this direction was the development of Doubly Asymptotic Approximations (DAA's) for treating the interaction of the structure with the surrounding fluid [10,14,6]. In such a treatment, the infinite fluid is represented by a boundary on the structure's wet surface that yields correct fluid behavior at high and low frequencies and effects a smooth transition between. A related advance was the development of staggered solution procedures [8,18], which provide efficient and stable means for integrating separately the structure and DAA equations in time. The DAA method was combined with a staggered solution algorithm to produce the USA (Underwater Shock Analysis) Code [5], which has been found to be an efficient and versatile tool.
154
G. C Ruzicka, T.L Geers / Transient response analysis of multiple submerged structures
Unfortunately, the shock analysis of multiple structures constitutes an application where a standard DAA approach is inadequate, for the DAA does not properly account for the high-frequency components of multiply scattered waves, although the low-frequency components are treated correctly. This limitation has been overcome, however, by development of the USA--STAGS--CFA Code [7], which allows a contained-fluid region to be placed between the structural assembly and the DAA boundary. This region is merely a local portion of the surrounding fluid, and is modelled with acoustic finite elements. Because the contained fluid extends into the gaps between the submerged structures, it is possible to obtain with USA--STAGS-CFA a refined analysis of wave scattering in these regions. However, a price must be paid for this refinement. In addition to the burden of constructing a contained-fluid mesh and integrating the contained-fluid equations, the time integration is only conditionally stable, which limits the size of the integration increment.
This study In view of the increased modelling and computational demands of the USA--STAGS--CFAcode, it is natural to inquire if adequate solutions for multiple-structure problems can be obtained using the simpler USA--STAGSCode [3], for which the DAA boundary is placed directly on the structures' wet surfaces. The investigation of this question is the purpose of the present study, in which computational results for two-dimensional shock response analysis of simple multiple structures produced by the USA--STAGSand USA-STAGS-CFA Codes are compared. The complexities of these analyses are such that, to our knowledge, no analytical solutions for them have been obtained. Consequently, the more refined USA--STAGS-CFA results are used as benchmarks against which the USA-STAGS solutions are judged. The next section describes the governing equations for the finite-element/boundary-element methods studied, and also discusses the computational techniques used to solve the equations. The third section describes the two-dimensional analyses used to evaluate USA--STAGStreatment of multiple-structure problems, presents the results obtained, and states some conclusions drawn from the evaluation.
Governing equations The solution of underwater shock problems for multiple structures involves as many as three coupled fields: a structural field, a contained-fluid field of finite extent, and a DAA surface field that approximates the behavior of the surrounding infinite fluid. This section describes the governing field equations and solution methods that are embodied in the USA-STAGS and USA--STAGS--CFACodes. The intent is to provide a summary of the relevant theory; for a more extensive discussion, the reader is referred to the cited references.
Equations for the structure The structural field is treated with the STAGS Code [1], which employs the familiar set of finite-element equations (see, e.g., [19]).
Ms s + Cs s + Ks s
(1)
where x~ is the vector for the structural degrees-of-freedom (DOF's) and ~ is a load vector arising from external applied forces and from pressure exerted by the contained fluid. The matrix M s is a diagonal lumped mass matrix. The stiffness matrix K s is given by
Ks = ( n'on dV, ~K
(2)
G.C. Ruzicka, T.L. Geers/ Transientresponseanalysisof multiplesubmergedstructures
155
where V~ is the structural volume, the superscript t denotes matrix transposition, and D is a constitutive matrix relating stress and strain; the matrix B relates strain at an interior point to nodal displacements and is given by B = L X t, where L is the strain-displacement operator and X is a finite-element shape-function matrix that approximates the physical displacement xs(~, t) at spatial point ~ as
xs(~, t) ~ X t ( ~ ) X s ( t ) .
(3)
The damping matrix C~ in (1) is based on the Rayleigh damping relation C~ = aM~ + flK s (a and fl are constants), which ensures the existence of classical normal modes in the dry structure.
Equations for the contained fluid The contained fluid is an acoustic medium of finite extent that lies between the structure and the DAA boundary. The treatment of this medium described below follows that of [17] and [7]. For inviscid, irrotational motion, the fluid displacement xf can be derived from a displacement potential ~k as -
pxf = V~k,
(4)
where p is the fluid density. The pressure p is given by p =~,
(5)
and the governing equation is the wave equation = c2X72~k,
(6)
The constant c is the speed of sound in the fluid, obtained from c 2 = x/p, where x is the bulk modulus of the fluid. Equation (6) may be discretized by application of the finite-element approximation ap(~', t) --- Nt(~)~b(t)
(7)
where N is a shape-function vector and ~k is a vector of nodal values of ~. An equation for ~p may be derived with the Bubnov-Galerkin method, which requires that the weighted average of the residual error in (6) should vanish, i.e., f v N ( ~ - c:V 2~) d V = 0.
(8)
Application of the divergence theorem to (8), followed by insertion of the finite-element approximation (7), leads to
Q~ + c2ng/= c2b,
(9)
where Q and H are symmetric matrices given by
Q = f N N t dV, v,
H= fv,(VN)(vN)'
(10)
dV,
(11)
and the vector b is a boundary forcing vector to be described momentarily. The matrices Q and H are analogs of the mass and stiffness matrices, respectively, that appear in the structural field equations. As in common structural analysis practice, the response calculations are expedited through replacement of the consistent Q matrix defined in (10) with
156
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
a lumped diagonal matrix Q. The lumping is accomplished by placing the row sums of Q on the diagonal of Q. The boundary forcing vector in (9) is given by
b=fBN~ndW,
(12)
where Bf is the contained fluid boundary and n is the unit normal to the boundary, taken positive outward. The integral in (12) is readily simphfied to a more convenient matrix expression. First, observe that, from (4),
3~k
px~ . n,
3n
(13)
where x~ is the fluid displacement on the boundary. In a manner analogous to the discretization of the governing equation, the boundary displacements may be interpolated from nodal values as
xb(~, t) -- N d ( ~ ) x b ( t ) ,
(14)
where N b is a shape-function vector and x b is a vector of boundary displacements. The use of (14) and (13) in (12) leads to
b = pLfx b,
(15)
where Lf = -
~ NN~,Fb dB
(16)
Br
in which Fb is a diagonal matrix of direction cosines. It is interesting to note that, for nodes on a symmetry plane, b = 0. To see this, observe that the symmetry boundary condition is simply xfb. n = 0. Hence, from (13) and (12), b = 0 on that boundary.
Equations for the DAA boundary If a structure is immersed in a fluid of finite extent, the contained-fluid field just described, in conjunction with appropriate boundary conditions, suffices to define fluid behavior completely. On the other hand, if the fluid domain is infinite, the presence of a contained-fluid field alone is adequate only if the domain extends out far enough to avoid interference from waves reflected at the outer boundary. This so-called island approach has the advantage of being "exact", but is prohibitively expensive for all but short-time calculations. An alternative method is to surround an abbreviated contained-fluid mesh with a transmitting boundary, which allows scattered-wave energy to pass out of the contained-fluid field. The error incurred through use of a transmitting boundary is generally small if the boundary is well formulated.
Asymptotic behavior The transmitting boundary used in this study is based on the first-order Doubly Asymptotic which is exact in the limits of both low-frequency and high-frequency motions. It is appropriate to begin the derivation of the DAA 1 equations with descriptions of fluid behavior at these extremes. In the development that follows, the DAA boundary is subdivided into a mesh of boundary elements, whose behavior is referred to nodes located at the centroids of the elements.
Approximation (DAm1) [10,14,6],
G.C. Ruzicka, 7".L Geers / Transient response analysis of multiple submerged structures
15 7
The fluid motion may always be represented as the sum of an incident wave and a scattered wave. This may be expressed in computational vector form for pressures and normal fluid-particle displacements at the DAA control points as Pd =Pd
i
s +Pd,
i X d = X d -t- X ~ ,
(17)
where the superscripts i and s denote incident and scattered waves, respectively. As the incident wave is specified ab initio, it is only the effects of the scattered wave that must be approximated. At high frequencies, boundary pressures and displacements are related by the plane wave approximation (PWA) [16] P ds- -_
pcica.
(18)
The physical basis for this relation is that, in the high-frequency limit, acoustic wavelengths in the fluid are much shorter than the surface wavelengths characterizing the scattered wave, so each element of the boundary may be thought of as an infinite flat plate radiating plane waves outward. At low frequencies, the virtual mass approximation (VMA) applies [2] which is given by A d p ~ = MdJt a,
(19)
where A d is a diagonal matrix containing the areas of the DAA boundary elements. The symmetric matrix M d is the fluid mass matrix for computing the kinetic energy for irrotational flow of an incompressible fluid that is excited by motions normal to the DAA boundary [4]. The physical rationale for (19) is that, in the low-frequency limit, acoustic wavelengths in the fluid are much longer than the surface wavelengths characterizing the scattered wave, so the fluid behaves as an incompressible medium. DAA equation The complete first-order DAA equation may be written: • S
S
--
..S
n d p d + p c A d p d -- p C M d X d.
(20)
It is readily seen through frequency-domain analysis that this equation embodies the proper asymptotic behavior. Taking the Laplace transform of (20), one obtains sMap~ + pcAdp~ = s2pcnax~,
(21)
where the bar denotes transformed quantities and s is the transform variable. At high frequencies, or large s, (21) becomes p~ = soci~,
(22)
which yields (18) upon back-transformation. At low frequencies, (21) reduces to Ad~a = s2Md2a,
(23)
which becomes (19) upon back-transformation. Applicability of the DAA boundary It was mentioned earlier that use of the DAA boundary is questionable when it is placed directly on the wet surfaces of multiple structures. At low frequencies, the VMA can be validly applied to multiple structures simply by calculating a fluid mass matrix that couples all the structures through the fluid; this extension has been implemented in the USA--STAGS code. Unfortunately, the high-frequency PWA is not applicable to multiple structures because the PWA assumes that scattered waves emanating from the DAA boundary radiate out to infinity. This assumption is clearly violated for multiple structures because waves scattered by one
15 8
G.C. Ruzicka, T. L Geers / Transient response analysis of multiple submerged structures
structure can impinge on another structure and then return. One can argue heuristically, however, that USA--STAGS should give reasonable results if the high-frequency components of multiply scattered waves do not significantly affect response. In the USA-STAGS--erA approach, multiple scattering is accommodated within the contained fluid. This is clearly a more rigorous treatment than that of USA-STAGS, although the overall analysis is still approximate because the interaction with the infinite fluid is modelled with a DAA boundary. In this instance, however, all conditions for applicability of the DAA may be satisfied by fashioning the fluid mesh so that the DAA boundary is everywhere non-concave. Boundary
conditions
A complete description of the component fields requires specification of the boundary conditions that couple the fields. These boundary conditions ensure that force and displacement compatibility is satisfied at interfaces between the domains. method In this method, the DAA boundary is placed directly on the wet surface of the structure and only one interface is present. Force compatibility requires that the structural force vector be equivalent to the pressure-force exerted by the contained fluid, i.e., USA--STAGS
= - GsdAoPd,
(24)
where ~ is the vector of nodal forces exerted on the structure's wet surface and Gsd is a transformation matrix relating structural and DAA-boundary forces. Application of the contragradience principle [15] then yields the displacement compatibility equation
(25)
X d = GstdXs .
USA-STAGS-CFA
method
In this method, the DAA boundary is placed on the exterior surface of the contained fluid so that two interfaces are present, the structure-fluid interface and the f l u i d - D A A interface. At the former, force compatibility requires that the structural force vector be equivalent to the pressure-force exerted by the fluid, i.e., Is = - G s f a f P f ,
(26)
where Gsf is a transformation matrix relating structural and fluid surface forces, A f gives the tributary area on the structural wet surface for each fluid node, and pf is a vector of fluid surface pressures. Displacement compatibility on this interface may be enforced by entering the structural displacements at the appropriate locations in the vector x b appearing in (15). An alternative procedure may be used that eliminates the need to calculate and store the matrix Lf defined in (16). This merely involves application of the contragradience principle to obtain [15] bs = oA,as'fx
,
(27)
where bs is the contribution of the structure-fluid interface to the boundary forcing vector. On the fluid-DAA interface, the force compatibility relation is
Pd= GdfPf,
(28)
where Gaf is a transformation matrix relating DAA control-point pressures to fluid surface pressures. An analysis similar to that used in the derivation of (27) leads to bd = OadG~fxa,
(29)
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
159
where bd is the contribution of the DAA interface to the boundary forcing vector and A d is a diagonal matrix giving the tributary area on the DAA boundary of each fluid node. Assembled response equations
Based on the preceding development, dynamic fluid-structure response may be calculated by step-by-step numerical integration of ordinary differential equations in time. The form of these ordinary differential equations is determined by the computational approach selected. In the following, a USA-STAGS form and a USA--STAGS--CFAform are presented that, in the opinion of the authors, provide the greatest insight into the solution processes. However, a variety of conditions have dictated that neither of these forms be used in the USA-STAGS and USA--STAGS--CFA Codes. The computational approaches actually employed in the codes are described in [3] and [7]. USA-STAGS form
Here, two coupled sets of ordinary differential equations are obtained as follows. The first set is assembled by introducing (24) and the first of (17) into (1) to obtain i
s
Ms)2 S + CsYC~ + K~x~ = - GsdAd( Pd +Pd)"
(30)
The second set is obtained by introducing (25) and the second of (17) into (20) to produce M d p ~ -1- p c A a p ~ = p c M d (Gstd2s -- 2 1 ) .
(31)
These two sets are solved simultaneously by step-by-step numerical integration in time for the response vectors x~ and p,]. The data transfer between STAGS and USA in solving the two sets is illustrated in Fig. 1. USA--STAGS--CFA form
Here, three coupled sets of ordinary differential equations are obtained as follows. The first set is assembled by introducing (26) and (5) into (1) to obtain Ms.~ s + Cs.ic~ + K s x s = - GseAy~s,
(32)
where ~k~ denotes that part of ~ pertaining to nodes on the structure-fluid interface. The second set is obtained by partitioning ~k as ~kt = (~k t
q'ot
~k~),
(33)
where ~ko and ~ka denote the parts of ~k that pertain to interior nodes and to nodes on the fluid-DAA interface, respectively. The introduction of (27), (29) and the second of (17) into (9) then yields
(
AfC Xs
Q ~ + c2H~k = pc2 ~
0
A dG ,
~.
+
(34)
))
Finally, the third set is obtained by introducing the first of (17), (28), and (5) into (20) and
displacements Submerged Structure
DA~. Boundary
<-
forces
Fig. 1. Data transfer in the USA--STAGS Code.
160
G.C. Ruzicka, T L. Geers / Transient response analysis of multiple submerged structures
forces
displacements ">
;>
Fluid Volume
Submerged Structure
forces
DAA
<"
Boundary
displacements
Fig. 2. Data Transfer in the USA-STAGS-CFA Code.
integrating the resulting equation twice (with quiescent initial conditions) to produce p c M d x ~ = n d (Gdf~d -- p d ) "[- pCAd(Gdf l~d --/~d ),
(35)
where the numbers 1 and 2 over p~ denote single and double integration in time, respectively. The equations sets (32), (34) and (35) are solved simultaneously by step-by-step numerical integration in time for the response vectors x s, ff and x~. The data transfer among STAGS, CFA and USA in solving the sets is illustrated in Fig. 2.
Computational procedures The USA-STAGS and USA-STAGS-CFA codes employ staggered solution procedures. Staggered schemes have the advantage of dealing individually with the coefficient matrices that pertain to the individual component fields. These matrices tend to be much more manageable than the motley matrices usually generated when coupled-field equations are merged. In addition, staggering permits the optimum assignment of a time-integration algorithm to each equation set. Yet another advantage of staggering is that it allows the individual field processors to be isolated in separate software modules. The computational procedures used by USA-STAGS and USA--STAGS--CFA are extensively discussed in [3] and [7]. We are content here to state that the USA-STAGS time-integration procedure is unconditionally stable with respect to time increment for linear problems, but that the USA--STAGS--CFA procedure is only conditionally stable. The stability limit for the latter is roughly given by
At < O( l/c),
(36)
where l is the shortest distance between contained fluid nodes. As (36) is generally more stringent than the limit imposed by integration accuracy, USA--STAGSwill run successfully with fewer time steps than the number required by USA--STAGS-CFA for the same shock-response problem.
Two-dimensional analyses The USA--STAGS and USA-STAGS--CFA analysis methods are compared here on the basis of two-dimensional analyses of two identical cylindrical-shell units separated by a distance of one-half radius (Fig. 3). Each shell unit consists of an internal mass connected to a sandwich shell by many uniform and uniformly spaced springs. Parameters varied in the analyses are the circumferential bending stiffness of the shell and the fixed shell natural frequency of the mass-spring oscillator. The excitation consists of an incident step-wave oriented either side-on or end-on to the shell-unit pair. The multiple-shell-unit analyses are supplemented by analyses of corresponding single units. In addition to the USA--STAGSand USA-STAGS--CFA analyses, the single-shell studies incorporate an analytical form of the DAA approach based on the decomposition of shell response into Fourier harmonics [12]. The single-shell studies serve two purposes: they gauge the extent of fluid coupling in the multiple-shell problems, and they indicate the level of error incurred through discretization.
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submergedstructures
161
INCIDENT STEP-WAVE
INCIDENT STEP-WAVE=.
END-ON WAVE
SIDE-ON WAVE
Fig. 3. ConfigurationsUsed in the Multiple-StructureStudies. The earlier discussion under Applicability of the D A A Boundary suggests that, as oscillator natural frequency rises, agreement between USA-STAGS and USA-STAGS--CFA results should deteriorate. Also, it is reasonable to expect greater disagreement between the results for excitation by an end-on wave than that by a side-on wave because of shadowing effects present in the former case.
Structure and fluid models
The shell units are most conveniently described in terms of the dimensionless parameters shown in Fig. 4. The parameter I2 is the natural frequency of the spring-mass oscillator inside a fixed shell. The parameter 7 is the square root of the ratio of the bending stiffness of the sandwich shell to the stiffness of a uniform shell constructed by removing the core and fusing the inner and outer surface layers of combined thickness d. This parameter is needed in order to model the ring stiffeners that characterize actual pressure hulls [9,12]. The material of the inner and outer shell layers is characterized by its density Ps, Poisson's ratio v, and plate velocity c s, the last given by c s2 = E J p s ( 1 - ~,2), where Es is Young's modulus. The values given in Fig. 4 pertain to steel.
Structure models Each shell is modelled with a single ring of STAGS 410 shell elements, with 72 elements to a full ring. Symmetry constraints are applied as required to the shell edges, including axial constraints, to simulate plane-strain conditions. Problem symmetry is exploited by using half-ring models in the single-shell and end-on-excitation studies and by using a single full-ring model in the side-on-excitation studies. The identical oscillator springs are modelled with STAGS200 beam elements that extend from every node on the shell model to an oscillator-mass node at the centroid of the model. The axial stiffness of the beams is adjusted to produce the desired oscillator frequency, while the bending rigidity is zero. The axial stiffness of the beams on the symmetry diameter of the half-ring model is, of course, half the nominal value.
162
G.C. Ruzicka, T.L. Geers / Transientresponse analysis of multiple submerged structures Ps, E s '
v
d/2
7d (APPROX.)
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M
-
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=
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1
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10
Fig. 4. Model Parameters.
Contained-fluid models
Different finite-element grids for the contained-fluid field are required for the single-shell, end-on-wave, and side-on-wave studies (Figs. 5-7). The dement used is an eight-node isoparametric brick with a trilinear interpolation scheme; at present, this is the only fluid element available in CFA. The grid geometry for the single-shell problems is generated automatically. The grids for the two-shdl problems were first sketched by hand and then entered into the database with a digitizer.
I N C I D E N T STEP WAVE
MASS
BOUNDARY
Fig. 5. Single-Shell Configuration: Fluid and StructureMeshes.
G.C. Ruzicka, T . L Geers ~/ Transient response analysis of multiple submerged structures
INCIDENT .
~
S T E P - W ~ l
i i lllll X
163
~A~
MASS SHELL
SHELL
BOUNDARY Fig. 6. Two-Shell Configuration: Fluid and Structure Meshes for the End-on Incident Wave.
DAA models Each D A A mesh matched the corresponding structural or contained-fluid mesh at the interface. This is shown in Figs. 5-7.
Analysis procedures USA-STAGS analysis
The first step here is the execution of the STAGSC-1 Code, which generates the structural mass and stiffness matrices, as well as geometry data and other information needed by the USA Code. The remaining steps involve modules of the USA Code. The FLUMAS module accepts the
IN~ DAA
BOUNDARY
Fig. 7. Two-Shell Configuration: Fluid and Structure Meshes for the Side-on Incident Wave.
164
G. C Ruzicka, T.L Geers / Transient response analysis of multiple submerged structures
geometry of the DAA boundary and generates the fluid mass matrix M d and the DAA-structure transformation matrix Gsd. This is followed by execution of the AUGMAT module, which generates several auxiliary matrices used in solving the response equations. Finally, the TIMINT module performs the time integration. USA-STAGS--CFA
analysis
The first step here is identical to that for a USA-STAGS analysis. The USA-CFA versions of the FLUMAS, AUGMAT, and TIMINT modules are then executed in sequence. These modules are similar in function and input to their USA counterparts, the major difference being that geometry data describing the contained-fluid mesh, rather than the structural mesh, must be prepared and placed in a file to be passed to the VLUMASmodule as input.
Time integration The lengthiest calculations are the time-integration runs. The costs of these are inversely proportional to the time increment, which should therefore be set as large as possible within the constraints imposed by accuracy and stability. For the USA--STAGS analyses, where only accuracy considerations apply, an increment of 0.045a/c was selected on the basis of previous experience. For the USA-STAGS--CFAcalculations, the highest accuracy is achieved by setting the increment as close as possible to the Courant stability limit, which is about O.08a/c for the fluid grids used. However, coupled-system stability requirements reduced the increment to 0.045a/c for the single-shell studies and 0.025a/c for the two-shell studies. Another integration parameter in USA--STAGS--CFAcalculations is the numerical damping parameter for CFA, which is here taken as unity [7]. The response selected for evaluation and display are radial velocities at the front and back of each shell. All calculations were done on a VAX 11/780 computer.
Response results Unstiffened shells, no oscillator The first shell unit to be considered is an unstiffened shell (~, = 1) with the internal oscillator absent. The incident wave is a step wave, with only end-on excitation considered in the two-shell analysis. The single-shell results are shown in Fig. 8. In addition to the discrete-element and analytical DAA results, the figure shows exact solutions obtained using the residual potential method [9-12]. The exact and approximate results are in close agreement. It is appropriate to discuss two features of the velocity histories in Fig. 8. First, the asymptotic translational velocity of the shell considerably exceeds the fluid-particle velocity of the incident wave, which is given by ocV/P = 1, where P is the magnitude of the incident-wave pressure. This occurs because, with the internal oscillator removed, the shell is very positively buoyant, see, e.g., [9]. Second, the small "blips" in the histories are produced by extensional waves propagating around the shell that gradually lose energy via "creeping waves" in the fluid, (see, e.g., [11]). The two-shell results for end-on excitation, along with their single-shell counterparts, are shown in Figs. 9 and 10. With regard to the forward shell (Fig. 9), the presence of the rear shell has little effect on response at the front, but significantly affects response at the back. The nature of the latter effect is as follows. When the incident wave and the scattered wave generated by the forward shell reach the front region of the rear shell, that area moves rapidly inward, generating a rarefactive scattered wave that propagates back toward the forward shell. When this rarefactive wave reaches the back of the forward shell, it pulls that region radially outward, which increases velocity response markedly above that occurring at the back of the
G.C. Ruzicka. T.L. Geers / Transient response analysis of multiple submerged structures
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single shell. USA-STAGS satisfactorily captures this effect as it p r o d u c e s results in c l o s e a g r e e m e n t w i t h t h o s e of USA--STAGS-CFA. Figure 10 s h o w s calculated r e s p o n s e histories at the front and b a c k of the rear shell, a l o n g w i t h c o r r e s p o n d i n g single-shell results. In c o n s o n a n c e w i t h the results o f F i g u r e 9, the f o r w a r d shell has little effect o n the r e s p o n s e at the b a c k of the rear shell, but it significantly affects the r e s p o n s e at the front. Here, USA--STAGS d o e s n o t d o so well, p r o d u c i n g a v e l o c i t y h i s t o r y at the C~
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10.0
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
166
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f r o n t that is m a r k e d l y m o r e a b r u p t t h a n its USA--STAGS--CFA c o u n t e r p a r t mating peak response.
and slightly overesti-
Stiffened shell, low-frequency oscillator This shell unit consists o f a stiffened shell ( y = 10) c o n t a i n i n g a l o w - f r e q u e n c y w i t h f i x e d - b a s e n a t u r a l f r e q u e n c y ~2a/c = 0.2. S i d e - o n as w e l l as e n d - o n s t e p - w a v e s two-shell configuration.
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50.0
oscillator e x c i t e the
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
167
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ct/a Fig. 12. Two-Shell
Responses:
Velocity
Forward
Stiffened Shell, Low-Frequency
Oscillator, End-on
Incident Wave.
The single-shell results, shown in Fig. 11, show that here, as with the unstiffened shell, there is close agreement between the USA--STAGS and USA--STAGS--CFA results. The slow oscillation in Fig. 11 reflects the interaction, through the springs, of the oscillator mass and the combined mass of the shell and entrained external fluid. The frequency of this motion is somewhat higher than the oscillator fixed-base natural frequency, as befits reduced-mass oscillation. Because the shell unit considered here is neutrally buoyant, the oscillation takes place about the fluid-particle velocity of the incident wave. It is interesting to note that, early in the motion, the in ¢~"
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25.0
"~ . 0
i
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~5.0
I 4s.0
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~.0
50.0
ct/a Fig. 13. Two-Shell
Velocity Responses:
Rear Stiffened Shell, Low-Frequency
Oscillator, End-on
Incident Wave.
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
168
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ctla F i g . 14. T w o - S h e l l V e l o c i t y R e s p o n s e s : S t i f f e n e d Shell, L o w - F r e q u e n c y
Oscillator, Side-on Incident Wave.
response histories in Fig. 11 agree closely with their counterparts in Fig. 8. This results from the rather high flexibility of the shell wails (be they stiffened or unstiffened) and the softness of the oscillator springs, which means that early response is dominated by inertial and m e m b r a n e effects, which are identical in the stiffened and unstiffened shells. Figures 12 and 13 show response histories for the two-shell configuration excited by an end-on step-wave. In general, the earlier comments on Figs. 9 and 10 apply equally well here. Once again, the highest level of shell interaction through the fluid, and the greatest discrepancies between the USA-STAGS and USA-STAGS-CFA results occur where shell regions are in close proximity. Velocity histories for excitation by the side-on wave are shown in Fig. 14. As expected, the results are generally intermediate between those for the single-sheU and end-on-wave cases with regard to the extent of shell interaction and agreement between the USA-STAGS and USA-STAGS--CFA results. A somewhat disturbing feature of the stiffened-shell response histories is the presence of small-scale, high-frequency oscillations that become noticeable at ct/a -- 20. These oscillations suggest slowly growing numerical instability, which did not yield to treatment during the present effort; hence it constitutes an item for future work.
Stiffened shell, high frequency oscillator This shell unit consists of a stiffened shell (y = 10) containing a high-frequency oscillator with fixed-base natural frequency 12a/c = 1.0. Here too, side-on as well as end-on step-waves are applied. The single-shell results (Fig. 15) reveal a slight deterioration in agreement, relative to that exhibited in Fig. 11, between USA--STAGS and M o d a l - D A A results on the one hand, and USA-STAGS--CFA results on the other. This is consistent with an earlier study [12] comparing exact and D A A solutions, where it was found that DAA-based methods tend to exaggerate radiation damping. Here, the USA-STAGS--CFA results benefit from the two layers of containedfluid elements (Fig. 5), which reduce excessive damping by moving the D A A boundary away from the shell's surface.
G.C. Ruzicka, T1- Geers / Transient response analysis of multiple submerged structures
169
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T h e results for e x c i t a t i o n b y the e n d - o n w a v e (Figs. 16 a n d 17) reveal s i m i l a r d e t e r i o r a t i o n in a g r e e m e n t b e t w e e n the USA-STAGS a n d USA--STAGS-CFA results. A s in p r e v i o u s c o m p a r i s o n s , the best a g r e e m e n t occurs w h e r e shell i n t e r a c t i o n is least i m p o r t a n t , viz., at the front o f the f o r w a r d shell a n d at the b a c k o f the r e a r shell. A t the o t h e r two locations, d i s c r e p a n c i e s are p r o n o u n c e d , especially at the front of the r e a r shell (Fig. 17), where USA--STAGS fails to a c c o u n t for e a r l y - t i m e (high-frequency) s h a d o w i n g b y the f o r w a r d shell.
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170
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D i s c r e p a n c i e s b e t w e e n USA-STAGS a n d USA--STAGS--CFA results are also a p p a r e n t in the s i d e - o n w a v e results (Fig. 18), a l t h o u g h , in c o m p a r i s o n w i t h the e n d - o n - w a v e results, t h e y are m o d e s t . A s expected, a g r e e m e n t is best at the front o f each shell, a n d poorer at the b a c k of each shell, w h e r e s o m e m u l t i p l e - s c a t t e r i n g effects m a n i f e s t t h e m s e l v e s . A n interesting feature of the " b a c k " v e l o c i t y histories in Figs. 1 5 - 1 8 is a h i g h - f r e q u e n c y oscillation appearing in the USA--STAGS a n d M o d a l - D A A results d u r i n g the interval I < c t / a < 3 or 3 < c t / a < 5. T h e fact that it takes 20 M o d a I - D A A c i r c u m f e r e n t i a l h a r m o n i c s to capture this
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G.C. Ruzicka, T L. Geers / Transient response analysis of multiple submerged structures
171
oscillation in Figure 15 is evidence of the short-structural-wavelength nature of the phenomenon. The absence of this oscillation from the USA-STAGS-CFA results reveals an important drawback in the use of finite-element grids to propagate transient waves. The USA--STAGS method, on the other hand, does not suffer from this drawback, because the interaction of the structure with the infinite fluid is handled entirely with boundary elements.
Conclusion The USA--STAGS and USA--STAGS-CFA results for transiently excited, multiple, submerged structures are likely to agree satisfactorily when response is dominated by low-frequency components. Agreement is less likely, however, when intermediate- and high-frequency components are significant. In these situations, the use of USA--STAGS--CFAappears to be mandatory, although the presence of important high-frequency components requires refined meshing, which in turn incurs high computational costs. In most cases, it should be possible to assess the applicability of the simpler USA--STAGSassembly to a three-dimensional problem by comparing results obtained from the application of both USA--STAGS and USA-STAGS--CFA to related, two-dimensional evaluation problems. Such a procedure is strongly advised in production analyses.
Acknowledgements This study was performed under Contract DNA001-83-C-0244 with the Defense Nuclear Agency when both authors were at the Lockheed Palo Alto Research Laboratory. The authors express their appreciation for valuable advice and assistance to Drs. John A. DeRuntz and Carlos A. Felippa, who developed the software utilized in the study.
References [1] ALMROTH, B.O., et al., Structural Analysis of General Shells: User Instructions for STAGSC, LMSC-D633873, Lockheed Palo Alto Research Laboratory, Palo Alto, CA, 1980. [2] CrlERXOCK, G., "Transient flexural vibrations of ship-like structures exposed to underwater explosions", J. Acoust. Soc. Am. 48, pp. 170-180, 1970. [3] DI~RUNTZ,J.A. and F.A. BROGAN, Underwater Shock Analysis of Nonlinear Structures, A Reference Manual for the USA-STAGS Code (Version 3), DNA 5545F, Defense Nuclear Agency, Washington, DC, 1980. [4] DERuNTZ, J.A. and T.L GEERS, "Added mass computation by the boundary integral method", Int. J. Num. Methods Eng. 12, pp. 531-550, 1978. [5] DI~RUNTZ, J.A., et al., The Underwater Shock Analysis Code ( U S A - Version 3), DNA 5615F, Defense Nuclear Agency, Washington, DC 1980. [6] FELIPPA, C.A., "Top-down derivation of double asymptotic approximations for structure-fluid interaction analysis,', in Innovative Numerical Analysis for the Applied Engineering Sciences, edited by R.P. SHAW et al., University Press of Virginia, Charlottesville, VA, pp. 79-88, 1980. [7] FELIPPA, C.A. and J.A. DERur~TZ, "Finite element analysis of shock-induced hull cavitation", Comput. Methods Appl. Mech. Eng. 44, pp. 297-337, 1984. [8] FEUPPA, C.A., and K.C. PARK, "Staggered solution transient analysis procedures for coupled mechanical systems: formulation", Comput. Methods Appl. Mech. Eng. 24, pp. 61-111, 1980. [9] GEERS, T.L., "Excitation of an elastic cylindrical shell by a transient acoustic wave", J. Appl. Mech. 36, pp. 459-469, 1969. [10] GEERS, T.L., "Residual potential and approximate methods for three-dimensional fluid-structure interaction problems", J. Acoust. Soe. Am. 49, pp. 1505-1510, 1971. [11] GEERS, T.L., "Scattering of a transient acoustic wave by an elastic cylindrical shell", J. Acoust. Soc. Am. 37, pp. 1091-1106, 1972.
172
G.C. Ruzicka, T.L. Geers / Transient response analysis of multiple submerged structures
[12] GEERS,T.L., "Shock response analysis of submerged structures", Shock Vib. Bull. 44 (Supp. 3), pp. 17-32, 1974. [13] GEERS, T.L., "Transient response analysis of submerged structures", Finite Element Analysis of Transient Nonlinear Structural Behavior, edited by T. BELVTSCHKOet al., AMD-Vol. 14, American Society of Mechanical Engineers, New York, pp. 59-84, 1975. [14] GEERS,T.L., "Doubly asymptotic approximations for transient motions of submerged structures", J. Acoust. Soc. Am. 64, pp. 1500-1508, 1978. [15] GEERS, T.L. and G.C. RUZICKA, "Finite-element/boundary-element analysis of multiple structures excited by transient acoustic waves", Numerical Methods for Transient and Coupled Problems, edited by R.W. LEWIS et al., Pineridge Press, Swansea, UK, pp. 150-157, 1984. [16] MINDUN, R.D., and H.H. BLEICH, "Response of an elastic cylindrical shell to a transverse step shock wave", J. Appl. Mech. 20, pp. 189-195, 1953. [17] NEWTON, R.E., "Finite element analysis of shock-induced cavitation", Preprint 80-110, ASCE Spring Convention, Portland, Oregon, 1980. [18] PARK, K.C., et al., "Stabilization of staggered solution procedures for fluid-structure interaction analysis", Computational Methods for Fluid-Structure Interaction Problems, edited by T. BELYTSCHKOand T.L. GEERS, AMD-Vol. 26, American Society of Mechanical Engineers, New York, pp. 95-124, 1977. [19] ZIENKIEWICZ,O.C., The Finite Element Metho~ McGraw-Hill, London, 1977.