recent developments in fluid-structure interaction modeling and...

Recent Developments in Fluid-Structure Interaction Modeling and Analysis

K. C. Park Center for Aerospace Structures, andDepartment of Aerospace Engineering Sciences University of Colorado at Boulder, CO, USA

Research Collaborators: José A. González Pérez, Carlos Felippa, Roger Ohayon, Among others

Topology op(miza(on of deformable bodies with dissimilar interfaces GE Jeong, SK Youn, KC Park. Computers & Structures 198, 2018, 1-‐11 Minimum influence point method to construct fic((ous frame domain for trea(ng nonmatching interface meshes. GE Jeong, SK Youn, KC Park. Journal of Mechanical Science and Technology 32 (3), 2018, 1253-‐1260.

Virtual gap element approach for the treatment of non-‐matching interface using three-‐dimensional solid elements. YU Song, SK Youn, KC Park Computa(onal Mechanics 60 (4), 2017, 585-‐594

A gap element for trea(ng non-‐matching discrete interfaces YU Song, SK Youn, KC Park. Computa(onal Mechanics 56 (3), 2015, 551-‐563

A simple algorithm for localized construc(on of non-‐matching structural interfaces KC Park, CA Felippa, G Rebel Interna(onal Journal for Numerical Methods in Engineering 53 (9), 2002, 2117-‐2142

Nonmatching Interface

System Identification and Inverse Problems

Xue Yue and K. C. Park (2002), ”Modeling of Joints and Interfaces,” in : Modeling and Simula(on-‐Based Life Cycle Engineering, K. Chong, S. Saigal, S. Thynell and H. Morgan (des.), Spon Press, London, pp.60-‐75.

Reich, G.W., Park, K. C. and Namba, H. (2001), ”Health Monitoring of a Reinforced Concrete ContainmentVessel by Localized Methods,” Proc. of the Third Interna(onal Workshop on Structural Health Monitoring, Technomic Publishing Company, Inc., 2001 Reich, G.W. and Park, K. C. (2001), “A Theory for Strain-‐Based Structural System Iden(fica(on,” in: Journal of Applied Mechanics, 68(4), 521-‐527.

Reich, G.W. and Park, K. C., “On the Use of Substructural Transmission Zeros for Structural Health Monitoring,” AIAA Journal, Vol. 38, No. 6, 2000, 1040-‐1046.

Alvin, K. F. and Park, K. C., “Extrac(on of Substructural Flexibili(es from Global Frequencies and Mode Shapes,” AIAA Journal, vol. 37, no.11, 1999, p. 1444-‐1451.

Park, K. C., Reich, G. W. and Alvin, K. F. “Structural Damage Detec(on Using Localized Flexibili(es,” Journal of Intelligent Material Systems and Structures, Vol. 9, No. 11, 1998, pp. 911-‐919. Park, K. C. and Felippa, C. A., “A Flexibility-‐Based Inverse Algorithm for Iden(fica(on of Structural Joint Proper(es,” ASME Symposium on Computa(onal Methods on Inverse Problems, 15-‐20 November 1998, Anaheim, CA. Reich, G. W. and Park, K. C., “Structural Health Monitoring via Structural Localiza(on,” Proc. 1998 AIAA SDM Conference, Paper No. AIAA-‐98-‐1892, April 20-‐24 1998, Long Beach, CA. Park, K. C., Reich, G. W. and K. F. Alvin, “Damage Detec(on Using Localized Flexibili(es,” in : Structural Health Monitoring, Current Status and Perspec(ves, ed. F-‐K Chang, Technomic Pub., 1997, 125-‐139.

Parallel Computing

Gumaste, Udayan and Park, K. C. (2000), “Interfacing an explicit nonlinear finite element code with an implicit parallel solu(on algorithm,” to be presented at the Interna(onal Congress on Computa(onal Engineering Sciences, August 5-‐8, 2000, Los Angeles, CA. Gumaste, Udayan, Park, K. C. and Alvin, K. F. , “A Family of Implicit Par((oned Time Integra(on Algorithms for Parallel Analysis of Heterogeneous Structural Systems,” Computa(onal Mechanics: an Interna(onal Journal, 24 (2000) 6, 463-‐475. Park, K. C., Gumaste, Udayan, and Felippa, C. A., “A Localized Version of the Method of Lagrange Mul(pliers and its Applica(ons,” Computa(onal Mechanics: an Interna(onal Journal, 24 (2000) 6, 476-‐490.

Park, K. C., Jus(no, M. R, Jr. and Felippa, C. A., “An Algebraically Par((oned FETI Method for Parallel Structural Analysis: Algorithm Descrip(on,” Interna(onal Journal of Numerical Methods in Engineering, 40, 2717-‐2737 (1997). Jus(no, M. R, Jr., Park, K. C. and Felippa, C. A., “An Algebraically Par((oned FETI Method for Parallel Structural Analysis:Implementa(on and Numerical Performance Evalua(on,” Interna(onal Journal of Numerical Methods in Engineering, 40, 2739-‐2758 (1997).

Contact-Impact Problems Y. Miyazaki and K. C. Park, ”A formula(on of conserving impact system based on localized Lagrange mul(pliers,” Interna(onal Journal of Numerical Methods in Engineering, 2006. G. Rebel, K. C. Park and C. A. Felippa (2002), ”A Contact Formula(on Based on Localised Lagrange Mul(pliers: Formula(on and Applica(on to Two-‐dimensional Problems,” Interna(onal Journal of Numerical methods in Engineering, 2002; 54:263-‐297.

G. Rebel and K. C. Park, Applica(on of the Localised Lagrange Mul(plier Method to a 3D Contact Patch Test Proc. 2002 AIAA SDM Conference, Paper No. AIAA-‐2002-‐1577, 22-‐26 April 2002, Denver, CO.

Coupled (Multiphysics) Problems

Park, K. C., Felippa, C. A. and Ohayon, R., “Reduced-‐Order Par((oned Modeling of Coupled Systems: Formula(on and Computa(onal Algorithms,” Mul(-‐physics and Mul(-‐scale Computer Models in Non-‐linear Analysis and Op(mal Design of Engineering Structures Under Extreme Condi(ons (NATO ARW PST.ARW980268), ed. A. Ibrahimbegovic and B. Brank, University of Ljubliana, 2004, 267-‐289.

Park, K. C., Felippa, C. A. and Ohayon, R., “Par((oned Formula(on of Internal Fluid-‐Structure Interac(on Problems via Localized Lagrange Mul(pliers,” Computer Methods in Applied Mechanics and Engineering, 190(24-‐25), 2001, 2989-‐3007. Park, K.C., Felippa, C. A. and Ohayon, R. (2001), “Localized Formula(on of Mul(body Systems,” in: Computa(onal Aspects of Nonlinear Systems with Large Rigid Body Mo(on (ed. J. Ambrosio and M. Kleiber), NATO Science Series, IOS Press, p.253-‐274. Ross, M. R., Coupling and Simula(on of Acous(c Fluid-‐Structure Interac(on Systems Using Localized Lagrange Mul(pliers, Ph.D. Thesis, Department of Aerospace Engineering Science, University of Colorado, 2006.

Par((oned vibra(on analysis of internal fluid-‐structure interac(on problems JA González, KC Park, I Lee, CA Felippa, R Ohayon Interna(onal Journal for Numerical Methods in Engineering 92 (3), 2012, 268-‐300

Par((oned formula(on of internal and gravity waves interac(ng with flexible structures KC Park, R Ohayon, CA Felippa, JA González Computer Methods in Applied Mechanics and Engineering 199 (9-‐12), 2010, 723-‐733

Reduced-Order Modeling A component mode selec(on method based on a consistent perturba(on expansion of interface displacement SM Kim, JG Kim, KC Park, SW Chae Computer Methods in Applied Mechanics and Engineering 330, 2018, 578-‐597 D. Markovic and K. C. Park, Reduc(on of substructural interface degrees of freedom in flexibility-‐based component mode synthesis, to appear in Interna(onal Journal of Numerical Methods in Engineering, 2007. K. C. Park, ”Par((oned formula(on with localized Lagrange mul(pliers and its applica(ons,” in: Structural Dynamics (Eurodyn 2005), Millpress, Roterdam, 2005, pp. 67-‐76.

D. Markovic and K. C. Park, ”Reduc(on of Interface Degrees of Freedom in Flexibility-‐Based Component Mode Synthesis,” Proc. 5th EUROMECH Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 7-‐12, 2005, pp. 900-‐907.

Park K. C. and Park, Yong Hwa, ”Par((oned Component Mode Synthesis via A Flexibility Approach,” AIAA Journal, 2004, vol.42, no.6, 1236-‐1245.

Par11oned Transient Analysis A method for mul(dimensional wave propaga(on analysis via component-‐wise par((on of longitudinal and shear waves SS Cho, KC Park, H Huh Interna(onal Journal for Numerical Methods in Engineering 95 (3), 2013, 212-‐237

Inverse mass matrix via the method of localized lagrange mul(pliers JA González, R Kolman, SS Cho, CA Felippa, KC Park Interna(onal Journal for Numerical Methods in Engineering 113 (2), 2018, 277-‐295 Explicit Mul(step Time Integra(on for Discon(nuous Elas(c Stress Wave Propaga(on in Heterogeneous Solids. Submiked. S. S. Cho, R. Kolman, Jose A. Gonzalez and K. C. Park

More expected to appear . . .

Vibration Control H. Sakamoto, K. C. Park, and Y. Miyazaki, “Distributed and localized ac(ve vibra(on isola(on in membrane structures, submiked to Journal of Spacecral and Rockets, 2005.

Hiraku Sakamoto, K.C. Park and Yasuyuki Miyazaki, ”Distributed Localized Vibra(on Control of Membrane Structures Using Piezoelectric Actuators,”PaperNo. AIAA-‐2005-‐2114, Proc. the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (SDM), 18-‐21 April 2005, Aus(n, TX.

Park, K. C., Kim, N. I., and Reich, G.W., ”A Theory of Localized Vibra(on Control via Par((oned LQR Synthesis,” Paper No. 3984-‐63, Proc. 2000 Smart Structures and Materials Conference: Mathema(cs and Control in Smart Structures, Newport Beach, CA, March 6-‐9, 2000.

BEM-BEM and BEM-FEM ModelingJ. A. Gonz´alez, K. C. Park and C. A. Felippa, A formula(on based on localized Lagrange mul(pliers for BEM-‐FEM coupling in contact problems, submiked to Interna(onal Journal of Numerical Methods in Engineering,

J. A. Gonz´alez, K. C. Park and C. A. Felippa, ”Par((oned formula(on of fric(onal contact problems,” Comm. Num. Meth. Engr., Volume 22, Issue 4, 2006, 319-‐333 J. A. Gonzalez, K. C. Park and C. A. Felippa, FEM and BEM coupling in elastosta(cs using localized Lagrange mul(pliers, Interna(onal Journal of Numerical Methods in Engineering, Volume 69, Issue 10, 2007, 2058-‐2074. MISC Topics Eui-‐Il Jung, Youn-‐Sik Park and K. C. Park, ”Structural Dynamics Modifica(on via Reorienta(on of Modifica(on Elements, Finite Element Analysis and Design, 42(1),2005, 50-‐70.

Park, Y.H and Park, K. C., “Anchor Loss Evalua(on of MEMS Resonators -‐ I: Energy Loss Mechanism through Substrate Wave Propaga(on,” Journal of Microelectromechanical Systems, Vol. 13, No. 2, 2004, 238-‐247. Park, Y.H and Park, K. C., “Anchor Loss Evalua(on of MEMS Resonators -‐ II: Coupled Substrate-‐REsonator Simula(on and Valida(on,” Journal of Microelectromechanical Systems, Vol. 13, No. 2, 2004, 248-‐257.

Park, K. C., “Par((oned Solu(on of Reduced Integrated Finite Element Equa(ons,” Computers & Structures, 74 (2000) 281-‐292.

Part I: Internal Flow Problems

Interface Conditions for FSI Problem

K. C. Park’s Involvement in FSI began in 1976.

Summary of our 1977 Paper: Stabilize Staggered Solution of the External Acoustic-Structure Interaction Equation

By the following augmented form:

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K. C. Park returns to the wonderful world of FSI after the 20-year dormant period

Thus, an era of three musketeers began in earnest!

Essence of our 2001 CMAME Paper

Adoption of Primitive Variable (Displacement) via Continuum Modeling of Fluids (Gotten away from the displacement potential formulation)

Sparsity of both Inertia and Stiffness Matrices

Partitioned Formulation of Fluid and Structural Governing Equations via the Method of Localized Lagrange Multipliers

The Resulting FSI Equations are Amenable to FETI-Like Scalable Parallel Solution

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2009; 77:1072–1099Published online 28 August 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2443

The d’Alembert–Lagrange principal equations and applicationsto floating flexible systems

K. C. Park1,∗,†, Carlos A. Felippa1 and Roger Ohayon2

1Center for Aerospace Structures, University of Colorado, Boulder, CO 80309-0429, U.S.A.2Structural Mechanics and Coupled Systems Laboratory, Conservatoire National des Arts et Metiers (CNAM),

C.C. 353, 292 rue Saint-Martin, 75141 Paris Cedex 03, France

SUMMARY

This paper addresses the dynamics and quasi-statics of floating flexible structures as well as extensionsto unconstrained substructures and partitions of coupled mechanical systems. The principal solution isdefined as the state of self-equilibrated forces obtained as the particular solution of the rigid motion andinterface equilibrium equations. This solution is independent of the stress–strain constitutive propertiesas well as of the compatibility equations. For statically determinate systems, the principal solution is thefinal force solution. For statically indeterminate systems, the correction due to flexibility and compatibilityis orthogonal to the principal solution. The formulation is done in the context of d’Alembert’s principle,which supplies the d’Alembert–Lagrange principal equations for floating bodies. These are obtained bysummation of virtually working forces and moments acting on the floating systems. Applications ofthis approach are demonstrated on a set of dynamic and quasi-static example problems of increasinggenerality. Linkage to variational principles with an interface potential is eventually discussed as providingthe theoretical foundation for handling interacting semi-discrete subsystems linked by node-collocatedLagrange multipliers. Copyright � 2008 John Wiley & Sons, Ltd.

Received 11 January 2008; Revised 9 July 2008; Accepted 10 July 2008

KEY WORDS: dynamics; d’Alembert’s principle; floating systems; principal equations; Lagrange multi-pliers; partitioned analysis

1. INTRODUCTION

In presentations dealing with motions of fully unconstrained fluid and solid systems, the authorshave often been asked whether a floating, flexible body under dynamic loads is in an equilibriumstate, or whether an unsupported structure such as a flying aircraft maintains equilibrium as itmoves and deforms. Doubts on such a fundamental subject suggest a gap in the teaching of classical

∗Correspondence to: K. C. Park, Center for Aerospace Structures, University of Colorado, Boulder, CO 80309-0429,U.S.A.

†E-mail: [email protected]

Copyright � 2008 John Wiley & Sons, Ltd.

Partitioned formulation of internal and gravity waves interactingwith flexible structures

K.C. Park a,b,*, R. Ohayon c, C.A. Felippa b, J.A. González d

a Division of Ocean Systems Engineering, School of Mechanical Engineering, KAIST, Daejeon 305-701, Republic of Koreab Department of Aerospace Engineering Sciences, University of Colorado at Boulder, CO 80309-429, USAc Chair of Mechanics, Structural Mechanics and Coupled System Laboratory, Conservatoire National des Arts et Métiers (CNAM), 75003 Paris, Franced Departamento de Ingeniería del Diseño, Escuela Técnica Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain

a r t i c l e i n f o

Article history:Received 12 July 2009Received in revised form 4 October 2009Accepted 5 November 2009Available online 12 November 2009

Keywords:Internal acoustic and gravity wavesFluid–structure interactionPartitioned FSI formulation

a b s t r a c t

This paper presents a partitioned modeling of internal and gravity fluid waves that interact with flexiblestructures. The governing interaction model consists of three completely partitioned entities: fluid model,structural model, and interface model that acts as an internal constraint on the fluid–structure interfaceboundary. Thus, the proposed partitioned multi-physics modeling can employ two completely modularfluid and structure software modules plus an interface solver, hence amenable to partitioned solutionalgorithms. The interface discretization can exploit the nonmatching interface algorithm previouslydeveloped via the method of localized Lagrange multipliers. Also noted is that the present fluid modelcan make use of widely available finite element software for standard Poisson-type problems.

! 2009 Elsevier B.V. All rights reserved.

1. Introduction

Fluid–structure interaction (FSI) phenomena have recentlyemerged as one of the most widely encountered multi-physicsproblems in science and engineering. As a result, various special-ized FSI formulations have been developed and successfully ap-plied to problems involving internal fluid problems [1–22],external fluids problems [23–40], and recently biomechanics[41–46], among others. Interested readers may consult a reviewby Dowell and Hall [47] for general FSI problems viewed fromthe fluid mechanics context, by Tijsseling [48] for piping flow,and by de Boera et al. [49] for various interface coupling methods.From the viewpoint of formulation, modeling, discretization andnumerical solution, a wide range of computational procedureshave been developed over the past three decades. They range fromtightly-to-loosely coupled to locally partitioned [51–70]. For exam-ple, FSI problems of blood flow typically adopt tightly coupledformulation and solution procedures [41], whereas aeroelasticityproblems employ partitioned solution procedures [71]. The viewof present authors is that, as much as possible, the task formulti-physics simulation is facilitated by adopting partitionedsolution procedures. Among the beneficial sides of invokingpartitioned solution procedures, we mention substantial reduc-tions, both in development time and cost, of the development of

single-discipline oriented analysis software, upgrading ease andsimplified maintenance, and the efficient use of discipline-specificspecialists.

This has motivated us to undertake a series of critical revision ofFSI problems and, if necessary and/or possible, to reformulate FSIproblems such that the resulting form may facilitate the treatmentof partitioned solution procedures. Because of our background andexperience, we begin with the reformulation of a flexible structureinteracting with internal compressive fluids including gravityeffects while deferring reformulations of other FSI problems to alater exposition. It should be pointed out that we focus solely onFSI formulations with small displacements and the method of local-ized Lagrange multipliers, although not necessarily restricted to lin-ear, that leads naturally to partitioned solution procedures for theremainder of the paper. Readers interested in other formulationssuch as Eulerian–Lagrangian approach, fictitious/mortar elementapproach may consult recent articles [70,66,61] and referencestherein. To this end, we offer a review of existing FSI formulationsof internal waves with gravity and free surface interacting withflexible structures.

The governing equations of motion for inviscid internal fluidscontained by flexible structures often rely on the so-called excesspressure [75] or modified pressure in [76] defined as follows:‘‘if the absolute pressure occurs in the boundary conditions, ashappens if part of the boundary is an interface with another fluidsor if it is a free surface, . . .the effect of gravity reenters the prob-lem.” Thus, the modified pressure is the difference between thetotal pressure and the gravity-induced pressure. This concept is

0045-7825/$ - see front matter ! 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.cma.2009.11.005

* Corresponding author. Address: Division of Ocean Systems Engineering, Schoolof Mechanical Engineering, KAIST, Daejeon 305-701, Republic of Korea.

E-mail addresses: [email protected], [email protected] (K.C. Park).

Comput. Methods Appl. Mech. Engrg. 199 (2010) 723–733

Contents lists available at ScienceDirect

Comput. Methods Appl. Mech. Engrg.

journal homepage: www.elsevier .com/locate /cma

Professor José González expands the trio-membership to four

the gravitational acceleration; and k is the upward unit vector alongthe vertical direction, that is, the Z-direction.

It is emphasized that, in contrast to [13], both pac and pgr are ex-pressed in a Lagrangian frame. A first-order expansion of the firstPiola-Kirchhoff stress tensor T gives:

T ! "ð1þr % uf ÞPI3 þ Pðruf Þ! "ð1þr % uf ÞPI3 þ Pðdiag½ruf (Þ; ð12Þ

where the replacement of ðruf ÞT by diag½ðruf ÞT ( is effected byinviscid assumption in which case fluid experiences no resistanceto shearing strains.

At this juncture it should be mentioned that there are two pathsby which one can carry out variational process and subsequentlydiscretize the resulting variational equation to obtain the discreteequations of motion. One is to obtain r % T and substitute into(10)1. Then carry out integration by part only for terms involvingrpac . This is the path taken in [77,78]. In the present paper, we pro-ceed along the lines of solid mechanics and the integration by partsto arrive at the virtual energy density for fluid that is analogous tothe term ðr : !Þ used for finite element discretization in solidmechanics. To this end, by using the formula

ðr0 % TÞ % du ¼ r % ðT % duÞ " TT : r0du; ð13Þ

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V0

ðr0 % TÞ % duf dV0 ¼Z

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Z

V0

TT : r0duf dV0:

ð14Þ

In the above equation, the first free surface integral represents sur-face traction energy while the second one is the internal energy. Asnoted in the beginning of Section 2, we treat the fluid–structureinterface as internal constraint, not as a boundary condition for thefluid. This is another contrast with classical fluid formulations thattreat the fluid surface contacting the structure by a wall boundarycondition.

4.2. Surface energy model

On the free surface we have

P ¼ pa þ rsr % n ¼ pa þ q0gk % uf ; ð15Þ

where pa the atmospheric pressure acting on the surface of thefluid; rs represents the surface tension (for water rs ! 70dynes=cm); and the well-known Young–Laplace equation thatrelates the surface tension to the gravity force is used. It should benoted that the preceding model is valid in principle for flat sur-faces. As the surface of each discretized surface element may be as-sumed to be flat even though the overall surface may be curved, we

are permitted to employ the flat surface hypothesis. Substituting theabove relation into the first of (14) leads toZ

Cf

ðT % duf Þ % ndC ¼ "Z

Cf

paðn % duf ÞdC"Z

Cf

q0gðk % uf Þðn % duf ÞdC:

ð16Þ

It should be noted that the present surface tension energy does notaccount for surface curvature effects as detailed in Landau [74] andLighthill [75]. However, its effect is known to be within a few per-centage error for waves whose length exceeds 0.1 m.

4.3. Internal energy model

Inserting the constitutive relation (11) into the stress tensor(12) gives

TT ¼ "ð1þr % uf Þð"q0c2r % uf þ pgrÞI3 þ ðpac þ pgrÞdiagðruf Þ:ð17Þ

The internal fluid energy density (TT : rduf ) is thus obtained as

TT :rduf ¼"ð1þr %uf Þð"q0c2r %uf þpgrÞr % duf

þðpacþpgrÞX3

i¼1

@ui

@Xi% @dui

@Xið18Þ

pgr ¼ q0gðh" Z " uf % kÞ; pac ¼ "q0c2r % uf :

The internal energy is thus obtained by integrating over the fluidvolume asZ

Vf

TT :rduf dV

¼Z

Vf

ðr %uf Þðq0c2Þðr % duf ÞdV "Z

Vf

pgrðr % duf ÞdV

"Z

Vf

ðr %uf ÞðpgrÞðr % duf ÞdV þZ

Vf

X3

i¼1

@ui

@Xi

! "ðpgrÞ

@dui

@Xi

! "dV

"Z

Vf

ðr %uf ÞðpacÞðr % duf ÞdV

þZ

Vf

X3

i¼1

@ui

@Xi

! "ðpacÞ

@dui

@Xi

! "dV : ð19Þ

4.4. Density stratified fluids

If the fluid density qðzÞ varies along the z-axis, the second termin the foregoing equation becomes

Fig. 3. Partitioning of internal fluid–structure interaction system.

726 K.C. Park et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 723–733

Enforcement of the above interface constraint is realized by theclassical Lagrange multipliers method, resulting in a functionalform

pc ¼Z

Cint

kfscfs dS: ð3Þ

In passing, it should be noted that in most existing formulations[13,77,78] the fluctuation pressure on the boundary is used to formthe interface constraint functional

pcon ¼Z

Cint

pecfs dS; ð4Þ

for which pe denotes the fluctuation pressure, and the displace-ments ðuf ;usÞ are to be interpreted as perturbation quantities, notthe total fluid and structural displacements.

In the present work we introduce a localized partition or frameas shown in Fig. 2. As a consequence, fluid and structure do notinteract directly but with the reference interface. This may be ex-pressed as

Fluid interface constraint : cf ¼ ðuf $ ubÞ % n ¼ 0;Structure interface constraint : cs ¼ ðus $ ubÞ % n ¼ 0;

ð5Þ

where ub is the frame displacement treated as an independent dis-placement variable. It will be shown that the interface forces andmoment balance equations are obtained with respect to this framedisplacement, a feature that can be exploited both for solution reg-ularization and software modularity. The resulting constraint func-tional thus requires two independent Lagrange multipliers (seeFig. 3):

p‘ðuf ;us;ub; kf ; ksÞ ¼Z

Sf

kTf cf dSþ

Z

Ss

kTs cs dS: ð6Þ

The first variation of the fluid–structure interface constraint in-volves five variables:

dp‘ðuf ;us;ub;kf ;ksÞ ¼þZ

Sf

dkTf ðuf $ubÞ %ndSþ

Z

Ss

dkTs ðus$ubÞ %ndS

þZ

Sf

kTf duf %ndSþ

Z

Ss

kTs dus %ndS

$Z

Sf

ðkTf þ kT

s Þdub %ndS: ð7Þ

If the interface geometry is allowed to vary, the following termsmust be added as discussed in [13,77]:

@p‘

@ndn ¼

Z

Sf

kTf ðuf $ ubÞ % dndSþ

Z

Ss

kTs ðus $ ubÞ % dn dS

in which dn ¼ @n@u

du: ð8Þ

In the present paper we will replace n by an averaged value for eachdiscrete interface segment or interface element, nav . This normal isnot subject to variation. Consequently, (7) becomes

dp‘ðuf ;us;ub;kf ;ksÞ ¼þZ

Sf

dkTf ðuf $ubÞ %nav dSþ

Z

Ss

dkTs ðus$ubÞ %nav dS

þZ

Sf

kTf duf %nav dSþ

Z

Ss

kTs dus %nav dS

$Z

Sf

ðkTf þ kT

s Þdub %nav dS: ð9Þ

The preceding variational functional constitutes one of the threevariational expressions for the derivation of the partitioned fluid–structure interaction equation set. The remaining two are the vir-tual work of the fluid domain and that of the flexible structure do-main. Their derivations are discussed in the subsequent sections.

4. Variational formulation of internal acoustics and gravitywaves

4.1. Virtual work for fluid

The formulation of internal and gravity fluid waves have beenstudied by many investigators [73,72,75,13], among others. Forthe present purpose, we will assume the flow to be inviscid and be-gin with the following Lagrangian virtual work principle statedover the fluid volume Vf :

dPf ¼Z

Vf

fr0 % Tþ b0 $ q0 €uf g % duf dV0 ¼ 0; T ¼ J F$1 % r;

F ¼ rxf ¼ Iþruf ; xf ¼ Xþ uf ; X ¼ Xiþ Yjþ Zk;uf ¼ u1iþ u2jþ u3k ¼ uiþ vjþwk;

r0 ¼@

@Xiþ @

@Yjþ @

@Zk; J ¼ detðFÞ ' 1þr % uf : ð10Þ

In the above variational equation, we assume that both the pre-scribed traction and displacement boundary conditions are exactlysatisfied; T and r are the first Piola–Kirchhoff stress and the Cauchystress tensor, respectively; b0 is the body force; q0 is the mass den-sity; €uf is the fluid particle acceleration; X and uf refer to the initialconfiguration and the fluid displacement, respectively; and sub-script 0 denotes the initial configuration. In the above variationalform, we omitted the convective term and the viscosity term fromthe standard Navier-Stokes equations for the fluid as usually thecase with modeling of internal waves.

As we will see shortly, the starting variational equations bothfor the fluid (10)1 and for the structure are the same. It is in theuse of constitutive relations that will lead to fluid or solid model.In the present study we take the fluid stress tensor as modeled by

r ¼ $PI3; P ¼ pac þ pgr ; pac ¼ $q0ðzÞc2r % uf ;

pgr ¼ qðzÞgðh$ zÞ; z ¼ Z þ k % uf ; ð11Þ

where I3 is the (3 ( 3) identity matrix; P is the total pressure; pac isknown as the fluctuation pressure that causes acoustic waves; pgr isthe pressure due to gravity that causes gravity waves; pa is theatmospheric pressure; c is the speed of sound of the fluid; h is thedepth of the fluid measured from the free surface to the bottomof the fluid that is taken as the origin of the Z-coordinate, some-times referred to as hydraulic head; z is the vertical coordinate atthe fluid particle of interest; uf is the particle displacement; g isFig. 2. Localized treatment of the partition interface frame.

K.C. Park et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 723–733 725

Problem Description

Inertia force:Z

Vf

⇢0u · �uf

dV

Acoustic Sti↵ness: +Z

Vf

(r · uf

)(⇢0c2)(r · �u

f

) dV

Vaisala-Brunt sti↵ness: + 12

Z

Vf

⇢0(z)[N ]2�(k · uf

)]2 dV

Gravity geometric sti↵ness 1: �Z

Vf

(r · uf

)(pgr

)(r · �uf

) dV

Gravity geometric sti↵ness 2: +Z

Vf

3X

i=1

(@u

i

@X

i

)(pgr

)(@�u

i

@X

i

) dV

Acoustic geometric sti↵ness 1: �Z

Vf

(r · uf

)(pac

)(r · �uf

) dV

Acoustic geometric sti↵ness 2: +Z

Vf

3X

i=1

(@u

i

@X

i

)(pac

)(@�u

i

@X

i

) dV

Surface sti↵ness: +Z

Sf

⇢0g(k · uf

)(k · �uf

) dS

Body force: =Z

Vf

f(t) · �uf

dV

Atmospheric pressure: �Z

Sf

p

a

n · �uf

dS

Density stratification: +Z

Vf

⇢

00gZ

0(k · �uf

) dV

Vaisala-Brunt frequency: [N ] =

vuut� g

2

c

2(z)� g

⇢

00

⇢0

(26)

where the pressure due to gravity p

gr

and the acoustic pressure p

ac

are givenin (18). For those who are not familiar with fluid formulations that containinitial pressure terms in the foregoing formulation, we note that if a full New-ton solution iteration is adopted, then the initial pressures would be updatedduring the iteration process. On the other hand, if a modified Newton iterationis used, then the initial pressures would be from the last time step values.

The present partitioned equation(26) for internal and gravity waves possessesseveral noteworthy features:

• The present formulation embodies purely acoustic waves, purely gravitywaves, their combined e↵ects, depending on which of the sti↵ness terms areretained;

• All sti↵ness terms are quadratic in uf

, which leads to a symmetric sti↵nessmatrix;

• In keeping with the geometric sti↵ness concept employed in structural mod-

13

eling of initial stresses, the present equation brings along the gravity pressureacting as an initial stress for modeling acoustic waves;

• When the acoustic pressure(pac

) is su�ciently large, it acts as an initialacoustic geometric pressure. This may serve well for modeling of nonlinearwaves whose group velocity is su�ciently di↵erent from the speed of fluid.

• It should be noted that the above fluid equation can account for cavitationmodels which can be important for containment vessels, especially carryinggaseous fluids. This can be realized by noting p

ac

= �⇢c

2r · uf

for whichr · u

f

takes on the positive value instead of negative value for compressionstate.

• As discussed in [18], the preceding fluid displacement model may be trans-formed into a pressure-based model by replacing p

ac

from the relation:

p

ac

= �⇢c

2r · uf

, subject to: curl(uf

) = 0 (27)

• The present formulation models the density stratification as indicated bythe Vaisala-Brunt sti↵ness term. When there is no noticeable stratification,viz., ⇢0 ⇡ 0, the term can be ignored as g2/c2 ⇡ 8.3 ⇤ 10�4 for air and g

2/c

2 ⇡4.3 ⇤ 10�5.

4.7 Comparison with Existing Formulations

In the work of Andrianarison and Ohayon [78] based on the Lighthill model ofgravity and compressibility interaction contributions [75], they reported thefollowing formulations (see Eq. 19 therein):

Inertia force:Z

Vf

⇢0u · �uf

dV

Acoustic Sti↵ness: +Z

Vf

(r · uf

)(⇢0c2)(r · �u

f

) dV

Partial Vaisala-Brunt sti↵ness: + 12

Z

Vf

⇢0(z)[�⇢

0g]�(k · u

f

)]2 dV

Gravity gradient: +Z

Vf

⇢0 �[(g · uf

)(r · uf

)] dV

Surface sti↵ness: +Z

Sf

⇢0(g · �uf

)(k · �uf

) dS

Body force: =Z

Vf

f(t) · �uf

dV

(28)

14

Previous Formulation##Our 2010 Formulation$$

## In previous formula(ons, the geometric s(ffness matrices due to fluid pressure are incorporated into the structural ini(al stress s(ffness matrix via boundary integral mapping, thus effec(vely internally coupling the fluid pressure into the structural equa(ons.

$$ The structural equations do not involve fluid pressure, thus accomplishing a complete modular partitioned formulation.

dard isoparametric basis functions and the pressure is sampled atthe Barlow points. That is, for constant-strain elements at the ele-ment centroid. The resulting discretization of the variational fluidEq. (26) can be stated as

d ~Pf ¼ duf ðf f #Mf €uf # Kf ðuf ;pac; pgrÞ uf Þ ð42Þ

It should be noted that the fluid stiffness matrix, Kf ðuf ;pac;pgrÞ, con-sists of the acoustic stiffness, Väisälä-Brunt stiffness for stratifyingfluids, geometric stiffness due to gravity pressure, and geometricstiffness due to acoustic pressure.

6.3. Discretization of structural equation

As stated, it is a standard practice to obtain the discrete versionof the linearized variational equation for structure (38) as

d ~Plins ¼ dusðfs #Ms €us # KsðrÞ usÞ ð43Þ

where the stiffness matrix, KðrÞ, consists of the material and geo-metric stiffness matrices as discussed in (38).

6.4. Discrete partitioned fluid–structure interaction equations

The coupled partitioned fluid–structure interaction model cannow be constructed by the following variational statement:

dPtotal ¼ dp‘ þ dPf þ dPs ¼ 0 ð1Þ

Inserting the discrete variational internal constraint (41), the dis-crete variational fluid Eq. (42) and the discrete variational structuralEq. (43) into the foregoing equation, the stationarity of the resultingexpression yields the following equation set:

Kf þMfd2

dt2 0 ~BTf 0 0

0 Ks þMsd2

dt2 0 ~BTs 0

~Bf 0 0 0 #~Lf

0 ~Bs 0 0 #~Ls

0 0 #~LTf #~LT

s 0

2

666666666664

3

777777777775

uf

us

kf

ks

ub

2

6666666664

3

7777777775

¼

f f

fs

0

0

0

2

6666666664

3

7777777775

ð44Þ

7. Vibration and transient analysis by present partitionedfluid–structure interaction equations

Efficient algorithms exist for the transient analysis of the abovepartitioned multi-physics models [96,97,59,60,63] and for vibra-tion analysis including reduced-order models [98,99]. While wedefer detailed aspects of computational procedures and numericalexperiments for a later exposition, we briefly discuss several spe-cial analyses that can accrue from the above formulation (44).

7.1. Vibration analysis of FSI systems

Eq. (44) can be specialized to vibration formulation by taking

d2

dt2 ¼ #x2; with fs ¼ f f ¼ 0 ð45Þ

whose substitutions leads to

Kf #x2Mf 0 ~BTf 0 0

0 Ks #x2Ms 0 ~BTs 0

~Bf 0 0 0 #~Lf

0 ~Bs 0 0 #~Ls

0 0 #~LTf #~LT

s 0

2

666666664

3

777777775

uf

us

kf

ks

ub

2

66666664

3

77777775

¼

0

0

0

0

0

2

66666664

3

77777775

ð46Þ

An efficient flexibility-based vibration analysis technique includingsubstructuring that is well suited to treat the above vibration modelis discussed in [98–100].

7.2. Transient analysis of FSI systems

There are three modes of transient analysis utilizing the presentpartitioned FSI formulation (44): explicit-explicit (meaning explicitintegration for both fluid and structural partitioned equations), ex-plicit–implicit and implicit–implicit integration. We will describethe implicit–implicit integration procedure and show that the ex-plicit-explicit and explicit–implicit procedure follows by extrapo-lating the stiffness force terms. For illustration purposes, weemploy the implicit-mid-point rule for both the fluid and struc-tural equations of motion:

_unþ12 ¼ _un þ d€unþ1

2; d ¼ 12Mt

unþ12 ¼ un þ d _unþ1

2

+

unþ12 ¼ hnþ1

2u þ d2 €unþ1

2; hnþ12

u ¼ un þ d _un

ð47Þ

where Mt is the step size. Once unþ12 is obtained, unþ1 can be ob-

tained from

unþ1 ¼ 2unþ12 # un ð48Þ

Substituting into (44) we obtain the following time-discretizedequation:

ð 1d2 Mf þ Kn

f Þ 0 ~BTf 0 0

0 ð 1d2 Ms þ Kn

s Þ 0 ~BTs 0

~Bf 0 0 0 #~Lf

0 ~Bs 0 0 #~Ls

0 0 #~LTf #~LT

s 0

2

666666664

3

777777775

uf

us

kf

ks

ub

2

6666664

3

7777775

nþ12

¼

f f þ 1d2 Mf huf

fs þ 1d2 Mshus

000

2

6666664

3

7777775

nþ12

ð49Þ

where the stiffness matrices (Kf ;Ks) are approximated by using thedisplacement, pressures and stresses at the nth step values.

The numerical solution of the above discrete equation canbe effected by employing a parallel solution algorithm de-scribed in [60]. Alternatively, one may employ a more matureFETI-DP or its allied methods [62], by solving for ðuf ;usÞ first,then projecting out the frame displacement ub except theso-called cross points interface degrees of freedom, and thelocalized Lagrange multipliers are transforming the presentlocalized Lagrange multipliers to the classical global Lagrangemultipliers as detailed in [57]. There exist a plethora of alliedmethods labeled as semi-implicit algorithm (see, e.g., Sy andMurea [101]) that do need to satisfy the interface compatibilityconstraints at each time step. This may present fruitful avenuefor further study.

7.2.1. Explicit–implicit transient analysis procedureFor explicit–implicit procedure, i.e., integrating the fluid equa-

tions by an explicit integration formula and the structural equa-tions by an implicit formula, all one needs to do is to transfer thefluid stiffness force term to the right-hand side with the displace-ment replaced by a predictor. This is illustrated in the equationbelow.

730 K.C. Park et al. / Comput. Methods Appl. Mech. Engrg. 199 (2010) 723–733

What the new formulations accomplished is to remove fluid pressure-dependentStructural stiffness expression

To the localized Lagrange multipliers, λf :

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2012; 92:268–300Published online 12 June 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4336

Partitioned vibration analysis of internal fluid–structureinteraction problems

José A. González1, K. C. Park2,4,*,†, I. Lee3, C. A. Felippa4 and R. Ohayon5

1Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, E-41092 Seville, Spain2Division of Ocean Systems Engineering, KAIST, Daejeon 305-701, Korea

3Division of Aerospace Engineering, KAIST, Daejeon 305-701, Korea4Department of Aerospace Engineering Sciences and Center for Aerospace Structures, University of Colorado, Campus

Box 429, Boulder, CO 80309, USA5Chair of Mechanics, Structural Mechanics and Coupled System Laboratory, Conservatoire National des Arts et

Métiers (CNAM), 75003 Paris, France

SUMMARY

A partitioned, continuum-based, internal fluid–structure interaction (FSI) formulation is developed for mod-eling combined sloshing, acoustic waves, and the presence of an initial pressurized state. The presentformulation and its computer implementation use the method of localized Lagrange multipliers to treatboth matching and non-matching interfaces. It is shown that, with the context of continuum Lagrangiankinematics, the fluid sloshing and acoustic stiffness terms originate from an initial pressure term akin to thatresponsible for geometric stiffness effects in solid mechanics. The present formulation is applicable to bothlinearized vibration analysis and nonlinear FSI transient analysis provided that a convected kinematics isadopted for updating the mesh geometry in a finite element discretization. Numerical examples illustrate thecapability of the present procedure for solving coupled vibration and nonlinear sloshing problems. Copyright© 2012 John Wiley & Sons, Ltd.

Received 11 September 2011; Revised 3 January 2012; Accepted 18 March 2012

KEY WORDS: acoustic waves; gravity waves; sloshing; vibration; fluid–structure interaction; partitionedanalysis; localized Lagrange multipliers

1. INTRODUCTION

The problems addressed in this paper have important engineering applications. Examples are liqui-fied natural gas carriers, liquid propellant launchers, fuel tanks in airplanes, satellites and automo-biles, large containers under seismic action, and dynamics of trapped water on the deck of offshorevessels and platforms. According to Ibrahim [1], the first reported work on sloshing was by Hough[2], who investigated the dynamics of a rotating ellipsoidal shell containing fluid. Since that earlywork, sloshing phenomena have remained a major design consideration in the aerospace and navalarchitecture fields. Interested readers may consult recent monographs and texts [1, 3–5] and reviewarticles [6–8], among others. It should be noted that the bulk of the sloshing problems considered sofar deal with rigid containers.

For sloshing motions of fluid in partially filled flexible containers, various formulations and solu-tion methods have been proposed, which include the following: velocity potential for the fluid andmodal superposition for the structure [9, 10], Boundary integral for the fluid and modal superpo-sition/FEM for the structure [11–16], Eulerian–Lagrangian equations for the fluid and the finite

*Correspondence to: K. C. Park, Division of Ocean Systems Engineering, KAIST, Daejeon 305-701, Korea.†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

Professor José González Expands the trio-membership to four – cont’d

COUPLED INTERNAL FLUID–STRUCTURE INTERACTION PROBLEMS 273

4.3. Fluid energy functional

The energy functional for the fluid is the total potential energy, written as

…f.uf/DZVf

Uf.Jf/ dV !…extf .uf/ (7)

where the volumetric strain-energy defined as

Uf.Jf/D1

2!f.Jf ! 1/2 (8)

is only function of the volumetric deformation with a volumetric stiffness !f . The potential energydue to external forces, written in d’Alembert form with the inertial forces introduced as modifiedbody forces, is expressed as

…extf .uf/D

ZVf

!"0f g! "0f Ruf

"" uf dV C

Z@Vf

Tf " uf dA (9)

where Ruf is the fluid acceleration, Tf is the surface traction vector, Vf represents the fluid domain,and @Vf is its physical boundary.

The second Piola–Kirchhoff stress tensor is obtained from the strain-energy function

Sf D [email protected]/

@CfD pfJfC!1f (10)

where

pf [email protected]/

@JfD !f.Jf ! 1/ (11)

is the acoustic fluid pressure. The Cauchy stress tensor is then obtained from Sf using the pushforward operation

! f D J!1f FfSfFTf (12)

and expressed in the deformed configuration. For an inviscid fluid, ! f presents only hydrostaticcomponent and can be expressed as

! f D pfI, (13)

a diagonal tensor in the current configuration.The principle of virtual work comes from the stationarity condition of (7), which, when written

in the initial configuration, reads as

ı…f.uf , ıuf/DZVf

Sf W ıEf dV ! ı…extf .uf , ıuf/ (14)

ı…extf .uf , ıuf/D

ZVf

!"0f g! "0f Ruf

"" ıuf dV C

Z@Vf

Tf " ıuf dA (15)

expressed in terms of the varied Green–Lagrangian strain Ef with its conjugate stress measure Sf

and a variation of external energy containing the external traction vector Tf D tf dadA per unit of

initial area.The next step is to proceed with the linearization of (14) around the initial hydrostatic equilibrium

state, to derive the equations of motion for the fluid

Dı…f.uf , ıuf/DZVf

ıEf WCf WDEf dV CZVf

STf W#rT0ufr0ıuf

$dV !Dı…ext

f (16)

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2012; 92:268–300DOI: 10.1002/nme

Distinct Feature of our 2012 Work: Nonlinear Continuum Constitutive Laws




…f.uf/DZVf



Uf.Jf/D1

2!f.Jf ! 1/2 (8)


…extf .uf/D

ZVf

!"0f g! "0f Ruf

"" uf dV C

Z@Vf

Tf " uf dA (9)




@CfD pfJfC!1f (10)

where


@JfD !f.Jf ! 1/ (11)




! f D pfI, (13)






ZVf

!"0f g! "0f Ruf

"" ıuf dV C

Z@Vf

Tf " ıuf dA (15)







STf W#rT0ufr0ıuf

$dV !Dı…ext

f (16)


Distinct Feature of our 2012 Work: Nonlinear Continuum Constitutive Laws – Cont’d





…f.uf/DZVf



Uf.Jf/D1

2!f.Jf ! 1/2 (8)


…extf .uf/D

ZVf

!"0f g! "0f Ruf

"" uf dV C

Z@Vf

Tf " uf dA (9)




@CfD pfJfC!1f (10)

where


@JfD !f.Jf ! 1/ (11)




! f D pfI, (13)






ZVf

!"0f g! "0f Ruf

"" ıuf dV C

Z@Vf

Tf " ıuf dA (15)







STf W#rT0ufr0ıuf

$dV !Dı…ext

f (16)


274 J. A. GONZÁLEZ ET AL.

where it is important to note that uf now represents a displacement from equilibrium, as representedin Figure 3. We can identify in the first volume integral the constitutive term that gives place to theconstitutive stiffness matrix, with a Lagrangian constitutive tensor given by

Cf D Cp C C! (17)

Cp D pfJf

!C!1f ˝C!1f ! 2I f

", C! D !fJ 2f C!1f ˝C!1f

where Cp comes from the variation of kinematic variables and C! from the variation of pressure.Second volume integral represents the initial-stress term, which now includes the initial hydrostaticpressure configuration of the fluid.

The linearized equations of motion (16) can be expressed in the current configuration with asimilar form

Dı…f.uf , ıuf/DZvf

ıdf W cf Wdf dvCZvf

pTf IW

!rTufrıuf

"dv !Dı…ext

f (18)

using the spatial Cauchy stress ! f and deformation df work conjugate tensors, together with thefourth-order constitutive tensor

cf D cp C c! (19)

cp D pf ŒI˝ I! 2i " , c! D !fJfI˝ I

with I˝ ID ıij ıkl and i D 12 .ıikıjl C ıilıjk/. In the current configuration, the initial-stress term

contains the total Lagrangian pressure pTf D p0f C pf .

Linearization of the external virtual work (15) provides

Dı…extf .uf , ıuf/D !

Zvf

#f Ruf " ıuf dvCD#Z

@vf

tf " ıuf da$

(20)

where gravitational body forces, initially present in (15), are invariably independent of the motionand consequently do not contribute to the linearized virtual work. Hence, only inertia forces andexternal tractions are present in the linearized virtual work because of external forces.

The second term in (20), because of externals tractions, has to be extended first to the free surface†f according to the boundary conditions. Assuming that the fluid displacements are small and thatthe final configuration coincides with the reference configuration, derivatives of the normal vectorcan be neglected and the following approximation holds

D

#Z†f

tf " ıuf da$D !

Z†f

paDnf " ıuf da# 0. (21)

Finally, the constraint along the surface in contact with the structure $f is to present the samenormal displacement than the structure, combined with zero traction in the tangential direction dueto the inviscid property of the fluid.

4.4. Mean dilatation method for incompressibility

A purely kinematic finite element discretization of (16) or (18) is unfortunately not applicable tosimulations involving incompressible or quasi-incompressible behavior. It is well known that with-out further development, previous formulation is over-constrained, resulting in the phenomenonknown as volumetric locking. Well-known solutions to this problem are as follows: to impose theincompressibility condition by penalization or by using reduced integration methods [47–50], toadopt an augmented Lagrangian formulation enforcing quasi-incompressibility condition [51, 52],or to use u–p elements that satisfy the inf–sup condition [53, 54]. Among all these possibilities, atotal displacement formulation with variational treatment of near incompressibility is preferred inorder to facilitate the treatment of the interface without sacrificing the benefits of a theory.



Distinct Feature of our 2012 Work: Nonlinear Continuum Constitutive Laws – Concluded


in which acoustic pressure Np.e/f is constant inside the element. Next, recall the linearized virtualwork equation (27), which for an element (e) is

Dı….e/f D

ZV .e/f

df W cpW ıdf dV CZV .e/f

pTf IW

!rTufrıuf

"dV

CZV.e/f

!f.r ! uf/.r ! ıuf/ dV CZV.e/f

"0f Ruf ! ıuf dV (30)

and apply the mean dilatation approach. Because the mean acoustic pressure Npf D !f! Nr ! uf

"is

constant over the element volume, the third volume integral becomes

Dı….e/f D

ZV .e/f


pTf IW

!rTufrıuf

"dV

C !fV .e/f . Nr ! uf/.e/. Nr ! ıuf/

.e/CZV .e/f

"0f Ruf ! ıuf dV . (31)

This is a linearized approximation around the initial hydrostatic equilibrium state, assuming that thefluid deviations from equilibrium are small. Note that in the second initial-stress term appears thetotal pressure and that the third acoustic term is pure volumetric.

Element displacements are discretized as uf D Nu.e/f , where Nf collects the element shape func-tions while uef gathers nodal values of the element. The element stiffness matrix is then composedof three terms:

K.e/ DK.e/conCK.e/

geoCK.e/ac , (32)

K.e/con D

ZV.e/f

BTf CpBf dV (33)

K.e/ac D

!f

V.e/f

ZV.e/f

.r !Nf/T dV

ZV.e/f

.r !Nf/ dV (34)

K.e/geo D

ZV.e/f

pTf .rNf/

T.rNf/ dV . (35)

where Cp is the spatial constitutive matrix constructed from the fourth-order tensor cp and Bf is thedeformation matrix.

Here, K.e/con is the constitutive stiffness matrix, K.e/

ac is the acoustic stiffness matrix, whereas K.e/geo

is the geometrical stiffness matrix. By starting our analysis from an equilibrium configuration whereacoustic pressure pf is zero and total pressure pT

f coincides with initial pressure p0f , assuming smallacoustic-pressure oscillations, the constitutive constitutive stiffness matrix (33) can be neglected andthe geometrical stiffness (35) evaluated at the initial hydrostatic state p0f .

Spectral analysis of the resultant stiffness matrix for an eight-node, 24 degree-of-freedom (DOF),brick element with a regular (cubic-like) geometry under exact 2 " 2 " 2 integration reveals fullrank of 18 with six zero-energy modes corresponding to rigid-body motions. The acoustic stiffnessmatrix (34) contributes to the rank with one purely volumetric mode, whereas the geometrical stiff-ness matrix (35) brings the remaining 17 non-zero-energy modes that include shear, torsion, andfilling modes.

On the other hand, inertia forces are modeled using the classical element mass matrix

M.e/ DZV .e/f

"0f NTf Nf dV . (36)

Upon assembling the element matrices, we arrive to the matrix form of the first variation for thecomplete fluid mesh

ı…f D ıuTf ¹Mf Ruf CKfuf $ ffº (37)




Dı….e/f D

ZV .e/f


pTf IW

!rTufrıuf

"dV

CZV.e/f




"is


Dı….e/f D

ZV .e/f


pTf IW

!rTufrıuf

"dV


.e/CZV .e/f

"0f Ruf ! ıuf dV . (31)



K.e/ DK.e/conCK.e/

geoCK.e/ac , (32)

K.e/con D

ZV.e/f

BTf CpBf dV (33)

K.e/ac D

!f

V.e/f

ZV.e/f

.r !Nf/T dV

ZV.e/f

.r !Nf/ dV (34)

K.e/geo D

ZV.e/f

pTf .rNf/

T.rNf/ dV . (35)








M.e/ DZV .e/f







Dı….e/f D

ZV .e/f


pTf IW

!rTufrıuf

"dV

CZV.e/f




"is


Dı….e/f D

ZV .e/f


pTf IW

!rTufrıuf

"dV


.e/CZV .e/f

"0f Ruf ! ıuf dV . (31)



K.e/ DK.e/conCK.e/

geoCK.e/ac , (32)

K.e/con D

ZV.e/f

BTf CpBf dV (33)

K.e/ac D

!f

V.e/f

ZV.e/f

.r !Nf/T dV

ZV.e/f

.r !Nf/ dV (34)

K.e/geo D

ZV.e/f

pTf .rNf/

T.rNf/ dV . (35)








M.e/ DZV .e/f






Linearizing the kinematics, the inverse of the fluid dilatation Jf can be approximated as J!1f !.1"r # uf/ and substituted back in (3) to provide a linearized version of mass conservation

!f D !0f .1"r # uf/. (4)

On the other hand, for a general equation of state, the speed of sound (cf ) relates the variation ofpressure to variation of density in adiabatic conditions, that is,

pTf D p0f C c2f

!!f " !0f

"C # # # (5)

Neglecting all terms involving squares and higher powers of!!f " !0f

", this relation can be

substituted in (4) to express the total fluid pressure in Lagrangian form as

pTf D p0f " "fr # uf (6)

an equation of state, relating Lagrangian pressure with displacements, where "f D !0f c2f is the fluidvolumetric stiffness. This approximation is well known and has been extensively used to model fluidmotion in the framework of Lagrangian kinematics (e.g., [40–45]).

4.2. Fluid problem in strong form

The strong form of the fluid equations is presented in the following for a pure volumetric con-tinuum by using two alternative descriptions. These are the description with respect to the initialconfiguration Vf and current configuration vf of the fluid domain. For the description with respectto the initial configuration, the second Piola–Kirchhoff stress tensor Sf is used together with theGreen–Lagrangian strain tensor Ef , and for the current configuration, the Cauchy stress tensor ! f

combined with the symmetric deformation tensor df is preferred.Partial differential equations to be satisfied on the fluid domain consist of kinematical relation,

local balance of momentum, and the constitutive equation, given by

Initial configuration: Current configuration:

Kinematics: Ef D 12.Cf " I/ df D

1

2

!ruf CrTuf

"

Equilibrium: r # .FfSf/C !0f gD !0f Ruf r # ! f C !fgD !f Ruf

Constitutive: Sf D 2 @Uf .Jf /@Cf

! f D J!1f FfSfFTf

where Cf D FTf Ff is the right Cauchy–Green strain tensor and Uf.Jf/ the volumetric strain-energy.

Additionally, the boundary conditions are extended to the complete fluid surface @Vf that can bedivided in two different regions: the free surface †f and the surface in contact with the structure #f

with @Vf D #f [†f as represented in Figure 2.When integrating over the free surface †f , the normal pressure is constant and equal to the

atmospheric pressure pa; thus, we ignore any pressure discontinuity across the air-liquid interfacedue to surface tension. This approximation holds for waves except ripples with extremely shortwavelength [46].

Also, normal displacements and tangential tractions have to be prescribed on #f , leading to thefollowing boundary conditions:

tf D "panf on †f

uf # nf D Nun on #f

tf # "f D 0 on #f

with nf and "f representing the boundary normal and tangential vectors and tf the surface tractionvector in the current configuration.

The necessary variational formulation will be described in the following sections based on areferential and spatial description.



3. Using previous nodal forces together with their total position!xnf , xns

", apply the ZMR to obtain

the location of the new frame nodes xnb DM0

!xnf , xns

".

4. Construct a frame mesh with isoparametric elements by using the previous frame nodes.5. Project the current position of the fluid and the structure active nodes on the frame elements

forming a new group of pairs. For each pair, calculate the position !p of the interface node inthe frame element, the normal in that position n.!p/, and shape functions Nb.!p/.

Once the frame is defined, the FSI problem is solved for time step tnC1, obtaining new positionsxnC1f and xnC1s . The frame definition algorithm can then be repeated for a new time step.

9. PARTITIONED FLUID–STRUCTURE FORMULATION

The partitioned FSI model can now be constructed following the variational statement given in(2) for the discrete form of the total energy functional variation ı…. Inserting the discrete varia-tional internal constraint (59), the discrete variational fluid equation (37) and the discrete variationalstructural equation (40) into the foregoing equation gives the variational form

ı…D ıuTf ¹Mf Ruf CKfuf CBf"f " ffº

C ıuTs ¹Ms RusCKsusCBs"s " fsº

C ı"Tf

®BT

f uf "Lfub " hf

¯C ı"T

s

®BT

s us "Lsub " hs

¯

" ıuTb

®LT

f "f CLTs"s

¯(77)

in which vectors hf D!BT

f Xf "LfXb"

and hs D!BT

s Xs "LsXb"

are function of the initial con-figuration. Making (77) stationary with respect to the primary vaariables yields the partitioned,semidiscrete equations of motion

2666664

Kf CMfd2

dt20 Bf 0 0

0 KsCMsd2

dt20 Bs 0

BTf 0 0 0 "Lf

0 BTs 0 0 "Ls

0 0 "LTf "LT

s 0

3777775

8ˆ<ˆ:

uf

us

"f

"s

ub

9>>>=>>>;D

8ˆ<ˆ:

fffshf

hs

0

9>>>=>>>;

. (78)

The first two rows of the foregoing matrix equation are the discrete equilibrium equations andthe fluid and structure partitions, respectively, with terms Bf"f and Bs"s representing the inter-action forces transacted through the frame. The third and fourth equations impose fluid–frame andstructure–frame displacement compatibility, respectively. The last equation states the equilibrium ofthe frame.

9.1. Vibration analysis

The equation of motion (78) can be specialized to small, unforced oscillations of frequency ! aboutan equilibrium configuration by replacing the time-differentiation operator d2

dt2with"!2 and setting

external forces to zero:266664

Kf "!2Mf 0 Bf 0 00 Ks "!2Ms 0 Bs 0

BTf 0 0 0 "Lf

0 BTs 0 0 "Ls

0 0 "LTf "LT

s 0

377775

8ˆ<ˆ:

uf

us

"f

"s

ub

9>>>=>>>;D

8ˆ<ˆ:

00000

9>>>=>>>;

. (79)

This is a generalized, symmetric, algebraic eigenvalue problem from which frequencies and modeshapes of the coupled system can be determined [34].



The mode shapes and corresponding sloshing frequencies for this case are presented in Figure 14with contour colors representing elevation. These sloshing mode shapes are more complicated thanfor the cubic container and thus require a more refined finite element mesh in the circumferentialdirection.

10.2.2. Sloshing in rigid containers: transient analysis. This example investigates the accuracylevel of the finite element approximation proposed for the fluid in Section 4 by comparing it withthe analytical response of a well-known benchmark [69]. The problem involves the forced sloshingof a liquid inside a rectangular 2D rigid tank subjected to harmonic horizontal base excitation. Thecomparison analytical solution is obtained from potential theory.

A water-filled tank has the rectangular geometry depicted in Figure 15. The height-to-baseaspect ratio is H=B D 1=2. The forced swaying motion imposed to the tank base is sinusoidal:xs D as cos.!st /, where as is the amplitude of motion (maximum displacement of the rigid walls

Table IV. Analytical/numerical comparison of the first four sloshing fre-quencies for symmetric (S) and antisymmetric (AS) modes of a cylindrical

container with rigid walls.

Sloshing freq. Analytical (Hz) Computed (Hz)

Mode (1,S) 0.98 0.99Mode (2,S) 1.32 1.41Mode (1,AS) 0.67 0.67Mode (2,AS) 0.87 0.88

Figure 14. Symmetric and antisymmetric sloshing modes and natural frequencies in a cylindrical containerwith rigid walls. S, symmetric; AS, antisymmetric.

Figure 15. Model of the tank excited horizontally with prescribed harmonic motion of amplitude as andfrequency !s. Time records of the free-surface elevation are obtained at the left wall of the rigid container

(point A).



Figure 8. Sketch of a 2D cavity with three rigid walls closed by a flexible beam on the top.

Table I. Comparison of analytical and numerical acoustic frequenciesobtained for a 2D acoustic cavity with rigid walls.

Acoustic freq. Analytical (Hz) Computed (Hz)

Mode (1,0) 37.5 37.5Mode (0,1) 93.7 94.3Mode (1,1) 100.9 101.3Mode (2,1) 120.1 120.1Mode (1,2) 191.2 195.6Mode (2,2) 210.9 205.1

The fluid domain is meshed using 8! 20 fluid elements, corresponding to 320 fluid DOFs. Eachfluid element is square with a side length of 1m. The structure on the top is modeled using 20 beamelements with two DOFs per node (vertical displacement and rotation) and connected to the fluidusing localized Lagrangian multipliers.

The natural frequencies and structural modes that describe the dynamic behavior of the simplysupported beam vibrating in vacuo are shown in Figure 10.

10.1.1. Case 1: Infinitely rigid beam. For later comparison with the flexible case, we first considerthe limit Is!1, whence the top beam acts as another rigid wall, transforming the problem into anacoustic cavity with rigid walls. This limit problem has a well-known analytical solution. Resonancecan be expected at frequencies for which the corresponding wavelengths match the dimensions ofthe cavity, that is,

!2l ,m D cfkl ,m .l ,mD 1, 2, 3, : : :/ (85)

where !l ,m is the resonant frequency and kl ,m D!"

l Lx

#2C"m L´

#2$1=2is the wave number.

Such frequencies are computed analytically using (85) and represented in Table I together withthe numerical results obtained performing a vibration analysis using the technique described inSection 9.1. These first six resonant frequencies have been selected for contour plotting in Figure 9,with pictures showing horizontal and vertical displacements of the fluid, respectively, together withfluid pressure level that is constant inside each element as required by the mean dilatation method.

As expected, the approximation error of the acoustic modes in the horizontal direction is lowerthan in the vertical direction because of the coarser vertical discretization. For the displayedfrequencies, the error is less than 3%. Observe that acoustic mode shapes for the selected frequenciesare correctly captured.

10.1.2. Case 2: Flexible beam. Next, we study the effect of placing a flexible wall, with thedynamic characteristics summarized in Figure 10, on top of the cavity. The fundamental frequency


Looking at Future Fluid-Structure Interaction (with focus on sloshing-structure, internal acoustic-structure, external acoustic-structure interaction) has come a long way.

The 1960 – 1980: Potential-based Poisson Equation for fluids

The 1990 – 2000: Mixture of Poisson equation and pressure/velocity formulation fluids

Past Decade: Fully nonlinear continuum formulation for both compressible and incompressible fluids

It is likely that fluid pressure-embedded FEM models for structures will be retired.

Just as structural mechanics community needed to re-interpret continuum mechanicssolutions in terms of bars, beams, plates, shells, etc., FSI solution based on continuum Mechanics-based solutions would have to be re-interpret the results in terms of Poissonequations and classical wave equations. This will take time.

Looking at Future – Cont’d For FSI problems relying on Navier-Stokes fluids and nonlinear solids interaction models, two schools of approaches have been employed in a parallel mode:

Fully coupled non-partitioned modeling of fluid and solids, or

Completely partitioned formulation approaches

Consensus on categories of problems for which one or the other approach is advantageous has not been reached – a thorough balanced study is needed.

Our crystal ball predicts FSI would thrive for:

deep ocean energy extraction; biomechanics, in particular, hemodynamics; wind and ocean current energy; membraneous structures deployed in the air and in ocean deep ocean habitat construction…

Part II: External Flow Problems

A Survey of External Acoustic-Structure Interaction Models

K. C. Park1,2 and Carlos A. Felippa1

1 Department of Aerospace Engineering Sciences, Center for Aerospace Structures, University of Colorado, Boulder, CO 80309-0429, USA, Email:[email protected].

2 Ocean Systems Engineering KAIST Daejeon, Korea Email: [email protected]

V International conference on Coupled Problems in Science and Engineering 17-19 June 2013, Ibiza, Spain

Two Interrelated Problems in External Acoustic Problems

Inverse Problem: Acoustic Signal Detection of Submerged Vehicles

Structural Integrity: Structural Response subjected to Near-Field Explosion - Present Focus Problem

Ultimately, one day these two problems should be treated by unified models and solution methods

Contributors:

Moonseok Lee (Hyundai Motor Co.) Youn-sik Park and Youngjin Park(KAIST) C. A. Felippa (University of Colorado) Heekyu Woo and Young Shin (KAIST) Joe Gonzalez (Univ. Seville, Spain)

Roger Ohayon (CNAM, Paris) Acknowledgments: WCU Visiting Professorship at KAIST

Related Literature: Baker and Copson(1949): My favorite theoretical text

G. Carrier(1951): Cylindrical shells excited by incident waves Junger and Feit(1950s): sound scattering by thin elastic shells Mindlin and Bleich(1953): Perhaps the first plane wave approximation Huang (1969): A successful solution of retarded potential for a sphere

Geers (1978): The seminal paper on doubly asymptotic approximation (DAA)

Felippa (1980): A systematic derivation for early-time response

Geers and Felippa (1983): Application of DAA to vibration problems

Astley et al (1998): Wave envelope elements that treats the scattering in addition to radiation in applying retarded potential.

Two computational approaches to external acoustic-structure interactions •  Solve the wave equation with approximate infinite radiation boundary conditions; •  Approximate Kirchhoff’s retarded potential that properly incorporates the infinite radiation boundary conditions. •  There are, of course, a host of approaches that try to exploit the desirable features of the two views.

(adopted in existing models)

(via filtering concept)

Theoretical Foundation for Present Approximations

Series expansion of the retarded operator,

where

in

has been responsible for unstable form unless a modification is incorporated.

Possible stabilization: Employ advanced potential!

Key Idea for Present Approximations

Modifications for Transient or Early-Time Response

(a) Consistency for Impulse Response compared with analytical solutions; (b) Incorporation of classical results wherein early-time response is dominated by plane waves;

The preceding requirements are met fortuitously if one approximates in the following terms

by replacing with

Impulse Response Consistency

Present model satisfies the early-time consistency independent of the choice of the weighting parameter, χ

Determination of Weighting Parameter, χ

1.  Plot the characteristic roots of the exact analytical solution for a sphere

2. Assume χ in the following form:

And determine the best fitting constants,

Analytical Pressure Characteristic Root Loci for a Sphere

Determination of Weighting Parametric Matrix, χ•  It specializes to the modal form of χn when applied to spherical geometries •  It should be robust with respect to computational errors for general geometries A general matrix form of χ that satisfies the above requirements is found to be

The case of S=0 is symbolically equivalent to DAA2 with curvature Correction which is found to be prone to instability for n = 0 mode.

Summary of Models Proposed So Far

Computer Implementation of Interaction Problems

Problems that can be treated by the present models

Governing Interaction Equations

Application

Benchmark Test: elastic sphere subjected to Incident acoustic excitations

•  Transient responses of a submerged spherical shell excited by cosine-type impulse pressure –  A spherical shell surrounded with fluid medium. –  In water medium –  h/a=0.01, ρs/ρ=7.7, cs/c=3.7

O

h , ,sE vρ

cos ( )Ip tθδ= −

r v w

a

θ P

Q

PQR

Numerical Simulation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4

Tc/a

Incident Pressure

PI =7π -1/2Exp[72(t-0.5)]

simulation

simulation

Conclusions §  A stable approximate model for external acoustic and structural interaction problems has been derived by employing a combination of retarded and advanced potentials. §  The maximum order of “regular” approximation is found to be two. §  The next talk offers our latest attempt to improve the second-order models for external acoustic-structure interaction models. Reference: Moonseok Lee, Youn-Sik Park, Youngjin Park, K.C. Park, New approximations of external acoustic-structural interactions: Derivation and evaluation, Computer Methods in Applied Mechanics and Engineering, Vol. 198 (15-16) pp.1368-1388, 2009

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