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Direct differentiation of the quasi-incompressible fluid formulation of fluid–

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DOI: 10.1007/s40571-016-0123-6

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Comp. Part. Mech.DOI 10.1007/s40571-016-0123-6

Direct differentiation of the quasi-incompressible fluidformulation of fluid–structure interaction using the PFEM

Minjie Zhu1 · Michael H. Scott1

Received: 27 March 2016 / Revised: 25 June 2016 / Accepted: 7 July 2016© OWZ 2016

Abstract Accurate and efficient response sensitivities forfluid–structure interaction (FSI) simulations are importantfor assessing the uncertain response of coastal and off-shorestructures to hydrodynamic loading. To compute gradientsefficiently via the direct differentiation method (DDM) forthe fully incompressible fluid formulation, approximationsof the sensitivity equations are necessary, leading to inac-curacies of the computed gradients when the geometry ofthe fluid mesh changes rapidly between successive timesteps or the fluid viscosity is nonzero. To maintain accu-racy of the sensitivity computations, a quasi-incompressiblefluid is assumed for the response analysis of FSI using theparticle finite elementmethod andDDM is applied to this for-mulation, resulting in linearized equations for the responsesensitivity that are consistent with those used to computethe response. Both the response and the response sensi-tivity can be solved using the same unified fractional stepmethod. FSI simulations show that although the responseusing the quasi-incompressible and incompressible fluid for-mulations is similar, only the quasi-incompressible approachgives accurate response sensitivity for viscous, turbulentflows regardless of time step size.

Keywords Particle finite element method · Fluid-structureinteraction · Sensitivity analysis

B Michael H. Scottmichael.scott@oregonstate.edu

Minjie Zhuzhum@oregonstate.edu

1 School of Civil and Construction Engineering,101 Kearney Hall, Corvallis, OR 97331, USA

1 Introduction

Efforts to improve design and mitigation strategies for struc-tures subject to hydrodynamic loads induced by tsunamis andwind storms have increased due to recent natural disasterssuch as the 2011 Great East Japan earthquake and tsunamiand Superstorm Sandy of 2012 [4,18]. To characterize therange of load effects, it is imperative to assess the sensi-tivity of structural response to stochastic wave loading anduncertain structural properties in fluid–structure interaction(FSI) simulations. The sensitivity has important implica-tions for gradient-based applications such as reliability andoptimization [10,11], the design of coastal infrastructure,and assessing the probability of failure of buildings andbridges in tsunami and storm events as part of an over-archingperformance-based engineering framework [5].

For simulating FSI, the particle finite element method(PFEM) [22] has been shown to be an effective approachbecause it uses the Lagrangian formulation for fluids, whichis the same formulation utilized in the finite element analysisof solids. As a result, a monolithic system of equations iscreated for the simultaneous solution of the fluid and struc-tural response. This alleviates the need to couple disparatecomputational fluid and structural modules through a stag-gered approach in order to simulate FSI response. Throughthe monolithic approach, compatibility and equilibrium aresatisfied naturally along the interfaces between the fluid andstructural domains. Therefore, the PFEM becomes a goodcandidate as the starting point to develop analytical sensitiv-ity methods for FSI simulations.

There are several methods for calculating the sensitivityof a simulated response. The finite difference method (FDM)repeats the simulation with a perturbed value for each para-meter and does not require additional finite element analysisimplementation as perturbations of parameters and differ-

123

Comp. Part. Mech.

encing of response quantities can be handled with pre- andpost-processing. The accuracy of the resulting finite differ-ence approximation depends on the size of the perturbationwhere the results are not accurate for large perturbations andare prone to numerical round-off error for very small pertur-bations. Due to the need for repeated simulations, the FDMapproach can become inefficient when the model is large,which is common for FSI simulations, and when there are alarge number of parameters.

A more accurate and efficient approach to gradient com-putations is the direct differentiation method (DDM), wherederivatives of the governing equations are implementedalongside the equations that govern the simulated response.At the one-time expense of derivation and implementa-tion, as well as additional storage, the DDM calculatesthe response sensitivity efficiently as the simulation pro-ceeds. This eliminates the need for repeated simulations asare required for finite difference calculation of the gradi-ents. DDM sensitivity analysis for structural response undermechanical loads has been well developed and extendedto material and geometric nonlinear formulations of frameelement response [7,13,25,26]. The DDM has also beenapplied to composites processing [2] and fire attack on struc-tures [12].

In addition to the FDM andDDM, there are other methodsfor computing finite element response sensitivities. First isthe complex perturbationmethod (CPM)which finds the sen-sitivity from the imaginary part of complex numbers used forthe response computations [15]. Although the CPM providesvery accurate sensitivities, it requires perturbation and re-analysis for each uncertain parameter in the structural modelin the same manner as the FDM. Another alternative is theadjoint structure method (ASM) [16] which yields analyticalsensitivities in a similar manner to the DDM, but is limitedto problems with path-independent constitutive response.

Returning to the focus of this paper, the DDM has beenapplied to FSI simulations using the PFEMwith incompress-ible fluid and nonlinear structures [31]. With this approach,the incompressible fractional step method (FSM) was usedto solve the FSI sensitivity equations using an approximationof the derivatives of the FSI geometric nonlinear response.While this preserved efficiency of the DDM computations,accuracy is lost due to the simplification of geometricallynonlinear fluid response, particularly for large simulationtime steps, �t .

The objective of this paper is to develop DDM sensitivityequations for FSI using a quasi-incompressible fluid formu-lation that leads to a unified FSM solver for the fluid andstructural response. The developed sensitivity equations willhave a similar format to the response equations and can besolved by the same unified solver so that accuracy and effi-ciency of theDDMaremaintained. FSI response based on thequasi-incompressible formulationwill be introduced, includ-

ing the governing equations, the element formulation, and thediscrete response equations. Then, the DDM is applied to thequasi-incompressible formulation in order to obtain the sen-sitivity of the discrete FSI equations. Lastly, the unified solverfor both response and sensitivity equations is introduced.Numerical examples of sloshing fluid interacting with a softbeam and breaking dam on nonlinear obstacle are shown tocompare the results of the quasi-incompressible DDM withfinite difference computations and also to compare the DDMsensitivity obtained from the quasi-incompressible DDMwith that obtained via the incompressible formulation.

2 A unified PFEM response computation

The PFEM response analysis of FSI is introduced with theassumption of quasi-incompressible fluid flow and nonlin-ear structural response. This formulation results in a unifiedFSM that can be used for efficient computation of both theresponse and the response sensitivity to changes in uncer-tain parameters. Although the presentation will focus on aparticular fluid element, the methods described herein aregenerally applicable to any element formulation.

2.1 Governing equations

Conservation of linear momentum for points in both thefluid and structural domains can be written in the updatedLagrangian formulation as

ρvi = ∂σi j

∂x j+ ρbi (1)

where ρ is the material density; vi , xi , and bi are veloc-ity, coordinates, and body acceleration vectors, respectively;and σi j is the Cauchy stress tensor. Neumann boundary con-ditions prescribe normal stresses on the surface, Γt ,

σi j n j = ti , (2)

where ti is the surface traction and n j is the unit normal vectorto the boundary surface. Dirichlet boundary conditions areimposed on coordinates and velocities on the surface, Γv,

xi = xpi , vi = vpi , (3)

where xpi and vpi are the prescribed coordinates and velocities,

respectively.For structures, the stress–strain relationship is assumed to

be a general nonlinear function of displacement and velocity

σi j = σi j (xi , vi ) (4)

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Comp. Part. Mech.

where it is straightforward to incorporate hysteresis andviscosity [27]. For Newtonian fluids, the Cauchy stress isdecomposed into spherical and deviatoric portions

σi j = si j − pδi j , (5)

where δi j is the Kronecker delta and p = σi i/3 is the aver-age stress (pressure) which is positive for compression. Thedeviatoric stress, si j , is defined in terms of the strain rate by

si j = 2μ

(εi j − 1

3εvδi j

), (6)

whereμ is the viscosity and εi j is the strain rate tensor whichis defined in terms of the velocity gradient

εi j = 1

2

(∂vi

∂x j+ ∂v j

∂xi

). (7)

Likewise, the volumetric strain rate is

εv = εi i = ∂vi

∂xi. (8)

Conservation of mass in the structural domain can be writtenin the form

ρ J = ρ0, (9)

where ρ0 is the density in the undeformed configuration andJ is the determinant of the deformation tensor as definedin [17]. Conservation of mass for quasi-incompressible flowcan be expressed in the differential form [14,21,24,34],

1

κp + εv = 0 (10)

where κ is the bulk modulus. Nonlinear structural responseis defined by the governing equations of linear momen-tum, Eq. (1); a general constitutive relationship, Eq. (4); andmass conservation Eq. (9). Likewise, the equations of linearmomentum, Eq. (1); constitution, Eq. (5); and conservationof mass, Eq. (10) define the quasi-incompressible Newtonianfluid response.

2.2 Unified discrete FSI equations

The governing equations of fluid and solid response can besolved through a single, unified system of equations. In theparticular case of Newtonian fluid and elastic solid response,the equations can be unified in their continuous form [3]and then discretized using finite element methods. It is notpossible, however, to unify Newtonian fluid equations withsolid response in their continuous formswhen accounting for

material nonlinearity of the solid. To resolve this problem andallow for FSI with nonlinear structural response, the discreterather than continuous equations are unified for computingthe FSI response

Mv + Kv − Gp + Fints = Fext (11)

Mpp + GTv = 0 (12)

where v and p are the vectors of nodal velocities over allelements in the model and pressure over fluid elements. Fur-ther details on the unified discrete equations are found in[32]; however, to facilitate the subsequent derivation of FSIsensitivity equations, the fluid internal force vector Gp isseparated from the structural internal force vector Fint

s . Themass matrix,M is assembled from element contributions

(13)

where A is the finite element assembly operator. The matrixK in Eq. (11) is the structural damping or fluid viscous matri-ces, which both are related to viscous effects of the modelbut defined differently. The structural damping matrix can bedefined as a linear combination of mass and stiffness matri-ces, i.e., Rayleigh damping [6]. The fluid viscous matrix isalso assembled over all elements

(14)

where the matrix D contains material properties of the New-tonian fluid

D = μ

⎡⎢⎢⎢⎢⎢⎢⎣

43 − 2

3 − 23 0 0 0

− 23

43 − 2

3 0 0 0− 2

3 − 23

43 0 0 0

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎦

(15)

and the matrix B contains derivatives of the element shapefunctions

BTa =

⎡⎢⎢⎢⎣

∂Nva

∂x10 0 ∂Nv

a∂x2

0 ∂Nva

∂x3

0 ∂Nva

∂x20 ∂Nv

a∂x1

∂Nva

∂x30

0 0 ∂Nva

∂x30 ∂Nv

a∂x2

∂Nva

∂x1

⎤⎥⎥⎥⎦ ,

B = [B1 B2 · · · Bnv

](16)

The gradient operator G is applied on the pressure field

(17)

123

Comp. Part. Mech.

where the vector m is defined as

m = [1 1 1 0 0 0

]T. (18)

The vector Fints represents the static resisting forces of the

structure and may also include nonlinear damping effectsthat cannot be represented by the linear damping matrix, K,in Eq. (11). Since structural constitutive equations are notexplicitly defined in the unified discrete equations (Eqs. 11,12), any structural elements may be used [33]. The externalforce vector Fext is composed of traction and body forces

(19)

The pressure mass matrix, Mp from Eq. (12), is defined bythe bulk modulus

(20)

The shape functions for the velocity Nv and the pressure Np

are defined as

Nv = [Nv1 Nv

2 · · · Nvnv

], Nv

a = N va I, a = 1, 2, . . . , np

(21)

Np = [N p1 N p

2 · · · N pnp

](22)

where nv and np are the numbers of velocity and pres-sure nodes and I is the identity matrix whose size dependson the dimensions of the model, either I2 (2 × 2) or I3(3 × 3). The Newtonian fluid in the quasi-incompressibleformulation requires the pressure to be stabilized. TheLBB (Ladyzenskaja-Babuska-Brezzi)-stable MINI elementis used for the fluid domain, as described in Appendix 1.

2.3 Linearization and time integration

To utilize a wide range of root-finding algorithms, the unifieddiscrete Eqs. (11) and (12) can be written in residual form

rv = Fext − Mv − Kv + Gp − Fints (23)

rp = −Mpp − GTv, (24)

where the solution corresponds to rv and rp equal to zero.In addition to the spatial discretization encapsulated by theresidual equations, time discretization is necessary in orderto advance the simulation. To this end, backward Euler timeintegration is used to relate the velocity to acceleration anddisplacement

v = v − vn�t

(25)

u = un + �tv, (26)

where vn and un are the nodal velocities and displacementsof last time step at time interval �t . In addition, the timederivative of pressure is approximated using the backwardEuler formula

p = p − pn�t

, (27)

where pn is the pressure at the previous time step.Newton-like methods require the Jacobian of the residual

vectors with respect to the unknowns, i.e., the algorithmictangent that accounts for the spatial and time discretizationof the simulation. Each algorithmic tangent matrix, KT, iscomputed from derivatives of Eqs. (23) and (24) with respectto velocity and pressure unknowns

KTvv = ∂rv∂v

= M�t

+ K + �t∂Fint

s

∂u(28)

KTvp = ∂rv∂p

= −G (29)

KTpv = ∂rp∂v

= GT (30)

KTpp = ∂rp∂p

= Mp

�t(31)

where ∂Fints∂u is the tangent stiffnessmatrix of the structure. The

following system of linear equations is then used to solve forthe increments of velocity and pressure for the unified FSIresponse equations in each Newton iteration

[KTvv KTvpKTpv KTpp

] [δvδp

]=

[rvrp

](32)

To solve Eq. (32), the FSM for incompressible fluid usuallyinvolves three steps of velocity prediction, pressure solution,and velocity correction. The second step of pressure solu-tion requiresKTvv to be diagonal or at least mostly diagonal,since an inversion of the matrixKTvv is involved. The incom-pressible FSM [23,29] modifies the matrix KTvv to meetthe requirement by assuming small fluid viscosity (ignor-ingK) and small FSI interface, and lumping the mass matrixM.

To circumvent the limitation, instead of solving pressurethrough an inversion of KTvv , a unified FSM [32] has beenproposed to solve velocity through an inversion of KTpp ,which is provided by the pressure mass matrix Mp definedin Eq. (20). This matrix can always be lumped as in Eq. (63)without further assumptions. The matrix KTvv can then bekept in its original form. Therefore, the steps in the incom-pressible FSM are reversed in the unified FSM as follows:

123

Comp. Part. Mech.

1. Compute predictor pressures by ignoring the divergenceof velocities in the mass equation;

2. Solve unified velocities through the tangent stiffnessmatrix with the contribution from the mass equation; and

3. Correct pressures using the velocities computed in step 2.

The unified FSM couples the advantages of the quasi-incompressible fluid formulation with the FSM and it doesnot distinguish between the fluid and structural domains [32].This allows for more efficient solution of FSI problems withviscous flow regardless of time step, and it also facilitates thecomputation of response sensitivity, as will be described inthe forthcoming sections.

With the unified FSM, the left-hand side algorithmic tan-gent inEqs. (28)–(31) and the right-hand side residual vectorsdefined in Eqs. (23) and (24) are computed at each iterationof a time step based on the current nodal coordinates. Whenthe convergence criterion for the nonlinear iteration is satis-fied, the simulation will proceed to the next time step aftercomputing the sensitivity of the FSI response, as describedin the following section.

3 A unified PFEM sensitivity computation

The application of theDDM to unified FSI equations requiresdifferentiation of the discrete equations that govern the fluidand structural response. For large displacement applicationssuch as FSI, additional terms that arise from updating theconfiguration at each iteration must be taken into account inthe derivation of sensitivity equations. To preserve the formatof the tangent matrix used to solve for the FSI response withan incompressible fluid, these geometric terms are movedto the right-hand side of the equations and approximatedusing the values from the previous time step [31]. Althoughthis approximation is efficient because the same FSM solvercan be used for the response sensitivity as for the ordi-nary response, errors of the computed gradients appear whenthe configuration changes rapidly or the fluid viscosity isnonzero. In this section, the unified FSI sensitivity equationsare derived with these additional geometric terms retained inthe tangent matrix for better accuracy of the resulting gradi-ents. In addition, efficiency is maintained as the unified FSMsolver can be employed for the response sensitivity withoutapproximations.

An uncertain parameter of the FSI model is representedby the symbol θ , which could be a constitutive property,such as fluid density or material yield strength, or geometricdimension, such as the cross-section width of a beam ele-ment. Taking the derivative of the discrete unifiedmomentum

equation (Eq. 11) with respect to θ gives

M∂ v∂θ

+ K∂v∂θ

− G∂p∂θ

+ ∂Fints

∂u∂u∂θ

+ H∂u∂θ

= ∂Fext

∂θ

∣∣∣∣u

− ∂Fints

∂θ

∣∣∣∣u

− ∂M∂θ

∣∣∣∣uv − ∂K

∂θ

∣∣∣∣uv + ∂G

∂θ

∣∣∣∣up

(33)

where ∂u/∂θ , ∂v/∂θ , ∂ v/∂θ , and ∂p/∂θ are sensitivities ofthe displacement, velocity, acceleration, and pressure vec-tors, respectively. Similarly, the derivative of the unifieddiscrete mass equation (Eq. 12) with respect to θ is

Mp∂p∂θ

+ GT ∂v∂θ

+ T∂u∂θ

= − ∂Mp

∂θ

∣∣∣∣up − ∂GT

∂θ

∣∣∣∣uv (34)

On the right-hand side, conditional derivatives, denoted |u,are taken with ∂u/∂θ equal to zero [28]. The matricesH andT on the left-hand side are partial derivatives that account forgeometric nonlinearity of the FSI response

H = ∂(Mv)∂u

+ ∂(Kv)∂u

− ∂(Gp)

∂u− ∂Fext

∂u(35)

T = ∂(Mpp)

∂u+ ∂(GTv)

∂u(36)

These terms affect the FSI response sensitivity but do notdepend on the uncertain parameter, θ . Further details on thesederivatives for the MINI element are found in [31] and herein Appendix 2.

In addition to the spatial discretization, consistent differ-entiation of the time discretized equations is also requiredfor the DDM [8]. This ensures that the developed sensitiv-ity equations have the same numerical conditioning as theunderlying response equations for the given level of spatialand temporal discretization.

The backward Euler approximations of Eqs. (25) and (26)are differentiated with respect to θ in order to obtain thesensitivity of the accelerations and displacements

∂ v∂θ

= 1

�t

(∂v∂θ

− ∂vn∂θ

)(37)

∂u∂θ

= ∂un∂θ

+ �t∂v∂θ

(38)

where ∂vn/∂θ and ∂un/∂θ are the velocity and displacementsensitivity computed at the previous time step. Likewise, thesensitivity of the pressure rate defined in Eq. (27) is

∂p∂θ

= 1

�t

(∂p∂θ

− ∂pn∂θ

)(39)

with ∂pn/∂θ the pressure sensitivity at the previous time step.By substitution of Eqs. (37), (38), and (39) into Eqs. (33) and

123

Comp. Part. Mech.

(34), the following system of linear equations is obtained forthe sensitivity of velocity and pressure at the current timestep

[KTvv + �tH KTvpKTpv + �tT KTpp

][∂v∂θ

∂p∂θ

]=

[svsp

](40)

The right-hand side “sensitivity” vectors are defined as

sv = M�t

∂vn∂θ

− H∂un∂θ

− ∂Fints

∂u∂un∂θ

+ ∂Fext

∂θ

∣∣∣∣u

− ∂Fints

∂θ

∣∣∣∣u

− ∂M∂θ

∣∣∣∣uv − ∂K

∂θ

∣∣∣∣uv + ∂G

∂θ

∣∣∣∣up (41)

sp = Mp

�t

∂pn∂θ

− T∂un∂θ

− ∂Mp

∂θ

∣∣∣∣up − ∂GT

∂θ

∣∣∣∣uv (42)

The conditional derivatives in sv and sp are evaluatedthrough partial derivatives of the element response quanti-ties with respect to the parameter θ and are assembled fromelement contributions. The closed-form expressions of theMINI element shown in Appendix 1 facilitate the implemen-tation of derivatives for the element matrices and vectors forassembly in the sensitivity equations. For example, the deriv-ative ofMp, defined in Eq. (63), with respect to bulkmodulus(θ ≡ κ) is

∂Mp

∂κ= −A

6κ2

⎡⎣1 0 00 1 00 0 1

⎤⎦ (43)

For Newtonian fluids, only two additional parameters can beuncertain: the viscosity μ in K and the density ρf in M. Itis noted thatG does not contain any constitutive parameters;therefore, its derivative ∂GT/∂θ |u is always zero in Eqs. (41)and (42). Additional details on these derivatives for FSImod-els are found in [31]. For a structural model withmaterial andgeometric nonlinear response, many more parameters couldcontribute to ∂Fs

int/∂θ∣∣u, the conditional derivative of the sta-

tic resisting force vector [7,25,26]. In this case, the elementmatrices and vectors are obtained by numerical integrationrather than in closed-form; however, this is not a limitationfor the DDM.

The system of linear equations in Eq. (40) is analogousto the system in Eq. (32) but with added terms �tH and�tT resulting from the inclusion of geometric nonlinearityin the derivation of sensitivity equations. As shown in thenext section, the inclusion of these terms in the system ofsensitivity equations does not compromise the accuracy andefficiency of the unified FSM in computing gradients of theFSI response via the DDM.

4 Numerical solution of the unified sensitivity andresponse FSI equations

The incompressible FSM lumps the mass matrix M andignores the fluid viscous matrixK in the matrixKTvv in orderto solve Eq. (32). However, the geometric term�tH added toKTvv in Eq. (40) can neither be lumped nor ignored becauseit accounts for geometric nonlinearity. This implies that theincompressible FSM cannot be used for the efficient solutionof FSI response sensitivity via Eq. (40) because the matrix(KTvv + �tH) becomes too expensive to invert.

On the other hand, the unified FSM [32] can solveEq. (40) without approximation because the matrix KTpp isnot affected by geometric nonlinearity and it can be lumpedwithout further assumptions beyond those made for New-tonian fluid flow. The unified FSM is then able to keep thematricesM,H, T, andK in their original form in the systemof sensitivity equations (Eq. 40), making it a suitable methodfor efficient solution of the unified sensitivity equations.

Following a procedure nearly identical to that for the uni-fied FSM summarized in this paper and with further detailsin [32], the steps for solving sensitivity equations are listedbelow:

1. Compute predictor pressure sensitivity by ignoring thedivergence of velocity sensitivity in the mass sensitivityequation;

2. Solve unified velocity sensitivity through the sensitivitytangent matrix with the contribution from the mass sen-sitivity equation; and

3. Correct pressure sensitivity using the velocity sensitivitycomputed in step 2.

The first step of the unified FSM for quasi-incompressibleFSI response sensitivity analysis is to compute the predictorpressure sensitivity as

∂p∂θ

∗= K−1

Tppsp. (44)

This is a fast operation as KTpp is diagonal. The second stepis to solve for velocity sensitivity by condensing the pressuresensitivity in Eq. (40)

((KTvv + �tH

) − KTvpK−1Tpp

(KTpv + �tT

)) ∂v∂θ

= sv − KTvp∂p∂θ

∗(45)

The last step is to correct the predictor pressure sensitivityusing velocity sensitivity computed from Eq. (45)

∂p∂θ

= ∂p∂θ

∗− K−1

Tpp

(KTpv + �tT

) ∂v∂θ

(46)

123

Comp. Part. Mech.

0 0.02 0.04 0.06 0.08 0.1

0

0.05

0.1

y (m)

x(m

)WaterWallBeam

Fig. 1 Model for elastic structure interacting with sloshing fluid

This three-step process is repeated for each parameter, θ , inthe FSI model before advancing to the next time step. Thecomputational cost of finding the sensitivity of velocity andpressurewith respect to a single parameter is roughly equal toone unified FSM iteration within a simulation time step. Thisrepresents a significant computational savings over methodsthat repeat the simulation for each parameter in order to com-pute the sensitivity. Additional details on the computationalcost of the DDM are found in [28].

The unified FSM has been implemented in the OpenSeesfinite element software framework [19,20] as an extensionof its existing PFEM implementations [30,31]. Through itsimplementation in OpenSees, DDM analysis of FSI usingthe PFEM is able to draw upon the various DDM implemen-tations of material and geometric nonlinear structural finiteelements for gradient-based FSI applications. The follow-ing numerical examples validate the implementation of theforegoing DDM equations.

5 Sloshing wave interacting with a soft beam

The model for this example, shown in Fig. 1, is an opentank with fixed boundaries on the left side and bottom and aflexible beam on the right side with length L = 0.1 m. Theelastic modulus, the width of the square cross-section, andthe density of the beam are E = 1 MPa, b = 0.012 m, andρs = 2500 kg/m3, respectively. The density, viscosity, andbulk modulus of the fluid are ρf = 1000 kg/m3, μ = 0.1kg/ms and κ = 2.15× 109 Pa, respectively. The out of planethickness of the fluid domain is assumed equal to the beamwidth b and the height of the fluid is h = 0.08m. Amesh sizeof 0.0025m is used togenerate 4126elements and8385nodesincluding pressure, velocity, and bubble nodes as shown inAppendix 1.

The beam is modeled as a corotational mesh of elasticbeam-columnelements [25] allowing for large displacementsof the beam in addition to geometric nonlinearity of the fluid

domain. The simulation time step is �t = 0.001 s. Snap-shots for the fluid sloshing against the soft beam are shownin Fig. 2. The deflection at the tip of the beam and the fluidpressure at its base are shown in Fig. 3. The simulated resultsobtained from the incompressible and quasi-incompressiblemethods agree because the quasi-incompressible approachincorporates a real value for the bulk modulus and the fluidflow is not turbulent.

To verify the DDM implementation for quasi-incompressible flow, gradients of the PFEM response are compared withthose obtained from finite difference computations. In thelimit as the parameter perturbation decreases, the gradientscomputed by finite differences should converge to the ana-lytical derivative found via the DDM

lim�θ→0

u(θ + �θ) − u(θ)

�θ= ∂u

∂θ(47)

where �θ is the magnitude of parameter perturbation. Aforward finite difference is shown in Eq. (47); however, back-ward and central finite differences can also be utilized.

The FDM is run with a small perturbation ε = 1e − 8,where ε = �θ/θ , for tip deflection, u, as shown in Fig. 4a, bfor parameters of the beam elastic modulus, E , and fluiddensity, ρf . The results show that the FDM matches theDDM implementation of the quasi-incompressible formula-tion; however, due to geometric nonlinearity in both the fluidand structural domains (Fig. 2), and viscous fluid flow, theDDM results computed by the incompressible formulationdiverge from the quasi-incompressible results. This is due tothe approximation made in the incompressible formulationwhere the matrices H, T, and K are neglected on the left-hand side of Eq. (40) when computing the DDM sensitivity.The displacement sensitivity computed by the DDM remainsbounded and on the same order of magnitude as the ordinarydisplacement response shown in Fig. 3a, i.e., E ∂u

∂E is O(u),implying that the DDM sensitivity computation is well con-ditioned. Similar trends are shown in Fig. 4c, d for sensitivityof the base pressure, p, with respect to elastic modulus andfluid density.

6 Breaking dam on nonlinear obstacle

The example of a breaking dam on an obstacle is a commonverification example for FSI [9,14,24]. A nonlinear materialis used for the obstacle here in order to highlight nonlinearFSI (Fig. 5). The characteristic length, L , of the initial reser-voir is equal to 0.146m. The obstacle height is 20b/3 = 0.08m, where b is its cross-section width b = 0.012 m, and ismodeled as a corotational mesh of displacement-based beamelements. Two Gauss points are used in each element wherecross-sections are discretized by 10 fibers, each with the uni-

123

Comp. Part. Mech.

0

0.05

0.1

y(m

)Time = 0.1 sec Time = 0.2 sec Time = 0.3 sec

0 0.05 0.10

0.05

0.1

x (m)

y(m

)

Time = 0.4 sec

0 0.05 0.1x (m)

Time = 0.5 sec

0 0.05 0.1x (m)

Time = 1.0 sec

Fig. 2 Snapshots of fluid sloshing against a soft beam at different time steps

(a) (b)

Fig. 3 Comparison between incompressible and quasi-incompressible solutions of tip deflection u and base pressure p

axial bilinear stress–strain response shown in Fig. 6a. Thestress–strain behavior is bilinear and path-dependent, andis characterized by parameters of yield strength Fy = 5e4Pa, initial elastic modulus E0 = 106 Pa, and strain hard-ening ratio b = 0.02. The material density is ρs = 2500kg/m3, while the fluid density is ρf = 1000 kg/m3 and itsviscosity is μ = 1.0 kg/ms. The bulk modulus of the fluid isκ = 2.15 × 109 Pa. A mesh size of 0.005 m generates 2654

elements and 6751 nodes including pressure, velocity, andbubble nodes. The simulation time step is �t = 0.001 s.

The tip deflection of the nonlinear obstacle using theincompressible and quasi-incompressible fluid formulationsis shown in Fig. 6b. The slight difference of the deflection isdue to the small compressibility of the quasi-incompressiblefluid which absorbs strain energy that would otherwise gointo deformation of the beam. Impact occurs at about 0.15 s,

123

Comp. Part. Mech.

(a) (b)

(c) (d)

Fig. 4 Comparison between incompressible and quasi-incompressible solutions of the sensitivity of tip deflection u and base pressure p withrespect to elastic modulus E and fluid density ρf

Fig. 5 Model of breaking damon nonlinear obstacle

the yielding of the nonlinear obstacle happens at about 0.19 s,and the steady-state deflection of approximately 3 mm isreached after about 1.0 s. The simulation snapshots in Fig. 7

show significant geometric nonlinearity of both the fluid andstructural domains.

The sensitivity solutions of the quasi-incompressibleDDM are compared with the incompressible DDM and ver-

123

Comp. Part. Mech.

(a) (b)

Fig. 6 Comparison between incompressible and quasi-incompressible solutions of tip deflection u and the stress–strain relationship of the nonlinearmaterial

Fig. 7 Snapshots for breakingdam on nonlinear obstacle atdifferent time steps

(a) (b)

Fig. 8 Comparison between incompressible and quasi-incompressible solutions of the sensitivity of the tip deflection u with respect to E and Fy

123

Comp. Part. Mech.

ified by the FDM with ε =1e−12 as shown in Fig. 8. A verysmall ε is used so that the mesh is the same for all time stepsin the repeated analyses required by the FDM. The sensitivityof the tip deflection to the elastic modulus, E , and the yieldstress, Fy, become nonzero at the time of impact and beamyielding, respectively. Due to the large changes in the fluidmesh and large displacements of the beam, the DDM sensi-tivity computed by the incompressible formulation beginsto diverge from the quasi-incompressible formulation atapproximately t = 0.25 s. The divergence owes to neglectingthe fluid viscosity matrix K and approximation of the geo-metric matricesH andT from Eq. (40). The DDM sensitivityobtained from the quasi-incompressible formulation remainsstable and well conditioned throughout the simulation andmatches the finite difference computations. This further ver-ifies the implementation of the quasi-incompressible DDMfor highly changing fluid meshes and viscous fluid flow.

7 Conclusion

The direct differentiation method (DDM) is applied to thequasi-incompressible fluid formulation for sensitivity analy-sis of fluid–structure interaction (FSI) using the particlefinite element method (PFEM). The advantage of the quasi-incompressible formulation for FSI sensitivity analysis isthat it includes geometric nonlinearity and fluid viscosityfor more accurate sensitivity computations. This approach isalso efficient since the response sensitivity can be computed,without any approximations, using the same unified FSMsolver as that used for the response. The examples validatethe currentmethodwith finite difference solutions. In cases ofhigh geometric nonlinearity and nonzero fluid viscosity, theincompressible formulation was shown to diverge from thecorrect results after the sensitivity becomes nonzero, whilethe quasi-incompressible formulation yields accurate resultsfor the response sensitivity throughout the simulations.

Acknowledgments This material is based on the work supported bythe National Science Foundation under Grant No. 0847055. Any opin-ions, findings, and conclusions or recommendations expressed in thismaterial are those of the authors and do not necessarily reflect the viewsof the National Science Foundation.

Appendix 1: MINI element

The MINI fluid element adds a bubble node for velocity atits center in order to satisfy the LBB condition [1].

Using the nodes in Fig. 9, the shape functions for velocityand pressure in a 2D MINI element are defined as

N v1 =L1, N v

2 = L2, N v3 = L3, N v

4 = 27L1L2L3

(48)

N p1 =L1, N p

2 = L2, N p3 = L3 (49)

where L1, L2, and L3 are area coordinates [34]

L1 + L2 + L3 = 1 (50)

The spatial derivatives of the velocity and pressure shapefunctions used in the viscous matrix and gradient operatordefined in Eqs. (14) and (17) are evaluated as

∂N va

∂xi= ∂N p

a

∂xi= 1

A

[cada

], a = 1, 2, 3 (51)

and

∂N 4v

∂xi= 27

A

[c1L2L3 + c2L3L1 + c3L1L2

d1L2L3 + d2L3L1 + d3L1L2

](52)

where A is twice the element area

A = x2y3 − x3y2 + x3y1 − x1y3 + x1y2 − x2y1 (53)

and the intermediate variables are

c1 = y2 − y3, c2 = y3 − y1, c3 = y1 − y2

d1 = x3 − x2, d2 = x1 − x3, d3 = x2 − x1(54)

The current coordinates xi and yi of each corner node aredetermined from the nodal displacements relative to theirinitial coordinates, x0i and y0i , at the start of the simulation

x1 = x01 + u1, x2 = x02 + u2, x3 = x03 + u3

y1 = y01 + w1, y2 = y02 + w2, y3 = y03 + w3(55)

where ui and wi are the horizontal and vertical nodal dis-placements. Using the shape functions of the MINI element,the mass matrix in Eq. (13) can be expressed in closed-formas

Mab = ρA

12I2, a = b, Mab = ρA

24I2, a �= b (56)

M4b = Ma4 = 3ρA

40I2, M44 = 81ρA

560I2, (57)

while the fluid viscous matrix in Eq. (14) is

Kab = μ

6A

[4cacb + 3dadb 3dacb − 2cadb3cadb − 2dacb 4dadb + 3cacb

],

a, b = 1, 2, 3 (58)

K4b = Ka4 = 0 (59)

K44 = 27μ

40A

[4

∑(ca)2 + 3

∑(da)2

∑(cada)∑

(cada) 4∑

(da)2 + 3∑

(ca)2

]

(60)

123

Comp. Part. Mech.

Fig. 9 Pressure and velocitynodes of the MINI fluid element

(a) (b)

The gradient operator in Eq. (17) is expressed as

Gab = 1

6

[cada

], G4b = − 9

40

[cbdb

], a, b = 1, 2, 3

(61)

and the external force vector in Eq. (19) is

Fexta = ρA

6b, Fext

4 = 9ρA

40b, a = 1, 2, 3 (62)

Finally, the pressure mass matrix in Eq. (20) is evaluated andlumped

Mp = A

6κ

⎡⎣1 0 00 1 00 0 1

⎤⎦ (63)

The expressions in Eqs. (56)–(63) will be used to show howthe sensitivity equations are obtained from differentiation ofthe response equations.

Appendix 2: Evaluation of geometric derivatives

The geometric derivatives in Eqs. (33) and (34) are evaluatedexactly using the MINI element formulation. This appendixdemonstrates evaluation of the derivatives in matricesH andT but can be extended to any other fluid elements. First, thederivatives of the element area A defined in Eq. (53) areevaluated

∂A

∂u1= c1,

∂A

∂u2= c2,

∂A

∂u3= c3 (64)

∂A

∂w1= d1,

∂A

∂w2= d2,

∂A

∂w3= d3 (65)

Then, the derivatives of the variables defined in Eq. (54) areevaluated, for example,

∂c1∂u1

= 0,∂c1∂u2

= 0,∂c1∂u3

= 0 (66)

∂c1∂w1

= 0,∂c1∂w2

= 1,∂c1∂w3

= −1 (67)

The derivatives of c2, c3, d1, d2, and d3 can be obtainedsimilarly.With the derivatives in Eqs. (64)–(67), the matricesH andT in Eqs. (35) and (36) can be evaluated on a columnbycolumn basis. For the 2D MINI element, both H and T havesix columns, each of which corresponds to the evaluationof the partial derivatives in Eqs. (35) and (36) with respectto u1, u2, u3, w1, w2, and w3 for columns one through six,respectively. For instance, the first column of the matricesHand T corresponds to the displacement u1

H1 = ∂M∂u1

v + ∂K∂u1

v − ∂G∂u1

p − ∂Fext

∂u1(68)

T1 = ∂Mp

∂u1p + ∂GT

∂u1v (69)

where

∂M∂u1

= ∂M∂A

∂A

∂u1,

∂Fext

∂u1= ∂Fext

∂A

∂A

∂u1,

∂Mp

∂u1= ∂Mp

∂A

∂A

∂u1(70)

∂K∂u1

= ∂K∂A

∂A

∂u1+

3∑a=1

(∂K∂ca

∂ca∂u1

)

+3∑

a=1

(∂K∂da

∂da∂u1

)(71)

123

Comp. Part. Mech.

∂G∂u1

=3∑

a=1

(∂G∂ca

∂ca∂u1

)+

3∑a=1

(∂G∂da

∂da∂u1

)(72)

∂GT

∂u1=

3∑a=1

(∂GT

∂ca

∂ca∂u1

)+

3∑a=1

(∂GT

∂da

∂da∂u1

)(73)

The derivatives ∂M∂A ,

∂K∂A ,

∂K∂ca

, etc. are evaluated based onEqs. (56)–(63).

References

1. Arnold D, Brezzi F, Fortin M (1984) A stable finite element for thestokes equations. Math Stat 21(4):337–344

2. Bebamzadeh A, Haukaas T, Vaziri R, Poursartip A, Fernlund G(2010)Application of response sensitivity in composite processing.J Compos Mater 44(15):1821–1840

3. Becker P, Idelsohn S, Oñate E (2015) A unified monolithicapproach for multi-fluid flows and fluidstructure interaction usingthe particle finite element method with fixed mesh. Comput Mech55(6):1091–1104

4. Chock G, Carden L, Robertson I, Olsen M, Yu G (2013)Tohoku tsunami-induced building failure analysis with implica-tions for U.S. tsunami and seismic design codes. Earthq Spectra29(S1):S99–126

5. Chock GYK, Robertson I, Riggs HR (2011) Tsunami struc-tural design provisions for a new update of building codes andperformance-based engineering. In: ASCE solutions of coastal dis-asters (COPRI). Anchorage, pp 423–435

6. Chopra AK (2007) Dynamics of structures: theory and applica-tions to earthquake engineering, 3rd edn. Prentice-Hall, EnglewoodCliffs

7. Conte JP, Barbato M, Spacone E (2004) Finite element responsesensitivity analysis using force-based frame models. Int J NumerMethods Eng 59(13):1781–1820

8. Conte JP, Vijalapura PK, Meghalla M (2003) Consistent finite-element response sensitivity analysis. J Eng Mech 129(12):1380–1393

9. Hübner B, Walhorn E, Dinkler D (2004) A monolithic approach tofluid-structure interaction using space-time finite elements. Com-put Methods Appl Mech Eng 193(23–26):2087–2104

10. Fujimura K, Kiureghian AD (2007) Tail-equivalent linearizationmethod for nonlinear random vibration. Prob Eng Mech 22(1):63–76

11. Gu Q, Barbato M, Conte J, Gill P, McKenna F (2012) Opensees-snopt framework for finite-element-based optimization of struc-tural and geotechnical systems. J Struct Eng 138(6):822–834

12. Guo Q, Jeffers A (2014) Direct differentiation method for responsesensitivity analysis of structures in fire. Eng Struct 77:172–180

13. Haukaas T, Scott MH (2006) Shape sensitivities in the reliabilityanalysis of nonlinear frame structures. Comput Struct 84(15–16):964–977

14. Idelsohn S,Marti J, Limache A, Oñate E (2008) Unified lagrangianformulation for elastic solids and incompressible fluids: applica-tion to fluid-structure interaction problems via the pfem. ComputMethods Appl Mech Eng 197:1762–1776

15. Kiran R, Li L, Khandelwal K (2016) Complex perturbationmethodfor sensitivity analysis of nonlinear trusses. J Struct Eng. To appear

16. Kleiber M, Antunez H, Hien T, Kowalczyk P (1997) Parametersensitivity in nonlinear mechanics. Wiley, New York

17. Mase GT, Smelser RE, Mase GE (2009) Continuummechanics forengineers, 3rd edn. CRC Press, Boca Raton

18. McAllister T (2014) The performance of essential facilities inSuperstorm Sandy. In: ASCE structures congress. Boston, pp2269–2281

19. McKenna F, Fenves GL, Scott, MH (2000) Open system for earth-quake engineering simulation. University of California, Berkeley.http://opensees.berkeley.edu

20. McKenna F, ScottMH, Fenves GL (2010) Nonlinear finite-elementanalysis software architecture using object composition. J ComputCivil Eng 24(1):95–107

21. Oñate E, Franci A, Carbonell JM (2014) Lagrangian formula-tion for finite element analysis of quasi-incompressible fluids withreduced mass losses. Int J Numer Methods Fluids 74(10):699–731

22. Oñate E, Idelsohn S, Pin FD, Aubry R (2004) The particle finiteelement method. An overview. Int J Comput Methods 1(2):267–307

23. Oñate E, Idelsohn SR, Celigueta MA, Rossi R (2007) Advancesin the particle finite element method for the analysis of fluid-multibody interaction and bed erosion in free surface flows.Comput Methods Appl Mech Eng 197(19–20):1777–1800

24. Ryzhakov P, Rossi R, Idelsohn S, Oñate E (2010) A mono-lithic Lagrangian approach for fluid-structure interaction problems.Comput Mech 46:883–899

25. Scott MH, Filippou FC (2007) Exact response gradients forlarge displacement nonlinear beam-column elements. J Struct Eng133(2):155–165

26. Scott MH, Franchin P, Fenves GL, Filippou FC (2004) Responsesensitivity for nonlinear beam-column elements. J Struct Eng130(9):1281–1288

27. Simo JC, Hughes TJR (1998) Computational inelasticity. Springer,New York

28. ZhangY,DerKiureghianA (1993)Dynamic response sensitivity ofinelastic structures. Comput Methods Appl Mech Eng 108:23–36

29. Zhu M, Scott MH (2014) Improved fractional step method forsimulating fluid-structure interaction using the pfem. Int J NumerMethods Eng 99(12):925–944

30. Zhu M, Scott MH (2014) Modeling fluid-structure interaction bythe particle finite element method in OpenSees. Comput Struct132:12–21

31. Zhu M, Scott MH (2016) Direct differentiation of the particlefinite element method for fluid-structure interaction. J Struct Eng142(3):04015159

32. Zhu M, Scott MH (2016) Unified fractional step method forlagrangian analysis of quasi-incompressible fluid and nonlinearstructure interaction using the pfem. Int J Numer Methods Eng.doi:10.1002/nme.5321

33. ZienkiewiczO, Taylor R (2005) The finite elementmethod for solidand structural mechanics, vol 2, 6th edn. Elsevier, Oxford

34. Zienkiewicz O, Taylor R, Zhu J (2005) The finite element method:its basis and fundamentals, vol 1, 6th edn. Elsevier, Butterworth-Heinemann, Oxford

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