Towards development of enhanced fully-Lagrangian mesh-free computational methods for fluid-structure interaction *

Abbas Khayyer, Hitoshi Gotoh, Hosein Falahaty, Yuma Shimizu

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Towards development of enhanced fully-Lagrangian mesh-free computational methods for fluid-structure interaction*

Abbas Khayyer, Hitoshi Gotoh, Hosein Falahaty, Yuma Shimizu

Simulation of incompressible fluid flow-elastic structure interactions is targeted by using fully-Lagrangian mesh-free computational methods. A projection-based fluid model (moving particle semi-implicit (MPS)) is coupled with either a Newtonian or a Hamiltonian Lagrangian structure model (MPS or HMPS) in a mathematically-physically consistent manner. The fluid model is founded on the solution of Navier-Stokes and continuity equations. The structure models are configured either in the framework of Newtonian mechanics on the basis of conservation of linear and angular momenta, or Hamiltonian mechanics on the basis of variational principle for incompressible elastodynamics. A set of enhanced schemes are incorporated for projection-based fluid model (Enhanced MPS), thus, the developedcoupled solvers for fluid structure interaction (FSI) are referred to as Enhanced MPS-MPS and Enhanced MPS-HMPS. Besides, two smoothed particle hydrodynamics (SPH)-based FSI solvers, being developed by theauthors, are considered and their potential applicability and comparable performance are briefly discussed in comparison with MPS-based FSI solvers. The SPH-based FSI solvers are established through coupling of projection-based incompressible SPH (ISPH) fluid model and SPH-based Newtonian/Hamiltonian structure models, leading to Enhanced ISPH-SPH and Enhanced ISPH-HSPH. A comparative study is carried out on the performances of the FSI solvers through a set of benchmark tests, including hydrostatic water column on an elastic plate, high speed impact of an elastic aluminum beam, hydroelastic slamming of a marine panel and dam break with elastic gate.

Fluid structure interaction (FSI), projection-based method, moving particle semi-implicit, incompressible smoothed particle hydrodynamics (ISPH), Hamiltonian MPS (HMPS), Hamiltonian SPH (HSPH)

Introduction

Fluid structure interactions (FSI) are often en- countered in various engineering fields, including coastal and ocean engineering. A great number of problems in coastal/ocean engineering involve highly interactive systems corresponding to violent fluid flows and deformable structures. Examples include tsunami/storm surge impact on coastal structures, wave interactions with floating structures, hydroelas- tic slamming of marine vessels. In such cases, precise information on spatial/temporal distributions of hydro- dynamic pressures and structural stresses/responses will be of crucial importance. Hence, accurate simu- lations of FSI problems corresponding to incompressi- ble fluid flows interacting with deformable elastic structures are of significant importance in coastal/ ocean engineering.

In view of intrinsic difficulties of numerical modeling of FSI problems usually encountered in coastal/ocean engineering, e.g., existence of violent free-surface flows as well as presence of large/abrupt hydrodynamics loads and consequently large structu- ral deformations, Lagrangian mesh-free methods (e.g., smoothedparticlehydrodynamics(SPH)[1,2]or moving particle semi-implicit (MPS)[3]) appear to be capable candidates for computational modeling of such im- portant phenomena. Accordingly, in several studies, simulations of FSI problems have been targeted by taking advantage of mesh-free methods[4-15]. In speci- fic, a number of fully-Lagrangian mesh-free coupled FSI solvers have been developed during the last decade[8-15]. However, most of the coupled FSI solvers have been founded on fully-explicit computations in which the stability of calculations is guaranteed through incorporation of numerical stabilizers that often require tuning and usually result in an unphysi- cal gap between fluid and structure. In addition, the instabilities associated with the state of stress in the structure models have been conventionally treated by using artificial terms rather than robust, mathemati- cally-sound and physically-consistent schemes e.g., Lagrangian kernel.

Considering the aforementioned challenges, a number of studies have targeted the development of fully-Lagrangian computational methods based on coupling between projection-based fluid models and Lagrangian structure models for simulation of FSI problems[12-15]. One of the distinct advantages that projection-based particle methods bring about for coupled FSI solvers corresponds to their prediction- correction solution process on the basis of Helmholtz- Leray decomposition[16]. Accordingly, continuity of normal stresses, velocities and volume can be gua- ranteed in a momentum-conservative manner[15]and without any artificial numerical treatment. Projection- based particle methods, such as ISPH or MPS can be coupled with Lagrangian structure models described in either Newtonian or Hamiltonian frameworks. The choice of Newtonian/Hamiltonian would depend on several aspects including the FSI problem of interest. Theoretically, Hamiltonian and Newtonian mechanics are equivalent. Numerically, however, Hamiltonian framework may provide advantages in strictly preser- ving conserved physical quantities such as energy, linear and angular momenta[17]as well as providing more flexibility for extensions to model complex physical systems[18]. Although the achievement of stable and precise numerical results in a Hamiltonian framework may require more careful considera- tions[19].

In this paper, four fully-Lagrangian mesh-free FSI solvers resulting from consistent coupling of projection-based fluid models and Lagrangian struc- ture models are considered. Two MPS-based FSI solvers are first introduced and then comparatively applied for simulation of four FSI benchmark tests. The fluid models are founded on the projection-based solutions of Navier-Stokes and continuity equations through a semi-implicit procedure. The Lagrangian structure models are established in the frameworks of either Newtonian, e.g., Kondo et al.[20], or Hamiltonian mechanics, e.g., Suzuki et al.[21]. The coupling between projection-based fluid model (MPS) and Lagrangian structural models (MPS or HMPS) is conducted in a mathematically-physically consistent manner via a careful attention to the concept of projection-based methods, i.e., Helmholtz-Leray de- composition[16]. A set of previously developed enhan- ced schemes are incorporated for projection-based fluid models[22-25], hence, the developed FSI solvers are referred to as Enhanced MPS-MPS and Enhanced MPS-HMPS FSI solvers. Besides, enhanced versions of SPH-based FSI solvers, currently being developed by authors, are considered and their potential applica- bility and comparable performance are portrayed and briefly discussed in comparison with MPS-based FSI solvers for one of the benchmark tests. The SPH- based FSI solvers are established through consistent coupling of incompressible SPH (ISPH)[26]and Newtonian or Hamiltonian SPH-based formulations of structural dynamics (leading to Enhanced ISPH-SPH and Enhanced ISPH-HSPH, respectively). The SPH- based FSI solvers, currently being developed by the authors, are not introduced in this paper and will be presented, validated and applied for simulation of FSI problems in our future papers. It should be noted that the fundamentals of MPS and SPH are comprehen- sively described in papers by Gotoh and Okayasu[23]and Gotoh et al.[27], Liu and Liu[28]and Violeau[29]. In addition, there have been excellent and comprehensive review papers on SPH[30, 31], MPS[32]as well as their applications for FSI[33].

The paper is organized as follows. In Section 1, MPS-based FSI solvers are concisely introduced. In Section 2, first the performances of MPS and HMPS structure models are verified in reproduction of the dynamic response of a free oscillating cantilever plate[34]. Then in Section 2.2, MPS-based FSI solvers are first verified through the simulation of hydrostatic water column on an elastic plate[4, 7]and then further verified in simulations of high speed impact of an elastic aluminum beam on undisturbed water sur- face[10, 35](Section 2.3), hydroelastic slamming of a marine panel[36-38](Section 2.4) and dam break with elastic gate[8](Section 2.5). Validations and compari- sons are conducted both qualitatively and quantita- tively in terms of reproduced hydrodynamic pressure fields as well as structural stresses and deflections. Finally, in Section 3, a set of concluding remarks are presented.

1. MPS-based FSI solvers

In this section, general descriptions are presented for the governing equations of MPS-based fluid and structure models incorporated in the developed FSI solvers. In addition, differential operator models applied for discretization of governing equations of structural dynamics corresponding to structure models are explained in sufficient details. In Section 1.3, the fluid-structure coupling scheme is briefly discussed.

1.1 Fluid model

The fluid model is founded on the solution of continuity and Navier-Stokes equations (Eqs. (1) and (2)) in the framework of projection-based methods, i.e., MPS or ISPH.

The MPS-based fluid model (as well as ISPH- based one) benefits from a set of enhanced schemes, i.e., the so-called higher-order source term of PPE (HS), higher-order Laplacian of PPE (HL), error com- pensating source of PPE (ECS), gradient correction (GC) and dynamic stabilization (DS). A fifth order Wendland kernel is utilized in all performed simu- lations, similar to that applied in Khayyer et al.[24, 25]. The mentioned schemes are described compre- hensively by Gotoh and Khayyer[22], Gotoh and Okayasu[23]and Khayyer et al.[24, 25].

1.2 Structure models

The fully-Lagrangian structure models are confi- gured either in the framework of Newtonian mecha- nics (so-called MPS-based or ISPH-based structure model) or Hamiltonian one (so-called HMPS or HSPH).

1.2.1 Newtonian dynamics - MPS structure model

The structure model in the context of Newtonian mechanics is set by MPS-based (or SPH-based) dis- cretization of equations for conservation of linear and angular momenta in a Lagrangian form[20]. The equa- tions for conservation of linear and angular momenta are described as:

Fig. 1 (Color online) Verification of MPS-based and HMSP- based structure models: (a) Set up of the benchmark test, (b) Snapshots corresponding to the reproduced stress fields at by MPS and HMPS structure models, respectively, (c) Time histories of the deflec- tion of the plate in the case of particle diameter , dynamic response of a free oscillating cantilever plate[34]

1.2.2 Hamiltonian dynamics-HMPS structure model

Fig. 2 (Color online) Snapshots of particles together with stress/pressure fields at reproduced by En- hancedMPS-MPS,MPS-HMPS,ISPH-SPHand ISPH- HSPH,?hydrostatic water column on elastic plate[4, 6, 7]

where and refer to the target structure particle and its neighboring structure particle, “” signifies double product of two tensors, denotes deforma- tion gradient tensor and represents the first Piola- Kirchhoff stress tensor whichis obtained based on Eq. (13)

where

With respect to the Hamiltonian-based formula- tion, a symplectic scheme as St?rmer-Verlet is used for time integration of Hamiltonian structure models.

1.3 Fluid-structure coupling

2. Verifications and comparisons

2.1 Dynamic response of a free oscillating cantilever elastic plate

In Fig.1(c), the simulated time histories of deflections of the plate?s free end, simulated by MPS andHMPS structure models, areshowntobeingood agreements with the analytical solution. In terms of amplitude, the displacements reproduced by HMPS are slightly more consistent with analytical solution in comparison to that of MPS, however, in terms of the period of oscillations, a slightly larger phase lag is observed in time history of HMPS with respect to that by MPS. Table 1 presents the root mean square error (RMSE) and normalized root mean square error (NRMSE) corresponding to the simulation results of dynamic response of the plate by MPS and HMPS structure models, respectively. The MPS structure modelhasresultedinsmallerRMSEduetosmaller phase lag in time history.

Table 1RMSE and NRMSE for dynamic response of the free oscillating cantilever plate by MPS and HMPS structure models

2.2 Hydrostatic water column on elastic plate

In order to investigate the robustness and accu- racyoffluid-structurecouplingscheme,theprojec- tion-based FSI solvers, i.e., Enhanced MPS-MPS and Enhanced MPS-HMPS, are employed in simulation of a simple but computationally challenging benchmark test corresponding to a hydrostatic water column on an elastic aluminum plate[4, 7].

After a few initial oscillations, the system will finally approach an equilibrium state. According to the theoretical solution[4, 44], at the equilibrium state, the magnitude of static displacement at the mid-span of aluminum plate due to the loading of 2 m high water column should be equal to-6.85′10-5m[4, 44].

Figure 3(b) presents the simulated time histories of plate’s midpoint displacement simulated by En- hanced ISPH-SPH and Enhanced ISPH-HSPH with respect to those of MPS-based FSI solvers. The results by the SPH and MPS-based solvers are comparable, despite slight overestimations by SPH-based ones.

2.3 High speed impact of elastic wedge on undistur- bed water surface

The Enhanced MPS-MPS and MPS-HMPS FSI solvers are applied for simulation of a high speed impact of an elastic aluminum beam wedge on undisturbed water surface, for which semi-analytical solutions exist[10, 35].

2.4 Hydroelastic slamming of a marine panel

For further validation of the developed FSI solvers, hydroelastic slamming of a homogeneous marinepanel,correspondingtotheexperimentsby Allen[36], is considered. These tests were conducted using a servo-hydraulic slam testing system (SSTS) with a set of different panels. In the present study, the hydroelastic slamming corresponding to a Solid Glass-fiber single skin panel[36-38]is reproduced.

Fig. 8(Color online) Qualitative comparison in between experimental photos and their corresponding snapshots by Enhanced MPS-MPS and Enhanced MPS-HMPS FSI solversdam break with elastic gate[8]

Fig. 9(Color online) Time histories of horizontal/vertical dis- placements of the free end of elastic gate by Enhanced MPS-MPS and Enhanced MPS-HMPSdam break with elastic gate[8]

2.5 Dam break with elastic gate

Figure 8 presents typical snapshots correspon- ding to the simulation of dam break with elastic gate by using Enhanced MPS-MPS and Enhanced MPS- HMPS together with their corresponding experimental photos[8].According to Fig. 8, both solvers have provided qualitatively accurate pressure/stress fields at fluid and structure partitions. The deformations of elastic gate are well consistent with fluid flow without formation of an unphysical gap between fluid and structure particles at the fluid-structure interface.

Figure 9 presents time histories of the horizontal and vertical displacements at the end point of rubber gate. From the presented figure, the reproduced displacements by Enhanced MPS-HMPS FSI solver are larger with respect to those by Enhanced MPS- MPS and more consistent with the experiment[8].

3. Concluding remarks

The paper aims at highlighting the potential capa- bilities of fully-Lagrangian mesh-free computational methods, established through consistent coupling of projection-based fluid models with Lagrangian struc- ture models, in providing stable/accurate simulation results for incompressible Fluid flow-elastic Structure Interactions (FSI). A distinct feature of proposed FSI solvers corresponds to their mathematically sound and physically consistent formulations, in absence of any artificial numerical stabilizers with tuning parameters.

The main considered projection-based fluid and Lagrangian structure models are discretized within the framework of MPS[3]through consideration of either Newtonian or Hamiltonian formulations of structural dynamics. A set of enhanced schemes[23-25], previously developed by authors, are incorporated to improve the accuracy of MPS fluid model. Thus, the coupled FSI solvers are referred to as Enhanced MPS-MPS and Enhanced MPS-HMPS (Hamiltonian MPS). Besides, potential applicability and comparable performance of enhanced versions of two SPH-based FSI solvers, currently being developed by authors, are briefly discussed. Followed by a simple validation of MPS- based structure models, the Enhanced MPS-MPS, MPS-HMPS, ISPH-SPH and ISPH-HSPH FSI solvers are applied for simulation of hydrostatic water column on an elastic plate[4, 7]. The fully-Lagrangian mesh- free FSI solvers with projection-based fluid models are allshown to possess appropriate and comparable levels of stability and accuracy, without the need for any artificial numerical stabilization schemes that may suffer from absence of a rigorous physical back- ground.

The Enhanced MPS-MPS and MPS-HMPS FSI solvers are employed in simulation of FSI benchmark tests including high speed impact of an elastic alumi- num beam on undisturbed water surface[10,35], hydroe- lastic slamming of a homogeneous marine panel[36-38]and dam break with an elastic gate[8]. Both MPS-based FSI solvers are shown to possess acceptable stability and accuracy. Despite significant potential advantages of Hamiltonian framework for describing the structural dynamics, at least for the considered benchmark test cases, almost comparable stability and accuracy are achieved for both HMPS and MPS as well as coupled MPS-HMPS and MPS-MPS FSI solvers. In future we need to extend this comparative study by considering highly non-linear elastic defor- mations as well as test cases of longer physical time in which preservations of conservative properties tend to become more challenging.

Other future works comprise of further valida- tions[45], extensions[46], enhancements, conservation studies[47]considerations of convergence tests, appli- cations to more violent fluid-structure interactions with turbulence[48]and air phase[49]modelings, leading to development of reliable multi-physics, multi-scale particle-based computational methods for practical scientific and engineering applications.

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(October 22, 2017, Accepted December 14, 2017)

?China Ship Scientific Research Center 2018

Abbas Khayyer (1979-), Male, Ph. D.,

Associate Professor

Abbas Khayyer,

E-mail: khayyer@particle.kuciv.kyoto-u.ac.jp