A corrected transport velocity formulation (CTVF) based solver is developed to study the deformation of elastic structures under hydrodynamic loads. CTVF is adopted to handle both ﬂuid and structural dynamics. The current solver’s advantage is to eliminate inhomogeneous particle distribution in ﬂuid ﬂow and tensile instability in elastic structures. A ghost particle-based approach is used to handle the coupling between the ﬂuid and solid phases. The proposed scheme is veriﬁed by simulating a few test cases, which we validate with exact analytical solutions. As an application, we simulate an elastic plate deformation due to dam-breaking ﬂuid. Keywords: Smoothed particle hydrodynamics; Fluid-structure interaction; Transport velocity formulation I.INTRODUCTION Fluid-structure interaction (FSI) is a common engineering problem that is seen in daily life. Some examples include the deformation of the wind turbine blade due to the ﬂuid ﬂow, the ﬂow traversal due to the deﬂected blade, blood ﬂow in heart value, coastal engineering, and vortex-induced vibration [28], [5]. An accurate study of FSI can allow us to optimize the systems where FSI is dominant. However, studying the FSI phenomena through experiments or analytical techniques is complex due to its nonlinear behavior. Mesh-based schemes such as ﬁnite element method (FEM) [14] and ﬁnite volume method (FVM) [ 11] have been used for the last few decades in modeling the FSI problems. However, mesh-based methods are not favorable when dealing with free surface ﬂow problems or problems involving large deformation of the structure. This is due to explicit free surface tracking, and mesh distortion [ 16] while dealing with large deformation solids. Therefore, meshless methods are preferred while handling FSI problems involving free surfaces, multiphase ﬂows, and large deformation in solids. The smoothed particle hydrodynamics (SPH) and material point method (MPM) are more commonly used to model the ﬂuid phase. While the solids are modeled with SPH or Reproducing Kernel Particle Methods (RKPM), or the Discrete Element Method (DEM) [ 9], [13]. These meshless techniques have been coupled for the past two decades to model the ﬂuid-structure interaction. A few schemes with SPH and MPM are SPH-DEM [ 29], SPH-TLSPH [ 22], SPH-RKPM [ 19], SPH-Peridynamics [ 26], MPM-DEM [ 23]. For more, see the review by [ 12]. In the current work, we use SPH to handle FSI problems. In SPH, ﬂuids are modeled primarily using two approaches, one by assuming they are weakly compressible and another by considering them incompressible. Though SPH is successful in modeling ﬂuid ﬂows, it suffers from particle pairing and irreg- ular particle distribution problems resulting in poor functionapproximation [ 21] and unstable simulations. To homogenize the particles, [ 30] adjusts the particle positions after each timestep. Then, the particle properties are adjusted to the new position using Taylor series approximation. Later, Adami [ 2] proposed a transport velocity formulation (TVF) scheme where the particles are moved with a ”transport velocity” rather than the momentum velocity. TVF is applied to internal ﬂuid ﬂow problems. Zhang [ 32] extended the TVF formulation and called it the generalized TVF (GTVF) and applied it to free surface problems and elastic solids. However, while extending TVF, GTVF has missing terms and is sensitive to the particle shifting technique (PST). [ 3] proposed a corrected TVF (CTVF) scheme where the missing terms in TVF are incorporated and is robust to the PST. While handling elastic solids with SPH, the classical SPH [ 8] faces tensile instability and results in numerical fracture. [8] has proposed adding artiﬁcial stress, [ 7] introduces stress- points in addition to the existing particles, [ 24] modiﬁes the equation of state to eliminate the tensile instability. CTVF can eliminate tensile instability and handle higher Poisson ratios. While CTVF is also robust to different PST. FSI in SPH is modeled by several works, such as, WCSPH-Total Lagrangian SPH (TLSPH) [ 31], WCSPH-Updated Lagrangian SPH (ULSPH) [4], ISPH-TLSPH[22]. Handling FSI problems with the transport velocity formu- lation framework is advantageous as it can solve the tensile instability issue in solid dynamics and inhomogeneous particle distribution in ﬂuids. In the current work, we handle FSI problems by the CTVF method, where both ﬂuids and solid phases are modeled using CTVF alone. To validate the proposed method, we consider three numerical test cases. A uniformly distributed load over a clamped beam (UDL) problem is considered to validate the elastic dynamics of CTVF. An aluminum plate over a hydrostatic tank for FSI validation is considered. Finally, it is applied to a ﬂuid ﬂow hitting an elastic plate. Here, the deformation of the elastic plate is compared against the experimental results. A convergence analysis is undertaken for both UDL and elastic deformation under hydrodynamic load problems. II. METHODOLOGY We follow CTVF formulation to model the ﬂuid and solid phase. Following CTVF, the particles are moved with a transport velocity rather than the momentum velocity, with which we get a homogenized particle distribution as well as it eliminates the tensile instability. In the next two sections we show the discretized equations of both ﬂuid and solid phase. Please see [3] for more details. A.Discrete equations of the ﬂuid medium The governing equations of the ﬂuid are conservation of mass and momentum. Following the weakly compressible SPH 1 scheme, we use an equation of state to complete the system. The SPH discretization of the continuity equation (1)and the EDAC based [ 20] pressure evolution equation (2)respectively are, ~da dt=X bmb b(a~uab+ ((~uu))ab)raWab;(1) ~dpa dt=X bmb b (paac2 s)uab+pa~uab(p(~uu))ab+ 4edacpapb (a+b)(r2 ab+ 0:01h2 ab)rab raWab:(2) Where~d dtis the material derivative, with ~ubeing the transport velocity of the particles, uab=uaub. Similarly, the discretized momentum equation for ﬂuids is written as, ~dua dt=X bmbpa 2a+pb 2 b IAa 2a+Ab 2 b+abI raWab +uaX bmb b~uabraWab +X bmb4rWabrab (a+b)(r2 ab+ 0:01h2 ab)uab+ga+Fa FSI ma(3) where Aa=aua (~ uaua),Iis the identity matrix, is the kinematic viscosity of the ﬂuid and [ 17] formulation is used to discretize the viscosity term. abis the artiﬁcial viscosity [ 15] to maintain the stability of the numerical scheme. It is given as, ab=(  habcab ab abuabrab<0; 0 uabrab0;(4) where, ab=uabrab r2 ab+ 0:01h2 ab; (5) where, rab=rarb,hab= (ha+hb)=2,ab= (a+ b)=2,cab= (ca+cb)=2, and is the artiﬁcial viscosity parameter. Fa FSIis the force due to the interaction with elastic structure. This force modeling is explained in section II-D. We utilize the ghost particle approach proposed in [ 1] to handle the boundaries. B.Solid phase modeling Similar to discretized ﬂuid governing equation, the elastic structure equations are (1), and the momentum equation, ~dua dt=X bmbpa 2a+pb 2 b I0 a 2a+0 b 2 b+abI raWab +ga+Fa FSI ma;(6) while the additional stress terms, Ain momentum equation of solid mechanics are not considered as it has a negligible effect. An equation of state is utilized to link the pressure with density following a weakly compressible SPH scheme. Jaumann stress rate equation is solved to evolve the shear stress, given as, ~d0 a dt= 2G(_a1 3_aI) + 0 a T a+ a0 a(7)The SPH discretization of the gradient of velocity is given as, rua=X bmb b(uaub) (raWab); (8) where is the outer product. With the strain and rotation tensors as, _ij=1 2@ui @xj+@uj @xi ; (9) and ijis the rotation tensor, ij=1 2@ui @xj@uj @xi : (10) C.Transport velocity computation The particles in the current scheme are moved with the transport velocity, dra dt=~ ua: (11) The transport velocity is updated using, ~ ua(t+ t) =ua(t) + t~dua dt+dua dt ct (12) Wheredua dt cis the homogenization acceleration which ensures that the particle positions are homogeneous. In the current work we have used Sun’s [ 25] PST to homogenization the ﬂuid medium while iterative PST (IPST)[ 10] for the solid phase. According to [3] the force formulation of [25] is, dua dt c=M(2h)c0 tX b 1 +RWab W(x)n raWabVb; (13) whereRis an adjustment factor to handle the tensile instability, andMis the mach number of the ﬂow. Vbis the volume of thebthparticle,c0is the speed of sound. The acceleration is modiﬁed to account for particles on the free surface. Here, R= 0:2andn= 4 are used. Please see [ 3] for detailed explanation of PST. D.Fluid-structure interaction Coupling is handled in a straight forward way in SPH. While modelling the ﬂuid phase and treating the ﬂuid-structure interactions, the structure particles are assumed to be boundary particles. From the boundary handling given in Adami [ 1], we compute the pressure of the boundary particles from the extrapolated equation as, ps=fpfWsf+ (gas)ffrsfWsf fWsf: (14) Here, asis the acceleration of the structure particles. The subscriptfdenotes the ﬂuid particles and sdenotes the structure particles. By the extrapolated pressure, hydrodynamic density of structure properties are computed. Please note that the pressure we set here are only pertaining to the FSI force and does not correspond to the real pressure or density of the structure particles. By utilizing the previously set hydrodynamic properties on the structure, the interaction force is computed using, Fi FSI=miX amapi 2 i+pa 2a+ ia riW(xia) (15) whereiis ﬂuid particle, ais structure particle. 2 III. RESULTS AND DISCUSSION A. Uniformly distributed loading (UDL) on a clamped beam In the ﬁrst test case, we validate the structural part of the current solver. We chose a homogeneous elastic plate clamped on both ends acted upon by a uniformly distributed load ( q= 20 Nm-1) as shown in ﬁg. 1. The beam’s length (L) and height (H) is 0.2 m and 0:012m, respectively. The mechanical properties Figure 1:The schematic of a clamped elastic beam being acted upon by a uniformly distributed load. of the plate are set as E= 107Pa in Young’s modulus, = 0in Poisson’s ratio and = 1000 kgm-3) in density. The numerical solution of the y-displacement at the center of the beam is compared against the analytical counterpart. The analytical solution for the deﬂection of a uniformly distributed beam clamped at both ends is given by L 2 =qL4 384D; (16) where,Dis deﬁned asEh3 12(1()2). We consider three particle resolutions such that, 10,15, and 20particles along the beam’s width are used. We run for a total physical time of 2seconds. Figure 2 depicts the time history of y-displacement of the beam center for different particle resolutions computed using the current solver compared against the analytical solution. From ﬁg. 2, we can see that the current solver can accurately predict the displacement of the clamped beam. Convergence of the current scheme is captured in ﬁg. 2, and the computational results are within a reasonable variation of the analytical solution with the variation of the particle spacing. B. Hydrostatic water column on an elastic plate In this example we study the deformation of an elastic plate due to the hydrostatic water column. We utilise the current example to examine the accuracy and convergence of the current solver. The schematic of ﬂuid with the elastic beam is shown in ﬁg. 3 along with the initial pressure distribution in the ﬂuid. The ﬁgure includes the dimensions as well. The material properties of the beam are, a density of 2700 kgm-3, with an Young’s modulus of 67:5GPa, and a Poisson ratio of 0:34. The material properties of the ﬂuid are, a density of 1000 kgm-3, with a dynamic viscosity of 0kgm-1s-1. We consider two particle resolutions such that we get 10,15and20particles along the width directing of the beam. We run the simulation for a total physical time of 3seconds. The y-displacement at the center of the beam is compared against the analytical with the current numerical solver for quantitative validation. Here, the beam deﬂection computed using an analytical expression results in a deﬂectiond=6:85105m. 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 time (seconds)0.000025 0.000020 0.000015 0.000010 0.000005 0.000000Y - amplitude (m)N = 10 N = 15 N = 20 AnalyticalFigure 2:Time variation of the y-displacement of the center of the beam for three different resolutions, compared against the analytical result. Figure 3:Schematic of the hydrostatic water column on an elastic plate. Fluid particle color represents pressure. Figure 4 shows the particle plot of the ﬂuid along with the elastic solid at time 2seconds with color of the ﬂuid particles describing the pressure. This snapshot corresponds to the highest particle resolution i.e., 20particles along the width direction. From the ﬁg. 4, we can see that the current solver produces a smooth pressure distribution demonstrating the stability of the current solver. Figure 5 depicts the time history of y-displacement of the beam center for different particle resolutions computed using the current solver compared against the analytical solution. From ﬁg. 5 we can see that the current solver is able to predict the displacement of the clamped beam within the vicinity of the analytical results. Also as the particle spacing is reduced, the beam displacement is converging as well. C. Water impact onto an elastic plate In this case, we study the deformation of the elastic plate due to the impact of a dam breaking ﬂow. Figure 6 shows the initial positions of ﬂuid and the structure inside the dam, including the dimensions. Following [ 27] we set the material 3 Figure 4:Snapshot of the ﬂuid and the elastic structure at time 0.5 sec including the pressure of the ﬂuid. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 time0.00012 0.00010 0.00008 0.00006 0.00004 0.00002 0.00000y - amplitudeAnalytical N=25 SPH-VCPM (Ng et al. 2020) N=10 N=15 N=20 Figure 5:The mid-span deﬂection of the structure under hydrodynamic loading with time for different resolutions, compared against the analytical and the numerical result of [18] properties of the elastic plate, a density of 2500 kgm-3, with an Young’s modulus of 106Pa, and a Poisson ratio of 0. The material properties of the ﬂuid are, a density of 1000 kgm-3, with a dynamic viscosity of 0 kgm-1s-1. A particle spacing of 5104m is taken, resulting in a total of 182911 particles, which includes ﬂuid, structure and solid wall. We run a total physical time of 0:7seconds. Here the ﬂuid is initially released which attains a certain velocity by the time it impacts the structure. The structure will obstruct the ﬂuid making it rise and the ﬂuid will deform the elastic plate. The ﬂuid will rise and hit the other end of the dam, following it comes back and hits the structure from the back. For a quantitative validation, we compare the current solver results to the other numerical techniques. Figure 6:Schematic of the dam-break ﬂow impacting an elastic plate. All dimensions are in meters. Figure 7 shows the snapshots of the ﬂuid and the elastic structure at different time instances. From ﬁg. 7, we can see that the ﬂuid after hitting the structure rises and hits the other end of the tank and travels back to hit the structure again. The time variation of the x-displacement of the elastic structure is compared against other numerical results [ 27], [6]. From the ﬁg. 8 we can see that the displacement computed by the current solver is with in a vicinity of the other results produced. 4 (a) (b) (c) Figure 7:(a) Snapshot of the ﬂuid and the structure at different time stamps - Water impact onto an elastic plate. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 time0.02 0.01 0.000.010.020.030.04x - amplitude Sun et al 2019, MPS-DEM Bogaers 2016, QN-LS Current solver Figure 8:Time histories of horizontal displacement of the free end of the elastic structure compared against the numerical results of [27], [6]- Water impact onto an elastic plate.IV. CONCLUSIONS CTVF is able eliminate several issues SPH faces while solving ﬂuid and solid problems. Through particle shifting techniques as well as incorporating the missing terms CTVF is able to produce better approxmiation in simulating ﬂuid prob- lmes, also it is adaptable to different PST techniqes. Similarly with PST the newly CTVF eliminates tensile instability while solving the elastic dynamics probnlems. With the adavantages in mind, we solve both ﬂuid and solid phases in simulating FSI problems with CTVF with this approach we are able to handle FSI problems. We validated the current schemes by solving a UDL problem to test the structure equations and an aluminium plate over a hydrostatic tank where an analytical solution is available is utilized to validate the FSI part of the current solver. The current solver is applied to wave front arising due to a dam break hitting a steady elastic plate, here the deformation of the elastic plate is compared against the experimental results. A convergence analysis is undertaken for both fundamental benchmarks, UDL and hydrostatic tank. For the future work, we plan to extend the current solver being extended to handle the anisotropic structures and 3D FSI problems.