ABSTRACT
Improving the accuracy and order of convergence in the smoothed particle hydrodynamics method is still one of the challenges of the researchers. In this research, the first-order convergence scheme for the first spatial derivative in the continuity and conservation equations of linear momentum has been implemented in the DualSPHysics open-source code, in order to increase the accuracy and the order of convergence. By achieving more than three orders of improvement in the accuracy value, after implementing this method to simulate the Taylor-Green vortex benchmark problem in the authors' previous research, in this study, the simulation of the wavemaker benchmark problem by standard solvers and improved and compared with the analytical results through the calculation of the relative error of the velocity values at a certain position of the solution domain have been evaluated. The existence of stationary and moving solid boundary conditions as well as the free surface is one of the main challenges of the new problem. By using the improved solver and adjusting the effective parameters such as artificial viscosity value, density diffusion coefficient, neighborhood radius calculation coefficient and particle displacement coefficient, it is observed that the error reduction process is improved and its first-order convergence is achieved.
 
INTRODUCTION
Among the studies addressing particle approximation consistency, Taylor series expansion has been employed to correct the smoothing function approximation and its derivatives to enhance convergence rates (Liu and Liu, 2006). The use of a renormalization matrix to correct the gradient of the smoothing function and achieve first-order consistency represents another improvement, particularly for non-uniform particle distributions (Fourtakas et al., 2019; Sun et al., 2019). Among mesh-free methods, SPH has gained attention due to its advances in solving engineering problems (Violeau and Rogers, 2016; Manenti et al., 2019; Amicarelli et al., 2020; Khayyer, Rogers and Zhang, 2022). Nevertheless, significant non-physical pressure oscillations have been observed with this method (Gotoh and Khayyer, 2018; Wang et al., 2019). Extensive research is being conducted to enhance its stability, including pressure field consistency (Antuono et al., 2021; You et al., 2021), high-order methods (King, Lind and Nasar, 2020; Nasar et al., 2021), spatial anisotropy of particles (Rastelli et al., 2022), and performance improvements (Valdez-Balderas et al., 2013; Park et al., 2020). In the context of WCSPH, methods like Riemann-SPH (Inutsuka, 2002; Gao et al., 2012) and delta-SPH (Marrone et al., 2011; Antuono, Colagrossi and Marrone, 2012; Fourtakas et al., 2019) have been proposed to enhance the accuracy and stability of SPH, particularly to address non-physical pressure oscillations. The delta-SPH method reduces high-frequency numerical oscillations in the pressure field by adding an appropriate numerical dissipation term to the continuity equation. Moreover, efforts have been made to improve its boundary accuracy due to the method's inefficiencies in this regard (Han et al., 2013; Long et al., 2017; Zhang et al., 2018). Fatehi and colleagues (Fatehi et al., 2019) proposed a method using artificial time-stepping and another by adding a compensatory term to the momentum equation to achieve a zero-divergence velocity field. By comparing these with WCSPH for three incompressible problems, they achieved smoother pressure fields with less divergence. Since the conventional SPH method is considered a low-accuracy method, researchers' efforts have led to the development of various modified or improved versions of the SPH particle approximation schemes (Lind, Rogers and Stansby, 2020).
The objective of this research is to employ the gradient correction method for the smoothing function to reduce errors and enhance the convergence of solutions by implementing the differential form of the pressure gradient in the open-source software DualSPHysics. After verifying the improvement in results for the Taylor-Green vortex benchmark problem (Ravanbakhsh, Faghih and Fatehi, 2023), this study proceeds with the simulation of a wave-maker problem, which involves free surface boundary conditions as well as stationary and moving walls. The application of various criteria for free surface detection and the dynamic conditions governing it, along with the calculation of pressure on solid particles, are among the challenges addressed in this paper.
MATERIALS AND METHODS 
Various studies have been conducted to determine the suitable term to add to the continuity equation. An expression known as the density diffusion term, given by equation (11), has been used to reduce oscillations in the pressure/density field (Molteni and Colagrossi, 2009). The normalization tensor prevents the appearance of errors in the calculation of spatial derivatives for particles that do not have a uniform distribution around themselves by eliminating lower-order truncation errors (Fatehi and Manzari, 2012). The asymmetric form of equation (17) ensures first-order convergence and significantly reduces the error in the first-order spatial derivative based on particle approximation. To calculate the pressure on the wall, in the absence of shear stress in the fluid, equation (18) is used (Fatehi and Manzari, 2012). Since the normal component of acceleration is taken into account, the no-slip condition is not required, making it applicable for free-slip conditions as well. To detect the free surface, four criteria given in equation (19) are used, and in the code, simultaneous fulfillment of these criteria is used to apply the corresponding boundary conditions. In this problem, a piston-type wave generator on the left side of the solution domain (as shown in Figure (1)) creates a second-order regular wave through its oscillatory motion. The initial condition for the piston speed is zero, and the bottom of the solution domain is considered stationary. On the right side of the domain, to prevent wave reflections, wave damping is applied using equation (23) to absorb the wave energy.
RESULTS 
The most important parameters affecting the results are artificial viscosity value, DDT coefficient, neighborhood radius calculation coefficient, particle displacement coefficient, and time step calculation coefficient. Despite the satisfactory graphical results of Figure (2), for the distribution of particles and velocity vectors compared to the standard solver, quantitative evaluation of the results is necessary to calculate the error; Also, to summarize and compare the results, the SS symbol indicating the standard software solver has been introduced. The parameters of Table (3) are determined based on the fact that the simulation run after passing through the initial transient state must continue for 15 seconds or 5 cycles after 5 seconds without divergence to ensure the stability of the solution. Considering that the analytical results of this problem are available (Altomare et al., 2017); Error evaluation is done by calculating the norm of a relative error based on equation (22), about the value of horizontal and vertical velocities at a point with coordinates (6, 0, 0.4).
 DISCUSSION AND CONCLUSION
In this study, by using the differential discretization method of the gradient operator in the linear momentum conservation equation and using the normalizing matrix in this equation and the divergence operator of the continuity equation, the new wavemaker problem in the open-source software DualSPHysics is similar to was developed and the degree of convergence was improved compared to the standard software method and reached one. However, the challenge of the new problem is the existence of stationary, moving and free surface solid boundaries. Using various criteria for detecting the free surface and determining their values, as well as setting effective parameters is a time-consuming step for simulation convergence.
REFERENCES 
Altomare, C., Domínguez, J.M., Crespo, A.J.C., González-Cao, J., Suzuki, T., Gómez-Gesteira, M., Troch, P., 2017. Long-crested wave generation and absorption for SPH-based DualSPHysics model. Coast. Eng. 127, pp. 37–54. https://doi.org/10.1016/j.coastaleng.2017.06.004
Amicarelli, A., Manenti, S., Albano, R., Agate, G., Paggi, M., Longoni, L., Mirauda, D., Ziane, L., Viccione, G., Todeschini, S., Sole, A., Baldini, L.M., Brambilla, D., Papini, M., Khellaf, M.C., Tagliafierro, B., Sarno, L., Pirovano, G., 2020. SPHERA v.9.0.0: A Computational Fluid Dynamics research code, based on the Smoothed Particle Hydrodynamics mesh-less method. Comput. Phys. Commun. 250, p.107157. https://doi.org/10.1016/j.cpc.2020.107157
Antuono, M., Colagrossi, A., Marrone, S., 2012. Numerical diffusive terms in weakly-compressible SPH schemes. Comput. Phys. Commun. 183, pp.2570–2580. https://doi.org/10.1016/j.cpc.2012.07.006
Antuono, M., Colagrossi, A., Marrone, S., Lugni, C., 2011. Propagation of gravity waves through an SPH scheme with numerical diffusive terms. Comput. Phys. Commun. 182, pp.866–877. https://doi.org/10.1016/j.cpc.2010.12.012
Antuono, M., Sun, P.N., Marrone, S., Colagrossi, A., 2021. The δ-ALE-SPH model: An arbitrary Lagrangian-Eulerian framework for the δ-SPH model with particle shifting technique. Comput. Fluids 216, p.104806. https://doi.org/10.1016/j.compfluid.2020.104806
Bonet, J., Lok, T.S.L., 1999. Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations. Comput. Methods Appl. Mech. Eng. 180, pp.97–115. https://doi.org/10.1016/S0045-7825(99)00051-1
Bouscasse, B., Colagrossi, A., Marrone, S., Antuono, M., 2013. Nonlinear water wave interaction with floating bodies in SPH. J. Fluids Struct. 42, pp.112–129. https://doi.org/10.1016/j.jfluidstructs.2013.05.010
Chen, B.F., Nokes, R., 2005. Time-independent finite difference analysis of fully non-linear and viscous fluid sloshing in a rectangular tank. J. Comput. Phys. 209, 47–81. https://doi.org/10.1016/j.jcp.2005.03.006
De Chowdhury, S., Sannasiraj, S.A., 2014. Numerical simulation of 2D sloshing waves using SPH with diffusive terms. Appl. Ocean Res. 47,pp. 219–240. https://doi.org/10.1016/j.apor.2014.06.004
Domínguez, J.M., Fourtakas, G., Altomare, C., Canelas, R.B., Tafuni, A., García-Feal, O., Martínez-Estévez, I., Mokos, A., Vacondio, R., Crespo, A.J.C., Rogers, B.D., Stansby, P.K., Gómez-Gesteira, M., 2021. DualSPHysics: from fluid dynamics to multiphysics problems. Comput. Part. Mech. 9, pp.867–895. https://doi.org/10.1007/s40571-021-00404-2
Farzin, S., Fatehi, R., Hassanzadeh, Y., 2019. Position explicit and iterative implicit consistent incompressible SPH methods for free surface flow. Comput. Fluids 179, pp.52–66. https://doi.org/10.1016/j.compfluid.2018.10.010
Fatehi, R., Manzari, M.T., 2012. A consistent and fast weakly compressible smoothed particle hydrodynamics with a new wall boundary condition. Int. J. Numer. Methods Fluids 68, pp.905–921. https://doi.org/https://doi.org/10.1002/fld.2586
Fatehi, R., Rahmat, A., Tofighi, N., Yildiz, M., Shadloo, M.S., 2019. Density-based smoothed particle hydrodynamics methods for incompressible flows. Comput. Fluids 185, pp.22–33. https://doi.org/10.1016/j.compfluid.2019.02.018
Fourtakas, G., Dominguez, J.M., Vacondio, R., Rogers, B.D., 2019. Local uniform stencil (LUST) boundary condition for arbitrary 3-D boundaries in parallel smoothed particle hydrodynamics (SPH) models. Comput. Fluids 190, pp.346–361. https://doi.org/10.1016/j.compfluid.2019.06.009
Gao, R., Ren, B., Wang, G., Wang, Y., 2012. Numerical modelling of regular wave slamming on subface of open-piled structures with the corrected SPH method. Appl. Ocean Res. 34, pp.173–186. https://doi.org/10.1016/j.apor.2011.08.002
Gingold, R.A., Monaghan, J.J., 1982. Kernel estimates as a basis for general particle methods in hydrodynamics. J. Comput. Phys. 46, pp.429–453. https://doi.org/10.1016/0021-9991(82)90025-0
Gotoh, H., Khayyer, A., 2018. On the state-of-the-art of particle methods for coastal and ocean engineering. Coast. Eng. J. 60, pp.79–103. https://doi.org/10.1080/21664250.2018.1436243
Han, Y.W., Qiang, H.F., Zhao, J.L., Gao, W.R., 2013. A new repulsive model for solid boundary condition in smoothed particle hydrodynamics. Wuli Xuebao/Acta Phys. Sin. 62. https://doi.org/10.7498/aps.62.044702
Hashemi, M.R., Fatehi, R., Manzari, M.T., 2012. A modified SPH method for simulating motion of rigid bodies in Newtonian fluid flows. Int. J. Non. Linear. Mech. 47, pp.626–638. https://doi.org/10.1016/j.ijnonlinmec.2011.10.007
House, D., Keyser, J.C., 2020. Smoothed Particle Hydrodynamics, in: Foundations of Physically Based Modeling and Animation. A K Peters/CRC Press, Boca Raton : Taylor & Francis, a CRC title, part of the, pp. 305–314. https://doi.org/10.1201/9781315373140-26
Inutsuka, S.I., 2002. Reformulation of smoothed particle hydrodynamics with Riemann solver. J. Comput. Phys. 179, pp.238–267. https://doi.org/10.1006/jcph.2002.7053
Khayyer, A., Rogers, B.D., Zhang, A.M., 2022. Preface: Special Issue on Advances and Applications of SPH in Ocean Engineering. Appl. Ocean Res. 118, p.103028. https://doi.org/10.1016/j.apor.2021.103028
King, J.R.C., Lind, S.J., Nasar, A.M.A., 2020. High order difference schemes using the local anisotropic basis function method. J. Comput. Phys. 415, pp.109549. https://doi.org/10.1016/j.jcp.2020.109549
Lind, S.J., Rogers, B.D., Stansby, P.K., 2020. Review of smoothed particle hydrodynamics: Towards converged Lagrangian flow modelling: Smoothed Particle Hydrodynamics review. Proc. R. Soc. A Math. Phys. Eng. Sci. 476. https://doi.org/10.1098/rspa.2019.0801
Liu, M., Shao, J., Chang, J., 2012. On the treatment of solid boundary in smoothed particle hydrodynamics. Sci. China Technol. Sci. 55, pp.244–254. https://doi.org/10.1007/s11431-011-4663-y
Liu, M.B., Liu, G.R., 2006. Restoring particle consistency in smoothed particle hydrodynamics. Appl. Numer. Math. 56, pp.19–36. https://doi.org/10.1016/j.apnum.2005.02.012
Long, T., Hu, D., Wan, D., Zhuang, C., Yang, G., 2017. An arbitrary boundary with ghost particles incorporated in coupled FEM–SPH model for FSI problems. J. Comput. Phys. 350, pp.166–183. https://doi.org/10.1016/j.jcp.2017.08.044
Lucy, L.B., 1977. A numerical approach to the testing of the fission hypothesis. Astron. J. 82, p.1013. https://doi.org/10.1086/112164
Manenti, S., Wang, D., Domínguez, J.M., Li, S., Amicarelli, A., Albano, R., 2019. SPH modeling of water-related natural hazards. Water (Switzerland) 11, 1875. https://doi.org/10.3390/w11091875
Marrone, S., Antuono, M., Colagrossi, A., Colicchio, G., Le Touzé, D., Graziani, G., 2011. δ-SPH model for simulating violent impact flows. Comput. Methods Appl. Mech. Eng. 200, pp.1526–1542. https://doi.org/10.1016/j.cma.2010.12.016
Meringolo, D.D., Marrone, S., Colagrossi, A., Liu, Y., 2019. A dynamic δ-SPH model: How to get rid of diffusive parameter tuning. Comput. Fluids 179, pp.334–355. https://doi.org/10.1016/j.compfluid.2018.11.012
Molteni, D., Colagrossi, A., 2009. A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH. Comput. Phys. Commun. 180, pp.861–872. https://doi.org/10.1016/j.cpc.2008.12.004
Monaghan, J.J., 1994. Simulating Free Surface Flows with SPH. J. Comput. Phys. 110, pp.399–406. https://doi.org/10.1006/jcph.1994.1034
Monaghan, J.J., 1992. Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys. 30, pp.543–574. https://doi.org/10.1146/annurev.aa.30.090192.002551
Nasar, A.M.A., Fourtakas, G., Lind, S.J., King, J.R.C., Rogers, B.D., Stansby, P.K., 2021. High-order consistent SPH with the pressure projection method in 2-D and 3-D. J. Comput. Phys. 444, p.110563. https://doi.org/10.1016/j.jcp.2021.110563
Park, S.H., Jo, Y.B., Ahn, Y., Choi, H.Y., Choi, T.S., Park, S.S., Yoo, H.S., Kim, J.W., Kim, E.S., 2020. Development of Multi-GPU–Based Smoothed Particle Hydrodynamics Code for Nuclear Thermal Hydraulics and Safety: Potential and Challenges. Front. Energy Res. 8. https://doi.org/10.3389/fenrg.2020.00086
Price, D., 2005. Smoothed Particle Hydrodynamics. Reports Prog. Phys. 68, pp.1703–1759. https://doi.org/10.1088/0034-4885/68/8/R01
Randles, P.W., Libersky, L.D., 1996. Smoothed particle hydrodynamics: Some recent improvements and applications. Comput. Methods Appl. Mech. Eng. 139, pp.375–408. https://doi.org/10.1016/S0045-7825(96)01090-0
Rastelli, P., Vacondio, R., Marongiu, J.C., Fourtakas, G., Rogers, B.D., 2022. Implicit iterative particle shifting for meshless numerical schemes using kernel basis functions. Comput. Methods Appl. Mech. Eng. 393, p.114716. https://doi.org/10.1016/j.cma.2022.114716
Ravanbakhsh, H., Faghih, A.R., Fatehi, R., 2023. Implementation of improved spatial derivative discretization in DualSPHysics: simulation and convergence study. Comput. Part. Mech. 10, pp.1685–1696. https://doi.org/10.1007/s40571-023-00582-1
Sun, P.N., Colagrossi, A., Marrone, S., Antuono, M., Zhang, A.-M., 2019. A consistent approach to particle shifting in the δ-Plus-SPH model. Comput. Methods Appl. Mech. Eng. 348, pp.912–934. https://doi.org/10.1016/j.cma.2019.01.045
Sun, P. N., Colagrossi, A., Marrone, S., Antuono, M., Zhang, A.M., 2018. Multi-resolution Delta-plus-SPH with tensile instability control: Towards high Reynolds number flows. Comput. Phys. Commun. 224, pp.63–80. https://doi.org/10.1016/j.cpc.2017.11.016
Sun, P.N., Colagrossi, A., Marrone, S., Zhang, A.M., 2017. The δplus-SPH model: Simple procedures for a further improvement of the SPH scheme. Comput. Methods Appl. Mech. Eng. 315, pp.25–49. https://doi.org/10.1016/j.cma.2016.10.028
Sun, Peng Nan, Colagrossi, A., Zhang, A.M., 2018. Numerical simulation of the self-propulsive motion of a fishlike swimming foil using the δ+-SPH model. Theor. Appl. Mech. Lett. 8, pp.115–125. https://doi.org/10.1016/j.taml.2018.02.007
Takeda, H., Miyama, S.M., Sekiya, M., 1994. Numerical Simulation of Viscous Flow by Smoothed Particle Hydrodynamics. Prog. Theor. Phys. 92, pp.939–960. https://doi.org/10.1143/ptp/92.5.939
Toma, M., Chan-Akeley, R., Arias, J., Kurgansky, G.D., Mao, W., 2021. Fluid–structure interaction analyses of biological systems using smoothed-particle hydrodynamics. Biology (Basel). 10, pp.1–12. https://doi.org/10.3390/biology10030185
Vacondio, R., Altomare, C., De Leffe, M., Hu, X., Le Touzé, D., Lind, S., Marongiu, J.C., Marrone, S., Rogers, B.D., Souto-Iglesias, A., 2021. Grand challenges for Smoothed Particle Hydrodynamics numerical schemes. Comput. Part. Mech. 8, pp.575–588. https://doi.org/10.1007/s40571-020-00354-1
Valdez-Balderas, D., Domínguez, J.M., Rogers, B.D., Crespo, A.J.C., 2013. Towards accelerating smoothed particle hydrodynamics simulations for free-surface flows on multi-GPU clusters. J. Parallel Distrib. Comput. 73, pp.1483–1493. https://doi.org/10.1016/j.jpdc.2012.07.010
Valizadeh, A., Monaghan, J.J., 2015. A study of solid wall models for weakly compressible SPH. J. Comput. Phys. 300, pp.5–19. https://doi.org/10.1016/j.jcp.2015.07.033
Violeau, D., Rogers, B.D., 2016. Smoothed particle hydrodynamics (SPH) for free-surface flows: Past, present and future. J. Hydraul. Res. 54, pp.1–26. https://doi.org/10.1080/00221686.2015.1119209
Wang, L., Khayyer, A., Gotoh, H., Jiang, Q., Zhang, C., 2019. Enhancement of pressure calculation in projection-based particle methods by incorporation of background mesh scheme. Appl. Ocean Res. 86, pp.320–339. https://doi.org/10.1016/j.apor.2019.01.017
Wu, G.X., Ma, Q.W., Eatock Taylor, R., 1998. Numerical simulation of sloshing waves in a 3D tank based on a finite element method. Appl. Ocean Res. 20, pp.337–355. https://doi.org/10.1016/S0141-1187(98)00030-3
Ye, T., Pan, D., Huang, C., Liu, M., 2019. Smoothed particle hydrodynamics (SPH) for complex fluid flows: Recent developments in methodology and applications. Phys. Fluids, 31. https://doi.org/10.1063/1.5068697
You, Y., Khayyer, A., Zheng, X., Gotoh, H., Ma, Q., 2021. Enhancement of δ-SPH for ocean engineering applications through incorporation of a background mesh scheme. Appl. Ocean Res. 110, pp.102508. https://doi.org/10.1016/j.apor.2020.102508
Zhang, D.H., Shi, Y.X., Huang, C., Si, Y.L., Huang, B., Li, W., 2018. SPH method with applications of oscillating wave surge converter. Ocean Eng. 152, pp.273–285. https://doi.org/10.1016/j.oceaneng.2018.01.057.