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==Applications==
Through the past years, the MPS method has been applied in a wide range of engineering applications including Nuclear Engineering (e.g. [https://dx.doi.org/10.1016/S0029-5493(98)00270-2 Koshizuka et al., 1999]; Koshizuka and Oka, 2001; [https://dx.doi.org/10.1016/j.nucengdes.2005.01.011 Xie et al., 2005]), Coastal Engineering (e.g. [https://dx.doi.org/10.1142/S0578563405001239 Gotoh et al., 2005]; [https://dx.doi.org/10.1016/j.coastaleng.2005.10.007 Gotoh and Sakai, 2006]), Environmental Hydraulics (e.g. [https://dx.doi.org/10.1002/fld.2132 Shakibaeina and Jin, 2009]; [http://ascelibrary.org/doi/10.1061/%28ASCE%29HY.1943-7900.0001275 Nabian and Farhadi, 2016]), Ocean Engineering ([https://dx.doi.org/10.1016/j.oceaneng.2005.12.012 Shibata and Koshizuka, 2007]; [https://doi.org/10.1007%2Fs00773-007-0260-y Sueyoshi et al., 2008]; [https://doi.org/10.1016/j.oceaneng.2019.04.054 Xu and Jin, 2019]; [https://doi.org/10.1016/j.oceaneng.2022.111569 Zuo et al. 2022]), Structural Engineering (e.g. [https://doi.org/10.1007%2Fs004660000216 Chikazawa et al., 2001]), Mechanical Engineering (e.g. [https://dx.doi.org/10.1016/S0017-9310(02)00011-X Heo et al., 2002]; [https://archive.today/20130223115811/http://link.aip.org/link/?PHFLE6/21/032106/1 Sun et al., 2009]), Bioengineering (e.g. [http://www.jamstec.go.jp/esc/publication/journal/jes_vol.5/pdf/JES5_21-Tsubota.pdf Tsubota et al., 2006]) and Chemical Engineering (e.g. [https://dx.doi.org/10.1016/j.ces.2008.10.034 Sun et al., 2009]; [https://doi.org/10.1016/j.ces.2018.07.016 Xu and Jin, 2018]; [https://doi.org/10.1063/5.0067111 Xu et al. 2021]).
==Improvements==
Improved versions of MPS method have been proposed for enhancement of numerical stability (e.g. [https://archive.today/20130105131748/http://www3.interscience.wiley.com/journal/2910/abstract Koshizuka et al., 1998]; [https://dx.doi.org/10.1002/fld.1106 Zhang et al., 2005]; [https://dx.doi.org/10.1016/j.fluiddyn.2005.12.002 Ataie-Ashtiani and Farhadi, 2006];[https://dx.doi.org/10.1002/fld.2132 Shakibaeina and Jin, 2009]; [https://doi.org/10.1016/j.cma.2019.112771 Jandaghian and Shakibaeinia, 2020]), momentum conservation (e.g. Hamiltonian MPS by [https://dx.doi.org/10.1016/j.cma.2006.12.006 Suzuki et al., 2007]; Corrected MPS by [https://dx.doi.org/10.1142/S0578563408001788 Khayyer and Gotoh, 2008]; Enhanced MPS by [https://doi.org/10.1016/j.cma.2019.112771 Jandaghian and Shakibaeinia, 2020]), mechanical energy conservation (e.g. Hamiltonian MPS by [https://dx.doi.org/10.1016/j.cma.2006.12.006 Suzuki et al., 2007]), pressure calculation (e.g. [https://dx.doi.org/10.1016/j.coastaleng.2008.10.004 Khayyer and Gotoh, 2009], [https://dx.doi.org/10.1002/fld.2207 Kondo and Koshizuka, 2010], [https://dx.doi.org/10.1016/j.apor.2010.01.001 Khayyer and Gotoh, 2010],[https://onlinelibrary.wiley.com/doi/full/10.1002/fld.5022 Xu, 2021]; [https://doi.org/10.1016/j.compfluid.2019.104235 Xu and Jin, 2019]), and for simulation of multiphase and granular flows ([http://ascelibrary.org/doi/10.1061/%28ASCE%29HY.1943-7900.0001275 Nabian and Farhadi 2016]; [https://doi.org/10.1007/s11440-022-01766-4 Xu and Li, 2022]; [https://doi.org/10.1017/jfm.2021.320 Xu and Jin, 2021]).
== References ==
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* Z. Sun, G. Xi and X. Chen, "Mechanism study of deformation and mass transfer for binary droplet collisions with particle method," Phys. Fluids, Vol 21, 032106, 2009.
* K. Tsubota, S. Wada, H. Kamada, Y. Kitagawa, R. Lima and T. Yamaguchi, "A Particle Method for Blood Flow Simulation – Application to Flowing Red Blood Cells and Platelets–," Journal of the Earth Simulator, Vol 5, pp. 2–7, 2006.
* T. Xu, Y. C. Jin, Simulation the convective mixing of CO2 in geological formations with a meshless model. Chemical Engineering Science, 192, 187-198, 2018.
* H. Xie, S. Koshizuka and Y. Oka, "Simulation of drop deposition process in annular mist flow using three-dimensional particle method," Nuclear Engineering and Design, Vol 235, pp. 1687–1697, 2005.
* S. Zhang, K. Morita, K. Fukuda and N. Shirakawa, "An improved MPS method for numerical simulations of convective heat transfer problems," Int. J. Numer. Meth. Fluid, 51, 31–47, 2005.
* T. Xu, Explicit calculation for the pressure Poisson equation to simulate incompressible fluid flows in a mesh‐free method. International Journal for Numerical Methods in Fluids, 93(10), 3034-3052, 2021.
* T. Xu, Y. C. Jin, Improvement of a projection-based particle method in free-surface flows by improved Laplacian model and stabilization techniques. Computers & Fluids, 191, 104235, 2019.
* T. Xu, Y. C. Jin, Modeling impact pressure on the surface of porous structure by macroscopic mesh-free method. Ocean Engineering, 182, 1-13, 2019.
* J. Zuo, T. Xu, D. Z. Zhu, H. Gu, Impact pressure of dam-break waves on a vertical wall with various downstream conditions by an explicit mesh-free method. Ocean Engineering, 256, 111569, 2022.
* T. Xu, S. S. Li, Development of a non-local partial Peridynamic explicit mesh-free incompressible method and its validation for simulating dry dense granular flows. Acta Geotechnica, 1-20, 2022.
* T. Xu, Y. C. Jin, Y. C. Tai, Mesh-free peridynamic coupled simulation of impacting collapse of a granular column with various heights. Physics of Fluids, 33(10), 103321, 2021.
* T. Xu, Y. C. Jin, Two-dimensional continuum modelling granular column collapse by non-local peridynamics in a mesh-free method with rheology. Journal of Fluid Mechanics, 917, A51, 2021.
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