Abstract
A corrective smooth particle method (CSPM) within smooth particle hydrodynamics (SPH) is used to study the deformation of an aircraft structure under highvelocity waterditching impact load. The CSPMSPH method features a new approach for the prediction of twoway fluid–structure interaction coupling. Results indicate that the implementation is well suited for modeling the deformation of structures under highvelocity impact into water as evident from the predicted stress and strain localizations in the aircraft structure as well as the integrity of the impacted interfaces, which show no artificial particle penetrations. To reduce the simulation time, a heterogeneous particle size distribution over a complex threedimensional geometry is used. The variable particle size is achieved from a finite element mesh with variable element size and, as a result, variable nodal (i.e., SPH particle) spacing. To further accelerate the simulations, the SPH code is ported to a graphics processing unit using the OpenACC standard. The implementation and simulation results are described and discussed in this paper.
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Acknowledgements
This work is based upon a project supported by the US National Science Foundation under grant no. CMMI1650641. The authors gratefully acknowledge this support.
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Appendices
Appendix A
The % time spent in subroutines ordered from top to bottom as identified using PGPROF:

1.
Subroutine Direct_find (neighbor search subroutine, 38% of total execution time)

2.
Subroutine Int_Force (force calculation subroutine, 23% of total execution time)

3.
Subroutine Time_Intg (time integration subroutine, 12% of total execution time)

4.
Main program SPH (main program, 9% of total execution time)

5.
Subroutine Cont_Density (continuity subroutine including the CSPM modification, 8% of total execution time)

6.
Subroutine H_Upgrade (update smoothing length subroutine, 5% of total execution time)

7.
Other subroutines (5% of total execution time)
In this work, subroutines 1–5 were ported to GPU.
Appendix B

(i)
Neighbor particles search within computational domain:

(ii)
Force calculation:

(iii)
Time integration:

(iv)
Continuity:

(v)
SPH main program:
A loop from the continuity subroutine is presented below to better illustrate how the OpenACC data and kernels directives can be used to run the loop in parallel on GPU. The “reduction” and “private” clauses ensure that there are no race conditions while accessing the summation over the scalar “vcc” using the GPU threads. Additionally, “copyin” clauses show the arrays data input from CPU (host) to the device (GPU).
In above loop, “x”, “vx”, “rho”, “drhodt”, “pair_i”, “pair_j”, “niac”, “dwdx”, and “dim” represent particle position, velocity, density, time rate of density, neighbor particle “i” interacting with particle “j”, neighbor particle “j” interacting with particle “i”, gradient of kernel function, total number of interacting pairs, and domain dimension, respectively.
Supplementary material
Movies showing the evolution of pressure, equivalent plastic strain, and von Mises stress during aircraft water ditching at an angle of \(60{^{\circ }}\).
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Eghtesad, A., Knezevic, M. A new approach to fluid–structure interaction within graphics hardware accelerated smooth particle hydrodynamics considering heterogeneous particle size distribution. Comp. Part. Mech. 5, 387–409 (2018) doi:10.1007/s4057101701761
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Keywords
 Smooth particle hydrodynamics
 Heterogeneous particle size distribution
 Fluid–structure interaction
 Graphics processing unit
 OpenACC