### “An Open-Source Python-Based Boundary-Element Method Code for the Three-Dimensional, Zero-Froude, Infinite-Depth, Water-Wave Diffraction-Radiation Problem”

**Authors:**Savin Viswanathan, Christian Holden, Olav Egeland and Marilena Greco,

**Affiliation:**NTNU and NTNU, Department of Marine Technology

**Reference:**2021, Vol 42, No 2, pp. 47-81.

**Keywords:**Wave-body interaction, 3D boundary-element method, frequency-domain hydrodynamic analysis, diffraction-radiation loads

**Abstract:**In this paper, a new open-source implementation of the lower-order, 3-D Boundary Element Method (BEM) of solution to the deep-water, zero Froude-number wave-body interaction problem is described. A validation case for OMHyD, the new open-source package, is included, where the outputs are compared to results obtained using the widely used frequency-domain hydrodynamic analysis package ANSYS-AQWA. The theory behind the solution to the diffraction-radiation problem is re-visited using the Green function method. The Hess and Smith panel method is then extended to the case of a floating object using the image-source to impose the wall condition at the free-surface, and a wavy Green function component to account for presence of free-surface waves. An algorithm for computer implementation of the procedure is developed and subsequently implemented in PYTHON. The wavy part of the Green function is determined using a verified and published FORTRAN code by Teleste and Noblesse, wrapped for PYTHON using the Fortran to Python (F2PY) interface. Results are presented for the various stages of implementation viz. panelling, body in infinite fluid domain, effect of the free-surface, and effect of surface-waves. The hydrodynamic coefficients obtained from this preliminary frequency-domain analysis are shown to be in satisfactory agreement with ANSYS-AQWA results. Conclusions are drawn based on the performance of the code, followed by suggestions for further improvement by including the removal of irregular-frequencies, multi-body interactions, and bottom interference, which are not considered in the present implementation.

PDF (1855 Kb) DOI: 10.4173/mic.2021.2.2

**DOI forward links to this article:**

[1] Matthieu Ancellin, Pierre Marchand and Frederic Dias (2024), doi:10.3390/en17020372 |

[2] Bo Yu and Ruijiang Jing (2024), doi:10.1016/j.cpc.2024.109185 |

**References:**

[1] A calculation method for finite depth Free-Surface green function. (2015). International Journal of Naval Architecture and Ocean Engineering, doi:10.1515/ijnaoe-2015-0026

[2] Chakrabarti, S.K. (2001). Application and verification of deepwater green function for water waves, Journal of ship research. 45(3):187--196. https://onepetro.org/journal-paper/SNAME-JSR-2001-45-3-187.

[3] Chiang, C.M. (1983). The Applied Dynamics of Ocean Surface Waves, World Scientific. doi:10.1142/0752

[4] Faltinsen, O.M. (1990). Sea Loads on Ships and Offshore Structures, Cambridge University Press. https://www.cambridge.org/no/academic/subjects/engineering/engineering-design-kinematics-and-robotics/sea-loads-ships-and-offshore-structures?format=PB&isbn=9780521458702.

[5] Faltinsen, O.M. and Michelsen, F.C. (1975). Motions of large structures in waves at zero Froude number, Technical report. https://www.studocu.com/no/document/norges-teknisk-naturvitenskaplige-universitet/akademisk-skriving/essay/motions-of-large-structures-in-waves-at-zero-froude-number/6109623/view.

[6] Fossen, T.I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control, Wiley.

[7] Garrison, C. (1978). Hydrodynamic loading of large offshore structures: Three dimensional source distribution method, In O.Zienkicwicz, R.Lewis, and K.Stagg, editors, Numer. Methods in Offshore Eng., volume3, pages 97--140. John Wiley & Sons.

[8] Guha, A. (2012). Development of a computer program for three dimensional frequency domain analysis of zero speed first order wave body interaction, Master's thesis. https://oaktrust.library.tamu.edu/handle/1969.1/148193.

[9] Guha, A. and Falzarano, J. (2013). Development of a Computer Program for Three Dimensional Analysis of Zero Speed First Order Wave Body Interaction in Frequency Domain, In Volume 5: Ocean Engineering. American Society of Mechanical Engineers. doi:10.1115/OMAE2013-11601

[10] Guha, A., Somayajula, A., and Falzarano, J. (2016). Time domain simulation of large amplitude motions in shallow water, In 21st SNAME Offshore Symposium, Society of Naval Architects and Marine Engineers, Houston, February. 2016. http://www.doe.iitm.ac.in/abhilash/wp-content/uploads/sites/50/2019/05/Guha-Somayajula-Falzarano-2016-Time-domain-simulation-of-large-amplitude-motions-in-shallow-water.pdf.

[11] Hess, J.L. and Smith, A. (1962). Calculation of non-lifting potential flow about arbitrary three-dimensional bodies, Technical report, Douglas Aircraft Co., Long Beach CA. https://apps.dtic.mil/dtic/tr/fulltext/u2/282255.pdf.

[12] Hess, J.L. and Smith, A. M.O. (1967). Calculation of potential flow about arbitrary bodies, Progress in Aerospace Sciences. 8:1 -- 138. doi:10.1016/0376-0421(67)90003-6

[13] Hess, J.L. and Wilcox, D.C. (1969). Progress in the solution of the problem of a three-dimensional body oscillating in the presence of a free surface, Technical report, McDonnell Douglas Corp., Long Beach, CA Douglas Aircraft Div.. https://apps.dtic.mil/dtic/tr/fulltext/u2/690203.pdf.

[14] John, F. (1949). On the motion of floating bodies I, Communications on Pure and Applied Mathematics. 2(1):13--57. doi:10.1002/cpa.3160020102

[15] John, F. (1950). On the motion of floating bodies II, Simple harmonic motions. Communications on Pure and Applied Mathematics. doi:10.1002/cpa.3160030106

[16] Katz, J. and Plotkin, A. (2001). Low-Speed Aerodynamics, 2001. doi:10.1017/cbo9780511810329

[17] Kim, C.H. (2008). Nonlinear Waves and Offshore Structures, volume27 of Advanced Series on Ocean Engineering, World Scientific. doi:10.1142/4906

[18] Kleinman, R.E. (1982). On the mathematical theory of the motion of floating bodies-An Update, Technical report, David W Taylor Naval Ship Research and Development Center, Bethsada MD. https://apps.dtic.mil/dtic/tr/fulltext/u2/a121433.pdf.

[19] Lamb, S.H. (1879). Hydrodynamics, Cambridge University Press. https://www.cambridge.org/no/academic/subjects/mathematics/fluid-dynamics-and-solid-mechanics/hydrodynamics-6th-edition?format=PB&isbn=9780521458689.

[20] Lau, S.M. and Hearn, G.E. (1989). Suppression of irregular frequency effects in fluidâ€“structure interaction problems using a combined boundary integral equation method, International Journal for Numerical Methods in Fluids. doi:10.1002/fld.1650090703

[21] Linton, C.M. (1999). Rapidly convergent representations for Green's functions for Laplace's equation, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. doi:10.1098/rspa.1999.0379

[22] Linton, C.M. and McIver, P. (2001). Handbook of mathematical techniques for wave/structure interactions, 2001. doi:10.1201/9781420036060

[23] Malenica, S. and Chen, X.B. (1998). On the Irregular Frequencies Appearing in Wave Diffraction-Radiation Solutions, International Journal of Offshore and Polar Engineering. https://www.onepetro.org/journal-paper/ISOPE-98-08-2-110.

[24] McTaggart, K.A. (2002). Three dimensional ship hydrodynamic coefficients using the zero forward speed green function, Defence R & D Canada-Atlantic. https://pdfs.semanticscholar.org/77f1/889f836a37df015c2fdae599804d0b40da6c.pdf.

[25] Milgram, J.H. (2003). Lecture notes on numerical marine hydrodynamics: Numerical methods in incompressible fluid mechanics, 2003. https://ocw.mit.edu/courses/mechanical-engineering/2-29-numerical-marine-hydrodynamics-13-024-spring-2003/lecture-notes/.

[26] Newman, J. and Sclavounos, P. (1988). The computation of wave loads on large offshore structures, In BOSS'88. pages 605--622. https://www.wamit.com/Publications/boss88.pdf.

[27] Newman, J.N. (1977). Marine Hydrodynamics, MIT Press. https://mitpress.mit.edu/books/marine-hydrodynamics-40th-anniversary-edition.

[28] Newman, J.N. (1985). Algorithms for the free-surface Green function, Journal of Engineering Mathematics. doi:10.1007/BF00055041

[29] Newman, J.N. (1992). Panel methods in marine hydrodynamics, In 11th Australasian Fluid Mechanics Conference. 1992. https://people.eng.unimelb.edu.au/imarusic/proceedings/11/Newman.pdf.

[30] Newman, J.N. and Lee, C. (2013). WAMIT User Manual Version 7, 0. Technical report.

[31] Ohmatsu, S. (1975). On the irregular frequencies in the theory of oscillating bodies in a free surface, Paper 48 of the Ship Research Institute, Ship Dynamics Division, Tokyo, Japan. http://resolver.tudelft.nl/uuid:cf6d35d4-d22c-4eed-b4ba-8dd9d79c6fbe.

[32] OpenModelica. (2020). User guide, Open Source Modelica Consortium. https://www.openmodelica.org/doc/OpenModelicaUsersGuide/latest/.

[33] Penalba, M., Kelly, T., and Ringwood, J. (2017). Using nemoh for modelling wave energy converters: A comparative study with wamit, In Proc. of the 12th European Wave and Tidal Energy Conference (EWTEC2017), Cork, Ireland, volume27. pages 631--1. http://mural.maynoothuniversity.ie/12466/.

[34] Peterson, P. (2005). F2PY user guide and reference manual, 2005. http://pds8.egloos.com/pds/200802/20/11/f2py_usersguide.pdf.

[35] Pidcock, M.K. (1985). The calculation of Green's functions in three dimensional hydrodynamic gravity wave problems, International Journal for Numerical Methods in Fluids. doi:10.1002/fld.1650051004

[36] Sarpkaya, T. and Isaacson, M. (1981). Mechanics of wave forces on offshore structures, 1981. doi:10.1016/0261-7277(82)90028-6

[37] Telste, J.G. and Noblesse, F. (1986). Numerical evaluation of the green function of water-wave radiation and diffraction, Journal of Ship Research. 30(02):69--84. https://www.onepetro.org/journal-paper/SNAME-JSR-1986-30-2-69.

[38] Thorne, R.C. (1953). Multipole expansions in the theory of surface waves, Mathematical Proceedings of the Cambridge Philosophical Society. 49(04):707----716. doi:10.1017/S0305004100028905

[39] Viswanathan, S. and Holden, C. (2019). Towards the development of an ocean engineering library for openmodelica, In Proc. ASME/OMAE Conf., volume 7B: Ocean Engineering, OMAE2019-95054. Glasgow, Scotland, UK. doi:10.1115/OMAE2019-95054

[40] Wehausen, J.V. (1971). The motion of floating bodies, Annual review of fluid mechanics. 3(1):237--268. doi:10.1146/annurev.fl.03.010171.001321

[41] Wehausen, J.V. and Laitone, E.V. (1960). Surface Waves, In Fluid Dynamics, pages 446--778. Springer, Berlin, Heidelberg. doi:10.1007/978-3-642-45944-3_6

[42] Wu, H., Zhang, C., Zhu, Y., Li, W., Wan, D., and Noblesse, F. (2017). A global approximation to the Green function for diffraction radiation of water waves, European Journal of Mechanics, B/Fluids. doi:10.1016/j.euromechflu.2017.02.008

[43] Zhu, X. (1994). Irregular Frequency Removal from the Boundary Integral Equation for the Wave-body Problem, M.s, Massachusetts Institute of Technology. https://dspace.mit.edu/bitstream/handle/1721.1/11691/32279180-MIT.pdf?sequence=.

**BibTeX:**

@article{MIC-2021-2-2,

title={{An Open-Source Python-Based Boundary-Element Method Code for the Three-Dimensional, Zero-Froude, Infinite-Depth, Water-Wave Diffraction-Radiation Problem}},

author={Viswanathan, Savin and Holden, Christian and Egeland, Olav and Greco, Marilena},

journal={Modeling, Identification and Control},

volume={42},

number={2},

pages={47--81},

year={2021},

doi={10.4173/mic.2021.2.2},

publisher={Norwegian Society of Automatic Control}

};