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Method for real-time simulation of haptic interaction with deformable objects using GPU-based parallel computing and homogeneous hexahedral elements


This paper proposes a method for simulating real-time haptic interaction with deformable objects. The deformable model consists of regular hexahedrons of a single type. This homogeneity is exploited to improve the efficiency in deformation computations. Model boundaries are approximated using a moving-least-squares function reflecting the deformation results of the hexahedrons. A method for adaptively approximating the model boundaries is presented for efficient collision handling in the haptic loop. The proposed method can simulate a model of 16,481 nodes in less than 1 ms, which is a significant improvement over the previous methods in the literature. Small gap between the model boundary and the hexahedrons can cause errors in the proposed method. Numerical examples considering the characteristics of human tissues show that the errors are less than just-noticeable difference of human.

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  1. 1.

    Zhang J, Zhong Y, Gu C (2018) Deformable models for surgical simulation: a survey. IEEE Rev Biomed Eng 11:143–164

    Google Scholar 

  2. 2.

    Peterlik I, Nouicer M, Duriez C, Cotin S, Kheddar A (2011) Constraint-based haptic rendering of multirate compliant mechanisms. IEEE Trans Haptics 4(3):175–187

    Google Scholar 

  3. 3.

    Tian Y, Yang Y, Guo X, Prabhakaran B (2014) Haptic simulation of needle-tissue interaction based on shape matching. In: Proceedings of IEEE international symposium on haptic, audio and visual environments and games (HAVE 2014), pp 7–12

  4. 4.

    Knott TC, Kuhlen TW (2016) Accurate and adaptive contact modeling for multi-rate multi-point haptic rendering of static and deformable environments. Comput Graph 57:68–80

    Google Scholar 

  5. 5.

    Zhang X, Sun W, Song A (2014) Layered rhombus-chain-connected model for real-time haptic rendering. Artif Intell Rev 41(1):49–65

    Google Scholar 

  6. 6.

    Wang D, Shi Y, Liu S, Zhang Y, Xiao J (2014) Haptic simulation of organ deformation and hybrid contacts in dental operations. IEEE Trans Haptics 7(1):48–60

    Google Scholar 

  7. 7.

    Zhang J, Zhong Y, Gu C (2019) Neural network modelling of soft tissue deformation for surgical simulation. Artif Intell Med 97:61–70

    Google Scholar 

  8. 8.

    Barbič J, James DL (2008) Six-dof haptic rendering of contact between geometrically complex reduced deformable models. IEEE Trans Haptics 1(1):39–52

    Google Scholar 

  9. 9.

    Ryckelynck D, Chinesta F, Cueto E, Ammar A (2006) On the a priori model reduction: overview and recent developments. Arch Comput Methods Eng 13(1):91–128

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Niroomandi S, Alfaro I, Cueto E, Chinesta F (2008) Real-time deformable models of non-linear tissues by model reduction techniques. Comput Methods Programs Biomed 91(3):223–231

    Google Scholar 

  11. 11.

    Niroomandi S, Alfaro I, Cueto E, Chinesta F (2012) Accounting for large deformations in real-time simulations of soft tissues based on reduced-order models. Comput Methods Programs Biomed 105(1):1–12

    Google Scholar 

  12. 12.

    González D, Alfaro I, Quesada C, Cueto E, Chinesta F (2015) Computational vademecums for the real-time simulation of haptic collision between nonlinear solids. Comput Methods Appl Mech Eng 283(1):210–223

    Google Scholar 

  13. 13.

    Taylor ZA, Cheng M, Ourselin S (2008) High-speed nonlinear finite element analysis for surgical simulation using graphics processing units. IEEE Trans Med Imaging 27(5):650–663

    Google Scholar 

  14. 14.

    Joldes GR, Wittek A, Miller K (2010) Real-time nonlinear finite element computations on GPU–Application to neurosurgical simulation. Comput Methods Appl Mech Eng 199:3305–3314

    MATH  Google Scholar 

  15. 15.

    Mafi R, Sirouspour S, Mahdavikhah B, Moody B, Elizeh K, Kinsman A, Nicolici N (2010) A parallel computing platform for real-time haptic interaction with deformable bodies. IEEE Trans Haptics 3(3):211–223

    Google Scholar 

  16. 16.

    Mahdavikhah B, Mafi R, Sirouspour S, Nicolici N (2014) A Multiple-FPGA parallel computing architecture for real-time simulation of soft-object deformation. ACM Trans Embed Comput Syst (TECS) 13(4):81

    Google Scholar 

  17. 17.

    Courtecuisse H, Allard J, Kerfriden P, Bordas SP, Cotin S, Duriez C (2014) Real-time simulation of contact and cutting of heterogeneous soft-tissues. Med Image Anal 18(2):394–410

    Google Scholar 

  18. 18.

    Jia S, Zhang W, Yu X, Pan Z (2015) CPU–GPU mixed implementation of virtual node method for real-time interactive cutting of deformable objects using OpenCL. Int J Comput Assist Radiol Surg 10(9):1477–1491

    Google Scholar 

  19. 19.

    Jia S, Zhang W, Yu X, Pan Z (2018) CPU–GPU Parallel framework for real-time interactive cutting of adaptive octree-based deformable objects. Comput Graph Forum 37(1):45–59

    Google Scholar 

  20. 20.

    Weber D, Bender J, Schnoes M, Stork A, Fellner D (2013) Efficient GPU data structures and methods to solve sparse linear systems in dynamics applications. In: Computer graphics forum, vol 32, no 1. Blackwell Publishing Ltd, Oxford

  21. 21.

    Kumar AV, Padmanabhan S, Burla R (2008) Implicit boundary method for finite element analysis using non-conforming mesh or grid. Int J Numer Meth Eng 74(9):1421–1447

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Kumar AV, Burla R, Padmansbhan S, Gu L (2008) Finite element analysis using nonconforming mesh. J Comput Inf Sci Eng 8(3):031005

    Google Scholar 

  23. 23.

    Dick C, Georgii J, Westermann R (2011) A hexahedral multigrid approach for simulating cuts in deformable objects. IEEE Trans Vis Comput Graph 17(11):1663–1675

    Google Scholar 

  24. 24.

    Wu J, Dick C, Westermann R (2013) Efficient collision detection for composite finite element simulation of cuts in deformable bodies. Vis Comput 29(6–8):739–749

    Google Scholar 

  25. 25.

    Müller M, Dorsey J, McMillan L, Jagnow R, Cutler B (2002) Stable real-time deformations. In: Proceedings of the 2002 ACM SIGGRAPH/eurographics symposium on computer animation, pp 49–54

  26. 26.

    Barnes JM, Przybyla L, Weaver VM (2017) Tissue mechanics regulate brain development, homeostasis and disease. J Cell Sci 130(1):71–82

    Google Scholar 

  27. 27.

    Fierz B, Spillmann J, Harders M (2011) Element-wise mixed implicit-explicit integration for stable dynamic simulation of deformable objects. In: Proceedings of the 2011 ACM SIGGRAPH/eurographics symposium on computer animation, pp 257–266

  28. 28.

    Xie H, Liu H, Luo S, Seneviratne LD, Althoefer K (2013) Fiber optics tactile array probe for tissue palpation during minimally invasive surgery. In: 2013 IEEE/RSJ international conference on intelligent robots and systems, pp 2539–2544

  29. 29.

    Müller M, Heidelberger B, Teschner M, Gross M (2005) Meshless deformations based on shape matching. ACM Trans Graph (TOG) 24(3):471–478

    Google Scholar 

  30. 30.

    Fried I (1972) Condition of finite element matrices generated from nonuniform meshes. AIAA J 10(2):219–221

    MATH  Google Scholar 

  31. 31.

    Lim YJ, Deo D, Singh TP, Jones DB, De S (2009) In situ measurement and modeling of biomechanical response of human cadaveric soft tissues for physics-based surgical simulation. Surg Endosc 23(6):1298–1307

    Google Scholar 

  32. 32.

    Anuradha C, Ramakrishna B, Venkatramani S (2012) Formula for calculating standard liver volume in Indians. Indian J Gastroenterol 31(1):15–19

    Google Scholar 

  33. 33.

    Atluri SN, Cho JY, Kim HG (1999) Analysis of thin beams, using the meshless local Petrov–Galerkin method, with generalized moving least squares interpolations. Comput Mech 24(5):334–347

    MATH  Google Scholar 

  34. 34.

    Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37(155):141–158

    MathSciNet  MATH  Google Scholar 

  35. 35.

    De S, Bathe KJ (2001) Displacement/pressure mixed interpolation in the method of finite spheres. Int J Numer Meth Eng 51(3):275–292

    MATH  Google Scholar 

  36. 36.

    Steven WS (1997) The scientist and engineer’s guide to digital signal processing. California Technical Pub

  37. 37.

    The Stanford 3D Scanning Repository.

  38. 38.

    Paggetti G, Cizmeci B, Dillioglugil C, Steinbach E (2014) On the discrimination of stiffness during pressing and pinching of virtual spring. In: 2014 IEEE international symposium on haptic, audio and visual environments and games (HAVE) proceedings, pp 94–99

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This work was supported by the National Research Foundation of Korea (NRF) Grants funded by the Ministry of Science and ICT (Nos. NRF-2015R1A2A1A10054420, and NRF-2019R1H1A2080008) and the Brain Korea 21 PLUS Program.

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Correspondence to Doo Yong Lee.

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Byeon, S.P., Lee, D.Y. Method for real-time simulation of haptic interaction with deformable objects using GPU-based parallel computing and homogeneous hexahedral elements. Comput Mech 65, 1205–1218 (2020).

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  • Haptic simulation
  • Interactive simulation
  • Parallel computing
  • Finite-element method
  • Deformable object
  • Physics-based model