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RBF morphing techniques for simulation-based design optimization

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Abstract

Morphing an existing simulation mesh according to updated geometric parameters in the underlying computer-aided design model is a crucial technique within fully automatic design optimization. By avoiding costly automatic or even manual meshing, it enables the automatic and parallel generation and evaluation of new design variations, e.g., through finite element or computational fluid dynamics simulations. In this paper, we present a simple yet versatile method for high-quality mesh morphing. Building upon triharmonic radial basis functions, our shape deformations minimize distortion and thereby implicitly preserve shape quality. Moreover, the same unified code can morph tetrahedral, hexahedral, or arbitrary polyhedral meshes. We compare our method to other recently proposed techniques and show that ours yields superior results in most cases. We analyze how to explicitly prevent inverted mesh elements by successively splitting the deformation into smaller steps. Finally, we investigate the performance of different linear solvers as well as the use of an incremental least squares solver for the sake of improved scalability.

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References

  1. Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Croz JD, Greenbaum A, Hammarling S, McKenney A, Sorensen D (1999) LAPACK users’ guide, third edn. SIAM, Philadelphia, PA

  2. Angelidis A, Cani MP, Wyvill G, King S (2006) Swirling-Sweepers: constant volume modeling. Graph Models 68(4):324–332

    Article  MATH  Google Scholar 

  3. Baker TJ (2001) Mesh movement and metamorphosis. In: Proceedings of the 10th international meshing roundtable, pp 387–396

  4. Bechmann D (1994) Space deformation models survey. Comput Graph 18(4):571–586

    Article  Google Scholar 

  5. de Boer A, van der Schoot M, Bijl H (2007) Mesh deformation based on radial basis function interpolation. Comput Struct 85:784–795

    Article  Google Scholar 

  6. Botsch M, Kobbelt L (2005) Real-time shape editing using radial basis functions. Comput Graph Forum 24(3):611–621

    Article  Google Scholar 

  7. Brewer M, Diachin LF, Knupp P, Leurent T, Melander D (2003) The Mesquite mesh quality improvement toolkit. In: Proceedings of the 12th international meshing roundtable, pp 239–250

  8. Carr JC, Beatson RK, Cherrie JB, Mitchell TJ, Fright WR, McCallum BC, Evans TR (2001) Reconstruction and representation of 3D objects with radial basis functions. In: Proceedings of ACM SIGGRAPH, ACM, New York, pp 67–76

  9. Duchon J (1977) Spline minimizing rotation-invariant semi-norms in Sobolev spaces. In: Schempp W, Zeller K (eds) Constructive theory of functions of several variables, no. 571 in Lecture Notes in Mathematics, Springer, Berlin, pp 85–100

  10. Fasshauer GE (2007) Meshfree approximation methods with MATLAB. World Scientific Publishing, Singapore

  11. Floater MS, Kos G, Reimers M (2005) Mean value coordinates in 3D. Comput Aided Geom D 22:623–631

    Article  MATH  MathSciNet  Google Scholar 

  12. von Funck W, Theisel H, Seidel HP (2006) Vector field-based shape deformations. ACM T Graphic 25(3):1118–1125

    Article  Google Scholar 

  13. Gain J, Bechmann D (2008) A survey of spatial deformation from a user-centered perspective. ACM T Graphic 27, 107:1–107:21

    Google Scholar 

  14. Gain J, Dodgson N (2001) Preventing self-intersection under free-form deformation. IEEE Trans Vis Comput Graph 7(4):289–298

    Article  Google Scholar 

  15. Harmon D, Panozzo D, Sorkine O, Zorin D (2011) Interference aware geometric modeling. ACM Trans Graph 30(6):137:1–137:10

    Google Scholar 

  16. Helenbrook BT (2003) Mesh deformation using the biharmonic operator. Int J Numer Meth Eng 56:1007–1021

    Article  MATH  Google Scholar 

  17. Hormann K, Sukumar N (2008) Maximum entropy coordinates for arbitrary polytopes. Comput Graph Forum 27(5):1513–1520

    Article  Google Scholar 

  18. Intel (2013) Intel Math Kernel Library version 11.0. http://software.intel.com/en-us/intel-mkl

  19. Jakobsson S, Amoignon O (2007) Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization. Comput Fluids 36(6):1119–1136

    Article  MATH  Google Scholar 

  20. Joshi P, Meyer M, DeRose T, Green B, Sanocki T (2007) Harmonic coordinates for character articulation. ACM Trans Graph 26(3)

  21. Knupp P (2000) Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part I. Int J Numer Meth Eng 48(3):401–420

    Article  MATH  Google Scholar 

  22. Knupp P (2008) Updating meshes on deforming domains: an application of the target-matrix paradigm. Commun Num Method Eng 24(6):467–476

    MATH  MathSciNet  Google Scholar 

  23. Martin S, Kaufmann P, Botsch M, Wicke M, Gross M (2008) Polyhedral finite elements using harmonic basis functions. Comput Graph Forum 27(5):1521–1529

    Article  Google Scholar 

  24. Martinez Esturo J, Rössl C, Fröhlich S, Botsch M, Theisel H (2011) Pose correction by space–time integration. In: Proceedings of vision, modeling, visualization, pp 33–40

  25. Michler AK (2011) Aircraft control surface deflection using RBF-based mesh deformation. Int J Numer Meth Eng 88(10):986–1007

    Article  MATH  Google Scholar 

  26. Open CASCADE (2012) Open CASCADE technology, 3D modeling and numerical simulation. http://www.opencascade.org/

  27. Samareh JA (1999) A survey of shape parameterization techniques. Tech. Rep. NASA/CP-1999-209136/PT1, NASA Langley Research Center

  28. Samet H (1994) The design and analysis of spatial data structures. Addison Wesley, Reading, MA

  29. Sederberg TW, Parry SR (1986) Free-form deformation of solid geometric models. In: Proceedings of ACM SIGGRAPH, ACM, New York, pp 151–159

  30. Shontz SM, Vavasis SA (2003) A mesh warping algorithm based on weighted Laplacian smoothing. In: Proceedings of the 12th international meshing roundtable, pp 147–158

  31. Shontz SM, Vavasis SA (2010) Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes. BIT Numer Math 50(4):863–884

    Article  MATH  MathSciNet  Google Scholar 

  32. Shontz SM, Vavasis SA (2012) A robust solution procedure for hyperelastic solids with large boundary deformation. Eng Comput 28(2):135–147

    Article  Google Scholar 

  33. Sibson R (1981) A brief description of natural neighbor interpolation, vol 21. In: Barnett V (ed) Interpreting multivariate data. Wiley and Sons, New York

  34. Staten ML, Canann SA, Owen SJ (1999) BMS weep: locating interior nodes during sweeping. Eng Comput 15(3):212–218

    Article  MATH  Google Scholar 

  35. Staten ML, Owen SJ, Shontz SM, Salinger AG, Coffey TS (2011) A comparison of mesh morphing methods for 3D shape optimization. In: Proceedings of the 20th international meshing roundtable, pp 293–311

  36. Sukumar N (2004) Construction of polygonal interpolants: a maximum entropy approach. Int J Numer Meth Eng 61(12):2159–2181

    Article  MATH  MathSciNet  Google Scholar 

  37. Sukumar N, Malsch EA (2006) Recent advances in the construction of polygonal finite element interpolants. Arch Comput Method E 13(1):129–163

    Article  MATH  MathSciNet  Google Scholar 

  38. Tomov S, Nath R, Ltaief H, Dongarra J (2010) Dense linear algebra solvers for multicore with GPU accelerators. In: 2010 IEEE international symposium on parallel and distributed processing, workshops and PhD forum (IPDPSW), pp 1–8

  39. Wachspress EL (1975) A rational finite element basis. Academic Press, New York

  40. Walder C, Schölkopf B, Chapelle O (2006) Implicit surface modelling with a globally regularised basis of compact support. Comput Graph Forum 25(3):635–644

    Article  Google Scholar 

  41. Wendland H (2005) Scattered data approximation. Cambridge University Press, Cambridge

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Acknowledgments

The authors kindly thank Matthew Staten from Sandia National Laboratories for providing us with the Bore, Pipe, and Courier models from [35]. We also thank the anonymous reviewers for their useful comments and suggestions. Daniel Sieger gratefully acknowledges the financial support from Honda Research Institute Europe (HRI-EU). Mario Botsch is supported by the German National Research Foundation (DFG CoE 277: CITEC).

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Authors and Affiliations

  1. Bielefeld University, Postfach 100 131, 33501, Bielefeld, Germany

    Daniel Sieger & Mario Botsch

  2. Honda Research Institute Europe GmbH, Carl-Legien-Str. 30, 63073, Offenbach/Main, Germany

    Stefan Menzel

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  1. Daniel Sieger
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  2. Stefan Menzel
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  3. Mario Botsch
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Correspondence to Daniel Sieger.

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Sieger, D., Menzel, S. & Botsch, M. RBF morphing techniques for simulation-based design optimization. Engineering with Computers 30, 161–174 (2014). https://doi.org/10.1007/s00366-013-0330-1

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