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Rapid Enriched Simulation Application Development with PUMA

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Scientific Computing and Algorithms in Industrial Simulations

Abstract

In this paper we describe the functionality and the main components of the PUMA software toolkit. PUMA is designed to allow for the rapid development of simulation applications using generalized finite element techniques based on the partition of unity method (PUM). Unlike classical finite element methods (FEM) a PUM can directly utilize user insight, domain-specific information and physics-based basis functions to reduce the computational cost substantially and thereby allows for the rapid evaluation of novel models. We discuss the basic building blocks of the PUMA software framework and present some examples showcasing the capabilities of PUMA and its ease of use.

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Notes

  1. 1.

    Note that only for the largest extraction radius the SIFs are computed from local approximation spaces without any enrichments.

References

  1. M.S. Alnæs, J. Blechta, J. Hake et al., The FEniCS project version 1.5. Arch. Numer. Softw. 3, 9–23 (2015)

    Google Scholar 

  2. U. Ayachit, The ParaView Guide: A Parallel Visualization Application (Kitware, New York, 2015)

    Google Scholar 

  3. I. Babuška, U. Banerjee, Stable generalized finite element method (SGFEM). Comput. Methods Appl. Mech. Eng. 201–204, 91–111 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. I. Babuška, G. Caloz, J.E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J. Numer. Anal. 31, 945–981 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  5. I. Babuška, J.M. Melenk, The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 139, 289–314 (1996). Special Issue on Meshless Methods

    Google Scholar 

  6. I. Babuška, J.M. Melenk, The partition of unity method. Int. J. Numer. Methods Eng. 40, 727–758 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Belytschko, T. Black, Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45, 601–620 (1999)

    Article  MATH  Google Scholar 

  8. G.L. Bernstein, F. Kjolstad, Why new programming languages for simulation? ACM Trans. Graph. 35, 20e:1–20e:3 (2016)

    Google Scholar 

  9. G.L. Bernstein, C. Shah, C. Lemire et al., Ebb: a DSL for physical simulation on CPUs and GPUs. ACM Trans. Graph. 35, 21:1–21:12 (2016)

    Google Scholar 

  10. G.P. Cherepanov, The propagation of cracks in a continuous medium. J. Appl. Math. Mech. 31, 503–512 (1967)

    Article  MATH  Google Scholar 

  11. T.-P. Fries, T. Belytschko, The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84, 253–304 (2010)

    MathSciNet  MATH  Google Scholar 

  12. M. Griebel, M.A. Schweitzer, A particle-partition of unity method for the solution of elliptic, parabolic and hyperbolic PDE. SIAM J. Sci. Comput. 22, 853–890 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Griebel, M.A. Schweitzer, A particle-partition of unity method—Part II: efficient cover construction and reliable integration. SIAM J. Sci. Comput. 23, 1655–1682 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Griebel, M.A. Schweitzer, A particle-partition of unity method—Part III: a multilevel solver. SIAM J. Sci. Comput. 24, 377–409 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Griebel, M.A. Schweitzer, A particle-partition of unity method—Part V: boundary conditions, in Geometric Analysis and Nonlinear Partial Differential Equations, ed. by S. Hildebrandt, H. Karcher (Springer, Berlin, 2002), pp. 517–540

    Google Scholar 

  16. V. Gupta, C. Duarte, I. Babuška, U. Banerjee, Stable GFEM (SGFEM): improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics. Comput. Methods Appl. Mech. Eng. 289, 355–386 (2015)

    Article  MathSciNet  Google Scholar 

  17. F. Hecht, New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Huerta, T. Belytschko, T. Fernández-Méndez, T. Rabczuk, Meshfree Methods. Encyclopedia of Computational Mechanics, vol. 1, chap. 10 (Wiley, New York, 2004), pp. 279–309

    Google Scholar 

  19. M.J. Hunsweck, Y. Shen, A.J. Lew, A finite element approach to the simulation of hydraulic fractures with lag. Int. J. Numer. Anal. Methods Geomech. 37, 993–1015 (2013)

    Article  Google Scholar 

  20. F. Kjolstad, S. Kamil, J. Ragan-Kelley et al., Simit: a language for physical simulation. ACM Trans. Graph. 35 20:1–20:21 (2016)

    Google Scholar 

  21. A. Logg, K.-A. Mardal, G.N. Wells et al., Automated Solution of Differential Equations by the Finite Element Method (Springer, Berlin, 2012)

    Book  MATH  Google Scholar 

  22. N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46, 131–150 (1999)

    Article  MATH  Google Scholar 

  23. J.R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379–386 (1968)

    Article  Google Scholar 

  24. M.A. Schweitzer, A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 29 (Springer, Berlin, 2003)

    Google Scholar 

  25. M.A. Schweitzer, Meshfree and Generalized Finite Element Methods. Habilitation, Institute for Numerical Simulation, University of Bonn (2008)

    Google Scholar 

  26. M.A. Schweitzer, An adaptive hp-version of the multilevel particle–partition of unity method. Comput. Methods Appl. Mech. Eng. 198, 1260–1272 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. M.A. Schweitzer, An algebraic treatment of essential boundary conditions in the particle–partition of unity method. SIAM J. Sci. Comput. 31, 1581–1602 (2009)

    Article  MATH  Google Scholar 

  28. M.A. Schweitzer, Multilevel particle–partition of unity method. Numer. Math. 118, 307–328 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. M.A. Schweitzer, Stable enrichment and local preconditioning in the particle–partition of unity method. Numer. Math. 118, 137–170 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. M.A. Schweitzer, Generalizations of the finite element method. Cent. Eur. J. Math. 10, 3–24 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. M.A. Schweitzer, Variational mass lumping in the partition of unity method. SIAM J. Sci. Comput. 35, A1073–A1097 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. M.A. Schweitzer, S. Wu, Evaluation of local multiscale approximation spaces for partition of unity methods, in Meshfree Methods for Partial Differential Equations VIII, ed. by M. Griebel, M.A. Schweitzer. Lecture Notes in Science and Engineering, vol. 115 (2017), pp. 163–194

    Google Scholar 

  33. M.A. Schweitzer, A. Ziegenhagel, Embedding enriched partition of unity approximations in finite element simulations, in Meshfree Methods for Partial Differential Equations VIII, ed. by M. Griebel, M.A. Schweitzer. Lecture Notes in Science and Engineering, vol. 115 (2017), pp. 195–204

    Google Scholar 

  34. D. Shepard, A two-dimensional interpolation function for irregularly spaced data, in Proceedings 1968 ACM National Conference (ACM, New York, 1968), pp. 517–524

    Google Scholar 

  35. T. Strouboulis, I. Babuška, K. Copps, The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Eng. 181, 43–69 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  36. T. Strouboulis, K. Copps, I. Babuška, The generalized finite element method. Comput. Methods Appl. Mech. Eng. 190, 4081–4193 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. B. Szabó, I. Babuška, Finite Element Analysis (Wiley, New York, 1991)

    MATH  Google Scholar 

  38. Q. Zhang, U. Banerjee, I. Babuška, Higher order stable generalized finite element method. Numer. Math. 128, 1–29 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Fraunhofer Institute for Algorithms and Scientific Computing SCAI, Schloss Birlinghoven, 53757, Sankt Augustin, Germany

    Marc Alexander Schweitzer & Albert Ziegenhagel

  2. Institute for Numerical Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Wegelerstr. 6, 53115, Bonn, Germany

    Marc Alexander Schweitzer

Authors
  1. Marc Alexander Schweitzer
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  2. Albert Ziegenhagel
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Correspondence to Marc Alexander Schweitzer .

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Editors and Affiliations

  1. Schloss Birlinghoven, Fraunhofer-Institut SCAI, Sankt Augustin, Germany

    Michael Griebel

  2. Schloss Birlinghoven, Fraunhofer-Institut SCAI, Sankt Augustin, Germany

    Anton Schüller

  3. Schloss Birlinghoven, Fraunhofer-Institut SCAI, Sankt Augustin, Germany

    Marc Alexander Schweitzer

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Schweitzer, M.A., Ziegenhagel, A. (2017). Rapid Enriched Simulation Application Development with PUMA. In: Griebel, M., Schüller, A., Schweitzer, M. (eds) Scientific Computing and Algorithms in Industrial Simulations. Springer, Cham. https://doi.org/10.1007/978-3-319-62458-7_11

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