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Surface Representation with Closest Point Projection in the X-FEM

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Abstract

At present, the eXtended Finite Element Method (X-FEM) is a common generalization of the classical finite element method for solving problems of deformable solid mechanics in the presence of large cracks. The main advantage of this method is the possibility of using computational meshes that are not consistent with the crack geometry and the feasibility of accurately taking into account the singular asymptotics of the solution near the front of the crack. One of the key elements of this method is the way that the middle surface of the crack is represented in the algorithm. It is traditional to use an implicit surface representation based on the level set method. Such an approach is efficient and robust and allows the calculation in the case of propagating cracks. In the paper, we propose a version of the X-FEM approach which uses the closest point projection (CPP) method to represent the middle surface of the crack and provides, in our opinion, several advantages over the common X-FEM. The paper presents a short overview of the classical X-FEM technique. The proposed algorithm and its distinctions from the conventional one are described in detail and its advantages are formulated. We consider the problems of the numerical integration of functions defined on the surface described by the closest point projector method, the local recovery of the level set functions, and the computation of local frames at a point of the surface or its edge. The algorithmic details of the X-FEM enriched with a surface representation based on the projection of the closest point are described. In conclusion, we present some numerical results that demonstrate the algorithmic features of the method and performance capabilities of the proposed algorithm.

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REFERENCES

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

    Article  Google Scholar 

  2. N. Moes, J. Dolbow, and T. Belytschko, “A finite element method for crack growth without remeshing,” Int. J. Numer. Methods Eng. 46, 131–150 (1999).

    Article  MathSciNet  Google Scholar 

  3. T. P. Fries and T. Belytschko, “The extended/generalized finite element method: an overview of the method and its applications,” Int. J. Numer. Methods Eng. 84, 253–304 (2012).

    MathSciNet  MATH  Google Scholar 

  4. T. P. Fries, N. Moës, and A. Zilian, “The extended finite element method. Special issue,” Int. J. Numer. Methods Eng. 86, 403–666 (2011).

    Article  Google Scholar 

  5. A. Gravouil, N. Moës, and T. Belytschko, “Non-planar 3D crack growth by the extended finite element and level sets. Part II: Level set update,” Int. J. Numer. Methods Eng. 53, 2569–2586 (2002).

    Article  Google Scholar 

  6. N. Moës, A. Gravouil, and T. Belytschko, “Non-planar 3D crack growth by the extended fiite element and level sets. Part I: Mechanical model,” Int. J. Numer. Methods Eng. 53, 2549–2568 (2002).

    Article  Google Scholar 

  7. S. J. Osher and R. P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, New York, 2002).

    MATH  Google Scholar 

  8. M. Stolarska, D. Chopp, N. Moës, and T. Belytschko, “Modelling crack growth by level sets in the extended finite element method,” Int. J. Numer. Methods Eng. 51, 943–960 (2001).

    Article  Google Scholar 

  9. C. B. Macdonald, J. Brandman, and S. J. Ruuth, “Solving eigenvalue problems on curved surfaces using the Closest Point Method,” J. Comput. Phys. 230, 7944–7956 (2011).

    MathSciNet  MATH  Google Scholar 

  10. G. Ventura, E. Budyn, and T. Belytschko, “Vector level sets for description of propagating cracks in finite elements,” Int. J. Numer. Methods Eng. 58, 1571–1592 (2003).

    Article  Google Scholar 

  11. M. Berger, A. Tagliasacchi, L. Seversky, P. Alliez, J. Levine, A. Sharf, and C. Silva, “State of the art in surface reconstruction from point clouds,” in Proceedings of the EUROGRAPHICS 2014, STAR - State of The Art Report, Ed. by S. Lefebvre and M. Spagnuolo (2014).

  12. A. V. Ivanov and E. B. Savenkov, “Simulation and visualization of the dynamics of a surface with a movable boundary on a stationary unstructured mesh,” Nauch. Vizualiz. 9 (2), 64–81 (2017).

    Google Scholar 

  13. T. März and C. B. Macdonald, “Calculus on surfaces with general closest point functions,” SIAM J. Numer. Anal. 50, 3303–3328 (2012).

    Article  MathSciNet  Google Scholar 

  14. O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method (Butterworth-Heinemann, Oxford, 2005).

    MATH  Google Scholar 

  15. A. F. Bower, Applied Mechanics of Solids (CRC, Boca Raton, FL, 2009).

    Book  Google Scholar 

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Funding

This work was supported by the Russian Science Foundation, grant no. 15-11-00021.

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Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

    E. B. Savenkov, V. E. Borisov & B. V. Kritskiy

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  1. E. B. Savenkov
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  2. V. E. Borisov
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  3. B. V. Kritskiy
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Correspondence to E. B. Savenkov, V. E. Borisov or B. V. Kritskiy.

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The authors declare that they have no conflicts of interest.

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Translated by I. Tselishcheva

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Savenkov, E.B., Borisov, V.E. & Kritskiy, B.V. Surface Representation with Closest Point Projection in the X-FEM. Math Models Comput Simul 12, 36–52 (2020). https://doi.org/10.1134/S207004822001007X

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  • DOI: https://doi.org/10.1134/S207004822001007X

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