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Publikacje
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[59430] Artykuł:

AN IMPROVED XFEM FOR THE POISSON EQUATION WITH DISCONTINUOUS COEFFICIENTS

Czasopismo: Archive of Mechanical Engineering   Tom: 64, Zeszyt: 1, Strony: 123-144
ISSN:  2300-1895
Opublikowano: 2017
 
  Autorzy / Redaktorzy / Twórcy
Imię i nazwisko Wydział Katedra Do oświadczenia
nr 3
Grupa
przynależności
Dyscyplina
naukowa
Procent
udziału
Liczba
punktów
do oceny pracownika
Liczba
punktów wg
kryteriów ewaluacji
Paweł Stąpór orcid logo WZiMKKatedra Informatyki i Matematyki Stosowanej**Niespoza "N" jednostkiInżynieria mechaniczna10015.00.00  

Grupa MNiSW:  Publikacja w recenzowanym czasopiśmie wymienionym w wykazie ministra MNiSzW (część B)
Punkty MNiSW: 15


DOI LogoDOI     Web of Science LogoYADDA/CEON    
Słowa kluczowe:

równanie Poissona  słaba nieciągłość  rozszerzona metoda elementów skończonych  XFEM 


Keywords:

Poisson equation  weak discontinuity  XFEM 



Abstract:

Discontinuous coefficients in the Poisson equation lead to the weak discontinuity
in the solution, e.g. the gradient in the field quantity exhibits a rapid change across
an interface. In the real world, discontinuities are frequently found (cracks, material
interfaces, voids, phase-change phenomena) and their mathematical model can be
represented by Poisson type equation. In this study, the extended finite element method
(XFEM) is used to solve the formulated discontinuous problem. The XFEM solution
introduce the discontinuity through nodal enrichment function, and controls it by
additional degrees of freedom. This allows one to make the finite element mesh
independent of discontinuity location. The quality of the solution depends mainly
on the assumed enrichment basis functions. In the paper, a new set of enrichments
are proposed in the solution of the Poisson equation with discontinuous coefficients.
The global and local error estimates are used in order to assess the quality of the
solution. The stability of the solution is investigated using the condition number of the
stiffness matrix. The solutions obtained with standard and new enrichment functions
are compared and discussed.



B   I   B   L   I   O   G   R   A   F   I   A
[1] T.P. Fries and H.G. Matthies. Classification and overview of meshfree methods. Informatikbericht Nr.: 2003-3. Technical University Braunschweig, Brunswick, Germany, 2004.
[2] M.A. Schweitzer. Meshfree and generalized finite element methods. Postdoctoral dissertation. Mathematisch–Naturwissenschaftlichen Fakultat der Rheinischen Friedrich- -Wilhelms--Universitat, Bonn, Germany, 2008.
[3] Vinh Phu Nguyen, C. Anitescu, S. Bordas, and T. Rabczuk. Isogeometric analysis: An overview and computer implementation aspects. Mathematics and Computers in Simulation, 117:89–116, 2015. doi: 10.1016/j.matcom.2015.05.008.
[4] T. Belytschko and T. Black. Elastic crack growth in finite elements with minimal remeshing. International journal for numerical methods in engineering, 45(5):601–620, 1999.
[5] R. Merle and J. Dolbow. Solving thermal and phase change problems with the eXtended finite element method. Computational mechanics, 28(5):339–350, 2002. doi: 10.1007/s00466-002-0298-y.
[6] J. Chessa, P. Smolinski, and T. Belytschko. The extended finite element method (XFEM) for solidification problems. International Journal for Numerical Methods in Engineering, 53(8):1959–1977, 2002. doi: 10.1002/nme.386.
[7] P. Stapór. The XFEM for nonlinear thermal and phase change problems. International Journal of Numerical Methods for Heat & Fluid Flow, 25(2):400–421, 2015. doi: 10.1108/HFF-02-2014-0052.
[8] J.Y. Wu and F.B. Li. An improved stable XFEM (Is-XFEM) with a novel enrichment function for the computational modeling of cohesive cracks. Computer Methods in Applied Mechanics and Engineering, 295:77–107, 2015. doi: 10.1016/j.cma.2015.06.018.
[9] P. Hansbo, M.G. Larson, and S. Zahedi. A cut finite element method for a stokes interface problem. Applied Numerical Mathematics, 85:90–114, 2014. doi: 10.1016/j.apnum.2014.06.009.
[10] E. Wadbro, S. Zahedi, G. Kreiss, and M. Berggren. A uniformly well-conditioned, unfitted nitsche method for interface problems. BIT Numerical Mathematics, 53(3):791–820, 2013. doi: 10.1007/s10543-012-0417-x.
[11] I. Babuška and U. Banerjee. Stable generalized finite element method (SGFEM). Computer Methods in Applied Mechanics and Engineering, 201:91–111, 2012. doi: 10.1016/j.cma.2011.09.012.
[12] K. Kergrene, I. Babuška, and U. Banerjee. Stable generalized finite element method and associated iterative schemes
application to interface problems. Computer Methods in Applied Mechanics and Engineering, 305:1–36, 2016. doi: 10.1016/j.cma.2016.02.030.
[13] G. Zi and T. Belytschko. New crack-tip elements for xfem and applications to cohesive cracks. International Journal for Numerical Methods in Engineering, 57(15):2221–2240, 2003. doi: 10.1002/nme.849.
[14] G. Ventura, E. Budyn, and T. Belytschko. Vector level sets for description of propagating cracks in finite elements. International Journal for Numerical Methods in Engineering, 58(10):1571–1592, 2003. doi: 10.1002/nme.829.
[15] J.E. Tarancón, A.Vercher, E. Giner, and F.J. Fuenmayor. Enhanced blending elements for XFEM applied to linear elastic fracture mechanics. International Journal for Numerical Methods in Engineering, 77(1):126–148, 2009. doi: 10.1002/nme.2402.
[16] T.P. Fries. A corrected XFEM approximation without problems in blending elements. International Journal for Numerical Methods in Engineering, 75(5):503–532, 2008. doi: 10.1002/nme.2259.
[17] P. Stąpór. Application of xfem with shifted-basis approximation to computation of stress intensity factors. Archive of Mechanical Engineering, 58(4):447–483, 2011. doi: 10.2478/v10180-011-0028-0.
[18] N. Moës, M. Cloirec, P. Cartraud, and J.-F. Remacle. A computational approach to handle complex microstructure geometries. Computer methods in applied mechanics and engineering, 192(28):3163–3177, 2003. doi: 10.1016/S0045-7825(03)00346-3.
[19] J. Dolbow, N. Moës, and T. Belytschko. Discontinuous enrichment in finite elements with a partition of unity method. Finite elements in analysis and design, 36(3):235–260, 2000. doi:10.1016/S0168-874X(00)00035-4.
[20] B.A. Saxby. High-order XFEM with applications to two-phase flows. PhD thesis, The University of Manchester, Manchester, UK, 2014. www.escholar.manchester.ac.uk/uk-ac-manscw:234445.