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Abstract

In the paper, the extended finite element method (XFEM) is combined with a recovery procedure in the analysis of the discontinuous Poisson problem. The model considers the weak as well as the strong discontinuity. Computationally efficient low-order finite elements provided good convergence are used. The combination of the XFEM with a recovery procedure allows for optimal convergence rates in the gradient i.e. as the same order as the primary solution. The discontinuity is modelled independently of the finite element mesh using a step-enrichment and level set approach. The results show improved gradient prediction locally for the interface element and globally for the entire domain.

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Bibliography

[1] P. Stąpór. An improved XFEM for the Poisson equation with discontinuous coefficients. Archive of Mechanical Engineering, 64(1):123–144, 2017. doi: 10.1515/meceng-2017-0008.
[2] T. Grätsch and K.-J. Bathe. A posteriori error estimation techniques in practical finite element analysis. Computers & Structures, 83(4-5):235–265, 2005. doi: 10.1016/j.compstruc.2004.08.011.
[3] M. Ainsworth and J.T. Oden. A posteriori error estimation in finite element analysis. Computer Methods in Applied Mechanics and Engineering, 142(1-2):1–88, 1997. doi: 10.1016/S0045-7825(96)01107-3.
[4] P.J. Payen and K.-J. Bathe. A stress improvement procedure. Computers & Structures, 112-113:311–326, 2012. doi: 10.1016/j.compstruc.2012.07.006.
[5] 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. doi: 10.1002/(SICI)1097-0207(19990620)45:5601::AID-NME598>3.0.CO;2-S.
[6] P. Stąpór. Application of XFEM with shifted-basis approximation to computation of stress intensity factors. The Archive of Mechanical Engineering, 58(4):447–483, 2011. doi: 10.2478/v10180-011-0028-0.
[7] D. Belsley, R.E.Welsch, and E.Kuh. The Condition Number. Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. John Wiley & Sons, Inc., Hoboken, New Jersey, 1980.
[8] S. Hou and X.-D. Liu. A numerical method for solving variable coeffiecient elliptic equation with interfaces. Jurnal of Computational Physics, 202(2):411–445, 2005. doi: 10.1016/j.jcp.2004.07.016.
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Authors and Affiliations

Paweł Stąpór
1

  1. Faculty of Management and Computer Modelling, Kielce University of Technology, Kielce, Poland.
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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.

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Bibliography

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[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.
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[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.
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Authors and Affiliations

Paweł Stąpór
1

  1. Faculty of Management and Computer Modelling, Kielce University of Technology, Poland
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Abstract

This paper presents an extended finite element method applied to solve phase change problems taking into account natural convection in the liquid phase. It is assumed that the transition from one state to another, e.g., during the solidification of pure metals, is discontinuous and that the physical properties of the phases vary across the interface. According to the classical Stefan condition, the location, topology and rate of the interface changes are determined by the jump in the heat flux. The incompressible Navier-Stokes equations with the Boussinesq approximation of the natural convection flow are solved for the liquid phase. The no-slip condition for velocity and the melting/freezing condition for temperature are imposed on the interface using penalty method. The fractional four-step method is employed for analysing conjugate heat transfer and unsteady viscous flow. The phase interface is tracked by the level set method defined on the same finite element mesh. A new combination of extended basis functions is proposed to approximate the discontinuity in the derivative of the temperature, velocity and the pressure fields. The single-mesh approach is demonstrated using three two-dimensional benchmark problems. The results are compared with the numerical and experimental data obtained by other authors.

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Bibliography

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[14] 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.
[15] 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.
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[33] 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(28-30):3163–3177, 2003. doi: 10.1016/S0045-7825(03)00346-3.
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[35] 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.
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Authors and Affiliations

Paweł Stąpór
1

  1. Faculty of Management and Computer Modelling, Kielce University of Technology, Kielce, Poland.
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Abstract

The essential parameters for structure integrity assessment in Linear Elastic Fracture Mechanics (LEFM) are Stress Intensity Factors (SIFs). The estimation of SIFs can be done by analytical or numerical techniques. The analytical estimation of SIFs is limited to simple structures with non-complicated boundaries, loads and supports. An effective numerical technique for analyzing problems with singular fields, such as fracture mechanics problems, is the extended finite element method (XFEM). In the paper, XFEM is applied to compute an actual stress field in a two-dimensional cracked body. The XFEM is based on the idea of enriching the approximation in the vicinity of the discontinuity. As a result, the numerical model consists of three types of elements: non-enriched elements, fully enriched elements (the domain of whom is cut by a discontinuity), and partially enriched elements (the so-called blending elements). In a blending element, some but not all of the nodes are enriched, which adds to the approximation parasitic term. The error caused by the parasitic terms is partly responsible for the degradation of the convergence rate. It also limits the accuracy of the method. Eliminating blending elements from approximation space and replacing them with standard elements, together with applying shifted-basis enrichment, makes it possible to avoid the problem. The numerical examples show improvements in results when compared with the standard XFEM approach.

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Authors and Affiliations

Paweł Stąpór
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Abstract

In the paper, the authors present the shakedown analysis of the plate structures pre-loaded beyond the elastoplastic range. Two cases of loading are considered, namely: the structure is subjected to the action of two independent sets ofloads with constant points of application or one parameter set of loads moves slowly according to an a priori described program. As a result, the safe loading boundary or the shakedown load parameter are calculated, respectively, by means of the finite element method (FEM). Three examples confirmed the effectiveness of the proposed algorithms of analysis.
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Authors and Affiliations

Czesław Cichoń
Paweł Stąpór

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