22 research outputs found
Enhanced error estimator based on a nearly equilibrated moving least squares recovery technique for FEM and XFEM
In this paper a new technique aimed to obtain accurate estimates of the error
in energy norm using a moving least squares (MLS) recovery-based procedure is
presented. We explore the capabilities of a recovery technique based on an
enhanced MLS fitting, which directly provides continuous interpolated fields,
to obtain estimates of the error in energy norm as an alternative to the
superconvergent patch recovery (SPR). Boundary equilibrium is enforced using a
nearest point approach that modifies the MLS functional. Lagrange multipliers
are used to impose a nearly exact satisfaction of the internal equilibrium
equation. The numerical results show the high accuracy of the proposed error
estimator
Efficient recovery-based error estimation for the smoothed finite element method for smooth and singular linear elasticity
[EN] An error control technique aimed to assess the quality of smoothed finite element approximations is presented in this paper. Finite element techniques based on strain smoothing appeared in 2007 were shown to provide significant advantages compared to conventional finite element approximations. In particular, a widely cited strength of such methods is improved accuracy for the same computational cost. Yet, few attempts have been made to directly assess the quality of the results obtained during the simulation by evaluating an estimate of the discretization error. Here we propose a recovery type error estimator based on an enhanced recovery technique. The salient features of the recovery are: enforcement of local equilibrium and, for singular problems a ¿smooth + singular¿ decomposition of the recovered stress. We evaluate the proposed estimator on a number of test cases from linear elastic structural mechanics and obtain efficient error estimations whose effectivities, both at local and global levels, are improved compared to recovery procedures not implementing these features.Stephane Bordas would like to thank the partial financial support of the Royal Academy of Engineering and of the Leverhulme Trust for his Senior Research Fellowship Towards the next generation surgical simulators as well as the financial support for Octavio A. Gonzalez-Estrada and Stephane Bordas from the UK Engineering Physical Science Research Council (EPSRC) under grant EP/G042705/1 Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method. Stephane Bordas also thanks partial financial support of the European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) and the FP7 Initial Training Network Funding under grant number 289361 "Integrating Numerical Simulation and Geometric Design Technology, INSIST". This work has been carried out within the framework of the research project DPI2010-20542 of the Ministerio de Ciencia e Innovacion (Spain). The financial support from Universitat Politecnica de Valencia, PROMETEO/2012/023 and Generalitat Valenciana are also acknowledged.González Estrada, OA.; Natarajan, S.; J.J. Ródenas; Nguyen-Xuan, H.; Bordas, S. (2013). Efficient recovery-based error estimation for the smoothed finite element method for smooth and singular linear elasticity. Computational Mechanics. 52(1):37-52. https://doi.org/10.1007/s00466-012-0795-6S3752521Liu GR, Dai KY, Nguyen TT (2006) A smoothed finite element method for mechanics problems. Comput Mech 39(6): 859–877. doi: 10.1007/s00466-006-0075-4Liu GR, Nguyen TT, Dai KY, Lam KY (2007) Theoretical aspects of the smoothed finite element method (SFEM). Int J Numer Methods Eng 71(8): 902–930Nguyen-Xuan H, Bordas SPA, Nguyen-Dang H (2008) Smooth finite element methods: convergence, accuracy and properties. Int J Numer Methods Eng 74(2): 175–208. doi: 10.1002/nmeBordas SPA, Natarajan S (2010) On the approximation in the smoothed finite element method (SFEM). Int J Numer Methods Eng 81(5): 660–670. doi: 10.1002/nmeZhang HH, Liu SJ, Li LX (2008) On the smoothed finite element method. Int J Numer Methods Eng 76(8): 1285–1295. doi: 10.1002/nme.2460Nguyen-Thoi T, Liu G, Lam K, Zhang G. (2009) A face-based smoothed finite element method (FS-FEM) for 3D linear and nonlinear solid mechanics using 4-node tetrahedral elements. Int J Numer Methods Eng 78: 324–353Liu G, Nguyen-Thoi T, Lam K (2009) An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids. J Sound Vib 320: 1100–1130Liu G, Nguyen-Thoi T, Nguyen-Xuan H, Lam K (2009) A node based smoothed finite element method (NS-FEM) for upper bound solution to solid mechanics problems. Comput Struct 87: 14–26Liu G. Smoothed Finite Element Methods. CRC Press, 2010Liu G, Nguyen-Xuan H, Nguyen-Thoi T (2010) A theoretical study on the smoothed FEM (SFEM) models: Properties, accuracy and convergence rates. Int J Numer Methods Biomed Eng 84: 1222–1256Nguyen T, Liu G, Dai K, Lam K (2007) smoothed finite element method. Tsinghua Sci Technol 12: 497–508Hung NX, Bordas S, Hung N (2009) Addressing volumetric locking and instabilities by selective integration in smoothed finite element. Commun Numer Methods Eng 25: 19–34Nguyen-Xuan H, Rabczuk T, Bordas S, Debongnie JF (2008) A smoothed finite element method for plate analysis. Comput Methods Appl Mech Eng 197: 1184–1203Nguyen NT, Rabczuk T, Nguyen-Xuan H, Bordas S (2008) A smoothed finite element method for shell analysis. Comput Methods Appl Mech Eng 198: 165–177Bordas SPA, Rabczuk T, Hung NX, Nguyen VP, Natarajan S, Bog T, óuan DM, Hiep NV (2010) Strain smoothing in FEM and XFEM. Comput Struct 88(23–24): 1419–1443. doi: 10.1016/j.compstruc.2008.07.006Bordas SP, Natarajan S, Kerfriden P, Augarde CE, Mahapatra DR, Rabczuk T, Pont SD (2011) On the performance of strain smoothing for óuadratic and enriched finite element approximations (XFEM/GFEM/PUFEM). Int J Numer Methods Biomed Eng 86: 637–666Liu G, Nguyen-Thoi T, Nguyen-Xuan H, Dai K, Lam K (2009) On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM). Int J Numer Methods Eng 77: 1863–1869. doi: 10.1002/nme.2587Strouboulis T, Zhang L, Wang D, Babuška I. (2006) A posteriori error estimation for generalized finite element methods. Comput Methods Appl Mech Eng 195(9–12): 852–879Bordas SPA, Duflot M (2007) Derivative recovery and a posteriori error estimate for extended finite elements. Comput Methods Appl Mech Eng 196(35–36): 3381–3399Xiao óZ, Karihaloo BL (2004) Statically admissible stress recovery using the moving least sóuares technique. In: Topping BHV, Soares CAM (eds) Progress in computational structures technology. Saxe-Coburg Publications, Stirling, pp 111–138Ródenas JJ, González-Estrada OA, Tarancón JE, Fuenmayor FJ (2008) A recovery-type error estimator for the extended finite element method based on singular + smooth stress field splitting. Int J Numer Methods Eng 76(4): 545–571. doi: 10.1002/nme.2313Panetier J, Ladevèze P, Chamoin L (2010) Strict and effective bounds in goal-oriented error estimation applied to fracture mechanics problems solved with XFEM. Int J Numer Methods Eng 81(6): 671–700Barros FB, Proenca SPB, de Barcellos CS (2004) On error estimator and p-adaptivity in the generalized finite element method. Int J Numer Methods Eng 60(14):2373–2398. doi: 10.1002/nme.1048Nguyen-Thoi T, Liu G, Nguyen-Xuan H, Nguyen-Tran C (2011) Adaptive analysis using the node-based smoothed finite element method (NS-FEM). Int J Numer Methods Biomed Eng 27(2): 198–218. doi: 10.1002/cnmGonzález-Estrada OA, Ródenas JJ, Bordas SPA, Duflot M, Kerfriden P, Giner E (2012) On the role of enrichment and statical admissibility of recovered fields in a-posteriori error estimation for enriched finite element methods. Eng Comput 29(8)Zienkiewicz OC, Zhu JZ (1987) A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Methods Eng 24(2): 337–357Ródenas JJ, González-Estrada OA, Díez P, Fuenmayor FJ (2010) Accurate recovery-based upper error bounds for the extended finite element framework. Comput Methods Appl Mech Eng 199(37–40): 2607–2621Williams ML (1952) Stress singularities resulting from various boundary conditions in angular corners of plate in extension. J Appl Mech 19: 526–534Szabó BA, Babuška I (1991) Finite element analysis. Wiley, New YorkBarber JR. (2010) Elasticity. Series: solid mechanics and its application, 3rd edn. Springer, DordrechtChen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerki mesh-free methods. Int J Numer Methods Eng 50: 435–466Yoo J, Moran B, Chen J (2004) Stabilized conforming nodal integration in the natural element method. Int J Numer Methods Eng 60: 861–890Zienkiewicz OC, Zhu JZ (1992) The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int J Numer Methods Eng 33(7): 1331–1364Zienkiewicz OC, Zhu JZ (1992) The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity. Int J Numer Methods Eng 33(7): 1365–1382Wiberg NE, Abdulwahab F (1993) Patch recovery based on superconvergent derivatives and eóuilibrium. Int J Numer Methods Eng 36(16): 2703–2724. doi: 10.1002/nme.1620361603Blacker T, Belytschko T (1994) Superconvergent patch recovery with eóuilibrium and conjoint interpolant enhancements. Int J Numer Methods Eng 37(3): 517–536Stein E, Ramm E, Rannacher R (2003) Error-controlled adaptive finite elements in solid mechanics. Wiley, ChichesterDuflot M, Bordas SPA (2008) A posteriori error estimation for extended finite elements by an extended global recovery. Int J Numer Methods Eng 76: 1123–1138. doi: 10.1002/nmeBordas SPA, Duflot M, Le P (2008) A simple error estimator for extended finite elements. Commun Numer Methods Eng 24(11): 961–971Ródenas JJ, Tur M, Fuenmayor FJ, Vercher A (2007) Improvement of the superconvergent patch recovery technique by the use of constraint eóuations: the SPR-C technique. Int J Numer Methods Eng 70(6): 705–727. doi: 10.1002/nme.1903Díez P, Ródenas JJ, Zienkiewicz OC (2007) Eóuilibrated patch recovery error estimates: simple and accurate upper bounds of the error. Int J Numer Methods Eng 69(10): 2075–2098. doi: 10.1002/nmeYau J, Wang S, Corten H (1980) A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J Appl Mech 47(2): 335–341Ródenas JJ, González-Estrada OA, Fuenmayor FJ, Chinesta F (2010) Upper bounds of the error in X-FEM based on a moving least sóuares (MLS) recovery technique. In: Khalili N, Valliappan S, Li ó, Russell A (eds) 9th World congress on computational mechanics (WCCM9). 4th Asian Pacific Congress on computational methods (APCOM2010). Centre for Infrastructure Engineering and SafetyRódenas JJ, González-Estrada OA, Díez P, Fuenmayor FJ (2007) Upper bounds of the error in the extended finite element method by using an eóuilibrated-stress patch recovery technique. In: International conference on adaptive modeling and simulation (ADMOS 2007). International Center for Numerical Methods in Engineering (CIMNE), pp 210–213Menk A, Bordas S (2010) Numerically determined enrichment function for the extended finite element method and applications to bi-material anisotropic fracture and polycrystals. Int J Numer Methods Eng 83: 805–828Menk A, Bordas S (2011) Crack growth calculations in solder joints based on microstructural phenomena with X-FEM. Comput Mater Sci 3: 1145–1156Ródenas JJ (2001) Error de discretización en el cálculo de sensibilidades mediante el método de los elementos finitos. PhD Thesis, Universidad Politécnica de ValenciaAinsworth M, Oden JT (2000) A posteriori error estimation in finite element analysis. Wiley, Chicheste
Mechanistic Studies of CX Bond Activation at Transition-Metal Centers
This chapter surveys mechanistic computational studies on the activation of CX single bonds mediated by transition-metal centers. X can be based on any element of groups 13-17 and 'mechanistic' indicates that the transition state for CX bond activation has been located. Bond activation itself refers to the cleavage of a CX bond such that both partners may (potentially) act as neutral one-electron donors to the metal center. This definition covers oxidative addition, typically via a concerted and also by SN2 or radical processes; also included are σ-bond metathesis, as well as CX cleavage promoted by Lewis acidic and Lewis basic species. The reactivity of neutral transition atoms and cations (often modeling gas-phase experimental reactivity) and of molecular complexes (of relevance to solution-phase reactivity and homogeneous catalysis) is discussed.</p
Mechanistic Studies of CX Bond Activation at Transition-Metal Centers
This chapter surveys mechanistic computational studies on the activation of CX single bonds mediated by transition-metal centers. X can be based on any element of groups 13-17 and 'mechanistic' indicates that the transition state for CX bond activation has been located. Bond activation itself refers to the cleavage of a CX bond such that both partners may (potentially) act as neutral one-electron donors to the metal center. This definition covers oxidative addition, typically via a concerted and also by SN2 or radical processes; also included are σ-bond metathesis, as well as CX cleavage promoted by Lewis acidic and Lewis basic species. The reactivity of neutral transition atoms and cations (often modeling gas-phase experimental reactivity) and of molecular complexes (of relevance to solution-phase reactivity and homogeneous catalysis) is discussed.</p
Mechanistic Studies of CX Bond Activation at Transition-Metal Centers
This chapter surveys mechanistic computational studies on the activation of CX single bonds mediated by transition-metal centers. X can be based on any element of groups 13-17 and 'mechanistic' indicates that the transition state for CX bond activation has been located. Bond activation itself refers to the cleavage of a CX bond such that both partners may (potentially) act as neutral one-electron donors to the metal center. This definition covers oxidative addition, typically via a concerted and also by SN2 or radical processes; also included are σ-bond metathesis, as well as CX cleavage promoted by Lewis acidic and Lewis basic species. The reactivity of neutral transition atoms and cations (often modeling gas-phase experimental reactivity) and of molecular complexes (of relevance to solution-phase reactivity and homogeneous catalysis) is discussed.</p
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Cs diffusion mechanisms in UO investigated by SIMS, TEM, and atomistic simulations
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