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. 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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. 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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. 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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

Computational Study of an Iron(II) Polypyridine Electrocatalyst for CO2 Reduction: Key Roles for Intramolecular Interactions in CO2 Binding and Proton Transfer.

A solar-driven conversion of CO2 into fuels by artificial photosynthesis would not only mitigate the greenhouse effect but also provide an alternative to obtain fuels in a renewable fashion. To this end, the new iron polypyridine catalyst [Fe(bpyNHEtPY2Me)L2]2+ (L = H2O, CH3CN) was recently developed for the electrochemical reduction of CO2 to CO. In this study, we performed density functional theory (DFT) electronic structure calculations to shed light on a possible pathway for CO2 reduction and the origin of the selectivity between CO2 reduction versus the hydrogen evolution reaction. The metal center remains Lewis acidic throughout the reduction process due to ligand loss and mainly ligand-based reduction stabilized by antiferromagnetic coupling to a high-spin Fe(II) center. This results in a high barrier for hydride formation but a facile addition and activation of CO2 via an η2 coordination and stabilizing hydrogen bonding by the amine group. The second unoccupied equatorial coordination site opens up the possibility for an intramolecular protonation with a coordinated water ligand. This facilitates protonation because not only CO2 but also the proton source H2O is activated and properly aligned for a proton transfer due to the Fe-OH2 bond; consequently, both protonation steps are facile. The moderate ligand field allows a rapid ligand exchange for a second intramolecular protonation step and facilitates an exergonic CO release. The lower selectivity of the related [Fe(bpyOHPY2Me)L2]2+ complex can be related to its more acidic second coordination sphere, which enables an intramolecular proton transfer that is kinetically competitive with CO2 addition

Cs diffusion mechanisms in UO2_2 investigated by SIMS, TEM, and atomistic simulations

International audienceExperimental investigations and atomistic simulations are combined to study the cesium diffusion processes at high temperature in UO2. After 133Cs implantation in UO2 samples, diffusion coefficients are determined using the depth profile evolution after annealing as measured by secondary ion mass spectrometry. An activation energy of 1.8 ± 0.2 eV is subsequently deduced in the 1300-1600 °C temperature range. Experimental results are compared to nudged elastic band simulations performed for different atomic paths including several types of uranium vacancy defects. Activation energies ranging from 0.49 up to 2.34 eV are derived, showing the influence of the defect (both in terms of type and concentration) on the Cs diffusion process. Finally, molecular dynamics simulations are performed, allowing the identification of preferential Cs trajectories that corroborate experimental observations

Bioinspired design of redox-active ligands for multielectron catalysis: effects of positioning pyrazine reservoirs on cobalt for electro- and photocatalytic generation of hydrogen from water.

Mononuclear metalloenzymes in nature can function in cooperation with precisely positioned redox-active organic cofactors in order to carry out multielectron catalysis. Inspired by the finely tuned redox management of these bioinorganic systems, we present the design, synthesis, and experimental and theoretical characterization of a homologous series of cobalt complexes bearing redox-active pyrazines. These donor moieties are locked into key positions within a pentadentate ligand scaffold in order to evaluate the effects of positioning redox non-innocent ligands on hydrogen evolution catalysis. Both metal- and ligand-centered redox features are observed in organic as well as aqueous solutions over a range of pH values, and comparison with analogs bearing redox-inactive zinc(ii) allows for assignments of ligand-based redox events. Varying the geometric placement of redox non-innocent pyrazine donors on isostructural pentadentate ligand platforms results in marked effects on observed cobalt-catalyzed proton reduction activity. Electrocatalytic hydrogen evolution from weak acids in acetonitrile solution, under diffusion-limited conditions, reveals that the pyrazine donor of axial isomer 1-Co behaves as an unproductive electron sink, resulting in high overpotentials for proton reduction, whereas the equatorial pyrazine isomer complex 2-Co is significantly more active for hydrogen generation at lower voltages. Addition of a second equatorial pyrazine in complex 3-Co further minimizes overpotentials required for catalysis. The equatorial derivative 2-Co is also superior to its axial 1-Co congener for electrocatalytic and visible-light photocatalytic hydrogen generation in biologically relevant, neutral pH aqueous media. Density functional theory calculations (B3LYP-D2) indicate that the first reduction of catalyst isomers 1-Co, 2-Co, and 3-Co is largely metal-centered while the second reduction occurs at pyrazine. Taken together, the data establish that proper positioning of non-innocent pyrazine ligands on a single cobalt center is indeed critical for promoting efficient hydrogen catalysis in aqueous media, akin to optimally positioned redox-active cofactors in metalloenzymes. In a broader sense, these findings highlight the significance of electronic structure considerations in the design of effective electron-hole reservoirs for multielectron transformations

Bioinspired design of redox-active ligands for multielectron catalysis: effects of positioning pyrazine reservoirs on cobalt for electro- and photocatalytic generation of hydrogen from water.

Mononuclear metalloenzymes in nature can function in cooperation with precisely positioned redox-active organic cofactors in order to carry out multielectron catalysis. Inspired by the finely tuned redox management of these bioinorganic systems, we present the design, synthesis, and experimental and theoretical characterization of a homologous series of cobalt complexes bearing redox-active pyrazines. These donor moieties are locked into key positions within a pentadentate ligand scaffold in order to evaluate the effects of positioning redox non-innocent ligands on hydrogen evolution catalysis. Both metal- and ligand-centered redox features are observed in organic as well as aqueous solutions over a range of pH values, and comparison with analogs bearing redox-inactive zinc(ii) allows for assignments of ligand-based redox events. Varying the geometric placement of redox non-innocent pyrazine donors on isostructural pentadentate ligand platforms results in marked effects on observed cobalt-catalyzed proton reduction activity. Electrocatalytic hydrogen evolution from weak acids in acetonitrile solution, under diffusion-limited conditions, reveals that the pyrazine donor of axial isomer 1-Co behaves as an unproductive electron sink, resulting in high overpotentials for proton reduction, whereas the equatorial pyrazine isomer complex 2-Co is significantly more active for hydrogen generation at lower voltages. Addition of a second equatorial pyrazine in complex 3-Co further minimizes overpotentials required for catalysis. The equatorial derivative 2-Co is also superior to its axial 1-Co congener for electrocatalytic and visible-light photocatalytic hydrogen generation in biologically relevant, neutral pH aqueous media. Density functional theory calculations (B3LYP-D2) indicate that the first reduction of catalyst isomers 1-Co, 2-Co, and 3-Co is largely metal-centered while the second reduction occurs at pyrazine. Taken together, the data establish that proper positioning of non-innocent pyrazine ligands on a single cobalt center is indeed critical for promoting efficient hydrogen catalysis in aqueous media, akin to optimally positioned redox-active cofactors in metalloenzymes. In a broader sense, these findings highlight the significance of electronic structure considerations in the design of effective electron-hole reservoirs for multielectron transformations

Photochemical Reactions of Fluorinated Pyridines at Half-Sandwich Rhodium Complexes: Competing Pathways of Reaction

Irradiation of CpRh(PMe3)(C2H4) (1; Cp = η5-C5H5) in the presence of pentafluoropyridine in hexane solution at low temperature yields an isolable η2-C,C-coordinated pentafluoropyridine complex, CpRh(PMe3)(η2-C,C-C5NF4) (2). The molecular structure of 2 was determined by single-crystal X-ray diffraction, showing coordination by C3–C4, unlike previous structures of pentafluoropyridine complexes that show N-coordination. Corresponding experiments with 2,3,5,6-tetrafluoropyridine yield the C–H oxidative addition product CpRh(PMe3)(C5NF4)H (3). In contrast, UV irradiation of 1 in hexane, in the presence of 4-substituted tetrafluoropyridines C5NF4X, where X = NMe2, OMe, results in elimination of C2H4 and HF to form the metallacycles CpRh(PMe3)(κ2-C,C-CH2N(CH3)C5NF3) (4) and CpRh(PMe3)(κ2-C,C-CH2OC5NF3) (5), respectively. The X-ray structure of 4 shows a planar RhCCNC-five-membered ring. Complexes 2–5 may also be formed by thermal reaction of CpRh(PMe3)(Ph)H with the respective pyridines at 50 °C