### Citations

329 | Meshless methods: An overview and recent developments - Belytschko, Krongauz, et al. - 1996 |

323 |
Element-free Galerkin methods
- Belytschko, Lu, et al.
- 1994
(Show Context)
Citation Context ...s, nold is the number of old nodes or Gauss points in the support of xnew and ψj (xnew) are MLS shape functions, the detailed derivation of which can be found in many papers on meshless methods, e.g. =-=[8]-=-. 3 NUMERICAL EXAMPLES Numerical examples are now given to demonstrate the correct implementation and performance of the full meshless modelling approach whose components have been described above. Al... |

96 | Error estimation procedures in the finite element method and applications. - Ladeveze, Leguillon - 1983 |

55 |
Meshfree Particle Methods
- Li, WK
- 2004
(Show Context)
Citation Context ...]. In these approaches a singular weight function is used. However careful choices are necessary for the arrangements of the domains of influence of nodes otherwise the methods acncan become unstable =-=[55]-=-. It is clearly recognized in FE modelling that adaptive procedures based on robust error estimates are necessary to remove user-bias and to automate analyses, and mature procedures are now available,... |

54 |
Nodal integration of the element-free Galerkin method
- Beissel, Belytschko
- 1996
(Show Context)
Citation Context ...delineate the quadrature cells. As will be shown below, this allows for a straightforward refinement algorithm to be used. An alternative to quadrature is nodal integration which is first proposed in =-=[83]-=- for linear elasto-static problems, in order to avoid the cell based integration in EFGM. Instability due to underintegration is treated by adding the square of the residual in the equilibrium equatio... |

51 | A new implementation of the element free Galerkin method - Lu, Belytschko, et al. - 1994 |

50 |
Information theory and statistical mechanics.
- ET
- 1957
(Show Context)
Citation Context ... formulations in which imposition of essential boundary conditions is as straightforward as in the FEM. Sukumar [41] describes the link between concepts of informational entropy [42], maximum entropy =-=[43, 44]-=- and global approximations to a function. Non-local and non-interpolating characteristics of these shape functions are highlighted in [45] where the (more useful) local max-ent formulation is introduc... |

44 | Local maximum-entropy approximation schemes: A seamless bridge between finite elements and meshfree methods
- Arroyo, Ortiz
(Show Context)
Citation Context ...n concepts of informational entropy [42], maximum entropy [43, 44] and global approximations to a function. Non-local and non-interpolating characteristics of these shape functions are highlighted in =-=[45]-=- where the (more useful) local max-ent formulation is introduced and incorporated into meshless modelling of linear and non-linear elasticity. Compact support shape functions are derived using Gaussia... |

41 |
Crack propagation by element-free Galerkin methods, Engng Fract Mech
- Belytschko, Lu, et al.
- 1995
(Show Context)
Citation Context ...nd essential boundary conditions are enforced using Lagrange multipliers. The EFGM has been used to model a variety of physics, e.g. 2D linear elasticity [8–10], static and dynamic fracture mechanics =-=[11, 12]-=-, plate and shell analysis [13–15], vibration [7, 16, 17], electromagnetics [18], heat transfer [8, 19–21], metal forming [22, 23], biomechanics [24, 25] and geomechanics [26]. While the EFGM is super... |

40 |
Imposing essential boundary conditions in mesh-free methods. submitted to Elsevier Science,
- Fernandez-Mendez, Huerta
- 2003
(Show Context)
Citation Context ... of essential boundary conditions. The use of Lagrange multipliers for the latter also increases the dimension of the final system of equations and the stiffness matrix is no longer positive definite =-=[28]-=-. Other examples of meshfree approaches applied to cracking problems can be found in [29–31]. A related meshfree method is the finite point method which also uses MLS techniques but adds stabilization... |

35 |
An Introduction to Programming the Meshless Element Free Galerking Method
- Dolbow, Belytschko
- 1998
(Show Context)
Citation Context ... is a scaling parameter. ci is the maximum distance to the neighbouring nodes. Two-dimensional shape functions require 2D weight functions, and here we use the tensor product of one-dimensional forms =-=[7, 10]-=- with a square nodal domain of influence. The size of the influence domain is dmk, where k = x, y in the x and y directions respectively. The weight function of a node i at point x is then written as ... |

34 | Construction of polygonal interpolants: A maximum entropy approach
- Sukumar
(Show Context)
Citation Context ...hless methods have been those based on maximum entropy (max-ent) concepts. These lead to formulations in which imposition of essential boundary conditions is as straightforward as in the FEM. Sukumar =-=[41]-=- describes the link between concepts of informational entropy [42], maximum entropy [43, 44] and global approximations to a function. Non-local and non-interpolating characteristics of these shape fun... |

30 |
Computational Methods for Plasticity: Theory and Applications
- Neto, Peric, et al.
- 2008
(Show Context)
Citation Context ... 2D to demonstrate the robustness of the basic code, before moving on to examples which demonstrate the full features of the modelling. 3.1 Strip footing collapse The same problem is also analyzed in =-=[100]-=- to determine a limiting load with finite element analysis. The geometry of the problem, with boundary conditions and material properties are given in Figure 4. The analytical solution for the same pr... |

27 |
A simple error estimator and adaptive procedure for practical engineering analysis
- OC, JZ
- 1987
(Show Context)
Citation Context ...r EFGM proposed in [70] and as described above, to nonlinear problems modelled with the EFGM including finite deformation and elasto-plasticity. The error in energy norm is used, as first proposed in =-=[98]-=- as a practical error estimator for engineering analysis for linear elasto-static problems. This also follows the approach in [91] also mentioned above. For the nonlinear case the exact incremental er... |

25 |
Tabbara M. Dynamic fracture using element-free Galerkin methods
- Belytschko
- 1996
(Show Context)
Citation Context ...nd essential boundary conditions are enforced using Lagrange multipliers. The EFGM has been used to model a variety of physics, e.g. 2D linear elasticity [8–10], static and dynamic fracture mechanics =-=[11, 12]-=-, plate and shell analysis [13–15], vibration [7, 16, 17], electromagnetics [18], heat transfer [8, 19–21], metal forming [22, 23], biomechanics [24, 25] and geomechanics [26]. While the EFGM is super... |

25 | Analysis of Thin Shells by the Element-Free Galerkin Method - Krysl, Belytschko - 1996 |

23 |
Stabilized conforming nodal integration for Galerkin meshfree methods.
- JS, CT, et al.
- 2000
(Show Context)
Citation Context ...bility due to underintegration is treated by adding the square of the residual in the equilibrium equation. Stabilized confirming nodal integration for meshfree Galerkin methods is first described in =-=[84]-=-, using strain smoothing as a stabilization as compared to the residual in [83]. Strain smoothing avoids the need to calculate shape function derivatives at node and has been extended to nonlinear pro... |

22 | Analysis of thin plates by the Element-Free Galerkin method - Krysl, Belytschko - 1996 |

20 | A model study of the quality of a posteriori error estimators for linear elliptic problems. Error estimation in the interior of patchwise uniform grids of triangles', Comput. Methods - Babuika, Strouboulis, et al. - 1994 |

19 | Overview and construction of meshfree basis functions: From moving least squares to entropy approximants
- Sukumar, Wright
(Show Context)
Citation Context ...nd incorporated into meshless modelling of linear and non-linear elasticity. Compact support shape functions are derived using Gaussian weight functions (or priors) in [45], work which is extended in =-=[46]-=- for any weight function (or generalized prior). First-order consistent max-ent shape functions [46] are then extended to second order in [47] and max-ent is used in [48] for the automatic calculation... |

16 | An edge-based smoothed finite element method for primal-dual shakedown analysis of structures, Int
- Tran, Liu, et al.
(Show Context)
Citation Context ...grange multipliers. The EFGM has been used to model a variety of physics, e.g. 2D linear elasticity [8–10], static and dynamic fracture mechanics [11, 12], plate and shell analysis [13–15], vibration =-=[7, 16, 17]-=-, electromagnetics [18], heat transfer [8, 19–21], metal forming [22, 23], biomechanics [24, 25] and geomechanics [26]. While the EFGM is superior to the FEM in terms of accuracy and convergence, and ... |

16 | Adaptivity for structured meshfree particle methods
- Rabczuk, Belytschko
- 2005
(Show Context)
Citation Context ...] where comparison is instead made with a field calculated using a first order Taylor series expansion with a four quadrant criterion, and the two approaches are compared by the same authors in [74]. =-=[75]-=- describes an error estimate for the EFGM based on a Taylor series with a higher order derivative and a structured grid is used instead of a cloud of points, which makes the implementation very straig... |

15 | Element-free Galerkin method for electromagnetic field computations
- Cingoski, Miyamoto, et al.
- 1998
(Show Context)
Citation Context ...as been used to model a variety of physics, e.g. 2D linear elasticity [8–10], static and dynamic fracture mechanics [11, 12], plate and shell analysis [13–15], vibration [7, 16, 17], electromagnetics =-=[18]-=-, heat transfer [8, 19–21], metal forming [22, 23], biomechanics [24, 25] and geomechanics [26]. While the EFGM is superior to the FEM in terms of accuracy and convergence, and there are no issues of ... |

15 |
New boundary condition treatments for meshless computation of contact problems.
- JS, HP
- 2000
(Show Context)
Citation Context ... use of so-called interpolatory MLS, which satisfies the Kronecker delta property and is used for imposition of the essential boundary condition in the EFGM in [53] and also in other meshless methods =-=[54]-=-. In these approaches a singular weight function is used. However careful choices are necessary for the arrangements of the domains of influence of nodes otherwise the methods acncan become unstable [... |

14 |
An improved element free Galerkine formulation.
- Kaljevic, Saigal
- 1997
(Show Context)
Citation Context ...tion which is not considered here is the use of so-called interpolatory MLS, which satisfies the Kronecker delta property and is used for imposition of the essential boundary condition in the EFGM in =-=[53]-=- and also in other meshless methods [54]. In these approaches a singular weight function is used. However careful choices are necessary for the arrangements of the domains of influence of nodes otherw... |

13 | A three dimensional large deformation meshfree method for arbitrary evolving cracks - Rabczuk, Belytschko - 2007 |

12 | Error estimation and mesh optimization for classical finite elements - Ladevèze, Pelle, et al. - 1991 |

11 | Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery,” - Xiao, Karihaloo - 2006 |

10 | A geometrically non-linear three dimensional cohesive crack method for reinforced concrete structures. Eng Fract Mech 2008; 75 - Rabczuk, Zi, et al. |

10 |
A Finite Point Method For Elasticity Problems
- Onate, Perazzo, et al.
- 2001
(Show Context)
Citation Context ...]. A related meshfree method is the finite point method which also uses MLS techniques but adds stabilization and was developed first for fluid mechanics [32] but later was applied to solid mechanics =-=[33]-=-. Smoothed finite element methods (SFEMs) are often cited as competitors to meshless methods such as the EFGM. In SFEMs a modified or constructed strain field is used, which involves only the shape fu... |

10 |
Recovery procedures in error estimation and adaptivity. Part I: Adaptivity in linear problems,” Computer Methods in Applied Mechanics and Engineering
- Zienkiewicz, Boroomand, et al.
- 1999
(Show Context)
Citation Context ...tatic problems. This also follows the approach in [91] also mentioned above. For the nonlinear case the exact incremental error in energy norm for solution step n for the problem domain Ω is equal to =-=[99]-=- ‖e‖ = ∫ Ω ∣∣∣(τn (x)− τhn (x))T (∆εn (x)−∆εhn (x))∣∣∣ dΩ 12 , (36) where τn (x) and τhn (x) are the exact and approximate Kirchhoff stresses respectively at point x for solution step n, while ∆ε... |

9 |
An Introduction to Meshfree Methods and their Programming
- GR, YT
- 2005
(Show Context)
Citation Context ... accuracy of stresses and complexity of adaptive procedures become important. It is also not ideal for certain classes of problems, e.g. crack growth, moving boundaries and large deformation problems =-=[1]-=- and various refinements are necessary, XFEM for fracture for instance. Indeed in this area major advances have been made [2–4]. With mesh-based numerical methods, one cannot avoid the issues of eleme... |

9 | New advances on a posteriori error on constitutive relation in finite element analysis - Ladevèze, Rougeot - 1997 |

8 |
Nguyen-Dang H. Smooth finite element methods: convergence, accuracy and properties
- Nguyen-Xuan, Bordas
(Show Context)
Citation Context ...EM) in [35], n-sided polygonal 2 (nSFEM) [36], node based (NS-FEM) [37], edge based (ES-FEM) for 2D [7], and face based (FS-FEM) for 3D [38]. Its properties, convergence and accuracy are discussed in =-=[39]-=- and it has been used in adaptive analysis [40]. Returning to the EFGM, a recent and welcome advance in shape function formulations for meshless methods have been those based on maximum entropy (max-e... |

8 |
On the control of the load increments for a proper description of multiple delamination in a domain decomposition framework. International Journal for Numerical Methods in Engineering 2010
- Allix, Kerfriden, et al.
(Show Context)
Citation Context ...recovery-based method, however it should be noted that there is considerable work on more advanced error estimators [60] and of discretization errors due to choice of (pseudo- in this case) time step =-=[97]-=-. Here we extend the use of the recovery type error estimation for EFGM proposed in [70] and as described above, to nonlinear problems modelled with the EFGM including finite deformation and elasto-pl... |

7 | Constitutive relation error estimators for (visco)plastic finite element analysis with softening - Ladevèze, Moës, et al. |

7 |
A new a posteriori error estimation for nonlinear time-dependent finite element analysis
- Ladevèze, Moës
(Show Context)
Citation Context ... with large deformation. here we are taking a relatively simple approach by using a recovery-based method, however it should be noted that there is considerable work on more advanced error estimators =-=[60]-=- and of discretization errors due to choice of (pseudo- in this case) time step [97]. Here we extend the use of the recovery type error estimation for EFGM proposed in [70] and as described above, to ... |

7 |
Rheinboldt WC. A-posteriori error estimates for finite-element method.
- Babuska
- 1978
(Show Context)
Citation Context ...isfy the constitutive relation, from which the error measure is drawn. These are also extended to nonlinear problems in [59–62]. Residual based error estimation methods for FEs were first proposed in =-=[63]-=- which are further divided into explicit and implicit depending on the use of the residual. Details of explicit methods can be found in many references, e.g. [64, 65] and the implicit in [66–68]. An i... |

7 |
CT. Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods
- JS, SP, et al.
(Show Context)
Citation Context ...train smoothing as a stabilization as compared to the residual in [83]. Strain smoothing avoids the need to calculate shape function derivatives at node and has been extended to nonlinear problems in =-=[85]-=-. 2.2 Finite deformation and elasto-plasticity There are relatively few changes required to model finite deformation and elasto-plasticity with a meshless approach as opposed to one using finite eleme... |

7 |
Comparison of Two Algorithms for the Computation of Fourth-Order Isotropic Tensor Functions,”
- Miehe
- 1998
(Show Context)
Citation Context ...(σ)il δjk. (29) Here Dalg is the small strain consistent or algorithmic tangent, L is the derivative of the symmetric secondorder tensor with respect to its component, the detail of which is given in =-=[88]-=-, while S is known as the non-symmetric stress corrector and δij is the Kronecker delta. 2.3 Error estimation and adaptivity in the EFGM Adaptive refinement algorithms when applied to finite elements ... |

7 |
On adaptive strategies for large deformations of elasto-plastic solids at finite strains: computational issues and industrial applications
- Peric, Vaz, et al.
- 1999
(Show Context)
Citation Context ...enkiewicz-Zhu (Z2) error estimator (based on the superconvergent patch recovery method (SPRM)), has been widely used for materially and geometrically nonlinear FE problems. An example can be found in =-=[91]-=- for industrial metal forming problems with elasto-plasticity and finite strains, with estimators based on energy norms, plastic dissipation and rate of plastic work. In [92] the SPRM is used in large... |

6 | Belytschko T. A meshfree thin shell method for non-linear dynamic fracture. International journal for numerical methods in engineering 2007 - Rabczuk, Areias |

6 |
second order, non-negative meshfree approximants selected by maximum entropy
- CJ, Arroyo, et al.
(Show Context)
Citation Context ... functions (or priors) in [45], work which is extended in [46] for any weight function (or generalized prior). First-order consistent max-ent shape functions [46] are then extended to second order in =-=[47]-=- and max-ent is used in [48] for the automatic calculation of the nodal domain of influence within a meshless method. Other recent examples of the use of max-ent in meshless methods can be found in [4... |

6 |
On the optimum support size in meshfree methods: A variational adaptivity approach with maximum-entropy approximants
- Rosolen, Millán, et al.
(Show Context)
Citation Context ...5], work which is extended in [46] for any weight function (or generalized prior). First-order consistent max-ent shape functions [46] are then extended to second order in [47] and max-ent is used in =-=[48]-=- for the automatic calculation of the nodal domain of influence within a meshless method. Other recent examples of the use of max-ent in meshless methods can be found in [49–52]. Another option which ... |

5 |
Mathematical foundation of information theory
- AI
- 1957
(Show Context)
Citation Context ...oncepts. These lead to formulations in which imposition of essential boundary conditions is as straightforward as in the FEM. Sukumar [41] describes the link between concepts of informational entropy =-=[42]-=-, maximum entropy [43, 44] and global approximations to a function. Non-local and non-interpolating characteristics of these shape functions are highlighted in [45] where the (more useful) local max-e... |

5 |
An adaptive procedure based on background cells for meshfree methods,
- Liu, Tu
- 2002
(Show Context)
Citation Context ...ud of points, which makes the implementation very straightforward. Error estimation based on the gradient of strain energy density is proposed in [76] and an adaptive analysis for EFGM is proposed in =-=[77]-=- based on background cells, error being estimated based on two different integration orders and a refinement algorithm based on local Delaunay triangulation is also proposed. In [78], the approach fro... |

5 |
H-adaptive FE analysis of elasto-plastic non-homogeneous soil with large deformation
- Hu, MF
- 1998
(Show Context)
Citation Context ...viatoric stress is used to calculate the error in energy norm due to the incompressible nature of the materials. The work in [94] is further extended in [95] for 3D complex forging simulations and in =-=[96]-=- the SPRM with error based on an L2 norm in strain is used in analyses involving non-homogeneous soil with large deformation. here we are taking a relatively simple approach by using a recovery-based ... |

4 | Classification and overview of meshfree methods. - TP, HG - 2004 |

4 |
Zienkiewicz OC, Taylor RL. A finite point method in computational mechanics – applications to convective transport and fluid flow. Int J Numer Methods Eng
- Oñate, Idelsohn
- 1996
(Show Context)
Citation Context ...ed to cracking problems can be found in [29–31]. A related meshfree method is the finite point method which also uses MLS techniques but adds stabilization and was developed first for fluid mechanics =-=[32]-=- but later was applied to solid mechanics [33]. Smoothed finite element methods (SFEMs) are often cited as competitors to meshless methods such as the EFGM. In SFEMs a modified or constructed strain f... |

4 |
Theoretical aspects of the smoothed finite element method (SFEM).
- GR, Nguyen-Thoi, et al.
- 2007
(Show Context)
Citation Context ... is also changed to get stability and convergence. The SFEM was first proposed in [34] for linear elastic problems. Based on its strain smoothing formulation it is divided into cell based (CS-FEM) in =-=[35]-=-, n-sided polygonal 2 (nSFEM) [36], node based (NS-FEM) [37], edge based (ES-FEM) for 2D [7], and face based (FS-FEM) for 3D [38]. Its properties, convergence and accuracy are discussed in [39] and it... |

4 | A higher order method based on local maximum entropy approximation - González, Cueto, et al. |

4 | Strouboulis T., Upadhyay C.S., Gangaraj SK., Copps K., Validation of a posteriori error estimators by numerical approach - Babuska - 1994 |

4 |
Hansbo P, Runesson K. Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity
- Larsson
(Show Context)
Citation Context ...explicit methods can be found in many references, e.g. [64, 65] and the implicit in [66–68]. An interesting avenue of research in error estimation can also be found in the works of Larsson & Runesson =-=[69]-=-. Error estimation and adaptivity have yet to become widely used in meshless methods partly because of the current lack of a firm mathematical basis for error estimation as has been developed for FEs.... |

4 | An adaptive approach with the element-free-Galerkin method - Hussler-Combe, Korn - 1998 |

3 | An extended arbitrary lagrangian–eulerian finite element method for large deformation of solid mechanics. Finite Elements in Analysis and Design 2008 - Khoei, Anahid, et al. |

3 | Towards meshless methods for surgical simulation
- Horton, Wittek, et al.
- 2006
(Show Context)
Citation Context ... [8–10], static and dynamic fracture mechanics [11, 12], plate and shell analysis [13–15], vibration [7, 16, 17], electromagnetics [18], heat transfer [8, 19–21], metal forming [22, 23], biomechanics =-=[24, 25]-=- and geomechanics [26]. While the EFGM is superior to the FEM in terms of accuracy and convergence, and there are no issues of volumetric locking [27], MLS shape functions are computationally more exp... |

3 | Application of particle methods to static fracture of reinforced concrete structures - Rabczuk, Belytschko |

3 |
A smoothed finite element method for mechanics problems.
- GR, KY, et al.
- 2007
(Show Context)
Citation Context ... SFEMs a modified or constructed strain field is used, which involves only the shape functions. The Galerkin weak form is also changed to get stability and convergence. The SFEM was first proposed in =-=[34]-=- for linear elastic problems. Based on its strain smoothing formulation it is divided into cell based (CS-FEM) in [35], n-sided polygonal 2 (nSFEM) [36], node based (NS-FEM) [37], edge based (ES-FEM) ... |

3 | Thin shell analysis from scattered points with maximum-entropy approximants - Millán, Rosolen, et al. |

3 | Constitutive relation errors for f.e. analysis considering (visco-) plasticity and damage - Ladevèze |

3 | Upadhyay CS. A model study of the quality of a posteriori error estimators for finite element solutions of linear elliptic problems, with particular reference to the behavior near the boundary. International Journal for Numerical Methods in Engineering - Babuška, Strouboulis - 1997 |

3 |
Modeling and discretization errors in hyperelasto-(visco-)plasticity with a view to hierarchical modeling
- Larsson, Runesson
(Show Context)
Citation Context ...Ω ∣∣∣σp (x)T D−1σp (x)∣∣∣ dΩ 12 . (35) We should here distinguish between the above discretisation-based error estimator, and the model error approach exemplified by the work of Larsson & Runesson =-=[89, 90]-=-. 2.3.2 Error estimation in nonlinear problems For finite element modelling the Zienkiewicz-Zhu (Z2) error estimator (based on the superconvergent patch recovery method (SPRM)), has been widely used f... |

2 |
An element free Galerkin method for the free vibration analysis of composite laminates of complicated shape. Composite Structures 2003; 59(2):279
- XL, GR, et al.
(Show Context)
Citation Context ...grange multipliers. The EFGM has been used to model a variety of physics, e.g. 2D linear elasticity [8–10], static and dynamic fracture mechanics [11, 12], plate and shell analysis [13–15], vibration =-=[7, 16, 17]-=-, electromagnetics [18], heat transfer [8, 19–21], metal forming [22, 23], biomechanics [24, 25] and geomechanics [26]. While the EFGM is superior to the FEM in terms of accuracy and convergence, and ... |

2 | Meshless element free Galerkin method for unsteady nonlinear heat transfer problems” - Singh, Singh, et al. - 2007 |

2 |
Application of the element free Galerkin method to the simulation of plane strain rolling. Eur J Mech A—Solid 2004;23:77
- Xiong, JMC, et al.
(Show Context)
Citation Context ...g. 2D linear elasticity [8–10], static and dynamic fracture mechanics [11, 12], plate and shell analysis [13–15], vibration [7, 16, 17], electromagnetics [18], heat transfer [8, 19–21], metal forming =-=[22, 23]-=-, biomechanics [24, 25] and geomechanics [26]. While the EFGM is superior to the FEM in terms of accuracy and convergence, and there are no issues of volumetric locking [27], MLS shape functions are c... |

2 |
Meshfree implementation of individualized active cardiac dynamics. Computerized Medical Imaging and Graphics 2010; 34(1):91–103
- KCL, Wang, et al.
(Show Context)
Citation Context ... [8–10], static and dynamic fracture mechanics [11, 12], plate and shell analysis [13–15], vibration [7, 16, 17], electromagnetics [18], heat transfer [8, 19–21], metal forming [22, 23], biomechanics =-=[24, 25]-=- and geomechanics [26]. While the EFGM is superior to the FEM in terms of accuracy and convergence, and there are no issues of volumetric locking [27], MLS shape functions are computationally more exp... |

2 |
de Borst R, Heeres O. Conditions for locking-free elasto-plastic analyses in the Element-Free Galerkin method
- Askes
- 1999
(Show Context)
Citation Context ... 19–21], metal forming [22, 23], biomechanics [24, 25] and geomechanics [26]. While the EFGM is superior to the FEM in terms of accuracy and convergence, and there are no issues of volumetric locking =-=[27]-=-, MLS shape functions are computationally more expensive and complicate the imposition of essential boundary conditions. The use of Lagrange multipliers for the latter also increases the dimension of ... |

2 | Belytschko T. Simulations of instability in dynamic fracture by the cracking particles method. Engineering Fracture Mechanics 2009; 76(6):730–741 - Rabczuk, Song |

2 |
and forced vibration analysis using the smoothed finite element method (SFEM
- Dai, Free
(Show Context)
Citation Context ...and convergence. The SFEM was first proposed in [34] for linear elastic problems. Based on its strain smoothing formulation it is divided into cell based (CS-FEM) in [35], n-sided polygonal 2 (nSFEM) =-=[36]-=-, node based (NS-FEM) [37], edge based (ES-FEM) for 2D [7], and face based (FS-FEM) for 3D [38]. Its properties, convergence and accuracy are discussed in [39] and it has been used in adaptive analysi... |

2 |
Nguyen-Xuan H, Nguyen-Tran C. Adaptive analysis using the node-based smoothed finite element method
- Nguyen-Thoi, GR
(Show Context)
Citation Context ...node based (NS-FEM) [37], edge based (ES-FEM) for 2D [7], and face based (FS-FEM) for 3D [38]. Its properties, convergence and accuracy are discussed in [39] and it has been used in adaptive analysis =-=[40]-=-. Returning to the EFGM, a recent and welcome advance in shape function formulations for meshless methods have been those based on maximum entropy (max-ent) concepts. These lead to formulations in whi... |

2 | A posteriori constitutive relation error estimators for nonlinear finite element analysis and adaptive control - Ladevèze, Moës - 1998 |

2 |
An error indicator for the element free Galerkin method
- Gavete, Falcn, et al.
(Show Context)
Citation Context ...istribution but with reduced domains of influence. In [71] error estimation and adaptivity 3 is performed using stress gradients, thus requiring second derivatives of the meshless shape functions. In =-=[72]-=- a computationally efficient method for error estimation based on tessellation is proposed for the EFGM, i.e. error at a Gauss point is calculated as the difference between the EFGM stress or strain a... |

2 |
A numerical comparison of two different approximations of the error in a meshless method
- Gavete, JL, et al.
(Show Context)
Citation Context ...essellation is proposed for the EFGM, i.e. error at a Gauss point is calculated as the difference between the EFGM stress or strain and that calculated at the nearest node. A slight change is made in =-=[73]-=- where comparison is instead made with a field calculated using a first order Taylor series expansion with a four quadrant criterion, and the two approaches are compared by the same authors in [74]. [... |

2 |
A gradient-based adaptation procedure and its implementation in the element-free Galerkin method
- Luo, Häussler-Combe
(Show Context)
Citation Context ...ivative and a structured grid is used instead of a cloud of points, which makes the implementation very straightforward. Error estimation based on the gradient of strain energy density is proposed in =-=[76]-=- and an adaptive analysis for EFGM is proposed in [77] based on background cells, error being estimated based on two different integration orders and a refinement algorithm based on local Delaunay tri... |

2 |
On error estimation and adaptive refinement for element free Galerkin method: Part II: adaptive refinement
- CK, CE
(Show Context)
Citation Context ...is also proposed. In [78], the approach from [70] is used for error estimation and adaptive analysis in crack propagation problems. A Zienkiewicz-Zhu type recovery type error estimator is proposed in =-=[79]-=-; two methods are used for stress recovery including that due to Chung & Belytschko [70] with the discrete MLS over a stationary least square fitting and it is found that the approach of [70] is more ... |

2 |
H-adaptive mesh refinement for shear band localization in elasto-plasticity cosserat continuum
- AR, AR, et al.
(Show Context)
Citation Context ...4 Footing loaded on a slope The fourth example is another footing, this time on a block of elasto-plastic material resembling a natural slope. This problem also appears in a number of references e.g. =-=[99, 101, 102, 104, 105]-=- and the modelling of problems which are dominated by shear bands are covered using meshless methods in [106–108]. The 20 (a) (b) (c) (d) Figure 13: Step by step discretization (a) 1st (b) 2nd (c) 3rd... |

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Element-free Galerkin method for free vibration of rectangular plates with interior elastic point supports and elastically restrained edges
- Wang, Wang, et al.
- 2010
(Show Context)
Citation Context ...grange multipliers. The EFGM has been used to model a variety of physics, e.g. 2D linear elasticity [8–10], static and dynamic fracture mechanics [11, 12], plate and shell analysis [13–15], vibration =-=[7, 16, 17]-=-, electromagnetics [18], heat transfer [8, 19–21], metal forming [22, 23], biomechanics [24, 25] and geomechanics [26]. While the EFGM is superior to the FEM in terms of accuracy and convergence, and ... |

1 | Parallel EFG algorithm for heat transfer problems - PK |

1 | Heat transfer analysis of composite slabs using meshless element free Galerkin method - Singh |

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Guoqun Z, Ping L. A nonlinear numerical analysis for metal-forming process using the rigid-(visco)plastic element-free Galerkin method
- Yanjin, Xin
- 2009
(Show Context)
Citation Context ...g. 2D linear elasticity [8–10], static and dynamic fracture mechanics [11, 12], plate and shell analysis [13–15], vibration [7, 16, 17], electromagnetics [18], heat transfer [8, 19–21], metal forming =-=[22, 23]-=-, biomechanics [24, 25] and geomechanics [26]. While the EFGM is superior to the FEM in terms of accuracy and convergence, and there are no issues of volumetric locking [27], MLS shape functions are c... |

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Large deformation analysis of pile foundation using meshless method
- You, Liang, et al.
- 2008
(Show Context)
Citation Context ...c fracture mechanics [11, 12], plate and shell analysis [13–15], vibration [7, 16, 17], electromagnetics [18], heat transfer [8, 19–21], metal forming [22, 23], biomechanics [24, 25] and geomechanics =-=[26]-=-. While the EFGM is superior to the FEM in terms of accuracy and convergence, and there are no issues of volumetric locking [27], MLS shape functions are computationally more expensive and complicate ... |

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Rabczuk T, Nguyen-Xuan H. A node-based smoothed finite element method (NS-FEM) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes
- Nguyen-Thoi, Vu-Do
(Show Context)
Citation Context ...was first proposed in [34] for linear elastic problems. Based on its strain smoothing formulation it is divided into cell based (CS-FEM) in [35], n-sided polygonal 2 (nSFEM) [36], node based (NS-FEM) =-=[37]-=-, edge based (ES-FEM) for 2D [7], and face based (FS-FEM) for 3D [38]. Its properties, convergence and accuracy are discussed in [39] and it has been used in adaptive analysis [40]. Returning to the E... |

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Zhang GY. A face-based smoothed finite element method (FSFEM) for 3d linear and geometrically non-linear solid mechanics problems using 4-node tetrahedral elements. International Journal for Numerical Methods in Engineering 2009
- Nguyen-Thoi, GR, et al.
(Show Context)
Citation Context ...strain smoothing formulation it is divided into cell based (CS-FEM) in [35], n-sided polygonal 2 (nSFEM) [36], node based (NS-FEM) [37], edge based (ES-FEM) for 2D [7], and face based (FS-FEM) for 3D =-=[38]-=-. Its properties, convergence and accuracy are discussed in [39] and it has been used in adaptive analysis [40]. Returning to the EFGM, a recent and welcome advance in shape function formulations for ... |

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Information theory and statistical mechanics-II
- ET
- 1957
(Show Context)
Citation Context ... formulations in which imposition of essential boundary conditions is as straightforward as in the FEM. Sukumar [41] describes the link between concepts of informational entropy [42], maximum entropy =-=[43, 44]-=- and global approximations to a function. Non-local and non-interpolating characteristics of these shape functions are highlighted in [45] where the (more useful) local max-ent formulation is introduc... |

1 | Puso M, Sukumar N. Maximum-entropy meshfree method for incompressible media problems - Ortiz |

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De S R Gago JP, Zienkiewicz OC, Babuška I. A posteriori error analysis and adaptive processes in the finite element method: Part I-error analysis. International Journal for Numerical Methods in Engineering
- DW
- 1983
(Show Context)
Citation Context ...ethods for FEs were first proposed in [63] which are further divided into explicit and implicit depending on the use of the residual. Details of explicit methods can be found in many references, e.g. =-=[64, 65]-=- and the implicit in [66–68]. An interesting avenue of research in error estimation can also be found in the works of Larsson & Runesson [69]. Error estimation and adaptivity have yet to become widely... |

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A Posteriori Error Estimation in Finite Element Analysis
- JTO
- 2000
(Show Context)
Citation Context ...ethods for FEs were first proposed in [63] which are further divided into explicit and implicit depending on the use of the residual. Details of explicit methods can be found in many references, e.g. =-=[64, 65]-=- and the implicit in [66–68]. An interesting avenue of research in error estimation can also be found in the works of Larsson & Runesson [69]. Error estimation and adaptivity have yet to become widely... |

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Belytschko T. An error estimate
- HJ
- 1998
(Show Context)
Citation Context ...of the current lack of a firm mathematical basis for error estimation as has been developed for FEs. However there is evidence of a strong interest as indicated in the following. Chung and Belytschko =-=[70]-=- describe possibly the first error estimator for the EFGM using the difference between raw EFGM results and projected stresses from the same nodal distribution but with reduced domains of influence. I... |

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A procedure for approximation of the error
- Gavete, JL, et al.
(Show Context)
Citation Context ...in [73] where comparison is instead made with a field calculated using a first order Taylor series expansion with a four quadrant criterion, and the two approaches are compared by the same authors in =-=[74]-=-. [75] describes an error estimate for the EFGM based on a Taylor series with a higher order derivative and a structured grid is used instead of a cloud of points, which makes the implementation very ... |

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Adaptive crack propagation analysis with the element-free Galerkin method
- GH, HJ, et al.
(Show Context)
Citation Context ...GM is proposed in [77] based on background cells, error being estimated based on two different integration orders and a refinement algorithm based on local Delaunay triangulation is also proposed. In =-=[78]-=-, the approach from [70] is used for error estimation and adaptive analysis in crack propagation problems. A Zienkiewicz-Zhu type recovery type error estimator is proposed in [79]; two methods are use... |

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Adaptive element-free Galerkin method applied to the limit analysis of plates
- CV, Gilbert
(Show Context)
Citation Context ...ki‖ dmk and dmk = dmax · cki and cki is calculated as ci = maxj∈Sj ‖x− xi‖ . (19) Sj is a set of surrounding nodes, found here using the information from the Voronoi diagram of the nodal distribution =-=[81, 82]-=- Sj = [j : V (j) ∩ V (i) 6= ∅] (20) where V (i) is the Voronoi cell of node i. For illustrative purposes a sample 2D grid of nodes is shown in Figure 1(a), its Voronoi diagram is shown in Figure 1(b) ... |

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Novel numerical procedures for limit analysis of structures - meshfree methods and mathematical programming
- CV
- 2010
(Show Context)
Citation Context ...ki‖ dmk and dmk = dmax · cki and cki is calculated as ci = maxj∈Sj ‖x− xi‖ . (19) Sj is a set of surrounding nodes, found here using the information from the Voronoi diagram of the nodal distribution =-=[81, 82]-=- Sj = [j : V (j) ∩ V (i) 6= ∅] (20) where V (i) is the Voronoi cell of node i. For illustrative purposes a sample 2D grid of nodes is shown in Figure 1(a), its Voronoi diagram is shown in Figure 1(b) ... |

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AM, Bazant ZP. Errors caused by non-work-conjugate stress and strain measures and necessary corrections in finite element programs
- Ji, Waas
(Show Context)
Citation Context ...X are the coordinates of a point in the current and reference configurations respectively. The work-conjugate stress and strain measures used in this paper are logarithmic strain and Kirchhoff stress =-=[86]-=-, which are given as [87] ε = 1 2 lnb, τ = Jσ, (22) where b is the left Cauchy-Green tensor and J is the determinant of the deformation gradient F. The updated deformation gradient Fn at the end of in... |

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Augarde CE. 70-line 3D finite deformation elastoplastic finite-element code
- WM, RS
(Show Context)
Citation Context ...a point in the current and reference configurations respectively. The work-conjugate stress and strain measures used in this paper are logarithmic strain and Kirchhoff stress [86], which are given as =-=[87]-=- ε = 1 2 lnb, τ = Jσ, (22) where b is the left Cauchy-Green tensor and J is the determinant of the deformation gradient F. The updated deformation gradient Fn at the end of increment n is written as F... |

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Runesson K, Larsson F. Calibration of a class of non-linear viscoelasticity models with adaptive error control
- Johansson
(Show Context)
Citation Context ...Ω ∣∣∣σp (x)T D−1σp (x)∣∣∣ dΩ 12 . (35) We should here distinguish between the above discretisation-based error estimator, and the model error approach exemplified by the work of Larsson & Runesson =-=[89, 90]-=-. 2.3.2 Error estimation in nonlinear problems For finite element modelling the Zienkiewicz-Zhu (Z2) error estimator (based on the superconvergent patch recovery method (SPRM)), has been widely used f... |

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A study of applying the superconvergent patch recovery (SPR) method to large deformation problem
- Hai, Hisashi
- 2000
(Show Context)
Citation Context ...n example can be found in [91] for industrial metal forming problems with elasto-plasticity and finite strains, with estimators based on energy norms, plastic dissipation and rate of plastic work. In =-=[92]-=- the SPRM is used in large deformation problems with hyperelasticity as a material model. An adaptive strategy with error estimation based on an L2 norm of strain is used in 3D analysis in [93] with l... |

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Adaptive mesh refinement and error estimate for 3-D seismic analysis of liquefiable soil considering large deformation. Journal of natural disaster science 2004
- Xiaowei, Tadanobu
(Show Context)
Citation Context ...rk. In [92] the SPRM is used in large deformation problems with hyperelasticity as a material model. An adaptive strategy with error estimation based on an L2 norm of strain is used in 3D analysis in =-=[93]-=- with large deformation 11 to model liquefaction phenomena. The Z2 error estimator is used in the case of viscoplasticity problems, with application to metal forming processes in [94]. In this case th... |

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Error estimators for viscoplastic materials: application to forming processes
- JC
- 1995
(Show Context)
Citation Context ...n 3D analysis in [93] with large deformation 11 to model liquefaction phenomena. The Z2 error estimator is used in the case of viscoplasticity problems, with application to metal forming processes in =-=[94]-=-. In this case the deviatoric stress is used to calculate the error in energy norm due to the incompressible nature of the materials. The work in [94] is further extended in [95] for 3D complex forgin... |

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Coupez T, Fourment L. Adaptive remeshing based on a posteriori error estimation for forging simulation
- Boussetta
(Show Context)
Citation Context ... forming processes in [94]. In this case the deviatoric stress is used to calculate the error in energy norm due to the incompressible nature of the materials. The work in [94] is further extended in =-=[95]-=- for 3D complex forging simulations and in [96] the SPRM with error based on an L2 norm in strain is used in analyses involving non-homogeneous soil with large deformation. here we are taking a relati... |

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Progressive adaptivity formulation in Galerkin meshfree method
- Lu
- 2001
(Show Context)
Citation Context ...he steps in the adaptive analysis curve occur at points of refinement and are not physically meaningful, although they provide a useful reminder of the adaptive process and follow the presentation in =-=[101]-=-. The effect of re-discretization at refinement is to change the net equilibrium reaction for the given displacement (as one would expect between different refinements in FEA for instance). Figure 8: ... |