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...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...

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...]. 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,...

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...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...

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... 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...

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...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...

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...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...

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... 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...

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... 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 ...

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...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...

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... 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...

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...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...

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...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...

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...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...

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...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...

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...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 ...

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...] 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...

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...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 ...

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... 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 [...

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...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...

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...]. 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...

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...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...

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... 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...

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...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...

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...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...

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... 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 ...

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...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...

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...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...

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...(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 ...

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...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...

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... 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...

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...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 ...

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...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...

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...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...

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...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 ...

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...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...

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... 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...

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...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....

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... [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...

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... 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) ...

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...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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...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 ...

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...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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... [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...

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... 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 ...

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...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...

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...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...

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...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...

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...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]. [...

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...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...

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...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 ...

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...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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...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 ...

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...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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...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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...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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...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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... 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...

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...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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...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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...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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...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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...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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...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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...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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...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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...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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...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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...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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...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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...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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... 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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...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: ...