Element-free Galerkin Formulation Research Articles

Element Free Galerkin (EFG) Formulation in Solving Kinematic Wave Equation for Hydrological Modelling

Meshfree methods such as Element Free Galerkin Method (EFG) are the cutting edge of numerical methods as it does not require user to perform meshing which often is time consuming and tedious. The method being "meshfree" for the user can theoretically generate more accurate models which follow more complex boundary conditions such as natural rivers etc. The time saving and potential of higher accuracy in modelling makes EFG formulation an interesting research topic. EFG formulation is used to solve kinematic wave equations which is detailed in this paper. Galerkin weighted residual method using EFG’s shape function parameters is used to discretize the partial differential equation (PDE) of kinematic wave equations. To discretize the equations, a forward difference scheme is used for temporal discretization and Picard direct substitution method is employed to solve the nonlinear system at each time step. The formulations are validated by making a comparison between well-established numerical techniques against actual recorded datasets. After which it is found that EFG method agrees with conventional established techniques, thus validating the formulations as useful. To further investigate the usefulness of this formulation, an optimal performance study was also conducted to determine the most optimal shape function parameters. The optimal shape function values improved the performance to where it was found that established methods such as FDM were surpassed and on par with FEM. This shows that EFG is a good alternative for hydrological modelling.

Solving heat conduction problems with phase-change under the heat source term approach and the element-free Galerkin formulation

The present work has been conducted in order to introduce an alternative solution for heat transfer problems with phase-change and moving boundaries, via the element-free Galerkin (EFG) method. In this novel approach, the phase-change non-linearity is addressed under the heat source term methodology. For this purpose, the global weak-form of the internal energy balance for such a non-linear transient heat transfer problem has been developed by including the latent heat as a source vector. In order to prove the reliability of this approach, the EFG formulation has been adapted to the solution of a 1-D alloy solidification benchmark problem. The suitability and potentialities of this novel approach have also been depicted in the solution of a variable domain (moving boundary) solidification problem, by considering the start-up stage of a round billet steel continuous casting process as an applied example. Results have revealed that this technique can be successfully used to deal appropriately with the marked non-linearity related to the fusion latent heat release in heat transfer problems with phase change, and its straightforward implementation leads to both accurate and stable numerical results.

Implementation of a standard stream-upwind stabilization scheme in the element-free Galerkin based solution of advection-dominated heat transfer problems during solidification in direct chill casting processes

The present work has been conducted in order to propose an alternative solution for the heat transfer problem involved in direct-chill casting (DCC) processes, employing the element-free Galerkin (EFG) method. The internal energy balance has been solved on the basis of this global weak formulation under an Eulerian description. The EFG formulation has been adapted to a three-dimensional steady thermal problem on an aluminum alloy slab. Additionally, the advection dominated problem involved in the casting direction during the liquid-solid phase change has been stabilized by extending adequately the implementation of the stream upwind Petrov–Galerkin (SUPG) scheme to the EFG formulation. A detailed explanation concerning the suitable numerical procedure to achieve such an extension, has also been provided. The feasibility and reliability of this novel approach has been depicted by comparison with both finite element method (FEM) and finite volume method (FVM) based solutions. The results have demonstrated that a proper extension of the SUPG scheme to the EFG method allows the achievement of stable numerical solutions, when predicting the three-dimensional temperature distribution during the phase-change heat transfer phenomena involved in slab DCC processes.

Meshfree Continuum Damage Model for Twill Composites

This paper presents continuum damage mechanics analysis of twill woven composites. Element Free Galerkin formulation is utilized and enriched with mathematical representation of twill composite in a way that includes details of the wrap/weft/matrix without the requirement of a detailed mesh of individual components. Continuum damage mechanics formulation is developed within the meshfree context and applied to the twill composite.

Modeling the viscoplastic flow behavior of a 20MnCr5 steel grade deformed under hot-working conditions, employing a meshless technique

The present work has been conducted in order to develop a novel approach to predict the inhomogeneous flow of a 20MnCr5 steel during an axisymmetric hot compression test, by using the element-free Galerkin (EFG) method under a differential constitutive description. The governing equations have been solved on the basis of the EFG global weak formulation. A detailed explanation concerning the characteristics inherent to the application of the EFG method to this problem, has also been provided. Furthermore, a return mapping algorithm for the solution of associative von Mises inelastic problems, has been formulated for the differential constitutive description employed in this communication, namely, a differential return mapping algorithm (DRMA). The feasibility and suitability of the EFG method for solving the axial compression problem has been shown by comparing its results with a FEM based solution employing a simple constitutive description. The reliability of the proposed DRMA has been demonstrated by a comparison of a homogeneous deformation numerical test with the experimental and direct integration results reported in a previous communication. Finally, the EFG model has been used to predict the stress, strain and volume fraction recrystallized distributions under steady and transient nominal strain rate and temperature deformation conditions. These parametric studies have been carried out by considering the differential constitutive description, but also a conventional integrated constitutive model of a 20MnCr5 steel. The results have revealed the suitability of the EFG formulation under the proposed DRMA, for predicting the performance of an axisymmetric hot compression test. The differences between the use of differential and integrated constitutive descriptions on the performance of hot-working processes under inhomogeneous deformation conditions, have also been evidenced in this research work.

Element-free Galerkin formulation for solving transient heat transfer problems of direct chill casting processes

ABSTRACTThe present work has been conducted in order to develop a novel solution for the transient heat transfer problem during the startup phase of a direct-chill casting (DCC) process by using the element-free Galerkin method (EFGM). The material domain evolution is conditioned by a bottom block, which represents the material moving boundary. The governing equations have been solved on the basis of this global weak formulation under a Lagrangian description during the startup phase, and coupled with an Eulerian description once the block has reached the lower boundary of the computational domain along the casting direction. Finally, the EFG formulation has been adapted to an axisymmetric temperature transient problem over an aluminum alloy round billet. The feasibility and suitability of this novel approach has been verified by comparison with the numerical techniques proposed and the results reported by other researchers. The results have revealed that this technique could be used successfully in the solution of the DCC heat transfer problem during the startup phase.

Stress Analysis of a Continuous Casting Process, on the Basis of the Element‐Free Galerkin Formulation

The present work has been conducted in order to develop an alternative solution to the thermal‐mechanical problem in a conventional continuous casting (CC) process using the element‐free Galerkin (EFG) method. The linear momentum balance equation has been developed on the basis of this global weak formulation. The resulting equation system has been adapted to an axisymmetric stress–strain distribution over a round billet into the mold region. The constitutive laws, nonlinear aspects, path‐dependent variables, and boundary conditions have been specified. Finally, the model has been solved under a mixed Eulerian–Lagrangian description in order to estimate the mechanical behavior of the billet during the CC process. A brief discussion on the importance of including the field forces into the analysis is also presented. The results have revealed that this technique could be used successfully in the modeling of continuous casting thermal‐mechanical problems. The solution by means of this formulation has exhibited a remarkable sensitivity with the casting parameters and the approach implemented to compute inelastic strains during the δ ferrite–austenite transformation.

Element Free Galerkin Formulation for Problems in Composite Micromechanics

The heterogeneity and anisotropy of structural composites make the application of the standard mesh-based methods using the meshing of interfacial region between matrix and fibers a difficult task. The objective of this study is to present the EFG formulation for problems of composite micromechanics. It is expected that the tediousness and approximations involved in mesh generation, and hence inaccuracies in the results can be avoided using the new meshless techniques such as the Element Free Galerkin (EFG) method. The theoretical methodologies, computer implementations and practical application are carried out. Periodic boundary conditions of the unit cell under tensile load are set up. The method of Lagrange multipliers is introduced for the treatment of material discontinuity at the fiber-matrix interface in which both the displacement continuity and traction reciprocity are satisfied. The EFG method is formulated for the generalized plane strain problems. Examples are presented to illustrate the effectiveness of the proposed micromechanical model, and it is validated by comparing the results with available numerical solutions.

Element-Free Galerkin Formulation by Moving Least Squares for Internal Energy Balance in a Continuous Casting Process

The present work has been conducted in order to develop an alternative solution to the heat transfer problem in a conventional continuous casting (CC) process using an Element-Free Galerkin (EFG) technique with shape functions formulated by a moving least squares approximation. The Internal Energy Balance Equation (Eulerian Description) has been developed with this weak formulation in order to solve the heat transfer problem. The EFG features and the shape functions construction have also been described. The resulting equation system has been adapted to a 2-D steady-state temperature distribution over the longitudinal thickness of a solidifying wide slab into the mold region. The transport laws, the nonlinear aspects, and the boundary conditions have been specified. Finally, the model has been solved in order to estimate the thermal distribution and the solidified shell evolution. The results have revealed that this technique could be used successfully in the modeling of CC heat transfer problems.

Subspace analysis to alleviate the volumetric locking in the 3D solid-shell EFG method

The main objective of this work is to reduce the volumetric locking pathology at the Element Free Galerkin (EFG) 3D solid-shell meshless method. For this purpose, a subspace analysis is performed in order to extract the linearly independent incompressible deformation modes that can be reproduced by different background integration cells. Additional deformation modes reproduced by a cell configuration produces more flexibility for the meshless formulation. The Enhanced Assumed Strain (EAS) method is blended with the EFG formulation in such a way that additional (mathematical) variables are included in the formulation increasing its flexibility for nearly incompressible materials. Three numerical examples show the effect of the EAS formulation in alleviating the locking pathology for an almost incompressible material condition.

Element Free Galerkin Formulation of Composite Beams with Longitudinal and Transverse Partial Interactions

Partial interaction between two different materials of composite beam is consequent from longitudinal slip and transverse uplift effects at interfacial surface. The behaviour has yielded higher order differential equation as compared to beam that interacts fully. In this paper, a meshless approach for the analysis of composite beam with partial interaction by Element Free Galerkin (EFG) method is formulated and investigated. Discretized solely by nodes, the Moving Least Square (MLS) method is adopted for the EFG shape functions formulation and the variational approach is chosen in developing its Galerkin weak form. The weak form essential boundary conditions are enforced by Lagrange multiplier, where comparable results are obtained between developed EFG code and established analytical solutions. In addition, influence of various weight functions on shape function smoothness of EFG code is explored.

Higher-Order Stress-Strain Theory for Damage Modeling Implemented in an Element-free Galerkin Formulation

Gradient theories have found wide applications in modeling of strain softening phenomena. This paper presents a higher order stress-strain theory to describe the damage behavior of strain softening materials. In contrast to most conventional gradient approaches for damage modeling, the present higher order theory considers strain gradients and their conjugate higher-order stress such that stable numerical solutions may be achieved. We have described the derivation of the required constitutive relationships, the governing equations and its weak form for this higher-order theory. The constitutive coefficients were obtained from a granular media approach such that the internal length scale parameter reflects the natural granularity of the underlying microstructure. The weak form was discretized using an element-free Galerkin (EFG) formulation that readily admits approximation functions of higher-order continuity. We have also discussed the implementation of essential boundary conditions and linearization of the derived discrete equations. Finally, the applicability of the derived model is demonstrated through two examples with different imperfections designed to initiate dislocation bands and shear bands, respectively.

Continuum damage growth analysis using element free Galerkin method

This paper presents an elasto-plastic element free Galerkin formulation based on Newton-Raphson algorithm for damage growth analysis. Isotropic ductile damage evolution law is used. A study has been carried out in this paper using the proposed element free Galerkin method to understand the effect of initial damage and its growth on structural response of single and bi-material problems. Asimple method is adopted for enforcing EBCs by scaling the function approximation using a scaling matrix, when non-singular weight functions are used over the entire domain of the problem definition. Numerical examples comprising of oneand two-dimensional problems are presented to illustrate the effectiveness of the proposed method in analysis of uniform and non-uniform damage evolution problems. Effect of material discontinuity on damage growth analysis is also presented.

Suite of meshless algorithms for accurate computation of soft tissue deformation for surgical simulation.

The ability to predict patient-specific soft tissue deformations is key for computer-integrated surgery systems and the core enabling technology for a new era of personalized medicine. Element-Free Galerkin (EFG) methods are better suited for solving soft tissue deformation problems than the finite element method (FEM) due to their capability of handling large deformation while also eliminating the necessity of creating a complex predefined mesh. Nevertheless, meshless methods based on EFG formulation, exhibit three major limitations: (i) meshless shape functions using higher order basis cannot always be computed for arbitrarily distributed nodes (irregular node placement is crucial for facilitating automated discretization of complex geometries); (ii) imposition of the Essential Boundary Conditions (EBC) is not straightforward; and, (iii) numerical (Gauss) integration in space is not exact as meshless shape functions are not polynomial. This paper presents a suite of Meshless Total Lagrangian Explicit Dynamics (MTLED) algorithms incorporating a Modified Moving Least Squares (MMLS) method for interpolating scattered data both for visualization and for numerical computations of soft tissue deformation, a novel way of imposing EBC for explicit time integration, and an adaptive numerical integration procedure within the Meshless Total Lagrangian Explicit Dynamics algorithm. The appropriateness and effectiveness of the proposed methods is demonstrated using comparisons with the established non-linear procedures from commercial finite element software ABAQUS and experiments with very large deformations. To demonstrate the translational benefits of MTLED we also present a realistic brain-shift computation.

On a multi-scale element-free Galerkin method for the Stokes problem

In this paper, a multi-scale element-free Galerkin method is presented for the Stokes problem. The new method is based on the Hughes’ variational multi-scale formulation, and arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. In this method, an unresolved model is obtained in which unresolved scales are incorporated analytically through the bubble functions. Modeling of the unresolved scales corrects the lack of the stability of the standard element-free Galerkin formulation and the resulting stabilized formulation possesses superior properties like that of the streamline upwind/Petrov–Galerkin (SUPG) method and the Galerkin/least-squares (GLS) method. The method allows equal order basis for pressure and velocity because it violates the celebrated Babuska–Brezzi condition. A significant feature of the present method is that the structure of the stabilization tensor τ appears naturally via the solution of the fine-scale problem. Numerical results for example problems confirm that this method has some excellent properties, such as better stability and accuracy.

Mesh free Galerkin method based on natural neighbors and conformal mapping

In this work, a robust mesh free method has been presented for the analysis of two dimensional problems. An efficient natural neighbor algorithm for construction of polygonal support domains has been used in this method that takes into account the nonuniform nodal discretization in the element free Galerkin formulation. The use of natural neighbors for determining the compact support is shown to overcome some of the shortcomings of the conventional distance metric based methods. For nonuniform nodal discretization there is a need for evaluating weights that have anisotropic compact supports. The smoothness and conformance of the weight function to the support domain obtained from natural neighbor algorithm is achieved through an efficient conformal mapping procedure such as Schwarz-Christoffel mapping. Numerical examples demonstrate that the proposed mesh free method gives good estimates of the stress/strain fields.

Development of a EFG formulation for damage analysis of reinforced concrete beams

In this work the element-free Galerkin (EFG) method is used for damage analysis of reinforced concrete beams. The chosen damage model is that due to Mazars whereby the concrete is represented by an elastic medium with progressive damage. As usual in the EFG method, moving least squares (MLS) approximations are considered. The essential boundary conditions are included by means of Lagrange multipliers. A method is proposed to accurately represent concentrated loads, inhomogeneities or geometry. The proposed method is, basically, a domain decomposition technique where, at the interface, a weak form of continuity is enforced with Lagrange multipliers. Numerical tests are carried out on a reinforced concrete beam for which experimental as well as other numerical results are available in the literature.

Development of a EFG formulation for damage analysis of reinforced concrete beams

In this work the element-free Galerkin (EFG) method is used for damage analysis of reinforced concrete beams. The chosen damage model is that due to Mazars whereby the concrete is represented by an elastic medium with progressive damage. As usual in the EFG method, moving least squares (MLS) approximations are considered. The essential boundary conditions are included by means of Lagrange multipliers. A method is proposed to accurately represent concentrated loads, inhomogeneities or geometry. The proposed method is, basically, a domain decomposition technique where, at the interface, a weak form of continuity is enforced with Lagrange multipliers. Numerical tests are carried out on a reinforced concrete beam for which experimental as well as other numerical results are available in the literature.

An enriched meshless method for non‐linear fracture mechanics

AbstractThis paper presents an enriched meshless method for fracture analysis of cracks in homogeneous, isotropic, non‐linear‐elastic, two‐dimensional solids, subject to mode‐I loading conditions. The method involves an element‐free Galerkin formulation and two new enriched basis functions (Types I and II) to capture the Hutchinson–Rice–Rosengren singularity field in non‐linear fracture mechanics. The Type I enriched basis function can be viewed as a generalized enriched basis function, which degenerates to the linear‐elastic basis function when the material hardening exponent is unity. The Type II enriched basis function entails further improvements of the Type I basis function by adding trigonometric functions. Four numerical examples are presented to illustrate the proposed method. The boundary layer analysis indicates that the crack‐tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered, whether for the near‐tip asymptotic field or for the far‐tip elastic field. Numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the J‐integral for the applied load intensities and material properties considered. Also, the crack‐mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results. Furthermore, the meshless results show excellent agreement with the experimental measurements, indicating that the new basis functions are also capable of capturing elastic–plastic deformations at a stress concentration effectively. Copyright © 2003 John Wiley & Sons, Ltd.

Stochastic meshless analysis of elastic?plastic cracked structures

This paper presents a stochastic mesh-free method for probabilistic fracture-mechanics analysis of nonlinear cracked structures. The method involves enriched element-free Galerkin formulation for calculating the J-integral; statistical models of uncertainties in load, material properties, and crack geometry; and the first-order reliability method (FORM) for predicting probabilistic fracture response and reliability of cracked structures. The sensitivity of fracture parameters with respect to crack size, required for probabilistic analysis, is calculated using a virtual crack extension technique. Numerical examples based on mode-I fracture problems have been presented to illustrate the proposed method. The results from sensitivity analysis indicate that the maximum difference between sensitivity of the J-integral calculated using the proposed method and reference solutions obtained by the finite-difference method is about three percent. The results from reliability analysis show that the probability of fracture initiation using the proposed sensitivity and meshless-based FORM are very accurate when compared with either the finite-element-based Monte Carlo simulation or finite-element-based FORM. Since all gradients are calculated analytically, the reliability analysis of cracks can be performed efficiently using meshless methods.