Element Free Galerkin Approach Research Articles

A novel EFG meshless-ANN approach for static analysis of FGM plates based on the higher-order theory

An Element Free Galerkin (EFG) meshless formulation and solutions using higher order shear deformation theory with nine degrees of freedom for the static analysis of Functionally Graded Material (FGM) plates are provided. This technique estimates the shape function using Moving Least Squares (MLS) method. The proposed method is validated by comparing the present findings with those in the literature. A novel Artificial Neural Network (ANN) model is developed to forecast the deflection of FGM plates within less computational time. Detailed parametric and convergent studies reveal that the proposed EFG solution and the ANN technique are more efficient than their conventional counterparts. The validation and comparison of the generated results in the present investigation with the other analysis methods revealed that the EFG method and ANN model give more accurate results than the FEM and other meshless methods. The current EFG-ANN model reduces computing time by 99.94% when compared to the EFG approach. Also, the accuracy is enhanced using the EFG approach with HSDT9 for the FGM plate.

Exponentially Fitted Element-Free Galerkin Approach for Nonlinear Singularly Perturbed Problems

As it is well recognized that conventional numerical schemes are inefficient in approximating the solutions of the singularly perturbed problems (SPP) in the boundary layer region, in the present work, an effort has been made to propose a robust and efficient numerical approach known as element-free Galerkin (EFG) technique to capture these solutions with a high precision of accuracy. Since a lot of weight functions exist in the literature which plays a crucial role in the moving least square (MLS) approximations for generating the shape functions and hence affect the accuracy of the numerical solution, in the present work, due emphasis has been given to propose a robust weight function for the element-free Galerkin scheme for SPP. The key feature of nonrequirement of elements or node connectivity of the EFG method has also been utilized by proposing a way to generate nonuniformly distributed nodes. In order to verify the computational consistency and robustness of the proposed scheme, a variety of linear and nonlinear numerical examples have been considered and L ∞ errors have been presented. Comparison of the EFG solutions with those available in the literature depicts the superiority of the proposed scheme.

An element-free Galerkin approach for rigid–flexible coupling dynamics in 2D state

Based on an improved element-free Galerkin (EFG) method and the geometric nonlinear elastic theory, an element-free numerical approach for rigid–flexible coupling dynamics of a rotating hub–beam system is implemented. Using the Hamilton principle, the coupled nonlinear integral-differential governing equations are derived, in which the dynamic stiffening effect is expressed by the longitudinal shrinking of the beam induced by the transverse displacement. By the EFG method where the global interpolating moving least squares (IMLS) and generalized moving least squares (IGMLS) were used for discretizing the longitudinal and transverse deformation variables, respectively, the spatially element-free discretized dynamic equations of the system are obtained and the Newmark scheme was selected as the time integration method for the numerical computation. The superiority of the global interpolating property makes the ease for imposing both displacement and derivative boundary conditions. The transverse bending vibration and structural dynamics of the flexible beam in non-inertial system are analyzed in the numerical simulations and influence parameters of the EFG method are also fully discussed. Simulation results verify the feasibility of the improved EFG approach for rigid–flexible coupling dynamics.

Evaluation of mechanical properties using spherical ball indentation and coupled finite element–element-free galerkin approach

ABSTRACTIn the present work, the mechanical properties of pressure tube material (Zr-Nb2.5) are evaluated using the coupled finite element–element-free Galerkin approach. Penalty approach is used to impose contact constraints and non-penetration condition at the interface. An efficient node-to-segment algorithm is employed to model the contact behavior. An updated Lagrangian approach is used to model the large deformation. Loading and unloading response of the indentation process is analyzed using von-Mises and Gurson-Tvergaard-Needleman (GTN) plasticity models. In multiple indentations, the indentation depth is progressively increased up to a maximum specified limit with partial unloading. Load-indentation depth curves are used to extract the flow properties of the material.

An Element-free Galerkin (EFG) scaled boundary method

The scaled boundary method is a semi-analytical method of analysis which can be used in computational mechanics. This method uses a normalised radial coordinate system, introducing shape functions to weaken the governing equations in the circumferential direction and solving the resulting differential equations analytically in the radial direction. This paper presents a new Element-free Galerkin (EFG) scaled boundary method in which the EFG approach is used in the circumferential direction. The proposed model is verified by application to a number of standard problems of elasticity. The numerical solutions show that the new method has higher accuracy (for any particular number of nodes) and better convergence than scaled boundary finite element methods, and an accurate smooth stress field can be obtained directly without the necessity of using a stress recovery procedure.

Three-dimensional simulation of fully coupled hydro-mechanical behavior of saturated porous media using Element Free Galerkin (EFG) method

Meshless methods are a relatively new type of numerical methods that have attracted the attention of many researchers over the past years. So far, a number of meshless methods have been developed and applied to solve problems in various fields of engineering, including solid mechanics and geotechnical problems. The Element-Free Galerkin (EFG) method is adopted in this study for solving the governing partial differential equations of equilibrium and continuity of pore fluid flow for numerical simulation of coupled hydro-mechanical problems. For this purpose, the weak form of the governing equations is derived by applying the weighted residual method and Galerkin technique. The penalty method is used for imposition of essential boundary conditions. The spatial discretization of the resulting constrained Galerkin weak forms is based on using the same EFG shape functions for displacements and pore water pressures, and a fully implicit scheme is used for discretization in the time domain. The performance of the developed EFG-based code is evaluated by comparing the results of the model with analytical solutions. After verification, the developed code is utilized for analyzing two- and three-dimensional consolidation problems considering different drainage and loading conditions. The numerical results indicate that the robust 3D mesh-free code developed in this paper is capable of simulating fully coupled hydro-mechanical problems without spurious oscillations if numerical parameters of the EFG approach are properly selected.

Determination of coefficients of crack tip asymptotic fields by an element‐free Galerkin scaled boundary method

ABSTRACTThis study applies a new scaled boundary element‐free Galerkin (EFG) method to determine the coefficients of the Williams’ series expansion for the elastic crack tip asymptotic stress field. Scaled boundary methods can calculate the coefficients of Williams’ series without any special crack tip treatment. Unlike the conventional scaled boundary finite element method, in this paper an EFG approach is used in the circumferential direction of the scaled boundary model. The EFG method offers high accuracy, rapid convergence and smooth stress solutions without post‐processing, and no mesh generation is required. The proposed method is verified through numerical examples comparing the new approach with analytical, reference and conventional scaled boundary FEM solutions, and results are obtained which indicate that the new approach converges even more rapidly than the conventional scaled boundary finite element method.

Computation of Higher Order Spatial Derivatives in the Multiscale Expansion of Electromagnetic-Field Problems

This paper deals with the evaluation of higher order spatial derivatives in electromagnetic-field problems with heterogeneous media handled through a multiscale expansion technique. A comparison is performed on the higher derivative computation, considering an element-free Galerkin approach, and bicubic spline interpolations as well as least-square approximations of piecewise finite-element solutions.

Shape sensitivity analysis and shape optimization in planar elasticity using the element-free Galerkin method

This paper demonstrates that the element-free Galerkin (EFG) method can be successfully used in shape design sensitivity analysis and shape optimization for problems in 2D elasticity. The continuum-based variational equations for displacement sensitivities are derived and are subsequently discretized. This approach allows one to avoid differentiating the EFG shape functions. The present formulation, that employs a penalty method for imposing the essential boundary conditions, can be easily extended to 3D and/or non-linear problems. Numerical examples are presented to show the capabilities of the current approach for calculating sensitivities. The flexibility of the EFG method, that eliminates the element connectivity requirement of the finite element method (FEM), permits solving shape optimization problems without re-meshing. The problem of shape optimization of a fillet is used to demonstrate this fact. Smoother stresses and better accuracy for points close to the boundary allow for a better EFG solution compared to published results using the FEM, the boundary element method (BEM) or the boundary contour method (BCM). Furthermore, for the EFG approach, grid-optimization appears unnecessary.