Conventional Element-free Galerkin Method Research Articles

Numerical Investigation of Thermoelastic Crack Interactions in Various Materials using a Novel Enrichment Approach

Crack interaction studies play a crucial role in understanding and predicting the fracture behaviour of various engineering components subjected to thermomechanical loads. The present work investigates the interaction effect of multiple cracks in different types of material subjected to thermoelastic loadings using Element free Galerkin method (EFGM). These materials include isotropic material, orthotropic material, functionally graded material, and layered material. These all materials are subjected to thermoelastic loads in presence of multiple cracks to investigate the effect of crack interactions. A novel modified Intrinsic enrichment has been proposed to precisely capture the interaction effect and stress fields in the presence of multiple cracks. The proposed algorithm has been tested for a benchmark problem and it produced better stress fields in comparison to the conventional EFGM procedure. Stress intensity factors corresponding to variations in crack parameters have been evaluated concerning with the primary crack. Results reveal that presence of multiple cracks alters the crack tip stress fields owing to the interaction effect i.e shielding or amplification. Additionally, parameters such as crack length, crack orientation, distance between cracks, and domain properties greatly influence the stress intensity factor of the primary crack. These parameters exhibit varying behaviour under distinct circumstances, and their effects have been thoroughly analysed. Current work provides valuable insights into the effects of crack interactions in different media under thermoelastic loadings, thereby ensuring the structural integrity and durability of these materials for practical applications.

An efficient meshfree framework for simulation of crack tip stress fields in two-dimensional graded media subjected to thermoelastic loads

In the present work a novel framework of algorithm has been proposed for enhancing the versatility of the element-free Galerkin method for modeling of fracture in graded media subjected to thermoelastic loadings. The proposed work restructures the conventional element free Galerkin method (EFGM) at multiple level through generic MATLAB algorithms thereby developing an efficient and robust method for simulating crack tip stress fields in graded material subjected to thermoelastic loads. The proposed algorithm utilizes varying nodal density criterion in conjunction with optimum Gaussian quadrature around these nodes for numerical integration of the governing equations. The proposed algorithm also incorporates blended approximation which is effective for simulating crack tip stress fields in both convex and non-convex problem domains. Nonequilibrium formulation has been selected for evaluating stress intensity factors (SIFs) for the fracture problems in graded media. A few problems of cracked domain subjected to thermoelastic loads have been modeled and simulated to illustrate the effectiveness of proposed framework of algorithms. Further, three component level problems having non-convex domains have also been discussed to establish the robustness and modeling capabilities of the proposed work. A good agreement between the simulated results and reference data demonstrates the algorithm’s competence and robustness for modeling thermoelastic fracture in graded materials. The proposed methodology improves the computational efficiency and flexibility of the element-free Galerkin method for modeling thermoelastic fracture in graded media. Moreover, enhancement of computing speed with the proposed methodology adds to its modeling potential.

Accelerated element-free Galerkin method for analysis of fracture problems

The work presents a modified form of element-free Galerkin method (EFGM) based on a procedure which allows selection of equal number of nodes in the support domain. This procedure is further improved by selecting domain nodes depending on the region of the problem geometry. The optimum number of support domain nodes is obtained by performing an optimization of selectable EFGM parameters namely total number of nodes in problem geometry, Gauss quadrature and random or desired number of nodes in support domain. Taguchi L-16 orthogonal array is used to obtain optimized values of these parameters. The optimized EFGM parameters provide good accuracy with lesser number of nodal points and reduce the computational time of conventional EFGM by an average of 85%. Interaction integral technique has been used in order to extract the stress intensity factors for the simulated problems. The worth of so presented accelerated element-free Galerkin method (AEFG) is established by simulating various cases of fracture problems and the results so obtained are concurrent with those available in literature.

Coupling of Adaptive Element Free Galerkin Method with Variational Multiscale Method for Two-Dimensional Transient Heat Conduction Problem

AbstractIn this paper, the coupling of adaptive element free Galerkin method (EFGM) with variational multiscale method is developed to solve two-dimensional transient heat conduction problem. The EFGM is chosen to combine with the adaptive analysis since this method only use a set of particles to generate the discrete equations instead of mesh construction as in finite element method (FEM). This property makes additional nodes can be easily inserted at the high gradient regions to improve the numerical results. The refinement criteria and refinement strategy will be clearly shown for the adaptive analysis procedure. The performance of this coupling method is first validated by solving two-dimensional large localized gradient problem with available theoretical solutions. Next, this method will also be used to solve two-dimensional transient heat conduction problem. The accuracy and efficiency of the method is demonstrated by comparing the results between conventional EFGM and the proposed method. The results obtained show that this method is able to produce higher accuracy results compared with the standard element free Galerkin method.

Coupling of Adaptive Element Free Galerkin Method with Variational Multiscale Method for Two-Dimensional Sine-Gordon Equation

In this paper, a coupling of adaptive element free Galerkin method with variational multiscale method is used to solve sine-Gordon equation in two-dimensional for the first time. Meshfree method is used where no mesh regeneration is needed comparing to finite element method. Therefore, this property facilitates the insertion of nodes which is triggered by the adaptive refinement procedure. Additional new nodes will be inserted at the high gradient regions to improve the numerical solutions. The adaptive analysis such as the refinement criteria and refinement strategy will be shown as well as the development of the modified moving least squares approximation. The performance of the proposed method is validated by solving two numerical problems. The first problem is two-dimensional large localized gradient problem with available analytical solution and the second problem is sine-Gordon equation. Numerical results proved that this method can obtain higher accuracy results compared with the conventional element free Galerkin method.

Global interpolating meshless shape function based on generalized moving least-square for structural dynamic analysis

A global interpolating meshless shape function based on the generalized moving least-square (GMLS) is formulated by the transformation technique. Both the shape function and its derivatives meet the Kronecker delta function property. With the interpolating GMLS (IGMLS) shape function, an improved element-free Galerkin (EFG) method is proposed for the structural dynamic analysis. Compared with the conventional EFG method, the obvious advantage of the proposed method is that the essential boundary conditions including both displacements and derivatives can be imposed by the straightforward way. Meanwhile, it can greatly improve the ill-condition feature of the standard GMLS approximation, and provide good accuracy at low cost. The dynamic analyses of the Euler beam and Kirchhoff plate are performed to demonstrate the feasibility and effectiveness of the improved method. The comparison between the numerical results of the conventional method and the improved method shows that the proposed method has better stability, higher accuracy, and less time consumption.

Enriched Element-Free Galerkin Method for Fracture Analysis of Functionally Graded Piezoelectric Materials

A new method using the enriched element-free Galerkin method (EEFGM) to model functionally graded piezoelectric materials (FGPMs) with cracks was presented. To improve the solution accuracy, extended terms were introduced into the approximation function of the conventional element-free Galerkin method (EFGM) to describe the displacement and electric fields near the crack. Compared with the conventional EFGM, the new approach requires smaller domain to describe the crack-tip singular field. Additionally, the domain of the nodes was not affected by the crack. Therefore, the visibility method and the diffraction method were no longer needed. The mechanical response of FGPM was discussed, when its material parameters changed exponentially in a certain direction. The modifiedJ-integrals for FGPM were deduced, whose results were compared with the results of the conventional EFGM and the analytical solution. Numerical example results illustrated that this method is feasible and precise.

Analyzing modified equal width (MEW) wave equation using the improved element-free Galerkin method

Abstract The element-free Galerkin (EFG) method is a promising method for solving partial differential equations in which trial and test functions employed in the discretization process result from moving least-squares (MLS) approximation. In this paper, by employing the improved moving least-squares (IMLS) approximation, we derive formulae for an improved element-free Galerkin (IEFG) method for the modified equal width (MEW) wave equation. A variation of the method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method. Therefore, the IEFG method may result a better computing speed. In this paper, the effectiveness of the IEFG method for modified equal width (MEW) wave equation is investigated by numerical examples.

A NEW ELEMENT-FREE GALERKIN METHOD BASED ON IMPROVED COMPLEX VARIABLE MOVING LEAST-SQUARES APPROXIMATION FOR ELASTICITY

In this paper, by constructing a new functional, an improved complex variable moving least-squares (ICVMLS) approximation is presented. Based on element-free Galerkin (EFG) method and the ICVMLS approximation, a new complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems is presented. Galerkin weak form is used to obtain the discretized equations and the essential boundary conditions are applied with Lagrange multiplier. Then the formulae of the new CVEFG method for two-dimensional elasticity problems are obtained. Compared with the conventional EFG method, the new CVEFG method has greater computational precision and efficiency. For the purposes of demonstration, some selected numerical examples are solved using the ICVEFG method.

An improved complex variable element-free Galerkin method for two-dimensional elasticity problems

In this paper, the improved complex variable moving least-squares (ICVMLS) approximation is presented. The ICVMLS approximation has an explicit physics meaning. Compared with the complex variable moving least-squares (CVMLS) approximations presented by Cheng and Ren, the ICVMLS approximation has a great computational precision and efficiency. Based on the element-free Galerkin (EFG) method and the ICVMLS approximation, the improved complex variable element-free Galerkin (ICVEFG) method is presented for two-dimensional elasticity problems, and the corresponding formulae are obtained. Compared with the conventional EFG method, the ICVEFG method has a great computational accuracy and efficiency. For the purpose of demonstration, three selected numerical examples are solved using the ICVEFG method.

New Implementation Method for Essential Boundary Condition to Extended Element-Free Galerkin Method: Application to Nonlinear Problem

A new method has been proposed for implementing essential boundary conditions to the Element-Free Galerkin Method (EFGM) without using the Lagrange multiplier. Furthermore, the performance of the proposed method has been investigated for a nonlinear Poisson problem. The results of computations show that, as interpolation functions become closer to delta functions, the accuracy of the solution is improved on the boundary. In addition, the accuracy of the proposed method is higher than that of the conventional EFGM. Therefore, it might be concluded that the proposed method is useful for solving the nonlinear Poisson problem.

Analyzing three-dimensional potential problems with the improved element-free Galerkin method

The potential problem is one of the most important partial differential equations in engineering mathematics. A potential problem is a function that satisfies a given partial differential equation and particular boundary conditions. It is independent of time and involves only space coordinates, as in Poisson’s equation or the Laplace equation with Dirichlet, Neumann, or mixed conditions. When potential problems are very complex, both in their field variable variation and boundary conditions, they usually cannot be solved by analytical solutions. The element-free Galerkin (EFG) method is a promising method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by employing improved moving least-squares (IMLS) approximation, we derive the formulas for an improved element-free Galerkin (IEFG) method for three-dimensional potential problems. Because there are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method, fewer nodes are selected in the entire domain than in the conventional EFG method, the IEFG method should result in a higher computing speed.

Improved element-free Galerkin method for two-dimensional potential problems

Potential difficulties arise in connection with various physical and engineering problems in which the functions satisfy a given partial differential equation and particular boundary conditions. These problems are independent of time and involve only space coordinates, as in Poisson's equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, they usually cannot be solved with analytical solutions. The element-free Galerkin (EFG) method is a meshless method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by using the weighted orthogonal basis function to construct the MLS interpolants, we derive the formulae of an improved EFG (IEFG) method for two-dimensional potential problems. There are fewer coefficients in the improved MLS (IMLS) approximation than in the MLS approximation, and in the IEFG method fewer nodes are selected in the entire domain than in the conventional EFG method. Hence, the IEFG method should result in a higher computing speed.

Analyzing 2D fracture problems with the improved element-free Galerkin method

This paper presents an improved moving least-squares (IMLS) approximation in which the orthogonal function system with a weight function is used as the basis function. The IMLS approximation has greater computational efficiency and precision than the existing moving least-squares (MLS) approximation, and does not lead to an ill-conditioned system of equations. By combining the element-free Galerkin (EFG) method and the IMLS approximation, an improved element-free Galerkin (IEFG) method for two-dimensional elasticity is derived. There are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method that is formed with the IMLS approximation fewer nodes are selected in the entire domain than are selected in the conventional EFG method. Hence, the IEFG method should result in a higher computing speed. For two-dimensional fracture problems, the enriched basis function is used at the tip of the crack to give an enriched IEFG method. When the enriched IEFG method is used, the singularity of the stresses at the tip of the crack can be shown better than that in the IEFG method. To provide a demonstration, numerical examples are solved using the IEFG method and the enriched IEFG method.

Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems

In this paper, the element-free Galerkin (EFG) method and improved moving least-squares (IMLS) approximation are combined. An improved FEG (IEFG) method for two-dimensional elasticity is discussed, and the coupling of the IEFG method and the boundary element method (BEM) is presented. In the IMLS approximation, an orthogonal function system with a weight function is used as the basis function. The IMLS approximation has greater computational efficiency and precision than the existing moving least-squares (MLS) approximation, and does not lead to an ill-conditioned system of equations. There are fewer coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method that is formed with the IMLS approximation fewer nodes are selected in the entire domain than are selected using the conventional EFG method. Hence, the IEFG method should result in a higher computing speed. Based on the IMLS approximation and the IEFG method, a direct coupling of the IEFG method and the BEM is discussed for two-dimensional elasticity problems, and the corresponding formulae of the coupled method are obtained. The coupled method does not need a new sub-domain between the IEFG method and the BEM sub-domains. Selected numerical examples are solved using the coupled method.

Meshfree analyses of cable-reinforced membrane structures by ALE–EFG method

Many large membrane structures have been constructed in these days and large membrane structures are often stiffened by cables for reinforcement of the strength. In the analysis of cable-reinforced membrane structures, there are several complicated problems, such as folding of membrane by cable, sliding of cable on membrane surface and so on. As these problems are hardly analyzed by the finite element method, in this study, a meshfree method based on the element free Galerkin (EFG) method is applied to overcome these problems. In the conventional EFG method, the problem that contains discontinuous slope of displacement cannot be analyzed because the moving least square approximation yields C 1 continuous displacement field without any modification. Additionally, sliding between cable and membrane surface must be considered. In this study, the arbitrary Lagrangian–Eulerian formulation and simple technique using patches with the penalty method are combined and incorporated into the conventional EFG method. The proposed method is applied to the analyses of cable-reinforced membrane structures to demonstrate the potential of the method. Satisfactory results are obtained and the validity of the method is clarified.

Demonstration of a simple method to satisfy homogenous boundary conditions in element‐free Galerkin method through the vibration problem of Timoshenko beam

AbstractA simple method to satisfy the homogeneous boundary condition in the element‐free Galerkin method is proposed in this paper. Effectiveness of the method is demonstrated, through free vibration problems of Timoshenko beams. Several case studies involving different slenderness ratios and end conditions are carried out. The results of the present study are compared with those obtained using classical Rayleigh–Ritz and conventional element‐free Galerkin method wherein Lagrangian multipliers are employed to satisfy boundary conditions. Eventhough, a one‐dimensional problem is considered in the present paper, the validity of the proposed method for the two‐dimensional problems is demonstrated theoretically. Copyright © 2003 John Wiley & Sons, Ltd.

Meshless formulation for shear-locking free bending elements

An improved version of the Element-free Galerkin method (EFGM) is presented here for addressing the problem of transverse shear locking in shear-deformable beams with a high length over thickness ratio. Based upon Timoshenko's theory of thick beams, it has been recognized that shear locking will be completely eliminated if the rotation field is constructed to match the field of slope, given by the first derivative of displacement. This criterion is applied directly to the most commonly implemented version of EFGM. However in the numerical process to integrate strain energy, the second derivative of the standard Moving Least Square (MLS) shape functions must be evaluated, thus requiring at least a <TEX>$C^1$</TEX> continuity of MLS shape functions instead of <TEX>$C^0$</TEX> continuity in the conventional EFGM. Yet this hindrance is overcome effortlessly by only using at least a <TEX>$C^1$</TEX> weight function. One-dimensional quartic spline weight function with <TEX>$C^2$</TEX> continuity is therefore adopted for this purpose. Various numerical results in this work indicate that the modified version of the EFGM does not exhibit transverse shear locking, reduces stress oscillations, produces fast convergence, and provides a surprisingly high degree of accuracy even with coarse domain discretizations.

Mesh Free Analyses of Structures with Discontinuous Field using EFGM.

In this study, the element free Galerkin method is applied to the structural analyses which has material or geometrical discontinuity. The conventional element free Galerkin method, the first mesh free method proposed by Belytschko et al. in 1994, has an advantage that the continuous displacement derivatives are obtained due to the moving least squares interpolation for the approximation of the field function. Therefore, in order to evaluate discontinuous derivatives, the special treatments should be given at the discontinuity and the penalty method is adopted in this study. The proposed method is firstly applied to several structural analyses including material discontinuity and the appropriate size of domain of influence and the effect of nodal distribution as well as the accuracy of the method is investigated. Finally, it is applied to a geometrically nonlinear analysis of membrane structure, which has discontinuous slopes, and the effectiveness of analyses using the present method is demonstrated.