Galerkin Weak Form Research Articles

N-sided polygonal smoothed finite element method (nSFEM) with non-matching meshes and their applications for brittle fracture problems

In this work, an n-sided polygonal smoothed finite element method (nSFEM) based on gradient smoothing technique is presented for phase field fracture modelling. The phase field approach approximates the crack topology in a diffuse way through an exponential function. A coupled elastic displacement and phase field variational formulation is established by using a Smoothed Galerkin Weak form. Meanwhile, non-matching meshes for local refinement are considered in brittle fracture problems. Non-matching meshes at their interfaces are automatically converted to matching meshes by using present arbitrary sided polygonal elements. In other words, no interface constraint is needed here. In order to validate the feasibility of present nSFEM for brittle fracture problem with non-matching meshes, several numerical examples are employed. The nSFEM can discretize the computational domain in a very flexible manner. Arbitrary sided polygonal elements and non-matching meshes are adopted for comparison. Numerical results validate that the nSFEM shows great potential for phase field fracture problems due to its flexibility in dealing with complex elements shape and non-matching meshes.

A three-dimensional formulation for levy-type transversely loaded cross-ply shell panels

In this study, a formulation is developed for investigation of three-dimensional edge stresses in cylindrical composite shell panels under distributed transverse loading. The formulation is obtained by employing the Galerkin weak formulation of the equilibrium equations. The obtained governing equations for the panel are solved via an analytical method for Levy-type boundary conditions including free and clamped edges. The presented formulation is capable of prediction of three-dimensional and boundary layer stresses in the composite panels subjected to transverse loading. The edge stresses in free edge, clamped edge and simple edge of transversally loaded cylindrical composite panel are studied. Comparison of numerical results with the available results of exact elasticity solution in the literature confirms the accuracy and the efficiency of the formulation.

A truly meshfree method for solving acoustic problems using local weak form and radial basis functions

It has been known that numerical solutions to Helmholtz problems obtained using several numerical methodologies (e.g. finite element approach) are always plagued by the pollution error effect particularly for high wave numbers, giving rise to erroneous results of numerical acoustic wave. To mitigate this effect, a truly meshfree technique using a local weak form and radial basis functions is employed to analyze acoustic problems. On the basis of the local Petrov–Galerkin weak form, numerical integration is performed over the quadrature domains related to all field nodes, instead of the mesh grid required in the global Galerkin weak form like finite element approach and element-free Galerkin method. Hence, the present methodology is totally independent of the mesh grid either in forming shape functions, or in the integration procedure for system matrices. Also, owing to the use of radial point interpolation by passing all the relevant nodes, the constructed shape functions hold the practical feature of the Kronecker delta function, resulting in easy enforcement of the essential boundary condition as in finite element approach. The results of several numerical examples have shown that with same group of field nodes the present methodology can lead to much more accurate solutions than finite element approach, in particular for relatively high frequencies, and can also generate comparable solutions in comparison to other global Galerkin meshfree techniques.

The dimension splitting reproducing kernel particle method for three‐dimensional potential problems

SummaryIn this paper, the dimension splitting reproducing kernel particle method (DSRKPM) for three‐dimensional (3D) potential problems is presented. In the DSRKPM, a 3D potential problem can be transformed into a series of two‐dimensional (2D) ones in the dimension splitting direction. The reproducing kernel particle method (RKPM) is used to solve each 2D problem, the essential boundary conditions are imposed by penalty method, and the discretized equation is obtained from Galerkin weak form of potential problems. Finite difference method is used in the dimension splitting direction. Then, by combining a series of the equations of the RKPM for solving 2D problems, the final equation of the DSRKPM for 3D potential problems is obtained. Five example problems on regular or irregular domains are selected to show that the DSRKPM has higher computational efficiency than the RKPM and the improved element‐free Galerkin method for 3D potential problems.

An interpolating meshless method for the numerical simulation of the time-fractional diffusion equations with error estimates

Purpose This paper aims to present an interpolating element-free Galerkin (IEFG) method for the numerical study of the time-fractional diffusion equation, and then discuss the stability and convergence of the numerical solutions. Design/methodology/approach In the time-fractional diffusion equation, the time fractional derivatives are approximated by L1 method, and the shape functions are constructed by the interpolating moving least-squares (IMLS) method. The final system equations are obtained by using the Galerkin weak form. Because the shape functions have the interpolating property, the unknowns can be solved by the iterative method after imposing the essential boundary condition directly. Findings Both theoretical and numerical results show that the IEFG method for the time-fractional diffusion equation has high accuracy. The stability of the fully discrete scheme of the method on the time step is stable unconditionally with a high convergence rate. Originality/value This work will provide an interpolating meshless method to study the numerical solutions of the time-fractional diffusion equation using the IEFG method.

A new Fragile Points Method (FPM) in computational mechanics, based on the concepts of Point Stiffnesses and Numerical Flux Corrections

In this paper, a new method, named the Fragile Points Method (FPM), is developed for computer modeling in engineering and sciences. In the FPM, simple, local, polynomial, discontinuous and Point-based trial and test functions are proposed based on randomly scattered points in the problem domain. The local discontinuous polynomial trial and test functions are postulated by using the Generalized Finite Difference method. These functions are only piece-wise continuous over the global domain. By implementing the Point-based trial and test functions into the Galerkin weak form, we define the concept of Point Stiffnesses as the contribution of each Point in the problem domain to the global stiffness matrix. However, due to the discontinuity of trial and test functions in the domain, directly using the Galerkin weak form leads to inconsistency. To resolve this, Numerical Flux Corrections, which are frequently used in Discontinuous Galerkin methods are further employed in the FPM. The resulting global stiffness matrix is symmetric and sparse, which is advantageous for large-scale engineering computations. Several numerical examples of 1D and 2D Poisson equations are given in this paper to demonstrate the high accuracy, robustness and convergence of the FPM. Because of the locality and discontinuity of the Point-based trial and test functions, this method can be easily extended to model extreme problems in mechanics, such as fragility, rupture, fracture, damage, and fragmentation. These extreme problems will be discussed in our future studies.

Contact Analysis of Functionally Graded Materials Using Smoothed Finite Element Methods

In the paper, the smoothed finite element method (S-FEM) based on linear triangular elements is used to solve 2D solid contact problems for functionally graded materials. Both conforming and nonconforming contacts algorithms are developed using modified Coulomb friction contact models including tangential strength and normal adhesion. Based on the smoothed Galerkin weak form, the system stiffness matrices are created using the formulation procedures of node-based S-FEM (NS-FEM) and edge-based S-FEM (ES-FEM), and the contact interface equations are discretized by contact point-pairs. Then these discretized system equations are converted into a form of linear complementarity problems (LCPs), which can be further solved efficiently using the Lemke method. The singular value decomposition method is used to deal with the singularity of the stiffness matrices in the procedure constructing the standard LCP, which can greatly improve the stability and accuracy of the numerical results. Numerical examples are presented to investigate the effects of the various parameters of functionally graded materials and comparisons have been made with reference solutions and the standard FEM. The numerical results demonstrate that the strain energy solutions of ES-FEM have higher convergence rate and accuracy compared with that of NS-FEM and FEM for functionally graded materials through the present contact analysis approach.

Size-dependent nonlinear analysis and damping responses of FG-CNTRC micro-plates

This paper presents a nonlinear numerical model, which is based on the modified couple stress theory (MCST), and trigonometric shear deformation theory coupled with isogeometric analysis. The present approach captures the small scale effects on the geometrically nonlinear behaviors of functionally graded carbon nanotube reinforced composite (FG-CNTRC) micro-plate with four patterns distribution. The equations of motion are established based on a Galerkin weak form associated with von-Kármán nonlinear strains. The MCST utilizes only one material length scale parameter to predict the size effect in FG-CNTRC micro-plate, for which its material properties are derived from an extended rule of mixture. The solutions of nonlinear static equation are obtained by using the Newton–Raphson technique and the Newmark time iteration procedure in association with Picard method is assigned to get responses of the nonlinear dynamic problems. In addition, the Rayleigh damping is applied to consider the influence of damping characteristic on the oscillation of FG-CNTRC micro-plates. Comparisons are performed to verify the proposed approach. Afterward, the numerical examples are used to show the effects of the distribution of carbon nanotubes (CNT), their volume fraction, the material length scale parameter and the boundary conditions on the nonlinear static and dynamic behaviors of FG-CNTRC micro-plates.

A Moving Kriging Interpolation Meshfree Method Based on Naturally Stabilized Nodal Integration Scheme for Plate Analysis

A moving Kriging interpolation (MKI) meshfree method based on naturally stabilized nodal integration (NSNI) scheme is presented to study static, free vibration and buckling behaviors of isotropic Reissner–Mindlin plates. Gradient strains are directly computed at nodes similar to the direct nodal integration (DNI). Outstanding features of the current approach are to alleviate instability solutions in the DNI and to decrease computational cost significantly when compared with the traditional high-order Gauss quadrature scheme. The NSNI is a naturally implicit gradient expansion and does not employ a divergence theorem for strain fields as addressed in the stabilized conforming nodal integration method. The present formulation is derived from the Galerkin weak form and avoids a naturally shear-locking phenomenon without using any other techniques. Thanks to satisfied Kronecker delta function property of MKI shape function, essential boundary conditions (BCs) are easily and directly enforced similar to the finite element method. A variety of numerical examples with various geometries, stiffness ratios and BCs are studied to verify the effectiveness of the present approach.

An efficient gradient smoothing meshfree formulation for the fourth-order phase field modeling of brittle fracture

A reproducing kernel gradient smoothing meshfree formulation is proposed for the fourth-order phase field modeling of brittle fracture. In order to circumvent the complexity and lower efficiency of meshfree gradient computation, a reproducing kernel gradient smoothing formulation is presented with particular reference to quadratic basis functions. Both first- and second-order meshfree smoothed gradients are discussed, in which the first-order smoothed gradients are expressed as a reproducing kernel form and meet the quadratic consistency conditions of Galerkin weak form, and the second-order smoothed gradients are then computed through performing a direct differentiation on the first-order smoothed gradients. It turns out that the resulting second-order smoothed gradients satisfy the standard gradient reproducing conditions of meshfree approximations. Subsequently, the smoothed gradients of meshfree shape functions are employed to discretize the Galerkin weak forms of equilibrium and crack phase field equations, where the numerical integration and smoothed gradient construction are specially emphasized. The tensile–compressive strain decomposition is adopted to prevent compressive crack evolution. Numerical results demonstrate the effectiveness of the proposed gradient smoothing meshfree formulation for the fourth-order phase field modeling of brittle fracture.

The improved element-free Galerkin method for three-dimensional elastoplasticity problems

In this study, the improved element-free Galerkin (IEFG) method is presented to the three-dimensional elastoplasticity problems. The improved least-squares (IMLS) approximation is used to obtain the shape function, the Galerkin weak form of three-dimensional elastoplasticity problems considering the nonlinear stress–strain relationship is used to form the discrete equation system, and penalty method is applied to the displacement boundary conditions, then the formula of the IEFG method for three-dimensional elastoplasticity problems are obtained. Some numerical examples are given to discuss the convergence of the IEFG method in this paper and the influences of the weight function, the scaling parameter, the penalty factor, the node distribution and the step number on the computational accuracy of the numerical solutions of the IEFG method. Comparing with the element-free Galerkin (EFG) method, the method in this study has a greater computational efficiency.

A Gradient Weighted Finite Element Method (GW-FEM) for Static and Quasi-Static Electromagnetic Field Computation

This paper presented a Gradient Weighted Finite Element Method (GW-FEM) for solving electromagnetic problems. First, the analysis domain is discretized into a set of triangular or tetrahedral elements which are easy to automatically generate. Then, Gradient Weighted influence domains are further constructed by the center element with all the adjacent elements. The Galerkin Weak form is evaluated based on these influence domains. The GW-FEM is employed here for the solution of static and quasi-static electromagnetic problems by using linear triangular or tetrahedral elements. All the properties of GW-FEM are proved theoretically and analyzed in detail. Consistency between four benchmark results is obtained by GW-FEM and analytical results verify the accuracy, stability, and potential of this method. It turns out that GW-FEM possesses potentials in the applications of electromagnetic problems.

Statistical analysis of composites reinforced with randomly distributed fibers using a meshless method

This paper adopts a meshless method that combines the moving least-squares approximation and Galerkin weak form to investigate the mechanical properties of unidirectional fiber-reinforced composites under a lateral load. The degree of nonuniformity is adopted to quantify the spatial distribution of randomness during simulation, and then the numerical implementation is developed to generate a representative volume element (RVE) model with random distribution of fibers. A statistical analysis is carried out by using three descriptors, that is, inter-fiber distance, second-order intensity function and radial distribution function. Numerical examples are presented to illustrate the accuracy of the proposed multiscale meshless method, and excellent agreement is achieved in comparison with experiments and the finite element method. Finally, the performance of the proposed numerical technique is evaluated by considering the effects of RVE size, node influence domain, degree of nonuniformity and fiber volume fraction separately.

A smoothed enriched meshfree Galerkin method with two-level nesting triangular sub-domains for stress intensity factors at crack tips

A smoothed enriched meshfree Galerkin method with two-level nesting triangular sub-domains (NSEMM) is presented for linear elastic fracture mechanics. In the present method, displacement fields are constructed by using the linear MLS approximation in linear region away from the crack tip and the enriched MLS approximation in enriched region around the crack tip respectively. To overcome the discontinuous problem of the coupling approximations and accelerate meshfree computations, two-level nesting triangular sub-domains based on the triangular background integration cells are designed and used to construct the smoothing strain. Each two-level nesting sub-domains includes the triangular background cell itself and four equal-area triangular sub-cells. The discretized system equations are created by the Generalized smoothed Galerkin (GS-Garlerkin) weak form and Richardson extrapolation method. The values of SIFs are evaluated by using the domain form of interaction integral. Numerical examples have been simulated to demonstrate the effectiveness and robustness of the present method for crack problems with complicated configurations, and very satisfactory results have been obtained. The NSEMM takes advantages of both EFGM and generalized strain smoothing technique such as lower computational cost, high accurate and easy to implementation.

A three-field smoothed formulation for partitioned fluid–structure interaction via nonlinear block-Gauss–Seidel procedure

This article presents a three-field smoothed formulation for partitioned fluid–structure interaction problems. The cell-based smoothed finite element method (CS-FEM) is adopted to solve the Navier–Stoke equations under the arbitrary Lagrangian–Eulerian description. To do this, the number and numbering of smoothing cells are deployed in line with those of Gaussian points in each four-node quadrilateral element for smoothed Galerkin weak form. In the meantime, CS-FEM is directly applied to nonlinear solid dynamics. A simple post-smoothing algorithm is introduced to enhance the quality of the updated mesh. Nonlinear block-Gauss–Seidel procedure is subsequently employed for the coupling of all smoothed individual fields. Two popular examples are presented to demonstrate the applicability and robustness of the proposed methodology.

An edge-based smoothed finite element method for wave scattering by an obstacle in elastic media

Scattering of a time-harmonic plane elastic wave by a rigid obstacle embedded in an isotropic homogeneous elastic medium has wide applications in science and engineering. This paper presents a novel method to study elastic wave scattering problems satisfying the Navier equation and Helmholtz equations with coupled boundary obtained by Helmholtz decomposition. We derive first smoothed Galerkin weak forms of Navier equation and Helmholtz equations to create effective smoothed finite element method (S-FEM) models. On the top of a three-noded triangular mesh, an edge-based S-FEM (ES-FEM-T3) combining with the perfectly matched layer (PML) technique is then established for solving the elastic wave scattering in bounded domains with PML to eliminate wave reflections. Some numerical experiments demonstrate that ES-FEM-T3 is more stable and accurate than the standard FEM for the elastic wave scattering.

2.5D electric resistivity forward modeling with element-free Galerkin method

An element-free Galerkin method (EFGM) is proposed to perform the direct current (DC) resistivity modeling problems. The advantage of this method is the absence of elements, which makes nodes free from the elemental restraint. In this method, the moving least square (MLS) approximation is employed to construct the shape function. And the cubic spline function with high continuity is used as a weight function. Then, the boundary value problem and the corresponding variation problem of DC resistivity modeling problems are discussed. The discrete equations of two and a half dimensional (2.5D) DC resistivity modeling problems are obtained by using the Galerkin weak form formulation. Thereby, a 1D multilayered geoelectric model is adopted to verify the validity of the proposed EFGM. And the further tests of complicated geoelectric models illustrate that the correctness, good adaptability and flexibility of EFGM compared with the finite element method (FEM). The EFGM can conveniently simulate the undulating terrain and improve the simulation accuracy by refining nodes in local domain. In addition, the discretization of test models shows that EFGM is easy to handle the complicated geometrical geoelectric models by using arbitrary and irregular nodes. All these tests demonstrate that the EFGM is suitable to perform forward modeling for complicated geoelectric models.

A Meshless Method Based on the Nonsingular Weight Functions for Elastoplastic Large Deformation Problems

In this study, based on a nonsingular weight function, the improved element-free Galerkin (IEFG) method is presented for solving elastoplastic large deformation problems. By using the improved interpolating moving least-squares (IMLS) method to form the approximation function, and using Galerkin weak form based on total Lagrange formulation of elastoplastic large deformation problems to form the discretilized equations, which is solved with the Newton–Raphson iteration method, we obtain the formulae of the IEFG method for elastoplastic large deformation problems. In numerical examples, the influences of the penalty factor, scale parameter of influence domain and weight functions on the computational accuracy are analyzed, and the numerical solutions show that the IEFG method for elastoplastic large deformation problems has higher computational efficiency and accuracy.

Interpolating element-free Galerkin method for viscoelasticity problems

In this paper, based on the improved interpolating moving least-square (IMLS) approximation, the interpolating element-free Galerkin (IEFG) method for two-dimensional viscoelasticity problems is presented. The shape function constructed by the IMLS approximation can overcome the shortcomings that the shape function of the moving least-squares (MLS) can-not satisfy the property of Kronecker function, so the essential boundary conditions can be directly applied to the IEFG method. Under a similar computational precision, compared with the meshless method based on the MLS approximation, the meshless method using the IMLS approximation has a high computational efficiency. Using the IMLS approximation to form the shape function and adopting the Galerkin weak form of the two-dimensional viscoelasticity problem to obtain the final discretized equation, the formulae for two-dimensional viscoelasticity problem are derived by the IEFG method. The IEFG method has some advantages over the conventional element-free Galerkin (EFG) method, such as the concise formulae and direct application of the essential boundary conditions, For the IEFG method of two-dimensional viscoelasticity problems proposed in this paper, three numerical examples and one engineering example are given. The convergence of the method is analyzed by considering the effects of the scale parameters of influence domains and the node distribution on the computational precision of the solutions. It is shown that when <i>d</i><sub>max</sub> = 1.01−2.00, the method in this paper has a good convergence. The numerical results from the IEFG method are compared with those from the EFG method and from the finite element method or analytical solution. We can see that the IEFG method in this paper is effective. The results of the examples show that the IEFG method has the advantage in improving the computational efficiency of the EFG method under a similar computational accuracy. And the engineering example shows that the IEFG method can not only has higher computational precision, but also improve the computational efficiency.

The improved element-free Galerkin method based on the nonsingular weight functions for elastic large deformation problems

In this paper, the interpolating moving least-squares (IMLS) method based on a nonsingular weight function is applied to obtain the approximation function. The penalty method is applied to impose the displacement boundary condition, and Galerkin weak form of elastic large deformation problems based on total Lagrange formulation is used to form the final equations which is solved with the Newton–Raphson iteration method, then the improved element-free Galerkin (IEFG) method based on a nonsingular weight function for elastic large deformation problems is presented. The IMLS method can overcome the disadvantage of singular weight functions in the traditional MLS method, then the IEFG method in this paper has high computational accuracy and efficiency, which are shown by numerical examples of elastic large deformation problems. And the influences of the weight functions, scale parameter of influence domain, step number and penalty factor on the numerical results are discussed.