Galerkin Weak Form Research Articles

The dimension splitting element-free Galerkin method for 3D transient heat conduction problems

By transforming a 3D problem into some related 2D problems, the dimension splitting element-free Galerkin (DSEFG) method is proposed to solve 3D transient heat conduction problems. The improved element-free Galerkin (IEFG) method is used for 2D transient heat conduction problems, and the finite difference method is applied in the splitting direction. The discretized system equation is obtained based on the Galerkin weak form of 2D problem; the essential boundary conditions are imposed with the penalty method; and the finite difference method is employed in the time domain. Four exemplary problems are chosen to verify the efficiency of the DSEFG method. The numerical solutions show that the efficiency and precision of the DSEFG method are greater than ones of the IEFG method for 3D problems.

An element-free smoothed radial point interpolation method (EFS-RPIM) for 2D and 3D solid mechanics problems

This paper presents a novel element-free smoothed radial point interpolation method (EFS-RPIM) for solving 2D and 3D solid mechanics problems. The idea of the present technique is that field nodes and smoothing cells (SCs) used for smoothing operations are created independently and without using a background grid, which saves tedious mesh generation efforts and makes the pre-process more flexible. In the formulation, we use the generalized smoothed Galerkin (GS-Galerkin) weak-form that requires only discrete values of shape functions that can be created using the RPIM. By varying the amount of nodes and SCs as well as their ratio, the accuracy can be improved and upper bound or lower bound solutions can be obtained by design. The SCs can be of regular or irregular polygons. In this work we tested triangular, quadrangle, n-sided polygon and tetrahedron as examples. Stability condition is examined and some criteria are found to avoid the presence of spurious zero-energy modes. This paper is the first time to create GS-Galerkin weak-form models without using a background mesh that tied with nodes, and hence the EFS-RPIM is a true meshfree approach. The proposed EFS-RPIM is so far the only technique that can offer both upper and lower bound solutions. Numerical results show that the EFS-RPIM gives accurate results and desirable convergence rate when comparing with the standard finite element method (FEM) and the cell-based smoothed FEM (CS-FEM).

An improved pseudospectral meshless radial point interpolation (PSMRPI) method for 3D wave equation with variable coefficients

In this paper, a pseudospectral meshless radial point interpolation (PSMRPI) technique is applied to the three-dimensional wave equation with variable coefficients subject to given appropriate initial and Dirichlet boundary conditions. The present method is a kind of combination of meshless methods and spectral collocation techniques. The point interpolation method along with the radial basis functions is used to construct the shape functions as the basis functions in the frame of the spectral collocation methods. These basis functions will have Kronecker delta function property, as well as unitary possession. In the proposed method, operational matrices of higher order derivatives are constructed and then applied. The merit of this innovative method is that, it does not require any kind of integration locally or globally over sub-domains, as it is essential in meshless methods based on Galerkin weak forms, such as element-free Galerkin and meshless local Petrov–Galerkin methods. Therefore, computational cost of PSMRPI method is low. Further, it is proved that the procedure is stable with respect to the time variable over some conditions on the 3D wave model, and the convergence of the technique is revealed. These latest claims are also shown in the numerical examples, which demonstrate that PSMRPI provides excellent rate of convergence.

Numerical simulation of a modified anomalous diffusion process with nonlinear source term by a local weak form meshless method

In the current work, a modified anomalous diffusion process in two dimensional space with nonlinear source term is investigated numerically. The process is modeled as a two dimensional nonlinear time-fractional sub-diffusion equation in sense of Riemann–Liouville fractional derivatives. An efficient and accurate computational technique is performed for solving the governing problem. In the proposed method, firstly, a second-order accurate formulation is proposed to discretize the problem in the temporal direction. It has been proved that the proposed time discretization scheme is unconditionally stable. Then a meshfree method based on a combination of the local Petrov–Galerkin weak form and a collocation approach is implemented for spatial discretization. In our implementation both regularly and irregularly distributed field nodes are used to represent the primary spatial domain and its boundary. The radial point interpolation method is used for constructing meshfree shape functions on the distributed field nodes. Some numerical experiments have been carried out to verify the accuracy and performance of the proposed method. The numerical findings demonstrate efficiency and high accuracy of the technique and confirm the theoretical prediction.

A hybrid improved complex variable element-free Galerkin method for three-dimensional advection-diffusion problems

In this paper, combining the dimension splitting method with the improved complex variable element-free Galerkin method, a hybrid improved complex variable element-free Galerkin (H-ICVEFG) method is presented for three-dimensional advection-diffusion problems. Using the dimension splitting method, a three-dimensional advection-diffusion problem is transformed into a series of two-dimensional ones which can be solved with the improved complex variable element-free Galerkin (ICVEFG) method. In the ICVEFG method, the improved complex variable moving least-squares (ICVMLS) approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the one-dimensional direction. And Galerkin weak form of three-dimensional advection-diffusion problems is used to obtain the final discretized equations. Then the H-ICVEFG method for three-dimensional advection-diffusion problems is presented. Numerical examples are provided to discuss the influences of the weight functions, the effects of the scale parameter, the penalty factor, the number of nodes, the step number and the time step on the numerical solutions. And the advantages of the H-ICVEFG method with higher computational accuracy and efficiency are shown.

A peridynamic failure analysis of fiber-reinforced composite laminates using finite element discontinuous Galerkin approximations

This paper presents a discontinuous Galerkin weak form for bond-based peridynamic models to predict the damage of fiber-reinforced composite laminates. To represent the anisotropy of a laminate in a peridynamic model, a lamina is simplified as a transversely isotropic medium under a plane stress condition. The laminated structure is modeled by stacking the surface mesh layers along the thickness direction according to the laminate sequence. To avoid a mesh dependence on either the fiber orientation or the discretization, the spherical harmonic expansion theory is employed to construct a function for the micro-elastic modulus in terms of the bond-fiber angle. The laminate material is decomposed into an isotropic matrix material part and a transversely isotropic material part. The bond stiffness can be evaluated using the engineering material constants, based on the equivalence between the elastic energy density in the peridynamic theory and the elastic energy density in the classic continuum mechanics theory. Benchmark tests are conducted to verify the proposed model. Numerical results illustrate that the convergence of simulations with different horizon sizes and meshes can be achieved. In terms of damage analysis, the proposed model can capture the dynamic process of the complex coupling of the inner-layer and delamination damage modes.

An edge-based smoothed finite element method for nonlinear magnetostatic and eddy current analysis

Abstract In this work, an edge-based smoothed finite element method (ES-FEM) is presented to solve electromagnetic field problems. First, the analysis domain is discretized into a set of tetrahedron cells that can be much easily generated automatically for complicated domains, and edge-based smoothing domains are further constructed associated with edges of tetrahedron elements. The Smoothed Galerkin Weak form is then evaluated based on edge-based smoothing domains. The ES-FEM is implemented here to solve nonlinear magnetostatic and eddy current problems, and to observe its properties using linear tetrahedral elements, in which relatively poor result is carried with standard finite element method (FEM). Numerical examples demonstrate that results obtained by the present ES-FEM are much more accurate than ones by standard FEM, and agree well with the experimental ones. It possesses potentials in the successful applications of electromagnetic problems.

Analysis of heat transfer problems using a novel low-order FEM based on gradient weighted operation

In this work, a novel numerical method is formulated to improve the accuracy and efficiency of the finite element method (FEM) when using low-order elements in analysis of heat transfer problems. In this method, the problem domain is discretized using linear triangular or tetrahedral elements. For each independent element, a supporting domain that consists of element itself and its adjacent elements sharing common edges/faces is constructed. Based on the Shepard interpolation, a weighted temperature gradient is then obtained, which will be considered to the generalized Galerkin weak form to form the discretized system equations. In present method, gradient weighted coefficients αi are introduced to reconstruct the temperature gradient field. And the key idea of present method is just the temperature gradient reconstruction through gradient weighted operation. A series of numerical examples are studied to examine the performance of present method in heat transfer analysis. Results show that, with a proper value of αi, present method can achieve much better accuracy, higher convergence rate and is more efficient compared with the standard FEM.

High precision solution for thermo-elastic equations using stable node-based smoothed finite element method

The stable node-based smoothed finite element method (SNS-FEM) is presented to analyze thermo-elastic equations with various boundary conditions. The node-based smoothing domains are constructed to implement numerical integrations and then, the smoothed Galerkin weak form is utilized to obtain the discretized system equations. In the formulation, both the smoothed strains and the strain variance items over the node-based smoothing domains are used to perform the integration. Several numerical examples are studied in detail to investigate the performance of the SNS-FEM by comparison with the original NS-FEM and traditional FEM. Results show that the present SNS-FEM can provide very high precision solution for thermo-elastic equations. In addition, the SNS-FEM is more efficient than original NS-FEM and the standard FEM.

Stable finite element analysis of viscous dusty plasma

PurposeThe purpose of this paper is to provide an analysis of dust acoustic (solitary) waves including viscosity. Specifically, the authors consider a dusty unmagnetized plasma system consisting of negatively charged dust and Boltzmann electrons and ions.Design/methodology/approachIn this paper, a Petrov–Galerkin weak form with upwinding is adopted. Nonlinearity of ion and electron number density in terms of an electrostatic potential is included. A fully implicit time integration is used (backward Euler method), which requires the first derivative of the weak form. A three-field formulation is proposed, with the dust number density, the electrostatic potential and the dust velocity being the unknown fields.FindingsIn this study, two numerical examples are introduced and results show great promise for the proposed formulation as a predictive tool in viscous dusty plasmas. Presence of solitary waves was demonstrated. Dusty plasma vortices are predicted in 2D and 3D, as mentioned in the specialized literature.Research limitations/implicationsWe observed some dependence on step size, which is due to the simple time-stepping scheme. This can be solved with a higher order integration scheme, which implies an added cost to the solution.Practical implicationsDusty plasmas are found in astrophysics (Saturn rings) and electronics industry at several scales and have high impact as a contaminant.Originality/valueTo the authors’ knowledge, this is the first paper with a simulation of dusty plasma including vortices.

An efficient size-dependent computational approach for functionally graded isotropic and sandwich microplates based on modified couple stress theory and moving Kriging-based meshfree method

We present a non-classical model for bending, free vibration and buckling analyses of functionally graded (FG) isotropic and sandwich microplates based on the modified couple stress theory (MCST) and the refined higher order shear deformation theory. Unlike the classical higher order shear deformation theory, the present model retains four variables and contains a single material length scale parameter which can be considered as the size-dependent effect. Effective material properties of FG isotropic and sandwich microplates are calculated by a rule of mixture. The discrete system equations are derived from the Galerkin weak form and are then solved using the moving Kriging meshfree method. Due to satisfying the Kronecker delta function property of moving Kriging integration (MKI) shape function, essential boundary conditions are directly enforced by the same way as the standard finite element method. In addition, a simple rotation-free technique originated from isogeometric analysis is used to eliminate the slopes in clamped plates. The effects of material length scale parameter, volume fraction, geometrical parameters and boundary conditions are investigated to conduct the effectiveness of the present approach.

A moving Kriging meshfree method with naturally stabilized nodal integration for analysis of functionally graded material sandwich plates

This paper presents a moving Kriging meshfree method based on a naturally stabilized nodal integration (NSNI) for bending, free vibration and buckling analyses of isotropic and sandwiched functionally graded plates within the framework of higher-order shear deformation theories. A key feature of the present formulation is to develop a NSNI technique for the moving Kriging meshfree method. Using this scheme, the strains are directly evaluated at the same nodes as the direct nodal integration (DNI). Importantly, the computational approach alleviates instability solutions in the DNI and significantly decreases the computational cost from using the traditional high-order Gauss quadrature. Being different from the stabilized conforming nodal integration scheme which uses the divergence theorem to evaluate the strains by boundary integrations, the NSNI adopts a naturally implicit gradient expansion. The NSNI is then integrated into the Galerkin weak form for deriving the discrete system equations. Due to satisfying the Kronecker delta function property of the moving Kriging integration shape function, the enforcement of essential boundary conditions in the present method is similar to the finite element method. Through numerical examples, the effects of geometries, stiffness ratios, volume fraction and boundary conditions are studied to prove the efficiency of the present approach.

Analyzing the nonlinear p-Laplacian problem with the improved element-free Galerkin method

In this paper, the improved element-free Galerkin (IEFG) method is developed for the numerical analysis of the nonlinear p-Laplacian problem. In this method, the improved moving least squares approximation is adopted to form meshless shape functions, the Galerkin weak form is used to derive the nonlinear system of algebraic equations, and the subroutine ‘fsolve’ of MATLAB is adopted to deal with the strong nonlinearity. The computational formulas of the IEFG method for the p-Laplacian problem are obtained in detail. Numerical results show that the IEFG method for the p-Laplacian problem is convergent and has high computational accuracy.

A parameter identification method for continuous-time nonlinear systems and its realization on a Miura-origami structure

Abstract Many mechanical systems are nonlinear and often high-dimensional. Constructing accurate models for continuous-time nonlinear systems calls for effectively identifying their parameters, whereas measurement noise and sensitivity to initial conditions make the identification challenging. This paper proposes a new parameter identification method for ordinary differential equations based on the idea of B-Spline Galerkin finite element. In this approach, the system’s solution is globally constructed by a set of B-Splines. With Galerkin weak formulation, instead of taking analytical derivatives on basis functions, the differential terms are eliminated through integration by parts so that the measurement noise will not be amplified. Then least square algorithms can be adopted for solving the optimization problem to estimate the parameters. By solving two intractable testbed problems, the coupled Chua’s circuits and the Tank reactor equations, we show that the new approach is effective and efficient in dealing with systems with high-dimensionality, complex nonlinearity, discontinuous input and output, and noisy data without specific pre-processing. In addition, this method is employed to identify the geometrical and mechanical parameters of a Miura-origami structure under base excitation, which possesses complex global nonlinearity, exhibits chaotic responses, and suffers from significant measurement noise. The proposed method gains success in dealing with this system; based on the identified parameters, the corresponding constituent force-displacement relation and the simulation results agree well with the experiments.

Analyzing wave propagation problems with the improved complex variable element-free Galerkin method

Two-dimensional wave propagation problems in this paper are analyzed with improved complex variable element-free Galerkin (ICVEFG) method. Improved complex variable moving least-squares (ICVMLS) approximation is applied to construct the shape function, and modified Galerkin weak form of wave propagation problems is used for obtaining the final system equations, then all equations of ICVEFG method for wave propagation problems are obtained. The convergences of ICVEFG method about node distribution and time step are analyzed, and the influences of weight functions, scale parameter of the influence domain and penalty factor on the numerical solutions are discussed. Example problems show that ICVEFG method has higher computational speed with higher precision.

Multiscale cohesive zone modeling and simulation of high-speed impact, penetration, and fragmentation

In this work, a multiscale cohesive zone model (MCZM) is developed to simulate the high-speed penetration induced dynamic fracture process such as fragmentation in crystalline solids. This model describes bulk material as a local quasi-continuum medium which follows the Cauchy–Born rule while cohesive zone element is governed by an interface depletion potential, such that the cohesive zone constitutive descriptions are genetically consistent with that of bulk element. This multiscale method proved to be effective in describing material inhomogeneities and it is constructed and implemented in a cohesive finite element Galerkin weak formulation. Numerical simulations of high-speed penetration with different shape of penetrators, i.e., square, circle and parabola nose penetrators are performed. Results show that the proposed MCZM can successfully capture spall fracture, the penetration process and different characteristics of fragmentation under different shape of penetrators.

Numerical treatment for solving two-dimensional space-fractional advection–dispersion equation using meshless method

Space-fractional advection–dispersion equation (SFADE) can describe particle transport in a variety of fields more accurately than the classical models of integer-order derivative. Because of nonlocal property of integro-differential operator of space-fractional derivative, it is very challenging to deal with fractional model, and few have been reported in the literature. In this paper, a numerical analysis of the two-dimensional SFADE is carried out by the element-free Galerkin (EFG) method. The trial functions for the SFADE are constructed by the moving least-square (MLS) approximation. By the Galerkin weak form, the energy functional is formulated. Employing the energy functional minimization procedure, the final algebraic equations system is obtained. The Riemann–Liouville operator is discretized by the Grünwald formula. With center difference method, EFG method and Grünwald formula, the fully discrete approximation schemes for SFADE are established. Comparing with exact results and available results by other well-known methods, the computed approximate solutions are presented in the format of tables and graphs. The presented results demonstrate the validity, efficiency and accuracy of the proposed techniques. Furthermore, the error is computed and the proposed method has reasonable convergence rates in spatial and temporal discretizations.

Shear deformable plate elements based on exact elasticity solution

The 2-D approximation functions based on a general exact 3-D plate solution are used to derive locking-free, rectangular, 4-node Mindlin (i.e., first-order plate theory), Levinson (i.e., a third-order plate theory), and Full Interior plate finite elements. The general plate solution is defined by a biharmonic mid-surface function, which is chosen for the thick plate elements to be the same polynomial as used in the formulation of the well-known nonconforming thin Kirchhoff plate element. The displacement approximation that stems from the biharmonic polynomial satisfies the static equilibrium equations of the 2-D plate theories at hand, the 3-D Navier equations of elasticity, and the Kirchhoff constraints. Weak form Galerkin method is used for the development of the finite element model, and the matrices for linear bending, buckling and dynamic analyses are obtained through analytical integration. In linear buckling problems, the 2-D Full Interior and Levinson plates perform particularly well when compared to 3-D elasticity solutions. Natural frequencies obtained suggest that the optimal value of the shear correction factor of the Mindlin plate theory depends primarily on the boundary conditions imposed on the transverse deflection of the 3-D plate used to calibrate the shear correction factor.

A cell-based smoothed finite element method for multi-body contact analysis using linear complementarity formulation

In this paper, the cell-based smoothed finite element using quadrilateral elements (CS-FEM) is used for 2D contact problems which are converted into linear complementarity problems (LCPs), which can be solved efficiently using the Lemke method. The modified Coulomb friction contact model with tangential strength and normal adhesion is considered, which models sticking-slipping, contacting-departing, and bonding-debonding processes, in a unified formulation. Smoothed Galerkin weak-form with contact boundary is deduced, in which the stiffness is implemented using the CS-FEM with 1 smoothing domain (1SD), 2SD, 3SD, 4SD, 8SD, and 16SD for each element. Contact interface equations are discretized through contact point-pairs that are constructed using a master-slave surface algorithm. Intensive numerical examples are given to investigate the effects of contact parameters on contact behaviors and examine the effectiveness of the proposed approach. The numerical results of CS-FEM models are compared with that of FEM-Q4 model, which demonstrates that all CS-FEM models are softer than FEM-Q4 model. The strain energy solutions, obtained using several CS-FEM models, are monotonically decreasing with the number of the SDs for each element increasing. The upper bound solutions in strain energy can be obtained using a CS-FEM-1SD model in our examples, while the lower bound solutions are obtained using CS-FEM-16SD model or FEM-Q4, with FEM-Q4 solution being the lowest.

The dimension split element-free Galerkin method for three-dimensional potential problems

This paper presents the dimension split element-free Galerkin (DSEFG) method for three-dimensional potential problems, and the corresponding formulae are obtained. The main idea of the DSEFG method is that a three-dimensional potential problem can be transformed into a series of two-dimensional problems. For these two-dimensional problems, the improved moving least-squares (IMLS) approximation is applied to construct the shape function, which uses an orthogonal function system with a weight function as the basis functions. The Galerkin weak form is applied to obtain a discretized system equation, and the penalty method is employed to impose the essential boundary condition. The finite difference method is selected in the splitting direction. For the purposes of demonstration, some selected numerical examples are solved using the DSEFG method. The convergence study and error analysis of the DSEFG method are presented. The numerical examples show that the DSEFG method has greater computational precision and computational efficiency than the IEFG method.