The improved complex variable element-free Galerkin method for the analysis of Kirchhoff plates

In this paper, based on the element-free Galerkin (EFG) method and the improved complex variable moving least-square (ICVMLS) approximation, a new meshless method, which is the improved complex variable element-free Galerkin (ICVEFG) method for two-dimensional potential problems, is presented. In the method, the integral weak form of control equations is employed, and the Lagrange multiplier is used to apply the essential boundary conditions. Then the corresponding formulas of the ICVEFG method for two-dimensional potential problems are obtained. Compared with the complex variable moving least-square (CVMLS) approximation proposed by Cheng, the functional in the ICVMLS approximation has an explicit physical meaning. Furthermore, the ICVEFG method has greater computational precision and efficiency. Three numerical examples are given to show the validity of the proposed method.

Based on the complex variable moving least-square (CVMLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional viscoelasticity problems under the creep condition is presented in this paper. The Galerkin weak form is employed to obtain the equation system, and the penalty method is used to apply the essential boundary conditions, then the corresponding formulae of the CVEFG method for two-dimensional viscoelasticity problems under the creep condition are obtained. Compared with the element-free Galerkin (EFG) method, with the same node distribution, the CVEFG method has higher precision, and to obtain the similar precision, the CVEFG method has greater computational efficiency. Some numerical examples are given to demonstrate the validity and the efficiency of the method.

On the basis of the moving least-squares (MLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional elastic large deformation problems, such as large displacement and large rotation, is presented in this paper. The advantage of the CVMLS approximation is that the trial function of a two-dimensional problem is formed with one-dimensional basis function. Then the CVMLS approximation is applied to solve two-dimensional large deformation problems. The Galerkin weak form is employed to obtain the discretized system equation, the penalty method is used to apply the essential boundary conditions. And then the complex variable element-free Galerkin (CVEFG) method for two-dimensional large deformation problems based on the total lagrange formulation is formed, the corresponding formulae are obtained, and the Newton-Raphson iteration is used in the numerical implementation. Three numerical examples are given to show that this method in this paper is effective for elastic large deformation problems.

The element-free Galerkin (EFG) method is one of the widely used meshfree methods for solving partial differential equations. In the EFG method, shape functions are derived from a moving least-squares (MLS) approximation, which involves the inversion of a small matrix for every point of interest. To avoid the calculation of matrix inversion in the formulation of the shape functions, an improved MLS approximation is presented, where an orthogonal function system with a weight function is used. However, it can also lead to ill-conditioned or even singular system of equations. In this paper, aspects of the IMLS approximation are analyzed in detail. The reason why singularity problem occurs is studied. A novel technique based on matrix triangular process is proposed to solve this problem. It is shown that the EFG method with present technique is very effective in constructing shape functions. Numerical examples are illustrated to show the efficiency and accuracy of the proposed method. Although our study relies on monomial basis functions, it is more general than existing methods and can be extended to any basis functions.

An improved interpolating complex variable moving least squares (IICVMLS) method is proposed for numerical simulations of structures, in which a complete basis function and singular weight function are used to form a new basis function through the orthogonalization process. In this method, a new shape function which has the property of Kronecker [Formula: see text] function is derived to build the interpolating function. Based on the IICVMLS method, an improved interpolating complex variable element free Galerkin (IICVEFG) method is obtained for bending problem of Kirchhoff plates. In the IICVEFG method, the essential boundary conditions can be satisfied directly, and thus the final discrete matrix equation is more concise than that in the non-interpolating complex variable element free Galerkin methods. Hence, the proposed meshless method is more accurate and efficient than conventional complex variable meshless methods. Numerical examples of bending problem of Kirchhoff plates are presented to validate the advantages of the IICVEFG method.

Event-driven sorting algorithm-based Monte Carlo method with neighbour merging method for solving aerosol dynamics

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

The bending problem of free orthotropic rectangular thin plate (RTP) on two‐parameter elastic foundation under a concentrated load is studied by the symplectic superposition method. Firstly, the original bending problem is decomposed into three subproblems by analyzing load effects and boundary conditions, each of which is the bending problem of the plate with two opposite edges slidingly supported. In order to solve the three sub‐problems based on the separation of variables method in Hamiltonian system, the Hamiltonian system for the orthotropic RTP with two opposite edges slidingly supported is studied, and then the eigenvalues and eigenfunctions of the Hamiltonian operator are obtained by combining the separation of variables and symbolic computation. Secondly, according to the symplectic orthogonality and the completeness of the eigenfunctions, the general solution of the Hamiltonian system with the two opposite edges slidingly supported is obtained. Furthermore, the solutions in the form of series of the three subproblems are obtained respectively. Finally, the symplectic superposition solution of the original bending problem is obtained by superposing the solutions of the three subproblems and the correctness of the symplectic superposition solution is verified by two numerical examples.

An improved interpolating element-free Galerkin (IIEFG) method for elastoplasticity is proposed in this paper. In the IIEFG method, the shape functions are constructed by the improved interpolating moving least-squares (IIMLS) method, and the final system equations are obtained by using the Galerkin weak form of elastoplasticity. Compared with the interpolating moving least-squares (IMLS) method, the weight functions are not singular in the IIMLS method, in which the shape functions have the interpolating property. The IIMLS method has fewer unknown coefficients to be solved in the trial functions than the moving least-squares (MLS) approximation. Hence, the IIEFG method is able to directly enforce the displacement boundary condition and obtain numerical solutions with high computational accuracy and efficiency. To show advantages of the IIEFG method, some selected elastoplastic examples are given.

This study aims to interpret the non-linear steady-state heat conduction for temperature-dependent thermal conductivity ($k$ ($T$)) using Element-Free Galerkin (EFG) method. In this present study, a one-dimensional heat conduction problem with uniform heat generation was explicated. Moving Least Squares (MLS) approximants were applied to estimate the unknown function of temperature $T$ ($x$) with $T^h$ ($x$) using linear basis and weight functions. The variational method has been used to develop discrete equations. Essential boundary conditions are enforced by using the Penalty method. The results have been obtained for the one-dimensional model using essential MATLAB codes. The results obtained by the EFG method are compared with the analytical and finite-element method results. The results are also studied by increasing the number of nodes to study the convergence which indicated that EFG has good convergence behavior. The results have also been obtained for different values of the scaling parameter ($α_s$) and any values of αs between 1.8 and 2.0 were found suitable for providing better results in the EFG method.

In this paper, the complex variable reproducing kernel particle method (CVRKPM) for solving the bending problems of isotropic thin plates on elastic foundations is presented. In CVRKPM, one-dimensional basis function is used to obtain the shape function of a two-dimensional problem. CVRKPM is used to form the approximation function of the deflection of the thin plates resting on elastic foundation, the Galerkin weak form of thin plates on elastic foundation is employed to obtain the discretized system equations, the penalty method is used to apply the essential boundary conditions, and Winkler and Pasternak foundation models are used to consider the interface pressure between the plate and the foundation. Then the corresponding formulae of CVRKPM for thin plates on elastic foundations are presented in detail. Several numerical examples are given to discuss the efficiency and accuracy of CVRKPM in this paper, and the corresponding advantages of the present method are shown.

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.

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

In this paper, an improved interpolating moving least-square (IIMLS) method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker δ function. The weight function used in the IIMLS method is nonsingular. Then the IIMLS method can overcome the difficulties caused by the singularity of the weight function in the IMLS method. The number of unknown coefficients in the trial function of the IIMLS method is less than that of the moving least-square (MLS) approximation. Then by combining the IIMLS method with the Galerkin weak form of the potential problem, the improved interpolating element-free Galerkin (IIEFG) method for two-dimensional potential problems is presented. Compared with the conventional element-free Galerkin (EFG) method, the IIEFG method can directly use the essential boundary conditions. Then the IIEFG method has higher accuracy. For demonstration, three numerical examples are solved using the IIEFG method.

• A high-accuracy Deep Neural Network Potential (DNNP) for FeCrV-based Refractory Medium Entropy Alloys (RMEAs) is developed. • The DNNP successfully predicts phase separation, showing good agreement with experimental data. • The important role of Short-Range Ordering (SRO) in influencing dislocation density, yield strength, and mechanical properties of the RMEA is revealed. Refractory medium entropy alloys (RMEAs) have attracted significant attention in recent years due to their exceptional mechanical properties and high-temperature stability, making them suitable for a number of advanced applications. While computational modelling such as density functional theory (DFT) and molecular dynamics (MD) are powerful for investigating RMEAs, these traditional methods are often constrained by high computational cost and limited accuracy. In this work, a deep neural network potential (DNNP) was developed to address the complex compositional nature of FeCr 2 V-based RMEAs with varying levels of tungsten doping. The DNNP demonstrated high accuracy, comparable to that of DFT calculations. Utilizing the DNNP, high-accuracy MD simulations were conducted to examine large-scale effects, including short-range ordering (SRO), twining, and dislocation behaviour, on the mechanical properties of the RMEAs. The results indicate that in the SRO structure, the covalency of V-V, Cr-Cr, and V-W ordered atomic pairs enhances local bonding strength and increases the elastic modulus of the RMEA. As the simulation temperature increases, dislocation mobility improves while dislocation density decreases, thereby enhancing the ductility of the material. Above 823 K, the SRO structure demonstrates superior mechanical performance, which is attributed to the increased length of 1/2<111>, dislocations facilitated by the formation of Cr-Fe and Cr-Cr ordered twins. This work underscores the potential of DNNP and MD simulations in predicting and analyzing the mechanical properties of RMEAs, advancing their development for various applications.