Dynamic analysis of non-symmetric FG cylindrical shell under shock loading by using MLPG method

This study introduces a deep neural network approach that utilizes radial basis functions (RBFs) to solve forward and inverse problems in the process of diffusion. The input layer incorporates multiquadric (MQ) RBFs, symbolizing the radial distance between the boundary points on the spacetime boundary and the source points positioned outside the spacetime boundary. The output layer is the initial and boundary data given by analytical solutions of the diffusion equation. Utilizing the concept of the spacetime coordinates, the approximations for forward and backward diffusion problems involve assigning initial data on the bottom or top spacetime boundaries, respectively. As the need for discretization of the governing equation is eliminated, our straightforward approach uses only the provided boundary data and MQ RBFs. To validate the proposed method, various diffusion scenarios, including forward, backward, and inverse problems with noise, are examined. Results indicate that the method can achieve high-precision numerical solutions for solving diffusion problems. Notably, only 1/4 of the initial and boundary conditions are known, yet the method still yields precise results.

Local radial basis function-based differential quadrature (RBF-DQ) method was recently proposed by us. The method is a natural mesh-free approach. Like the conventional differential quadrature (DQ) method, it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots, which may be distributed randomly. However, different from the conventional DQ method, the weighting coefficients in present method are determined by taking the radial basis functions (RBFs) instead of high order polynomials as the test functions. The method works in a similar fashion as conventional finite difference schemes but with truly mesh-free property. In this paper, we mainly concentrate on the multiquadric (MQ) radial basis functions since they have exponential convergence. The effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied, and how the value of optimal c varies with the number of local support knots is also numerically demonstrated. The proposed method is validated by its application to solve incompressible Navier-Stokes equations. Excellent numerical results are obtained on an irregular knot distribution.

In this paper, based on the moving Kriging interpolation (MKI), the meshless interpolating local Petrov–Galerkin (ILPG) method is presented to solve two- and three-dimensional potential problems. In the present method, the shape function constructed by the MKI has the property of the Kronecker δ function. Then in the ILPG method the essential boundary conditions can be implemented directly. The discrete equations are obtained using the local symmetric weak form. The Heaviside step function is used as the test function in each sub-domain to avoid some domain integral in the symmetric weak form, which will greatly improve the effectiveness of the present method. The ILPG method in this paper is a truly meshless method, which does not require a mesh either for obtaining shape function or for numerical integration in the local weak form. Several numerical examples of potential problems show that the ILPG method has higher computational efficiency and convergence rate than the MLPG method.

Elastic buckling analysis of imperfect FGM cylindrical shells under axial compression in thermal environments is carried out, using two different models for geometrical imperfections. The material properties of the functionally graded shell are assumed to vary continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituent materials, also temperature dependency of the material properties is considered. Derivation of equations is based on classical shell theory using the Sanders nonlinear kinematic relations. The stability and compatibility equations for the imperfect FGM cylindrical shell are obtained, and the buckling analysis of shell is carried out using Galerkin’s method. The novelty of the present work is to obtain closed form solutions for critical buckling loads of the imperfect FGM cylindrical shells, which may be easily used in engineering design applications. The effects of shell geometry, volume fraction exponent, magnitude of initial imperfections, and environment temperature on the buckling load are investigated. The results reveal that initial geometrical imperfections and temperature dependency of the material properties play major roles in dictating the bifurcation point of the functionally graded cylindrical shells under the action of axial compressive loads. Also results show that for a particular value of environment temperature, critical buckling load is almost independent of volume fraction exponent.

In this study numerical solutions for the compressible fluid flow around obstacles, based on a meshless method with Radial Basis Functions (RBFs) are presented. The numerical solutions are obtained by a hybrid method that uses the first step of the boundary element method, namely a singular boundary integral equation, which is an equivalent mathematical model for the problem of the compressible fluid around an obstacle, and then a meshless method based on RBFs for solving this singular boundary integral equation. Two types of RBFs are used, namely the Multiquadric RBFs and Gauss-type RBFs. An analytical check is made in order to study the numerical solutions' accuracy. Using computer simulations, we compare our results with an analytical solution available for a specific case. We examine how different parameters used to obtain the numerical solutions affect their accuracy, and we compare these numerical solutions to determine which RBF is most suitable for this scenario. We also note that the shape parameters of RBFs have a great influence on numerical solutions accuracy and, through simulations, optimal values are found. For evaluating the integrals of singular kernels which appear, the truncation method is applied. The influence of the parameter used to evaluate these integrals is also highlighted in the paper.

The radial basis function (RBF) method, especially the multiquadric (MQ) function, was introduced in solving linear integral equations. The procedure of MQ method includes that the unknown function was firstly expressed in linear combination forms of RBFs, then the integral equation was transformed into collocation matrix of RBFs, and finally, solving the matrix equation and an approximation solution was obtained. Because of the superior interpolation performance of MQ, the method can acquire higher precision with fewer nodes and low computations which takes obvious advantages over thin plate splines (TPS) method. In implementation, two types of integration schemes as the Gauss quadrature formula and regional split technique were put forward. Numerical results showed that the MQ solution can achieve accuracy of1E-5. So, the MQ method is suitable and promising for integral equations.

Meshless and wavelets based complex quadrature of highly oscillatory integrals and the integrals with stationary points

In this study, a numerical meshless method is used to solve the weak form of the linear elastic equations in solid mechanics. Evaluation and comparison of the numerical meshless methods have been carried out via the radial point interpolation meshless method with multi-quadrics base functions (MQ-RPIM) and meshless local Petrov-Galerkin method (MLPG). Using these two methods, stress intensity factors in an elastic medium containing geometric discontinuities and cracks are estimated based on tensile and bending cyclic loading. The analysis domain has been identified via three-dimensional modeling of the notched and un-notched shafts with an initial surface semi-elliptical crack subjected to tensile or bending cyclic loadings. To enhance the accuracy of calculations, the RPIM meshless method is applied using polynomial and extended-enriched 3D base functions. Shape functions have been developed using standard and optimal parameters and values with Mono-Objective Function in PSO algorithm. In the MLPG meshless method with the extended-enriched functions, discretization is performed via direct and penalty factor methods, to reach more efficient results and meet the boundary conditions. Efficiency comparison of the selected numerical methods with the experimental findings and the numerical analysis of finite elements method indicates that in comparison with the MLPG method, MQ-RPIM enriched meshless method can be utilized with fewer nodes in the analysis domain while reaching the accuracy and convergence with lower stress intensity factors and gentler slope. However, the processing time of the MLPG meshless method is lower than that of the other methods.

Radial basis function (RBF) interpolation algorithm for data transfer on coupling interface is derived from the shape function, and the global D-Lanczos (GLDL) method is introduced to solve the displacement transfer matrix. In order to improve the computational efficiency, a local compactly supported RBF (CSRBF) interpolation algorithm is proposed based on the partition acceleration treatment, and a comparison of the algorithmic complexity is made among the global Multi-Quadrics (MQ), CSRBF and the local CSRBF. Numerical test shows the significant shortening of CPU time by using the local CSRBF interpolation algorithm.

The study introduces the formulation of the radial basis function-based finite element method (RBF-FEM) for bending, free vibration, and buckling analysis of Mindlin-Reissner laminated composite plates by first-order shear deformation theory. The method utilizes the radial basis functions to construct the shape functions from the finite nodes. Compared with the conventional finite element method (FEM), the present RBF-FEM obtains the shape functions with simplicity, especially when the number of element nodes is increasing. The proposed RBF-FEM eight-node element (RBF-FEM-8) is utilized. By applying the Gauss point reduction technique, the method could solve the thick and thin laminated composite plates with high accuracy, quick convergence, reducing mesh size in discretization, and could remove the shear locking phenomenon. Several numerical example studies of the thick and thin Mindlin-Reissner plates with the difference in geometries (including line and curve boundary edges), oriented layers, edge boundary conditions, Young’s modulus ratio, length-to-thickness ratio, are carried out. The results are then compared with other numerical methods and analytical solutions.

A global particular solution meshless approach for the four-sided lid-driven cavity flow problem in the presence of magnetic fields

Based on the multi-quadrics (MQ) radial basis function, the two-step discretization method will be used to solve Allen-Cahn (AC) equation in this paper. In the first step, we will use the Backward Euler Formula to solve the AC equation with Newton iterative, and then apply the global RBF method to solve the Semi discrete linearized problem on a small number of collocation points or on a coarse mesh. In the second step, finite difference (FD) and radial basis function-finite difference (RBF-FD) methods will be used to solve the problem on a fine mesh or on a large number of collocation points, respectively. The two-step method will be proved to be more feasible and efficient by some numerical examples. In addition, compared with FD method, the application of RBF-FD method will improve the accuracy of the numerical solution in the second step.

The implementation of the conforming radial point interpolation method (CRPIM) for spatial thick shell structures is presented in this paper. The formulation of the discrete system equations is derived from a stress-resultant geometrically exact theory of shear flexible shells based on the Cosserat surface. A discrete singularity-free mapping between the five degrees of freedom of the Cosserat surface and the normal formulation with six degrees of freedom is constructed by exploiting the geometry connection between the orthogonal group and the unit sphere. A radial basis function is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometries. The major advantage of the CRPIM is that the shape functions possess a delta function property and the interpolation function obtained passes through all the scattered points in the influence domain. Thus, essential boundary conditions can be easily imposed, as in finite element method. A range of shape parameters is studied to examine the performance of CRPIM for shells, and optimal values are proposed. The phenomena of shear locking and membrane locking are illustrated by presenting the membrane and shear energies as fractions of the total energy. Several benchmark problems for shells are analyzed to demonstrate the validity and efficiency of the present CRPIM. The convergence rate of the results using a Gaussian (EXP)radial basis is relatively high compared to those using a multi-quadric (MQ) radial basis for the shell problems.

Natural convection heat transfer within trapezoidal cavity has been studied numerically. The present study is attended to reviewed heat and fluid flow patterns for natural convection in a trapezoidal cavity with uniformly heated bottom wall. The left and right walls are maintained at uniform cold temperature and the top wall is assumed to be thermally insulated. The mathematical model is conducted by the mass, momentum and energy equations and solved by applying multiquadrics radial basis function (MQRBF) method. The MQRBF method is extended using an improved primitive variable formulation to solve the governing equations. The method was based on implicit Euler scheme for temporal discretization and MQRBF method for spatial discretization. The effects of Rayleigh number on the flow and heat transfer process in the trapezoidal cavity are investigated. Stream function, isothermal patterns and the Nusselt number are presented for the Rayleigh numbers varied as 103, 104 and 105. Comparing the numerical results of a test case obtained using the present method with those obtained using finite element and finite volume method shows very good agreement.

This paper presents a new method to construct response surface function and a new hybrid optimization method. For the response surface function, the radial basis function is used for a zeroth-order approximation, while new bases is proposed for the moving least squares method for a first-order approximation. For the new hybrid optimization method, the gradient-based algorithm and pattern search algorithm are integrated for robust and efficient optimization process. These methods are based on: (1) multi-point approximations of the objective and constraint functions; (2) a multi-quadric radial basis function for the zeroth-order function representation or radial basis function plus polynomial based moving least squares approximation for the first-order function approximation; and (3) a pattern search algorithm to impose a descent condition. Several numerical examples are presented to illustrate the accuracy and computational efficiency of the proposed method for both function approximation and design optimization. The examples for function approximation indicate that the multi-quadric radial basis function and the proposed radial basis function plus polynomial based moving least squares method can yield accurate estimates of arbitrary multivariate functions. Results also show that the hybrid method developed provides efficient and convergent solutions to both mathematical and structural optimization problems.