Discrete Least Squares Meshless Research Articles

Simplified numerical method for nonlocal static and dynamic analysis of a graphene nanoplate

In this paper, the dynamic analysis of single-layer rectangular armchair graphene nanoplates has been considered. The theory of nonlocal elasticity for small scale effects and the Kirchhoff's theory for plates have been used to obtain the dynamic equation of graphene nanoplates. Discrete Least Squares Meshless (DLSM) method has been used to examine the response of single-layer graphene nanoplates with various boundary conditions. The validation of the results has also been carried out using dynamic analysis of single-layer rectangular armchair graphene nanoplates under stationary loads. Results revealed that DLSM method is an efficient mean to solve the problems of structural mechanics in nano-dimensions. In addition, in nanostructures, the small scale effects have considerable impacts that should be considered as well. An increase in nonlocal coefficient increases the deflection. Higher nonlocal coefficient leads to higher deflection intensity and vibration amplitude.

Numerical solution of bed load transport equations using discrete least squares meshless (DLSM) method

The discrete least squares meshless (DLSM) method is extended in this paper for solving coupled bedload sediment transport equations. The mathematical formulation of this model consists of shallow water equations for the hydrodynamical component and an Exner equation expressing sediment continuity for the bedload transport. This method uses the moving least squares (MLS) function approximation to construct the shape functions and the minimizing least squares functional method to discretize the system of equations. The method can be viewed as a truly meshless method as it does not need any mesh for both field variable approximation and the construction of system matrices; it also provides the symmetric coefficient matrix. In the present work, several benchmark problems are studied and compared with the work of other researchers; the proposed method shows good accuracy, high convergence rate, and high efficiency, even for irregularly distributed nodes. At the end, a real test problem is performed to show and verify the main benefit and applicability of the proposed method to cope with complex geometry in practical problems.

Optimum two-dimensional crack modeling in discrete least-squares meshless method by charged system search algorithm

In this paper, a node adaptive rearrangement is presented based on the estimated error in various domains for some problems in the fracture mechanics by Discrete Least-Squares Meshless method (DLSM). This method is one of the approximate methods recently introduced and used in the various elds. The method is based on minimization of the least-squares functional with respect to the nodal parameters, and it uses moving least-squares method for calculating the shape functions. Due to the natural process of problem solving, after calculating the shape functions, the residuals are calculated and their values are considered as an objective function for rearrangement of the nodes. There are three popular methods for constructing shape functions in discontinuous domains, and here, the transparency method is utilized. Similar to other numerical methods, there are di erent procedures for re nement and improvement of the results; however, adaptive rearrangement can be employed without increasing the computational cost. In this paper, the Charged System Search (CSS) algorithm is used as a tool for adaptive rearrangement or repositioning process. Eciency and e ectiveness of the proposed adaptive rearrangement technique is tested by some benchmark two-dimensional crack examples with available analytical solution around crack tips.

Using micro-planes concrete damage model in discrete least squares meshless method for predicting the crack growth

In the present study, the author’s previously developed micro-planes concrete damage model has been implemented in the Discrete Least Squares Meshless method (DLSM) for assessment of the crack performance in concrete domains. DLSM is a true meshless method needing no partition in the local approximation as well as in the assembling of the local approximations in order to obtain a global solution. Instead it uses a collection of the un-structured nodal points in place of the elements and discretizes the governing differential equations with the discrete least squares approach. This can reduce considerably the pre-processing efforts of the calculations. However, DLSM like other meshless methods loses its efficiency when it is applied for the analysis of the solid domains dealing with the growing cracks. In recent years, some researchers have utilized some techniques to overcome this problem such as visibility, transparency and diffraction for precisely detecting the crack discontinuity in the case of a static non-growing crack, but these methods require some reciprocal corrective operations within each step of the calculation process extending the time of processing. In this study, however, a new, straightforward and general applicable method has been used for accommodating this issue using the advantages of the Moving Least Squares (MLS) shape functions constructed based on the Voronoi tessellation algorithm and micro-planes concrete damage constitutive model. Voronoi based MLS shape functions have been implemented for evaluating the crack effects accurately as a discontinuity during the analysis processing and micro-planes concrete damage approach used for proper detecting the crack growth trajectory. In the proposed method, the domain of interest is divided into the Voronoi cells based on the classical Voronoi tessellation approach, a cell for each node, and then the support domain of each node located inside the domain as well as on or near the crack faces is determined based on a neighboring criterion. During each step of the analysis for all the integration points, damages are computed according to the micro-planes concrete damage model and the crack path for the next step will be defined. After the end of the iterations in each step, both the stiffness tensor and internal load vector are modified based on the final stress fields obtained on and near the crack. The accuracy and efficiency of the proposed method have been investigated by solving some benchmark examples and comparing the obtained results with the analytical or valid finite element analyses’ results.

Error Estimate and Adaptive Refinement in Mixed Discrete Least Squares Meshless Method

The node moving and multistage node enrichment adaptive refinement procedures are extended in mixed discrete least squares meshless (MDLSM) method for efficient analysis of elasticity problems. In the formulation of MDLSM method, mixed formulation is accepted to avoid second-order differentiation of shape functions and to obtain displacements and stresses simultaneously. In the refinement procedures, a robust error estimator based on the value of the least square residuals functional of the governing differential equations and its boundaries at nodal points is used which is inherently available from the MDLSM formulation and can efficiently identify the zones with higher numerical errors. The results are compared with the refinement procedures in the irreducible formulation of discrete least squares meshless (DLSM) method and show the accuracy and efficiency of the proposed procedures. Also, the comparison of the error norms and convergence rate show the fidelity of the proposed adaptive refinement procedures in the MDLSM method.

Adaptive simulation of free surface flows with discrete least squares meshless (DLSM) method using a posteriori error estimator

PurposeThe purpose of this paper is to propose an adaptive refinement strategy based on a posteriori error estimate for the efficient simulation of free surface flows using discrete least squares meshless (DLSM) method.Design/methodology/approachA pressure projection method is employed to discretize the governing equations of mass and momentum conservation in a Lagrangian form. The semi‐discretized equations are then discretized in space using the DLSM method, in which the sum of squared residual of the governing equations and their boundary conditions are minimized with respect to the unknown nodal parameters.FindingsSince the position of the free surface is of great significant in free surface problems, a posteriori error estimator which automatically associates higher error to the nodes near the free surface is proposed and used along with a node moving refinement strategy to simulate the free surface problems more efficiently. To test the ability and efficiency of the proposed adaptive simulation method, two test problems, namely dam break and evolution of a water bubble, are solved and the results are presented and compared to those of analytical and experimental results.Originality/valueError estimate and adaptive refinement have been mostly used in confined and steady‐state flow. Here in this paper, a new attempt has been made to use these concepts in moving boundary problem.

Corrected discrete least-squares meshless method for simulating free surface flows

Abstract In this paper, a modified version of discrete least-squares meshless (DLSM) method is used to simulate free surface flows with moving boundaries. DLSM is a newly developed meshless approach in which a least-squares functional of the residuals of the governing differential equations and its boundary conditions at the nodal points is minimized with respect to the unknown nodal parameters. The meshless shape functions are also derived using the Moving Least Squares (MLS) method of function approximation. The method is, therefore, a truly meshless method in which no integration is required in the computations. Since the second order derivative of the MLS shape function are known to contain higher errors compared to the first derivative, a modified version of DLSM method referred to as corrected discrete least-squares meshless (corrected DLSM) is proposed in which the second order derivatives are evaluated more accurately and efficiently by combining the first order derivatives of MLS shape functions with a finite difference approximation of the second derivatives. The governing equations of fluid flow (Navier–Stokes) are solved by the proposed method using a two-step pressure projection method in a Lagrangian form. Three benchmark problems namely; dam break, underwater rigid landslide and Scott Russell wave generator problems are used to test the accuracy of the proposed approach. The results show that proposed corrected DLSM can be employed to simulate complex free surface flows more accurately.

A node enrichment adaptive refinement in Discrete Least Squares Meshless method for solution of elasticity problems

In this paper, an adaptive refinement procedure is proposed to be used with Discrete Least Squares Meshless (DLSM) method for accurate solution of planar elasticity problems. DLSM method is a newly introduced meshless method based on the least squares concept. The method is based on the minimization of a least squares functional defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at nodal points used to discretize the domain and its boundaries. A Moving Least Square (MLS) method is used to construct the shape function making the approach a fully least squares based approach. An error estimate and adaptive refinement strategy is proposed in this paper to increase the efficiency of the DLSM method. For this, a residual based error estimator is introduced and used to discover the region of higher errors. The proposed error estimator has the advantages of being available at the end of each analysis contributing to the efficiency of the proposed method. An enrichment method is then used by adding more nodes to the area of higher errors as indicated by the error estimator. A Voronoi diagram is used to locate the position of the nodes to be added to the current nodal configuration. Efficiency and effectiveness of the proposed procedure is examined by adaptively solving two benchmark problems. The results show the ability of the proposed strategy for accurate simulation of elasticity problems.

Steady‐state solution of incompressible Navier–Stokes equations using discrete least‐squares meshless method

AbstractA fractional step method for the solution of the steady state incompressible Navier–Stokes equations is proposed in this paper in conjunction with a meshless method, named discrete least‐squares meshless (DLSM). The proposed fractional step method is a first‐order accurate scheme, named semi‐incremental fractional step method, which is a general form of the previous first‐order fractional step methods, i.e. non‐incremental and incremental schemes. One of the most important advantages of the proposed scheme is its capability to use large time step sizes for the solution of incompressible Navier–Stokes equations. DLSM method uses moving least‐squares shape functions for function approximation and discrete least‐squares technique for discretization of the governing differential equations and their boundary conditions. As there is no need for a background mesh, the DLSM method can be called a truly meshless method and enjoys symmetric and positive‐definite properties. Several numerical examples are used to demonstrate the ability and the efficiency of the proposed scheme and the discrete least‐squares meshless method. The results are shown to compare favorably with those of the previously published works. Copyright © 2010 John Wiley & Sons, Ltd.

Node moving adaptive refinement strategy for planar elasticity problems using discrete least squares meshless method

In this paper, an error indicator and adaptive refinement procedure in conjunction with the Discrete Least Squares Meshless (DLSM) method is presented for the effective and efficient analysis of planar elasticity problems. The DLSM method is a truly meshless method, which has been used for the solution of different problems ranging from solid to fluid mechanics problems. The method is based on the minimization of the least squares functional with respect to the nodal parameters. The least squares functional is formed as the weighted summation of the residual of the differential equation and its boundary condition. The DLSM method enjoys from providing a natural error indicator defined as the value of the functional at nodal points, which can be efficiently used to identify zones of larger numerical errors. The proposed error indicator has the additional advantage of easy computation since most of its components are already available from the main DLSM computation for analysis. Since both the approximation method and discretization method are meshless, repositioning of the nodes can be easily exploited for refining the numerical solution without any geometrical difficulties usually encountered in mesh based methods. Here, a node moving strategy based on spring analogy is proposed to displace the nodal points to the areas indicated by the higher values of the error indicator. A Voronoi diagram is used to identify neighboring nodes that should be connected to each other using springs. The efficiency and effectiveness of proposed adaptive refinement technique is tested on some benchmark examples with available analytical solutions and the results are presented. The results show that the proposed adaptive refinement technique is quiet effective for the accurate and efficient solution of elasticity problems.

Discrete Least Squares Meshless (DLSM) method for simulation of steady state shallow water flows

In this paper, a meshless method, namely, Discrete Least Squares Meshless (DLSM) method is used for the solution of shallow water problems. In this method, the computational domain is discretized by some nodes and then a set of simultaneous equations are built using Moving Least Squares (MLS) shape functions and the least squares technique. The proposed method does not need any background mesh and therefore is a truly meshless method. The stability and accuracy of the DLSM method is improved using some sampling points. Some 2D benchmark problems and a two dimensional flow over an ogee spillway are used to illustrate the performance of the present DLSM method. Numerical results are compared with experimental or analytical ones. Both regular and irregular meshes of nodes are used to show the potential of the DLSM method in solving problems with more complex domains.

Efficient simulation of free surface flows with discrete least-squares meshless method using a priori error estimator

In this article, a priori error estimate is employed to improve the efficiency of simulating free surface flows with discrete least-squares meshless (DLSM) method. DLSM is a fully least-squares approach in which both function approximation and the discretisation of the governing differential equations is carried out using a least-squares concept. The meshless shape functions are derived using the moving least-squares (MLS) method of function approximation. The discretised equations are obtained via a discrete least-squares method in which the sum of the squared residuals are minimised with respect to unknown nodal parameters. The governing equations of mass and momentum conservation are solved in a Lagrangian form using a pressure projection method. The proposed simulation strategy is composed of error estimation and a node moving refinement method. Since in free surface problems, the position of the free surface is of primary interest, a priori error estimate is used which automatically associates higher error to the nodes near the free surface. The node moving refinement method is used to construct a nodal configuration with dense nodal arrangement near the free surface. Four test problems namely dam break, evolution of a water bubble, solitary wave propagation and wave run-up on slope are investigated to test the ability and efficiency of the proposed efficient simulation method.

Simulating free surface problems using Discrete Least Squares Meshless method

A Discrete Least Squares Meshless (DLSM) method is presented here for the simulation of incompressible free surface flows. The governing equations of the mass and momentum conservations are solved in a Lagrangian form using a pressure projection method. Since there are no particles in the outer region of the free surface, the particle density will drop significantly. Free surfaces are, therefore, resolved by tracking particles with highly reduced density. A fully least squares approach is used in both function approximation and the discretization of the governing differential equations in space. The meshless shape functions are derived using the Moving Least Squares (MLS) method of function approximation. The discretized equations are obtained via a discrete least squares method in which the sum of the squared residuals are minimized with respect to unknown nodal parameters. The method enjoys the advantage of producing symmetric, positive definite matrixes for the cases considered. The method can be viewed as a truly meshless method since it does not need any mesh for both field variable approximation and the construction of system matrices. Two free surface problems namely dam break and evolution of a drop with an initial known velocity are solved to test the accuracy of the proposed method. The results show the ability of the proposed method to solve complex fluid dynamic problems with moving free surface boundaries.

Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations

A meshfree method namely, discrete least squares meshless (DLSM) method, is presented in this paper for the solution of elliptic partial differential equations. In this method, computational domain is discretized by some nodes and the set of simultaneous algebraic equations are built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional is defined as the sum of squared residuals of the differential equation and its boundary condition calculated at a set of points called sampling points, generally different from nodal points. A moving least squares (MLS) technique is used to construct the shape functions. The proposed method automatically leads to symmetric and positive-definite system of equations. The proposed method does not need any background mesh and, therefore, it is a truly meshless method. The solutions of several one- and two-dimensional examples of elliptic partial differential equations are presented to illustrate the performance of the proposed method. Sensitivity analysis on the parameters of the method is also carried out and the results are presented.