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

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 meshless Pareto-optimal method for topology optimization

A meshless Galerkin Pareto-optimal method is proposed for topology optimization of continuum structures in this paper. The compactly supported radial basis function (CSRBF) is used to create shape functions. The shape function is constructed by meshfree approximations based on a set of unstructured field nodes. Considering the Pareto-optimality theory, the initial single objective topology optimization problem is transformed into multi-objective problem. The optimum solution is traced via the Pareto-optimal frontier in a computationally effective manner. The optimal problem does not need to be solved directly. Finally, several examples are used to prove the validity and effectiveness of the proposed approach.     

Adaptive multiple scale meshless simulation on springback analysis in sheet metal forming

Springback is one of the major considerations in the design of part shape, die geometry and processing parameters of sheet metal forming. In this study, an adaptive multiple scale meshless method is developed to predict the amount of springback, which occurs after unloading in sheet metal forming. A two-dimensional meshless continuum approach is applied to the bending deformation of plate/shell structures. The meshless method called reproducing kernel particle method (RKPM) is modified to develop the springback analysis algorithm using two scales. The effective strain is decomposed into two scales, high and low. The two scale decomposition is incorporated into non-linear elasto-plastic formulation to obtain high and low components of effective stresses. The high scale component of effective stress indicates the high stress gradient regions without posterior estimation. Enrichment nodes with a proper refinement scheme are inserted/deleted in those high stress regions to exactly calculate the stress distribution and thus accurately predict the amount of springback. The simulation results show that the algorithm can effectively locate the high stress gradient regions and can be utilized as an efficient indicator for the adaptive refinement technique for non-linear elasto-plastic deformation. The comparison of the amount of springback via the processing parameters between experiment, FEM (ABAQUS), meshless method and adaptive meshless method shows that the adaptive meshless solutions are the closest to experiment results.     

Buoyancy-driven fluid flow and heat transfer in a square cavity with a wavy baffle—Meshless numerical analysis

The meshless local Petrov–Galerkin method is implemented to simulate the buoyancy-driven flow and heat transfer in a differentially-heated enclosure having a baffle attached to its higher temperature side wall. To execute the proposed numerical treatment, the stream function–vorticity formulation is employed and a unity weighting function is applied for the weak form of the governing equations. In this meshless numerical approach, the field variables are approximated using the MLS interpolation technique. Being attested through comparing the results of two test case simulations with the results of either an analytical or a conventional numerical approach, the MLPG method is applied to investigate the buoyancy-driven flow and heat transfer in the baffled cavity. The present analyses involve a parametric study to implicate the effects of the baffle undulation number, amplitude, location on the wall, and the system Rayleigh number. The investigation reveals the eminent participation of the baffle in transferring heat from the hot wall. The analyses disclose an increase of the hot wall average Nusselt number by elevating the location of the baffle on the hot wall. This average Nusselt number descends with increasing the baffle amplitude. The cold wall average Nusselt number increases as the baffle number of undulation augments.     

An improved meshless Shepard and least squares method possessing the delta property and requiring no singular weight function

The meshless Shepard and least squares (MSLS) method and the meshless Shepard method are partition of unity based meshless interpolations which eliminate the problems by other meshless methods such as the difficulty in direct imposition of the essential boundary conditions. However, singular weight functions have to be used in both methods to enforce the approximation interpolatory, which leads to the loss of smoothness in approximation and locally oscillatory results. In this paper, an improved MSLS interpolation is developed by using dually defined nodal supports such that no singular weight function is required. The proposed interpolation satisfies the delta property at boundary nodes and the compatibility condition throughout the domain, and is capable of exactly reproducing the basis function. The computational cost of the present interpolation is much lower than the moving least-squares approximation which is probably the most widely used meshless interpolation at present.     

Solving acoustic nonlinear eigenvalue problems with a contour integral method

In this paper, a contour integral method (especially the block Sakurai–Sugiura method) is used to solve the eigenvalue problems governed by the Helmholtz equation, and formulated through two meshless methods. Singular value decomposition is employed to filter out the irrelevant eigenvalues. The accuracy and the ease of use of the proposed approach is illustrated with some numerical examples, and the choice of the contour integral method parameters is discussed. In particular, an application of the method on a sphere with realistic impedance boundary condition is performed and validated by comparison with results issued from a finite element method software.     

基于欧拉插值的最小二乘混合配点法在弹性力学平面问题中的应用

由于常规配点型无网格法存在求解不稳定、精度差和求解高阶导数等问题,提出了基于欧拉插值的最小二乘混合配点法。该方法同时以位移和应变作为未知量,通过欧拉插值将未知变量的导数表达出来,同时在插值中引入高斯权函数,并代入微分方程,从而形成以位移和应变为未知数的超定方程组,然后形成最小二乘意义下的法方程,法方程和相应的位移边界条件、应力边界条件一起形成定解体系。该方法不需要域积分,是一种真正的无网格法。一些典型的弹性力学平面问题表明本文方法具有良好的精度。     

Probabilistic fracture mechanics by Galerkin meshless methods – part II: reliability analysis

 This is the second in a series of two papers generated from a study on probabilistic meshless analysis of cracks. In this paper, a stochastic meshless method is presented for probabilistic fracture-mechanics analysis of linear-elastic cracked structures. The method involves an element-free Galerkin method for calculating fracture response characteristics; statistical models of uncertainties in load, material properties, and crack geometry; and the first-order reliability method for predicting probabilistic fracture response and reliability of cracked structures. The sensitivity of fracture parameters with respect to crack size, required for probabilistic analysis, is calculated using a virtual crack extension technique described in the companion paper [1]. Numerical examples based on mode-I and mixed-mode problems are presented to illustrate the proposed method. The results show that the predicted probability of fracture initiation based on the proposed formulation of the sensitivity of fracture parameter is accurate in comparison with the Monte Carlo simulation results. Since all gradients are calculated analytically, reliability analysis of cracks can be performed efficiently using meshless methods. Received 20 February 2001 / Accepted 19 December 2001     

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.     

Imposing boundary conditions in the meshless local Petrov-Galerkin method

A particular meshless method, named meshless local Petrov-Galerkin is investigated. To treat the essential boundary condition problem, an alternative approach is proposed. The basic idea is to merge the best features of two different methods of shape function generation: the moving least squares (MLS) and the radial basis functions with polynomial terms (RBFp). Whereas the MLS has lower computational cost, the RBFp imposes in a direct manner the essential boundary conditions. Thus, dividing the domain into different regions a hybrid method has been developed. Results show that it leads to a good trade-off between computational time and precision.     

Node‐by‐node meshless approach and its applications to structural analyses

The meshless method is expected to become an effective procedure for realizing a CAD/CAE seamless system for analyses ranging from modelling to computation, because time‐consuming mesh generation processes are not required. In the present study, a new meshless approach, referred to as the Node‐By‐Node Meshless method is proposed, in which only nodal data is utilized to discretize the governing equations, which are derived using either the principle of virtual work or the Galerkin method. In this method, three key methodologies are utilized: (i) nodal integration using stabilization terms, (ii) interpolation by the Moving Least‐Squares Method, and (iii) a node‐by‐node iterative solver. This paper presents the formulation of the proposed method along with numerical results obtained for two‐dimensional elastostatic and eigenvalue problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Dispersion and pollution of the improved meshless weighted least-square (IMWLS) solution for the Helmholtz equation

The meshless weighted least-square (MWLS) method is a meshless method based on the moving least-square (MLS) approximation. Compared with the Galerkin based meshless methods, the MWLS avoids numerical integrations, which improves the computational efficiency significantly. The MLS may form ill-conditioned system of equations, an accurate solution of which is difficult to obtain. In this paper, by using the weighted orthogonal basis function to construct the improved moving least-square (IMLS) approximation and the Lagrange multiplier method to enforce the Dirichlet boundary condition, we derive the formulas and perform the dispersion analysis for an improved meshless weighted least-square (IMWLS) method for two-dimensional (2D) Helmholtz problems. Results demonstrated that the IMWLS is more accurate and has advantages in handling dispersion. A 2D industrial model problem illustrated that the proposed method can easily reach higher frequency without losing accuracy.     

A local Heaviside weighted meshless method for two-dimensional solids using radial basis functions

 A meshless method is developed for the stress analysis of two-dimensional solids, based on a local weighted residual method with the Heaviside step function as the weighting function over a local subdomain. Trial functions are constructed using radial basis functions (RBF). The present method is a truly meshless method based only on a number of randomly located nodes. No domain integration is needed, no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. The behaviour of shape parameters of multiquadrics (MQ) has been systematically studied. Example problems in elastostatics are presented and compared with closed-form solutions and show that the proposed method is highly accurate and possesses no numerical difficulties. Received: 10 November 2002 / Accepted: 5 March 2003     

A novel meshless approach – Local Kriging (LoKriging) method with two-dimensional structural analysis

This paper develops a novel meshless approach, called Local Kriging (LoKriging) method, which is based on the local weak form of the partial differential governing equations and employs the Kriging interpolation to construct the meshless shape functions. Since the shape functions constructed by this interpolation have the delta function property based on the randomly distributed points, the essential boundary conditions can be implemented easily. The local weak form of the partial differential governing equations is obtained by the weighted residual method within the simple local quadrature domain. The spline function with high continuity is used as the weight function. The presently developed LoKriging method is a truly meshless method, as it does not require the mesh, either for the construction of the shape functions, or for the integration of the local weak form. Several numerical examples of two-dimensional static structural analysis are presented to illustrate the performance of the present LoKriging method. They show that the LoKriging method is highly efficient for the implementation and highly accurate for the computation.     

An implicit RBF meshless approach for time fractional diffusion equations

This paper aims to develop an implicit meshless approach based on the radial basis function (RBF) for numerical simulation of time fractional diffusion equations. The meshless RBF interpolation is firstly briefed. The discrete equations for two-dimensional time fractional diffusion equation (FDE) are obtained by using the meshless RBF shape functions and the strong-forms of the time FDE. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical examples with different problem domains and different nodal distributions are studied to validate and investigate accuracy and efficiency of the newly developed meshless approach. It has proven that the present meshless formulation is very effective for modeling and simulation of fractional differential equations.     

Meshless Galerkin least-squares method

Collocation method and Galerkin method have been dominant in the existing meshless methods. Galerkin-based meshless methods are computational intensive, whereas collocation-based meshless methods suffer from instability. A new efficient meshless method, meshless Galerkin lest-squares method (MGLS), is proposed in this paper to combine the advantages of Galerkin method and collocation method. The problem domain is divided into two subdomains, the interior domain and boundary domain. Galerkin method is applied in the boundary domain, whereas the least-squares method is applied in the interior domain.The proposed scheme elliminates the posibilities of spurious solutions as that in the least-square method if an incorrect boundary conditions are used. To investigate the accuracy and efficiency of the proposed method, a cantilevered beam and an infinite plate with a central circular hole are analyzed in detail and numerical results are compared with those obtained by Galerkin-based meshless method (GBMM), collocation-based meshless method (CBMM) and meshless weighted least squares method (MWLS). Numerical studies show that the accuracy of the proposed MGLS is much higher than that of CBMM and is close to, even better than, that of GBMM, while the computational cost is much less than that of GBMM.Acknowledgements The authors gratefully acknowledge the support of the National Natural Science Foundation of China with grant number 10172052.     

An enriched meshless method for non‐linear fracture mechanics

This paper presents an enriched meshless method for fracture analysis of cracks in homogeneous, isotropic, non‐linear‐elastic, two‐dimensional solids, subject to mode‐I loading conditions. The method involves an element‐free Galerkin formulation and two new enriched basis functions (Types I and II) to capture the Hutchinson–Rice–Rosengren singularity field in non‐linear fracture mechanics. The Type I enriched basis function can be viewed as a generalized enriched basis function, which degenerates to the linear‐elastic basis function when the material hardening exponent is unity. The Type II enriched basis function entails further improvements of the Type I basis function by adding trigonometric functions. Four numerical examples are presented to illustrate the proposed method. The boundary layer analysis indicates that the crack‐tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered, whether for the near‐tip asymptotic field or for the far‐tip elastic field. Numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the J‐integral for the applied load intensities and material properties considered. Also, the crack‐mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results. Furthermore, the meshless results show excellent agreement with the experimental measurements, indicating that the new basis functions are also capable of capturing elastic–plastic deformations at a stress concentration effectively. Copyright © 2003 John Wiley & Sons, Ltd.     

An efficient meshless method for fracture analysis of cracks

This paper presents an efficient meshless method for analyzing linear-elastic cracked structures subject to single- or mixed-mode loading conditions. The method involves an element-free Galerkin formulation in conjunction with an exact implementation of essential boundary conditions and a new weight function. The proposed method eliminates the shortcomings of Lagrange multipliers typically used in element-free Galerkin formulations. Numerical examples show that the proposed method yields accurate estimates of stress-intensity factors and near-tip stress field in two-dimensional cracked structures. Since the method is meshless and no element connectivity data are needed, the burdensome remeshing required by finite element method (FEM) is avoided. By sidestepping remeshing requirement, crack-propagation analysis can be dramatically simplified. Example problems on mixed-mode condition are presented to simulate crack propagation. The predicted crack trajectories by the proposed meshless method are in excellent agreement with the FEM or the experimental data. Received 6 March 2000     

Iterative domain decomposition meshless method modeling of incompressible viscous flows and conjugate heat transfer

We develop an effective domain decomposition meshless methodology for conjugate heat transfer problems modeled by convecting fully viscous incompressible fluid interacting with conducting solids. The meshless formulation for fluid flow modeling is based on a radial basis function interpolation using Hardy inverse Multiquadrics and a time-progression decoupling of the equations using a Helmholtz potential. The domain decomposition approach effectively reduces the conditioning numbers of the resulting algebraic systems, arising from convective and conduction modeling, while increasing efficiency of the solution process and decreasing memory requirements. Moreover, the domain decomposition approach is ideally suited for parallel computation. Numerical examples are presented to validate the approach by comparing the meshless solutions to finite volume method (FVM) solutions provided by a commercial CFD solver.     

Exact imposition of essential boundary condition and material interface continuity in Galerkin‐based meshless methods

Represented by the element free Galerkin method, the meshless methods based on the Galerkin variational procedure have made great progress in both research and application. Nevertheless, their shape functions free of the Kronecker delta property present great troubles in enforcing the essential boundary condition and the material continuity condition. The procedures based on the relaxed variational formulations, such as the Lagrange multiplier‐based methods and the penalty method, strongly depend on the problem in study, the interpolation scheme, or the artificial parameters. Some techniques for this issue developed for a particular method are hard to extend to other meshless methods. Under the framework of partition of unity and strict Galerkin variational formulation, this study, taking Poisson's boundary value problem for instance, proposes a unified way to treat exactly both the material interface and the nonhomogeneous essential boundary as in the finite element analysis, which is fit for any partition of unity‐based meshless methods. The solution of several typical examples suggests that compared with the Lagrange multiplier method and the penalty method, the proposed method can be always used safely to yield satisfactory results. Copyright © 2016 John Wiley & Sons, Ltd.