The least‐squares meshfree method

A new efficient meshfree method is presented in which the first‐order least‐squares method is employed instead of the Galerkin's method. In the meshfree methods based on the Galerkin formulation, the source of many difficulties is in the numerical integration. The current method, in this respect, has different characteristics and is expected to remove some of the integration‐related problems. It is demonstrated through numerical examples that the present formulation is highly robust to integration errors. Therefore, numerical integration can be performed with great ease and effectiveness using very simple algorithms. Copyright © 2001 John Wiley & Sons, Ltd.     

Least‐squares collocation meshless method

A finite point method, least‐squares collocation meshless method, is proposed. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted. Unlike the direct collocation method, the equilibrium conditions are satisfied not only at the collocation points but also at the auxiliary points in a least‐squares sense. The moving least‐squares interpolant is used to construct the trial functions. The computational effort required for the present method is in the same order as that required for the direct collocation, while the present method improves the accuracy of solution significantly. The proposed method does not require any mesh so that it is a truly meshless method. Three numerical examples are studied in detail, which show that the proposed method possesses high accuracy with low computational effort. Copyright © 2001 John Wiley & Sons, Ltd.     

An improved moving‐least‐squares reconstruction for immersed boundary method

The current work presents an improved immersed boundary method based on the ideas proposed by Vanella and Balaras (M. Vanella, E. Balaras, A moving‐least‐squares reconstruction for embedded‐boundary formulations, J. Comput. Phys. 228 (2009) 6617–6628). In the method, an improved moving‐least‐squares approximation is employed to build the transfer functions between the Lagrangian points and discrete Eulerian grid points. The main advantage of the improved method is that there is no need to obtain the inverse matrix, which effectively eliminates numerical instabilities caused by matrix inversion and reduces the computational cost significantly. Several different flow problems (Taylor‐Green decaying vortices, flows past a stationary circular cylinder and a sphere, and the sedimentation of a free‐falling sphere in viscous fluid) are simulated to validate the accuracy and efficiency of the method proposed in the present paper. The simulation results show good agreement with previous numerical and experimental results, indicating that the improved immersed boundary method is efficient and reliable in dealing with the fluid–solid interaction problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Essential boundary condition enforcement in meshless methods: boundary flux collocation method

Element‐free Galerkin (EFG) methods are based on a moving least‐squares (MLS) approximation, which has the property that shape functions do not satisfy the Kronecker delta function at nodal locations, and for this reason imposition of essential boundary conditions is difficult. In this paper, the relationship between corrected collocation and Lagrange multiplier method is revealed, and a new strategy that is accurate and very simple for enforcement of essential boundary conditions is presented. The accuracy and implementation of this new technique is illustrated for one‐dimensional elasticity and two‐dimensional potential field problems. Copyright © 2001 John Wiley & Sons, Ltd.     

A modified least‐squares mixed finite element with improved momentum balance

The main goal of this contribution is to provide an improved mixed finite element for quasi‐incompressible linear elasticity. Based on a classical least‐squares formulation, a modified weak form with displacements and stresses as process variables is derived. This weak form is the basis for a finite element with an advanced fulfillment of the momentum balance and therefore with a better performance. For the continuous approximation of stresses and displacements on the triangular and tetrahedral elements, lowest‐order Raviart–Thomas and linear standard Lagrange interpolations can be used. It is shown that coercivity and continuity of the resulting asymmetric bilinear form could be established with respect to appropriate norms. Further on, details about the implementation of the least‐squares mixed finite elements are given and some numerical examples are presented in order to demonstrate the performance of the proposed formulation. Copyright © 2009 John Wiley & Sons, Ltd.     

Moving least‐squares particle hydrodynamics II: conservation and boundaries

Previous work by the author has shown that the consistency of the SPH method can be improved to acceptable levels by substituting MLS interpolants for SPH interpolants, that the SPH inconsistency drives the tension instability and that imposition of consistency via MLS severely retards tension instability growth. The new method however was not conservative, and made no provision for boundary conditions. Conservation is an essential property in simulations where large localized mass, momentum or energy transfer occurs such as high‐velocity impact or explosion modeling. A new locally conservative MLS variant of SPH that naturally incorporates realistic boundary conditions is described. In order to provide for the boundary fluxes one must identify the boundary particles. A new, purely geometric boundary detection technique for assemblies of spherical particles is described. A comparison with SPH on a ball‐and‐plate impact simulation shows qualitative improvement. Copyright © 2000 John Wiley & Sons, Ltd.     

A study on the convergence of least‐squares meshfree method under inaccurate integration

In the authors' previous work, it has been shown through numerical examples that the least‐squares meshfree method (LSMFM) is highly robust to the integration errors while the Galerkin meshfree method is very sensitive to them. A mathematical study on the convergence of the solution of LSMFM under inaccurate integration is presented. New measures are introduced to take into account the integration errors in the error estimates. It is shown that, in LSMFM, solution errors are bounded by approximation errors even when integration is not accurate. Copyright © 2003 John Wiley & Sons, Ltd.     

Implementation of meshless LBIE method to the 2D non‐linear SG problem

In this paper the meshless local boundary integral equation (LBIE) method for numerically solving the non‐linear two‐dimensional sine‐Gordon (SG) equation is developed. The method is based on the LBIE with moving least‐squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables. The approximation functions are constructed entirely using a set of scattered nodes, and no element or connectivity of the nodes is needed for either the interpolation or the integration purposes. A time‐stepping method is employed to deal with the time derivative and a simple predictor–corrector scheme is performed to eliminate the non‐linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of method to deal with the unsteady non‐linear problems in large domains. Copyright © 2009 John Wiley & Sons, Ltd.     

A posteriori error estimates and an adaptive scheme of least‐squares meshfree method

A posteriori error estimates and an adaptive refinement scheme of first‐order least‐squares meshfree method (LSMFM) are presented. The error indicators are readily computed from the residual. For an elliptic problem, the error indicators are further improved by applying the Aubin–Nitsche method. It is demonstrated, through numerical examples, that the error indicators coherently reflect the actual error. In the proposed refinement scheme, Voronoi cells are used for inserting new nodes at appropriate positions. Numerical examples show that the adaptive first‐order LSMFM, which combines the proposed error indicators and nodal refinement scheme, is effectively applied to the localized problems such as the shock formation in fluid dynamics. Copyright © 2003 John Wiley & Sons, Ltd.     

Moving least‐squares approximation with discontinuous derivative basis functions for shell structures with slope discontinuities

Moving least‐squares approximation with discontinuous derivative basis functions (MLSA‐DBF) is introduced for analysis of shell structures with slope discontinuities. To deal with shells with arbitrary slope discontinuities, the Cartesian coordinate is introduced in the construction of MLSA on the shell surface. The possible causes of singularity in the moment matrix of MLSA on the shell surface with slope discontinuities are identified, and the Moore–Penrose pseudoinverse is used to obtain the generalized inverse of the singular moment matrix resulting from linear dependency and insufficient influence nodes in the MLSA. Following the proposed formulations for shear deformable shell structures with slope discontinuities in the Cartesian coordinates, several numerical examples are analyzed to demonstrate the performance, validity, accuracy, and convergence properties of the proposed MLSA‐DBF approach. Copyright © 2007 John Wiley & Sons, Ltd.     

High‐order finite volume schemes on unstructured grids using moving least‐squares reconstruction. Application to shallow water dynamics

This paper introduces the use of moving least‐squares (MLS) approximations for the development of high‐order finite volume discretizations on unstructured grids. The field variables and their successive derivatives can be accurately reconstructed using this mesh‐free technique in a general nodal arrangement. The methodology proposed is used in the construction of two numerical schemes for the shallow water equations on unstructured grids: a centred Lax–Wendroff method with added shock‐capturing dissipation, and a Godunov‐type upwind scheme, with linear and quadratic reconstructions. This class of mesh‐free techniques provides a robust and general approximation framework which represents an interesting alternative to the existing procedures, allowing, in addition, an accurate computation of the viscous fluxes. Copyright © 2005 John Wiley & Sons, Ltd.     

A least‐squares coupling method between a finite element code and a discrete element code

This paper presents a coupling method between a discrete element code CeaMka3D and a finite element code Sem. The coupling is based on a least‐squares method, which adds terms of forces to finite element code and imposes the velocity at coupling particles. For each coupling face, a small linear system with a constant matrix is solved. This method remains conservative in energy and shows good results in applications. Copyright © 2014 John Wiley & Sons, Ltd.     

Moving‐least‐squares‐particle hydrodynamics—I. Consistency and stability

The Smooth‐Particle‐Hydrodynamics (SPH) method is derived in a novel manner by means of a Galerkin approximation applied to the Lagrangian equations of continuum mechanics as in the finite‐element method. This derivation is modified to replace the SPH interpolant with the Moving‐Least‐Squares (MLS) interpolant of Lancaster and Saulkaskas, and define a new particle volume which ensures thermodynamic compatibility. A variable‐rank modification of the MLS interpolants which retains their desirable summation properties is introduced to remove the singularities that occur when divergent flow reduces the number of neighbours of a particle to less than the minimum required. A surprise benefit of the Galerkin SPH derivation is a theoretical justification of a common ad hoc technique for variable‐h SPH. The new MLSPH method is conservative if an anti‐symmetric quadrature rule for the stiffness matrix elements can be supplied. In this paper, a simple one‐point collocation rule is used to retain similarity with SPH, leading to a non‐conservative method. Several examples document how MLSPH renders dramatic improvements due to the linear consistency of its gradients on three canonical difficulties of the SPH method: spurious boundary effects, erroneous rates of strain and rotation and tension instability. Two of these examples are non‐linear Lagrangian patch tests with analytic solutions with which MLSPH agrees almost exactly. The examples also show that MLSPH is not absolutely stable if the problems are run to very long times. A linear stability analysis explains both why it is more stable than SPH and not yet absolutely stable and an argument is made that for realistic dynamic problems MLSPH is stable enough. The notion of coherent particles, for which the numerical stability is identical to the physical stability, is introduced. The new method is easily retrofitted into a generic SPH code and some observations on performance are made. Copyright © 1999 John Wiley & Sons, Ltd.     

A procedure for approximation of the error in the EFG method

In this paper, we present a procedure to estimate the error in elliptic equations using the element‐free Galerkin (EFG) method, whose evaluation is computationally simple and can be readily implemented in existing EFG codes. The estimation of the error works very well in all numerical examples for 2‐D potential problems that are presented here, for regular and irregular clouds of points. Moreover, it was demonstrated that this method is very simple in terms of economy and gives a good performance. The results show that the error in EFG approximation may be estimated via the error estimator described in this paper. The quality of the estimation of the error is demonstrated by numerical examples. The implemented procedure of error approximation allows the global energy norm error to be estimated and also gives a good evaluation of local errors. It can, thus, be combined with a full adaptive process of refinement or, more simply, provide guidance for redesign of cloud of points. Copyright © 2001 John Wiley & Sons, Ltd.     

Mixed discrete least squares meshless method for planar elasticity problems using regular and irregular nodal distributions

A Mixed Discrete Least Square Meshless (MDLSM) method is proposed for the solution of planar elasticity problems. In this approach, the differential equations governing the planar elasticity problems are written in terms of the stresses and displacements which are approximated independently using the same shape functions. Since the resulting governing equations are of the first order, both the displacement and stress boundary conditions are of the Dirichlet-type which is easily incorporated via a penalty method. Because least squares based algorithm of MDLSM method, the proposed method does not need to be satisfied by the LBB condition. The performance of the proposed method is tested on a benchmark example from theory of elasticity namely the problem of infinite plate with a circular hole and the results are presented and compared with those of the analytical solution and the solutions obtained using the irreducible DLSM formulation. The results indicate that the proposed MDLSM method is more accurate than the DLSM method. The results show that the numerical solutions of the MDLSM method can be obtained with lower computational cost and with higher accuracy. Also its performance is marginally affected by the irregularity of the nodal distribution.     

A posteriori error approximation in EFG method

Recently, considerable effort has been devoted to the development of the so‐called meshless methods. Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. One of the paths in the evolution of meshless methods has been the development of the element free Galerkin (EFG) method. In the EFG method, it is obviously important that the ‘a posteriori error’ should be approximated. An ‘a posteriori error’ approximation based on the moving least‐squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least‐squares approximations. Different selection strategies of the moving least‐squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error. The performance of the developed approximation of the error is illustrated by analysing different examples for two‐dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors. Copyright © 2003 John Wiley & Sons, Ltd.     

Point collocation methods using the fast moving least‐square reproducing kernel approximation

A pseudo‐spectral point collocation meshfree method is proposed. We apply a scheme of approximating derivatives based on the moving least‐square reproducing kernel approximations. Using approximated derivatives, we propose a new point collocation method. Unlike other meshfree methods that require direct calculation of derivatives for shape functions, with the proposed scheme, approximated derivatives are obtained in the process of calculating the shape function itself without further cost. Moreover, the scheme does not require the regularity of the window function, which ensures the regularity of shape functions. In this paper, we show the reproducing property and the convergence of interpolation for approximated derivatives of shape functions. As numerical examples of the proposed scheme, Poisson and Stokes problems are considered in various situations including the case of randomly generated node sets. In short, the proposed scheme is efficient and accurate even for complicated geometry such as the flow past a cylinder. Copyright © 2003 John Wiley & Sons, Ltd.     

The least‐squares meshfree method for elasto‐plasticity and its application to metal forming analysis

A new meshfree method for the analysis of elasto‐plastic deformation is presented. The method is based on the proposed first‐order least‐squares formulation for elasto‐plasticity and the moving least‐squares approximation. The least‐squares formulation for classical elasto‐plasticity and its extension to an incrementally objective formulation for finite deformation are proposed. In the formulation, equilibrium equation and flow rule are enforced in least‐squares sense, i.e. their squared residuals are minimized, and hardening law and loading/unloading condition are enforced pointwise at each integration point. The closest point projection method for the integration of rate‐form constitutive equation is inherently involved in the formulation, and thus the radial‐return mapping algorithm is not performed explicitly. The proposed formulation is a mixed‐type method since the residuals are represented in a form of first‐order differential system using displacement and stress components as nodal unknowns. Also the penalty schemes for the enforcement of boundary and frictional contact conditions are devised and the reshaping of nodal supports is introduced to avoid the difficulties due to the severe local deformation near contact interface. The proposed method does not employ structure of extrinsic cells for any purpose. Through some numerical examples of metal forming processes, the validity and effectiveness of the method are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

The least‐squares finite element method in elasticity—Part I: Plane stress or strain with drilling degrees of freedom

A new least‐squares finite element method (LSFEM) for plane elasticity problems is developed based on the first‐order displacement–stress–rotation formulation which includes two new first‐order compatibility constraints among the stresses and the drilling rotation. This LSFEM can accommodate all kinds of equal‐order interpolations. Numerical experiments on various examples including incompressible materials show that the method achieves an optimal rate of convergence for all variables. Copyright © 2001 John Wiley & Sons, Ltd.