The boundary element‐free method for elastoplastic implicit analysis

A meshless procedure, based on boundary integral equations, is proposed to analyze elastoplastic problems. To cope with non‐linear problems, the usual boundary element method introduces domain discretization cells, often considered a ‘drawback’ of the method. Here, to get rid of the standard element and cell, i.e. boundary and domain discretization, the orthogonal moving least squares (also known as improved moving least squares) method is used. The algorithm adopted to solve these particular inelastic non‐linear problems is a well‐established, criterion‐independent implicit procedure, previously developed by the authors. Comparative results are presented at the end to illustrate the effectiveness of the proposed techniques. Copyright © 2008 John Wiley & Sons, Ltd.     

Boundary element‐free method (BEFM) for two‐dimensional elastodynamic analysis using Laplace transform

In this paper, we present a direct meshless method of boundary integral equation (BIE), known as the boundary element‐free method (BEFM), for two‐dimensional (2D) elastodynamic problems that combines the BIE method for 2D elastodynamics in the Laplace‐transformed domain and the improved moving least‐squares (IMLS) approximation. The formulae for the BEFM for 2D elastodynamic problems are given, and the numerical procedures are also shown. The BEFM is a direct numerical method, in which the basic unknown quantities are the real solutions of the nodal variables, and the boundary conditions can be implemented directly and easily that leads to a greater computational precision. For the purpose of demonstration, some selected numerical examples are solved using the BEFM. Copyright © 2005 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.     

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.     

A Jacobian‐free‐based IIM for incompressible flows involving moving interfaces with Dirichlet boundary conditions

In this paper, a finite difference marker‐and‐cell (MAC) scheme is presented for the steady Stokes equations with moving interfaces and Dirichlet boundary condition. The moving interfaces are represented by Lagrangian control points and their position is updated implicitly using a Jacobian‐free approach within each time step. The forces at the moving interfaces are calculated from the position of the interfaces and interpolated using cubic splines and then applied to the fluid through the related jump conditions. The proposed Jacobian‐free Newton–generalized minimum residual (GMRES) method avoids the need to form and store the matrix explicitly in the computation of the inverse of the Jacobian and betters numerical stability. The Stokes equations are discretized on a MAC grid via a second‐order finite difference scheme with the incorporation of jump contributions and the resulting saddle point system is solved by the conjugate gradient Uzawa‐type method. Numerical results demonstrate very well the accuracy and effectiveness of the proposed method. The present algorithm has been applied to solve incompressible Navier–Stokes flows with moving interfaces. Copyright © 2010 John Wiley & Sons, Ltd.     

Efficient DG‐like formulation equipped with curved boundary edges for solving elasto‐acoustic scattering problems

A Discontinuous Galerkin (DG)‐based approach is proposed for computing the scattered field from an elastic bounded object immersed in an infinite homogeneous fluid medium. The proposed method possesses two distinctive features. First, it employs higher‐order polynomial‐shape functions needed to address the high‐frequency propagation regime. Second, it is equipped with curved boundary edges to provide an accurate representation of the fluid–structure interface. The most salient benefits resulting from the latter feature, as demonstrated by the numerical investigation, are the following: (i) an improvement by—at least—two orders of magnitude on the relative error and (ii) the disappearance of spurious resonance frequencies in the surrounding fluid medium. In addition, the reported numerical results reveal that when using cubic polynomials with less than three elements per wavelength, the proposed DG method computes the scattered field with a relative error below 1% for an elastic scatterer of about 30 wavelengths. This observation highlights the potential of the proposed solution methodology for efficiently solving mid‐frequency to high‐frequency elasto‐acoustic scattering problems. Copyright © 2014 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.     

A singular‐value decomposition (SVD)‐based generalized finite difference (GFD) method for close‐interaction moving boundary flow problems

In this paper, we present a study on a singular‐value decomposition (SVD)‐based generalized finite difference (GFD) method and a nodal selection scheme for moving body/boundary flow problems formulated on a hybrid Cartesian cum meshfree grid system. The present study shows that the SVD‐based method is more robust and accurate than the conventional least‐squares‐based GFD scheme. A nodal selection scheme is also introduced to overcome the problem of numerical instability associated with the clustering of computational nodes. Such nodal clustering occurs dynamically when moving bodies or boundaries approach within close proximity of each other, resulting in the overlap of their meshfree grids. The nodal scheme is applied to close‐interaction flow problems as exemplified by the squeezing action of a circular cylinder through a very narrow slot and the close proximity bypass interaction of two oscillating circular cylinders. Copyright © 2008 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.     

A high‐order accurate particle‐in‐cell method

We propose the use of high‐order weighted essentially non‐oscillatory interpolation and moving‐least‐squares approximation schemes alongside high‐order time integration to enable high‐order accurate particle‐in‐cell methods. The key insight is to view the unstructured set of particles as the underlying representation of the continuous fields; the grid used to evaluate integro–differential coupling terms is purely auxiliary. We also include a novel regularization term to avoid the accumulation of noise in the particle samples without harming the convergence rate. We include numerical examples for several model problems: advection–diffusion, shallow water, and incompressible Navier–Stokes in vorticity formulation. The implementation demonstrates fourth‐order convergence, shows very low numerical dissipation, and is competitive with high‐order Eulerian schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

This paper considers a 2‐D fracture analysis of anisotropic piezoelectric solids by a boundary element‐free method. A traction boundary integral equation (BIE) that only involves the singular terms of order 1/r is first derived using integration by parts. New variables, namely, the tangential derivative of the extended displacement (the extended displacement density) for the general boundary and the tangential derivative of the extended crack opening displacement (the extended displacement dislocation density), are introduced to the equation so that solution to curved crack problems is possible. This resulted equation can be directly applied to general boundary and crack surface, and no separate treatments are necessary for the upper and lower surfaces of the crack. The extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the moving least‐squares (MLS) approximation. The numerical scheme of the boundary element‐free method is established, and an effective numerical procedure is adopted to evaluate the singular integrals. The extended ‘stress intensity factors’ (SIFs) are computed for some selected example problems that contain straight or curved cracks, and good numerical results are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Boundary element‐free method (BEFM) and its application to two‐dimensional elasticity problems

In this study, we first discuss the moving least‐square approximation (MLS) method. In some cases, the MLS may form an ill‐conditioned system of equations so that the solution cannot be correctly obtained. Hence, in this paper, we propose an improved moving least‐square approximation (IMLS) method. In the IMLS method, the orthogonal function system with a weight function is used as the basis function. The IMLS has higher computational efficiency and precision than the MLS, and will not lead to an ill‐conditioned system of equations. Combining the boundary integral equation (BIE) method and the IMLS approximation method, a direct meshless BIE method, the boundary element‐free method (BEFM), for two‐dimensional elasticity is presented. Compared to other meshless BIE methods, BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied easily; hence, it has higher computational precision. For demonstration purpose, selected numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd.     

Combined three‐dimensional lattice Boltzmann method and discrete element method for modelling fluid–particle interactions with experimental assessment

A general algorithmic framework is established in this paper for numerical simulations of three‐dimensional fluid–particle interaction problems with a large number of moving particles in turbulent flows using a combined lattice Boltzmann method (LBM) and discrete element method (DEM). In this approach, the fluid field is solved by the extended three‐dimensional LBM with the incorporation of the Smagorinsky turbulence model, while particle interactions are modelled by the DEM. The hydrodynamic interactions between fluid and particles are realized through the extension of an existing two‐dimensional fluid–particle hydrodynamic interaction scheme. The main computational aspects comprise the lattice Boltzmann formulation for the solution of fluid flows, the incorporation of a large eddy simulation‐based turbulence model within the framework of the three‐dimensional LBM for turbulent flows, the moving boundary condition for hydrodynamic interactions between fluid and moving particles, and the discrete element modelling of particle‐particle interactions. To assess the solution accuracy of the proposed approach, a much simplified laboratory model of vacuum dredging systems for mineral recovery is employed. The numerical results are compared with the experimental data available. It shows that the overall correspondence between numerical results and experimental measurements is good and thus indicates, to a certain extent, the solution accuracy of the proposed methodology. Copyright © 2009 John Wiley & Sons, Ltd.     

An element‐free method for in‐plane notch problems with anisotropic materials

This study developed an element‐free Galerkin method (EFGM) to simulate notched anisotropic plates containing stress singularities at the notch tip. Two‐dimensional theoretical complex displacement functions are first deduced into the moving least‐squares interpolation. The interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the Galerkin method. In the numerical validation, an interface layer of the EFGM is used to combine the mesh between the traditional finite elements and the proposed singular notch EFGM. The H‐integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method. The comparisons indicate that the proposed method obtains more accurate results for the displacement, stress, and energy fields than those determined from the standard finite element method. Copyright © 2013 John Wiley & Sons, Ltd.     

A fast BE‐FE coupling scheme for partly immersed bodies

Fluid–structure coupled problems are investigated to predict the vibro‐acoustic behavior of submerged bodies. The finite element method is applied for the structural part, whereas the boundary element method is used for the fluid domain. The focus of this paper is on partly immersed bodies. The fluid problem is favorably modeled by a half‐space formulation. This way, the Dirichlet boundary condition on the free fluid surface is incorporated by a half‐space fundamental solution. A fast multipole implementation is presented for the half‐space problem. In case of a high density of the fluid, the forces due to the acoustic pressure, which act on the structure, cannot be neglected. Thus, a strong coupling scheme is applied. An iterative solver is used to handle the coupled system. The efficiency of the proposed approach is discussed using a realistic model problem. Copyright © 2009 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.     

Weighted radial basis collocation method for boundary value problems

This work introduces the weighted radial basis collocation method for boundary value problems. We first show that the employment of least‐squares functional with quadrature rules constitutes an approximation of the direct collocation method. Standard radial basis collocation method, however, yields a larger solution error near boundaries. The residuals in the least‐squares functional associated with domain and boundary can be better balanced if the boundary collocation equations are properly weighted. The error analysis shows unbalanced errors between domain, Neumann boundary, and Dirichlet boundary least‐squares terms. A weighted least‐squares functional and the corresponding weighted radial basis collocation method are then proposed for correction of unbalanced errors. It is shown that the proposed method with properly selected weights significantly enhances the numerical solution accuracy and convergence rates. Copyright © 2006 John Wiley & Sons, Ltd.     

The dimension splitting and improved complex variable element‐free Galerkin method for 3‐dimensional transient heat conduction problems

In this paper, by combining the dimension splitting method and the improved complex variable element‐free Galerkin method, the dimension splitting and improved complex variable element‐free Galerkin (DS‐ICVEFG) method is presented for 3‐dimensional (3D) transient heat conduction problems. Using the dimension splitting method, a 3D transient heat conduction problem is translated into a series of 2‐dimensional ones, which can be solved with the improved complex variable element‐free Galerkin (ICVEFG) method. In the ICVEFG method for each 2‐dimensional problem, the improved complex variable moving least‐square approximation is used to obtain the shape functions, and the penalty method is used to apply the essential boundary conditions. Finite difference method is used in the 1‐dimensional direction, and the Galerkin weak form of 3D transient heat conduction problem is used to obtain the final discretized equations. Then, the DS‐ICVEFG method for 3D transient heat conduction problems is presented. Four numerical examples are given to show that the new method has higher computational precision and efficiency.     

Complex variable moving least‐squares method: a meshless approximation technique

Based on the moving least‐squares (MLS) approximation, we propose a new approximation method—the complex variable moving least‐squares (CVMLS) approximation. With the CVMLS approximation, the trial function of a two‐dimensional problem is formed with a one‐dimensional basis function. The number of unknown coefficients in the trial function of the CVMLS approximation is less than in the trial function of the MLS approximation, and we can thus select fewer nodes in the meshless method that is formed from the CVMLS approximation than are required in the meshless method of the MLS approximation with no loss of precision. The meshless method that is derived from the CVMLS approximation also has a greater computational efficiency. From the CVMLS approximation, we propose a new meshless method for two‐dimensional elasticity problems—the complex variable meshless method (CVMM)—and the formulae of the CVMM for two‐dimensional elasticity problems are obtained. Compared with the conventional meshless method, the CVMM has a greater precision and computational efficiency. For the purposes of demonstration, some selected numerical examples are solved using the CVMM. Copyright © 2006 John Wiley & Sons, Ltd.     

Moving least‐square interpolants in the hybrid particle method

The hybrid particle method (HPM) is a particle‐based method for the solution of high‐speed dynamic structural problems. In the current formulation of the HPM, a moving least‐squares (MLS) interpolant is used to compute the derivatives of stress and velocity components. Compared with the use of the MLS interpolant at interior particles, the boundary particles require two additional treatments in order to compute the derivatives accurately. These are the rotation of the local co‐ordinate system and the imposition of boundary constraints, respectively. In this paper, it is first shown that the derivatives found by the MLS interpolant based on a complete polynomial are indifferent to the orientation of the co‐ordinate system. Secondly, it is shown that imposing boundary constraints is equivalent to employing ghost particles with proper values assigned at these particles. The latter can further be viewed as placing the boundary particle in the centre of a neighbourhood that is formed jointly by the original neighbouring particles and the ghost particles. The benefit of providing a symmetric or a full circle of neighbouring points is revealed by examining the error terms generated in approximating the derivatives of a Taylor polynomial by using a linear‐polynomial‐based MLS interpolant. Symmetric boundaries have mostly been treated by using ghost particles in various versions of the available particle methods that are based on the strong form of the conservation equations. In light of the equivalence of the respective treatments of imposing boundary constraints and adding ghost particles, an alternative treatment for symmetry boundaries is proposed that involves imposing only the symmetry boundary constraints for the HPM. Numerical results are presented to demonstrate the validity of the proposed approach for symmetric boundaries in an axisymmetric impact problem. Copyright © 2005 John Wiley & Sons, Ltd.