A linearly conforming point interpolation method (LC‐PIM) for three‐dimensional elasticity problems

Linearly conforming point interpolation method (LC‐PIM) is formulated for three‐dimensional elasticity problems. In this method, shape functions are generated using point interpolation method by adopting polynomial basis functions and local supporting nodes are selected based on the background cells. The shape functions so constructed have the Kronecker delta functions property and it allows straightforward imposition of point essential boundary conditions. Galerkin weak form is used for creating discretized system equations, and a nodal integration scheme with strain‐smoothing operation is used to perform the numerical integration. The present LC‐PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method using linear elements, the LC‐PIM can achieve better efficiency, and higher accuracy especially for stresses. Copyright © 2007 John Wiley & Sons, Ltd.     

A superconvergent point interpolation method (SC‐PIM) with piecewise linear strain field using triangular mesh

A superconvergent point interpolation method (SC‐PIM) is developed for mechanics problems by combining techniques of finite element method (FEM) and linearly conforming point interpolation method (LC‐PIM) using triangular mesh. In the SC‐PIM, point interpolation methods (PIM) are used for shape functions construction; and a strain field with a parameter α is assumed to be a linear combination of compatible stains and smoothed strains from LC‐PIM. We prove theoretically that SC‐PIM has a very nice bound property: the strain energy obtained from the SC‐PIM solution lies in between those from the compatible FEM solution and the LC‐PIM solution when the same mesh is used. We further provide a criterion for SC‐PIM to obtain upper and lower bound solutions. Intensive numerical studies are conducted to verify these theoretical results and show that (1) the upper and lower bound solutions can always be obtained using the present SC‐PIM; (2) there exists an α_exact∈(0, 1) at which the SC‐PIM can produce the exact solution in the energy norm; (3) for any α∈(0, 1) the SC‐PIM solution is of superconvergence, and α=0 is an easy way to obtain a very accurate and superconvergent solution in both energy and displacement norms; (4) a procedure is devised to find a α_prefer∈(0, 1) that produces a solution very close to the exact solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Retracted:Comments on ‘Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC‐PIM)’ by G. R. Liu and G. Y. Zhang,International Journal for Numerical Methods in Engineering, 2008

The following article from International Journal for Numerical Methods in Engineering, Comments on ‘Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC‐PIM)’ by G. R. Liu and G. Y. Zhang, published online on 19 June 2008 in Wiley InterScience (www.interscience.wiley.com), has been retracted by agreement between the authors, the journal Editor‐in‐Chief, Ted Belytschko and John Wiley & Sons, Ltd. The retraction has been agreed due to the article being published out of sequence with a related article and with insufficient citation details.     

A scaled boundary radial point interpolation method for 2‐D elasticity problems

The scaled boundary radial point interpolation method (SBRPIM), a new semi‐analytical technique, is introduced and applied to the analysis of the stress–strain problems. The proposed method combines the advantages of the scaled boundary finite element method and the boundary radial point interpolation method. In this method, no mesh is required, nodes are scattered only on the boundary of the domain, no fundamental solution is required, and as the shape functions constructed based on the radial point interpolation method possess the Kronecker delta function property, the boundary conditions of problems can be imposed accurately without additional efforts. The main ideas of the SBRPIM are introducing a new method based on boundary scattered nodes without the need to element connectivity information, satisfying Kronecker delta function property, and being capable to handle singular problems. The equations of the SBRPIM in stress–strain fields are outlined in this paper. Several benchmark examples of 2‐D elastostatic are analyzed to validate the accuracy and efficiency of the proposed method. It is found that the SBRPIM is very easy to implement and the obtained results of the proposed method show a very good agreement with the analytical solution. Copyright © 2017 John Wiley & Sons, Ltd.     

Retracted:Comments on ‘Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC‐PIM)’ by G. R. Liu and G. Y. Zhang,International Journal for Numerical Methods in Engineering,DOI: 10.1002/nme.2204

The following article from International Journal for Numerical Methods in Engineering, Comments on ‘Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC‐PIM)’ by G. R. Liu and G. Y. Zhang, published online on 15 August 2008 in Wiley InterScience (www.interscience.wiley.com), has been retracted by agreement between the authors, the journal Editor‐in‐Chief, Ted Belytschko and John Wiley & Sons, Ltd. The retraction has been agreed due to the article being published out of sequence with a related article and with insufficient citation details.     

A point interpolation method for two‐dimensional solids

A point interpolation method (PIM) is presented for stress analysis for two‐dimensional solids. In the PIM, the problem domain is represented by properly scattered points. A technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points. The PIM equations are then derived using variational principles. In the PIM, the essential boundary conditions can be implemented with ease as in the conventional finite element methods. The present PIM has been coded in FORTRAN. The validity and efficiency of the present PIM formulation are demonstrated through example problems. It is found that the present PIM is very easy to implement, and very flexible for obtained displacements and stresses of desired accuracy in solids. As the elements are not used for meshing the problem domain, the present PIM opens new avenues to develop adaptive analysis codes for stress analysis in solids and structures. Copyright © 2001 John Wiley & Sons, Ltd.     

A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations

A new algorithm is developed to improve the accuracy and efficiency of the material point method for problems involving extremely large tensile deformations and rotations. In the proposed procedure, particle domains are convected with the material motion more accurately than in the generalized interpolation material point method. This feature is crucial to eliminate instability in extension, which is a common shortcoming of most particle methods. Also, a novel alternative set of grid basis functions is proposed for efficiently calculating nodal force and consistent mass integrals on the grid. Specifically, by taking advantage of initially parallelogram‐shaped particle domains, and treating the deformation gradient as constant over the particle domain, the convected particle domain is a reshaped parallelogram in the deformed configuration. Accordingly, an alternative grid basis function over the particle domain is constructed by a standard 4‐node finite element interpolation on the parallelogram. Effectiveness of the proposed modifications is demonstrated using several large deformation solid mechanics problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Non‐linear version of stabilized conforming nodal integration for Galerkin mesh‐free methods

A stabilized conforming (SC) nodal integration, which meets the integration constraint in the Galerkin mesh‐free approximation, is generalized for non‐linear problems. Using a Lagrangian discretization, the integration constraints for SC nodal integration are imposed in the undeformed configuration. This is accomplished by introducing a Lagrangian strain smoothing to the deformation gradient, and by performing a nodal integration in the undeformed configuration. The proposed method is independent to the path dependency of the materials. An assumed strain method is employed to formulate the discrete equilibrium equations, and the smoothed deformation gradient serves as the stabilization mechanism in the nodally integrated variational equation. Eigenvalue analysis demonstrated that the proposed strain smoothing provides a stabilization to the nodally integrated discrete equations. By employing Lagrangian shape functions, the computation of smoothed gradient matrix for deformation gradient is only necessary in the initial stage, and it can be stored and reused in the subsequent load steps. A significant gain in computational efficiency is achieved, as well as enhanced accuracy, in comparison with the mesh‐free solution using Gauss integration. The performance of the proposed method is shown to be quite robust in dealing with non‐uniform discretization. Copyright © 2002 John Wiley & Sons, Ltd.     

A meshfree Hermite‐type radial point interpolation method for Kirchhoff plate problems

A meshfree computational method is proposed in this paper to solve Kirchhoff plate problems of various geometries. The deflection of the thin plate is approximated by using a Hermite‐type radial basis function approximation technique. The standard Galerkin method is adopted to discretize the governing partial differential equations which were derived from using the Kirchhoff's plate theory. The degrees of freedom for the slopes are included in the approximation to make the proposed method effective in enforcing essential boundary conditions. Numerical examples with different geometric shapes and various boundary conditions are given to verify the efficiency, accuracy, and robustness of the method. Copyright © 2005 John Wiley & Sons, Ltd.     

Mesh‐free radial point interpolation method for the buckling analysis of Mindlin plates subjected to in‐plane point loads

In this paper, a mesh‐free approach is employed for buckling analysis of Mindlin plates that are subjected to in‐plane point loads. The radial point interpolation method (RPIM) is used to approximate displacements based on nodes. Variational forms of the system equations are established. Two‐step solution procedures are implemented. The non‐uniform pre‐stress distribution of plate is first obtained using the RPIM based on a two‐dimensional (2D) elastic plane stress problem. This predetermined non‐uniform pre‐stress distribution is then used to compute buckling loads of plate using the RPIM based on Mindlin's plate assumption. The RPIM can easily handle any number and location of nodes in the plate domain for a desired computational accuracy without major difficulties in solving the initial stresses and buckling loads. Numerical examples considered here include circular and rectangular Mindlin plates that are subjected to in‐plane uniform and point loads with different aspect ratios and boundary conditions. The present results are validated against the available analytical and numerical solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

A convected particle least square interpolation material point method

Applying the convected particle domain interpolation (CPDI) to the material point method has many advantages over the original material point method, including significantly improved accuracy. However, in the large deformation regime, the CPDI still may not retain the expected convergence rate. The paper proposes an enhanced CPDI formulation based on least square reconstruction technique. The convected particle least square interpolation (CPLS) material point method assumes the velocity field inside the material point domain as nonconstant. This velocity field in the material point domain is mapped to the background grid nodes with a moving least squares reconstruction. In this paper, we apply the improved moving least squares method to avoid the instability of the conventional moving least squares method due to a singular matrix. The proposed algorithm can improve convergence rate, as illustrated by numerical examples using the method of manufactured solutions.     

A new support integration scheme for the weakform in mesh‐free methods

A new support integration technique is proposed, which is similar to those used in truly mesh‐free methods. The contribution of this paper is to exploit the divergence‐free condition for the support integrals to construct quadrature formulas that only require three integration points per particle in two dimensions. Numerical examples show that the proposed integration method can achieve results that agree with manufactured closed‐form solutions. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis and reduction of quadrature errors in the material point method (MPM)

The material point method (MPM) has demonstrated itself as a computationally effective particle method for solving solid mechanics problems involving large deformations and/or fragmentation of structures, which are sometimes problematic for finite element methods (FEMs). However, similar to most methods that employ mixed Lagrangian (particle) and Eulerian strategies, analysis of the method is not straightforward. The lack of an analysis framework for MPM, as is found in FEMs, makes it challenging to explain anomalies found in its employment and makes it difficult to propose methodology improvements with predictable outcomes. In this paper we present an analysis of the quadrature errors found in the computation of (material) internal force in MPM and use this analysis to direct proposed improvements. In particular, we demonstrate that lack of regularity in the grid functions used for representing the solution to the equations of motion can hamper spatial convergence of the method. We propose the use of a quadratic B‐spline basis for representing solutions on the grid, and we demonstrate computationally and explain theoretically why such a small change can have a significant impact on the reduction in the internal force quadrature error (and corresponding ‘grid crossing error’) often experienced when using MPM. Copyright © 2008 John Wiley & Sons, Ltd.     

A novel singular node‐based smoothed finite element method (NS‐FEM) for upper bound solutions of fracture problems

It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node‐based smoothed finite element method (NS‐FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS‐FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five‐node singular crack tip element is used within the framework of NS‐FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix‐mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS‐FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

A consistent point‐searching algorithm for solution interpolation in unstructured meshes consisting of 4‐node bilinear quadrilateral elements

To translate and transfer solution data between two totally different meshes (i.e. mesh 1 and mesh 2), a consistent point‐searching algorithm for solution interpolation in unstructured meshes consisting of 4‐node bilinear quadrilateral elements is presented in this paper. The proposed algorithm has the following significant advantages: (1) The use of a point‐searching strategy allows a point in one mesh to be accurately related to an element (containing this point) in another mesh. Thus, to translate/transfer the solution of any particular point from mesh 2 to mesh 1, only one element in mesh 2 needs to be inversely mapped. This certainly minimizes the number of elements, to which the inverse mapping is applied. In this regard, the present algorithm is very effective and efficient. (2) Analytical solutions to the local co‐ordinates of any point in a four‐node quadrilateral element, which are derived in a rigorous mathematical manner in the context of this paper, make it possible to carry out an inverse mapping process very effectively and efficiently. (3) The use of consistent interpolation enables the interpolated solution to be compatible with an original solution and, therefore guarantees the interpolated solution of extremely high accuracy. After the mathematical formulations of the algorithm are presented, the algorithm is tested and validated through a challenging problem. The related results from the test problem have demonstrated the generality, accuracy, effectiveness, efficiency and robustness of the proposed consistent point‐searching algorithm. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical solution of second‐order,two‐point boundary value problems using continuous genetic algorithms

Second‐order, two‐point boundary‐value problems are encountered in many engineering applications including the study of beam deflections, heat flow, and various dynamic systems. Two classical numerical techniques are widely used in the engineering community for the solution of such problems; the shooting method and finite difference method. These methods are suited for linear problems. However, when solving the non‐linear problems, these methods require some major modifications that include the use of some root‐finding technique. Furthermore, they require the use of other basic numerical techniques in order to obtain the solution. In this paper, the author introduces a novel method based on continuous genetic algorithms for numerically approximating a solution to this problem. The new method has the following characteristics; first, it does not require any modification while switching from the linear to the non‐linear case; as a result, it is of versatile nature. Second, this approach does not resort to more advanced mathematical tools and is thus easily accepted in the engineering application field. Third, the proposed methodology has an implicit parallel nature which points to its implementation on parallel machines. However, being a variant of the finite difference scheme with truncation error of the order O(h²), the method provides solutions with moderate accuracy. Numerical examples presented in the paper illustrate the applicability and generality of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.     

A spline strip kernel particle method and its application to two‐dimensional elasticity problems

In this paper we present a novel spline strip kernel particle method (SSKPM) that has been developed for solving a class of two‐dimensional (2D) elasticity problems. This new approach combines the concepts of the mesh‐free methods and the spline strip method. For the interpolation of the assumed displacement field, we employed the kernel particle shape functions in the transverse direction, and the B₃‐spline function in the longitudinal direction. The formulation is validated on several beam and semi‐infinite plate problems. The numerical results of these test problems are then compared with the existing solutions obtained by the exact or numerical methods. From this study we conclude that the SSKPM is a potential alternative to the classical finite strip method (FSM). Copyright © 2003 John Wiley & Sons, Ltd.     

Radial point interpolation collocation method (RPICM) for the solution of nonlinear poisson problems

This paper applies radial point interpolation collocation method (RPICM) for solving nonlinear Poisson equations arising in computational chemistry and physics. Thin plate spline (TPS) Radial basis functions are used in the work. A series of test examples are numerically analysed using the present method, including 2D Liouville equation, Bratu problem and Poisson-Boltzmann equation, in order to test the accuracy and efficiency of the proposed schemes. Several aspects have been numerically investigated, namely the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on Neumann boundary for the solution of PDEs with Neumann boundary conditions. Particular emphasis was on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that a good accuracy can be obtained. The h-convergence rates are also studied for RPICM with coarse and fine discretization models.     

A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory

This paper introduces a G space theory and a weakened weak form (W²) using the generalized gradient smoothing technique for a unified formulation of a wide class of compatible and incompatible methods. The W² formulation works for both finite element method settings and mesh‐free settings, and W² models can have special properties including softened behavior, upper bounds and ultra accuracy. Part I of this paper focuses on the theory and fundamentals for W² formulations. A normed G space is first defined to include both continuous and discontinuous functions allowing the use of much more types of methods/techniques to create shape functions for numerical models. Important properties and a set of useful inequalities for G spaces are then proven in the theory and analyzed in detail. These properties ensure that a numerical method developed based on the W² formulation will be spatially stable and convergent to the exact solutions, as long as the physical problem is well posed. The theory is applicable to any problems to which the standard weak formulation is applicable, and can offer numerical solutions with special properties including ‘close‐to‐exact’ stiffness, upper bounds and ultra accuracy. Copyright © 2009 John Wiley & Sons, Ltd.