Multiple crack weight for solution of multiple interacting cracks by meshless numerical methods

We devise a multiple crack weight (MCW) method for the accurate and effective solution of strongly interacting cracks by meshless numerical methods. The MCW method constructs weight functions around cracks so that they simultaneously characterize all the cracks present in the single nodal domain of influence. This approach reduces the number of nodes necessary to achieve sufficient accuracy and consequently it decreases the computational effort. Numerical examples demonstrate that the method allows an accurate solution of multiple cracks problems. Convergence of the method is analysed and discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

A review of meshless methods for laminated and functionally graded plates and shells

This review focuses mainly on the developments of element-free or meshless methods and their applications in the analysis of composite structures. This review is organized as follows: a brief introduction to shear deformation plate and shell theories for composite structures, covering the first-order and higher-order theories, is given in Section 2. A review of meshless methods is provided in Section 3, with main emphasis on the element-free Galerkin method and reproducing kernel particle method. The applications of meshless methods in the analysis of composite structures are discussed in Section 4, including static and dynamic analysis, free vibration, buckling, and non-linear analysis. Finally, the problems and difficulties in meshless methods and possible future research directions are addressed in Section 5.     

Quadrature for meshless methods

In this paper we discuss quadrature schemes for meshless methods. We consider the Neumann problem and derive an estimate for the energy norm error between the exact solution, u, and the quadrature approximate solution, u, in terms of a parameter, h, associated with the family of approximation spaces, and quantities η, τ, and ε that measure the errors in the stiffness matrix, in the boundary data, and in the right‐hand side vector, respectively, due to the quadrature. The major hypothesis in the estimate is that the quadrature stiffness matrix has zero row sums, a hypothesis that can be easily achieved by a simple correction of the diagonal elements. Copyright © 2008 John Wiley & Sons, Ltd.     

RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates

A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, are developed for the quasi-three-dimensional (3D) analysis of simply supported, multilayered composite and functionally graded material (FGM) plates. The strong and weak formulations of this 3D static problem are derived on the basis of the Reissner mixed variational theorem (RMVT) where the strong formulation consists of the Euler–Lagrange equations of the problem and its associated boundary conditions, and the weak formulation represents a weighted-residual integral in which the differentiation is equally distributed among the primary field variables and their variations. The early proposed DRK interpolation is used to construct the primary field variables where the Kronecker delta properties are satisfied, and the essential boundary conditions can be readily applied, exactly like the implementation in the finite element method. The system equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations, respectively, in combination with the DRK interpolation. In the illustrative examples, it is shown that the solutions obtained from these methods are in excellent agreement with the available 3D solutions, and their convergence rates are rapid.     

A new method for meshless integration in 2D and 3D Galerkin meshfree methods

A method for the evaluation of regular domain integrals without domain discretization is presented. In this method, a domain integral is transformed into a boundary integral and a 1D integral. The method is then utilized for the evaluation of domain integrals in meshless methods based on the weak form, such as the element-free Galerkin method and the meshless radial point interpolation method. The proposed technique results in truly meshless methods with better accuracy and efficiency in comparison with their original forms. Some examples, including linear and large-deformation problems, are also provided to demonstrate the usefulness of the proposed method.     

Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method

The nonlocal elasticity theory of Eringen is used to study bending, buckling and free vibration of Timoshenko nanobeams. A meshless method is used to obtain numerical solutions. Results are compared with available analytical solutions. Two different collocation techniques, global (RBF) and local (RBF-FD), are used with multi-quadrics radial basis functions.     

RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D free vibration analysis of multilayered composite and FGM plates

A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, are developed for the quasi-three-dimensional (3D) free vibration analysis of simply supported, multilayered composite and functionally graded material (FGM) plates. Based on the Reissner Mixed Variational Theorem (RMVT), the strong and weak formulations of this problem are derived, in which the material properties of each individual FGM layer, constituting the plate, are assumed to obey the power-law distributions of the volume fractions of the constituents. The system motion equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations, respectively, in combination with the DRK interpolation, in which the shape functions of the unknown functions satisfy the Kronecker delta properties, and the essential boundary conditions can be readily applied, exactly like the implementation in the finite element method. In the illustrative examples, the natural frequencies and their corresponding modal field variables varying along the thickness coordinate of the plate are studied. It is shown that the solutions obtained using these methods are in excellent agreement with the available 3D solutions, and their convergence rates are rapid.     

Analytical and numerical modeling of R curves for cracks with bridging zones

At the onset of fracture in materials with process zones, the fracture resistance, or R curve, rises as the process zone develops. After process zone development, crack propagation proceeds by steady state growth. By considering J integral contours inside and outside the process zone, the available energy can be partitioned into crack tip energy release rate and process zone energy. To model the rising R curve, however, required assumptions about damage mechanisms in the process zone and partitioning of its energy into released and recoverable energy. By considering process zones that are elastic fiber-bridging zones with softening regions caused by fiber breakage or damage, equations for rising R curves were derived as a function of crack tip toughness and bridging zone mechanics. The new methods were implemented into the Material Point Method for generalized numerical crack propagation simulations with bridging zones. The simulation method includes pure fracture mechanics and pure cohesive zone models as extreme special cases. The most realistic simulations for many materials will likely fall between these two extremes. The results guided comments on interpretation of experimental R curves.     

Application of the RBF meshless method to laminar flame propagation

In this work we explore the applicability of the RBF method to laminar flame propagation modeling. This problem is an interesting challenge for the RBF method since it involves the solution of two coupled nonlinear parabolic equations in temperature and mass fraction. We show the suitability of the method by solving unsteady flame propagation problems in one and two dimensions. We also apply the method to compute the shape of an anchored flame using both equispaced and non-equispaced nodes.     

Construction and analysis of meshless finite difference methods

The basic idea of meshless finite difference methods is to approximate a differential operator by means of directional difference quotients and their combinations, and then to analyze the stability and convergence of the solution using a discrete energy estimation. The practical computational steps of the meshless finite difference methods include: 1. If the equation contains the time factor then nodes are inserted for each time segment to generate several time layers, each of which corresponds to a spatial domain; 2. A set of points is scattered for each spatial domain and a subset with each inner point as its center is chosen within the point set so that the directional difference quotients of first or second orders are established on this subset in order to construct a finite difference scheme with respect to each point. In this paper the construction of meshless finite difference schemes for elliptic, parabolic and hyperbolic equations is considered. The schemes are analyzed for stability and convergence. Moreover practical issued with regards to point scattering and connecting are considered and the issues of adaptation and parallelism are also discussed.     

Topological aspects of meshless methods and nodal ordering for meshless discretizations

The meshless element‐free Galerkin method (EFGM) is considered and compared to the finite‐element method (FEM). In particular, topological aspects of meshless methods as the nodal connectivity and invertibility of matrices are studied and compared to those of the FE method. We define four associated graphs for meshless discretizations of EFGM and investigate their connectivity. The ways that the associated graphs for coupled FE‐EFG models might be defined are recommended. The associated graphs are used for nodal ordering of meshless models in order to reduce the bandwidth, profile, maximum frontwidth, and root‐mean‐square wavefront of the corresponding matrices. Finally, the associated graphs are numerically compared. Copyright © 2001 John Wiley & Sons, Ltd.     

A radial basis function approach in the meshless local Petrov-Galerkin method for Euler-Bernoulli beam problems

A meshless local Petrov-Galerkin (MLPG) method that uses radial basis functions rather than generalized moving least squares (GMLS) interpolations to develop the trial functions in the study of Euler-Bernoulli beam problems is presented. The use of radial basis functions (RBF) in meshless methods is demonstrated for C¹ problems for the first time. This interpolation choice yields a computationally simpler method as fewer matrix inversions and multiplications are required than when GMLS interpolations are used. Test functions are chosen as simple weight functions as in the conventional MLPG method. Patch tests, mixed boundary value problems, and problems with complex loading conditions are considered. The radial basis MLPG method yields accurate results for deflections, slopes, moments, and shear forces, and the accuracy of these results is better than that obtained using the conventional MLPG method.Lockheed Martin Space Operations     

Comparison of local weak and strong form meshless methods for 2-D diffusion equation

A comparison between weak form meshless local Petrov-Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square quadrature domain. Implicit timestepping is used. The methods are tested in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost.     

Coupled partition of unity method and improved meshless weighted least-square method for two-dimensional interior structure-acoustic problem

Conventional element based methods for modeling acoustic problems are limited to low-frequency applications due to the huge computational efforts. For high-frequency applications, probabilistic techniques, such as statistical energy analysis (SEA), are used. For medium-frequency range, currently no adequate and mature simulation methods exist. This paper discusses a two-dimensional (2D) coupled structure-acoustic problem by coupling the partition of unity method (PUM), for the solid domain, and, an improved meshless weighted least-square (IMWLS) method, for the fluid domain, which demonstrated that the coupled PUM-IMWLS method seems to be an excellent way to reach the medium-frequency accurately.     

Kinematically admissible meshless approximation using modified weight function

In meshless methods, in general, the shape functions do not satisfy Kronecker delta properties at nodal points. Therefore, imposing essential boundary conditions is not a trivial task as in FEM. In this regard, there has been a great deal of endeavor to find ways to impose essential boundary conditions. In this study, a new scheme for imposing essential boundary conditions is developed. Weight functions are modified by multiplying with auxiliary weight functions and the resulting shape functions satisfy Kronecker delta properties on the boundary nodes. In addition, the resulting shape functions possess linear interpolation features on the boundary segments where essential boundary conditions are prescribed. Therefore, the essential boundary conditions can be exactly satisfied with the new method. More importantly, the imposition of essential boundary conditions using the present method is infinitely easy as in the finite element method. Numerical examples show that the method also retains high convergence rate comparable to the Lagrange multiplier method. Copyright © 2001 John Wiley & Sons, Ltd.     

Blurred derivatives and meshless methods

In this work we first introduce and describe the concept of blurred derivatives. It is shown how they can be used both to approximate differential equations and to re‐express them in alternative ways. In particular, formulations in terms of functional integrals can be obtained using blurred derivatives and extended to non‐linear problems. Blurred derivatives are shown to provide higher flexibility for selection of approximation functions than strong and weak formulations. Some computational implementations of one‐dimensional problems are discussed and the relationship between several well‐known numerical methods is analysed. Finally a meshless numerical scheme for the Poisson equation is described in detail. Its performance is compared with linear finite elements and generalized finite differences on unstructured meshes of points. Copyright © 2002 John Wiley & Sons, Ltd.     

A meshless local Petrov-Galerkin scaled boundary method

The scaled boundary finite-element method is a new semi-analytical approach to computational mechanics developed by Wolf and Song. The method weakens the governing differential equations by introducing shape functions along the circumferential coordinate direction(s). The weakened set of ordinary differential equations is then solved analytically in the radial direction. The resulting approximation satisfies the governing differential equations very closely in the radial direction, and in a finite-element sense in the circumferential direction. This paper develops a meshless method for determining the shape functions in the circumferential direction based on the local Petrov-Galerkin approach. Increased smoothness and continuity of the shape functions is obtained, and the solution is shown to converge significantly faster than conventional scaled boundary finite elements when a comparable number of degrees of freedom are used. No stress recovery process is necessary, as sufficiently accurate stresses are obtained directly from the derivatives of the displacement field.     

Parallel computation of meshless methods for explicit dynamic analysis

A parallel computational implementation of modern meshless methods is presented for explicit dynamic analysis. The procedures are demonstrated by application of the Reproducing Kernel Particle Method (RKPM). Aspects of a coarse grain parallel paradigm are detailed for a Lagrangian formulation using model partitioning. Integration points are uniquely defined on separate processors and particle definitions are duplicated, as necessary, so that all support particles for each point are defined locally on the corresponding processor. Several partitioning schemes are considered and a reduced graph-based procedure is presented. Partitioning issues are discussed and procedures to accommodate essential boundary conditions in parallel are presented. Explicit MPI message passing statements are used for all communications among partitions on different processors. The effectiveness of the procedure is demonstrated by highly deformable inelastic example problems. Copyright © 2000 John Wiley & Sons, Ltd.