Efficient searching in meshfree methods

In many engineering problems, the meshfree methods (MMs) have been dynamically projected and increasingly advanced in order to overwhelm some hitches in the predictable numerical methods. Over the past three decades in many different application area, MMs have found their way ranging from solid mechanics analysis, fluid problems, vibration analysis, heat transfer and optimization to numerical solutions of all kinds of (partial) differential equations. As every technique has shortcomings, the meshfree method also has drawbacks like higher computational cost and imposition of boundary condition which can be overruled by coupling it with the finite element method (FEM). In the past two decades, coupled MMs and FEM have appeared into a new session of computational methods with significant achievement. In addition, a noteworthy amount of growth has been made in addressing the major deficiencies that were present in the conventional methods and MMs at the premature phases. The objective of the present work is to provide a comprehensive review of various coupling techniques used for interface elements of MMs and FEM and general discussion on shape function formulation of FE and element free Galerkin method (EFGM). Key contribution of coupling techniques for coupled EFGM and FEM to structure mechanics application as primary concern.

One of major difficulties in the implementation of meshless methods is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. As a consequence, the imposition of essential boundary conditions in meshless methods is quite awkward. In this paper, a displacement constraint equations method (DCEM) is proposed for the imposition of the essential boundary conditions, in which the essential boundary conditions is treated as a constraint to the discrete equations obtained from the Galerkin methods. Instead of using the methods of Lagrange multipliers and the penalty method, a procedure is proposed in which unknowns are partitioned into two subvectors, one consisting of unknowns on boundary Γu, and one consisting of the remaining unknowns. A simplified displacement constraint equations method (SDCEM) is also proposed, which results in a efficient scheme with sufficient accuracy for the imposition of the essential boundary conditions in meshless methods. The present method results in a symmetric, positive and banded stiffness matrix. Numerical results show that the accuracy of the present method is higher than that of the modified variational principles. The present method is a exact method for imposing essential boundary conditions in meshless methods, and can be used in Galerkin‐based meshless method, such as element‐free Galerkin methods, reproducing kernel particle method, meshless local Petrov–Galerkin method. Copyright © 2001 John Wiley & Sons, Ltd.

On the use of radial basis functions in a local weighted meshless method

This thesis is devoted to the numerical analysis of mesh-free methods and, in particular, to the study of the possible advantages of the EFG (Element Free Galerkin) mesh-free method against the well-known FE (Finite Element) method. More precisely, the EFG method and the FE method behavior are compared in two particular interesting problems: (1) analysis of volumetric locking in mechanical problems and (2) accurate resolution of transient convection dominated problems. In both cases the good properties and possibilities of mesh-free methods become apparent. <br/>However, in several situations the FE method is still more competitive: for instance, the computation of the FE shape functions and its integrals are less costly, and essential boundary conditions can be easily imposed. Thus, in order to take advantage of the good properties of both methods, a mixed interpolation combining FE and EFG is proposed. This formulation can be applied in two useful situations: (i) enrichment of finite elements with EFG, and (ii) coupling of FE and EFG. An a priori error estimate for the first one is presented and proved. Several examples show the applicability of the mixed interpolation in adaptive computations. Aquesta tesi està dedicada a l'anàlisi numèrica dels mètodes sense malla i, en particular, a l'estudi dels possibles avantatges del mètode EFG (Element Free Galerkin) davant del ben conegut MEF (Mètode dels Elements Finits). Concretament, es comparen el mètode EFG i el MEF en dos problemes concrets d'interès: (1) l'anàlisi del bloqueig volumètric en problemes mecànics i (2) la resolució precisa de problemes transitoris amb convecció dominant. Les bones propietats i possibilitats dels mètodes sense malla es fan evidents en tots dos casos.<br/>Tot i així, en varis aspectes el MEF resulta més competitiu: per exemple, el càlcul de les funcions de forma i de les seves integrals es menys costós, i les condicions de contorn essencials es poden imposar fàcilment. Amb l'objectiu d'aprofitar les bones qualitats dels dos mètodes, es proposa una interpolació mixta combinant elements finits y EFG, aplicable en dues situacions: (i) enriquiment d'elements finits amb EFG i (ii) acoblament d'elements finits i EFG. Per al primer cas, es presenta i demostra una cota a priori de l'error. L'aplicabilitat d'aquesta interpolació mixta en processos adaptatius es mostra amb varis exemples. Esta tesis está dedicada al análisis numérico de los métodos sin malla y, en particular, al estudio de las posibles ventajas del método EFG (Element Free Galerkin) frente al bien conocido MEF (Método de los Elementos Finitos). Concretamente, se comparan el método EFG y el MEF en dos problemas concretos de interés: (1) el análisis del bloqueo volumétrico en problemas mecánicos y (2) la resolución precisa de problemas transitorios con convección dominante. Las buenas propiedades y posibilidades de los métodos sin malla se hacen evidentes en ambos casos.<br/>Sin embargo, en varios aspectos el MEF resulta más competitivo: por ejemplo, el cálculo de las funciones de forma y sus integrales es menos costoso, y las condiciones de contorno esenciales se pueden imponer fácilmente. Con el objetivo de aprovechar las buenas cualidades de ambos métodos, se propone una interpolación mixta combinando elementos finitos y EFG, aplicable en dos situaciones: (i) enriquecimiento de elementos finitos con EFG, y (ii) acoplamiento de elementos finitos y EFG. Para el primer caso, se presenta y demuestra una cota a priori del error. La aplicabilidad de esta interpolación mixta en procesos adaptativos se muestra con varios ejemplos.

A topology optimization, based on a coupling method of finite element and meshless method, is proposed for continuum structure. A reasonable arrangement of the meshless domain can guarantee the accuracy of the meshless methods and meanwhile keep the computational efficiency of the finite element method. Besides, as the coupling method is adopted, the displacement boundary conditions are applied to finite elements nodes by standard finite element method. A dual-level density approximant is carried out to approximate and interpolate a unified and continuous density field. An unstable nodes phenomenon is observed in the interface domain, leading to nonconvergence of the equilibrium iterations. A smooth blending function and an energy convergence criterion are used to circumvent the convergent difficulty. Three benchmark problems of topology optimization are given to demonstrate the effectiveness of the proposed approach. Neither “islanding” phenomenon nor other common numerical instabilities of the SIMP method occur in this approach.

In recent years, a class of Galerkin-based meshfree or meshless methods have been developed that do not require a structured mesh to discretize the problem, such as the element-free Galerkin method, and the reproducing kernel particle method. These methods employ a moving least-squares (MLS) approximation method that allows resultant shape functions to be constructed entirely in terms of arbitrarily placed nodes. These methods employ a moving least-squares approximation method that allows resultant shape functions to be constructed entirely in terms of arbitrarily placed nodes. Meshless discretization presents significant advantages for modeling fracture propagation. By sidestepping remeshing requirements, crack-propagation analysis can be dramatically simplified. Since no element connectivity data are needed, the burdensome remeshing required by the FEM is avoided. A growing crack can be modeled by simply extending the free surfaces, which correspond to the crack. In addition, stochastic meshless method (SMM) facilitates to have different discretizations for capturing the spatial variability of the material properties and the structural response, without much difficulty in mapping between the two discretizations. However, the computational cost of a SMM typically exceeds the cost of a SFEM. Hence, it is advantageous to adopt MLS approximation for material property discretization and FEM for structural response computation. This paper presents a new coupled finite element-moving least squares technique for predicting probabilistic structural response and reliability of cracked structures.

For acoustic computations in the mid-frequency range the finite element method (FEM) is a well-known standard tool. Unfortunately, for increasing frequencies, i.e. higher wavenumbers, the FEM suffers from the so-called pollution effect which is mainly a consequence of the dispersion, meaning that the numerical wavenumber and the exact wavenumber disagree. Using meshfree methods as, e.g., the radial point interpolation method (RPIM) or the element-free Galerkin method (EFGM) can reduce this effect significantly. Moreover, meshfree methods allow the usage of shape functions that can be adapted to the differential equation to be solved. Consequently, an iterative method can be derived, which uses a standard meshfree method to compute a first approximation for the given problem. In a second step this approximation is taken to construct new shape functions that are able to better reproduce the wave-like character of the solution. If a few requirements are considered, this method leads to better results in a more efficient way. In this paper two meshfree methods, namely an iterative RPIM and an iterative EFGM, are examined. The methods are compared to the FEM and restrictions for an efficient applicability are shown.

Determination of Two-Dimensional Plastic Zone Shape and SIF at Crack-Tip Using RKPM

The Reproducing Kernel Particle Method (RKPM) is one of several so-called meshfree methods of structural analysis and has been applied in this paper to the gradient-based shape optimization of two-dimensional automotive components. As indicated by the term meshfree, no mesh is required, but rather a field of points, or particles, are distributed within the domain of the problem. Three standard linear examples are chosen to allow a comprehensive comparison between the optimization with the RKPM and the state-of-art optimization tool in Unigraphics Version 18 (UG V18). The results show that the optimization analysis with the RKPM is able to accommodate very large shape changes required in the optimization process without doing particle re-adaptation, and provides accurate solutions of the objective and constraint functions and their gradients, therefore facilitates a fast convergence of the gradient-based optimization algorithm.

It is common knowledge that conventional finite element method (FEM) has intrinsic limitations in analyzing large deformation problems like high-velocity impact, explosion, etc. because of mesh distortion and tangling; while these problems can be easily avoided by the meshfree method (MM), the latter involves greater computation time. Therefore, in this article, in order to simultaneously utilize the respective advantages of the two methods, a coupled simulation method between both FEM and MM was employed to analyze the high-velocity impact on ductile metal pipe with polymer coating. The impacted area with large deformation was discretized by SPH (smoothed particle hydrodynamics) particles, a classic meshfree model, and the remaining section was modeled by FEM meshes. By this method, the interfacial shear stresses between the coating and the substrate and the residual stresses beneath the contact points were studied, which would have referenced values in analyzing failure modes of components with similar composite structure. Then, the results were compared with sole FEM and MM too.

Meshfree analyses of cable-reinforced membrane structures by ALE–EFG method

Application of NEM in seepage analysis with a free surface

TO undertake a successful computation task w.r.t. the construction and operation process of engineering structures, one of the mostly concerned obstacles is the limited or incomplete capability to automatically generate regular grid or to re-mesh retorted grid for complex structure domain containing structural planes and undergoing finite deformation. This chapter presents the principle and basic algorithm of the “element free Galerkin method (EFGM)” in structural dynamics that belongs to the family of “mesh-free methods (MFMs)”. In EFGM, the basis functions no longer belong to standard PU (e.g. the shape functions of fundamental FEM), instead, they are created from the neighborhood field nodes in an “influence or support domain” of any shape by the technique of “moving least squares (MLS)” approximation. The nodal variables in the piecewise trial function may be solved from the algebraic equation set according to the virtual work principle or variational principle (weak form). In this manner, no rigorous restraint is imposed on the generation of scattered field nodes (points), which allows for a great simplification in the pre-process work of complex engineering structure. Although it has not outperformed FEMs in the routine problems of geotechnical and hydraulic structures, yet EFGM exhibits great expectation due to its flexibility in handling structural planes, crack growth, and other large deformation issues.

The analysis of laminated plates using distinct advanced discretization meshless techniques

Effect of an efficient numerical integration technique on the element-free Galerkin method