Natural Neighbour Galerkin Methods Research Articles

A smoothed natural neighbour Galerkin method for flexoelectric solids

A smoothed natural neighbour Galerkin method for flexoelectric solids

Nonlocal nonlinear analysis of functionally graded plates using natural neighbour Galerkin method

In the present work, flexural response of functionally graded plates subjected to transverse loads have been investigated using the meshless natural neighbor Galerkin method (NNGM). The plate formulation has been developed based on the Reddy’s (Mechanics of laminated composite plates and shells: theory and analysis, 2nd edition, CRC Press, Boca Raton, 2014) third-order shear deformation theory (TSDT) using the von Karman nonlinear strains. The governing equations of the TSDT have been derived accounting for the length scale/size effects considering the Eringen’s nonlocal stress-gradient model (Eringen in Microcontinuum filed theories—I: foundations and solids, Springer-Verlag, 1998). The C1 continuous shape functions have been computed using the sibson’s interpolant and generalizing a Bezier patch over the domain. The nonlocal nonlinear model of the resulting governing equations has been developed, and Newton’s iterative procedure is used for the solution of nonlinear algebraic equations. The mechanical properties of functionally graded plate are assumed to vary continuously through the thickness and obey a power-law distribution of the volume fraction of the constituents. The variation of volume fractions through the thickness have been computed using two different homogenization techniques, namely, the rule of mixtures and the Mori–Tanaka scheme. A detailed parametric study to show the effect of side-to-thickness ratio, power-law index, and nonlocal parameter on the load-deflection characteristics of plates have been presented. The central deflections obtained using (NNGM) have been compared with the results from literature based on finite element method. The results have been compared with the two homogenization schemes and also with results computed with the first-order shear deformation theory (FSDT) to show the accuracy of nonlocal nonlinear formulation based on TSDT.

Phase field modeling of fracture in Quasi-Brittle materials using natural neighbor Galerkin method

Recently developed phase field models of fracture require a diffusive crack representation based on an introduction of a crack phase field. We outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles. In this work, the numerical implementation of the phase field model for brittle fracture using natural neighbor Galerkin method is presented. Phase field method diffuses the sharp discontinuity by introducing a regularized crack functional. The notion of regularized crack functional provides a basis for defining suitable convex dissipation functions which govern the evolution of the crack phase field. The use of natural neighbors to dynamically decide the compact support at a nodal point makes it truly mesh free and nonlocal. The method uses smooth non-polynomial type Sibson interpolants, which are C0 at a given node and C∞ everywhere else. This allows to capture the interface region in a phase field model more accurately with only few nodes. For a given accuracy, there is an improvement in the computational time with the use of natural neighbor interpolants. Several benchmark problems were solved to investigate the efficiency of the numerical implementation of phase field model using natural neighbor Galerkin method.

An interpolation filter based on natural neighbor Galerkin method for salt and pepper noise restoration with adaptive size local filtering window

In this paper, we present a new interpolation filter for salt and pepper noise (SPN) restoration in digital images. The proposed filter is established on decision-based filters (DBF) and consists of two units: noise detection and noise restoration. In noise restoration unit, the natural neighbor Galerkin method (NNGM), which is a two-dimensional scattered data interpolation, is adopted for estimating the intensities of noisy pixels. For each detected noisy pixel, an adaptive size local filtering window is considered and NNGM interpolation is applied locally. Compared to the other state-of-the-art DBFs for SPN removal, which are based on interpolation methods, our proposed method has better performance in terms of both objective and subjective assessments. In our proposed method, the noise restoration unit requires no manual parameter tuning. Numerous experiments results demonstrate that our proposed filter removes SPN effectively and preserves the details and edges well.

Meshless natural neighbor Galerkin method for the bending and vibration analysis of composite plates

In the present work we present a meshless natural neighbor Galerkin method for the bending and vibration analysis of plates and laminates. The method has distinct advantages of geometric flexibility of meshless method. The compact support and the connectivity between the nodes forming the compact support are performed dynamically at the run time using the natural neighbor concept. By this method the nodal connectivity is imposed through nodal sets with reduced size, reducing significantly the computational effort in construction of the shape functions. Smooth non-polynomial type interpolation functions are used for the approximation of inplane and out of plane primary variables. The use of non-polynomial type interpolants has distinct advantage that the order of interpolation can be easily elevated through a degree elevation algorithm, thereby making them suitable also for higher order shear deformation theories. The evaluation of the integrals is made by use of Gaussian quadrature defined on background integration cells. The plate formulation is based on first order shear deformation plate theory. The application of natural neighbor Galerkin method formulation has been made for the bending and free vibration analysis of plates and laminates. Numerical examples are presented to demonstrate the efficacy of the present numerical method in calculating deflections, stresses and natural frequencies in comparison to the Finite element method, analytical methods and other meshless methods available in the literature.

Natural element method for radiative heat transfer in two-dimensional semitransparent medium

The natural element method (NEM) is employed for solving radiative heat transfer problem in a two-dimensional enclosure containing an absorbing, emitting and isotropically scattering medium. Medium boundaries are considered to be opaque, diffuse as well as gray. The NEM referred to as natural neighbor Galerkin method is a new technique in the field of computational mechanics and can be considered as a meshless numerical method. Unlike most of other meshless methods such as element-free Galerkin method or those based on radial basis functions, the shape functions used in NEM, which are based on the Voronoi tesselation of a set of nodes, are strictly interpolant and the essential boundary conditions can be imposed by directly substituting the corresponding terms in the system of equations. Three types of radiative heat transfer problems are addressed. For pure radiation, studies are made for three representative benchmark problems dealing with radiative equilibrium and non-radiative equilibrium. For coupled heat transfer, transient conduction and radiation problem in a rectangular geometry, steady conduction and radiation problem in a gray cylindrical ring are solved. For these problems considered, tests are presented for various parameters, such as the aspect ratio, conduction-radiation parameter, optical thickness, scattering albedo and surface emissivity. Results for NEM are compared with those for finite element method (FEM) generated by the authors and those reported in the literatures. By comparison, it is shown that NEM is efficient, accurate and stable, and can be used for solving radiative heat transfer problem in 2-D semitransparent media.

Application of the Gradient Smoothing Technique to the Natural Neighbour Galerkin Method for the Couple-Stress Elasticity

Summary The natural neighbour Galerkin method is tailored to solve boundary value problems of the couple-stress elasticity to model the size dependent behaviour of materials. This method is based on the displacement-based Galerkin approach, and the calculation of the global stiffness matrix is performed using gradient smoothing technique combined with the non-Sibsonian partition of unity approximation scheme. This method possesses the following properties: the complex C1-continuous approximation scheme is avoided without using either Lagrange multipliers or penalty parameters; no domain integrals involved in the assembly of the global stiffness matrix; and the imposition of essential boundary conditions is straightforward. The validity and accuracy of this method are investigated through numerical examples. The results show that strong size effects can be captured by the numerical method when the length of deformation field and the characteristic length of the material are comparable, and good agreements with analytical solutions are obtained.

Mixed Natural Neighbor Galerkin Method for 3D Couple-Stress Elasticity

Based on the constrained variational principle, the mixed multi-variable natural neighbor Galerkin method for couple-stress elasticity is proposed for three-dimensional problems. The displacements and micro-rotations are taken to be independent nodal degrees of freedom, and the geometrical constraints between them are enforced through Lagrange multipliers. The C0-continuous non-Sibsonain interpolation is used to obtain the discrete equations. The proposed method is tested by the numerical example and the results show that strong size effects are observed when the length of deformation field and the characteristic length of the material are comparable.

Natural neighbor Galerkin method for deflection analysis of inflatable structures

PurposeThe purpose of this paper is to extend the application of natural neighbor Galerkin Method (NNGM) to deflection analysis of inflatable structures such as new and modern textile structures under arbitrary conditions.Design/methodology/approachInflatable structures have a proper mechanical strength when they are inflated at different pressures. NNGM or natural element method (NEM) is defined as a new meshfree method based on the natural neighbor interpolation to analyze the deflections of these structures under arbitrary pressures, load and support conditions. The whole interpolation is built with regard to the natural neighbor nodes and Voronoi tessellation of the given point.FindingsThe performance of NNGM is investigated in the deflection analysis of inflatable tubes and panels. The excellent agreement between the presented modeling and analytical results and also finite element solutions and experiment are observed.Originality/valueDespite the wide usage of NNGM in many engineering problems, this comparison shows the other aspect of application of NNGM in the deflection analysis of inflatable structures, not previously examined.

Natural neighbour Galerkin computation of the vibration modes of fluid‐structure systems

PurposeThis paper seeks to extend the application of the natural neighbour Galerkin method to vibration analysis of fluid‐structure interaction problems.Design/methodology/approachThe natural element method (NEM) which is a meshless technique is used to simulate the vibration analysis of the fluid‐structure interaction systems. The method uses the natural neighbour interpolation for the construction of test and trial functions. Displacement variable is used for both the solid and the fluid domains, whereas the fluid displacement is written as the gradient of a potential function. Two classical examples are considered: free vibration of a flexible cavity filled with liquid and vibration of an open vessel containing liquid. The corresponding eigenvalue problems are solved and the results are compared with the finite element method (FEM) and analytical solutions to show the accuracy and convergence of the method.FindingsThe performance of the NEM is investigated in the computation of the vibration modes of the fluid‐structure interaction problems. Good agreement with analytical and FEM solutions are observed. Through the notable obtained results, it is found that the NEM can also be used in vibration analysis of fluid‐structure interaction problems as it has been successfully applied to some problems in solid and fluid mechanics during the recent years.Originality/valueIn spite of notable achievements in solving some problems in solid and fluid mechanics using NEM, the vibration analysis of fluid‐structure interaction problems, as considered in this paper, has not been investigated so far.

Recent advances in the meshless simulation of aluminium extrusion and other related forming processes

In this paper we review some recent results in the field of numerical simulation of extrusion and other forming processes obtained by the authors by using a meshless approach, together with a wide review of the existing bibliography on the topic. Three main alternatives exist in the literature, namely (updated) Lagrangian, Eulerian and arbitrary Lagrangian-Eulerian (ALE) methods. A review of the most important characteristics of each of these three approaches is here presented and their possible advantages are pointed out. Finally, an updated Lagrangian approach over a meshless approximation, based on a class of methods globally coined as natural element methods (also as natural neighbour Galerkin methods) is analysed and its relative advantages studied. Some numerical examples are included that clearly show the potential capabilities of the proposed method.

A natural neighbour Galerkin method with octree structure

We present in this paper a highly structured numerical method based on the employ of natural neighbour interpolation in a Galerkin framework. It employs an octree discretisation of the domain, thus being suitable for numerical simulations in voxelized domains, obtained after image processing, for instance, and medical image techniques in particular, with nearly no user intervention. The accuracy of the method and computational cost are also addressed in this paper. We present some examples that illustrate this behaviour.

A natural neighbour Galerkin method with quadtree structure

We describe in this paper a highly structured numerical method that allows for an important speedup in the calculations. The method is implemented in a bi-dimensional binary tree (quadtree or octree) structure in a partition of unity framework. The partition of unity is constructed by using natural neighbour interpolation. Data can be easily obtained from voxel or pixel-based images, as well as STL files or other CAD descriptions. The method described here possesses linear completeness at least and essential boundary conditions are implemented through the characteristic function method, by employing a special class of functions called R-functions. After the theoretical description of the method, some examples of its performance are presented and analysed. Copyright © 2005 John Wiley & Sons, Ltd.

Volumetric locking in natural neighbour Galerkin methods

AbstractThe behaviour of natural neighbour Galerkin mixed approximation is described at the incompressible limit. Traditional natural elements based on the Sibson mixed interpolation do not verify the LBB (or inf–sup) condition. Here, a study of the possible benefits of enriching the interpolation is considered using the partition of unity paradigm. Different enrichments were tested leading to different reproducing properties. Results concerning the numerical verification of the inf–sup tests are addressed. Also the convenience of using different approximations developed in this work is analysed. Enrichment with the polynomial field {1, xy} seems to verify the LBB condition. Its behaviour has proven to be very similar to the MINI Finite Element, widely used in the literature. Examples proving these results are provided. Copyright © 2004 John Wiley & Sons, Ltd.

Numerical integration in Natural Neighbour Galerkin methods

AbstractIn this paper, issues regarding numerical integration of the discrete system of equations arising from natural neighbour (natural element) Galerkin methods are addressed. The sources of error in the traditional Delaunay triangle‐based numerical integration are investigated. Two alternative numerical integration schemes are analysed. First, a ‘local’ approach in which nodal shape function supports are exactly decomposed into triangles and circle segments is shown not to give accurate enough results. Second, a stabilized nodal quadrature scheme is shown to render high levels of accuracy, while resulting specially appropriate in a Natural Neighbour Galerkin approximation method. The paper is completed with several examples showing the performance of the proposed techniques. Copyright © 2004 John Wiley & Sons, Ltd.

Updated Lagrangian free surface flow simulations with natural neighbour Galerkin methods

AbstractIn this paper, a new method to simulate free surface fluid flows within an updated Lagrangian framework is described. It is based on the use of a meshless technique coined as natural element method (NEM) or, more recently, as natural neighbour Galerkin method. The position of the flow front or the geometry of the fluid domain is handled by invoking the geometrical concept of α‐shape of the cloud of points, thus avoiding the explicit definition of the boundary of the domain as it evolves. This method also avoids the traditional need of remeshing typical in finite element simulations of this kind of processes. Three types of fluid behaviour have been considered, namely a purely Newtonian fluid, a non‐Newtonian short fibre‐reinforced thermoplastic, and finally a Norton–Hoff viscoplastic behaviour. Benchmark examples showing the performance of the technique are included in the paper. Copyright © 2004 John Wiley & Sons, Ltd.

Overview and recent advances in natural neighbour galerkin methods

In this paper, a survey of the most relevant advances in natural neighbour Galerkin methods is presented. In these methods (also known as natural element methods, NEM), the Sibson and the Laplace (non-Sibsonian) interpolation schemes are used as trial and test functions in a Galerkin procedure. Natural neighbour-based methods have certain unique features among the wide family of so-called meshless methods: a well-defined and robust approximation with no user-defined parameters on non-uniform grids, and the ability to exactly impose essential (Dirichlet) boundary conditions are particularly noteworthy. A comprehensive review of the method is conducted, including a description of the Sibson and the Laplace interpolants in two- and three-dimensions. Application of the NEM to linear and non-linear problems in solid as well as fluid mechanics is studied. Other issues that are pertinent to the vast majority of meshless methods, such as numerical quadrature, imposing essential boundary conditions, and the handling of secondary variables are also addressed. The paper is concluded with some benchmark computations that demonstrate the accuracy and the key advantages of this numerical method.

On the imposition of essential boundary conditions in natural neighbour Galerkin methods

AbstractIn this paper issues related to the imposition of essential boundary conditions in Natural Neighbour Galerkin methods are addressed. Both Sibson and non‐Sibson interpolants ability to exactly reproduce essential boundary conditions is investigated and a new analytical condition ensuring linear precision along explicitly described (i.e. CAD) boundaries in both two and three dimensions is presented. The paper is completed with some benchmark numerical examples. Copyright © 2003 John Wiley & Sons, Ltd.

Natural neighbour Galerkin methods

Natural neighbour Galerkin methods

Natural neighbour Galerkin methods

SUMMARY Natural neighbour co-ordinates (Sibson co-ordinates) is a well-known interpolation scheme for multivariate data tting and smoothing. The numerical implementation of natural neighbour co-ordinates in a Galerkin method is known as the natural element method (NEM). In the natural element method, natural neighbour co-ordinates are used to construct the trial and test functions. Recent studies on NEM have shown that natural neighbour co-ordinates, which are based on the Voronoi tessellation of a set of nodes, are an appealing choice to construct meshless interpolants for the solution of partial dierential equations. In Belikov et al. (Computational Mathematics and Mathematical Physics 1997; 37(1):9{15), a new interpolation scheme (non-Sibsonian interpolation) based on natural neighbours was proposed. In the present paper, the nonSibsonian interpolation scheme is reviewed and its performance in a Galerkin method for the solution of elliptic partial dierential equations that arise in linear elasticity is studied. A methodology to couple nite elements to NEM is also described. Two signicant advantages of the non-Sibson interpolant over the Sibson interpolant are revealed and numerically veried: the computational eciency of the non-Sibson algorithm in 2-dimensions, which is expected to carry over to 3-dimensions, and the ability to exactly impose essential boundary conditions on the boundaries of convex and non-convex domains. Copyright ? 2001 John Wiley & Sons, Ltd.