Example: biology

Overview of Meshless Methods - Compumag

Technical ArticleOverview of Meshless MethodsAbstract This article presents an Overview of the main develop -ments of the mesh-free idea. A review of the main publicationson the application of the Meshless Methods in ComputationalElectromagnetics is also INTRODUCTIONS everal Meshless Methods have been proposed over the lastdecade. Although these Methods usually all bear the genericlabel Meshless , not all are truly Meshless . Some, such as those basedon the Collocation Point technique, have no associated meshbutothers, such as those based on the Galerkin method, actuallydorequire an auxiliary mesh or cell structure. At the time of writing,the authors are not aware of any proposedformalclassification ofthese techniques. This paper is therefore not concerned with anyclassification of these Methods , instead its objective is topresentan Overview of the main developments of themesh-freeidea,followed by a review of the main publications on the applicationof Meshless Methods to Computational MESHLESSMETHODS- THEHISTORYA.

Technical Article Overview of Meshless Methods Abstract—This article presents an overview of the main develop-ments of the mesh-free idea. A …

Tags:

  Methods, Overview, Ment, Develop, Develop ment, Overview of meshless methods, Meshless

Information

Domain:

Source:

Link to this page:

Please notify us if you found a problem with this document:

Other abuse

Transcription of Overview of Meshless Methods - Compumag

1 Technical ArticleOverview of Meshless MethodsAbstract This article presents an Overview of the main develop -ments of the mesh-free idea. A review of the main publicationson the application of the Meshless Methods in ComputationalElectromagnetics is also INTRODUCTIONS everal Meshless Methods have been proposed over the lastdecade. Although these Methods usually all bear the genericlabel Meshless , not all are truly Meshless . Some, such as those basedon the Collocation Point technique, have no associated meshbutothers, such as those based on the Galerkin method, actuallydorequire an auxiliary mesh or cell structure. At the time of writing,the authors are not aware of any proposedformalclassification ofthese techniques. This paper is therefore not concerned with anyclassification of these Methods , instead its objective is topresentan Overview of the main developments of themesh-freeidea,followed by a review of the main publications on the applicationof Meshless Methods to Computational MESHLESSMETHODS- THEHISTORYA.

2 The Smoothed Particle HydrodynamicsThe advent of themesh freeidea dates back from 1977,with Monaghan and Gingold [1] and Lucy [2] developing aLagrangian method based on theKernel Estimatesmethod tomodel astrophysics problems. This method, namedSmoothedParticle Hydrodynamics(SPH), is aparticle methodbased onthe idea of replacing the fluid by a set of moving particles andtransforming the governing partial differential equations into thekernel estimates integrals [3].Despite the success of the SPH in modelling astrophysicsphenomena, it was only after the 90 s that the SPH was ap-plied to model others classes of problem. This consequentlyexposed some inherent characteristics of the method, such astensile stability, energy zero, lack of interpolation consistencyand difficulty in enforcing essential boundary conditions [4].

3 The latter two are consequences of using the SPH to modelbounded problems, since originally the SPH was formulated tomodel open problems. Over the past years many modificationsand correction functions have been proposed in an attempt torestore the consistency and consequently the accuracy of themethod [5]. The SPH method has been successfully appliedto a wide range of problems such as free surface, impact,magnetohydrodynamics, explosion phenomena, heat conductionand many other computational mechanics problems which werepresented and discussed in extensive reviews of SPH [6] [5].B. The Diffuse Element Method (DEM)The firstmesh-freemethod based on the Galerkin technique wasonly introduced over a decade after Monaghan and Gingold firstpublished the SPH method. TheDiffuse Element Method(DEM)was introduced by Nayroles and Touzot in 1991.

4 Many authorsstate that it was only after the Diffuse Element method that theidea of amesh-freetechnique began to attract the interest of theresearch community. The idea behind the DEM was to replacethe FEM interpolation within an element by theMoving LeastSquare(MLS) local interpolation [7].Fig. background cell The Element-Free Galerkin (EFG)In 1994 Belytschko and colleagues introduced theElement-FreeGalerkin Method(EFG) [8], an extended version of Nayroles smethod. The Element-Free Galerkin introduced a series of im-provements over the Diffuse Element Method formulation, suchas Proper determination of the approximation derivatives:In DEM the derivatives of the approximation functionUhare obtained by considering the coefficientsbof thepolynomial basispas constants, such thatdUh(x)dx=dpT(x)dxb(x)(1)In EFG thefullform of the derivatives is used, such thatdUh(x)dx=dpT(x)dxb(x) +pT(x)db(x)dx(2)Belytschko argued in his paper that neglecting the deriva-tives ofb(x)detracts significantly from the accuracy of themethod.

5 Imposing essential boundary conditions: The MLS trialfunction does not yield an interpolation, (x)6=Uh(x), therefore the essential boundary conditions are notdirectly satisfied. Belytschko showed that the DEM fails topass the patch test due to the fact that these conditions arenot properly fulfilled. In the EFG formulation, LagrangeMultipliers are used in the weak form to enforce theessential boundary conditions. Process for Numerical Integration: Meshless methodsbased on the Galerkin technique require numerical integra-tion of the weak form. In the Element-Free Galerkin anauxiliary cell structure, shown in Fig. 1, is used in order tocreate a structure to define the quadrature Element-Free Galerkin method was found to be more accu-rate than the Diffuse Element method, although the improve-ments implemented in the method increased its computationalcosts [8].

6 EFG is one of the most popular mesh-free Methods andits application has been extended to different classes of problemssuch as fracture and crack propagation, wave propagation ,acoustics and fluid authors state that the use of a background cell to performthe numerical integration eliminates themesh-freecharacteristicof the EFG, therefore the method is not truly Reproducing Kernel Particle MethodIn 1995 Liu proposed theReproducing kernel particle method(RKPM) [9] in an attempt to construct a procedure to correct thelack of consistency in the SPH method. The RKPM has beensuccessfully used in multiscale techniques, vibration analysis,fluid dynamics and many other applications. Later, a combina-tion of MLS and RKPM resulted in theMoving Least SquareReproducing Kernel Particle method[10] [11].

7 E. Finite Point MethodTheFinite Point methodwas proposed by On ate and colleaguesin 1996 [12] [13]. It was originally introduced to model fluidflow problems and later applied to model many other mechanicsproblems such as elasticity and plate bending. The method isformulated using the Collocation Point technique and any ofthe following approximation techniques,Least Square approxi-mation(LSQ),Weighted Least Square approximation(WLS) orMoving Least Squares(MLS) can be used to construct the Meshless Local Petrov-GalerkinThe Meshless Local Petrov-Galerkin introduced by Atluri andZhu in 1998 [14] presents a different approach in constructing amesh-freemethod. It is based on the idea of theLocal weakformwhich eliminates the need of the background cell and,consequently, performs the numerical integration in a mesh-free sense.

8 The MLPG uses the Petrov-Galerkin method inan attempt to simplify the integrand of theweak form. Themethod was originally formulated using the MLS technique andlater Atluri extended the MLPG formulation to other meshlessapproximation techniques. The freedom of choice for the testfunction in the Petrov-Galerkin method gives rise to differentMLPG schemes [15]. The MLPG and its different schemes havebeen applied to a wide range of problems such as Euler-BernoulliBeam Problems, solid mechanics, vibration analysis for solids,transient heat conduction, amongst many Radial Basis FunctionsRadial basis functions (RBF) were first applied to solve partialdifferential equations in 1991 by Kansa, when a technique basedon the direct Collocation method and the Multiquadric RBF wasused to model fluid dynamics [16] [17].

9 The direct Collocationprocedure used by Kansa is relatively simple to implement, how-ever it results in an asymmetric system of equations due to themix of governing equations and boundary conditions. Moreover,the use of Multiquadric RBF results in global approximation,which leads to a system of equations that is characterised byadense stiffness matrix. The RBF Hermite-Collocation was pro-posed as an attempt to avoid the asymmetric system of globally and compactly supported radial basis functionshave been used to solve PDEs and results have shown thatthe global RBF yielded better accuracy. However the compactlysupported stiffness matrix is sparse, while the global RBF resultsin a dense matrix that may become very expensive to solve in thecase of a large number of collocation points. Recently, anotherapproach based on the global RBFs has been proposed.

10 Themethod, namedLocal Multiquadric, suggests the construction ofthe approximation using sub-domains, causing the MultiquadricRBF to be truncated at the boundaries of the sub-domains,resulting in a local approximation and a sparse stiffness matrix[18].Radial basis functions have also been used in the BoundaryElement method formulation, such as in the Dual ReciprocityMethod (DRM), Method of Fundamental solution (MFS) andthe RBF Boundary Knot method (BKM). These Methods havebeen successfully applied to solve non-linear problems in Com-putational variational approach to solve the Boundary Value Partial(BVP) using compactly supported radial basis functions hasbeeninvestigated by [19] and another approach suggested the useof radial basis functions in the Meshless Local Petrov-Galerkinformulation [15].


Related search queries