Study On The Element-free Galerkin Method And Its Applications

Meshfree (or meshless) methods are a class of newly developing numerical methodsfor solving partial diferential equations. Currently, they have been attracted many re-searchers' attention. A group of meshfree methods have been proposed and developed.Among them, the element-free Galerkin (EFG) method is a very promising method andcurrently widely used in computational mechanics and other scientific and engineeringareas. In this paper, the EFG method will be further studied and four aspects will beconsidered, i.e., the construction of shape functions, the imposition of essential bound-ary conditions, combing with other numerical methods and the solvers for discrete linearequations. The new method proposed in this paper will be applied for solving2D linearelasticity problem and Poisson equations.Element-free Galerkin method is a method based on global weak form and movingleast square (MLS) approximation. One disadvantage of MLS approximation is that aset of linear algebraic equations must be solved for every point of interest. To avoid thecalculation of matrix inversion in the formulation of the shape functions, an improved MLS(IMLS) approximation in which the orthogonal function system with a weight function isused as the basis function. It has been shown that the IMLS approximation has greatercomputational efciency and precision than that of the MLS approximation. But it canalso lead to ill-conditioned or even singular system of equations. In this paper, aspectsof the IMLS approximation are analyzed in detail. The reason why singularity problemoccurs is studied. A novel technique based on matrix triangular process is proposed tosolve this problem. Our study is relied on monomial basis functions, but it can be extendedto any basis functions. It is shown that the EFG method with present technique is veryefective in constructing shape functions.A point interpolation meshless method based on radial basis functions is a methodcombines the Galerkin weak form and radial basis functions to form a meshfree radialpoint interpolation method (RPIM). In fact, it is a special case of the EFG method.In the RPIM method, it uses the radial point interpolation method to construct theshape functions. This is the major diference between the EFG method and the RPIMmethod. It has been studied that the RPIM method has more computational cost thanthe EFG method, due to the dimension of the moment matrix for every computationalpoint is much bigger than that obtained by the MLS approximation in the EFG method.In this paper, a novel weighted nodal-radial point interpolation meshless (WN-RPIM)method is proposed. In the new approach, the moment matrices are performed only atthe nodes to get nodal coefcients. At each computational point, the shape functions are obtained by weighting the nodal coefcients whose nodes are located in the supportdomain. The shape functions obtained by the new scheme preserves the Kronecker deltafunction property in certain conditions. The new method has much less time consumingthan the RPIM method, since the number of nodes is generally much smaller than thenumber of integration points. Some numerical examples are illustrate the efectiveness ofthe proposed method.Another disadvantage of the MLS approximation is that the shape functions ob-tained are lack of Kronecker delta function property. As a consequence, the impositionof essential boundary conditions in the EFG method is quite awkward. In this paper,some existing methods, which are used to impose the essential boundary conditions, aresummarized from four aspects, i.e., problem domain, variational principle, constructionof shape functions and discrete linear equations. A new method called coupled element-free Galerkin-radial point interpolation method, which can impose essential boundaryconditions directly, is also proposed.Since the EFG method is based on global weak form, it needs integrations in theglobal problem domain. This makes the EFG method quite burdensome. The meshlessGalerkin least-squares method is an improvement of the traditional Galerkin method andthe meshless least-squares method, it has less computational cost, high accuracy andsome other advantages. But there are difculties in the imposition of essential boundaryconditions since it uses the MLS approximation. Using radial basis function has someadvantages, such as simple form, space dimension independent and so on. In this paper,in combination of the advantages of these two methods, a new meshless method-meshlessGalerkin least-squares method based on radial basis functions is proposed. The newmethod can impose essential boundary conditions directly and has high accuracy.It has been studied that using augmented Lagrange multiplier method to imposethe essential boundary conditions in the EFG method has high computational accuracy.But this approach leads to an indefinite coefcient matrix. The discrete linear system iscalled saddle point problem. In this paper, the preconditioned GMRES method is usedto solve the discrete linear system of equations. In general, a preconditioner should beconstructed first. The preconditioner should reduce the number of iterations required forconvergence but not significantly increase the amount of computation required at eachiteration. In this paper, estimates for the eigenvalues of the preconditioned saddle pointmatrices with two existing block triangular preconditioners are further studied and twonew preconditioners are proposed. Some numerical experiments are also presented.