Research On Improved Interpolating Meshless Methods Based On Nonsingular Weight Functions

Meshless method is a new numerical method which is developed after the finiteelement method. The shape function of a meshless method is entirely constructed ona set of nodes in the problem domain or the boundary without considering the meshof the domain or the boundary. Then meshless method has some advantages such assimple pre-processing, high computing accuracy and avoiding volumetric locking.Now meshless method has been one of the important numerical methods for scienceand engineering computing.The moving least-squares (MLS) approximation is one of the most importantmethods to form the shape functions in meshless methods. And various meshlessmethods are presented based on the MLS approximation. A disadvantage of the MLSapproximation is that its shape function does not satisfy the property of Kronecker function, then the meshless methods based on the MLS approximation can notapply the essential boundary conditions directly and exactly. The shape function ofthe interpolating moving least-squares (IMLS) method presented by Lancastersatisfies the property of the Kronecker function, but the weight functions whichare singular at nodes must be used. Then in the IMLS method, it is complicated toobtain the derivatives of the shape functions at the nodes, and it is hard to realize thenumerical calculation. To overcome the disadvantages of the singularity of theweight functions in the IMLS method, a new improved interpolating movingleast-squares (IIMLS) method based on nonsingular weight functions is presented inthis thesis, the error estimates of the IIMLS method are studied, and the improvedinterpolating boundary element-free method and improved interpolating element-freeGalerkin method based on the IIMLS method are presented. The main contents ofthis paper are as follows.The IIMLS method based on nonsingular weight functions is presented. TheIIMLS method can overcome the difficulties caused by the singularity of the weightfunction in the IMLS method by Lancaster. The IIMLS method has fewerundetermined coefficients in the trial function than that in the conventional MLSapproximation, and the shape function of the IIMLS method satisfies the property ofKronecker function. The meshless methods based on the IIMLS method can apply the essential boundary condition directly and exactly, which makes highercomputational precision and efficiency.The error estimate of the IIMLS method based on the nonsingular weightfunctions are presented in n-dimensional space. The convergence rates of theapproximation and its derivatives of the IIMLS method are obtained. And thenumerical examples are given to confirm the theory.Using the IIMLS method to obtain the shape function, and combing theGalerkin weak form of potential problems, an improved interpolating element-freeGalerkin (IIEFG) method is presented for potential problems. Compared with theconventional EFG method, the IIEFG method has the advantages of fewerundetermined coefficients when obtaining the shape functions and applying theessential boundary conditions naturally and directly. Compared with theinterpolating element-free Galerkin (IEFG) method based on the IMLS method, theIIEFG method applies the nonsingular weight functions and has no difficultiesbecause of singular weight functions. Numerical examples are presented to show themethod in this thesis has higher computational precision and efficiency.Combining the IIMLS method with the boundary integral equation (BIE) ofpotential problems, an improved interpolating boundary element-free method(IIBEFM) is presented for potential problems. Compared with the original boundaryelement-free method (BEFM), the IIBEFM has the advantages of fewerundetermined coefficients when obtaining the shape functions and applying theboundary conditions directly and exactly. Compared with the original interpolatingboundary element-free method (IBEFM), the IIBEFM can overcome the difficultiescaused by the singular weight functions. And numerical examples are presented toshow the IIBEFM has higher computational precision.Using the IIMLS method to obtain the shape function, and combining theGalerkin weak form of elasticity problems, an IIEFG method is presented fortwo-dimensional elasticity problems. The IIEFG method has the advantages ofnonsingular weight function, fewer undetermined coefficients to obtain the shapefunction, and applying the essential boundary conditions exactly. Numericalexamples are given to show the IIEFG method has higher computational precision.Using the IIMLS method to obtain the shape function, and combining theincremental theory and the Galerkin weak form of elastoplasticity problems, anIIEFG method is presented for elastoplasticity problems. The IIEFG method for the elastoplasticity problems also has the advantages of nonsingular weight function,fewer undetermined coefficients to obtain the shape function, and applying theessential boundary conditions exactly. And numerical examples are given to showthe IIMLS methods for elastoplasticity has higher computational precision andefficiency.The researches on the IIMLS method and its mathematical theory in this thesiscan obtain certain foundation for further developments of new meshless methods andmathematical theories of meshless methods. And the researches on the interpolatingmeshless methods for linear and nonlinear problems in this thesis can provide newnumerical methods to solve some complicated nonlinear problems with highcomputational precision and efficiency.