Numerical development of an improved elementfree Galerkin method for engineering analysis
改進的無單元 Galerkin 法的數值發展及其工程應用
Student thesis: Doctoral Thesis
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Award date  2 Oct 2009 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(396fa6a9051245f788837f63e3de6047).html 

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Abstract
Numerical methods are indispensable in the successful simulation of physical problems, and various types of differential or partial differential governing equations have been derived for these physical phenomena. Similar to conventional numerical methods such as the finite element method (FEM), the finite difference method (FDM), and the boundary element method (BEM), the meshless method is a useful tool for solving the partial differential equations that govern physical phenomena.
The common feature of meshless methods is that they do not use predefined meshes, at least for field variable interpolation. The main difference between meshless methods and conventional numerical methods is the way in which the shape function is formulated. However, once the shape function has been obtained, the meshless methods, the FEM, and the BEM use the same procedure to form the equations to obtain the solution to a problem.
The elementfree Galerkin (EFG) method is a promising meshless method that does not require data about element connectivity and does not suffer much degradation in accuracy when the nodal arrangement is very irregular. The method has been demonstrated to be quite successful in elasticity, heat conduction, wave equation, and fatigue crack growth modeling.
A major feature of the EFG method is that it uses the moving leastsquares (MLS) approximation to construct the shape function. The MLS approximation was developed from the conventional leastsquares method, and in practical numerical processes essentially involves the application of the conventional method to every selected point. A disadvantage of the conventional method is that the final algebraic equations system is sometimes ill conditioned. Hence, it is necessary to solve the illconditioned algebraic equations system to achieve an adequate MLS approximation. However, it is difficult to determine which of the algebraic equations is ill conditioned to the extent that it is sometimes impossible to obtain a good solution or even a correct numerical solution. To overcome this problem, the improved moving leastsquares (IMLS) approximation has been developed to obtain the approximation function. In the IMLS, an orthogonal function system with a weight function is used as the basis function. With the IMLS approximation, the algebraic equation system is not ill conditioned and can be solved without having to derive the inverse matrix. There are also fewer coefficients in the IMLS approximation than in the MLS approximation, and hence the computation speed and the efficiency with which a solution is found are greater.
An improved elementfree Galerkin (IEFG) method has been created based on the IMLS approximation and the EFG method. As there are fewer coefficients in the IMLS than in the MLS approximation, fewer nodes are selected in the entire domain with the IEFG method than with the conventional EFG method which should result in a higher computation speed. Furthermore, the IEFG method has greater computational precision than the EFG method when the same nodes are selected in the domain.
In this thesis, an IEFG method is formulated for several engineering problems based on the IMLS approximation. A convergence study of the proposed method is carried out by analyzing the final function values under different discretization schemes and different scaling factors for the nodes of the study field. The main contents of the thesis are as follows.
First, the IEFG method for potential problem is presented and it is shown that compared with the EFG at the same level of precision, the IEFG has a greater computation speed. With both methods the solution converges when the number of nodes increases for various numbers of nodes with a certain , whereas, when the number of nodes is held unchanged and increase the value of , the finial results oscillate around the analytical solution. In additional, for some individual cases, use EFG and IEFG methods may get different results and precision. maxd
Forms of the IEFG method for transient heat conduction and the corresponding error estimates are then proposed. The convergence study and error analysis show that the error of the IEFG approximation is not only related to the radii of the weight functions and node number, but also to the step length.
Wave equations are then studied. Unlike transient heat conduction, the wave equations contain the term that represents the acceleration at point and two initial conditions must be prescribed: the initial displacement and the initial velocity. Several illustrative numerical results obtained by the IEFG method are presented alongside comparisons with the exact solutions and results obtained by the EFG method. In all cases, the wave equations with appropriate prescribed boundary conditions are solved. ttux
The IEFG method can also be applied to study elasticity, where the discrete equation is derived from the weak form of a variational equation in which penalty parameters are used to enforce the essential boundary conditions to obtain the corresponding formulae. Under certain , the relative error norm decreases as the number of nodes increases, and thus a higher completeness order of the basis function achieves a better convergence characteristic than a lower order. When the number of nodes is kept constant, the relative error norm decreases as increases. maxd
Finally, the IEFG method is used to study twodimensional elastodynamic problems. The Galerkin weak form for elastodynamic problems is employed to obtain the discretized system equations, and the Newmark time integration method is used for the time history analyses. In the solution process, the penalty method is employed to apply the essential boundary conditions to obtain the corresponding formulae of the IEFG method for elastiodynamic problems.
To show the efficiency of the proposed IEFG method, MATLAB codes of the method are written for various applications, and some numerical examples are provided to demonstrate its validity and efficiency.
 Numerical analysis, Meshfree methods (Numerical analysis), Galerkin methods