In this study we apply a Kansa--radial basis function (RBF) collocation method to 2D and 3D boundary value problems (BVPs) governed by high order partial differential equations (PDEs) of order 2N where N ≥ 3. As in such problems there are N boundary conditions (BCs), N distinct sets of boundary centers are needed. These could all be placed on the boundary with each set being different to the other or, alternatively, each set of boundary centers could be placed on a corresponding distinct curve surrounding the boundary of the problem. We apply these approaches to several 2D and 3D high order BVPs.

The complex-valued fundamental solution ln(z) can be decomposed into the radial basis function (RBF) and the angular basis function (ABF). The behavior of a screw dislocation is similar to the Arg(z) instead of sink or source. Hence, the ABF is well suited to model a screw dislocation. In this paper, not only the RBF in the null-field BIE but also the ABF for the screw dislocation are employed to study the interaction between a screw dislocation and an elastic elliptical inhomogeneity. This problem is decomposed into a free field with a screw dislocation and a boundary value problem containing an elliptical inhomogeneity. The boundary value problem is solved by using the RBF and the boundary integral equation. Since the geometric shape is an ellipse, the degenerate kernel is expanded to a series form under the elliptical coordinates, while the unknown boundary densities are expanded to eigenfunctions. By combining the degenerate kernel and the null-field integral equation, the boundary value problem can be easily solved. The inconsistency between Sendeckyj (1970b) and Gong & Meguid (1994) for the problem was also found by using the present approach. The error in (Gong and Meguid 1994) was also printed out. Finally, some examples are demonstrated to verify the validity of the present approach.

The plane elasticity problem with the line segments is studied by using BEM/BIEM. The degenerate scale is analytically examined and numerically performed. Different from the past result with complex variables, we propose a new idea to deal with the problem, and obtain the analytical solution of the degenerate scale. This analytical derivation can clearly show the appearing mechanism why the BEM/BIEM suffer the degenerate scale in the plane elasticity problem with the line segment. Double-degeneracy of degenerate boundary and degenerate scale in the BEM are numerically examined, respectively. The double-degeneracy mechanism is clearly displayed through numerical results by showing the number of zero singular values in the influence matrix. The degenerate scales for the plane elasticity problem have a relation with the degenerate scale of the Laplace problem. Following the result of single line segment, we can extend to multiple line segments. Finally, the analytical and numerical results show consistency.

In this paper, a meshfree boundary integral equation method is proposed to solve anti-plane problems containing a hole or a rigid inclusion under uniformly remote shear. The governing equation is the 2D Laplace equation. Only boundary nodes are required for the present method different from the conventional boundary element method that needs to generate the mesh. To avoid the calculating singular integral in the sense of the Cauchy principal value and determining the solid angle, a boundary integral equation of a local exact solution for a boundary point is introduced. This local exact solution must satisfy three conditions, including Laplace equation for the domain point, original boundary datum and its normal derivative on a boundary point. Therefore, a local exact solution is a linear combination of boundary data with corresponding shape functions which satisfy the 2D Laplace equation. By subtracting the boundary integral equation of a local exact solution from the boundary integral equation of original problem, the singular integral in the sense of the Cauchy principal value can be technically determined. Free of calculating the solid angle for the boundary point is also gained. Then, the Gaussian quadrature is employed only once for the above boundary integral equation, the boundary integral equation is nothing but an algebraic equation. This is the reason why only boundary nodes are required in the present method. Those boundary nodes are just the Gaussian quadrature points. Simultaneously, they are also the collocation points to obtain the simultaneous equation. This idea can also preserve the flexibility of numerical method, hence it is suitable for any geometry shape. Finally, some shapes of a hole or a rigid inclusion such as circle, ellipse, triangle and square are considered to examine the validity of the present meshfree boundary integral equation method.

本研究發展一套能處理乾濕結點變動的二維淺水方程無網格數值模式。相較於本文作者先前發展的數值模式，這個新的數值模式在計算上更有效率。
二維淺水方程式為一組非線性的時變方程式，求解過程中，需要對時間及空間做離散。在時間方面，我們採用預測-修正法(predictor-corrector)，為一種顯式數方法。而空間方面，需要求出比如水面高程、流速等各項物理量的偏導數，則採用加權多項式局部近似法(weighted-least-squares local polynomial approximation)，為一種無網格法。以往在處理乾濕交界處的乾濕結點變動時，由於乾點不具備水位這個物理量，故在建構水位的局部近似所需包含鄰近濕點的數量會一直變動，然後需要不斷重複做蒐尋鄰近濕點並求解反矩陣的動作，會相當耗費計算時間。在這個新的模式中，我們為每個乾濕交界處的乾結點設定了一個虛擬的水位，這樣不用重複進行前述的蒐尋鄰近濕點並求解反矩陣的動作，大幅減少計算所需的時間。
節省時間的幅度，則視計算域內乾濕交界線的長度而定。愈長，則節省時間的效果愈明顯。在所展示的算例中，新模式所花的時間約為舊模式的三分之一。

This study presents a bridge with functional bearing modelling as an rigid deck, a viscoelastoplastic pier, a viscoelastic bearing, and Coulomb’s dry friction interface and then the viscoelastoplastic analysis of the model under earthquake excitations is developed in this study. The model is recognized as four phase system capable of obtaining the exact solution of each phase. In the computation, the four computing modules according to the exact solution of each phase and four pull-back modules which determine the exact time of phase switching are arranged. Then we construct the computational algorithm based on the arrangement. Furthermore, the phase transition diagram, the history of pier’s plastic equivalent, and the history of the bearing’s equivalent sliding distance are established. The phase diagram qualitatively reflects the similar feature of the near-far earthquake. On the other hand, he history of the plastic equivalent and that of the equivalent sliding distance quantitatively gives the index to evaluate the plastic degradation of the pier and the fraction interface consumption of the bearing.

In this study, we investigated the vibration analysis of a finite bar with an external spring on one side and the support motion on the other side. Two analytical methods, the mode superposition method in conjunction with the quasi-static decomposition method and the method of characteristics using the diamond rule, were employed to solve this problem. Both advantages and disadvantages of two methods were discussed. It is interesting to find that the mode superposition method can capture the silent area in terms of sum of an infinite series while the method of characteristics using the diamond rule can exactly derive the dead zone.