Aiming at complex configuration and large deformation, an efficient hybrid method for dynamic mesh generation is presented in this paper, which is based on Radial Basis Functions (RBFs) and Delaunay graph mapping. Based on the computational mesh, a set of very coarse grid named as background grid is generated firstly, and then the computational mesh can be located at the background grid by Delaunay graph mapping technique. After that, the RBFs method is applied to deform the background grid by choosing partial mesh points on the boundary as the control points. Finally, Delaunay graph mapping method is used to relocate the computational mesh by employing area or volume weight coefficients. By applying different dynamic mesh methods to a moving NACA0012 airfoil, it can be found that the RBFs-Delaunay graph mapping hybrid method is as accurate as RBFs and is as efficient as Delaunay graph mapping technique. Numerical results show that the dynamic meshes for all test cases including one two-dimensional (2D) and two three-dimensional (3D) problems with different complexities, can be generated in an accurate and efficient manner by using the present hybrid method.

In [SIAM J. Sci. Comput., 35(2)(2013), A1049-A1072], a class of multi-domain hybrid DG and WENO methods for conservation laws was introduced. Recent applications of this method showed that numerical instability may encounter if the DG flux with Lagrangian interpolation is applied as the interface flux during the moment of conservative coupling. In this continuation paper, we present a more robust approach in the construction of DG flux at the coupling interface by using WENO procedures of reconstruction. Based on this approach, such numerical instability is overcome very well. In addition, the procedure of coupling a DG method with a WENO-FD scheme on hybrid meshes is disclosed in detail. Typical testing cases are employed to demonstrate the accuracy of this approach and the stability under the flexibility of using either WENO-FD flux or DG flux at the moment of requiring conservative coupling.

The responses to an incident plane SH wave on or near a surface irregularity which is embedded in an elastic half-plane are investigated. The surface irregularity represents a canyon, an alluvial valley or a hill. The wave function expansion method has been employed to solve surface irregularities, such as a semi-cylindrical canyon, a semi-cylindrical alluvial valley, or a semi-elliptical canyon and a semi-elliptical alluvial valley. These solutions to the scattering problem of SH wave can be used to test the accuracy of the other numerical methods. But solutions for surface irregularities with arbitrarily shapes cannot be found easily. A hybrid method combines the finite element method with series expansion is applied to solve scattering problems in this study. A subregion encloses the surface irregularity with a semi-circular auxiliary boundary can be meshed by the finite element method. By using the transfinite interpolation (TFI) produces excellent grid mesh on the subregion. The advantage of TFI is the flexibility to facilitate modeling of the subregion. On the other hand, the boundary data can be formulated by using a series representation with unknown coefficients. The Lamb's solution which satisfies the traction free condition and the radiation condition at infinity is implemented to be the basis function. The unknown coefficients can be obtained by satisfying the continuity conditions of the semi-circular auxiliary boundary between the subregion and the half-plane. The hybrid method that combines TFI with series expansion is successfully herein to solve the scattering problem by a surface irregularity. Numerical results in this study for special cases agree well with those in the published literatures. In this study, the steps and skills of hybrid method are described systematically and completely to solve the surface irregularity.

A new algorithm, called boosted hybrid method, is proposed for the simulation of chemical reaction systems with scale-separation in time and disparity in species population. For such stiff systems, the algorithm can automatically identify scale-separation in time and slow down the fast reactions while maintaining a good approximation to the original effective dynamics. This technique is called boosting. As disparity in species population may still exist in the boosted system, we propose a hybrid strategy based on coarse-graining methods, such as the tau-leaping method, to accelerate the reactions among large population species. The combination of the boosting strategy and the hybrid method allow for an efficient and adaptive simulation of complex chemical reactions. The new method does not need a priori knowledge of the system and can also be used for systems with hierarchical multiple time scales. Numerical experiments illustrate the versatility and efficiency of the method.

A concept of “static reconstruction” and “dynamic reconstruction” was introduced for higher-order (third-order or more) numerical methods in our previous work. Based on this concept, a class of hybrid DG/FV methods had been developed for one-dimensional conservation law using a “hybrid reconstruction” approach, and extended to two-dimensional scalar equations on triangular and Cartesian/triangular hybrid grids. In the hybrid DG/FV schemes, the lower-order derivatives of the piece-wise polynomial are computed locally in a cell by the traditional DG method (called as “dynamic reconstruction”), while the higher-order derivatives are re-constructed by the “static reconstruction” of the FV method, using the known lower-order derivatives in the cell itself and in its adjacent neighboring cells. In this paper, the hybrid DG/FV schemes are extended to two-dimensional Euler equations on triangular and Cartesian/triangular hybrid grids. Some typical test cases are presented to demonstrate the performance of the hybrid DG/FV methods, including the standard vortex evolution problem with exact solution, isentropic vortex/weak shock wave interaction, subsonic flows past a circular cylinder and a three-element airfoil (30P30N), transonic flow past a NACA0012 airfoil. The accuracy study shows that the hybrid DG/FV method achieves the desired third-order accuracy, and the applications demonstrate that they can capture the flow structure accurately, and can reduce the CPU time and memory requirement greatly than the traditional DG method with the same order of accuracy.

A hybrid differential transformation / finite difference scheme is used to analyze the complex nonlinear behavior of an electrostatically-actuated micro cantilever beam which high aspect ratios (length/width). The validity of the proposed method is confirmed by comparing the numerical results obtained for the tip displacement and pull-in voltage of the cantilever beam with the analytical and experimental results presented in the literature. The hybrid scheme is then applied to analyze both the steady-state and the dynamic deflection behavior of the cantilever beam as a function of the applied voltage. Overall, the results confirm that the hybrid method provides an accurate and computationally-efficient means of analyzing the complex nonlinear behavior of both the current micro cantilever beam system and other micro-scale electrostatically-actuated structures.