In this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.

The paper presents the size-dependent behaviors of the carbon nanotubes under electrostatic actuation using the modified couple stress theory and homotopy perturbation method. Due to the less accuracy of the classical elasticity theorems, the modified couple stress theory is applied in order to capture the size-dependent properties of the carbon nanotubes. Both of the static and dynamic behaviors under static DC and step DC voltages are discussed. The effects of various dimensions and boundary conditions on the deflection and pull-in voltages of the carbon nanotubes are to be investigated in detail via application of the homotopy perturbation method to solve the nonlinear governing equations semi-analytically.

In order to overcome the possible singularity associated with the Point Interpolation Method (PIM), the Radial Point Interpolation Method (RPIM) was proposed by G. R. Liu. Radial basis functions (RBF) was used in RPIM as basis functions for interpolation. All these radial basis functions include shape parameters. The choice of these shape parameters has been and stays a problematic theme in RBF approximation and interpolation theory. The object of this study is to contribute to the analysis of how these shape parameters affect the accuracy of the radial PIM. The RPIM is studied based on the global Galerkin weak form performed using two integration technics: classical Gaussian integration and the strain smoothing integration scheme. The numerical performance of this method is tested on their behavior on curve fitting, and on three elastic mechanical problems with regular or irregular nodes distributions. A range of recommended shape parameters is obtained from the analysis of different error indexes and also the condition number of the matrix system. All resulting RPIM methods perform very well in terms of numerical computation. The Smoothed Radial Point Interpolation Method (SRPIM) shows a higher accuracy, especially in a situation of distorted node scheme.

In this paper, we extend using the Runge-Kutta discontinuous Galerkin method together with the front tracking method to simulate the compressible two-medium flow on unstructured meshes. A Riemann problem is constructed in the normal direction in the material interfacial region, with the goal of obtaining a compact, robust and efficient procedure to track the explicit sharp interface precisely. Extensive numerical tests including the gas-gas and gas-liquid flows are provided to show the proposed methodologies possess the capability of enhancing the resolutions nearby the discontinuities inside of the single medium flow and the interfacial vicinities of the two-medium flow in many occasions.

In this paper, we analyze a nonconforming finite element method for the computation of transmission eigenvalues and the corresponding eigenfunctions. The error estimates of the eigenvalue and eigenfunction approximation are given, respectively. Finally, some numerical examples are provided to validate the theoretical results.

Semi-Lagrangian (S-L) methods have no CFL stability constraint, and are more stable than the Eulerian methods. In the literature, the S-L method for the level-set re-initialization equation was complicated, which may be unnecessary. Since the re-initialization procedure is auxiliary, we propose to use the first-order S-L scheme coupled with a projection technique to improve the accuracy at the grid points just adjacent to the interface. Standard second-order S-L method is used for evolving the level-set convection equation. The implementation is simple, including on the block-structured adaptive mesh. The efficiency of the S-L method is demonstrated by extensive numerical examples including passive convection of interfaces with corners/kinks/large deformation under given velocity fields, a geometrical flow with topological changes, simulations of bubble/ droplet dynamics in incompressible two-phase flows. In terms of accuracy it is comparable to the other existing methods.

In this paper, a method is proposed for extracting fracture parameters in anisotropic thermoelasticity cracking via interaction integral method within the framework of extended finite element method (XFEM). The proposed method is applied to linear thermoelastic crack problems. The numerical results of the stress intensity factors (SIFs) are presented and compared with those reported in related references. The good agreement of the results obtained by the developed method with those obtained by other numerical solutions proves the applicability of the proposed approach and confirms its capability of efficiently extracting thermoelasticity fracture parameters in anisotropic materials.

In this paper, He's homotopy perturbation method is utilized to obtain the analytical solution for the nonlinear natural frequency of functionally graded nanobeam. The functionally graded nanobeam is modeled using the Eringen's nonlocal elasticity theory based on Euler-Bernoulli beam theory with von Karman nonlinearity relation. The boundary conditions of problem are considered with both sides simply supported and simply supported-clamped. The Galerkin's method is utilized to decrease the nonlinear partial differential equation to a nonlinear second-order ordinary differential equation. Based on numerical results, homotopy perturbation method convergence is illustrated. According to obtained results, it is seen that the second term of the homotopy perturbation method gives extremely precise solution.

In this paper we develop explicit fast exponential Runge-Kutta methods for the numerical solutions of a class of parabolic equations. By incorporating the linear splitting technique into the explicit exponential Runge-Kutta schemes, we are able to greatly improve the numerical stability. The proposed numerical methods could be fast implemented through use of decompositions of compact spatial difference operators on a regular mesh together with discrete fast Fourier transform techniques. The exponential Runge-Kutta schemes are easy to be adopted in adaptive temporal approximations with variable time step sizes, as well as applied to stiff nonlinearity and boundary conditions of different types. Linear stabilities of the proposed schemes and their comparison with other schemes are presented. We also numerically demonstrate accuracy, stability and robustness of the proposed method through some typical model problems.

In this paper the fractional Euler-Lagrange equation is considered. The fractional equation with the left and right Caputo derivatives of order α ∈ (0,1] is transformed into its corresponding integral form. Next, we present a numerical solution of the integral form of the considered equation. On the basis of numerical results, the convergence of the proposed method is determined. Examples of numerical solutions of this equation are shown in the final part of this paper.

In this paper we extend the idea of interpolated coefficients for a semilinear problem to the quadratic triangular finite volume element method. At first, we introduce quadratic triangular finite volume element method with interpolated coefficients for a boundary value problem of semilinear elliptic equation. Next, we derive convergence estimate in $H^1$-norm, $L^2$-norm and $L^∞$-norm, respectively. Finally, an example is given to illustrate the effectiveness of the proposed method.

Similarity solution is investigated for the synchronous grouting of shield tunnel under the vertical non-axisymmetric displacement boundary condition in the paper. The synchronous grouting process of shield tunnel was simplified as the cylindrical expansion problem, which was based on the mechanism between the slurry and stratum of the synchronous grouting. The stress harmonic function on the horizontal and vertical ground surfaces is improved. Based on the virtual image technique, stress function solutions and Boussinesq's solution, elastic solution under the vertical non-axisymmetric displacement boundary condition on the vertical surface was proposed for synchronous grouting problems of shield tunnel. In addition, the maximum grouting pressure was also obtained to control the vertical displacement of horizontal ground surface. The validity of the proposed approach was proved by the numerical method. It can be known from the parameter analysis that larger vertical displacement of the horizontal ground surface was induced by smaller tunnel depth, smaller tunnel excavation radius, shorter limb distance, larger expansion pressure and smaller elastic modulus of soils.