We present an efficient Galerkin-characteristic finite element method for the numerical solution of convection-diffusion problems in three space dimensions. The modified method of characteristics is used to discretize the convective term in a finite element framework. Different types of finite elements are implemented on three-dimensional unstructured meshes. To allocate the departure points we consider an efficient search-locate algorithm for three-dimensional domains. The crucial step of interpolation in the convection step is carried out using the basis functions of the tetrahedron element where the departure point is located. The resulting semi-discretized system is then solved using an implicit time-stepping scheme. The combined method is unconditionally stable such as no Courant-Friedrichs-Lewy condition is required for the selection of time steps in the simulations. The performance of the proposed Galerkin-characteristic finite element method is verified for the transport of a Gaussian sphere in a three-dimensional rotational flow. We also apply the method for simulation of a transport problem in a three-dimensional pipeline flow. In these test problems, the method demonstrates its ability to accurately capture the three-dimensional transport features.

Singularity of Navier-Stokes equations is uncovered for the first time which explains the mechanism of transition of a smooth laminar flow to turbulence. It is found that when an inflection point is formed on the velocity profile in pressure driven flows, velocity discontinuity occurs at this point. Meanwhile, pressure pulse is produced at the discontinuity due to conservation of the total mechanical energy. This discontinuity makes the Navier-Stokes equations be singular and causes the flow to become indefinite. The analytical results show that the singularity of the Navier-Stokes equations is the cause of turbulent transition and the inherent mechanism of sustenance of fully developed turbulence. Since the velocity is not differentiable at the singularity, there exist no smooth and physically reasonable solutions of Navier-Stokes equations at high Reynolds number (beyond laminar flow). The negative spike of velocity and the pulse of pressure due to discontinuity have obtained agreement with experiments and simulations in literature qualitatively.

In this paper, we prove a lower convergence rate of the weak Galerkin finite element method for second order elliptic equations under a standard weak smoothness assumption. In all present literatures on weak Galerkin finite element methods for second order PDEs, the $H^2$ smoothness is compulsorily assumed for the real solution and hence a second order convergence is obtained. This lead to that the piecewise linear functions are excluded to construct finite element bases, although they behave very well in all numerical experiments. We intend to prove the $(1+s)$-order convergence rate under the $H^1$-smoothness assumption of the real solution and an additional $s>0$ regularity of the dual problem. Our strategy is that we firstly approximate the elliptic problem using the traditional finite element method with at least $H^2$ smooth bases, and then we apply the weak Galerkin method to approach this smooth approximating solution. Our result is an important supplementary for the weak Galerkin finite element method theory.

This paper investigates bifurcation and chaos of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) beam containing piezoelectric layer (PL) under combined electro-thermo-mechanical loads. We assumed that FG-CNTRC material properties were graded along thickness direction and determined them using mixtures' law. Governing equations of structures were derived according to the theory of Euler-Bernoulli beam, PL with thermal effects and von Kármán geometric nonlinearity. Next, the governing equations were transformed into second order nonlinear ordinary differential equations (SNODE) with cubic terms through Galerkin procedure and further into first order nonlinear ordinary differential equations (FNODE) through introducing additional state variables. Complex system dynamic behavior was qualitatively examined using fourth order Runge-Kutta method. The effects of different factors including applied voltage, volume fraction, temperature change, and distribution of carbon nanotubes (CNTs) on bifurcation and chaos of FG-CNTRC beams with PL were comprehensively studied.

In this paper, we present a new semi-analytical method based on the B-spline approximation for solving 2D time-dependent convection-diffusion-reaction equations to model transfer in the anisotropic inhomogeneous medium. The mathematical model is expressed as the initial-boundary value problem for quasi-linear parabolic equation with the second order elliptic spatial operator with mixed derivatives and variable coefficients. The time-stepping Crank-Nicolson scheme transforms the original equation into a sequence of quasi-linear elliptic partial differential equations. The approximate solution is sought as series over basis functions which are taken in the form of the tensor products of the  B-splines with centers distributed inside the solution domain. Due to the modification of spline basis, the final approximate solution satisfies the boundary conditions of the initial problem with any choice of the coefficients of the series. The numerical examples demonstrate the high accuracy of the proposed method in solving 2D convection-diffusion-reaction problems in single- and multi-connected domains.

A multiple-relaxation-time (MRT) lattice Boltzmann model (LBM) is used to study the relative permeabilities in porous media. In many simulations in the literature, usually the periodic boundary condition at inlet and outlet and a uniform pressure gradient were applied to measure the relative permeabilities. However, it is not consistent with the pressure or velocity boundary conditions in the real experiments and may lead to unphysical results. Here using the convective outflow and constant velocity boundary conditions at outlet and inlet, respectively, we can simulate the real experimental setup. Meanwhile, the distribution of the two phases at the outlet can be resolved. The effects of wettability, initial saturation, viscosity ratio $(M\in(1,50)),$ capillary number ($Ca \in(10^{-4},10^{-2})$) and micro two-phase distribution at the inlet on permeabilities are investigated comprehensively. It is found that generally speaking, the strong wetting, drainage, larger $Ca$, and larger $M$ may result in a larger relative permeability of the non-wetting phase. Different flow pattern, the lubrication effect of the wetting phase that attaches to the wall, and influence of stagnant pores may contribute to the feature. The study is helpful to further develop the LBM to simulate the real experimental process.

In this paper, we use the radial basis function meshless method to solve the irregular region interface problem. The key idea is to construct radial basis functions corresponding to different regions divided by the interfaces. This method avoids the difficulty of mesh generation, and is efficient in the numerical simulation of partial differential equations in irregular domain with variable matrix coefficients. The numerical error is effectively reduced by using the direct method to handle the interface jump conditions. Numerical simulation results show that the radial basis function meshless numerical method can effectively deal with various kinds of interface problems with irregular domains and sharp-edged interfaces, including Poisson equations, heat conduction equations and wave equations.

In this paper, a simple and robust multi-dimensional shock-capturing limiter is presented and applied to the high-order least square-based finite difference-finite volume (LSFD-FV) method for simulation of compressible inviscid flows on unstructured grids. The novel limiter considers all the spatial information of the neighbouring supporting cells of the reference cell instead of only the immediate neighbours. As a result, the local upper and lower bounds can be correctly defined based on the extrema at specific points within the supporting stencil and thereafter the numerical oscillations in the adjacent region to the discontinuities can be suppressed effectively. The main advantage of this limiting technique lies in its simple methodology and easy implementation. Furthermore, the multi-dimensional character and the point extrema-based methodology endow the proposed limiter with ability to solve multi-dimensional flow problems on unstructured grids. A series of smooth and non-smooth numerical examples in one, two and three dimensions on unstructured grids are tested. Numerical results obtained demonstrate the superior performance of the proposed limiter on simulating compressible inviscid flows.

The present study is devoted to developing a yield criterion that can model both the yielding asymmetry and plastic anisotropy of pressure-insensitive metals. First, a new isotropic yield criterion which can model the yielding asymmetry of pressure-insensitive metals is proposed. The main advantage of the proposed criterion is that it leads to a good approximation of yield loci calculated by the Taylor-Bishop-Hill crystal plasticity model. Further, the isotropic criterion is extended to orthotropy to take plastic anisotropy into account. The new anisotropic criterion is general and can be used in three-dimensional stresses. The coefficients of the criterion are determined by an error minimization procedure. Applications of the proposed theory to a hexagonal close packed (HCP) magnesium, a Cu-Al-Be shape memory alloy and a Ni3Al based intermetallic alloy show that the proposed theory can describe well the plastic anisotropy and yielding asymmetry of metals and the transformation onset of the shape memory alloy, showing excellent predictive ability and flexibility.

Analytical study on nonlinear hydroelastic waves beneath a very large floating structure or a thin ice sheet floating on deep water is presented. Adopting the special Cosserat theory of hyperelastic shells satisfying Kirchhoff's hypothesis to describe the floating sheet, we use the potential flow theory with the dynamic boundary condition expressing a balance among the hydrodynamic, surface tension, inertial, and elastic forces. For the case of incident progressive waves, the influences of different physical parameters on the hydroelastic waves are discussed with the aid of the homotopy analysis method. We compare the hydroelastic wave deflections based on nonlinear Plotnikov and Toland's model with those obtained by the corresponding linear Euler-Bernoulli model. It is found that the behaviors of both models are almost the same for small amplitudes, while the nonlinear plate deflections increase greatly at large amplitudes. Further, the graphical comparisons are presented to show the behavior of the angular frequency versus wave amplitudes.