We propose a family of nonconforming rectangular elements for the linear strain gradient elastic model. Optimal error estimates uniformly with respect to the small material parameter have been proved. Numerical results confirm the theoretical prediction.

This paper investigates two methods of coupling fluids across an interface, motivated by air-sea interaction in application codes. One method is for sequential configurations, where the air code module in invoked over some time interval prior to the sea module. The other method is for concurrent setups, in which the air and sea modules run in parallel. The focus is the temporal representation of air-sea fluxes. The methods we study conserve moments of the fluxes, with an arbitrary order of accuracy possible in time. Different step sizes are allowed for the two fluid codes. An $a$ $posteriori$ stability indicator is defined, which can be computed efficiently on-the-fly over each coupling interval. For a model of two coupled fluids with natural heat convection, using finite elements in space, we prove the sufficiency of our stability indicator. Under certain conditions, we also prove that stability can be enforced by iteration when the coupling interval is small enough. In particular, for solutions in a certain class, we show that the step size scaling is no worse than $\mathcal{O}(h)$ in three dimensions of space, where $h$ is a mesh parameter. This is a sharper result than what has been shown previously for related algorithms with finite element methods. Computational examples illustrate the behavior of the algorithms under a wide variety of configurations.

In this paper, we will investigate a multigrid algorithm for poroelasticity problem by a new finite element method with homogeneous boundary conditions in two dimensional space. We choose Nédélec edge element for the displacement variable and piecewise continuous polynomials for the pressure variable in the model problem. In constructing multigrid algorithm, a distributive Gauss-Seidel iteration method is applied. Numerical experiments shows that the finite element method achieves optimal convergence order and the multigrid algorithm is almost uniformly convergent to mesh size $h$ and parameter $\delta t$ on regular meshes.

We consider an optimal control problem which serves as a mathematical model for several problems in economics and management. The problem is the minimization of a continuous constrained functional governed by a linear parabolic diffusion-advection equation controlled in a coefficient in advection part. The additional constraint is non-negativity of a solution of state equation. We construct and analyze several mesh schemes approximating the formulated problem using finite difference methods in space and in time. All these approximations keep the positivity of the solutions to mesh state problem, either unconditionally or under some additional constraints to mesh steps. This allows us to remove corresponding constraint from the formulation of the discrete problem to simplify its implementation. Based on theoretical estimates and numerical results, we draw conclusions about the quality of the proposed mesh schemes.

The concrete aggregate model is considered as a type of weakly discontinuous problem consisting of three phases: aggregates which randomly distributed in different shapes, cement paste and internal transition zone (ITZ). Because of different shapes of aggregate and thin ITZs, a huge number of elements are often used in the finite element method (FEM) analysis. In order to ensure the accuracy of the numerical solutions near the interfaces, we need to use higher-order elements. The widely used FEM softwares such as ANSYS and ABAQUS all provide the option of quadratic elements. However, they have much higher computational complexity than the linear elements. The corresponding coefficient matrix of the system of equations is a highly ill-conditioned matrix due to the large difference between three phase materials, and the convergence rate of the commonly used solving methods will deteriorate. In this paper, two types of simple and efficient preconditioners are proposed for the system of equations of the concrete aggregate models on unstructured triangle meshes by using the resulting hierarchical structure and the properties of the diagonal block matrices. The main computational cost of these preconditioners is how to efficiently solve the system of equations by using linear elements, and thus we can provide some efficient and robust solvers by calling the existing geometric-based algebraic multigrid (GAMG) methods. Since the hierarchical basis functions are used, we need not present those algebraic criterions to judge the relationships between the unknown variables and the geometric node types, and the grid transfer operators are also trivial. This makes it easy to find the linear element matrix derived directly from the fine level matrix, and thus the overall efficiency is greatly improved. The numerical results have verified the efficiency of the resulting preconditioned conjugate gradient (PCG) methods which are applied to the solution of several typical aggregate models.

In this paper, we obtain some new results on bilateral generating functions of the modified Laguerre polynomials. We also get generating function relations between the modified Laguerre polynomials and the generalized Lauricella functions. Some special cases and important applications are also discussed.

In this paper, a finite element method based on the characteristic for the incompressible Navier-Stokes equations is proposed by introducing Runge-Kutta method. At first, coordinate transformation operation is performed to obtain the alternative Navier-Stokes equations without convection term. Then, instead of the classical characteristic-based split (CBS) method, we use the third-order Runge-Kutta method along the characteristic to carry out time discretization in order to improve calculation accuracy, and segregate the calculation of the pressure from that of the velocity based on the momentum-pressure Poisson equation method. Finally, some classical benchmark problems are used to validate the effectiveness of the present method. Compared with the classical method, the present method has lower dissipation, larger permissible time step, and higher time accuracy. The code can be downloaded at DOI: 10.13140/RG.2.2.36336.56329.

Air flow distribution in radial flow adsorber was numerically investigated using computational fluid dynamics (CFD) method, which was proved to be applicable to study the problem of non-uniform distribution in radial flow adsorber. Results showed that the degree of non-uniformity was more serious in desorption process than that is adsorption process. Therefore, it was considered that the non-uniform distribution of flow in a radial flow adsorber was mainly manifested in the desorption process. Optimum design of distributor parameters can improve the flow distribution in adsorber. Meanwhile, three different structures of distributor and the effect of breathing valve were analyzed. Results revealed that truncated cone is more effective than tubular and conical distributors in flow distribution. By inserting the truncated cone in central channel, desorption uniformity was increased by 6.56% and the breakthrough time of CO$_2$ was extended from 564s to 1138s in the adsorption process. The "dead zone" problem at the top of adsorber during the desorption process was solved by opening breathing valve, which prolonged the working life of adsorber and was proved to have less effect on the uniform of airflow.