A Generalized Upper and Lower Solution Method for Singular Discrete Boundary Value Problems

This paper presents new existence results for singular discrete boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign.

POSITIVE SOLUTIONS OF HIGHER DIMENSIONAL DISCRETE BOUNDARY VALUE PROBLEM

By means of the critical point theory, we prove the existence of positive solutions for a higher dimensional discrete boundary value problem. Our results generalize one of Agarwal in [2].

EXISTENCE OF SOLUTIONS TO A CLASS OF NONLINEAR n-DIMENSIONAL DISCRETE BOUNDARY VALUE PROBLEMS

In this paper,using the critical point theory,we obtain a new result on the existence of the solutions to a class of n-dimensional discrete boundary value problems.Results obtained extend or improve the existing ones.

SOLUTIONS TO DISCRETE MULTIPARAMETER PERIODIC BOUNDARY VALUE PROBLEMS INVOLVING THE p-LAPLACIAN VIA CRITICAL POINT THEORY

In this paper, we consider the existence of three nontrivial solutions for a discrete non-linear multiparameter periodic problem involving the p-Laplacian. By using the similar method for the Dirichlet boundary value problems in [C. Bonanno and P. Candito, Appl. Anal., 88（4） （2009）, pp. 605-616], we construct two new strong maximum principles and obtain that the boundary value problem has three positive solutions for λ and μ in some suitable intervals. The approaches we use are the critical point theory.

Discrete Transparent Boundary Conditions for Burgers’ Equation

This paper is devoted to numerically solving the Burgers’ equation in unbounded region, which describes the nonlinear wave propagation and diffusion effect. How to numerically and efficiently solve this problem remains a challenge due to the principal difficulties of not only the unboundedness but also nonlinearity. This paper aims to design the discrete transparent boundary conditions of Burgers’ equation to overcome the unboundedness. The presence of a nonlinear term in the equation makes it challenging to derive suitable artificial boundary conditions. To deal with the nonlinear term, a linear parabolic equation is obtained by applying the well-known Cole-Hopf transformation to the Burgers’ equation. Employing the Z-transformation, we then establish the discrete transparent boundary conditions for the linearized problem. Subsequently, the original equation is reduced to an initial boundary value problem, that can be efficiently solved by the finite difference method. The numerical analysis is reported rigorously. Some numerical results are given to demonstrate the accuracy and feasibility of the proposed method.

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS,Ⅱ

In his series of three papers we study singularly perturbed (SP) boundary valueproblems for equations of elliptic and parabolic type. For small values of the pertur-bation parameter parabolic boundary and interior layers appear in these problems.If classical discretisation methods are used, the solution of the finite differencescheme and the approximation of the diffusive flux do not converge uniformly withrespect to this parameter. Using the method of special, adapted grids, we canconstruct difference schemes that allow approximation of the solution and the nor-malised diffusive flux uniformly with respect to the small parameter.We also consider sillgularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly We study what problems ap-pear, when classical schemes are used for the approximation of the spatial deriva-tives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Diriclilet, Neumann and RDbin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions-

Modeling particle sedimentation in viscous fluids with a coupled immersed boundary method and discrete element method

Numerical techniques have increasingly been used to model fluid–particle two-phase flows. Coupling the immersed boundary method （IBM） and discrete element method （DEM） is one promising approach for modeling particulate flows. In this study, IBM was coupled with DEM to improve the reliability and accuracy of IBM for determining the positions of particles during the sedimentation process within viscous fluids. The required ratio of the particle diameter to the grid size （D/dx） was determined by comparing the simulation results with the analytical solution and experimental data. A dynamic mesh refinement model was utilised in the IBM model to refine the computational fluid dynamics grid near the particles. In addition, an optimum coupling interval between the IBM and DEM models was determined based on the experimental results of a single particle sedimentation within silicon oil at a Reynolds number of 1.5. The experimental results and the analytical solution were then utilised to validate the IBM–DEM model at Reynolds numbers of 4.1, 11.6, and 31.9. Finally, the validated model was utilised to investigate the sedimentation process for more than one particle by modeling the drafting-kissing-tumbling process and the Boycott phenomenon. Benchmark tests showed that the IBM–DEM technique preserves the advantages of DEM for tracking a group of particles, while the IBM provides a reliable and accurate approach for modeling the particle–fluid interaction.

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PEOBLEMS WITH PARABOLIC LAYERS,I

In this series of three papers we study singularly perturbed (SP) boundaryvalue problems for equations of eiliptic and parabolic type- For small values ofthe perturbation parameter parabolic boundary and interior layers appear in theseproblems. If classical discretisation methods are used, the solution of the finitedifference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, edapted grids,we can construct difference schemes that allow apprcximation of the solution andthe normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection-diffusion equations. Also for these problems we construct special finite differenceschemes, the solution of which converges ε-uniformly. We study what problems appear, when classical schemes are used for the approximation of the spatial derivatives. We compare the results with those obtained by the adapted approach. Re-sults of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, andthen we consider respectively (i) Problems for SP parabolic equations, for whichthe solution and the normalised diffusive fluxes are required; (ii) Problems for SPelliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic equation with discontinuous boundary conditions.

DISCRETE APPROXIMATIONS FOR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS WITH PARABOLIC LAYERS, III

In this series of three papers we study singularly perturbed (SP) boundary vaue problems for equations of elliptic and parabolic troe. For small values of the perturbation parameter parabolic boundary and interior layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux do not converge uniformly with respect to this parameter. Using the method of special, adapted grids,we can construct difference schemes that allow approkimation of the solution and the normalised diffusive flux uniformly with respect to the small parameter.We also consider singularly perturbed boundary value problems for convection diffusion equations. Also for these problems we construct special finite difference schemes, the solution of which converges E-uniformly We study what problems appear, when classical schemes are used for the approximation of the spatial deriva tives. We compare the results with those obtained by the adapted approach. Results of numerical experiments are discussed.In the three papers we first give an introduction on the general problem, and then we consider respectively (i) Problems for SP parabolic equations, for which the solution and the normalised diffusive fluxes are required; (ii) Problems for SP elliptic equations with boundary conditions of Dirichlet, Neumann and Robin type;(iii) Problems for SP parabolic eqllation with discontinuous boundaxy conditions

A survey on artificial boundary method Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday

The artificial boundary method is one of the most popular and effective numerical methods tor solving partial differential equations on unbounded domains, with more than thirty years development. The artificiM boundary method has reached maturity in recent years. It has been applied to various problems in scientific and engineering computations, and the theoretical issues such as the convergence and error estimates of the artificial boundary method have been solved gradually. This paper reviews the development and discusses different forms of the artificial boundary method.

Highly efficient computation method for hazard quantification of uncontained rotor failure

Uncontained Engine Rotor Failure(UERF)can cause a catastrophic failure of an aircraft,and the quantitative assessment of the hazards related to UERF is a very important part of safety analysis.However,the procedure for hazard quantification of UERF recommended by the Federal Aviation Administration in advisory circular AC20-128A is cumbersome,as it involves building auxiliary lines and curve projections.To improve the efficiency and general applicability of the risk angle calculation,a boundary discretization method is developed that involves discretizing the geometry of the target part/structure into node points and calculating the risk angles numerically by iterating a particular algorithm over each node point.The improved efficiency and excellent accuracy for the developed algorithm was validated through a comparison with manual solutions for the hazard quantification of the engine nacelle structures of a passenger aircraft using the guidance in AC20-128A.To further demonstrate the applicability of the boundary discretization method,the proposed algorithm was used to examine the influence of the target size and the distance between the target and rotor on the hazard probability.

Effect of sample size on the response of DEM samples with a realistic grading

This paper shows that for DEM simulations of triaxial tests using samples with a grading that is repre- sentative of a real soil, the sample size significantly influences the observed material response. Four DEM samples with identical initial states were produced： three cylindrical samples bounded by rigid wails and one bounded by a cubical periodic cell, When subjected to triaxial loading, the samples with rigid boundaries were more dilative, stiffer and reached a higher peak stress ratio than the sample enclosed by periodic boundaries. For the rigid-wall samples, dilatancy increased and stiffness decreased with increasing sample size, The periodic sample was effectively homogeneous, The void ratio increased and the contact density decreased close to the rigid walls, This heterogeneity reduced with increasing sample size. The positions of the critical state lines （CSLs） of the overall response in e-log p＇ space were sensitive to the sample size, although no difference was observed between their slopes. The critical states of the interior regions of the rigid-wall-bounded samples approached that of the homogeneous periodic sample with increasing sample size. The ultimate strength of the material at the critical state is independent of sample size.

Numerical simulation of polygonal particles moving in incompressible viscous fluids

A two-dimensional coupled lattice Boltzmann immersed boundary discrete element method is introduced for the simulation of polygonal particles moving in incompressible viscous fluids. A collision model of polygonal particles is used in the discrete element method. Instead of a collision model of circular particles, the collision model used in our method can deal with particles of more complex shape and efficiently simulate the effects of shape on particle–particle and particle–wall interactions. For two particles falling under gravity, because of the edges and corners, different collision patterns for circular and polygonal particles are found in our simulations. The complex vortexes generated near the corners of polygonal particles affect the flow field and lead to a difference in particle motions between circular and polygonal particles. For multiple particles falling under gravity, the polygonal particles easily become stuck owing to their corners and edges, while circular particles slip along contact areas. The present method provides an efficient approach for understanding the effects of particle shape on the dynamics of non-circular particles in fluids.

Numerical simulation of the sticking process of glass-microparticles to a fiat wall to represent pollutant-particles treatment in a multi-channel cyclone

Ultrafine particles are dangerous to human health and are usually difficult to separate from airflow because of their low inertia, which helps them to stick easily to surfaces because of adhesive forces. This characteristic provides opportunities for adhesive ultrafine particle separation by designing air-cleaning devices that exploit the sticking ability. To understand governing effects in such air-cleaning devices, which can be designed as multi-channel cyclones, the sticking of adhesive spherical glass particles under oblique impact has been investigated numerically by using the discrete element method. An adhesive dissipative contact model was applied by implementing different interaction forces for various-sized ultraflne pollutant particles. Normal loading is represented by the elastic Hertz contact model, whereas viscous damping is described by the modified nonlinear Tsuji model. The influence of deformation- dependent adhesive forces for a range of ultrafine particle sizes is illustrated during the sticking process. Dissipative oscillations during the sticking process were observed because of the influence of viscous damping forces.