Static response of a layered magneto-electro-elastic half-space structure under circular surface loading

The main aim of this paper was to find the correct method of calculating equations of heat and mass transfer for the adsorption process and to calculate it numerically in reasonable time and with proper accuracy. An adsorption heat pump with a silica gel adsorbent and water adsorbate is discussed. We developed a mathematical model of temperature and uptake changes in the adsorber/desorber comprising the set of heat and mass balance partial differential equations (PDEs), together with the initial and boundary conditions and solved it by the numerical method of lines (NMOL). Spatial discretization was performed with equally spaced axial nodes and the PDEs were reduced to a set of ordinary differential equations (ODEs). We focused on the comparison of results obtained when the set of heat and mass balance ODEs for an adsorber was solved using: (1) the Runge–Kutta fixed step size fourth-order method (RKfixed), (2) the Runge–Kutta–Fehlberg 4.5th-order method with a variable step size (RK45), and (3) the Gear Backward Differentiation Formulae numerical (Gear BDF) methods. In our experience, all three types of ODE numerical methods (RKfixed, RK45, and Gear BDF) can be applied in simple models to model an adsorber with attention on their limitations. The Gear BDF method usually requires much fewer steps than the RK45 method for almost the same calculating time. RK methods require many more steps to obtain results, and the calculating time depends on accuracy or defined time step. Moreover, one should pay attention to the number of nodes or possible oscillations.

In this work, the displacements and stresses at any point in a transversely isotropic layered half-space under multiple horizontal loads are studied. The transversely isotropic plane is assumed to be parallel to the surface plane, and uniformly distributed circular loads with different magnitudes, radii and orientations are applied to the pavement surface. Based on the cylindrical system of vector functions in the transformed domain, the governing equations are first decoupled into two sets of equations related to the LM-type and N-type respectively. Solutions for the multilayered half-space in the transformed domain are then derived by virtue of the propagator matrix method. Solutions in the physical domain are then expressed in terms of the Bessel function integration. The method of superposition is finally utilized for multiple loads. We remark that while the propagator matrix method has been frequently used to solve the vertical loading problem in layered half-spaces, which only involve the LM-type equation, the corresponding horizontal loading problem involving multiple circular loads in a transversely isotropic layered half-space has not been addressed in the literature. A computer program has been coded by the authors' research group and numerical results obtained from this program for the isotropic layered half-space have been verified with existing ones. Further presented in this paper are the results for the transversely isotropic layered half-space, with examples elucidating clearly the effect of material anisotropy on the responses, especially on pavement failure. It is also observed that, in terms of computation, the developed program is very accurate, efficient and flexible. For instance, our program can easily handle more than 10,000 field points with more than 1,000 pavement layers.

Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

Three-dimensional static analysis of Levy-type functionally graded plate with in-plane stiffness variation

The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field ${\bf V}$ in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along ${\bf V}$: scalar fields (first integrals), forms (such as symplectic tensors), vectors (CT generators, Lie’s invariance generators), integrals over subspaces (which are not limited to Poincaré’s absolute and relative invariants). Most classical results are easily understood as simple relations between these. It is shown first that to any field ${\bf V}$ can be associated a Hamiltonian structure of forms (if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added). Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton’s variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms (here a Hamiltonian structure) must include the characteristics of the set (here the trajectories of ${\bf V}$). Auxiliary vector fields are also carefully discussed: the hierarchy of invariance generators of (i) the vector field ${\bf V}$, of (ii) its trajectories, of (iii) any associated Hamiltonian structure of forms. An Appendix summarizes the manipulation of vectors and forms, as a tool for applied mathematics.

The nonstratified steady flow hydrodynamics of rectangular sedimentation tanks is simulated using a combination of the strip integral and finite element methods. The strip integral method is generally a forward marching scheme in which the partial differential equations of continuity and momentum are reduced to a set of ordinary differential equations in terms of certain preselected parameters. These parameters, along with a set of shape functions, describe the velocity distribution. In this study three shape functions are chosen, corresponding to the boundary layer, the potential core, and the free mixing and recirculation zone. The shape functions were: (1) Power law in the boundary layer; (2) uniform velocity in the potential core; and (3) Gaussian distribution in the free mixing zone. The shape functions are chosen to allow recirculation above the mixing zone. A modified mixing length approach is used to introduce the effect of recirculation into the solution. A Runge‐Kutta method is used to integrate the set of ordinary differential equations; a finite element solution is used in the withdrawal zone. The numerical results compare favorably with experimental velocity distributions obtained by laser anemometry and with available field data. The convective and mixing characteristics of the flow can be obtained from this model and used with a transport model to determine the removal efficiency of the tank.

Nonlinear dynamic thermoelastic response of rectangular FGM plates with longitudinal velocity

This article addresses an investigation of the entropy analysis of Williamson nanofluid flow in the presence of gyrotactic microorganisms by considering variable viscosity and thermal conductivity over a convectively heated bidirectionally stretchable surface. Heat and mass transfer phenomena have been incorporated by taking into account the thermal radiation, heat source or sink, viscous dissipation, Brownian motion, and thermophoretic effects. The representing equations are nonlinear coupled partial differential equations and these equations are shaped into a set of ordinary differential equations via a suitable similarity transformation. The arising set of ordinary differential equations was then worked out by adopting a well‐known scheme, namely the shooting method along with the Runge‐Kutta‐Felberge integration technique. The effects of flow and heat transfer controlling parameters on the solution variables are depicted and analyzed through the graphical presentation. The survey finds that magnifying viscosity parameter, Weissenberg number representing the non‐Newtonian Williamson parameter cause to retard the velocity field in both the directions and thermal conductivity parameter causes to reduce fluid temperature. The study also recognizes that enhancing magnetic parameters and thermal conductivity parameters slow down the heat transfer rate. The entropy production of the system is estimated through the Bejan number. It is noticeable that the Bejan number is eminently dependent on the heat generation parameter, thermal radiation parameter, viscosity parameter, thermal conductivity parameter, and Biot number. The skillful accomplishment of the present heat and mass transfer system is achieved through the exteriorized choice of the pertinent parameters.

The general aim is to promote the use of individual-based models (biological agent-based models) in teaching and learning contexts in life sciences and to make their progressive incorporation into academic curricula easier, complementing other existing modelling strategies more frequently used in the classroom. Modelling activities for the study of a predator–prey system for a mathematics classroom in the first year of an undergraduate program in biosystems engineering have been designed and implemented. These activities were designed to put two modelling approaches side by side, an individual-based model and a set of ordinary differential equations. In order to organize and display this, a system with wolves and sheep in a confined domain was considered and studied. With the teaching material elaborated and a computer to perform the numerical resolutions involved and the corresponding individual-based simulations, the students answered questions and completed exercises to achieve the learning goals set. Students’ responses regarding the modelling of biological systems and these two distinct methodologies applied to the study of a predator–prey system were collected via questionnaires, open-ended queries and face-to-face dialogues. Taking into account the positive responses of the students when they were doing these activities, it was clear that using a discrete individual-based model to deal with a predator–prey system jointly with a set of ordinary differential equations enriches the understanding of the modelling process, adds new insights and opens novel perspectives of what can be done with computational models versus other models. The complementary views given by the two modelling approaches were very well assessed by students.

A criterion of asymptotic stability of the trivial solution of an autonomous set of ordinary differential equations

In the past decade, beams that are made of functionally graded materials (FGM) have been employed in many engineering and biomedical application fields. In this paper, dynamic response of a FGM Timoshenko beam that is supported by an elastic foundation and is subjected to a moving mass, i.e., the effects of boundary flexibility, has been investigated. It is assumed that the material properties of the beam will change only in the thickness direction. The governing equations of motion are derived using Hamilton’s principles. The partial governing differential equations of motion are reduced to a set of ordinary differential equations by using Petrov-Galerkin method. Runge-Kutta numerical scheme is employed to solve the obtained set of ordinary differential equations. After verification of the results for some special cases with known sloutions the effect of various parameters such as the velocity of moving loads, the boundary flexibility, the power-law index on the vibration of the beam have been investigated. The special case of the solution of the problem was compared with the study of Mesut Simsek [2010] and H. P. Lee [1998] which showed excellent agreement. The results have also been compared to the similar beam without FGM and the advantages of FGM have been discussed.

In certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosolutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil'nikov bifurcation (Proctor & Weiss 1990). This establishes, for the first time, the existence of chaotic solutions of the equations governing two-dimensional magneto-convection. Moreover, in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modelled by a new third-order set of ordinary differential equations, which is shown here to have chaotic solutions that are created in a homoclinic explosion, in the same manner as the chaotic solutions of the Lorenz equations. Unlike the Lorenz equations, however, this model provides an accurate description of convection in the parameter regime where the chaotic solutions appear.

Creep buckling of a cylindrical shell under non-uniform external loads

This paper analyses the discrete hydraulic systems with emphasis on hydraulic parameter analysis for a pump-turbine power plant. The developed model could make hydro-electric power plants transient behavior studies more easy and accurate, especially for complicated systems. The simulation was developed based on analog mathematical models of transient phenomenon equations (Hydroacoustic modeling). In this method, the governing equations for transient flows are solved based on equivalent electrical circuits being defined for different equipments of the system. By using the Kirchhoff’s voltage law for the final equivalent circuit of the system a set of nonlinear ordinary differential equations is generated. The unsteady friction loss and visco-elastic behavior of pipe and fluid are considered in the pipelines model. A set of nonlinear ordinary differential equations is generated by using an implicit scheme. These equations of nonlinear ODEs are solved using a fourth order Runge-Kutta method. Further validation has been carried out with experimental values. Also a mathematical model for Francis pump-turbine has been developed and the transient behavior of two different cases of hydraulic systems including the pump-turbine has been investigated. The results were logically correct and acceptable. This method is very efficient for overall simulation of hydro power plants including turbine, generator and transformer.

This paper is concerned with the rapid solution of large sets of ordinary differential equations using parallel computing. Parallel computing facilities can be used for this purpose if, for example, due to structural properties it is possible to use problem partitioning and related decomposition methods. It is shown in this paper how such methods can be used to organise the overall computation task into multilevel hierarchies. A particular example is considered and results regarding the evaluation of the possible speed-up for different structures are given.