Finite element methods media

Yasui et al. [29] have used solution of wave equation based on finite element method for characterization of the acoustic field distribution. A unique feature of the work is that it also considers contribution of the vibrations occurring due to the reactor wall and have evaluated the effect of different types of the reactor walls or in other words the effect of material of construction of the sonochemical reactor. The work has also contributed to the understanding of the dependence of the attenuation coefficient due to the liquid medium on the contribution of the vibrations from the wall. It has been shown that as the attenuation coefficient increases, the influence of the acoustic emission from the vibrating wall becomes smaller and for very low values of the attenuation coefficient, the acoustic field in the reactor is very complex due to the strong acoustic emission from the wall. 

The above considerations referred to the practically important examples of more or less ordered heterogeneities. If we face random distribution, usually effective medium and percolation theory have to be referred to in order to evaluate the inhomogeneous situations properly. However, attention has to be paid to the fact that they often require nonrealistic approximations. For more details see Ref.300 In such cases numerical calculations, e.g., via finite element methods are more reliable. 

Mass balance of solid Mass balance of water Mass balance of air Momentum balance for the medium Internal energy balance for the medium The resulting system of Partial Differential Equations is solved numerically. Finite element method is used for the spatial discretization while finite differences are used for the temporal discretization. The discretization in time is linear and the implicit scheme uses two intermediate points, t and t between the initial 1 and final t limes. Finally, since the problems are nonlinear, the Newton-Raphson method has been adopted following an iterative scheme. 

The relationship between the concentration and the current can be obtained by solving Pick s equation under appropriate boundary conditions. Time-dependent diffusion equations can be easily solved numerically, for instance by finite-element methods. However, such techniques normally require medium or large pla tforms to obtain accurate results with reasonable run times. This is particularly true when dealing with structured devices. A different approach was used here. Analytical methods2> show that the time dependence of concentration profiles is mostly exponential in character. Solutions of Pick s equation of the form... 

The blister test was analyzed by the finite element method (16). To evaluate G, Andrews and Stevenson (17) have tentatively added the elastic energy computed by plate theory (far field) to that computed for an internal crack in an infinite medium (near field) for the same radius and pressure. But the closed form solution (analog to the JKR solution (3) for spheres, and the Kanninen solution (12) for DCB) is not yet known. 

We briefly recapitulate the model, detailed previously [11], which is based on the variable-metric total-energy approach by Gusev [12] and consists of a system comprising an inclusion embedded in a continuous medium. The inclusion behavior is described in atomistic detail whereas the continuum is modeled by a displacement-based finite-element method. Since the atomistic... 

A number of approaches to connect multiple-scale simulation in finite-element techniques have been published [31-34], They are able to describe macroscopically inhomogeneous strain (e.g., cracks)—even dynamic simulations have been performed [35]—but invariably require artificial constraints on the atomistic scale [36], Recently, an approach has been introduced that considers a system comprising an inclusion of arbitrary shape embedded in a continuous medium [20], The inclusion behavior is described in an atomistically detailed manner [37], whereas the continuum is modeled by a displacement-based Finite-Element method [38,39], The atomistic model provides state- and configuration-dependent material properties, inaccessible to continuum models, and the inclusion in the atomistic-continuum model acts as a magnifying glass into the molecular level of the material. 

Bogdanis, E., 2001. Modelling of heat and mass transport during drying of an elastic-viscoelastic medium and resolution by the finite element methods. Diss. University Pau, France. 

The capacitance of such a multilayered structure, as shown in Fig. 7.5, having low layer thickness is very difficult to study either by modeling or by finite element method analysis [96], However, if there is a monotonic increase in permittivity of the layers in the direction of the electrodes plane to outer medium, the total capacitance of the sensor can be evaluated with the contribution of each layer in series as [96] ... 

If the medium is slightly compressible that is typically the case for real materials, one might be tempted to employ the penalty method. In such a situation, the value of the bulk modulus is considerably higher than that of the shear modulus - exactly the case for rubber as was noted in the introduction. Then, the actual value of the bulk modulus is specified in the Hook s law. However, such displacement methods display the same problems as the improperly designed mixed finite element methods, and the problem comes down again to the improper ratio of the dof and the constraints. 

Due to the complexity of the mathematical treatment for cylindrical systems that include phenomena such as the presence of a diffusion boundary layer, a membrane that laminates the device surface and/or finite external medium, analytical solutions are difficult to obtain. Consequently, the study of drug release from cylindrical matrix systems using numerical methods is a common practice. Zhou and Wu analyzed in detail the release from cylindrical monolithic dispersion devices by using the finite element method [189]. 

The analysis of the electromagnetic field of a vertical magnetic dipole located either on the axis of cylindrical interfaces (formations of an infinite thickness) or in a medium with horizontal interfaces only allows us to investigate the influence of the borehole and the invasion zone, as well as the effect caused by a finite thickness of the formation. For such models application of the separation of variables method is the most natural approax h enabling us to present the field in a explicit form by known functions. It is a much more complicated problem when the vertical magnetic dipole is located on the borehole axis and the formation has a finite thickness. In this case the method of separation of variable cannot be used, since both cylindrical and horizontal interfaces are present and it is more appropriate to apply such numerical methods as integral equations or finite elements. 

In biochemistry, methods that discretize the Poisson differential operator over finite elements (FE) have long been in use. In general, FEM approaches do not directly use the molecular cavity surface. Nevertheless, as the whole space filled by the continuous medium is partitioned into locally homogeneous regions, a careful consideration of the portion of space occupied by the molecular solute has still to be performed. 

It has been achieved through the processing of an array of data, obtained as the result of solving the equations describing the turbulent motion of a continuous medium, and using the K-s turbulence model (Equations 2.1-2.12) with the method of finite elements on a nonuniform calculation grid. 

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