Finite element techniques

Taylor, C. and Hood, P., 1973. A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1, 73-100. 

U-V-P schemes belong to the general category of mixed finite element techniques (Zienkiewicz and Taylor, 1994). In these techniques both velocity and pressure in the governing equations of incompressible flow are regarded as primitive variables and are discretized as unknowns. The method is named after its most commonly used two-dimensional Cartesian version in which U, V and P represent velocity components and pressure, respectively. To describe this scheme we consider the governing equations of incompressible non-Newtonian flow (Equations (1.1) and (1.4), Chapter 1) expressed as... 

The most general method of tackling the problem is the use of the finite-element technique 8 to determine the temperature distribution at any time by using the finite difference equation in the form of equation 9.40. 

Finite-element techniques can cope with large, highly non-linear deformations, making it possible to model soft tissues such as skin. When relatively large areas of skin are replaced during plastic surgery, there is a problem that excessive distortion of the apphed skin will prevent adequate adhesion. Finite-element models can be used to determine, either by rapid trial-and-error modelhng or by mathematical optimisation, the best way of... 

Two remaining problems relating to the treatment of solvation include the slowness of Poisson-Boltzmann calculations, when these are used to treat electrostatic effects, and the difficulty of keeping buried, explicit solvent in equilibrium with the external solvent when, e.g., there are changes in nearby solute groups in an alchemical simulation. Faster methods for solving the Poisson-Boltzmann equation by means of parallel finite element techniques are becoming available, however.22 24... 

As a second illustration of the finite-element technique, we proceed as follows. We assume a rectangular mesh of three rows and three columns with uniform step sizes h = k = along the x and y axes, respectively. Further, assume that potentials on the boundary have values f/oi, f/02, t/o3. t io. t i4. t 20. U24, U o, t/34. U41, f/42, f/43, where by f/ j we refer to the top and bottom or left and right rows or columns as i,j = 0 and i,j = 4, respectively. The five-point sampling Laplace equation has the algebraic form... 

A final note is in order. The finite-difference and finite-element techniques are entirely equivalent from a mathematical point of view. What is different about these are the conceptualization of the problem and the resulting computational techniques to be employed. One method is not better than the other, although in particular circumstances one may clearly be superior. The point is that a modeler and modeling systems should account for both methods as well as others not mentioned here. 

We simulate these systems using standard finite element techniques (e.g.. Baker 1983) for solutions to the porous media conservation equations of mass, momentum, and energy on a rectilinear mesh using a code called BasinLab (Manning etal. 1987). 

To solve the preceding set of equations, Equation 5.62 is plugged into Equation 5.60. By separately determining the compaction properties of the fiber bed [32] an evolution equation for the pressure can be obtained. Because this is a moving boundary problem the derivative in the thickness direction can be rewritten [32] in terms of an instantaneous thickness. The pressure field can then be solved for by finite difference or finite element techniques. Once the pressure is obtained and the velocity computed, the energy and cured species conservation equations can be solved using the methodology outlined in Section 5.4.1. 

The main interest in finite element analysis from a testing point of view is that it requires the input of test data. The rise in the use of finite element techniques in recent years is the reason for the greatly increased demand for stress strain data presented in terms of relationships such as the Mooney-Rivlin equation given in Section 1 above. 

Connor J.J., Brebbia C.A., Finite Element Techniques for Fluid Flow, Newnes -Butterworths, London - Boston (1977). 

Instead of starting with a rigorous and mathematical development of the finite element technique, we proceed to present the finite element method through a solution of onedimensional applications. To illustrate the technique, we will first find a numerical solution to a heat conduction problem with a volumetric heat source... 

The numerical study of the Orr-Sommerfeld equations requires to discretize the dy operators in equation (25). As in [84], the spectral tau-Chebychev approximations are often used, though pseudo-spectral [85] or finite element techniques [86] may be chosen too. 

From a numerical viev point, rapid progress has been made in the last few years in studies generally devoted to the entry flow problem, together with the use of more and more realistic constitutive equations for the fluids. Consequently, more complexity was involved for the munerical problem, in relation to the nonlinearity induced by the rheological model in the governing equations. The use of nonlinear constitutive models required approximate methods for solving the equations, such as finite element techniques, even for isothermal and steady-state conditions related to a simple flow geometry. 

Thus the objective here is a generally applicable simulation of steady, two-dimensional, incompressible flow between rigid rolls with film splitting. The results reported are solutions of the full Navier-Stokes system including the physically required boundary conditions. The analysis is also extended to a shearthinning fluid. The solutions consist of velocity and pressure fields, free surface position and shape, and the sensitivities of these variables to parameter variations, valuable information not readily available from the conventional approach (10). The rate-of-strain, vorticity, and stress fields are also available from the solutions reported here although they are not portrayed. Moreover, the stability of the flow states represented by the solutions can also be found by additional finite element techniques (11), and the results of doing so will be reported in the future. 

After the discrete points on the interface are moved with the flow, the continuous interface is reconstructed by connecting these points by appropriate linear or triangular elements (i.e., a finite element technique). It is noticed that explicit front tracking is generally more complex than the advection of a maker function as in the VOF and LS methods, nevertheless this technique is also considered more accurate [224]. 

A computational design procedure of a thermoelectric power device using Functionally Graded Materials (FGM) is presented. A model of thermoelectric materials is presented for transport properties of heavily doped semiconductors, electron and phonon transport coefficients are calculated using band theory. And, a procedure of an elastic thermal stress analysis is presented on a functionally graded thermoelectric device by two-dimensional finite element technique. First, temperature distributions are calculated by two-dimensional non-linear finite element method based on expressions of thermoelectric phenomenon. Next, using temperature distributions, thermal stress distributions are computed by two-dimensional elastic finite element analysis. 

At one time the stress analysis required for the use of fracture mechanics would have posed a formidable problem for many practical applications. This situation is becoming progressively less of a problem as more sophisticated and better numerical methods for stress analysis become available (30). For example, the use of finite element techniques in the fracture mechanics analysis of adhesive bonds has been explored (31). 

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