Finite-element approach

Finite element methods [20,21] have replaced finite difference methods in many fields, especially in the area of partial differential equations. With the finite element approach, the continuum is divided into a number of finite elements that are assumed to be joined by a discrete number of points along their boundaries. A function is chosen to represent the variation of the quantity over each element in terms of the value of the quantity at the boundary points. Therefore a set of simultaneous equations can be obtained that will produce a large, banded matrix. 

The three primary advantages of the finite element approach over finite difference methods are [9] ... 

An alternative approach to the finite element approach is one, introduced as a concept by Courant as early as 1943 [197], in which the total energy functional, implicit in the finite element method, is directly minimized with respect to all nodal positions. The approach is conjugate to the finite element method and merely differs in its procedural approach. It parallels, however, methods often used in atomistic modeling schemes where the potential energy functional of a system (e. g., given by the force field ) is minimized with respect to the position of all (or at least many) atoms of the system. A simple example of this emerging technique is given below. 

Figure 15.2. Region of interest for computing potential based on Laplace or Poisson equations, where (a) a complete rectangular grid is established to cover the region, which may be adapted to finite-difference techniques using (b) a five-point method, or (c) a finite-element approach based on sampling functions.

As with finite differences, the finite-element approach can be recast, using vectors and matrices, in the form Au = b, with A and b known and u to be determined. There are two basic approaches. In the first case, referred to as collocation, substimtion of Eq. (15.6) in (15.1) leads to... 

Consistent with the notion of approximating polynomials as in the preceding paragraph, finite-element approaches attempt to simplify the solution process by carefully choosing polynomials so as to minimize the number of coefficients required to determine and simplify the matrix inversion problem. One choice is the Bezier polynomials, defined by... 

Ciano et al. (2006) have used a finite element approach to model a tubular cell 0.3 m long. The equations are available in Ciano et al. (2006). Table 7.2 shows the partial differential equations and the mesh characteristics. This model is computationally demanding and the equations have been solved by adopting an iterative procedure. Initial guess values for temperature and current density are assumed (current density is calculated by means of a lumped model, as the function of the average temperature and the cell voltage). Momentum equation and continuity equation are... 

Bruels, R.G.M., Sengers, B.G., Oomens, C.W.J., Bouten, C.V.C. and Baaijens, F.P.T. (2002) Predicting local cell deformations in engineered tissue constructs a multilevel finite element approach. Journal of Biomechanical Engineering 124, 198-206... 

DEVELOPMENT OF A FINITE ELEMENT APPROACH TO MECHANICS, TRANSPORT AND BIOSYNTHESIS IN TISSUE ENGINEERING... 

Development of a Finite Element Approach to Mechanics, Transport and... 

V.B. Shenoy et al An adaptive finite element approach to atomic-scale mechanics-the quasicontinuum method. J. Mech. Phys. Solids 47, 611-642 (1999)... 

From the perspective of variational principles, the idea embodied in the finite element approach is similar to that in other schemes identify that particular linear combination of basis functions that is best in a sense to be defined below. The approximate solution of interest is built up as linear combinations of basis functions, and the crucial question becomes how to formulate the original boundary value problem in terms of the relevant expansion coefficients. 

Shenoy V. B., Miller R., Tadmor E. B., Rodney D, Phillips R. and Ortiz M., An Adaptive Finite Element Approach to Atomic-Scale Mechanics - The Quasicontinuum Method J. Mech. Phys. Solids AT, 611 (1999). 

The main method actually used is based on the finite element approach (FEM) and its variations. Most numerical schemes are bases on discrete distribution models, not all of them taking into account possible non-isotropic phase distribution. When this distribution is not isotropic but complies with a certain law, the same observation of measurement process yields... 

The gas flows horizontally, contacting by downflowing liquid. The effective driving force for mass transfer is between that for counter- and cocurrent contactors. Crossflow scrubbers have low pressure drop and usually require a lower liquid/gas ratio than either counter- or cocurrent scrubbers. The time of contact between gas and liquid is relatively low, and crossflow units are not reconunended for most chemical absorptions. Design procedures follow a finite-element approach the scrubber volume is divided into cubes, each of which is assumed to reach equilibrium. 

Up to now the basis functions Ni x) are still arbitrary and not restricted to a finite element approach. In the finite element frame a suitable approximation is given by Lagrange- or Hermite-interpolation polynomials. 

For the spatial integration we use the combined discrete variable and finite element approach mentioned above. A formal solution for the time development of the discretized wave-functions ipk t) is, given by ... 

Ippolito S. J., Kalantar-Zadeh K., Powell D. A., andWlodaiskiW., A 3-dimensional finite element approach for simulating acoustic wave propagation in layered SAW devices, presented at Ultrasonics, IEEE Symposium, October 5-8, Honolulu Hawaii Publication. 

This chapter presents both experimaital and finite element approaches that are commonly used to assist in the design and failure analysis of composite bolted joints. The work presented mostly stems from the authors work in the field over the past decade. Results from both local (i.e. detailed) finite element and global (Le. efficient) closed form and finite element approaches are presented, and expmmental results from the open literature are used to both calibrate and validate the modelling procedures. The primary variable under consideration is bolt-hole clearance, which is chosen as it induces significant three-dimensional stresses into the joint, and significandy alters the bolt-load distribution in multi-bolt joints, and so provides a rigorous test case for analysis. 

V.K. Jain, PC. Pdndey, Finite element approach to the two dimensional analysis of electrochemical machining. Precis. Eng. 2 (1) (January 1980) 23-28. 

Giannopoulos, G. I., Kakavas, P. A. Anifantis, N. K. (2008). Evaluation of the Effective Mechanical Properties of Single-Walled Carbon Nanotubes Using a Spring Based Finite Element Approach. Comput Mater. Sci., 41(4), 561-569. 

Simulation of Particle Trajectory-Finite Element Approach... 

For the system given in Fig. 1 using the analytical and finite element approaches, the performance of the system is evaluated. The methodologies used for these two methods are explained in the previous section. The results of these two simulation approaches will be compared to experimental results given in the work of Petersson et al. [2]. The simulation system is the replica of their experimental study performed with 3,7, and 10 p beads. The results of the experimental study are adopted from work of Petersson et al. [2] and... 

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