Finite-element algorithm

Swarbrick, S. J. and Nassehi, V., 1992a. A new decoupled finite element algorithm for viscoelastic flow. Part 1 numerical algorithm and sample results. Int. J. Numer. Methods Fluids 14, 1367-1376,... 

The objective of this work has been the development of finite element algorithms for hydraulically coupled three-dimensional thermomechanical processes to be applied to geological problems. A further aim was the realization of couplings to existing finite element algorithms for... 

Many packages have been developed for finite element algorithms. In fact, a Google search of best finite element package revealed more than fifty packages available with names ranging from ALGOR to Zebulon. 

Fiber-reinforced systems have been modeled with use of an MC method to place parallel fibers into a polymer matrix, with a finite element algorithm (FEA) then being used to compute elastic properties (274). A generic meshing algorithm for use in FEA studies of nanoparticle reinforcement of polymers has been developed (275) and applied to the calculation of mechanical properties of whisker and platelet filled systems. The method should be applicable to void-containing low dielectric materials of such great utility in the semiconductor industry. 

Ghoneim and Chen(33) developed a viscoelastic-viscoplastic law based on the assumption that the total strain rate tensor can be decomposed into a viscoelastic and a viscoplastic component. A linear viscoelasticity model is used in conjunction with a modified plasticity model in which hardening is assumed to be a function of viscoplastic strains as well as the total strain rate. The resulting finite-element algorithm is then used to analyze the strain rate and pressure effects on the mechanical behavior of a viscoelastic-viscoplastic material. 

Felicelli et al. [54] used a fixed finite element algorithm to calculate macrosegregation and the formation of channels and freckles in Pb-Sn alloys. They assumed that the mushy zone is a porous medium with an isotropic permeability and considered superficial velocity components for the fluid velocities in the mushy region. The superflcial velocities are... 

The penalty finite element algorithm is employed to solve the above nonlinear system of the government... 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

Algorithms based on the last approach usually provide more flexible schemes than the other two methods and hence are briefly discussed in here. Hughes et al. (1986) and de Sampaio (1991) developed Petrov-Galerkin schemes based on equal order interpolations of field variables that used specially modified weight functions to generate stable finite element computations in incompressible flow. These schemes are shown to be the special cases of the method described in the following section developed by Zienkiewicz and Wu (1991). 

An example describing the application of this algorithm to the finite element modelling of free surface flow of a Maxwell fluid is given in Chapter 5. 

Iterative solution methods are more effective for problems arising in solid mechanics and are not a common feature of the finite element modelling of polymer processes. However, under certain conditions they may provide better computer economy than direct methods. In particular, these methods have an inherent compatibility with algorithms used for parallel processing and hence are potentially more suitable for three-dimensional flow modelling. In this chapter we focus on the direct methods commonly used in flow simulation models. 

The most important direct solution algorithms used in finite element computations are based on the Gaussian elimination method. 

The right-hand side in Equation (6.18) is known and hence its solution yields the error 5x in the original solution. The procedure can be iterated to improve the solution step-by-step. Note that implementation of this algorithm in the context of finite element computations may be very expensive. A significant advantage of the LU decomposition technique now becomes clear, because using this technique [A] can be decomposed only once and stored. Therefore in the solution of Equation (6.18) only the right-hand side needs to be calculated. 

Although only approximate analytical solutions to this partial differential equation have been available for x(s,D,r,t), accurate numerical solutions are now possible using finite element methods first introduced by Claverie and coworkers [46] and recently generalized to permit greater efficiency and stabihty [42,43] the algorithm SEDFIT [47] employs this procedure for obtaining the sedimentation coefficient distribution. 

Clearly, the extent of exotherm-generated temperature overshoot predicted by the Chiao and finite element models differs substantially. The finite element results were not markedly changed by refining the mesh size or the time increments, so the difference appears to be inherent in the numerical algorithms used. Such comparison is useful in further development of the codes, as it provides a means of pinpointing those model parameters or algorithms which underlie the numerical predictions. These points will be explored more fully in future work. 

Figure 5. Schematic of nonorthogonal transformations used in finite-element/Newton algorithms for calculating cellular interfaces, (a) Monge cartesian representation for almost planar interfaces, (b) Mixed cylindrical/cartesian mapping for representing deep cells.

AI methods may be used in various ways. The models may be used as a standalone application, e.g., in recent work on the design of microwave absorbers using particle swarm optimization (PSO).6 Alternatively, a computational tool, such as a finite element analysis or a quantum mechanical calculation, may be combined with an AI technique, such as an evolutionary algorithm. 

Holst, M.J. Baker, N.A. Wang, F., Adaptive multilevel finite element solution of the Poisson-Boltzmann equation I algorithms and examples, J. Comp. Chem. 2000, 21, 1319-1342... 

Unlike parameter optimization, the optimal control problem has degrees of freedom that increase linearly with the number of finite elements. Here, for problems with many finite elements, the decomposition strategy for SQP becomes less efficient. As an alternative, we discussed the application of Newton-type algorithms for unconstrained optimal control problems. Through the application of Riccati-like transformations, as well as parallel solvers for banded matrices, these problems can be solved very efficiently. However, the efficient solution of large optimal control problems with... 

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