Parallel finite element methods

If the boundaiy is parallel to a coordinate axis any derivative is evaluated as in the section on boundary value problems, using either a onesided, centered difference or a false boundary. If the boundary is more irregular and not parallel to a coordinate line then more complicated expressions are needed and the finite element method may be the better method. 

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. 

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... 

V. E. Taylor and B. Nour-Omid. A study of the factorization fill-in for a parallel implementation of the finite element method. Int. J. Numer. Meth. Eng., 37 3809-3823, 1994. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

Wasfy, A. C. West, and V. Modi, Parallel Finite Element Computation of Unsteady Incompressible Flows, Int. J. Numerical Methods in Fluids, Vol. 26 (1998) 17. 

At about the same time, another detailed theoretical study was presented with intended applications to the acceptor spectrum in germanium and GaAs up to relatively large values of B parallel to the three main crystal orientations [128]. Considering the VB s-o splitting in these crystals, the ITV VB was disregarded so that the calculations cannot be transposed to acceptors in silicon. The numerical method (the matrix method ) used to determine the eigenvalues in this study is non-variational and different from the finite-element method of Said et al. (ref. 51 in Chap 5). The results of these calculations will also be compared later with the experimental observations. 

The equations to be solved are similar to those in the previous section with some minor differences due the change in geometry (parallel-plate microchannel versus microtube). In the solution, slip boundary conditions given in Eqs. (I) and (2) are applied and finite element method is used to solve for the velocity profile and the temperature distribution. Then, from the temperature profile, the local Nu is determined. 

E.F. Van de Velde. Concurrent Scientific Computing. Springer-Verlag, New York, 1994. This text contains an excellent introduction to parallel numerical algorithms including sections on the finite difference and finite element methods. 

For other than flat plates, heat flux lines are seldom parallel, rarely steady. In transient heat flow, determination of the temperature at a given time and point within the load necessitates use of the finite element method. 

Bielak J, Ghattas O, Kim EJ (2005) Parallel octree-based finite element method fm large-scale earthquake ground motion simulatimi. Comput Model Eng Sci... 

Reliability estimation for dynamic or seismic loading is evolving. The probability of failure implies that it needs to be estimated just before failure developing various sources of nonlinearities. The finite element method (FEM) is commonly used by the deterministic community to study nonlinear problems where dynamic loadings are applied in time domain. This clearly indicates that an FEM-based general purpose reliability evaluation method, parallel to the deterministic analysis, is necessary. 

It may be useful to the reader to consider as an example the work by Adams et al (1978c) and some of their hitherto unpublished results. They used finite-element methods to examine the stresses in high-performance composites in symmetrical lap joints with parallel, bevelled, scarfed and stepped adherends. The composite adherends were assumed to be linearly elastic type II carbon fibre reinforced epoxy composites with a 60% fibre volume fraction. The mechanical properties of this material are given in Table 3. 

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. 

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