Parallel finite elements

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... 

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. 

Parallel finite element computations have been developed for a number of years mostly for elastic solids and structures. The static domain decomposition (DD) methodology is currently used almost exclusively for decomposing such elastic finite element domains in subdomains. This subdivision has two main purposes, namely (a) to distribute element computations to CPUs in an even manner and (b) to distribute system of equations evenly to CPUs for maximum efficiency in solution process. 

The elastic-plastic parallel finite element computational problem... 

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. 

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. 

D. Givoli, J. E. Flaherty, M. S. Shephard. Parallel adaptive finite element analysis of viscous flows based on a combined compressible-incompressible formulation. Int J Numer Meth Heat and Fluid Flow 7 880, 1997. 

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. 

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... 

A plane-parallel stellar atmosphere is a semi-infinite medium. In the numerical calculation, we divide it into n finite elements and 1 semi-infinite element. Let us define a node as a point between two elements. Node 0 is defined as the boundary between the surface element and the vacuum. In total, we have n+1 nodes. The distribution of any physical quantity is represented by a vector of n+1 dimensions with its values at the n+1 nodes as elements. The mean intensity of radiation J is written in the ordinary expression as... 

Fig. 5.2 Comparison of creep behavior and time-dependent change in fiber and matrix stress predicted using a 1-D concentric cylinder model (ROM model) (solid lines) and a 2-D finite element analysis (dashed lines). In both approaches it was assumed that a unidirectional creep specimen was instantaneously loaded parallel to the fibers to a constant creep stress. The analyses, which assumed a creep temperature of 1200°C, were conducted assuming 40 vol.% SCS-6 SiC fibers in a hot-pressed SijN4 matrix. The constituents were assumed to undergo steady-state creep only, with perfect interfacial bonding. For the FEM analysis, Poisson s ratio was 0.17 for the fibers and 0.27 for the matrix, (a) Total composite strain (axial), (b) composite creep rate, and (c) transient redistribution in axial stress in the fibers and matrix (the initial loading transient has been ignored). Although the fibers and matrix were assumed to exhibit only steady-state creep behavior, the transient redistribution in stress gives rise to the transient creep response shown in parts (a) and (b). After Wu et al 1...

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. 

N. Baker, D. Sept, M. Holst, and J. A. McCammon, The adaptive multilevel finite element solution of the Poisson-Boltzmann equation on massively parallel computers, IBM J. Res. Dev. in press (2001). 

At the inlet to the finite element domain, the flow is parallel so the equations of the lubrication approximation are used to specify the inlet velocity profile. These equations are integrated from -oo to the inlet, generating an equation relating the flow rate to the inlet pressure. The remaining boundary conditions are as shown in Figure 3. The only complexity here is that the fluid traction, n T, at the free surface has to be specified as a boundary condition on the momentum equation. A force balance there gives, in dimensionless form,... 

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