The generalized finite element method (GFEM) was introduced in Reference 1 as a combination of the standard FEM and the partition of unity method The standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions) The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation The essential boundary conditions can be imposed exactly as in the standard FEM Adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM Copyright © 2000 John Wiley & Sons, Ltd

The generalized finite element method: an example of its implementation and illustration of its performance

Describes Uintah, a component-based visual problem-solving environment (PSE) that is designed to specifically address the unique problems of massively parallel computation on tera-scale computing platforms. Uintah supports the entire life-cycle of scientific applications by allowing scientific programmers to quickly and easily develop new techniques, debug new implementations and apply known algorithms to solve novel problems. Uintah is built on three principles: (1) as much as possible, the complexities of parallel execution should be handled for the scientist, (2) the software should be reusable at the component level, and (3) scientists should be able to dynamically steer and visualize their simulation results as the simulation executes. To provide this functionality, Uintah builds upon the best features of the SCIRun (Scientific Computing and Imaging Run-time) PSE and the DoE (Department of Energy) Common Component Architecture (CCA).

Uintah: a massively parallel problem solving environment

ReALE: A reconnection-based arbitrary-Lagrangian-Eulerian method

https://coaps.fsu.edu/pub/eric/OCP5930/Papers/Remapping.pdf

Incremental Remapping as a Transport/Advection Algorithm

A new multiscale enrichment method based on the partition of unity (MEPU) method is presented. It is a synthesis of mathematical homogenization theory and the partition of unity method. Its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible. MEPU is perfectly suited for enriching the coarse scale continuum descriptions (PDEs) with fine scale features and the quasi-continuum formulations with relevant atomistic data. Numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasi-continuum formulations. Copyright © 2005 John Wiley & Sons, Ltd.

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Multiscale enrichment based on partition of unity

The Implicit Continuous-fluid Eulerian (ICE) method is a finite-volume scheme that is stable for any value of the Courant number based on the sound speed. In the incompressible limit, the ICE method becomes essentially identical to the Marker and Cell (MAC) method, so the two schemes are closely related. In this article, the classical ICE method is extended to multiple interpenetrating phases, and employed with a single control volume (nonstaggered) mesh framework. The incompressible limit is preserved, so that problems involving equations of state, or those exhibiting constant material densities, can be addressed with the same computer code. The scheme reduces properly to a single-fluid method, enabling benchmarking using well-known test cases. Thus, the numerical issues focus only on those aspects unique to problems having multiple density, velocity and temperature fields. The discussion begins with a derivation of the exact, ensemble-averaged equations. Examples of the most basic closures axe given, and the well-posedness of the equations is demonstrated. The numerical method is described in operator notation, and the discretization is sketched. The flow patterns in a bubble column are computed as an incompressible flow example. For a compressible flow example, the expansion and compression of a bubble formed by high-explosivemore » gases under water is shown. In each case, comparison to experimental data is made.« less

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A cell-centered ICE method for multiphase flow simulations

Compatible Fluxes for van Leer Advection

A formalism for developing multiphase turbulence models is introduced by analogy to the phenomenological method used for single-phase turbulence. A sample model developed using the formalism is given in detail. The procedure begins with ensemble averaging of the exact conservation equations, with closure accomplished by using a combination of analytical and experimental results from the literature. The resulting model is applicable to a wide range of common multiphase flows including gas-solid, liquid-solid and gas-liquid (bubbly) flows. The model is positioned for ready extension to three-phase turbulence, or for use in two-phase turbulence in which one phase is accounted for in multiple size classes, representing polydispersivity. The formalism is expected to suggest directions toward a more fundamentally based theory, similar to the way that early work in single-phase turbulence has led to the spectral theory. The approach is unique in that a portion of the total energy decay rate is ascribed to each phase, as is dictated by the exact averaged equations, and results in a transport equation for energy decay rate associated with each phase. What follows is a straightforward definition of a turbulent viscosity for each phase, and accounts for the effect of exchange of fluctuational energy among phases on the turbulent shear viscosity. The model also accounts for the effect of slip momentum transfer among the phases on the production of turbulence kinetic energy and on the tensor character of the Reynolds stress. Collisional effects, when appropriate, are included by superposition. The model reduces to a standard form in limit of a single, pure material, and is expected to do a credible job of describing multiphase turbulent flows in a wide variety of regimes using a single set of coefficients.

/pdf/toward-a-general-theory-for-multiphase-turbulence-part-i-1v6pt8lau1.pdf

Toward a General Theory for Multiphase Turbulence Part I: Development and Gauging of the Model Equations

The effect of two-body forces on the structure of dynamic waves in fluidized beds is studied, with particular emphasis on expansion waves. Averaged equations of motion are used for the study, so the media appear to be interpenetrating continua. Both inertial and viscous two-body effects are considered for incompressible materials fluidized by an incompressible fluid. Inertial effects are included in the averaged momentum exchange force, using exact (classical) results for the potential flow generated by the motion of one submerged body relative to another body. Viscous effects are represented, in the limit of zero relative Reynolds number, by solutions to Stokes equations for the two-body problem. For simple one-dimensional motion the inertial force is repulsive always, giving a positive compressibility to the dispersed field total density; the force is of such a magnitude that the single-pressure continuum equations are unconditionally hyperbolic. The corresponding 1-D viscous force is attractive when the bodies move apart, and therefore introduces a negative compressibility to the dispersed field. Competition between the two-body inertial and viscous forces ultimately determines the nature of dynamic waves in a given fluidization system.

B. A. Kashiwa

Papers

A cell-centered ICE method for multiphase flow simulations

Compatible Fluxes for van Leer Advection

Toward a General Theory for Multiphase Turbulence Part I: Development and Gauging of the Model Equations

A study of two-body forces in fluidization