G.F. Carey and Y. Shen
A least‐squares finite element analysis of viscous fluidflow together with a trajectory integration technique fortracers is formulated and provides a mechanism for…
Abstract
A least‐squares finite element analysis of viscous fluid flow together with a trajectory integration technique for tracers is formulated and provides a mechanism for investigating mixing. Tracer integration is carried out using an improved Heun predictor‐corrector. Results from our supporting numerical studies on the CRAY and Connection Machine (CM) closely resemble the patterns of mixing observed in experiments. A “box‐counting” scheme and other measures to characterize the level of mixing are developed and investigated. This measure is utilized in numerical experiments to determine an optimal forcing frequency for mixing by periodic boundary motion in a rectangular enclosure. Some details concerning the numerical schemes and vector‐parallel implementation are also included.
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G.F. CAREY and M. SHARMA
A flux‐upwind finite element method is developed for the carrier transport equations in semiconductor device modeling. Our approach is motivated by the streamline upwind methods…
Abstract
A flux‐upwind finite element method is developed for the carrier transport equations in semiconductor device modeling. Our approach is motivated by the streamline upwind methods that have proven effective in fluid mechanics. The procedure reduces precisely to the Scharfetter‐Gummel approach in one dimension. In higher‐dimensions, however, it differs from this classical technique and is shown here to generate more accurate solutions with less numerical dissipation. Numerical results are presented for representative MOSFET and p‐n junction devices to illustrate this point. Both upwind techniques have been implemented in conjunction with an adaptive finite element refinement procedure for better layer resolution and yield a more stable algorithm.
Presents a review on implementing finite element methods on supercomputers, workstations and PCs and gives main trends in hardware and software developments. An appendix included…
Abstract
Presents a review on implementing finite element methods on supercomputers, workstations and PCs and gives main trends in hardware and software developments. An appendix included at the end of the paper presents a bibliography on the subjects retrospectively to 1985 and approximately 1,100 references are listed.
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J.W. Peterson, B.T. Murray and G.F. Carey
The purpose of this paper is to consider double‐diffusive convection in a heated porous medium saturated with a fluid. Of particular interest is the case where the fluid has a…
Abstract
Purpose
The purpose of this paper is to consider double‐diffusive convection in a heated porous medium saturated with a fluid. Of particular interest is the case where the fluid has a stabilizing concentration gradient and small diffusivity.
Design/methodology/approach
A fully‐coupled stabilized finite element scheme and adaptive mesh refinement (AMR) methodology are introduced to solve the resulting coupled multiphysics application and resolve fine scale solution features. The code is written on top of the open source finite element library LibMesh, and is suitable for parallel, high‐performance simulations of large‐scale problems.
Findings
The stabilized adaptive finite element scheme is used to compute steady and unsteady onset of convection in a generalized Horton‐Rogers‐Lapwood problem in both two and three‐dimensional domains. A detailed study confirming the applicability of AMR in obtaining the predicted dependence of solutal Nusselt number on Lewis number is given. A semi‐permeable barrier version of the generalized HRL problem is also studied and is believed to present an interesting benchmark for AMR codes owing to the different boundary and internal layers present in the problem. Finally, some representative adaptive results in a complex 3D heated‐pipe geometry are presented.
Originality/value
This work demonstrates the feasibility of stabilized, adaptive finite element schemes for computing simple double‐diffusive flow models, and it represents an easily‐generalizable starting point for more complex calculations since it is based on a highly‐general finite element library. The complementary nature of h‐adaptivity and stabilized finite element techniques for this class of problem is demonstrated using particularly simple error indicators and stabilization parameters. Finally, an interesting double‐diffusive convection benchmark problem having a semi‐permeable barrier is suggested.
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This paper gives a bibliographical review of the finite element and boundary element parallel processing techniques from the theoretical and application points of view. Topics…
Abstract
This paper gives a bibliographical review of the finite element and boundary element parallel processing techniques from the theoretical and application points of view. Topics include: theory – domain decomposition/partitioning, load balancing, parallel solvers/algorithms, parallel mesh generation, adaptive methods, and visualization/graphics; applications – structural mechanics problems, dynamic problems, material/geometrical non‐linear problems, contact problems, fracture mechanics, field problems, coupled problems, sensitivity and optimization, and other problems; hardware and software environments – hardware environments, programming techniques, and software development and presentations. The bibliography at the end of this paper contains 850 references to papers, conference proceedings and theses/dissertations dealing with presented subjects that were published between 1996 and 2002.
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X‐ray lithography is an important technique in micro fabrication used to obtain structures and devices with a high aspect ratio. The X‐ray exposure takes place in a system…
Abstract
X‐ray lithography is an important technique in micro fabrication used to obtain structures and devices with a high aspect ratio. The X‐ray exposure takes place in a system composed of a mask and a photoresist deposited on a substrate (with a gap between mask and resist). Predictions of the temperature distribution in three dimensions in the different layers (mask, gap, photoresist and substrate) and of the potential temperature rise are essential for determining the effect of high flux X‐ray exposure on distortions in the photoresist due to thermal expansion. In this study, we develop a three‐dimensional numerical method for obtaining the temperature profile in an X‐ray irradiation process by using a hybrid finite element‐finite difference scheme for solving three‐dimensional parabolic equations on thin layers. A domain decomposition algorithm is then obtained based on a parallel Gaussian elimination for solving block tridiagonal linear systems. The method is illustrated by a numerical method.
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M.B. Davis and G.F. Carey
The Rayleigh‐Benard‐Marangoni problem for natural convection in a rectangular cavity with thermocapillary forces on a free surface is investigated using a stream…
Abstract
The Rayleigh‐Benard‐Marangoni problem for natural convection in a rectangular cavity with thermocapillary forces on a free surface is investigated using a stream function‐vorticity formulation. The nonlinear system is iteratively decoupled and high‐degree p finite elements are used for the discretization of the physical domain. The linear systems arising from the discretization at each iteration are solved using a spectral multilevel scheme, which is a natural preconditioner for high‐p (spectral) elements. The spectral multilevel solver lends itself to parallelization in an element‐by‐element (EBE) framework. Simulation results are presented and compared to previously published results. The multilevel efficiency is compared to previous results for the driven cavity problem. Parallel performance studies are presented for the Cray T3E distributed memory architecture.
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Marcin Kamiński and Graham F. Carey
To generalize the traditional 2nd order stochastic perturbation technique for input random variables and fields and to demonstrate for flow problems.
Abstract
Purpose
To generalize the traditional 2nd order stochastic perturbation technique for input random variables and fields and to demonstrate for flow problems.
Design/methodology/approach
The methodology is based on an n‐th order expansion (perturbation) for input random parameters and state functions around their expected value to recover probabilistic moments of the response. A finite element formulation permits stochastic simulations on irregular meshes for practical applications.
Findings
The methodology permits approximation of expected values and covariances of quantities such as the fluid pressure and flow velocity using both symbolic and discrete FEM computations. It is applied to inviscid irrotational flow, Poiseulle flow and viscous Couette flow with randomly perturbed boundary conditions, channel height and fluid viscosity to illustrate the scheme.
Research limitations/implications
The focus of the present work is on the basic concepts as a foundation for extension to engineering applications. The formulation for the viscous incompressible problem can be implemented by extending a 3D viscous primitive variable finite element code as outlined in the paper. For the case where the physical parameters are temperature dependent this will necessitate solution of highly non‐linear stochastic differential equations.
Practical implications
Techniques presented here provide an efficient approach for numerical analyses of heat transfer and fluid flow problems, where input design parameters and/or physical quantities may have small random fluctuations. Such an analysis provides a basis for stochastic computational reliability analysis.
Originality/value
The mathematical formulation and computational implementation of the generalized perturbation‐based stochastic finite element method (SFEM) is the main contribution of the paper.
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Andrey B. Andreev and Todor D. Todorov
To study and to analyze a second order finite‐element boundary‐flux approximation using isoparametric numerical integration.
Abstract
Purpose
To study and to analyze a second order finite‐element boundary‐flux approximation using isoparametric numerical integration.
Design/methodology/approach
The numerical finite‐element integration is the main method used in this research. Since a domain with curved boundary is considered we apply an isoparametric approach. The lumped flux formulation is another method of approach in this paper.
Findings
This research study presents a careful analysis of the combined effect of the numerical integration and isoparametric FEM on the boundary‐flux error. Some L2‐norm estimates are proved for the approximate solutions of the problem under consideration.
Research limitations/implications
The authors offer a general study within the framework of the boundary‐flux approximation theory, which completes the results of published works in this scientific field of research.
Practical implications
A useful application is to employ appropriate quadrature formulae without violating the precision of the boundary‐flux FEM. The lumped mass approximation is also an important practical approach to the problem in question.
Originality/value
The paper presents an entire investigation in FE boundary‐flux approximation theory, in particular, elements of arbitrary degree and domains with curved boundaries. The work is addressed to the possible related fields of interest of postgraduate students and specialists in fluid mechanics and numerical analysis.
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Kemelli C. Estacio, Graham F. Carey and Norberto Mangiavacchi
The purpose of this paper is to develop a novel unstructured simulation approach for injection molding processes described by the Hele‐Shaw model.
Abstract
Purpose
The purpose of this paper is to develop a novel unstructured simulation approach for injection molding processes described by the Hele‐Shaw model.
Design/methodology/approach
The scheme involves dual dynamic meshes with active and inactive cells determined from an initial background pointset. The quasi‐static pressure solution in each timestep for this evolving unstructured mesh system is approximated using a control volume finite element method formulation coupled to a corresponding modified volume of fluid method. The flow is considered to be isothermal and non‐Newtonian.
Findings
Supporting numerical tests and performance studies for polystyrene described by Carreau, Cross, Ellis and Power‐law fluid models are conducted. Results for the present method are shown to be comparable to those from other methods for both Newtonian fluid and polystyrene fluid injected in different mold geometries.
Research limitations/implications
With respect to the methodology, the background pointset infers a mesh that is dynamically reconstructed here, and there are a number of efficiency issues and improvements that would be relevant to industrial applications. For instance, one can use the pointset to construct special bases and invoke a so‐called “meshless” scheme using the basis. This would require some interesting strategies to deal with the dynamic point enrichment of the moving front that could benefit from the present front treatment strategy. There are also issues related to mass conservation and fill‐time errors that might be addressed by introducing suitable projections. The general question of “rate of convergence” of these schemes requires analysis. Numerical results here suggest first‐order accuracy and are consistent with the approximations made, but theoretical results are not available yet for these methods.
Originality/value
This novel unstructured simulation approach involves dual meshes with active and inactive cells determined from an initial background pointset: local active dual patches are constructed “on‐the‐fly” for each “active point” to form a dynamic virtual mesh of active elements that evolves with the moving interface.