Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures

We design and investigate a sequential discontinuous Galerkin method to approximate two-phase immiscible incompressible flows in heterogeneous porous media with discontinuous capillary pressures. The nonlinear interface conditions are enforced weakly through an adequate design of the penalties on interelement jumps of the pressure and the saturation. An accurate reconstruction of the total velocity is considered in the Raviart-Thomas(-Nedelec) finite element spaces, together with diffusivity-dependent weighted averages to cope with degeneracies in the saturation equation and with media heterogeneities. The proposed method is assessed on one-dimensional test cases exhibiting rough solutions, degeneracies, and capillary barriers. Stable and accurate solutions are obtained without limiters. (C) 2010 Elsevier B.V. All rights reserved.

On the Optimality of Finite Constellations from Algebraic Lattices

A construction technique of finite point constellations in n-dimensional spaces from ideals in rings of algebraic integers is described. An algorithm is presented to find constellations with minimum average energy from a given lattice. For comparison, a numerical table of lattice constellations and group codes is computed for spaces of dimension two, three, and four. © 2001.

Stripe-tetragonal phase transition in the two-dimensional Ising model with dipole interactions: Partition function zeros approach

We have performed multicanonical simulations to study the critical behavior of the two-dimensional Ising model with dipole interactions. This study concerns the thermodynamic phase transitions in the range of the interaction delta where the phase characterized by striped configurations of width h = 1 is observed. Controversial results obtained from local update algorithms have been reported for this region, including the claimed existence of a second-order phase transition line that becomes first order above a tricritical point located somewhere between delta = 0.85 and 1. Our analysis relies on the complex partition function zeros obtained with high statistics from multicanonical simulations. Finite size scaling relations for the leading partition function zeros yield critical exponents. that are clearly consistent with a single second-order phase transition line, thus excluding such a tricritical point in that region of the phase diagram. This conclusion is further supported by analysis of the specific heat and susceptibility of the orientational order parameter.

Numerical assessment of stability of interface discontinuous finite element pressure spaces

The stability of two recently developed pressure spaces has been assessed numerically: The space proposed by Ausas et al. [R.F. Ausas, F.S. Sousa, G.C. Buscaglia, An improved finite element space for discontinuous pressures, Comput. Methods Appl. Mech. Engrg. 199 (2010) 1019-1031], which is capable of representing discontinuous pressures, and the space proposed by Coppola-Owen and Codina [A.H. Coppola-Owen, R. Codina, Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions, Int. J. Numer. Methods Fluids, 49 (2005) 1287-1304], which can represent discontinuities in pressure gradients. We assess the stability of these spaces by numerically computing the inf-sup constants of several meshes. The inf-sup constant results as the solution of a generalized eigenvalue problems. Both spaces are in this way confirmed to be stable in their original form. An application of the same numerical assessment tool to the stabilized equal-order P-1/P-1 formulation is then reported. An interesting finding is that the stabilization coefficient can be safely set to zero in an arbitrary band of elements without compromising the formulation's stability. An analogous result is also reported for the mini-element P-1(+)/P-1 when the velocity bubbles are removed in an arbitrary band of elements. (C) 2012 Elsevier B.V. All rights reserved.

Finite-element discretization of 3D energy-transport equations for semiconductors

In this thesis a mathematical model was derived that describes the charge and energy transport in semiconductor devices like transistors. Moreover, numerical simulations of these physical processes are performed. In order to accomplish this, methods of theoretical physics, functional analysis, numerical mathematics and computer programming are applied. After an introduction to the status quo of semiconductor device simulation methods and a brief review of historical facts up to now, the attention is shifted to the construction of a model, which serves as the basis of the subsequent derivations in the thesis. Thereby the starting point is an important equation of the theory of dilute gases. From this equation the model equations are derived and specified by means of a series expansion method. This is done in a multi-stage derivation process, which is mainly taken from a scientific paper and which does not constitute the focus of this thesis. In the following phase we specify the mathematical setting and make precise the model assumptions. Thereby we make use of methods of functional analysis. Since the equations we deal with are coupled, we are concerned with a nonstandard problem. In contrary, the theory of scalar elliptic equations is established meanwhile. Subsequently, we are preoccupied with the numerical discretization of the equations. A special finite-element method is used for the discretization. This special approach has to be done in order to make the numerical results appropriate for practical application. By a series of transformations from the discrete model we derive a system of algebraic equations that are eligible for numerical evaluation. Using self-made computer programs we solve the equations to get approximate solutions. These programs are based on new and specialized iteration procedures that are developed and thoroughly tested within the frame of this research work. Due to their importance and their novel status, they are explained and demonstrated in detail. We compare these new iterations with a standard method that is complemented by a feature to fit in the current context. A further innovation is the computation of solutions in three-dimensional domains, which are still rare. Special attention is paid to applicability of the 3D simulation tools. The programs are designed to have justifiable working complexity. The simulation results of some models of contemporary semiconductor devices are shown and detailed comments on the results are given. Eventually, we make a prospect on future development and enhancements of the models and of the algorithms that we used.

Three dimensional finite element simulation of polymer melting and flow in a single-screw extruder : optimization of screw channel geometry

Single-screw extrusion is one of the widely used processing methods in plastics industry, which was the third largest manufacturing industry in the United States in 2007 [5]. In order to optimize the single-screw extrusion process, tremendous efforts have been devoted for development of accurate models in the last fifty years, especially for polymer melting in screw extruders. This has led to a good qualitative understanding of the melting process; however, quantitative predictions of melting from various models often have a large error in comparison to the experimental data. Thus, even nowadays, process parameters and the geometry of the extruder channel for the single-screw extrusion are determined by trial and error. Since new polymers are developed frequently, finding the optimum parameters to extrude these polymers by trial and error is costly and time consuming. In order to reduce the time and experimental work required for optimizing the process parameters and the geometry of the extruder channel for a given polymer, the main goal of this research was to perform a coordinated experimental and numerical investigation of melting in screw extrusion. In this work, a full three-dimensional finite element simulation of the two-phase flow in the melting and metering zones of a single-screw extruder was performed by solving the conservation equations for mass, momentum, and energy. The only attempt for such a three-dimensional simulation of melting in screw extruder was more than twenty years back. However, that work had only a limited success because of the capability of computers and mathematical algorithms available at that time. The dramatic improvement of computational power and mathematical knowledge now make it possible to run full 3-D simulations of two-phase flow in single-screw extruders on a desktop PC. In order to verify the numerical predictions from the full 3-D simulations of two-phase flow in single-screw extruders, a detailed experimental study was performed. This experimental study included Maddock screw-freezing experiments, Screw Simulator experiments and material characterization experiments. Maddock screw-freezing experiments were performed in order to visualize the melting profile along the single-screw extruder channel with different screw geometry configurations. These melting profiles were compared with the simulation results. Screw Simulator experiments were performed to collect the shear stress and melting flux data for various polymers. Cone and plate viscometer experiments were performed to obtain the shear viscosity data which is needed in the simulations. An optimization code was developed to optimize two screw geometry parameters, namely, screw lead (pitch) and depth in the metering section of a single-screw extruder, such that the output rate of the extruder was maximized without exceeding the maximum temperature value specified at the exit of the extruder. This optimization code used a mesh partitioning technique in order to obtain the flow domain. The simulations in this flow domain was performed using the code developed to simulate the two-phase flow in single-screw extruders.

Microcanonical finite-size scaling in second-order phase transitions with diverging specific heat

A microcanonical finite-size ansatz in terms of quantities measurable in a finite lattice allows extending phenomenological renormalization the so-called quotients method to the microcanonical ensemble. The ansatz is tested numerically in two models where the canonical specific heat diverges at criticality, thus implying Fisher renormalization of the critical exponents: the three-dimensional ferromagnetic Ising model and the two-dimensional four-state Potts model (where large logarithmic corrections are known to occur in the canonical ensemble). A recently proposed microcanonical cluster method allows simulating systems as large as L = 1024 Potts or L= 128 (Ising). The quotients method provides accurate determinations of the anomalous dimension, η, and of the (Fisher-renormalized) thermal ν exponent. While in the Ising model the numerical agreement with our theoretical expectations is very good, in the Potts case, we need to carefully incorporate logarithmic corrections to the microcanonical ansatz in order to rationalize our data.

Henselian Discrete Valued Fields Admitting One-Dimensional Local Class Field Theory

2000 Mathematics Subject Classification: 11S31 12E15 12F10 12J20.

Eulerian-Lagrangian two phase debris flow model

The main objective of this work is to develop a quasi three-dimensional numerical model to simulate stony debris flows, considering a continuum fluid phase, composed by water and fine sediments, and a non-continuum phase including large particles, such as pebbles and boulders. Large particles are treated in a Lagrangian frame of reference using the Discrete Element Method, the fluid phase is based on the Eulerian approach, using the Finite Element Method to solve the depth-averaged Navier-Stokes equations in two horizontal dimensions. The particle’s equations of motion are in three dimensions. The model simulates particle-particle collisions and wall-particle collisions, taking into account that particles are immersed in a fluid. Bingham and Cross rheological models are used for the continuum phase. Both formulations provide very stable results, even in the range of very low shear rates. Bingham formulation is better able to simulate the stopping stage of the fluid when applied shear stresses are low. Results of numerical simulations have been compared with data from laboratory experiments on a flume-fan prototype. Results show that the model is capable of simulating the motion of big particles moving in the fluid flow, handling dense particulate flows and avoiding overlap among particles. An application to simulate debris flow events that occurred in Northern Venezuela in 1999 shows that the model could replicate the main boulder accumulation areas that were surveyed by the USGS. Uniqueness of this research is the integration of mud flow and stony debris movement in a single modeling tool that can be used for planning and management of debris flow prone areas.

Eulerian-Lagrangian Two Phase Debris Flow Model

The main objective of this work is to develop a quasi three-dimensional numerical model to simulate stony debris flows, considering a continuum fluid phase, composed by water and fine sediments, and a non-continuum phase including large particles, such as pebbles and boulders. Large particles are treated in a Lagrangian frame of reference using the Discrete Element Method, the fluid phase is based on the Eulerian approach, using the Finite Element Method to solve the depth-averaged Navier–Stokes equations in two horizontal dimensions. The particle’s equations of motion are in three dimensions. The model simulates particle-particle collisions and wall-particle collisions, taking into account that particles are immersed in a fluid. Bingham and Cross rheological models are used for the continuum phase. Both formulations provide very stable results, even in the range of very low shear rates. Bingham formulation is better able to simulate the stopping stage of the fluid when applied shear stresses are low. Results of numerical simulations have been compared with data from laboratory experiments on a flume-fan prototype. Results show that the model is capable of simulating the motion of big particles moving in the fluid flow, handling dense particulate flows and avoiding overlap among particles. An application to simulate debris flow events that occurred in Northern Venezuela in 1999 shows that the model could replicate the main boulder accumulation areas that were surveyed by the USGS. Uniqueness of this research is the integration of mud flow and stony debris movement in a single modeling tool that can be used for planning and management of debris flow prone areas.

DEVELOPMENT OF A 2-DIMENSIONAL FINITE VOLUME MODEL TO ASSESS HYDRODYNAMIC AND MICROBIAL CONTROLS ON DNAPL DISSOLUTION AND DETOXIFICATION

Tetrachloroethene (PCE) and trichloroethene (TCE) form dense non-aqueous phase liquids (DNAPLs), which are persistent groundwater contaminants. DNAPL dissolution can be "bioenhanced" via dissolved contaminant biodegradation at the DNAPL-water interface. This research hypothesized that: (1) competitive interactions between different dehalorespiring strains can significantly impact the bioenhancement effect, and extent of PCE dechlorination; and (2) hydrodynamics will affect the outcome of competition and the potential for bioenhancement and detoxification. A two-dimensional coupled flowtransport model was developed, with a DNAPL pool source and multiple microbial species. In the scenario presented, Dehalococcoides mccartyi 195 competes with Desulfuromonas michiganensis for the electron acceptors PCE and TCE. Simulations under biostimulation and low velocity (vx) conditions suggest that the bioenhancement with Dsm. michiganensis alone was modestly increased by Dhc. mccartyi 195. However, the presence of Dhc. mccartyi 195 enhanced the extent of PCE transformation. Hydrodynamic conditions impacted the results by changing the dominant population under low and high vx conditions.

Effect Of Platform Connection And Abutment Material On Stress Distribution In Single Anterior Implant-supported Restorations: A Nonlinear 3-dimensional Finite Element Analysis.

Although various abutment connections and materials have recently been introduced, insufficient data exist regarding the effect of stress distribution on their mechanical performance. The purpose of this study was to investigate the effect of different abutment materials and platform connections on stress distribution in single anterior implant-supported restorations with the finite element method. Nine experimental groups were modeled from the combination of 3 platform connections (external hexagon, internal hexagon, and Morse tapered) and 3 abutment materials (titanium, zirconia, and hybrid) as follows: external hexagon-titanium, external hexagon-zirconia, external hexagon-hybrid, internal hexagon-titanium, internal hexagon-zirconia, internal hexagon-hybrid, Morse tapered-titanium, Morse tapered-zirconia, and Morse tapered-hybrid. Finite element models consisted of a 4×13-mm implant, anatomic abutment, and lithium disilicate central incisor crown cemented over the abutment. The 49 N occlusal loading was applied in 6 steps to simulate the incisal guidance. Equivalent von Mises stress (σvM) was used for both the qualitative and quantitative evaluation of the implant and abutment in all the groups and the maximum (σmax) and minimum (σmin) principal stresses for the numerical comparison of the zirconia parts. The highest abutment σvM occurred in the Morse-tapered groups and the lowest in the external hexagon-hybrid, internal hexagon-titanium, and internal hexagon-hybrid groups. The σmax and σmin values were lower in the hybrid groups than in the zirconia groups. The stress distribution concentrated in the abutment-implant interface in all the groups, regardless of the platform connection or abutment material. The platform connection influenced the stress on abutments more than the abutment material. The stress values for implants were similar among different platform connections, but greater stress concentrations were observed in internal connections.

Assessment Of Robustness On Analysis Using Headspace Solid-phase Microextraction And Comprehensive Two-dimensional Gas Chromatography Through Experimental Designs.

Plackett-Burman experimental design was applied for the robustness assessment of GC×GC-qMS (Comprehensive Two-Dimensional Gas Chromatography with Fast Quadrupolar Mass Spectrometric Detection) in quantitative and qualitative analysis of volatiles compounds from chocolate samples isolated by headspace solid-phase microextraction (HS-SPME). The influence of small changes around the nominal level of six factors deemed as important on peak areas (carrier gas flow rate, modulation period, temperature of ionic source, MS photomultiplier power, injector temperature and interface temperature) and of four factors considered as potentially influential on spectral quality (minimum and maximum limits of the scanned mass ranges, ions source temperature and photomultiplier power). The analytes selected for the study were 2,3,5-trimethylpyrazine, 2-octanone, octanal, 2-pentyl-furan, 2,3,5,6-tetramethylpyrazine, and 2-nonanone e nonanal. The factors pointed out as important on the robustness of the system were photomultiplier power for quantitative analysis and lower limit of mass scanning range for qualitative analysis.

An Extension of the Invariance Principle for a Class of Differential Equations with Finite Delay

An extension of the uniform invariance principle for ordinary differential equations with finite delay is developed. The uniform invariance principle allows the derivative of the auxiliary scalar function V to be positive in some bounded sets of the state space while the classical invariance principle assumes that. V <= 0. As a consequence, the uniform invariance principle can deal with a larger class of problems. The main difficulty to prove an invariance principle for functional differential equations is the fact that flows are defined on an infinite dimensional space and, in such spaces, bounded solutions may not be precompact. This difficulty is overcome by imposing the vector field taking bounded sets into bounded sets.

Three-dimensional finite element thermal analysis of dental tissues irradiated with Er,Cr:YSGG laser

In the present study, a finite element model of a half-sectioned molar tooth was developed in order to understand the thermal behavior of dental hard tissues (both enamel and dentin) under laser irradiation. The model was validated by comparing it with an in vitro experiment where a sound molar tooth was irradiated by an Er,Cr:YSGG pulsed laser. The numerical tooth model was conceived to simulate the in vitro experiment, reproducing the dimensions and physical conditions of the typical molar sound tooth, considering laser energy absorption and calculating the heat transfer through the dental tissues in three dimensions. The numerical assay considered the same three laser energy densities at the same wavelength (2.79 mu m) used in the experiment. A thermographic camera was used to perform the in vitro experiment, in which an Er, Cr: YSGG laser (2.79 mu m) was used to irradiate tooth samples and the infrared images obtained were stored and analyzed. The temperature increments in both the finite element model and the in vitro experiment were compared. The distribution of temperature inside the tooth versus time plotted for two critical points showed a relatively good agreement between the results of the experiment and model. The three dimensional model allows one to understand how the heat propagates through the dentin and enamel and to relate the amount of energy applied, width of the laser pulses, and temperature inside the tooth. (C) 2008 American Institute of Physics. [DOI: 10.1063/1.2953526]