Plasma equilibrium evolution at the resistive diffusion timescale

The aim of this work is to propose an efficient numerical approximation of high frequency pulses propagating in nonlinear dispersive optical media. We consider the nonlinear Maxwell's equations with instantaneous nonlinearity. We first derive a physically and asymptotically equivalent model that is semi-linear. Then, for a large class of semi-linear systems, we describe the solution in terms of profiles. These profiles are solution of a singular equation involving one more variable describing the phase of the solution. We introduce a discretization of this equation using finite differences in space and time and an appropriate Fourier basis (with few elements) for the phase. The main point is that accurate solution of the nonlinear Maxwell equation can be computed with a mesh size of order of the wave length. This approximation is asymptotic-preserving in the sense that a multi-scale expansion can be performed on the discrete solution and the result of this expansion is a discretization of the continuous limit. In order to improve the computational delay, computations are performed in a window moving at the group velocity of the pulse. The second harmonic generation is used as an example to illustrate the proposed methodology. However, the numerical method proposed for this benchmark study can be applied to many other cases of nonlinear optics with high frequency pulses.

In this study; specially orthotropic laminated plates are modeled and equilibrium equations are obtained. These equilibrium equations are solved by finite difference method. A computer code is developed for this purpose. Depending on the finite difference mesh size the correct plate mid-point deflection value is determined and displacement of the mid-plane are obtained. Specially orthotropic laminated plates having Navier SS-1 boundary conditions at the four edges are analyzed for various loading conditions. Some examples which are taken from literature are solved and it is observed that our results are in good agreement with them

We consider the non stationary flow of a viscous incompressible fluid in a rigid homogeneous porous medium provided with mixed boundary conditions. Since the boundary pressure can present high variations, the permeability of the medium also depends on the pressure, so that the problem is nonlinear. We propose a discretization of this equation that combines Euler's implicit scheme in time and spectral methods in space. We prove optimal a priori error estimates and present some numerical experiments which confirm the interest of the discretization.

My work developed a kernel density estimator that well resolves typical structures of probability densities, which was demonstrated on a newly compiled marine data set of organic carbon-13 isotope ratios (δ13CPOC). All work was conducted within the emerging field of marine data science. I identified classical data science, a general understanding of ocean science, communication skills, and confidence as requirements for marine data scientists. In the beginning of my work, the existing δ13CPOC data consisted of about 500 data points in the global ocean. I expanded the existing data set to 4732 data points in a first version, and additionally to 6952 in a second. Both are published at PANGAEA along with meta information such as measurement location, time, and method, and interpolations. I have published a description of the temporal and geographic distribution of the first version at Earth System Science Data. I designed the development of the kernel density estimator algorithm on the existing concept of computing it as a solution of the diffusion equation. My algorithm uses finite differences in space and equidistant time steps with an implicit Euler method, and approximates the optimal smoothing parameter by two pilot steps. Compared to other well-known kernel density estimators, my algorithm produces reliable approximations of multimodal and boundary-close distributions on artificial and real marine data and is robust to noise. My implementation is published as a Python package on Zenodo, its description is submitted to Geoscientific Model Development. I was able to show that my kernel density estimator reliably evalu- ates ocean data and thus lays a foundation for calibrating Earth system models. At the same time, I was able to contribute to the definition and establishment of the field of Marine Data Science.

Computation of steady and oscillatory convection in saturated porous media

This paper presents an overview of two‐stage decoupling preconditioning techniques employed in the implicit parallel accurate reservoir simulator (IPARS) computational framework for modelling multi‐component multi‐phase flow in porous media. The underlying discretization method is implicit Euler in time and mixed finite elements or cell‐centred finite differences in space. IPARS permits rigorous, physically representative coupling of different physical and numerical flow models in different parts of the domain and accounts for structural discontinuities; the framework currently includes eight physical models. For simplicity of exposition, we have restricted our discussion to a two‐phase oil–water model and a three‐phase black oil model. Our decoupling approach involves extracting a pressure equation from the fully coupled linearized system thus allowing for a more accurate preconditioning of a discrete elliptic problem of lower dimension. Copyright © 2001 John Wiley & Sons, Ltd.

This paper describes parallel histogram modification techniques with embedded morphological preprocessing methods within the CNN-UM framework. The procedure is formulated in terms of nonlinear partial differential equations (PDE) and approximated through finite differences in space, resulting in coupled nonlinear ordinary differential equations (ODE). The I/O mapping of the system (containing both local and global couplings) can be calculated by a complex analogic (analog and logic) algorithm executed on a stored program nonlinear array processor, called the cellular nonlinear network universal machine (CNN-UM3). We describe and illustrate how the implementation of the algorithm results in an adaptive multi-thresholding scheme when histogram modification is combined with embedded morphological processing at a finite (low) number of gray-scale levels. This has obvious advantages if the further processing steps are segmentation and/or recognition. Experimental results processing real-life and echocardiography images are measured on different hardware/software platforms, including a 64 × 64 CNN-UM chip (ACE4k6,17).

Numerical evaluation of a fracture acidizing treatment in a three-dimensional fractured carbonate reservoir

The present thesis describes the development of a computational method for the numerical simulation of compressible flows in aerodynamics. The main goal is to study high-order discretization schemes that can potentially deliver a better computational efficiency. The spatial part of the conservation equations is discretized with the finite difference method. In particular, Summation By Parts (SBP) finite difference operators and treatment of boundary conditions via Simultaneous Approximation Terms (SAT) have been investigated. An SBP finite difference operator is essentially a (high-order) centered finite difference scheme with a specific closure at the boundary. The SAT are penalty-like terms that enforce the boundary conditions weakly and are used to augment the SBP schemes. The combination of SBP operators with SAT boundary terms constitutes schemes that, for the linearized equations governing a smooth flow, are provably stable (thus convergent) schemes. The discretization in time is carried out with high-order schemes as well. Both implicit and explicit time integration schemes have been investigated. The system of non-linear equations that arises when using the implicit time integration schemes is solved via (a damped) Newton's method. On the target grid, the initial guess required for Newton's method is obtained by performing grid sequencing combined a series of explicit relaxation iterations on each grid level. The flow equations and the equation of the turbulence model (if present) are solved in a fully-coupled manner. Numerical simulations for inviscid flow, laminar flow, as well as simulations based on DNS, LES and RANS have been performed. Results show that the described computational method is particularly well-suited for smooth external flows about relatively simple geometries, solved with the DNS and LES approach. For these types of flows the gain in efficiency obtained by using high-order schemes is clear (more than a factor of 5). For RANS, however, the comparison with a state-of-the-art second order finite volume code suggests that solving the turbulence and the flow equations fully coupled with implicit schemes may not be the most efficient iteration strategy. On the other hand, the numerical simulations of non-smooth/internal flows highlighted the need of further development of the SBP-SAT framework.

A novel meshless numerical simulation of oil-water two-phase flow with gravity and capillary forces in three-dimensional porous media

Propagation of a short pulse through human breast tissues is studied by numerically solving the diffuse equation. Different numerical methods, such as Monte Carlo and finite difference time domain, have been used to study the short pulse laser propagation through biological tissues. In this paper, we use boundary integral method to study the laser pulse–tissue interaction. The diffuse equation is used to examine the propagation of laser into biological tissues, and boundary integral method is used to alter this equation to the integral form and the result is solved by using boundary element method. To verify the precision of a boundary element method code, we compared the obtained results with those obtained by finite difference time domain method. In addition, the effects of different optical parameters of breast tissues, i.e., reduced scattering and absorption coefficients, on time evolution of a diffusely reflected pulse are studied.

A three-dimensional unsteady compressible viscous flow solver has been developed, and applied to a low speed centrifugal compressor configuration. Good agreement with experiments have been obtained for a compressor tested at NASA Lewis Research Center. Comparisons with measured surface pressures, and performance data are given. For this compressor, stall was found to occur in the diffuser region. It is demonstrated that this stall may be eliminated, and stable operation may be restored by the use of bleed valves located on the diffuser walls. INTRODUCTION In many army applications (tanks and rotorcraft) small gas turbine engines are often used. The mass flow rate in these engines is small (below 10 Ibs/sec). The compression is provided by one or two centrifugal compressor stages, sometimes in combination with an axial compressor stage. Centrifugal compressors are well suited for handling such low mass flow rates. They are rugged and compact. However, the geometric complexity of the blades and the twisting blade passages give rise to a host of aerodynamic problems such as unsteadiness, establishment of cross flows, flow separation at the trailing edge of the compressor blade, separation over diffuser guide vanes, and unsteady meandering motion of a 'tip vortex' that originates in the vicinity of the gap between the rotor and the casing. In many instances, this unsteadiness leads to rotating stall and surge. Considerable interest exists in the engine community in understanding and eventually controlling these phenomena, particularly compressor stall and surge. Computational fluid dynamics methods provide an efficient way of studying the complex flow phenomena within centrifugal compressors. Thanks to the availability of post-processing and scientific visualization software, the computed flow fields can be extensively analyzed. It is also convenient to test within a numerical model various control strategies such as bleeding, energizing of incoming flow using jets, excitation /wiggling of guide vanes, etc. Passive mechanisms such as slots, slats and flaps can also be easily studied within the scope of a numerical method. These simulations are thus a useful and economical first step to detailed flow control experiments. Finally, CFD simulations provide a wealth of numerical and flow physics information which are useful in developing reduced order models. For these reasons, a 3-D CFD code capable of analyzing centrifugal compressor configurations has been developed by the present investigators. Rather than solving the flow equations in a rotating coordinate system, the present calculations are carried out in an inertial system, with grid velocity terms used to simulate a rotor. Inflow and outflow boundary conditions appropriate for internal flows in general and centrifugal compressors in particular are used. This paper is organized as follows. The flow solution algorithm is first described. Next, results in the form of velocity fields, surface pressure data and performance map are presented for a NASA low-speed centrifugal compressor. The performance of this compressor for off-design low mass flow ratio conditions is subsequently studied. Methods for improving the compressor performance during such conditions are discussed. NUMERICAL FORMULATION The 3-D unsteady compressible Reynolds averaged Navier-Stokes equations are solved numerically using a time marching scheme. This involves solving the governing equations at each time step by marching in time from an initial flow condition with appropriate boundary conditions. The governing equations are cast in a curvilinear body-fitted coordinate system (£,T|,£) and solved using an Alternating Direction Implicit (ADI) scheme 1. This implicit scheme has been validated for a number of fixed wing and rotary wing configurations both in steady and unsteady flight conditions^'*. The partial derivatives in the governing equations are calculated using finite difference approximations. The Graduate Research Assistant, Student Member AIAA. * Regents' professor, Associate Fellow AIAA. Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. inviscid flux terms are calculated using a third order upwind scheme and the viscous stress terms are calculated using standard central difference representations. The details of the upwind formulation are given by Roe. The time derivative is approximated using a two point first order backward difference. The finite difference representation of the governing Navier-Stokes equations at time level 'n' is components at the inflow boundary were set to zero, assuming that there was no swirl at the inlet. It is possible within the framework of the present boundary conditions to prescribe swirl, non-uniform total pressure and total temperature at the inlet. At every time step, the speed of sound a and the normal component of velocity un were found at the inflow boundary from the relations: AT 2a 2a — u = n — u n = I 8 f R + S S + S , T (1) Interior where Aq* is the change in the flow property vector q during adjacent time levels and At is the time step. The operators 8£, 5r| and 8^ are the standard central difference operators. The viscous terms are evaluated explicitly and the inviscid flux vectors F, G and H are calculated implicitly at time level n+1. A three factor approximate factorization is used to factorize the implicit coefficient matrix operator into block tri-diagonal matrices. The factorized system of equations is solved using the Thomas algorithm. Turbulence effects are modeled using the SpalartAllmaras model. INITIAL AND BOUNDARY CONDITIONS The centrifugal compressor is shown in figure 1. The boundary grids are also shown for a single blade passage. The volume grid is made of three segments: inlet region, the rotor, and the diffuser. The following boundary conditions were applied: Periodicity: In the present study, the flow through only a single blade passage has been studied, assuming blade-toblade periodicity. At the periodic boundaries, the flow properties are computed by averaging the properties on either side of the periodic boundary. Solid walls: At the solid walls, the no slip boundary condition was used. The velocity on the walls of the inlet and the diffuser were set to zero, while the velocity for grid points on the compressor blades and the shaft were set equal to Q x r. The pressure, density and temperature values at the solid surfaces were extrapolated from the interior using relationships such as 9p/3n=dT/9n=3p/3n=0, where n represents the direction normal to the solid surface. Inflow Boundary: At this boundary, the stagnation temperature T0 and the total pressure p0 were assumed known. In the simulation of a compressor drawing the air from the atmosphere, these quantities may be set based on ambient conditions. In a multistage compressor analysis, these quantities will be known at the downstream boundary of the previous stage. The tangential (vand w-) (2)

Finite difference solutions of reaction diffusion equations with continuous time delays

In this paper, we study general solutions to the Stokes equations in hydrodynamics and equations of equilibrium in the elasticity theory, obtain new forms for the representation of general solutions, and construct a numerical method for calculating the Stokes flows of a viscous fluid. By the representation of the solutions to the Stokes equations we mean the expression of these solutions in terms of harmonic functions. The new formulas include the representation in terms of the first order operator and a class of 27 representations. The symmetries of fluid flows with low rates and of elastic media equilibria have been found. The suggested numerical method involves the above-mentioned representations and a boundary integral equation of a particular type. It is essentially different from the known methods of boundary integral equations based on multipole solutions to the Stokes equations [1‐3]. The manner in which the solutions to the Stokes equations and to the equations of the elasticity theory are related to the solutions to the Laplace equations is considered in detail in [4] for the case of plane problems.

Some local areas are potentially to be divided into very small size grids so as to improve the simulation precision during the inhomogeneous microwave system. The implicit time scheme is very appropriate for small grids due to its unconditional stability. In this work, based on the previous research on the implicit hybridizable discontinuous Galerkin time-domain (imHDGTD) method, we mainly study the effect of different grids on this method for Maxwell's equation. According to the comparative consequences with the classical discontinuous Galerkin time-domain (DGTD) method, numerical simulations validate the better effectiveness of imHDGTD when using some refined grids. As a consequence, the proposed imHDGTD way has a superior property towards intricate grid models.