Showing papers on "Finite difference published in 1999"

The nature of mathematical modeling

Abstract: Preface 1. Introduction Part I. Analytical Models: 2. Ordinary differential and difference equations 3. Partial differential equations 4. Variational principles 5. Random systems Part II. Numerical Models: 6. Finite differences: ordinary difference equations 7. Finite differences: partial differential equations 8. Finite elements 9. Cellular automata and lattice gases Part III. Observational Models: 10. Function fitting 11. Transforms 12. Architectures 13. Optimization and search 14. Clustering and density estimation 15. Filtering and state estimation 16. Linear and nonlinear time series Appendix 1. Graphical and mathematical software Appendix 2. Network programming Appendix 3. Benchmarking Appendix 4. Problem solutions Bibliography.

The nature of mathematical modeling

Abstract: Preface 1. Introduction Part I. Analytical Models: 2. Ordinary differential and difference equations 3. Partial differential equations 4. Variational principles 5. Random systems Part II. Numerical Models: 6. Finite differences: ordinary difference equations 7. Finite differences: partial differential equations 8. Finite elements 9. Cellular automata and lattice gases Part III. Observational Models: 10. Function fitting 11. Transforms 12. Architectures 13. Optimization and search 14. Clustering and density estimation 15. Filtering and state estimation 16. Linear and nonlinear time series Appendix 1. Graphical and mathematical software Appendix 2. Network programming Appendix 3. Benchmarking Appendix 4. Problem solutions Bibliography.

On particle weighted methods and smooth particle hydrodynamics

Abstract: This paper deal with designing of weighted particle approximation of conservation laws. New ideas concerning the use of variable smoothing length, renormalization and the use of Godunov type finite difference fluxes in particle methods are introduced and discussed in connection with standard implementation of the SPH method. A detailed analysis of boundary conditions approximation is also provided.

Numerical analysis of hygro-thermal behaviour and damage of concrete at high temperature

Abstract: A computational analysis of hygro-thermal and mechanical behaviour of concrete structures at high temperature is presented. The evaluation of thermal, hygral and mechanical performance of this material, including damage effects, needs the knowledge of the heat and mass transfer processes. These are simulated within the framework of a coupled model where non-linearities due to high temperatures are accounted for. The constitutive equations are discussed in some detail. The discretization of the governing equations is carried out by Finite Elements in space and Finite Differences in time. Copyright © 1999 John Wiley & Sons, Ltd.

Fluid Flow Phenomena: A Numerical Toolkit

Abstract: This book deals with the simulation of the incompressible Navier-Stokes equations for laminar and turbulent flows. The book is limited to explaining and employing the finite difference method. It furnishes a large number of source codes which permit to play with the Navier-Stokes equations and to understand the complex physics related to fluid mechanics. Numerical simulations are useful tools to understand the complexity of the flows, which often is difficult to derive from laboratory experiments. This book, then, can be very useful to scholars doing laboratory experiments, since they often do not have extra time to study the large variety of numerical methods; furthermore they cannot spend more time in transferring one of the methods into a computer language. By means of numerical simulations, for example, insights into the vorticity field can be obtained which are difficult to obtain by measurements. This book can be used by graduate as well as undergraduate students while reading books on theoretical fluid mechanics; it teaches how to simulate the dynamics of flow fields on personal computers. This will provide a better way of understanding the theory. Two chapters on Large Eddy Simulations have been included, since this is a methodology that in the near future will allow more universal turbulence models for practical applications. The direct simulation of the Navier-Stokes equations (DNS) is simple by finite-differences, that are satisfactory to reproduce the dynamics of turbulent flows. A large part of the book is devoted to the study of homogeneous and wall turbulent flows. In the second chapter the elementary concept of finite difference is given to solve parabolic and elliptical partial differential equations. In successive chapters the 1D, 2D, and 3D Navier-Stokes equations are solved in Cartesian and cylindrical coordinates. Finally, Large Eddy Simulations are performed to check the importance of the subgrid scale models. Results for turbulent and laminar flows are discussed, with particular emphasis on vortex dynamics. This volume will be of interest to graduate students and researchers wanting to compare experiments and numerical simulations, and to workers in the mechanical and aeronautic industries.

A thermal model for high speed motorized spindles

Abstract: Lack of a more complete understanding of the system characteristics, particularly thermal effects, severely limits the reliability of high speed spindles to support manufacturing. High speed spindles are notorious for their sudden catastrophic failures without alarming signs at high speeds due to thermal problems. In this paper, a finite difference thermal model is developed to characterize the power distribution of a high speed motorized spindle, in particular the characterization of heat transfer and heat sinks. Without loss of generality, this model is built upon and verified by a custom-built high performance motorized milling spindle of 32 KW and maximum speed of 25 000 rpm (1.5 million DN).

Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II

Abstract: We approximate quasi-linear parabolic SPDEs substituting the derivatives with finite differences. We investigate the resulting implicit and explicit schemes. For the implicit scheme we estimate the rate of Lp convergence of the approximations and we also prove their almost sure convergence when the nonlinear terms are Lipschitz continuous. When the nonlinear terms are not Lipschitz continuous we obtain convergence in probability provided pathwise uniqueness for the equation holds. For the explicit scheme we get these results under an additional condition on the mesh sizes in time and space.

Boussinesq modeling of a rip current system

Abstract: In this study, we use a time domain numerical model based on the fully nonlinear extended Boussinesq equations [Wei et al., 1995] to investigate surface wave transformation and breaking-induced nearshore circulation. The energy dissipation due to wave breaking is modeled by introducing an eddy viscosity term into the momentum equations, with the viscosity strongly localized on the front face of the breaking waves. Wave run-up on the beach is simulated using a moving shoreline technique. We employ quasi fourth-order finite difference schemes to solve the governing equations. Satisfactory agreement is found between the numerical results and the laboratory measurements of Haller et al. [1997], including wave height, mean water level, and longshore and cross-shore velocity components. The model results reveal the temporal and spatial variability of the wave-induced nearshore circulation, and the instability of the rip current in agreement with the physical experiment. Insights into the vorticity associated with the rip current and wave diffraction by underlying vortices are obtained.

Positivity-Preserving Numerical Schemes for Lubrication-Type Equations

Abstract: Lubrication equations are fourth order degenerate diffusion equations of the form $h_t + abla \cdot (f(h) abla \Delta h) = 0$, describing thin films or liquid layers driven by surface tension. Recent studies of singularities in which $h\to 0$ at a point, describing rupture of the fluid layer, show that such equations exhibit complex dynamics which can be difficult to simulate accurately. In particular, one must ensure that the numerical approximation of the interface does not show a false premature rupture. Generic finite difference schemes have the potential to manifest such instabilities especially when underresolved. We present new numerical methods, in one and two space dimensions, that preserve positivity of the solution, regardless of the spatial resolution, whenever the PDE has such a property. We also show that the schemes can preserve positivity even when the PDE itself is only known to be nonnegativity preserving. We prove that positivity-preserving finite difference schemes have unique positive solutions at all times. We prove stability and convergence of both positivity-preserving and generic methods, in one and two space dimensions, to positive solutions of the PDE, showing that the generic methods also preserve positivity and have global solutions for sufficiently fine meshes. We generalize the positivity-preserving property to a finite element framework and show, via concrete examples, how this leads to the design of other positivity-preserving schemes.

Compact finite difference schemes on non-uniform meshes. Application to direct numerical simulations of compressible flows

Abstract: In this paper, the development of a fourth- (respectively third-) order compact scheme for the approximation of first (respectively second) derivatives on non-uniform meshes is studied. A full inclusion of metrics in the coefficients of the compact scheme is proposed, instead of methods using Jacobian transformation. In the second part, an analysis of the numerical scheme is presented. A numerical analysis of truncation errors, a Fourier analysis completed by stability calculations in terms of both semi- and fully discrete eigenvalue problems are presented. In those eigenvalue problems, the pure convection equation for the first derivative, and the pure diffusion equation for the second derivative are considered. The last part of this paper is dedicated to an application of the numerical method to the simulation of a compressible flow requiring variable mesh size: the direct numerical simulation of compressible turbulent channel flow. Present results are compared with both experimental and other numerical (DNS) data in the literature. The effects of compressibility and acoustic waves on the turbulent flow structure are discussed.

Selective laser sintering of an amorphous polymer : simulations and experiments

Abstract: Thermal and powder densification modelling of the selective laser sintering of amorphous polycarbonate is reported. Three strategies have been investigated: analytical, adaptive mesh finite difference and fixed mesh finite element. A comparison between the three and experimental results is used to evaluate their ability reliably to predict the behaviour of the physical process. The finite difference and finite element approaches are the only ones that automatically deal with the non-linearities of the physical process that arise from the variation in the thermal properties of the polymer with density during sintering, but the analytical model has some value, provided appropriate mean values are used for thermal properties. Analysis shows that the densification and linear accuracies due to sintering are most sensitive to changes in the activation energy and heat capacity of the polymer, with a second level of sensitivities that includes powder bed density and powder layer thickness. Simulations of ...

Test of two methods for faulting in finite-difference calculations

Abstract: Tests of two fault boundary conditions show that each converges with second order accuracy as the finite-difference grid is refined. The first method uses split nodes so that there are disjoint grids that interact via surface traction. The 3D version described here is a generalization of a method I have used extensively in 2D; it is as accurate as the 2D version. The second method represents fault slip as inelastic strain in a fault zone. Offset of stress from its elastic value is seismic moment density. Implementation of this method is quite simple in a finite-difference scheme using velocity and stress as dependent variables.

Fully non‐linear free‐surface simulations by a 3D viscous numerical wave tank

Abstract: A finite difference scheme using a modified marker-and-cell (MAC) method is applied to investigate the characteristics of non-linear wave motions and their interactions with a stationary three-dimensional body inside a numerical wave tank (NWT). The Navier-Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co-ordinate system. The viscous stresses and surface tension are neglected in the dynamic free-surface condition, and the fully non-linear kinematic free-surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS-MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential-based fully non-linear NWT, and the second-order diffraction computation

Asymptotic methods and perturbation theory

Abstract: I Preface. 1 Ordinary Differential Equations. 2 Difference Equations. 3 Approximate Solution of Linear Differential Equations. 4 Approximate Solution of Nonlinear Equations. 5 Approximate Solution of Difference Equations. 6 Asymptotic Expansion of Integrals. 7 Perturbation Series. 8 Summation of Series. 9 Boundary Layer Theory. 10 WKB Theory. 11 Multiple Scales Analysis. Appendix, References, Index

Numerical methods for stochastic parabolic PDEs

Abstract: We develop a convergence theory for finite difference approximations of reaction diffusion equations forced by an additive space–time white noise. Special care is taken to develop the estimates in terms of non-smooth initial data and over a long time interval, motivated by an abstract approximation theory of ergodic properties developed by Shardlow& Stuart.

The Orthogonal Decomposition Theorems for Mimetic Finite Difference Methods

Abstract: Accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus provide reliable finite difference methods for approximating the solutions to a wide class of partial differential equations. These methods mimic many fundamental properties of the underlying physical problem including conservation laws, symmetries in the solution, and the nondivergence of particular vector fields (i.e., they are divergence free) and should satisfy a discrete version of the orthogonal decomposition theorem. This theorem plays a fundamental role in the theory of generalized solutions and in the numerical solution of physical models, including the Navier--Stokes equations and in electrodynamics. We are deriving mimetic finite difference approximations of the divergence, gradient, and curl that satisfy discrete analogs of the integral identities satisfied by the differential operators. We first define the natural discrete divergence, gradient, and curl operators based on coordinate invariant definitions, such as Gauss's theorem, for the divergence. Next we use the formal adjoints of these natural operators to derive compatible divergence, gradient, and curl operators with complementary domains and ranges of values. In this paper we prove that these operators satisfy discrete analogs of the orthogonal decomposition theorem and demonstrate how a discrete vector can be decomposed into two orthogonal vectors in a unique way, satisfying a discrete analog of the formula $\vec{A} = \ggrad \, \varphi + \curl \, \vec{B}$. We also present a numerical example to illustrate the numerical procedure and calculate the convergence rate of the method for a spiral vector field.

New spectral Lanczos decomposition method for induction modeling in arbitrary 3-D geometry

Abstract: Traditional resistivity tools are designed to function in vertical wells. In horizontal well environments, the interpretation of resistivity logs becomes much more difficult because of the nature of 3-D effects such as highly deviated bed boundaries and invasion. The ability to model these 3-D effects numerically can greatly facilitate the understanding of tool response in different formation geometries. Three-dimensional modeling of induction tools requires solving Maxwell's equations in a discrete setting, either finite element or finite difference. The solutions of resulting discretized equations are computationally expensive, typically on the order of 30 to 60 minutes per log point on a workstation. This is unacceptable if the 3-D modeling code is to be used in interpreting induction logs. In this paper we propose a new approach for solutions to Maxwell's equations. The new method is based on the spectral Lanczos decomposition method (SLDM) with Krylov subspaces generated from the inverse powers of the Maxwell operator. This new approach significantly speeds up the convergence of standard SLDM for the solution of Maxwell's equations while retaining the advantages of standard SLDM such as the ability of solving for multiple frequencies and eliminate completely spurious modes. The cost of evaluating powers of the matrix inverse of the stiffness operator is effectively equivalent to the cost of solving a scalar Poisson's equation. This is achieved by a decomposition of the stiffness operator into the curl-free and divergence-free projections. The solution of the projections can be computed by discrete Fourier transforms (DFT) and preconditioned conjugate gradient iterations. The convergence rate of the new method improves as frequency decreases, which makes it more attractive for low-frequency applications. We apply the new solution technique to model induction logging in geophysical prospecting applications, giving rise to two orders of magnitude convergence improvement over the standard Krylov subspace approach and more than an order of magnitude speed-up in terms of overall execution. This makes it feasible to routinely use 3-D modeling for model-based interpretation, a break-through in induction logging and interpretation.

Symmetry without Symmetry: Numerical Simulation of Axisymmetric Systems using Cartesian Grids

Abstract: We present a new technique for the numerical simulation of axisymmetric systems. This technique avoids the coordinate singularities which often arise when cylindrical or polar-spherical coordinate finite difference grids are used, particularly in simulating tensor partial differential equations like those of 3+1 numerical relativity. For a system axisymmetric about the z axis, the basic idea is to use a 3-dimensional Cartesian (x,y,z) coordinate grid which covers (say) the y=0 plane, but is only one finite-difference-molecule--width thick in the y direction. The field variables in the central y=0 grid plane can be updated using normal (x,y,z)--coordinate finite differencing, while those in the y eq 0 grid planes can be computed from those in the central plane by using the axisymmetry assumption and interpolation. We demonstrate the effectiveness of the approach on a set of fully nonlinear test computations in 3+1 numerical general relativity, involving both black holes and collapsing gravitational waves.

Post-modern Electromagnetics: Using Intelligent MaXwell Solvers

Abstract: Geometry, differential and integral forms function analysis optimization generating matrix equations Maxwell theory computational electromagnetics generalizing finite differences MMP - a general boundary method implementation applications.

Simulation-Based Optimization with Stochastic Approximation Using Common Random Numbers

Abstract: The method of Common Random Numbers is a technique used to reduce the variance of difference estimates in simulation optimization problems. These differences are commonly used to estimate gradients of objective functions as part of the process of determining optimal values for parameters of a simulated system. Asymptotic results exist which show that using the Common Random Numbers method in the iterative Finite Difference Stochastic Approximation optimization algorithm (FDSA) can increase the optimal rate of convergence of the algorithm from the typical rate of k-1/3 to the faster k-1/2, where k is the algorithm's iteration number. Simultaneous Perturbation Stochastic Approximation (SPSA) is a newer and often much more efficient optimization algorithm, and we will show that this algorithm, too, converges faster when the Common Random Numbers method is used. We will also provide multivariate asymptotic covariance matrices for both the SPSA and FDSA errors.

Three-dimensional finite difference viscoelastic wave modelling including surface topography

Abstract: Summary I have undertaken 3-D finite difference (FD) modelling of seismic scattering fromfree-surface topography. Exact free-surface boundary conditions for arbitrary 3-D topographies have been derived for the particle velocities. The boundary conditions are combined with a velocity–stress formulation of the full viscoelastic wave equations. A curved grid represents the physical medium and its upper boundary represents the free-surface topography. The wave equations are numerically discretized by an eighth-order FD method on a staggered grid in space, and a leap-frog technique and the Crank–Nicholson method in time. I simulate scattering from teleseismic P waves by using plane incident wave fronts and real topography from a 60 × 60 km area that includes the NORESS array of seismic receiver stations in southeastern Norway. Synthetic snapshots and seismograms of the wavefield show clear conversion from P to Rg (short-period fundamental-mode Rayleigh) waves in areas of rough topography, which is consistent with numerous observations. By parallelization on fast supercomputers, it is possible to model higher frequencies and/or larger areas than before.