Showing papers in "Communications in Computational Physics in 2008"

Spectral-element and adjoint methods in seismology

Abstract: We provide an introduction to the use of the spectral-element method (SEM) in seismology. Following a brief review of the basic equations that govern seismic wave propagation, we discuss in some detail how these equations may be solved numerically based upon the SEM to address the forward problem in seismology. Examples of synthetic seismograms calculated based upon the SEM are compared to data recorded by the Global Seismographic Network. Finally, we discuss the challenge of using the remaining differences between the data and the synthetic seismograms to constrain better Earth models and source descriptions. This leads naturally to adjoint methods, which provide a practical approach to this formidable computational challenge and enables seismologists to tackle the inverse problem.

Two-relaxation-time Lattice Boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions

Abstract: We develop a two-relaxation-time (TRT) Lattice Boltzmann model for hydrodynamic equations with variable source terms based on equivalent equilibrium functions. A special parametrization of the free relaxation parameter is derived. It controls, in addition to the non-dimensional hydrodynamic numbers, any TRT macroscopic steady solution and governs the spatial discretization of transient flows. In this framework, the multi-reflection approach [16, 18] is generalized and extended for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) boundary conditions. We propose second and third-order accurate boundary schemes and adapt them for corners. The boundary schemes are analyzed for exactness of the parametrization, uniqueness of their steady solutions, support of staggered invariants and for the effective accuracy in case of time dependent boundary conditions and transient flow. When the boundary scheme obeys the parametrization properly, the derived permeability values become independent of the selected viscosity for any porous structure and can be computed efficiently. The linear interpolations [5, 46] are improved with respect to this property. PACS: 47.10.ad, 47.56+r, 02.60-x

A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations

Abstract: In this review article we discuss different techniques to solve numerically the time-dependent Schrodinger equation on unbounded domains. We present in de- tail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimen- sions and the cubic nonlinear case.

Lumped parameter outflow models for 1-D blood flow simulations: Effect on pulse waves and parameter estimation

Abstract: Several lumped parameter, or zero-dimensional (0-D), models of the microcirculation are coupled in the time domain to the nonlinear, one-dimensional (1-D) equations of blood flow in large arteries. A linear analysis of the coupled system, together with in vivo observations, shows that: (i) an inflow resistance that matches the characteristic impedance of the terminal arteries is required to avoid non-physiological wave reflections; (ii) periodic mean pressures and flow distributions in large arteries depend on arterial and peripheral resistances, but not on the compliances and inertias of the system, which only affect instantaneous pressure and flow waveforms; (iii) peripheral inertias have a minor effect on pulse waveforms under normal conditions; and (iv) the time constant of the diastolic pressure decay is the same in any 1-D model artery, if viscous dissipation can be neglected in these arteries, and it depends on all the peripheral compliances and resistances of the system. Following this analysis, we propose an algorithm to accurately estimate peripheral resistances and compliances from in vivo data. This algorithm is verified against numerical data simulated using a 1-D model network of the 55 largest human arteries, in which the parameters of the peripheral windkessel outflow models are known a priori. Pressure and flow waveforms in the aorta and the first generation of bifurcations are reproduced with relative root-mean-square errors smaller than 3%. AMS subject classifications: 92C35, 35L45

Study of simple hydrodynamic solutions with the two-relaxation-times lattice Boltzmann scheme

Abstract: For simple hydrodynamic solutions, where the pressure and the velocity are polynomial functions of the coordinates, exact microscopic solutions are constructed for the two-relaxation-time (TRT) Lattice Boltzmann model with variable forcing and supported by exact boundary schemes. We show how simple numerical and analytical solutions can be interrelated for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) multi-reflection (MR) type schemes. Special care is taken to adapt them for corners, to examine the uniqueness of the obtained steady solutions and staggered invariants, to validate their exact parametrization by the non-dimensional hydrodynamic and a “kinetic” (collision) number. We also present an inlet/outlet “constant mass flux” condition. We show, both analytically and numerically, that the kinetic boundary schemes may result in the appearance of Knudsen layers which are beyond the methodology of the Chapman-Enskog analysis. Time dependent Dirichlet boundary conditions are investigated for pulsatile flow driven by an oscillating pressure drop or forcing. Analytical approximations are constructed in order to extend the pulsatile solution for compressible regimes. PACS: 47.10.ad, 47.56+r, 02.60-x

Galerkin method for wave equations with uncertain coefficients

Abstract: Polynomial chaos methods (and generalized polynomial chaos methods) have been extensively applied to analyze PDEs that contain uncertainties. However this approach is rarely applied to hyperbolic systems. In this paper we analyze the properties of the resulting deterministic system of equations obtained by stochastic Galerkin projection. We consider a simple model of a scalar wave equation with random wave speed. We show that when uncertainty causes the change of characteristic directions, the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues. A consistent method of imposing the boundary conditions is proposed and its convergence is established. Numerical examples are presented to support the analysis. AMS subject classifications: 65C20, 65C30

Language change and social networks

Abstract: Social networks play an important role in determining the dynamics and outcome of language change. Early empirical studies only examine small-scale local social networks, and focus on the relationship between the individual speakers’ linguistic behaviors and their characteristics in the network. In contrast, computer models can provide an efficient tool to consider large-scale networks with different structures and discuss the long-term effect of individuals’ learning and interaction on language change. This paper presents an agent-based computer model which simulates language change as a process of innovation diffusion, to address the threshold problem of language change. In the model, the population is implemented as a network of agents with age differences and different learning abilities, and the population is changing, with new agents born periodically to replace old ones. Four typical types of networks and their effect on the diffusion dynamics are examined. When the functional bias is sufficiently high, innovations always diffuse to the whole population in a linear manner in regular and small-world networks, but diffuse quickly in a sharp S-curve in random and scale-free networks. The success rate of diffusion is higher in regular and small-world networks than in random and scale-free networks. In addition, the model shows that as long as the population contains a small number of statistical learners who can learn and use both linguistic variants statistically according to the impact of these variants in the input, there is a very high probability for linguistic innovations with only small functional advantage to overcome the threshold of diffusion. AMS subject classifications: 91.D30 PACS: 89.65.-s, 89.75.Hc

Effectiveness of implicit methods for stiff stochastic differential equations

Abstract: In this paper we study the behavior of a family of implicit numerical methods applied to stochastic differential equations with multiple time scales. We show by a combination of analytical arguments and numerical examples that implicit methods in general fail to capture the effective dynamics at the slow time scale. This is due to the fact that such implicit methods cannot correctly capture non-Dirac invariant distributions when the time step size is much larger than the relaxation time of the system. AMS subject classifications: 65L20, 65C30, 37M25

Numerical study of the unsteady aerodynamics of freely falling plates

Abstract: The aerodynamics of freely falling objects is one of the most interesting flow mechanics problems. In a recent study, Andersen, Pesavento, and Wang [J. Fluid Mech., vol. 541, pp. 65-90 (2005)] presented the quantitative comparison between the experimental measurement and numerical computation. The rich dynamical behavior, such as fluttering and tumbling motion, was analyzed. However, obvious discrepancies between the experimental measurement and numerical simulations still exist. In the current study, a similar numerical computation will be conducted using a newly developed unified coordinate gas-kinetic method [J. Comput. Phys, vol. 222, pp. 155175 (2007)]. In order to clarify some early conclusions, both elliptic and rectangular falling plates will be studied. Under the experimental condition, the numerical solution shows that the averaged translational velocity for both rectangular and elliptical plates are almost identical during the tumbling motion. However, the plate rotation depends strongly on the shape of the plates. In this study, the details of fluid forces and torques on the plates and plates movement trajectories will be presented and compared with the experimental measurements. PACS: 47.11.-j, 47.85.Gj

Numerical Study of Heat and Moisture Transfer in Textile Materials by a Finite Volume Method

Abstract: This paper focuses on the numerical study of heat and moisture transfer in clothing assemblies, based on a multi-component and multiphase flow model which includes heat/moisture convection and conduction/diffusion as well as phase change. A splitting semi-implicit finite volume method is proposed for solving a set of nonlinear convection-diffusion-reaction equations, in which the calculation of liquid water content absorbed by fiber is decoupled from the rest of the computation. The method maintains the conservation of air, vapor and heat flux (energy). Four types of clothing assemblies are investigated and comparison with experimental measurements are also presented. AMS subject classifications: 78M20, 65N22, 65N06

A general moving mesh framework in 3D and its application for simulating the mixture of multi-phase flows

Abstract: In this paper, we present an adaptive moving mesh algorithm for meshes of unstructured polyhedra in three space dimensions. The algorithm automatically adjusts the size of the elements with time and position in the physical domain to re- solve the relevant scales in multiscale physical systems while minimizing computa- tional costs. The algorithm is a generalization of the moving mesh methods based on harmonic mappings developed by Li et al. (J. Comput. Phys., 170 (2001), pp. 562- 588, and 177 (2002), pp. 365-393). To make 3D moving mesh simulations possible, the key is to develop an efficient mesh redistribution procedure so that this part will cost as little as possible comparing with the solution evolution part. Since the mesh redistribution procedure normally requires to solve large size matrix equations, we will describe a procedure to decouple the matrix equation to a much simpler block- tridiagonal type which can be efficiently solved by a particularly designed multi-grid method. To demonstrate the performance of the proposed 3D moving mesh strategy, the algorithm is implemented in finite element simulations of fluid-fluid interface in- teractions in multiphase flows. To demonstrate the main ideas, we consider the for- mation of drops by using an energetic variational phase field model which describes the motion of mixtures of two incompressible fluids. Numerical results on two- and three-dimensional simulations will be presented. AMS subject classifications: 65M20, 65M50, 65M60

Plasma Edge Kinetic-MHD Modeling in Tokamaks Using Kepler Workflow for Code Coupling, Data Management and Visualization

Abstract: A new predictive computer simulation tool targeting the development of the H-mode pedestal at the plasma edge in tokamaks and the triggering and dynamics of edge localized modes (ELMs) is presented in this report. This tool brings together, in a coordinated and effective manner, several first-principles physics simulation codes, stability analysis packages, and data processing and visualization tools. A Kepler workflow is used in order to carry out an edge plasma simulation that loosely couples the kinetic code, XGC0, with an ideal MHD linear stability analysis code, ELITE, and an extended MHD initial value code such as M3D or NIMROD. XGC0 includes the neoclassical ion-electron-neutral dynamics needed to simulate pedestal growth near the separatrix. The Kepler workflow processes the XGC0 simulation results into simple images that can be selected and displayed via the Dashboard, a monitoring tool implemented in AJAX allowing the scientist to track computational resources, examine running and archived jobs, and view key physics data, all within a standard Web browser. The XGC0 simulation is monitored for the conditions needed to trigger an ELM crash by periodically assessing the edge plasma pressure and current density profiles using the ELITE code. If an ELM crash is triggered, the Kepler workflow launches the M3D code on a moderate-size Opteron cluster to simulate the nonlinear ELM crash and to compute the relaxation of plasma profiles after the crash. This process is monitored through periodic outputs of plasma fluid quantities that are automatically visualized with AVS/Express and may be displayed on the Dashboard. Finally, the Kepler workflow archives all data outputs and processed images using HPSS, as well as provenance information about the software and hardware used to create the simulation. The complete process of preparing, executing and monitoring a coupled-code simulation of the edge pressure pedestal buildup and the ELM cycle using the Kepler scientific workflow system is described in this paper.

RLC equivalent circuit synthesis method for structure-preserved reduced-order model of interconnect in VLSI

Abstract: This paper aims to explore RLC equivalent circuit synthesis method for reduced-order models of interconnect circuits obtained by Krylov subspace based model order reduction (MOR) methods. To guarantee pure RLC equivalent circuits can be synthesized, both the structures of input and output incidence matrices and the block structure of the circuit matrices should be preserved in the reduced-order models. Block structure preserving MOR methods have been well established. In this paper, we propose an embeddable Input-Output structure Preserving Order Reduction (IOPOR) technique to further preserve the structures of input and output incidence matrices. By combining block structure preserving MOR methods and IOPOR technique, we develop an RLC equivalent circuit synthesis method RLCSYN (RLC SYNthesis). Inline diagonalization and regularization techniques are specifically proposed to enhance the robustness of inductance synthesis. The pure RLC model, high modeling accuracy, passivity guaranteed property and SPICE simulation robustness make RLCSYN more applicable in interconnect analysis, either for digital IC design or mixed signal IC simulation. AMS subject classifications: 94C05, 93A15, 68U07, 68U20, 41A21

A Generalized Peierls-Nabarro Model for Curved Dislocations Using Discrete Fourier Transform

Abstract: In this paper, we present a generalized Peierls-Nabarro model for curved dislocations using the discrete Fourier transform. In our model, the total energy is expressed in terms of the disregistry at the discrete lattice sites on the slip plane, and the elastic energy is obtained efficiently within the continuum framework using the discrete Fourier transform. Our model directly incorporates into the total energy both the Peierls energy for the motion of straight dislocations and the second Peierls energy for kink migration. The discreteness in both the elastic energy and the misfit energy, the full long-range elastic interaction for curved dislocations, and the changes of core and kink profiles with respect to the location of the dislocation or the kink are all included in our model. The model is presented for crystals with simple cubic lattice. Simulation results on the dislocation structure, Peierls energies and Peierls stresses of both straight and kinked dislocations are reported. These results qualitatively agree with those from experiments and atomistic simulations. AMS subject classifications: 35Q72, 65D05, 74C99, 74G65, 74S25

Spectral Vanishing Viscosity Stabilized LES of the Ahmed Body Turbulent Wake

Abstract: The paper addressesthe Large-EddySimulation (LES)of the turbulent wake of the Ahmed car model. To this end we use a Fourier-Chebyshev multi-domain solver and the LES capability is implemented through the use of the Spectral Vanishing Vis- cosity (SVV) method, completed with a near-wall correction. A "pseudo-penalization" technique is used to model the bluff body. Comparisons of the present SVV-LES results with the experiments and also with a more classical Finite Volume LES are provided.

Point sources identification problems for heat equations

Abstract: We considered the point source identification problems for heat equations from noisy observation data taken at the minimum number of spatially fixed measurement points. We aim to identify the unknown number of sources and their locations along with their strengths. In our previous work, we proved that minimum measurement points needed under the noise-free setting. In this paper, we extend the proof to cover the noisy cases over a border class of source functions. We show that if the regularization parameter is chosen properly, the problem can be transformed into a poles identification problem. A reconstruction scheme is proposed on the basis of the developed theoretical results. Numerical demonstrations in 2D and 3D conclude the paper. AMS subject classifications: 35R30 (35K20)

Colocated finite volume schemes for fluid flows

Abstract: In this paper, the authors present two different time-discretization schemes combined with a finite-volume space discretization for the incompressible Navier-Stokes equations. They show that the collocated space discretization, usually used with a projection method, can also be extended to other time discretizations over previous projection methods, like the splitting methods. The main advantage of this splitting method is that the pressure approximation is truly of second order. In the first and second parts of this article, two different time-discretization schemes are considered with a collocated space discretization and it is explained how the unknown can be correctly coupled. Numerical simulations are presented in the last part. Tests of spatial and time accuracies are made. The numerical results when the Navier-Stokes problem is solved are shown, in the model problem of the driven cavity.

The discrete orthogonal polynomial least squares method for approximation and solving partial differential equations

Abstract: We investigate numerical approximations based on polynomials that are orthogonal with respect to a weighted discrete inner product and develop an algorithm for solving time dependent differential equations. We focus on the family of super Gaussian weight functions and derive a criterion for the choice of parameters that provides good accuracy and stability for the time evolution of partial differential equations. Our results show that this approach circumvents the problems related to the Runge phenomenon on equally spaced nodes and provides high accuracy in space. For time stability, small corrections near the ends of the interval are computed using local polynomial interpolation. Several numerical experiments illustrate the performance of the method. AMS subject classifications: 41A10, 65D15, 65M70

Multilevel preconditioners for the interior penalty discontinuous Galerkin method II- Quantitative studies

Abstract: This paper is concerned with preconditioners for interior penalty discontin- uous Galerkin discretizations of second-order elliptic boundary value problems. We extend earlier related results in (7) in the following sense. Several concrete realizations of splitting the nonconforming trial spaces into a conforming and (remaining) noncon- forming part are identified and shown to give rise to uniformly bounded condition numbers. These asymptotic results are complemented by numerical tests that shed some light on their respective quantitative behavior.

Full Wave Simulations of Lower Hybrid Waves in Toroidal Geometry with Non-Maxwellian Electrons

Abstract: Analysis of the propagation of waves in the lower hybrid range of frequen- cies in the past has been done using ray tracing and the WKB approximation. Ad- vances in algorithms and the availability of massively parallel computer architectures has permitted the solving of the Maxwell-Vlasov system for wave propagation directly (Wright et al., Phys. Plasmas (2004), 11, 2473-2479). These simulations have shown that the bridging of the spectral gap (the difference between the high injected phase velocities and the slower phase velocity at which damping on electrons occurs) can be explained by the diffraction effects captured in the full wave algorithm - an effect missing in WKB based approaches. However, these full wave calculations were done with a Maxwellian electron distribution and the presence of RF power induces quasi- linear velocity space diffusion that causes distortions away from an Maxwellian. With sufficient power, a flattened region or plateau is formed between the point of most efficient damping on electrons at about 2-3 vthe and where collisional and quasilin- ear diffusion balance. To address this discrepancy and better model experiment, we have implemented (Valeo et al., "Full-wave Simulations of LH wave propagation in toroidal plasma with non-Maxwellian electron distributions", 18th Topical Conference on Radio Frequency Power in Plasmas, AIP Conference Proceedings (2007)) a non- Maxwellian dielectric in our full wave solver. We will show how these effects modify the electron absorption relative to what is found for a Maxwellian distribution. AMS subject classifications: 35Q60, 65L60, 74J05, 82D10

Stability analysis of the immersed boundary method for a two-dimensional membrane with bending rigidity

Abstract: In this paper, we analyze the stability of the Immersed Boundary Method applied to a membrane-fluid system with a plasma membrane immersed in an incompressible viscous fluid. We show that for small deformations, the planar rest state is stable for a membrane with bending rigidity. The smoothed version, using a standard regularization technique for the singular force, is also shown to be stable. Furthermore, we show that the coupled fluid-membrane system is stiff and smoothing helps to reduce the stiffness. Compared to the system of elastic fibers immersed in an incompressible fluid, membrane with bending rigidity consist of a wider range of decay rates. Therefore numerical instability could occur more easily for an explicit method when the time step size is not sufficiently small, even though the continuous problem is stable.

Sequential multiscale modeling using sparse representation

Abstract: Carlos J. Garcia-Cervera, ∗ Weiqing Ren, † Jianfeng Lu, ‡ and Weinan E § Mathematics Department, University of California, Santa Barbara, CA 93106. Courant Institute of Mathematical Sciences, New York University, New York, NY 10012. Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544. Department of Mathematics and PACM, Princeton University, Princeton, NJ 08544. (Dated: June 8, 2007)

Numerical simulation of fluid membranes in two-dimensional space

Abstract: The membrane’s dynamics is very important for cells. A membrane in 2dimensional space can be seen as an incompressible closed curve in a plane or a cylindrical surface in 3-dimensional space. In this paper, we design a second-order accurate numerical algorithm to simulate the shape transformation of the membrane. In the algorithm, we use the tangent angles to present the curve and avoid the difficulties from the constraint of curve’s incompressible condition. A lot of interesting phenomena are obtained. Some of them are very like the life processes of cells, such as exocytosis and endocytosis. Furthermore, we can see the relation between two dynamic models clearly. At last, considering the influence of the inner incompressible fluids partially, we add a constraint: the area circled by the membrane maintain invariable. The numerical results show the dynamic motions of a curve remaining its local arc length and inner area constant. AMS subject classifications: 65M06, 65M12, 92C17

Vector addition theorem and its diagonalization

Abstract: The conventional vector addition theorem is written in a compact notation. Then a new and succinct derivation of the vector addition theorem is presented that is as close to the derivation of the scalar addition theorem. Newly derived expressions in this new derivation are used to diagonalize the vector addition theorem. The diagonal form of the vector addition theorem is important in the design of fast algorithms for computational wave physics such as computational electromagnetics and computational acoustics. AMS subject classifications: 43A45, 43A65, 43A90, 47L90, 78M99 PACS: 03.50.De, 85.40.Bh, 89.20.Ff

Free Energy Calculations for DNA Near Surfaces Using an Ellipsoidal Geometry.

Abstract: The change in some thermodynamic quantities such as Gibbs' free energy, entropy and enthalpy of the binding of two DNA strands (forming a double helix), while one is tethered to a surface and are analytically calculated. These particles are submerged in an electrolytic solution; the ionic strength of the media allows the linearized version of the Poisson-Boltzmann equation (from the theory of the double layer interaction) to properly describe the interactions [13]. There is experimental and computational evidence that an ion penetrable ellipsoid is an adequate model for the single strand and the double helix [22-25]. The analytic solution provides simple calculations useful for DNA chip design. The predicted electrostatic effects suggest the feasibility of electronic control and detection of DNA hybridization in the fast growing area of DNA recognition.

Lipschitz and total-variational regularization for blind deconvolution

Abstract: In [3], Chan and Wong proposed to use total variational regularization for both images and point spread functions in blind deconvolution. Their experimental results show that the detail of the restored images cannot be recovered. In this paper, we consider images in Lipschitz spaces, and propose to use Lipschitz regularization for images and total variational regularization for point spread functions in blind deconvolution. Our experimental results show that such combination of Lipschitz and total variational regularization methods can recover both images and point spread functions quite well.

A Multiple Temperature Kinetic Model and its Application to Near Continuum Flows

Abstract: In an early approach, we proposed a kinetic model with multiple transla- tional temperature (K. Xu, H. Liu and J. Jiang, Phys. Fluids 19, 016101 (2007)). Based on this model, the stress strain relationship in the Navier-Stokes (NS) equations is re- placed by the translational temperature relaxation terms. The kinetic model has been successfully used in both continuum and near continuum flow computations. In this paper, we will further validate the multiple translational temperature kinetic model to flow problems in multiple dimensions. First, a generalized boundary condition incor- porating the physics of Knudsen layer will be introduced into the model. Second, the direct particle collision with the wall will be considered as well for the further modifi- cation of particle collision time, subsequently a new effective viscosity coefficient will be defined. In order to apply the kinetic model to near continuum flow computations, the gas-kinetic scheme will be constructed. The first example is the pressure-driven Poiseuille flow at Knudsen number 0.1, where the anomalous phenomena between the results of the NS equations and the Direct Simulation Monte Carlo(DSMC) method will be resolved through the multiple temperature model. The so-called Burnett-order effects can be captured as well by algebraic temperature relaxation terms. Another test case is the force-driven Poiseuille flow at various Knudsen numbers. With the effec- tive viscosity approach and the generalized second-order slip boundary condition, the Knudsen minimum can be accurately obtained. The current study indicates that it is useful to use multiple temperature concept to model the non-equilibrium state in near continuum flow limit. In the continuum flow regime, the multiple temperature model will automatically recover the single temperature NS equations due to the efficient energy exchange in different directions.

Diagonalizations of vector and tensor addition theorems

Abstract: Based on the generalizations of the Funk-Hecke formula and the Rayleigh plan-wave expansion formula, an alternative and succinct derivation of the addition theorem for general tensor field is obtained. This new derivation facilitates the diagonalization of the tensor addition theorem. In order to complete this derivation, we have carried out the evaluation of the generalization of the Gaunt coefficient for tensor fields. Since vector fields (special case of tensor fields) are very useful in practice, we discuss vector multipole fields and vector addition theorem in details. The work is important in multiple scattering and fast algorithms in wave physics. AMS subject classifications: 43A90, 78A45, 78A25, 78M15 PACS: 02.70.Pt., 03.50.De, 89.20.Ff