Numerical Boundary Conditions Research Articles

Stability analysis of numerical boundary conditions in domain decomposition algorithms

This paper discusses numerical stability of a class of non-overlapping domain decomposition algorithms. Inherent shortcomings associated with different methods for proving stability are pointed out. Von Neumann analysis yields only a necessary condition for stability because it does not consider the overall effect of the boundary conditions between subdomains. Matrix analysis produces also a necessary condition for stability since the matrices of coefficients associated with the algorithms are not symmetric. The GKSO, on the other hand, produces necessary and sufficient conditions for stability. However, it is difficult to apply systematically this analysis to the algorithms.

Automatic finite element meshing of planar Voronoi tessellations

The concept of Voronoi tessellation has recently been extensively used in materials science, especially to model the geometrical features of random microstructures like aggregates of grains in polycrystals, patterns of intergranular cracks and composites. Solution of the underlying field equations usually requires use of numerical methods such as finite elements. The framework for automatic generation of quadrilateral finite element meshing of planar Voronoi tessellation is proposed in the paper, resulting in a powerful set of tools to be used in the rather wide field of micromechanics. As far as feasible, the implementation of features built in commercially available mesh generators was pursued. Additionally, the minimum geometric requirements for a “meshable” tessellation are outlined. Special attention is given to the meshes, which enable explicit modelling of grain boundary processes, such as for example contact (closure of cracks) or friction between grains. This is inline with numerical examples, which are oriented towards the fracture mechanics, in particular to the development of intergranular microcracks and/or their impact on the effective behaviour of the polycrystal. The examples were evaluated using the commercially available general-purpose finite element code abaqus. The usual continuum mechanics based numerical methods and boundary conditions were safely applied to aggregates of randomly oriented polycrystals with anisotropic elastic material behaviour as computational domains.

Quantifying Uncertainty in CFD

Quantifying Uncertainty in CFD

Noise from Fine-Scale Turbulence of Nonaxisymmetric Jets

The noise from the fine-scale turbulence of high-speed nonaxisymmetric jets is considered. A prediction method is developed by extending the work of Tam and Auriault. A set of improved numerical boundary conditions for use in nonaxisymmetricjet mean flow and turbulence calculation is developed. These new boundary conditions allow the computation to be carried out in a smaller computation domain. It is known that nonaxisymmetric mean flow has a significant impact on the radiated noise spectrum and directivity through refraction. In the Tam and Auriault theory, this effect is accounted for by means of the adjoint Green's function. Here the adjoint Green's function method is extended to nonaxisymmetric mean flows. The adjoint Green's function is first recast into the solution of a sound scattering problem. The sound scattering problem is then solved computationally by computational aeroacoustics methods. Extensive comparisons between calculated and experimentally measured jet noise spectra are presented. They include both rectangular and elliptic jets at supersonic and subsonic Mach numbers

Pricing Algorithms of Multivariate Path Dependent Options

Financial derivatives which are multivariate in nature are abundant in the financial markets. The underlying state variables may be the stock prices, interest rates, exchange rates, stochastic volatility, average of stock prices, extremum values of stock priors, etc. Option contracts whose life and payoff depend on the stochastic movement of the underlying asset prices are termed path dependent options. In this paper, we examine the pricing methods of several prototype path dependent options. These include options with sequential barriers, options with an external barrier and two-asset lookback options. The governing equations for the option prices are seen to resemble the diffusion type equations but with cross derivative terms, a feature which differs from the usual diffusion equations in engineering. Various techniques to reduce the complexity of the multivariate nature of these prototype option pricing models are discussed. It is illustrated that the dimensionality of a path dependent option model may be reduced by some ingenious choices of similarity variables. We also examine the design of pricing algorithms of these multivariate options, in particular, with regard to the treatment of discrete monitoring feature and the prescription of numerical boundary conditions. The possible generalizations of the numerical techniques presented in this paper to other models with more complicated path dependent payoff structures are also discussed.

On the influence of numerical boundary conditions

Our understanding about the behaviour of numerical solutions for evolutionary convection–diffusion equations is mainly based on analysis of infinite domains situations with stability given by von Neumann analysis. Almost all practical problems involve physical domains with boundaries. For evolution problems with Dirichlet boundary conditions, some algorithms can be used without alteration near a boundary. However, the application of higher order methods such as Quickest or second order upwinding introduces difficulty near an inflow boundary, since for interior points adjacent to the boundary there are insufficient upstream points for the high order scheme to be applied without alteration. For that reason such methods require a careful treatment on the inflow boundary, where additional numerical boundary conditions have to be introduced. The choice of numerical boundary conditions turns out to be crucial for stability. A test problem is described, showing the practical advantages of some numerical boundary conditions versus the others by comparison with an exact solution.

An analytical solution for stratified tidal lagoons: Application to the validation of a numerical open boundary condition

An analytical solution is presented for the case of a stratified, tidally forced lagoon. This solution, especially its energy and mass balances, is useful for the validation of numerical shallow water models under stratified, tidally forced conditions. The utility of the analytical solution for validation is demonstrated for a simple finite difference numerical model. The energy, mass, and their balances from the numerical model are shown to converge to the analytical solution with a power law dependence as spatial and temporal resolution are increased. The power law convergence of the numerical results to the analytical solution validates both the finite difference scheme and the open boundary conditions used for the tidal forcing. The open boundary conditions work well because they are consistent with the characteristics of the analytical solution.

A new type numerical model for action balance equation in simulating nearshore waves

Several current used wave numerical models are briefly described, the computing techniques of the source terms, numerical wave generation and boundary conditions in the action balance equation model are discussed. Not only the quadruplet wave-wave interactions, but also the triad wave-wave interactions are included in the model, so that nearshore waves could be simulated reasonably. The model is compared with the Boussinesq equation and the mild slope equation. The model is applied to calculating the distributions of wave height and wave period field in the Haian Bay area and to simulating the influences of the unsteady current and water level variation on the wave field. Finally, the developing tendency of the model is discussed.

Numerical simulation of tidal bore in Hangzhou Gulf and Qiantangjiang

AbstractIn this paper, a new numerical model, based on a set of non‐linear shallow water equations is developed for the simulation of the formation and evolution of tidal bore in the Hangzhou Gulf and Qiantangjiang river of China. The numerical method and boundary conditions are described in detail. The method is validated against analytical solutions and experimental data. Simulation of the actual tidal bore in Hangzhou Gulf and its propagation in the Qiantangjiang river are performed. Numerical results show that this proposed method is effective for the prediction of tidal bore and current flow at the entrance of Qiantangjiang river. Copyright © 2001 John Wiley & Sons, Ltd.

An implementation of the robust inviscid wall boundary condition in high-speed flow calculations

Boundary condition is one of the major factors to influence the numerical stability and solution accuracy in numerical analysis. One of the most important physical boundary conditions in the flowfield analysis is the wall boundary condition imposed on the body surface. To solve a two-dimensional Euler equation, totally four numerical wall boundary conditions should be prescribed. Two of them are supplied by the flow tangency condition. The other two conditions, therefore, should be prepared additionally in a suitable way. In this paper, four different sets of wall boundary conditions are proposed and then applied to solve high-speed flowfields around a quarter circle geometry. A two-dimensional compressible Euler solver is prepared based on the finite volume method. This solver hires three different upwind schemes; Steger-Warming’s flux vector splitting, Roe’s flux difference splitting, and Liou’s advection upstream splitting method. It is found that the way to specify the additional numerical wall boundary conditions strongly affects the overall stability and accuracy of the upwind schemes in high-speed flow calculation. The optimal wall boundary conditions should be also chosen very carefully depending on the numerical schemes used to solve the problem.

A note on finite difference discretizations for Poisson equation on a disk

AbstractA simple second‐order finite difference treatment of polar coordinate singularity for Poisson equation on a disk is presented. By manipulating the grid point locations, we can successfully avoid finding numerical boundary condition at the origin so that the resulting matrix is simpler than traditional schemes. The treatments for Dirichlet and Neumann boundary problems are slightly different by adjusting the radial mesh width. Furthermore, the present discretization can be applied with slight modifications to Helmholtz‐type equations. The numerical evidence shows that the second‐order accuracy can also be preserved. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 199–203, 2001

Accurate Projection Methods for the Incompressible Navier–Stokes Equations

This paper considers the accuracy of projection method approximations to the initial–boundary-value problem for the incompressible Navier–Stokes equations. The issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second-order methodology a decade and a half ago. It has been observed that while the velocity can be reliably computed to second-order accuracy in time and space, the pressure is typically only first-order accurate in the L∞-norm. This paper identifies the source of this problem in the interplay of the global pressure-update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second-order accurate, as demonstrated by a normal mode analysis and numerical experiments. In addition, a numerical method based on a gauge variable formulation of the incompressible Navier–Stokes equations, which provides another option for obtaining fully second-order convergence in both velocity and pressure, is discussed. The connection between the boundary conditions for projection methods and the gauge method is explained in detail.

Design of actuator profiles for minimum power consumption

The power consumption of piezoelectric actuators for active controlis an important parameter for the efficient performance of a smart structuralsystem. In this paper we address the issue of minimizing the maximum powerconsumption by the efficient layout of the actuators to achieve a specifiedmaximum displacement. The finite element method is used for the structuralresponse of piezoelectric composite plates under mechanical and electricalloading. A novel iterative design procedure based on the work done by theactuators on the host structure is presented. The voltage required to achievethe maximum displacement is calculated for changing actuator profiles and thepeak power consumed by the actuators is determined. Although the method isgeneral and can be applied to dynamic conditions, the scope of the presentpaper is restricted to static analysis only. Several numerical examples withsimple and complex boundary conditions are presented.

A Comparison of Transparent Boundary Conditions for the Fresnel Equation

We consider two numerical transparent boundary conditions that have been previously introduced in the literature. The first condition (BPP) was proposed by Baskakov and Popov (1991, Wave Motion14, 121–128) and Papadakis et al. (1992, J. Acoust. Soc. Am.92, 2030–2038) while the second (SDY) is that of Schmidt and Deuflhard (1995, Comput. Math. Appl.29, 53–76) and Schmidt and Yevick (1997, J. Comput. Phys.134, 96–107). The latter procedure is explicitly tailored to the form of the underlying numerical propagation scheme and is therefore unconditionally stable and highly precise. Here we present a new derivation of the SDY approach. As a result of this analysis, we obtain a simple modification of the BPP method that guarantees accuracy and stability for long propagation step lengths.

A novel hybridization of higher order finite element and boundary integral methods for electromagnetic scattering and radiation problems

A novel hybridization of the finite element (FE) and boundary integral methods is presented for an efficient and accurate numerical analysis of electromagnetic scattering and radiation problems. The proposed method derives an adaptive numerical absorbing boundary condition (ABC) for the finite element solution based on boundary integral equations. Unlike the standard finite element-boundary integral approach, the proposed method is free of interior resonance and produces a purely sparse system matrix, which can be solved very efficiently. Unlike the traditional finite element-absorbing boundary condition approach, the proposed method uses an arbitrarily shaped truncation boundary placed very close to the scatterer/radiator to minimize the computational domain; and more importantly, the method produces a solution that converges to the true solution of the problem. To demonstrate its great potential, the proposed method is implemented using higher order curvilinear vector elements. A mixed functional is designed to yield both electric and magnetic fields on an integration surface, without numerical differentiation, to be used in the calculation of the adaptive ABC. The required evaluation of boundary integrals is carried out using the multilevel fast multipole algorithm, which greatly reduces both the memory requirement and CPU time. The finite element equations are solved efficiently using the multifrontal algorithm. A mathematical analysis is conducted to study the convergence of the method. Finally, a number of numerical examples are given to illustrate its accuracy and efficiency.

On boundary conditions and numerical methods for the unsteady incompressible Navier–Stokes equations

Numerical methods and boundary conditions to solve the unsteady incompressible Navier–Stokes equations are discussed. It is shown, via the analysis of the governing partial differential equations and their discretized algebraic equations, that boundary conditions necessary to solve the incompressible Navier–Stokes equations are conditions either for the normal and tangential components of velocities or alternatively for the normal velocity and tangential vorticity components. In an explicit formulation, the solution of the resulting system of algebraic equations is got more efficiently by solving either the pressure or velocity Poisson equation. The boundary conditions necessary to solve these Poisson equations are provided from velocity boundary conditions and momentum equations. Results of numerical simulations of some selected unsteady flows are also presented.

Some steady vortex flows past a circular cylinder

Steady vortex flows past a circular cylinder are obtained numerically as solutions of the partial differential equation Δψ = f(ψ), f(ψ) = ω(1 − H(ω − α)), where H is the Heaviside function. Only symmetric solutions are considered so the flow may be thought of as that past a semicircular bump in a half-plane. The flow is transplanted by the complex logarithm to a semi-infinite strip. This strip is truncated at a finite height, a numerical boundary condition is used on the top, and the difference equations resulting from the five-point discretization for the Laplacian on a uniform grid are solved using Fourier methods and an iteration for the nonlinear equation. If the area of the vortex region is prescribed the magnitude of the vorticity ω is adjusted in an inner iteration to satisfy this area constraint.Three types of solutions are discussed: vortices attached to the cylinder, vortex patches standing off from the cylinder and strips of vorticity extending to infinity. Three families of each type of solution have been found. Equilibrium positions for point vortices, including the Föppl pair, are related to these families by continuation.

Comparison of Numerical Schemes Applied to CAA Problems

AbstractTo compare numerical methods and boundary conditions for computational aeroacoustics (CAA) two‐dimensional aeroacoustic benchmark problems are solved. The numerical methods are the dispersion relation preserving scheme (DRP) of Tam et al. [1] and the compact finite difference scheme proposed by Lele [2]. The acoustic boundary conditions applied are the non‐reflecting boundary condition of Thompson [4,5] and the asymptotic boundary condition of Tam [2]. The benchmark problems used describe the propagation of acoustic, vorticity and entropy waves relative to a mean flow with varied flow direction.

An active wave generating–absorbing boundary condition for VOF type numerical model

The objective of the present work is to discuss the implementation of an active wave generating–absorbing boundary condition for a numerical model based on the Volume Of Fluid (VOF) method for tracking free surfaces. First an overview of the development of VOF type models with special emphasis in the field of coastal engineering is given. A new type of numerical boundary condition for combined wave generation and absorption in the numerical model VOFbreak 2 is presented. The numerical boundary condition is based on an active wave absorption system that was first developed in the context of physical wave flume experiments, using a wave paddle. The method applies to regular and irregular waves. Velocities are measured at one location inside the computational domain. The reflected wave train is separated from the incident wave field in front of a structure by means of digital filtering and subsequent superposition of the measured velocity signals. The incident wave signal is corrected, so that the reflected wave is effectively absorbed at the boundary. The digital filters are derived theoretically and their practical design is discussed. The practical use of this numerical boundary condition is compared to the use of the absorption system in a physical wave flume. The effectiveness of the active wave generating–absorbing boundary condition finally is proved using analytical tests and numerical simulations with VOFbreak 2.

Domain-wall free energy of spin-glass models: numerical method and boundary conditions.

An efficient Monte Carlo method is extended to evaluate directly domain-wall free energy for randomly frustrated spin systems. Using the method, critical phenomena of spin-glass phase transition are investigated in the 4d+/-J Ising model under the replica boundary condition. Our values of the critical temperature and exponent, obtained by finite-size scaling, are in good agreement with those of the standard Monte Carlo and the series expansion studies. In addition, two exponents, the stiffness exponent and the fractal dimension of the domain wall, which characterize the ordered phase, are obtained. The latter value is larger than d-1, indicating that the domain wall is really rough in the 4d Ising spin-glass phase.