Non-reflecting Boundary Conditions Research Articles

Exact Non-Reflecting Boundary Conditions on Perturbed Domains and hp-Finite Elements

For exterior scattering problems one of the chief difficulties arises from the unbounded nature of the problem domain. Inhomogeneous obstacles may require a volumetric discretization, such as the Finite Element Method (FEM), and for this approach to be feasible the exterior domain must be truncated and an appropriate condition enforced at the far, artificial, boundary. An exact, non-reflecting boundary condition can be stated using the classical method if the Artificial Boundary's shape is quite specific: circular or elliptical. Recently, this approach has been generalized to permit quite general Artificial Boundaries which are shaped as perturbations of a circle resulting in the Enhanced DtN-FE method. In this paper we extend this method to a two-dimensional FEM featuring high-order polynomials in order to realize a high rate of convergence. This is more involved than simply specifying high-order test and trial functions as now the scatterer shape and Artificial Boundary must be faithfully represented. This entails boundary elements which conform (to high order) to the true boundary shapes. As we show, this can be accomplished and we realize an arbitrary order FEM without spurious reflections.

An energy absorbing far-field boundary condition for the elastic wave equation

We present an energy absorbing non-reflecting boundary condition of Clayton-Engquist type for the elastic wave equation together with a discretization which is stable for any ratio of compressional to shear wave speed. We prove stability for a second-order accurate finite-difference discretization of the elastic wave equation in three space dimensions together with a discretization of the proposed non-reflecting boundary condition. The stability proof is based on a discrete energy estimate and is valid for heterogeneous materials. The proof includes all six boundaries of the computational domain where special discretizations are needed at the edges and corners. The stability proof holds also when a free surface boundary condition is imposed on some sides of the computational domain. AMS subject classifications: 65M06, 65M12, 74B05, 86A15

High-order non-reflecting boundary conditions for the linearized 2-D Euler equations: No mean flow case

Higdon-type non-reflecting boundary conditions (NRBCs) are developed for the 2-D linearized Euler equations with Coriolis forces. This implementation is applied to a simplified form of the equations, with the NRBCs applied to all four sides of the domain. We demonstrate the validity of the NRBCs to high order. We close with a list of areas for further research.

VORTEX-GENERATED SOUND IN FLOW ABOUT SPINNING CYLINDERS

This study provides a computational insight into unsteady far-field and surface pressure developed by vortex interaction with multiple rigid bodies in lift conditions. Flows around a spinning cylinder and two cylinders in tandem were taken as simple but yet representative prototypes of flows around multi-element lifting devices. The unsteady Euler equations are solved in terms of propagating disturbances originating from deforming vortices in the mean flow developed in the neighborhood of cylinders. Numerical errors associated with the discretization and boundary conditions were kept small employing a high-order scheme with accurate nonreflecting boundary conditions. To model the interaction of vortices with rigid bodies using moderate amount of computational resources, we apply a single-grid approach and implement the bipolar coordinate transformation where applicable. We address here the amplification of sound by the mean flow with nonzero circulation, strong influence of vortex profile on the generated sound waves, and different degree of amplification of sound in lifting flows for localized and nonlocal vortices. We find that the flow about a cylinder not only amplifies the sound strength but it also shifts the sound directivity. Vortices deforming within the flow about cylinders placed in tandem and in the traverse layouts were examined.

Nonreflecting boundaries for ultrasound in fluctuating hydrodynamics of open systems

We present a formulation for nonreflecting boundaries in fluctuating hydrodynamics. Nonreflecting boundary conditions are designed to evacuate sound waves out of the computational domain, thus allowing one to deal with open systems and to avoid finite size effects associated with periodic boundaries. Thermodynamic consistency for the fluctuation of the total mass and momentum of the open system is ensured by a fluctuation-dissipation balance which controls the amplitude of the sound waves generated by stress fluctuations near the boundary. We consider equilibrium and out-of-equilibrium situations (forced sound) in liquid water at ambient conditions and argon ranging from gas to liquid densities. Nonreflecting boundaries for fluctuating hydrodynamics make feasible simulations of ultrasound in microfluidic devices.

Characteristic nonreflecting boundary conditions for open boundaries in lattice Boltzmann methods

A boundary condition for lattice Boltzmann methods, based on the movement of information through Euler characteristic directions, is developed. With respect to the similar conditions used in finite-difference or finite-volume implementations, some corrections are needed to compensate the isothermal compressible nature of standard lattice Boltzmann methods for fluid flow. The results show that the proposed method for inlets and outlets is highly nonreflecting, and mass conserving.

Finite element simulation of two-point elastic wave excitation method for damage detection in concrete structures

Two-point elastic wave excitation method is proposed to detect the damages (crack and deterioration). This method does not require any baseline signal to distinguish the damages. For verifying the proposed method, finite element simulations were carried out. Two models including 2-D plane and 3-D solid models were constructed, and non-reflecting boundary conditions were applied to the 3-D model to simulate a massive concrete structure. It is shown that cracks and deteriorations were successfully detected and their severities were well quantified through the finite element simulations. This baseline- free elastic wave method is useful for damage detection of concrete structures with or without nonreflecting boundaries.

Fractional calculus and non-reflecting boundary conditions in wave propagation

This paper shows why the fractional calculus naturally aris es for modeling the re- flection of linear and nonlinear waves at an interface throug h a boundary operator. The con- struction of such boundary conditions is difficult and requi res the use of advanced mathematical analysis theories related to fractional microlocal calcul us. Moreover, the numerical approxi- mation of such operators is nontrivial and leads to fundamen tal problems of numerical analysis and simulation. Finally, we provide an example of applicati on in computational ocean acous- tics. RESUME. Ce papier montre comment le calcul fractionnaire intervien t de maniere naturelle lorsque l'on essaie de modeliser la non reflexion d'ondes lineaires ou non lineaires par une interface par le biais d'un operateur aux limites. Des quest ions difficiles se posent sur la construction de telles conditions qui fait appel a la theori e de l'analyse microlocale fraction- naire. De plus, l'approximation numerique des operateurs s e trouve etre delicate et conduit a

Analytical and numerical investigation of the spectra of three-point difference operators

The properties of the spectra of discrete spatially one-dimensional problems of convection — diffusion type with constant coefficients and nonstandard boundary conditions are examined in the framework of stability of explicit algorithms for time-dependent problems of mathematical physics. An analytical method is proposed for finding isolated limit points of the operator spectrum. Limit points are determined for the difference transport equation with different versions of nonreflecting boundary conditions and for an approximation of the heat conduction equation on a grid with condensation near the boundary. Stability and other properties of the spectrum are also established numerically.

Modeling sound propagation in acoustic waveguides using a hybrid numerical method

Sound propagation in an acoustic waveguide is examined using a hybrid numerical technique. Here, the waveguide is assumed to be infinite in length with an arbitrary but uniform cross section. Placed centrally within the guide is a short component section with an irregular nonuniform shape. The hybrid method utilizes a wave based modal solution for a uniform section of the guide and, using either a mode matching or point collocation approach, matches this to a standard finite element based solution for the component section. Thus, one needs only to generate a transverse finite element mesh in uniform sections of the waveguide and this significantly reduces the number of degrees of freedom required. Moreover, utilizing a wave based solution removes the need to numerically enforce a nonreflecting boundary condition at infinity using a necessarily finite mesh, which is often encountered in studies that use only the standard finite element method. Accordingly, the component transmission loss may readily be computed and predictions are presented here for three examples: an expansion chamber, a converging-diverging duct, and a circular cylinder. Good agreement with analytic models is observed, and transmission loss predictions are also presented for multimode incident and transmitted sound fields.

Finite volume and WENO scheme in one-dimensional vascular system modelling

A finite volume and WENO model is developed, validated and applied to a one-dimensional numerical investigation of the vascular system. Suitable balance laws are spatially integrated using a finite volume WENO approach. Time integration is performed using a five steps, fourth order accurate Runge–Kutta strong stability preserving scheme. A detailed investigation of different sets of reflecting and non-reflecting boundary conditions is performed to achieve a correct representation of limited portions of the vascular system. Moreover, special attention is devoted to the numerical discretisation of the source term. Finally, as an application of the whole model, the effect of an abdominal aorta aneurysm on the blood flow is evaluated.

Near-field infinity-simulating boundary conditions for the heat equation

The numerical simulation of various physical phenomena in infinite domains poses great difficulties. Truncation of the domain is usually necessary in such cases. To ensure stability of the resulting problem on the restricted domain, appropriate boundary conditions should be applied. Here, we develop a boundary condition for the case in which the heat equation is satisfied outside the domain of interest with no restrictions on the equation inside. The condition employs a thin layer encasing the computational domain. The resulting condition, combined with techniques similar to those proposed by Jiang and Greengard [Jiang S, Greengard L (2004) Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension. Comp Math Appl 47:955–966.] promises to be more accurate and computationally efficient than previously described techniques.

On the Orbital Evolution of a Jovian Planet Embedded in a Self‐Gravitating Disk

We performed a series of hydrodynamic simulations to investigate the orbital evolution of a Jovian planet embedded in a protostellar disk. In order to take into account of the effect of the disk's self-gravity, we developed and adopted an ANTARES code which is based on a 2D Godunov scheme to obtain the exact Reimann solution for isothermal or polytropic gas, with nonreflecting boundary conditions. Our simulations indicate that in the study of runaway (type III) migration it is important to carry out a fully self-consistent treatment of the gravitational interaction between the disk and the embedded planet. Through a series of convergence tests, we show that adequate numerical resolution, especially within the planet's Roche lobe, critically determines the outcome of the simulations. We consider a variety of initial conditions and show that isolated, noneccentric protoplanets do not undergo type III migration. We attribute the difference between our and previous simulations to the contribution of a self-consistent representation of the disk's self-gravity. Nevertheless, type III migration cannot be completely suppressed, and its onset requires finite-amplitude perturbations such as that induced by planet-planet interaction. We determine the radial extent of type III migration as a function of the disk's self-gravity.

ACOUSTIC PERFORMANCE OF NONREFLECTING BOUNDARY CONDITIONS FOR A RANGE OF INCIDENT ANGLES

A detailed study on the accuracy and reflection behavior of nonreflection boundary conditions is presented. To this end, a selection of five distinct nonreflecting boundary conditions is evaluated and the mechanisms which dominate the reflection properties of boundary conditions are identified based on a rigorous quantification of reflection levels. It is shown that the reflection behavior is significantly affected by the incident angle. The relation between the boundary condition effectiveness and the incident angle is investigated and the reflection rates as a function of the incident angle are quantified. Furthermore, the obtained results are used to predict the reflections from boundary conditions in general applications. It is shown that these predictions are reliable for different test cases.

Numerical stabilities and boundary conditions in time-domain Eulerian simulations of acoustic wave propagations with and without background flow

A thorough numerical analysis is performed for time-domain simulation of acoustic wave propagations in the atmosphere, with the ground modeled as a porous medium. Two types of computational grid arrangement for the simulation, i.e., the staggered grid and the collocated grid, are considered. It is proved that the computational schemes based on these two grids are identical under certain finite differencing procedures. The numerical stability analysis is studied that applies to both of the grids. Non-reflecting, absorbing boundary conditions are used at the free-space boundary. Simulations on the collocated grid are then carried out for a model problem of sound propagation in the air/ground to confirm the equivalency of the two grids and to investigate the effectiveness of non-reflecting boundary conditions. The results are compared with the data in the literature and with bench-mark simulation, and very good agreements have been achieved.

On the intersection of sets of incoming and outgoing waves

In the neighborhood of a boundary point, the solution of a first-order symmetric homogenous hyperbolic system is conveniently decomposed into fundamental waves solutions that are readily classified as outgoing, incoming, and stationary, or tangential. Under a broad hypothesis, we show that the spans of the sets of outgoing and incoming waves have nontrivial intersection. Under these conditions, local, linear, perfectly nonreflecting local boundary conditions are shown to be an impossibility.

Simulation of aeroacoustic resonance in a deep cavity with grazing flow using a pressure-based solver

Aeroacoustic resonance in a cavity is caused by instability in the fluid flow and the geometry of the cavity. Aeroacoustic resonance occurs in many industrial applications, and usually causes noise and fatigue problems. The ability to predict and solve aeroacoustic resonance problems is of great interest to engineers. The unsteady Reynolds averaged Navier-Stokes and the partially resolved numerical simulation approaches with the standard k–ε turbulence model are employed to simulate this resonance. In order to obtain reliable results, the imposition of proper boundary conditions is necessary. In this paper, non-reflecting and Riemann invariant boundary conditions are applied within a pressure-based CFD solver. As reported in the literature for a similar study, the boundary-layer thickness affects the resonance in terms of its magnitude and frequency. Therefore, the effect of the boundary-layer thickness is investigated here. Two cases of deep-cavity resonance are simulated and compared to experimental results in terms of velocity patterns in a resonance cycle, resonance frequency and the sound pressure level.

A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases

We present spectral element (SE) and discontinuous Galerkin (DG) solutions of the Euler and compressible Navier–Stokes (NS) equations for stratified fluid flow which are of importance in nonhydrostatic mesoscale atmospheric modeling. We study three different forms of the governing equations using seven test cases. Three test cases involve flow over mountains which require the implementation of non-reflecting boundary conditions, while one test requires viscous terms (density current). Including viscous stresses into finite difference, finite element, or spectral element models poses no additional challenges; however, including these terms to either finite volume or discontinuous Galerkin models requires the introduction of additional machinery because these methods were originally designed for first-order operators. We use the local discontinuous Galerkin method to overcome this obstacle. The seven test cases show that all of our models yield good results. The main conclusion is that equation set 1 (non-conservation form) does not perform as well as sets 2 and 3 (conservation forms). For the density current (viscous), the SE and DG models using set 3 (mass and total energy) give less dissipative results than the other equation sets; based on these results we recommend set 3 for the development of future multiscale research codes. In addition, the fact that set 3 conserves both mass and energy up to machine precision motives us to pursue this equation set for the development of future mesoscale models. For the bubble and mountain tests, the DG models performed better. Based on these results and due to its conservation properties we recommend the DG method. In the worst case scenario, the DG models are 50% slower than the non-conservative SE models. In the best case scenario, the DG models are just as efficient as the conservative SE models.

Exact local nonreflecting boundary conditions for time‐dependent multiple scattering

AbstractIn [2, 3] a nonreflecting boundary condition(NBC) for time‐dependent multiple scattering was derived, which is local in time but nonlocal in space. Here, based on a high‐order local nonreflecting boundary condition (NBC) for single scattering [4], we seek a local NBC for time‐dependent multiple scattering, which is completely local both in space and time. To do so, we first develop a high order representation formula for a purely outgoing wave field, given its values and those of certain auxiliary functions needed for the artificial boundary condition. By combining that representation formula with a decomposition of the total scattered field into purely outgoing contributions, we obtain the first exact, completely local, NBC for time‐dependent multiple scattering. The accuracy and stability of this local NBC is evaluated by coupling it to standard finite element and finite difference methods. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Influence of support viscoelastic properties on the structural wave propagation

The dissipation of the structural vibration energy at viscoelastic supports is an efficient method of reducing modal resonances and consequent noise and fatigue related problems. The support stiffness has significant impact on the modal characteristics. The dissipation capabilities of the viscoelastic support depend on its stiffness. Methods to optimally tune this support stiffness are proposed in this study. The characteristic mechanical impedance for structural vibration is obtained from wave propagation analysis and non-reflecting boundary conditions. The wave propagation is analyzed near the supports installed at edges, middle of a structure, and for the tuned vibration absorber. The dependence of the optimal stiffness on the location and mass of the supports is identified. A simple analytical solution for optimal support stiffness for maximum dissipation of propagating vibration energy at supports is presented.