scholarly journals EVALUATION OF SEVERAL NONREFLECTING COMPUTATIONAL BOUNDARY CONDITIONS FOR DUCT ACOUSTICS

1995 ◽  
Vol 03 (04) ◽  
pp. 327-342 ◽  
Author(s):  
WILLIE R. WATSON ◽  
WILLIAM E. ZORUMSKI ◽  
STEVE L. HODGE
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Sound Propagation ◽  
Sparse Matrix ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Substantial Reduction ◽  
Sound Attenuation ◽  
Interior Solution ◽  
Duct Acoustics

Several nonreflecting computational boundary conditions that meet certain criteria and have potential applications to duct acoustics are evaluated for their effectiveness. The same interior solution scheme, grid, and order of approximation are used to evaluate each condition. Sparse matrix solution techniques are applied to solve the matrix equation resulting from the discretization. Modal series solutions for the sound attenuation in an infinite duct are used to evaluate the accuracy of each nonreflecting boundary condition. The evaluations are performed for sound propagation in a softwall duct, for several sources, sound frequencies, and duct lengths. It is shown that a recently developed nonlocal boundary condition leads to sound attenuation predictions considerably more accurate than the local ones considered. Results also show that this condition is more accurate for short ducts. This leads to a substantial reduction in the number of grid points when compared to other nonreflecting conditions.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

scholarly journals A NONLOCAL COMPUTATIONAL BOUNDARY CONDITION FOR DUCT ACOUSTICS

1995 ◽  
Vol 03 (01) ◽  
pp. 15-26 ◽  
Author(s):  
WILLIAM E. ZORUMSKI ◽  
WILLIE R. WATSON ◽  
STEVE L. HODGE
Keyword(s):  
Boundary Condition ◽  
Acoustic Waves ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Three Dimensional ◽  
Computational Domain ◽  
Radiation Boundary Condition ◽  
Constant Matrix ◽  
Duct Acoustics ◽  
Computational Solution

A nonlocal boundary condition is formulated for acoustic waves in ducts without flow. The ducts are two-dimensional with constant area, but with variable impedance wall lining. Extension of the formulation to three-dimensional and variable area ducts is straightforward in principle, but requires significantly more computation. The boundary condition simulates a nonreflecting wave field in an infinite duct. It is implemented by a constant matrix operator which is applied at the boundary of the computational domain. An efficient computational solution scheme is developed which allows calculations for high frequencies and long duct lengths. This computational solution utilizes the boundary condition to limit the computational space while preserving the radiation boundary condition. The boundary condition is tested for several sources. It is demonstrated that the boundary condition can be applied close to the sound sources, rendering the computational domain small. Computational solutions with the new nonlocal boundary condition are shown to be consistent with the known solutions for nonreflecting wave fields in an infinite uniform duct.

scholarly journals INVESTIGATION OF SPECTRUM CURVES FOR A STURM-LIOUVILLE PROBLEM WITH TWO-POINT NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
Vol 25 (1) ◽  
pp. 37-52
Author(s):  
Kristina Bingelė ◽  
Agnė Bankauskienė ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Characteristic Function ◽  
Boundary Conditions ◽  
Critical Points ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Nonlocal Boundary Conditions ◽  
Point Boundary ◽  
Sturm Liouville

The article investigates the Sturm–Liouville problem with one classical and another nonlocal two-point boundary condition. We analyze zeroes, poles and critical points of the characteristic function and how the properties of this function depend on parameters in nonlocal boundary condition. Properties of the Spectrum Curves are formulated and illustrated in figures for various values of parameter ξ.

NUMERICAL SIMULATION OF LOW-FREQUENCY AEROACOUSTICS OVER IRREGULAR TERRAIN USING A FINITE ELEMENT DISCRETIZATION OF THE PARABOLIC EQUATION

2002 ◽  
Vol 10 (01) ◽  
pp. 97-111 ◽  
Author(s):  
NIKOLAOS A. KAMPANIS
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Sound Propagation ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Low Frequency ◽  
Initial Boundary ◽  
Computational Domain ◽  
Element Discretization ◽  
Irregular Terrain

Environmental noise raises serious concerns in modern industrial societies. As a result, the study of sound propagation in the atmosphere over irregular terrain is a subject of current interest in aeroacoustics. We use the standard parabolic approximation of the Helmholtz equation to simulate the far-field, low-frequency sound propagation in a refracting atmosphere, over terrains with mild range-varying topography. At an artificial upper boundary of the computational domain, described in range and height coordinates, a nonlocal boundary condition is used to model the effect of a homogeneous, semi-infinite atmosphere. We define a curvilinear coordinate system fitting the irregular topography. We discretize the transformed initial-boundary value problem with a finite element technique in height and a conservative Crank–Nicolson scheme for marching in range. The underlying transformation of coordinates allows the effective coupling with the nonlocal boundary condition. The resulting discretization method is accurate and efficient for the numerical prediction of noise levels in the atmosphere.

AN EXACT RADIATION CONDITION FOR USE WITH THE A POSTERIORI PE METHOD

1994 ◽  
Vol 02 (02) ◽  
pp. 113-132 ◽  
Author(s):  
DAVID J. THOMSON ◽  
M. ELIZABETH MAYFIELD
Keyword(s):  
Boundary Condition ◽  
Sound Propagation ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Radiation Condition ◽  
Ocean Bottom ◽  
A Posteriori ◽  
Underwater Sound ◽  
Sea Bottom ◽  
Absorbing Layer

In the applications of the parabolic equation (PE) to underwater sound propagation, the radiation condition as z → ∞ is usually approximated numerically by appending an absorbing layer below the physical ocean bottom and imposing a simple pressure-release boundary condition at the base of the layer. A similar artifice is needed to prevent unphysical reflections from the top of the air layer in the applications of the PE to atmospheric sound propagation. In this paper, we replace this approximate boundary treatment for the standard PE with an exact, nonlocal boundary condition that can be applied along the sea-bottom or upper-atmosphere interface. Moreover, we make use of an exact relationship between the solution ψ of the standard PE and the solution p of the Helmholtz equation to postprocess the ψ field obtained using this nonlocal boundary condition into the p field. The effectiveness of this radiation condition and the a posteriori PE scheme is demonstrated for several examples that typify underwater and outdoor sound propagation in layered media.

scholarly journals STURM-LIOUVILLE PROBLEM FOR STATIONARY DIFFERENTIAL OPERATOR WITH NONLOCAL INTEGRAL BOUNDARY CONDITION

2005 ◽  
Vol 10 (4) ◽  
pp. 377-392 ◽  
Author(s):  
S. Pečiulyte ◽  
O. Štikoniene
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Integral Boundary Conditions ◽  
Liouville Problem ◽  
Complex Case ◽  
Integral Boundary ◽  
Sturm Liouville ◽  
Nonlocal Integral Boundary Conditions

The Sturm‐Liouville problem with various types of nonlocal integral boundary conditions is considered in this paper. In the first part of paper we investigate Sturm‐Liouville problem with two cases of nonlocal integral boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such problem in the complex case. In the second part we investigate real eigenvalues case. The spectrum depends of these problems on boundary condition parameters is analyzed. Qualitative behaviour of all eigenvalues subject to nonlocal boundary condition parameters is described. Šiame straipsnyje nagrinejamas Šturmo‐Liuvilio uždavinys su viena nelokaliaja integralinio tipo kraštine salyga. Pirmoje straipsnio dalyje tiriamas Šturmo‐Liuvilio uždavinys su dvieju tipu integraline nelokaliaja salyga. Irodytos tikriniu funkciju ir tikriniu reikšmiu bendrosios savybes komplesineje plokštumoje. Antroje dalyje plačiau ištirtas realiuju tikriniu reikšmiu atvejis. Straipsnyje nagrinejama kaip Šturmo‐Liuvilio uždavinio spektras priklauso nuo kraštiniu salygu parametru. Priklausomai nuo nelokaliuju kraštiniu salygu parametru, aprašytas kokybinis tikriniu reikšmiu pasiskirstymas.

A FINITE ELEMENT METHOD FOR THE PARABOLIC EQUATION IN AEROACOUSTICS COUPLED WITH A NONLOCAL BOUNDARY CONDITION FOR AN INHOMOGENEOUS ATMOSPHERE

2005 ◽  
Vol 13 (04) ◽  
pp. 569-584 ◽  
Author(s):  
NIKOLAOS A. KAMPANIS
Keyword(s):  
Finite Element Method ◽  
Boundary Condition ◽  
Finite Element ◽  
Parabolic Equation ◽  
Sound Propagation ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Computational Domain ◽  
Inhomogeneous Atmosphere ◽  
Element Method

The standard parabolic equation is used to simulate the far-field, low-frequency sound propagation over ground with mild range-varying topography. The atmosphere has a lower layer with a general, variable index of refraction. An unbounded upper layer with a squared refractive index varying linearly with height is considered and modeled by the nonlocal boundary condition of Dawson, Brooke and Thomson.1 A finite element/transformation of coordinates method is used to transform the initial-boundary value problem to one with a rectangular computational domain and then discretize it. The solution is marched in range by the Crank–Nicolson scheme. A discrete form of the nonlocal boundary condition, which is left unaffected by the transformation of coordinates, is employed in the finite element method. The fidelity of the overall method is shown in the numerical simulations performed for various cases of sound propagation in an inhomogeneous atmosphere over a ground with irregular topography.

scholarly journals An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

2020 ◽  
Vol 54 (4) ◽  
pp. 1373-1413 ◽  
Author(s):  
Huaiqian You ◽  
XinYang Lu ◽  
Nathaniel Task ◽  
Yue Yu
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Nonlocal Diffusion ◽  
Order Convergence ◽  
Flux Boundary Condition ◽  
Nonlocal Interactions ◽  
Local Case ◽  
Neumann Type ◽  
Type Boundary

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

scholarly journals Existence of Positive Solutions for Fractional Differential Equation with Nonlocal Boundary Condition

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Hongliang Gao ◽  
Xiaoling Han
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Positive Solutions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
The Fixed Point Theorem

By using the fixed point theorem, existence of positive solutions for fractional differential equation with nonlocal boundary conditionD0+αu(t)+a(t)f(t,u(t))=0,0<t<1,u(0)=0,u(1)=∑i=1∞αiu(ξi)is considered, where1<α≤2is a real number,D0+αis the standard Riemann-Liouville differentiation, andξi∈(0,1),  αi∈[0,∞)with∑i=1∞αiξiα-1<1,a(t)∈C([0,1],[0,∞)),  f(t,u)∈C([0,1]×[0,∞),[0,∞)).

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