A SLOPING BOUNDARY CONDITION FOR EFFICIENT PE CALCULATIONS IN RANGE–DEPENDENT ACOUSTIC MEDIA

1996 ◽  
Vol 04 (01) ◽  
pp. 11-27 ◽  
Author(s):  
GARY H. BROOKE ◽  
DAVID J. THOMSON ◽  
PHILIP M. WORT
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Boundary Conditions ◽  
Standard Test ◽  
Numerical Results ◽  
Normal Displacement ◽  
Test Cases ◽  
Approximate Form ◽  
Acoustic Media ◽  
Horizontal Interface

The traditional one-way parabolic equation (PE) formulation for range-dependent layered acoustic media is modified to include effects associated with the boundary conditions along a sloping interface. Essentially, the boundary condition for continuity of the normal displacement along a sloping interface is cast in an approximate form which does not depend on range but does contain terms up to second order in the derivatives with respect to depth. The new sloping-boundary condition is then applied along an "equivalent" horizontal interface within each range-independent step of the PE. Numerical results obtained for standard test cases indicate that the sloping-boundary condition, incorporated into a one-way PE, maintains the efficiency yet improves the accuracy of forward predictions.

scholarly journals DG-GMsFEM for Problems in Perforated Domains with Non-Homogeneous Boundary Conditions

Computation ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 75
Author(s):  
Valentin Alekseev ◽  
Maria Vasilyeva ◽  
Uygulaana Kalachikova ◽  
Eric T. Chung
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Homogeneous Boundary Condition ◽  
Multiscale Model ◽  
System Size ◽  
Numerical Results ◽  
Basis Functions ◽  
Perforated Domain ◽  
Perforated Domains ◽  
Grid Construction

Problems in perforated media are complex and require high resolution grid construction to capture complex irregular perforation boundaries leading to the large discrete system of equations. In this paper, we develop a multiscale model reduction technique based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM) for problems in perforated domains with non-homogeneous boundary conditions on perforations. This method implies division of the perforated domain into several non-overlapping subdomains constructing local multiscale basis functions for each. We use two types of multiscale basis functions, which are constructed by imposing suitable non-homogeneous boundary conditions on subdomain boundary and perforation boundary. The construction of these basis functions contains two steps: (1) snapshot space construction and (2) solution of local spectral problems for dimension reduction in the snapshot space. The presented method is used to solve different model problems: elliptic, parabolic, elastic, and thermoelastic equations with non-homogeneous boundary conditions on perforations. The concepts for coarse grid construction and definition of the local domains are presented and investigated numerically. Numerical results for two test cases with homogeneous and non-homogeneous boundary conditions are included, as well. For the case with homogeneous boundary conditions on perforations, results are shown using only local basis functions with non-homogeneous boundary condition on subdomain boundary and homogeneous boundary condition on perforation boundary. Both types of basis functions are needed in order to obtain accurate solutions, and they are shown for problems with non-homogeneous boundary conditions on perforations. The numerical results show that the proposed method provides good results with a significant reduction of the system size.

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

2008 ◽  
Vol 16 (01) ◽  
pp. 11-29 ◽  
Author(s):  
M. MESBAH ◽  
J. MEYERS ◽  
M. BAELMANS
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Incident Angle ◽  
Test Cases ◽  
Acoustic Performance ◽  
Nonreflecting Boundary Conditions ◽  
Selection Of

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.

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 The BV solution of the parabolic equation with degeneracy on the boundary

2016 ◽  
Vol 14 (1) ◽  
pp. 272-282
Author(s):  
Huashui Zhan ◽  
Shuping Chen
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Well Posedness ◽  
Test Functions ◽  
The Stability

AbstractConsider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

scholarly journals Stiffness Estimation and Equivalence of Boundary Conditions in FEM Models

2021 ◽  
Vol 11 (4) ◽  
pp. 1482
Author(s):  
Róbert Huňady ◽  
Pavol Lengvarský ◽  
Peter Pavelka ◽  
Adam Kaľavský ◽  
Jakub Mlotek
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Boundary Conditions ◽  
Model Updating ◽  
Element Model ◽  
Computational Time ◽  
Stiffness Coefficients ◽  
First Case ◽  
Numerical Approaches

The paper deals with methods of equivalence of boundary conditions in finite element models that are based on finite element model updating technique. The proposed methods are based on the determination of the stiffness parameters in the section plate or region, where the boundary condition or the removed part of the model is replaced by the bushing connector. Two methods for determining its elastic properties are described. In the first case, the stiffness coefficients are determined by a series of static finite element analyses that are used to obtain the response of the removed part to the six basic types of loads. The second method is a combination of experimental and numerical approaches. The natural frequencies obtained by the measurement are used in finite element (FE) optimization, in which the response of the model is tuned by changing the stiffness coefficients of the bushing. Both methods provide a good estimate of the stiffness at the region where the model is replaced by an equivalent boundary condition. This increases the accuracy of the numerical model and also saves computational time and capacity due to element reduction.

scholarly journals Bootstrapping boundary-localized interactions

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Connor Behan ◽  
Lorenzo Di Pietro ◽  
Edoardo Lauria ◽  
Balt C. van Rees
Keyword(s):  
Boundary Condition ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Parameter Space ◽  
Three Dimensional ◽  
Conformal Boundary ◽  
Dimensional Boundary ◽  
Dirichlet And Neumann Conditions ◽  
Boundary Fields ◽  
Boundary Dynamics

Abstract We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

scholarly journals Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Blow Up Analysis ◽  
Inner Absorption ◽  
Quasilinear Parabolic

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.

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.

Chaotic Vibration of a Two-dimensional Non-strictly Hyperbolic Equation

2018 ◽  
Vol 61 (4) ◽  
pp. 768-786 ◽  
Author(s):  
Liangliang Li ◽  
Jing Tian ◽  
Goong Chen
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Method Of Characteristics ◽  
Nonlinear Boundary ◽  
Chaotic Oscillation ◽  
Nonlinear Boundary Conditions ◽  
Rigorous Proof ◽  
Two Dimensional ◽  
Chaotic Vibration

AbstractThe study of chaotic vibration for multidimensional PDEs due to nonlinear boundary conditions is challenging. In this paper, we mainly investigate the chaotic oscillation of a two-dimensional non-strictly hyperbolic equation due to an energy-injecting boundary condition and a distributed self-regulating boundary condition. By using the method of characteristics, we give a rigorous proof of the onset of the chaotic vibration phenomenon of the zD non-strictly hyperbolic equation. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs. Numerical simulations are also provided.

Sign in / Sign up

Export Citation Format

Share Document