A FINITE ELEMENT CODE FOR THE NUMERICAL SOLUTION OF THE HELMHOLTZ EQUATION IN AXIALLY SYMMETRIC WAVEGUIDES WITH INTERFACES

1999 ◽  
Vol 07 (02) ◽  
pp. 83-110 ◽  
Author(s):  
NIKOLAOS A. KAMPANIS ◽  
VASSILIOS A. DOUGALIS
Keyword(s):  
Finite Element ◽  
Helmholtz Equation ◽  
Nonlocal Boundary Condition ◽  
Bottom Topography ◽  
Nonlocal Boundary ◽  
Small Scale ◽  
Far Field ◽  
Axially Symmetric ◽  
Fortran Code ◽  
Iterative Linear Solvers

We consider the Helmholtz equation in an axisymmetric cylindrical waveguide consisting of fluid layers overlying a rigid bottom. The medium may have range-dependent speed of sound and interface and bottom topography in the interior nonhomogeneous part of the waveguide, while in the far-field the interfaces and bottom are assumed to be horizontal and the problem separable. A nonlocal boundary condition based on the DtN map of the exterior problem is posed at the far-field artificial boundary. The problem is discretized by a standard Galerkin/finite element method and the resulting numerical scheme is implemented in a Fortran code that is interfaced with general mesh generation programs from the MODULEF finite element library and iterative linear solvers from QMRPACK. The code is tested on several small scale examples of acoustic propagation and scattering in the sea and its results are found to compare well with those of COUPLE.

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.

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.

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.

Finite Element Simulation of Laser Beam Welding Induced Residual Stresses and Distortions in a T-Joint Configuration for Aeronautic Structures

2009 ◽  
Author(s):  
Muhammad Zain-ul-abdein ◽  
Daniel Ne´lias ◽  
Jean-Franc¸ois Jullien ◽  
Dominique Deloison
Keyword(s):  
Finite Element ◽  
Residual Stresses ◽  
Laser Beam ◽  
Boundary Conditions ◽  
Laser Beam Welding ◽  
Temperature Fields ◽  
Heat Transfer Analysis ◽  
Small Scale ◽  
Joint Configuration ◽  
Simulation Results

Laser beam welding has found its application in the aircraft industry for the fabrication of fuselage panels in a T-joint configuration. However, the inconveniences like distortions and residual stresses are inevitable consequences of welding. The effort is made in this work to experimentally measure and numerically simulate the distortions induced by laser beam welding of a T-joint with industrially used thermal and mechanical boundary conditions on the thin sheets of aluminium 6056-T4. Several small scale experiments were carried out with various instrumentations to establish a database necessary to verify the simulation results. Finite element (FE) simulation is performed with Abaqus and the conical heat source is programmed in FORTRAN. Heat transfer analysis is performed to achieve the required weld pool geometry and temperature fields. Mechanical analysis is then performed with industrial loading and boundary conditions so as to predict the distortion and the residual stress pattern. A good agreement is found amongst the experimental and simulation results.

An Extrapolation Method to Compute Far-Field Pressures From Near-Field Pressures Obtained by Finite Element Method

1995 ◽  
Author(s):  
Jamal Assaad ◽  
Christian Bruneel ◽  
Jean-Michel Rouvaen ◽  
Régis Bossut
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Near Field ◽  
Radiation Condition ◽  
Directivity Pattern ◽  
Extrapolation Method ◽  
External Boundary ◽  
Finite Width ◽  
Far Field ◽  
Element Method

Abstract The finite element method is widely used for the modeling of piezoelectric transducers. With respect to the radiation loading, the fluid is meshed and terminated by an external nonreflecting surface. This reflecting surface can be made up with dipolar damping elements that absorb approximately the outgoing acoustic wave. In fact, with dipolar dampers the fluid mesh can be quite limited. This method can provides a direct computation of the near-field pressure inside the selected external boundary. This paper describes an original extrapolation method to compute far-field pressures from near-field pressures in the two-dimensional (2-D) case. In fact, using the 2-D Helmholtz equation and its solution obeying the Sommerfeld radiation condition, the far-field directivity pattern can be expressed in terms of the near-field directivity pattern. These developments are valid for any radiation problem in 2D. One test example is described which consists of a finite width planar source mounted in a rigid or a soft baffle. Experimental results concerning the far-field directivity pattern of lithium niobate bars (Y-cut) are also presented.

Accuracy of Failure Criteria Commonly Used for Ship Collision Simulations

2018 ◽  
Author(s):  
R. Villavicencio ◽  
Bin Liu ◽  
Kun Liu
Keyword(s):  
Finite Element ◽  
Failure Criteria ◽  
Small Scale ◽  
Finite Element Models ◽  
Impact Loads ◽  
Large Scatter ◽  
Element Analysis ◽  
Advantages And Disadvantages ◽  
Ship Collision ◽  
Double Hull

The paper summarises observations of the fracture response of small-scale double hull specimens subjected to quasi-static impact loads by means of simulations of the respective experiments. The collision scenarios are used to evaluate the discretisation of the finite element models, and the energy-responses given by various failure criteria commonly selected for collision assessments. Nine double hull specimens are considered in the analysis so that to discuss the advantages and disadvantages of the different failure criterion selected for the comparison. Since a large scatter is observed from the numerical results, a discussion on the reliability of finite element analysis is also provided based on the present study and other research works found in the literature.

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,∞)).

Sign in / Sign up

Export Citation Format

Share Document