Boundary Conditions and Iterative Schemes for the Helmholtz Equation in Unbounded Regions

2009 ◽  
pp. 127-158 ◽  
Author(s):  
E. Turkel
Keyword(s):  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Iterative Schemes

scholarly journals Atmospheric radiation boundary conditions for the Helmholtz equation

2018 ◽  
Vol 52 (3) ◽  
pp. 945-964 ◽  
Author(s):  
Hélène Barucq ◽  
Juliette Chabassier ◽  
Marc Duruflé ◽  
Laurent Gizon ◽  
Michael Leguèbe
Keyword(s):  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Acoustic Waves ◽  
Numerical Study ◽  
Outgoing Wave ◽  
Atmospheric Radiation ◽  
Angle Of Incidence ◽  
Radiation Boundary Conditions ◽  
Radiation Boundary ◽  
Dirichlet To Neumann Map

This work offers some contributions to the numerical study of acoustic waves propagating in the Sun and its atmosphere. The main goal is to provide boundary conditions for outgoing waves in the solar atmosphere where it is assumed that the sound speed is constant and the density decays exponentially with radius. Outgoing waves are governed by a Dirichlet-to-Neumann map which is obtained from the factorization of the Helmholtz equation expressed in spherical coordinates. For the purpose of extending the outgoing wave equation to axisymmetric or 3D cases, different approximations are implemented by using the frequency and/or the angle of incidence as parameters of interest. This results in boundary conditions called atmospheric radiation boundary conditions (ARBC) which are tested in ideal and realistic configurations. These ARBCs deliver accurate results and reduce the computational burden by a factor of two in helioseismology applications.

scholarly journals Problem with oblique derivative and mixed boundary conditions on the diameter for the Helmholtz equation in a semidisk

2018 ◽  
Author(s):  
N. Kapustin ◽  
E. Moiseev ◽  
A. Polosin ◽  
N. Popivanov
Keyword(s):  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions

scholarly journals A computing method for bending problem of thin plate on Pasternak foundation

2020 ◽  
Vol 12 (7) ◽  
pp. 168781402093933
Author(s):  
Jiarong Gan ◽  
Hong Yuan ◽  
Shanqing Li ◽  
Qifeng Peng ◽  
Huanliang Zhang
Keyword(s):  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Integral Equations ◽  
Thin Plate ◽  
Laplace Equation ◽  
Two Dimensional ◽  
Pasternak Foundation ◽  
Helmholtz Operator ◽  
Numerical Computing ◽  
Matlab Programming

The governing equation of the bending problem of simply supported thin plate on Pasternak foundation is degraded into two coupled lower order differential equations using the intermediate variable, which are a Helmholtz equation and a Laplace equation. A new solution of two-dimensional Helmholtz operator is proposed as shown in Appendix 1. The R-function and basic solutions of two-dimensional Helmholtz operator and Laplace operator are used to construct the corresponding quasi-Green function. The quasi-Green’s functions satisfy the homogeneous boundary conditions of the problem. The Helmholtz equation and Laplace equation are transformed into integral equations applying corresponding Green’s formula, the fundamental solution of the operator, and the boundary condition. A new boundary normalization equation is constructed to ensure the continuity of the integral kernels. The integral equations are discretized into the nonhomogeneous linear algebraic equations to proceed with numerical computing. Some numerical examples are given to verify the validity of the proposed method in calculating the problem with simple boundary conditions and polygonal boundary conditions. The required results are obtained through MATLAB programming. The convergence of the method is discussed. The comparison with the analytic solution shows a good agreement, and it demonstrates the feasibility and efficiency of the method in this article.

New full-wave phase-shift approach to solve the Helmholtz acoustic wave equation for modeling

Geophysics ◽  
2012 ◽  
Vol 77 (1) ◽  
pp. T11-T27 ◽  
Author(s):  
Kaushik Maji ◽  
Fuchun Gao ◽  
Sameera K. Abeykoon ◽  
Donald J. Kouri
Keyword(s):  
Integral Equation ◽  
Wave Equation ◽  
Phase Shift ◽  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Initial Conditions ◽  
Evanescent Waves ◽  
Absorbing Boundary Conditions ◽  
Operator Technique ◽  
Physical Boundary

We have developed a method of solving the Helmholtz equation based on a new way to generalize the “one-way” wave equation, impose correct boundary conditions, and eliminate exponentially growing evanescent waves. The full two-way nature of the Helmholtz equation is included, but the equation is converted into a pseudo one-way form in the framework of a generalized phase-shift structure consisting of two coupled first-order partial differential equations for wave propagation with depth. A new algorithm, based on the particular structure of the coupling between [Formula: see text] and [Formula: see text], is introduced to treat this problem by an explicit approach. More precisely, in a depth-marching strategy, the wave operator is decomposed into the sum of two matrices: The first one is a propagator in a reference velocity medium, whereas the second one is a perturbation term which takes into account the vertical and lateral variation of the velocity. The initial conditions are generated by solving the Lippmann-Schwinger integral equation formally, in a noniterative fashion. The approach corresponds essentially to “factoring out” the physical boundary conditions, thereby converting the inhomogeneous Lippmann-Schwinger integral equation of the second kind into a Volterra integral equation of the second kind. This procedure supplies artificial boundary conditions, along with a rigorous method for converting these solutions to those satisfying the correct, Lippmann-Scwinger (physical) boundary conditions. To make the solution numerically stable, the Feshbach projection operator technique is used to remove only the nonphysical exponentially growing evanescent waves, while retaining the exponentially decaying evanescent waves, along with all propagating waves. Suitable absorbing boundary conditions are implemented to deal with reflection or wraparound from boundaries. At the end, the Lippmann-Schwinger solutions are superposed to produce time snapshots of the propagating wave.

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