scholarly journals Exact nonreflecting boundary conditions for exterior wave equation problems

2014 ◽  
Vol 96 (110) ◽  
pp. 103-123 ◽  
Author(s):  
Silvia Falletta ◽  
Giovanni Monegato
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Time Domain ◽  
Region Of Interest ◽  
Boundary Integral ◽  
Implicit Time ◽  
Artificial Boundary ◽  
Equation Problem ◽  
Classical Wave Equation ◽  
The Time Domain

We consider the classical wave equation problem defined on the exterior of a bounded 2D space domain, possibly having far field sources. We consider this problem in the time domain, but also in the frequency domain. For its solution we propose to associate with it a boundary integral equation (BIE) defined on an artificial boundary surrounding the region of interest. This boundary condition is nonreflecting (or transparent) for both outgoing and incoming waves and it does not have to include necessarily the problem datum supports. The problem physical domain can even be a multi-domain, defined by the union of several disjoint domains. These domains can be convex or nonconvex. This transparent boundary condition is imposed pointwise on the chosen artificial boundary; therefore, its (space collocation) discretization can be coupled with a (space) finite difference or finite element method for the associated PDE problem. In the time-domain case, a classical (explicit or implicit) time integrator is also used. We present a consistency result for the BIE discretization and a sample of the intensive numerical testing we have performed.

Time-Domain Stability of Artificial Boundary Condition Coupled with Finite Element for Dynamic and Wave Problems in Unbounded Media

2019 ◽  
Vol 16 (04) ◽  
pp. 1850099 ◽  
Author(s):  
Mi Zhao ◽  
Huifang Li ◽  
Xiuli Du ◽  
Piguang Wang
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Time Domain ◽  
The Finite Element Method ◽  
Artificial Boundary ◽  
Artificial Boundary Condition ◽  
Unbounded Media ◽  
The Time Domain ◽  
Wave Problems ◽  
Domain Stability

The finite element modeling of the dynamic and wave problems in unbounded media requires an artificial boundary condition to simulate the truncated infinite domain. The Dirichlet-to-Neumann boundary condition has been transformed from frequency to time domain by using the rational function approximation and auxiliary variable technique. It is extended to three-dimensional layer problem here. The resulting artificial boundary condition is stable itself in time domain, whereas the time-domain instability of the artificial boundary condition coupled with the finite element method is found for the foundation vibration recently and for the wave propagation here. A simple and effective method that introduces the damping proportional to the stiffness matrix in the finite element method is given to cure such coupling instability completely. The stabilized damping is so small that it does not affect the solution accuracy nearly. The numerical examples show the instability phenomenon and indicate the effectiveness of the damping method. The time-domain stability studies here can be a reference for the other artificial boundary conditions.

scholarly journals An Advance-Tracing Boundary Element Method in the Time Domain for Solving the Radiating Problem of an Arbitrary Obstacle Accelerating Randomly

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Jui-Hsiang Kao
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Time Domain ◽  
Boundary Integral ◽  
Normal Velocity ◽  
Time Step ◽  
Random Acceleration ◽  
The Time Domain ◽  
First Time ◽  
Element Method

This research develops an Advance-Tracing Boundary Element Method in the time domain to calculate the waves that radiate from an immersed obstacle moving with random acceleration. The moving velocity of the immersed obstacle is multifrequency and is projected along the normal direction of every element on the obstacle. The projected normal velocity of every element is presented by the Fourier series and includes the advance-tracing time, which is equal to a quarter period of the moving velocity. The moving velocity is treated as a known boundary condition. The computing scheme is based on the boundary integral equation in the time domain, and the approach process is carried forward in a loop from the first time step to the last. At each time step, the radiated pressure on each element is updated until obtaining a convergent result. The Advance-Tracing Boundary Element Method is suitable for calculating the radiating problem from an arbitrary obstacle moving with random acceleration in the time domain and can be widely applied to the shape design of an immersed obstacle in order to attain security and confidentiality.

Gaussian wave packets in inhomogeneous media with curved interfaces

1987 ◽  
Vol 412 (1842) ◽  
pp. 93-123 ◽  
Keyword(s):  
Wave Equation ◽  
Large Scale ◽  
Evolution Equations ◽  
Wave Packets ◽  
Present Theory ◽  
Orthogonal Direction ◽  
Time Harmonic ◽  
Method Of Solution ◽  
Classical Wave Equation ◽  
The Time Domain

Time-dependent particle-like pulses are considered as asymptotic solutions of the classical wave equation. The wave packets are localized in space with gaussian envelopes. The pulse centres propagate along the rays of the wave equation, and the envelope parameters satisfy evolution equations very similar to the ray equations for time-harmonic disturb­ances. However, the present theory contains an extra degree of freedom not found in the time-harmonic theory. Explicit results are presented for media with constant velocity gradients, and interesting new phenomena are identified. For example, a pulse that is initially long in the direction of propagation and comparatively narrow in the orthogonal direction, maintains its initial spatial orientation even as the propagation direction rotates. The reflection and transmission of a pulse incident upon an interface are also discussed. The various theoretical results are illustrated by numerical simulations. This method of solution could be very useful for fast forward modelling in large-scale structures. It is formulated explicitly in the time domain and does not suffer from unphysical singularities at caustics.

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