On local nonreflecting boundary conditions for time dependent wave propagation

Chinese Annals of Mathematics (Impact Factor: 0.45). 09/2009; 30(5):589-606. DOI: 10.1007/s11401-009-0203-5


The simulation of wave phenomena in unbounded domains generally requires an artificial boundary to truncate the unbounded
exterior and limit the computation to a finite region. At the artificial boundary a boundary condition is then needed, which
allows the propagating waves to exit the computational domain without spurious reflection. In 1977, Engquist and Majda proposed
the first hierarchy of absorbing boundary conditions, which allows a systematic reduction of spurious reflection without moving
the artificial boundary farther away from the scatterer. Their pioneering work, which initiated an entire research area, is
reviewed here from a modern perspective. Recent developments such as high-order local conditions and their extension to multiple
scattering are also presented. Finally, the accuracy of high-order local conditions is demonstrated through numerical experiments.

KeywordsAbsorbing boundary conditions-Scattering problems-Wave equation
2000 MR Subject Classification65M99

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Available from: Marcus J. Grote, Jan 20, 2014
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    ABSTRACT: Capacitive micromachined ultrasonic transducers (CMUTs) offer several advantages such as wide bandwidth, high sensitivity and ease of fabrication over piezoelectric transducers. Recently, the CMUTs have emerged as an alternative technology in biomedical imaging applications. Consequently, the design and performance evaluation of CMUTs using finite element method (FEM) have become an important research field. This paper presents a method to compare, by means of a time-dependent simulation, transmission in the conventional, collapse-snapback and collapsed operation regime by a capacitive micromachined ultrasonic transducer (CMUT) using ANSYS 7.1. CMUTs were biased with a DC voltage and excited by an AC voltage. Depending on the amplitudes of these voltages relative to the device collapse and snapback voltages, they operated in the conventional, in the collapsed or in the collapse-snapback regimes. When the applied AC voltage was not negligible relative to the DC bias, it was not possible to fully characterize the dynamic behavior of a CMUT by performing harmonic analysis at the DC operating point. Instead, time- dependent analysis of the CMUT had to be performed by applying a time-varying voltage. We performed electrostatic and structural analyses sequentially to simulate the coupling between these domains. At each time step, the force on the membrane at the current membrane deformation due to the applied voltage was recalculated and this updated force was applied in the succeeding time step. A single membrane cell was immersed in a sphere of fluid to simulate the loading effect of water. The spherical boundary of the fluid was defined as an absorbing boundary to simulate an infinite medium. If the simulation time was prolonged, e.g. in order to model a steady-state response, the FLUID129 element, which used a second-order equation to approximate the absorbing boundary, became unstable. Furthermore, the approximate absorbing boundary required the radius of the surrounding sphere to be larger than 20 % of the largest wavelength of interest, which significantly increased computation time due to the large number of nodes in the model. We have implemented exact absorbing boundary conditions in our 2D axisymmetric model. This enabled us to have a highly efficient stable absorbing boundary and to reduce the radius of the fluid medium arbitrarily without loss of accuracy. Additional computation time due to the implementation of exact absorbing boundary is less than 10 % of the total computation time.
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    ABSTRACT: This paper reports on dynamic analysis of an immersed single capacitive micromachined ultrasonic transducer (CMUT) cell transmitting. A water loaded 24 μm circular silicon membrane of a transducer was modeled. The calculated collapse and snapback voltages were 80 V and 50 V, respectively. The resonance frequency, output pressure and nonlinearity of the CMUT in three regimes of operation were determined. These regimes were: a) the conventional regime in which the membrane does not make contact with the substrate, b) the collapsed regime in which the center of the membrane is in constant contact with the substrate, and c) the collapse-snapback regime in which the membrane intermittently makes contact with the substrate and releases. The average membrane displacement was compared as the CMUT was operated in these regimes. A displacement of 70 Å in the collapsed regime and 39 Å in conventional regime operation were predicted when a 5 V pulse was applied to the CMUT cell biased at 70 V. The CMUT showed a 2<sup>nd</sup> harmonic at -16 dB and -26 dB in conventional and collapsed regimes of operation, respectively. Collapse-snapback operation provided increased output pressure at the expense of a 3<sup>rd</sup> harmonic at -10 dB. Our simulations predicted that the average output pressure at the membrane could be 90 kPa/V with collapse-snapback operation compared to 4 kPa/V with conventional operation.
    Ultrasonics, 2003 IEEE Symposium on; 11/2003
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