Article

# On local nonreflecting boundary conditions for time dependent wave propagation

Chinese Annals of Mathematics (Impact Factor: 0.5). 01/2009; 30(5):589-606. DOI: 10.1007/s11401-009-0203-5

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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. - [Show abstract] [Hide abstract]

**ABSTRACT:**Energy transmitted along a waveguide decays less rapidly than in an unbounded medium. In this paper we study the efficiency of a PML in a time-dependent waveguide governed by the scalar wave equation. A straight forward application of a Neumann boundary condition can degrade accuracy in computations. To ensure accuracy, we propose extensions of the boundary condition to an auxiliary variable in the PML. We also present analysis proving stability of the constant coefficient PML, and energy estimates for the variable coefficients case. In the discrete setting, the modified boundary conditions are crucial in deriving discrete energy estimates analogous to the continuous energy estimates. Numerical stability and convergence of our numerical method follows. Finally we give a number of numerical examples, illustrating the stability of the layer and the high order accuracy of our proposed boundary conditions.Journal of Scientific Computing 12/2012; 53(3). · 1.71 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider the second order wave equation in an unbounded domain and propose an advanced perfectly matched layer (PML) technique for its efficient and reliable simulation. In doing so, we concentrate on the time domain case and use the finite-element (FE) method for the space discretization. Our un-split-PML formulation requires four auxiliary variables within the PML region in three space dimensions. For a reduced version (rPML), we present a long time stability proof based on an energy analysis. The numerical case studies and an application example demonstrate the good performance and long time stability of our formulation for treating open domain problems.Journal of Computational Physics 02/2013; 235(100):407-422. · 2.14 Impact Factor

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