Finite element calculation of wave propagation and excitation in periodic piezoelectric systems


ABSTRACT Many sensors and actuators in technical systems consist of quasiperiodic structures which are constructed by successive repetition of a base cell. Typical examples are piezoelectric composites used as ultrasonic transducers or surface acoustic wave (SAW) devices utilized in telecommunication systems. The precise numerical simulation of such devices including all physical effects is currently beyond the capacity of high end computation. Therewith, we have to restrict the numerical analysis to the periodic substructure and have to introduce special boundaries to account for the periodicity. Due to the fact, that wave propagation phenomena have to be considered for SAW applications the peri-odic boundary condition (PBC) has to be able to model each possible mode within the periodic structure. That means this condition must hold for each phase difference existing at the periodic boundaries. To fulfill this difficult criteria we have introduced the Floquet theorem to the PBCs offering two different solution strategies. The first method leads to a quadratic eigenvalue problem. Therein, a huge matrix in-cluding all nodes not belonging to the periodic boundaries has to be inverted due to a Schur-Complement Hofer et al. formulation. The application of the second method results in a general non symmetric eigenvalue prob-lem including the inner and the nodes at one periodic boundary. These techniques consider every kind of propagating mode automatically. The advantage of the latter scheme is that the matrices keep sparse. With the use of an efficient eigensolver utilizing the Arnoldi method, which calculates only the needed eigenvalues and takes advantage of the sparsity of the matrices, the second approach is much faster as the scheme with the Schur-Complement which has to deal with dense matrices . In the first part of this paper we describe the basic theory of our new PBC methods. In the second part we show simulation results for the eigensolution of a periodic SAW structure which can be expressed efficiently in a dispersion diagram. Such diagrams give valuable information to SAW designers such as wave velocity and reflectivity of electrodes and help them to analyze the interaction of surface acoustic waves (SAW) or leaky surface acoustic waves (LSAW) with radiating bulk waves. Finally, we calculate the charge distribution of SAW structures by solving the inhomogeneous piezoelectric partial differential equations with incorporated PBCs. Therewith, we determine the electrical admittance which character-izes the electrical behavior of a SAW device. Further on, the computation of a voltage excitation on arbitrary electrodes will be demonstrated using our newly developed PBCs.

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    ABSTRACT: Surface acoustic wave filters are widely used for frequency filtering in telecommunications. These devices mainly consist of a piezoelectric substrate with periodically arranged electrodes on the surface. The periodic structure of the electrodes subdivides the frequency domain into stop-bands and pass-bands. This means only piezoelectric waves excited at frequencies belonging to the pass-band-region can pass the devices undamped. The goal of the presented work is the numerical calculation of so-called “dispersion diagrams”, the relation between excitation frequency and a complex propagation parameter. The latter describes damping factor and phase shift per electrode. The mathematical model is governed by two main issues, the underlying periodic structure and the indefinite coupled field problem due to piezoelectric material equations. Applying Bloch-Floquet theory for infinite periodic geometries yields a unit-cell problem with quasi-periodic boundary conditions. We present two formulations for a frequency-dependent eigenvalue problem describing the dispersion relation. Reducing the unit-cell problem only to unknowns on the periodic boundary results in a small-sized quadratic eigenvalue problem which is solved by QZ-methods. The second method leads to a large-scaled generalized non-hermitian eigenvalue problem which is solved by Arnoldi methods. The effect of periodic perturbations in the underlying geometry is confirmed by numerical experiments. Moreover, we present simulations of high frequency SAW- filter structures as used in TV-sets and mobile phones.
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    ABSTRACT: The characteristic of wave propagation in finite elastic solids is a major issue for surface acoustic wave (SAW) devices, which are designed based on accurate analysis. As a result, it is essential in engineering applications to analyze the SAW propagation in finite piezoelectric solids with the objective of revealing the influence of structural changes. As the finite element analysis of solids is already sophisticated, the problem of SAW analysis is reduced to the calculation of eigenvalues of extremely large linear equations that involves hundreds of thousands or even millions of degree of freedom because of the higher frequency. The operations of large scale matrices need high computing speed and large memory. In fact, these demands are beyond the capacity of widely available computing resources. For these reasons, we need to make use of computer clusters and parallel finite element method to improve the computing efficiency and expand applications. Based on our existing finite element program, the PARPACK package is used to compute eigenvalues in the specified range; the compressed sparse row (CSR) storage format is used to replace the Symmetric Skyline (SSK) storage method; the PETSc package is used to solve linear equations. The program is parallelized on a Linux cluster to utilize the computing power. The parallel FEM program with 3D elements is used to analyze the wave propagation in an actual SAW device model with interdigital transducers.
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    ABSTRACT: Microfluidic biochips are biochemical laboratories on the microscale that are used for genotyping and sequencing in genomics, protein profiling in proteomics, and cytometry in cell analysis. There are basically two classes of such biochips: active devices, where the solute transport on a network of channels on the chip surface is realized by external forces, and passive chips, where this is done using a specific design of the geometry of the channel network. Among the active biochips, current interest focuses on devices whose operational principle is based on piezoelectrically driven surface acoustic waves (SAWs) generated by interdigital transducers placed on the chip surface. In this paper, we are concerned with the numerical simulation of such piezoelectrically agitated SAWs relying on a mathematical model that describes the coupling of the underlying piezoelectric and elastomechanical phenomena. Since the interdigital transducers usually operate at a fixed frequency, we focus on the time-harmonic case. Its variational formulation gives rise to a generalized saddle point problem for which a Fredholm alternative is shown to hold true. The discretization of the time-harmonic surface acoustic wave equations is taken care of by continuous, piecewise polynomial finite elements with respect to a nested hierarchy of simplicial triangulations of the computational domain. The resulting algebraic saddle point problems are solved by blockdiagonally preconditioned iterative solvers with preconditioners of BPX-type. Numerical results are given both for a test problem documenting the performance of the iterative solution process and for a realistic SAW device illustrating the properties of SAW propagation on piezoelectric materials.
    Computing and Visualization in Science 08/2007; 10(3):145-161.

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