Numerical simulation of piezoelectrically agitated surface acoustic waves on microfluidic biochips

Abstract

Abstract. Microfluidic biochips are biochemical laboratories on the microscale that are used for genotyping and sequencing in genomics, protein profiling in proteomics, and cytometry 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 generated by interdigital transducers placed on the chip surface.In this paper, we are concerned with the numerical simulation of such piezoelectrically agitated surface acoustic waves 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 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 block-diagonally 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 surface acoustic wave device illustrating the properties of surface acoustic wave propagation on piezoelectric materials.

Numerical simulation of surface acoustic wave actuated enantiomer separation by the finite element immersed boundary method

“…Denoting by u the mechanical displacement, by σ = σ(u, E) the stress tensor, and by D = D(u, E) the dielectric displacement, and given boundary data Φ E,D on Γ E,D , the pair (u, Φ) satisfies the following initial-boundary value problem for the piezoelectric equations (cf. [16])…”

Block preconditioning for saddle point systems with indefinite (1, 1) block

“…We note that in the Oseen problem, the matrix A is nonsymmetric. Similar ideas have been independently investigated by other researchers in order to develop block preconditioners for symmetric problems in other application areas; see, e.g., [12,15].…”

“…Linear systems of the type (1) will be referred to as (generalized) "saddle point systems with indefinite (1, 1) block." Such linear systems arise in various areas of scientific computing, including the solution of eigenvalue problems in fluid mechanics [8,13] and electromagnetics [2] by shift-and-invert algorithms, and in certain time-harmonic wave propagation problems [12,15]. We emphasize that while numerous effective solution algorithms exist for the case of a positive definite or semidefinite (1, 1) block (corresponding to either β ≤ 0 or β > 0 but smaller than the real part of the eigenvalue of A of smallest magnitude), see [3,7,9], relatively little has been done for the case where the (1, 1) block is indefinite.…”

Multiscale and Multiphysics Aspects in Modeling and Simulation of Surface Acoustic Wave Driven Microfluidic Biochips

“…In particular, in section 2 we will address in detail the multiphysics and multiscale aspects with regard to a proper modeling of the operational behavior of such biochips. Section 3 is devoted to the analysis of the model equations as given by a coupled system consisting of the linearized equations of piezoelectricity and the compressible Navier-Stokes equations, whereas section 4 deals with the development and implementation of efficient algorithmic tools for the numerical simulation [3,4,47]. The operational behavior can be substantially improved by optimal design.…”

“…Generalized saddle point problems such as (47) arise in the framework of stabilized Stokes systems [92,93] or in mixed finite element approximations of boundary value problems for elliptic equations and systems [25]. We refer to [20,32,83] for basic results and to [14,23,30,38,70] for efficient iterative solution techniques including multilevel preconditioning.…”