Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics

Microfluidic biochips are devices that are designed for high throughput screening and hybridization in genomics, protein profiling in proteomics, and cell analysis in cytometry. They are used in clinical diagnostics, pharmaceutics and forensics. The biochips consist of a lithographically produced network of channels and reservoirs on top of a glass or plastic plate. The idea is to transport the injected DNA or protein probes in the amount of nanoliters along the network to a reservoir where the chemical analysis is performed. Conventional biochips use external pumps to generate the fluid flow within the network. A more precise control of the fluid flow can be achieved by piezoelectrically agitated surface acoustic waves (SAW) generated by interdigital transducers on top of the chip, traveling across the surface and entering the fluid filled channels. The fluid and SAW interaction can be described by a mathematical model which consists of a coupling of the piezoelectric equations and the compressible Navier-Stokes equations featuring processes that occur on vastly different time scales. In this contribution, we follow a homogenization approach in order to cope with the multiscale behavior of the coupled system that enables a separate treatment of the fast and slowly varying processes. The resulting model equations are the basis for the numerical simulation which is taken care of by implicit time stepping and finite element discretizations in space. Finally, the need for a better efficiency and cost effectiveness of the SAW driven biochips in the sense of a significant speed-up and more favorable reliability of the hybridization process requires an improved design which will also be addressed in this contribution. In particular, the challenge to deal with the resulting large scale optimal control and optimization problems can be met by the application of projection based model reduction techniques.

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“…It is well-known that the the weak formulation of (60a)-(60d) admits a unique solution (cf., e.g., [78]). …”

Section: Path-following Barrier Methodsmentioning

confidence: 99%

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

Antil¹,

Hoppe²,

Linsenmann³

et al. 2012

Progress in Computational Physics (PiCP) Vol: 2 Coupled Fluid Flow in Energy, Biology and Environmental Research

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“…More details can be found in the books [2], [5], [7] or [12], [16], [22]. Let E be a reflexive Banach space.…”

Section: Basic Assumptionsmentioning

confidence: 99%

Reliable Solution of Parabolic Obstacle Problems with Respect to Uncertain Data

Lovíšek

2003

Applications of Mathematics

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“…A large number of such problems often arise in optimal structural design (see [1,2] for more details). For example, in order to widen a resonance-free frequency interval of some structure it is sufficient to maximize either its first natural frequency or the difference between the corresponding adjacent frequencies.…”

Section: Introductionmentioning

confidence: 99%

“…Optimization problems for eigenvalues of elliptic operators have been considered by many authors (see [1][2][3][4][5][6][7][8][9]). For surveys on such problems we refer the reader to [1][2][3].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Existence Criteria in Some Extremum Problems Involving Eigenvalues of Elliptic Operators

Goncharov¹,

Гончаров²

2016

J. Sib. Fed. Univ., Math. phys.

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Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics

Cited by 28 publications

References 36 publications

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

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

Reliable Solution of Parabolic Obstacle Problems with Respect to Uncertain Data

Existence Criteria in Some Extremum Problems Involving Eigenvalues of Elliptic Operators

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