Journal
INTERFACES AND FREE BOUNDARIES
Volume 19, Issue 1, Pages 1-30Publisher
EUROPEAN MATHEMATICAL SOC
DOI: 10.4171/IFB/375
Keywords
Electrowetting on dielectric; EWOD; contact line pinning; surface tension; sharp interface; optimal control of free boundary problems; mathematical program with equilibrium constraints; MPEC; PDE-constrained optimization; barycenter matching; trajectory tracking
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Funding
- DFG [HI 1466/5-1, SPP 1506, HI 1466/2-1]
- Research Center MATHEON through the Einstein Center for Mathematics Berlin [C-SE5]
- NSF [DMS-1521590, DMS-1411808]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1521590, 1411808] Funding Source: National Science Foundation
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A time-discrete spatially-continuous electrowetting on dielectric (EWOD) model with contact line pinning is considered as the state system in an optimal control framework. The pinning model is based on a complementarity condition. In addition to the physical variables describing velocity, pressure, and voltage, the solid-liquid-air interface, i.e., the contact line, arises as a geometric variable that evolves in time. Due to the complementarity condition, the resulting optimal control of a free boundary problem is thus a mathematical program with equilibrium constraints (MPEC) in function space. In order to cope with the geometric variable, a finite horizon model predictive control approach is proposed. Dual stationarity conditions are derived by applying a regularization procedure, exploiting techniques from PDE-constrained optimization, and then passing to the limit in the regularization parameters. Moreover, a function-space-based numerical procedure is developed by following the theoretical limit argument used in the derivation of the dual stationarity conditions. The performance of the algorithm is demonstrated by several examples; including barycenter matching and trajectory tracking.
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