Adjoint-based optimization of the actuator velocity profile in an inkjet print head

We consider the thermo-viscous acoustic flow inside the narrow channel and nozzle of an inkjet print head. We define a cost function to be the sum of the acoustic energy in the channel and the surface energy of the spherical cap of ink at the end of the nozzle. We derive the adjoint equations and obtain the sensitivity of this cost function to boundary forcing from the piezo-electric actuator opposite the nozzle. We use this forcing to eliminate residual oscillations after a droplet is ejected. We use a gradient-based optimization algorithm to find the time-varying boundary forcing that minimizes the cost function at various final times and for geometries with increasing complexity. For all geometries, the actuator first extracts fluid so that the ink/air interface becomes flat. This unavoidably sends an acoustic wave upstream, which reflects off the inlet manifold. The actuator subsequently moves to absorb this returning wave without reflection. The optimal boundary forcing and the final energy depend on the channel length, the actuator length, the forcing’s temporal resolution, and the available optimization time. The minimum time required to dampen residual oscillations is the time taken for waves to travel from the actuator to the inlet and back. For times greater than this, the total energy inside the microchannel can be reduced by a factor of 1000 compared to the uncontrolled case. This method is general and can be applied to other cost functions and initial conditions. Successful application of this method could lead more repeatable droplets at higher ejection frequencies.