This work develops an efficient and accurate new method for direct numerical simulation of laminar and turbulent flow past a circular cylinder with a deformable (compliant) surface. It studies the interaction of the incompressible flow with the compliant cylinder. From the outset, this is defined as an optimization problem, in which we seek to minimize aeroacoustic noise generated by dipole sound sources on the compliant surface at low Mach numbers. We build on a unique method developed in our lab for simulating turbulent flow in a channel with compliant walls. This method is accurate and efficient for large surface deformations. We adapt this method for the cylindrical polar coordinate system to study flow past a compliant cylinder. In this method, a time-dependent coordinate transformation is used to map the deformed flow domain to a regular computational domain. The governing Navier Stokes equations are formulated in the cylindrical polar form and not the contravariant form, as the latter is computationally expensive to simulate. The compliant surface is modeled by a simple spring-mass-damper system. As surface compliance is increased, a decrease in the peak lift coefficient for the compliant cylinder is observed both in the laminar 2D case at Re = 80, as well as the turbulent 3D case at Re = 300. On the other hand, the frequency of vortex shedding and the time-average drag both increase with surface compliance. This work also develops a new method for optimizing highly-constrained PDE systems by splitting up the governing equations into parallel linear programs, thus achieving scalability. It explores optimization of a single-phase fluid heat exchanger to minimize the power required to drive coolant through it by appropriately adjusting the channel width or channel porosity. The Stokes flow in the heat exchanger is modeled as a resistor network, while the flow rate and pressures in the flow are analogous to currents and voltages in the resistor network. The method developed and demonstrated on the resistor network problem extends naturally to the optimization of the variable channel width/porosity distribution in the heat exchanger model.