Mechanics as a Convex Quadratic Programming Problem With Application to Incompressible Flows

Gauss’s principle of least constraint transforms a dynamics problem into a pure minimization framework. We show that this minimization problem is a Strongly Convex Quadratic Programming (SCQP) problem whose necessary condition is Newton’s equation of motion. The principle of minimum pressure gradient (PMPG) is to incompressible flows what Gauss’s principle is to particle and rigid-body dynamics. The principle asserts that an incompressible flow evolves from one instant to another by minimizing the $L^{2}$ -norm of the pressure gradient force. That is, Navier-Stokes equation is the first-order necessary condition for minimizing the pressure gradient cost. Here, we show that the PMPG transforms the incompressible fluid mechanics problem into a pure minimization framework, allowing one to determine the evolution of the flow field by solely focusing on minimizing the cost—without directly invoking the Navier-Stokes equation. Moreover, we formulate the resulting minimization problems from Gauss’s principle and the PMPG as a SCQP problem—one of the most computationally tractable classes in nonlinear optimization, which has a rich literature with many efficient algorithms. This formulation eliminates the daunting task of solving the Poisson equation in pressure at each time step. Rather, it replaces it with a SCQP problem.