First-order system least squares for elastohydrodynamics with application to flow in compliant blood vessels

Mathematical modeling of compliant blood vessels generally involves the Navier-Stokes equations on the evolving fluid domain and constitutive structural equations on the tissue domain. Coupling these systems while accounting for the changing shape of the fluid domain is a major challenge in numerical simulation. Many techniques have been developed to model compliant vessels, but all suffer from disproportionate increase in computational cost as problem complexity increases (i.e., larger domains, more dimensions, and more variables.). Even the best standard methods result in computational cost that typically grows quadratically with the degrees of freedom. Using a least-squares formulation of the problem, elliptic grid generation for the changing fluid domain, and an algebraic multigrid solver for the linear system can overcome many shortcomings of standard techniques. Most notably, the computational cost of solving the problem increases linearly with the degrees of freedom and the associated functional provides an a posteriori error measurement. Least squares represents a systematic approach for formulating the original problem so that the numerical process becomes straightforward and optimal, and it avoids restrictions that often limit other methods. Results are presented for a two-dimensional test problem consisting of a Newtonian fluid with properties of blood in a linear-elastic vessel with properties of smooth muscle.