Adaptive boundary conditions for exterior stationary flows in three dimensions

Recently there has been an increasing interest for a better understanding of ultra low Reynolds number flows. In this context we present a new setup which allows to efficiently solve the stationary incompressible Navier-Stokes equations in an exterior domain in three dimensions numerically. The main point is that the necessity to truncate for numerical purposes the exterior domain to a finite sub-domain leads to the problem of finding so called "artificial boundary conditions" to replace the conditions at infinity. To solve this problem we provide a vector filed that describes the leading asymptotic behavior of the solution at large distances. This vector field depends explicitly on drag and lift which are determined in a self-consistent way as part of the solution process. When compared with other numerical schemes the size of the computational domain that is needed to obtain the hydrodynamic forces with a given precision is drastically reduced, which in turn leads to an overall gain in computational efficiency of typically several orders of magnitude.Comment: 17 pages, 3 tables, 11 figure

HPC-based uncertainty quantification for fluidstructure coupling in medical engineering

In recent decades biomedical studies with living probands (in vivo) and artificial experiments (in vitro) have been complemented more and more by computation and simulation (in silico). In silico techniques for medical engineering can give for example enhanced information for the diagnosis and risk stratification of cardiovascular disease, one of the most occurring causes of death in the developed countries. Other use cases for in silico methods are given by virtual prototyping and the simulation of possible surgery outcomes. High reliability is a requirement for cardiovascular diagnosis and risk stratification methods especially with surgical decision-making. Given uncertainties in the input data of a simulation, this implies a necessity to quantify the uncertainties in simulation results. Uncertainties can be propagated within a numerical simulation by methods of Uncertainty Quantification (UQ)

Second Order Adaptive Boundary Conditions for Exterior Flow Problems: Non-Symmetric Stationary Flows in Two Dimensions

Abstract.: We consider the problem of solving numerically the stationary incompressible Navier-Stokes equations in an exterior domain in two dimensions. For numerical purposes we truncate the domain to a finite sub-domain, which leads to the problem of finding so called "artificial boundary conditions” to replace the boundary conditions at infinity. To solve this problem we construct - by combining results from dynamical systems theory with matched asymptotic expansion techniques based on the old ideas of Goldstein and Van Dyke - a smooth divergence free vector field depending explicitly on drag and lift and describing the solution to second and dominant third order, asymptotically at large distances from the body. The resulting expression appears to be new, even on a formal level. This improves the method introduced by the authors in a previous paper and generalizes it to non-symmetric flows. The numerical scheme determines the boundary conditions and the forces on the body in a self-consistent way as an integral part of the solution process. When compared with our previous paper where first order asymptotic expressions were used on the boundary, the inclusion of second and third order asymptotic terms further reduces the computational cost for determining lift and drag to a given precision by typically another order of magnitud