18 research outputs found
Refactorization of Cauchy's method: a second-order partitioned method for fluid-thick structure interaction problems
This work focuses on the derivation and the analysis of a novel,
strongly-coupled partitioned method for fluid-structure interaction problems.
The flow is assumed to be viscous and incompressible, and the structure is
modeled using linear elastodynamics equations. We assume that the structure is
thick, i.e., modeled using the same number of spatial dimensions as fluid. Our
newly developed numerical method is based on generalized Robin boundary
conditions, as well as on the refactorization of the Cauchy's one-legged
`theta-like' method, written as a sequence of Backward Euler-Forward Euler
steps used to discretize the problem in time. This family of methods,
parametrized by theta, is B-stable for any theta in [0.5,1] and second-order
accurate for theta=0.5+O(tau), where tau is the time step. In the proposed
algorithm, the fluid and structure subproblems, discretized using the Backward
Euler scheme, are first solved iteratively until convergence. Then, the
variables are linearly extrapolated, equivalent to solving Forward Euler
problems. We prove that the iterative procedure is convergent, and that the
proposed method is stable provided theta in [0.5,1]. Numerical examples, based
on the finite element discretization in space, explore convergence rates using
different values of parameters in the problem, and compare our method to other
strongly-coupled partitioned schemes from the literature. We also compare our
method to both a monolithic and a non-iterative partitioned solver on a
benchmark problem with parameters within the physiological range of blood flow,
obtaining an excellent agreement with the monolithic scheme
Analysis of Nonlinear Spectral Eddy-Viscosity Models of Turbulence
Fluid turbulence is commonly modeled by the Navier-Stokes equations with a large Reynolds number. However, direct numerical simulations are not possible in practice, so that turbulence modeling is introduced. We study artificial spectral viscosity models that render the simulation of turbulence tractable. We show that the models are well posed and have solutions that converge, in certain parameter limits, to solutions of the Navier-Stokes equations. We also show, using the mathematical analyses, how effective choices for the parameters appearing in the models can be made. Finally, we consider temporal discretizations of the models and investigate their stability. © 2009 Springer Science+Business Media, LLC
Analysis of an optimal control problem for the three-dimensional coupled modified Navier–Stokes and Maxwell equations
AbstractThe mathematical formulation and analysis of an optimal control problem associated with a viscous, incompressible, electrically conducting fluid in a bounded three-dimensional domain with fixed perfectly conducting boundaries is considered. The objective of control is the matching of the velocity and magnetic fields to given target fields; control is effected through distributed mechanical force and current controls. The existence of optimal solutions is shown, the Gâteaux differentiability for the magnetohydrodynamic system with respect to controls is proved, and the optimality system is obtained
High accuracy method for magnetohydrodynamics system in elsasser variables
© 2015 by De Gruyter. A method has been developed recently by the third author, that allows for decoupling of the evolutionary full magnetohydrodynamics (MHD) system in the Elsässer variables. The method entails the implicit discretization of the subproblem terms and the explicit discretization of coupling terms, and was proven to be unconditionally stable. In this paper we build on that result by introducing a high-order accurate deferred correction method, which also decouples the MHD system. We perform the full numerical analysis of the method, proving the unconditional stability and second order accuracy of the two-step method. We also use a test problem to verify numerically the claimed convergence rate
Theory of the NS-omega Model
We study a recent regularization of the Navier-Stokes equations, the NS-ω model. This model has similarities to the NS-α model, but its structure is more amenable to be used as a basis for numerical simulations of turbulent flows. In this report we present the model and prove existence and uniqueness of strong solutions as well as convergence (modulo a subsequence) to a weak solution of the Navier-Stokes equations as the averaging radius decreases to zero. We then apply turbulence phenomenology to the model to obtain insight into its prediction
Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations
We consider an uncoupled, modular regularization algorithm for approximation of the
Navier-Stokes equations. The method is: Step 1.1: Advance the NSE
one time step, Step 1.1: Regularize to obtain the approximation at
the new time level. Previous analysis of this approach has been for specific time stepping
methods in Step 1.1 and simple stabilizations in Step 1.1. In this report we extend the mathematical support for uncoupled,
modular stabilization to (i) the more complex and better performing BDF2 time
discretization in Step 1.1, and (ii) more general (linear or
nonlinear) regularization operators in Step 1.1. We give a complete
stability analysis, derive conditions on the Step 1.1
regularization operator for which the combination has good stabilization effects,
characterize the numerical dissipation induced by Step 1.1, prove
an asymptotic error estimate incorporating the numerical error of the method used in Step
1.1 and the regularizations consistency error in Step 1.1 and provide numerical tests