Swimming of an assembly of rigid spheres at low Reynolds number

A matrix formulation is derived for the calculation of the swimming speed and the power required for swimming of an assembly of rigid spheres immersed in a viscous fluid of infinite extent. The spheres may have arbitrary radii and may interact with elastic forces. The analysis is based on the Stokes mobility matrix of the set of spheres, defined in low Reynolds number hydrodynamics. For small amplitude swimming optimization of the swimming speed at given power leads to an eigenvalue problem. The method allows straightforward calculation of the swimming performance of structures modeled as assemblies of interacting rigid spheres.Comment: 14 pages, 5 figure

Artificial boundary conditions for stationary Navier-Stokes flows past bodies in the half-plane

We discuss artificial boundary conditions for stationary Navier-Stokes flows past bodies in the half-plane, for a range of low Reynolds numbers. When truncating the half-plane to a finite domain for numerical purposes, artificial boundaries appear. We present an explicit Dirichlet condition for the velocity at these boundaries in terms of an asymptotic expansion for the solution to the problem. We show a substantial increase in accuracy of the computed values for drag and lift when compared with results for traditional boundary conditions. We also analyze the qualitative behavior of the solutions in terms of the streamlines of the flow. The new boundary conditions are universal in the sense that they depend on a given body only through one constant, which can be determined in a feed-back loop as part of the solution process

An inverse problem in Fluid Mechanics applied in Biomedicine

In this thesis, new advances are presented in inverse problems of Fluid Mechanics in steady state, with direct applications in the recovery of domain deformations and obstacles, and whose purpose is to contribute to the detection of aortic valve conditions (such as insufficiency or stenosis). The first main result of this thesis is an asymptotic approximation result between the obstacle detection problems and the recovery of a non-negative permeability parameter that assumes significantly large values in the regions with obstacles or the value 0 in other parts. This result is supported by numerical tests that confirm the approximation result. The second result of this thesis presents a logarithmic inequality for the identification problem of the permeability parameter on Navier-Stokes equations from local measurements of fluid velocity. Numerical tests on the recovery of smooth and non-smooth parameters by a minimization problem and adaptive refinement algorithms are also included. Finally, a parameter identification problem for the Oseen and Navier-Stokes equations is studied in order to recover a permeability parameter from local or global measurements of the fluid velocity. Several numerical experiments using Navier-Stokes flow illustrate the applicability of the method, for the localization of a simulated 2D cardiac valve from synthetic MRI and also recovering of the permeability parameter from 3D synthetic MRI

Well-posedness and Robust Preconditioners for the Discretized Fluid-Structure Interaction Systems

In this paper we develop a family of preconditioners for the linear algebraic systems arising from the arbitrary Lagrangian-Eulerian discretization of some fluid-structure interaction models. After the time discretization, we formulate the fluid-structure interaction equations as saddle point problems and prove the uniform well-posedness. Then we discretize the space dimension by finite element methods and prove their uniform well-posedness by two different approaches under appropriate assumptions. The uniform well-posedness makes it possible to design robust preconditioners for the discretized fluid-structure interaction systems. Numerical examples are presented to show the robustness and efficiency of these preconditioners.Comment: 1. Added two preconditioners into the analysis and implementation 2. Rerun all the numerical tests 3. changed title, abstract and corrected lots of typos and inconsistencies 4. added reference

Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed

Partitioned Algorithms for Fluid-Structure Interaction Problems in Haemodynamics

We consider the fluid-structure interaction problem arising in haemodynamic applications. The finite elasticity equations for the vessel are written in Lagrangian form, while the Navier-Stokes equations for the blood in Arbitrary Lagrangian Eulerian form. The resulting three fields problem (fluid/ structure/ fluid domain) is formalized via the introduction of three Lagrange multipliers and consistently discretized by p-th order backward differentiation formulae (BDFp). We focus on partitioned algorithms for its numerical solution, which consist in the successive solution of the three subproblems. We review several strategies that all rely on the exchange of Robin interface conditions and review their performances reported recently in the literature. We also analyze the stability of explicit partitioned procedures and convergence of iterative implicit partitioned procedures on a simple linear FSI problem for a general BDFp temporal discretization

Combining Boundary-Conforming Finite Element Meshes on Moving Domains Using a Sliding Mesh Approach

For most finite element simulations, boundary-conforming meshes have significant advantages in terms of accuracy or efficiency. This is particularly true for complex domains. However, with increased complexity of the domain, generating a boundary-conforming mesh becomes more difficult and time consuming. One might therefore decide to resort to an approach where individual boundary-conforming meshes are pieced together in a modular fashion to form a larger domain. This paper presents a stabilized finite element formulation for fluid and temperature equations on sliding meshes. It couples the solution fields of multiple subdomains whose boundaries slide along each other on common interfaces. Thus, the method allows to use highly tuned boundary-conforming meshes for each subdomain that are only coupled at the overlapping boundary interfaces. In contrast to standard overlapping or fictitious domain methods the coupling is broken down to few interfaces with reduced geometric dimension. The formulation consists of the following key ingredients: the coupling of the solution fields on the overlapping surfaces is imposed weakly using a stabilized version of Nitsche's method. It ensures mass and energy conservation at the common interfaces. Additionally, we allow to impose weak Dirichlet boundary conditions at the non-overlapping parts of the interfaces. We present a detailed numerical study for the resulting stabilized formulation. It shows optimal convergence behavior for both Newtonian and generalized Newtonian material models. Simulations of flow of plastic melt inside single-screw as well as twin-screw extruders demonstrate the applicability of the method to complex and relevant industrial applications

Shape optimization of channels for incompressible flows

Název práce: Optimalizace tvaru kanálu v úlohách nestlačitelného proudění Autor: Zuzana Záhorová Katedra: Katedra numerické matematiky Vedoucí diplomové práce: Doc. Dr. Ing. Eduard Rohan e-mail vedoucího: [email protected] Abstrakt: V předložené práci studujeme problém tvarové optimalizace pro úlohy vnitřního proudění ve 3D. Uvažováno je laminární, nestlačitelné, stacionární proudění popsané Navier- Stokesovými rovnicemi. Jsou popsány stabilizace Navier-Stokesových rovnic potřebné pro řešení úloh s nízkou viskozitou. Předloženy jsou teoretické poznatky týkající se problému tvarové optimalizace včetně důkazu existence řešení. Je popsána adjungovaná metoda pro řešení optimalizační úlohy. Odvozena je analytická analýza citlivosti. Představujeme postupy využité při výpočtech a numerický software pro řešení optimalizačních úloh. Jsou prezentovány výsledky pro stabilizované i nestabilizované řešení Navier-Stokesových rovnic. Představíme výsledky zahrnující lineární omezení geometrie oblasti. Klíčová slova: Nestlačitelné Navier-Stokesovy rovnice, SUPG/PSPG stabilizace, Adjun- govaná metoda, Analýza citlivosti Title: Shape optimization of channels for incompressible flows Author: Zuzana Záhorová Department:...Název práce: Optimalizace tvaru kanálu v úlohách nestlačitelného proudění Autor: Zuzana Záhorová Katedra: Katedra numerické matematiky Vedoucí diplomové práce: Doc. Dr. Ing. Eduard Rohan e-mail vedoucího: [email protected] Abstrakt: V předložené práci studujeme problém tvarové optimalizace pro úlohy vnitřního proudění ve 3D. Uvažováno je laminární, nestlačitelné, stacionární proudění popsané Navier- Stokesovými rovnicemi. Jsou popsány stabilizace Navier-Stokesových rovnic potřebné pro řešení úloh s nízkou viskozitou. Předloženy jsou teoretické poznatky týkající se problému tvarové optimalizace včetně důkazu existence řešení. Je popsána adjungovaná metoda pro řešení optimalizační úlohy. Odvozena je analytická analýza citlivosti. Představujeme postupy využité při výpočtech a numerický software pro řešení optimalizačních úloh. Jsou prezentovány výsledky pro stabilizované i nestabilizované řešení Navier-Stokesových rovnic. Představíme výsledky zahrnující lineární omezení geometrie oblasti. Klíčová slova: Nestlačitelné Navier-Stokesovy rovnice, SUPG/PSPG stabilizace, Adjun- govaná metoda, Analýza citlivosti Title: Shape optimization of channels for incompressible flows Author: Zuzana Záhorová Department:...Department of Numerical MathematicsKatedra numerické matematikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult