## Fachbereich Mathematik

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- Doctoral Thesis (3) (remove)

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The aim of the thesis is the numerical investigation of saturated, stationary, incompressible Newtonian flow in porous media when inertia is not negligible. We focus our attention to the Navier-Stokes system with two pressures derived by two-scale homogenization. The thesis is subdivided into five Chapters. After the introductory remarks on porous media, filtration laws and upscaling methods, the first chapter is closed by stating the basic terminology and mathematical fundamentals. In Chapter 2, we start by formulating the Navier-Stokes equations on a periodic porous medium. By two-scale expansions of the velocity and pressure, we formally derive the Navier-Stokes system with two pressures. For the sake of completeness, known existence and uniqueness results are repeated and a convergence proof is given. Finally, we consider Stokes and Navier-Stokes systems with two pressures with respect to their relation to Darcy's law. Chapter 3 and Chapter 4 are devoted to the numerical solution of the nonlinear two pressure system. Therefore, we follow two approaches. The first approach which is developed in Chapter 3 is based on a splitting of the Navier-Stokes system with two pressures into micro and macro problems. The splitting is achieved by Taylor expanding the permeability function or by discretely computing the permeability function. The problems to be solved are a series of Stokes and Navier-Stokes problems on the periodicity cell. The Stokes problems are solved by an Uzawa conjugate gradient method. The Navier-Stokes equations are linearized by a least-squares conjugate gradient method, which leads to the solution of a sequence of Stokes problems. The macro problem consists of solving a nonlinear uniformly elliptic equation of second order. The least-squares linearization is applied to the macro problem leading to a sequence of Poisson problems. All equations will be discretized by finite elements. Numerical results are presented at the end of Chapter 3. The second approach presented in Chapter 4 relies on the variational formulation in a certain Hilbert space setting of the Navier-Stokes system with two pressures. The nonlinear problem is again linearized by the least-squares conjugate gradient method. We obtain a sequence of Stokes systems with two pressures. For the latter systems, we propose a fast solution method which relies on pre-computing Stokes systems on the periodicity cell for finite element basis functions acting as right hand sides. Finally, numerical results are discussed. In Chapter 5 we are concerned with modeling and simulation of the pressing section of a paper machine. We state a two-dimensional model of a press nip which takes into account elasticity and flow phenomena. Nonlinear filtration laws are incorporated into the flow model. We present a numerical solution algorithm and the chapter is closed by a numerical investigation of the model with special focus on inertia effects.

Composite materials are used in many modern tools and engineering applications and
consist of two or more materials that are intermixed. Features like inclusions in a matrix
material are often very small compared to the overall structure. Volume elements that
are characteristic for the microstructure can be simulated and their elastic properties are
then used as a homogeneous material on the macroscopic scale.
Moulinec and Suquet [2] solve the so-called Lippmann-Schwinger equation, a reformulation of the equations of elasticity in periodic homogenization, using truncated
trigonometric polynomials on a tensor product grid as ansatz functions.
In this thesis, we generalize their approach to anisotropic lattices and extend it to
anisotropic translation invariant spaces. We discretize the partial differential equation
on these spaces and prove the convergence rate. The speed of convergence depends on
the smoothness of the coefficients and the regularity of the ansatz space. The spaces of
translates unify the ansatz of Moulinec and Suquet with de la Vallée Poussin means and
periodic Box splines, including the constant finite element discretization of Brisard and
Dormieux [1].
For finely resolved images, sampling on a coarser lattice reduces the computational
effort. We introduce mixing rules as the means to transfer fine-grid information to the
smaller lattice.
Finally, we show the effect of the anisotropic pattern, the space of translates, and the
convergence of the method, and mixing rules on two- and three-dimensional examples.
References
[1] S. Brisard and L. Dormieux. “FFT-based methods for the mechanics of composites:
A general variational framework”. In: Computational Materials Science 49.3 (2010),
pp. 663–671. doi: 10.1016/j.commatsci.2010.06.009.
[2] H. Moulinec and P. Suquet. “A numerical method for computing the overall response
of nonlinear composites with complex microstructure”. In: Computer Methods in
Applied Mechanics and Engineering 157.1-2 (1998), pp. 69–94. doi: 10.1016/s00457825(97)00218-1.

The thesis at hand deals with the numerical solution of multiscale problems arising in the modeling of processes in fluid and thermo dynamics. Many of these processes, governed by partial differential equations, are relevant in engineering, geoscience, and environmental studies. More precisely, this thesis discusses the efficient numerical computation of effective macroscopic thermal conductivity tensors of high-contrast composite materials. The term "high-contrast" refers to large variations in the conductivities of the constituents of the composite. Additionally, this thesis deals with the numerical solution of Brinkman's equations. This system of equations adequately models viscous flows in (highly) permeable media. It was introduced by Brinkman in 1947 to reduce the deviations between the measurements for flows in such media and the predictions according to Darcy's model.