## Fraunhofer (ITWM)

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- Design of pleated filters by computer simulations (2009)
- Four aspects are important in the design of hydraulic lters. We distinguish between two cost factors and two performance factors. Regarding performance, filter eciencynd lter capacity are of interest. Regarding cost, there are production considerations such as spatial restrictions, material cost and the cost of manufacturing the lter. The second type of cost is the operation cost, namely the pressure drop. Albeit simulations should and will ultimately deal with all 4 aspects, for the moment our work is focused on cost. The PleatGeo Module generates three-dimensional computer models of a single pleat of a hydraulic lter interactively. PleatDict computes the pressure drop that will result for the particular design by direct numerical simulation. The evaluation of a new pleat design takes only a few hours on a standard PC compared to days or weeks used for manufacturing and testing a new prototype of a hydraulic lter. The design parameters are the shape of the pleat, the permeabilities of one or several layers of lter media and the geometry of a supporting netting structure that is used to keep the out ow area open. Besides the underlying structure generation and CFD technology, we present some trends regarding the dependence of pressure drop on design parameters that can serve as guide lines for the design of hydraulic lters. Compared to earlier two-dimensional models, the three-dimensional models can include a support structure.

- An analysis of one regularization approach for solution of pure Neumann problem (2008)
- In this paper, the analysis of one approach for the regularization of pure Neumann problems for second order elliptical equations, e.g., Poisson’s equation and linear elasticity equations, is presented. The main topic under consideration is the behavior of the condition number of the regularized problem. A general framework for the analysis is presented. This allows to determine a form of regularization term which leads to the “natural” asymptotic of the condition number of the regularized problem with respect to mesh parameter. Some numerical results, which support theoretical analysis are presented as well. The main motivation for the presented research is to develop theoretical background for an efficient and robust implementation of the solver for pure Neumann problems for the linear elasticity equations. Such solvers usually are needed in a number of domain decomposition methods, e.g. FETI. Developed approaches are planed to be used in software, developing in ITWM, e.g. KneeMech simulation software.

- Finite volume discretization of equations describing nonlinear diffusion in Li-Ion batteries (2010)
- Numerical modeling of electrochemical process in Li-Ion battery is an emerging topic of great practical interest. In this work we present a Finite Volume discretization of electrochemical diffusive processes occurring during the operation of Li-Ion batteries. The system of equations is a nonlinear, time-dependent diffusive system, coupling the Li concentration and the electric potential. The system is formulated at length-scale at which two different types of domains are distinguished, one for the electrolyte and one for the active solid particles in the electrode. The domains can be of highly irregular shape, with electrolyte occupying the pore space of a porous electrode. The material parameters in each domain differ by several orders of magnitude and can be non-linear functions of Li ions concentration and/or the electrical potential. Moreover, special interface conditions are imposed at the boundary separating the electrolyte from the active solid particles. The field variables are discontinuous across such an interface and the coupling is highly non- linear, rendering direct iteration methods ineffective for such problems. We formulate a Newton iteration for an purely implicit Finite Volume discretization of the coupled system. A series of numerical examples are presented for different type of electrolyte/electrode configurations and material parameters. The convergence of the Newton method is characterized both as function of nonlinear material parameters as well as the nonlinearity in the interface conditions.

- Modeling of species and charge transport in Li-Ion Batteries based on non-equilibrium thermodynamics (2010)
- In order to improve the design of Li ion batteries the complex interplay of various physical phenomena in the active particles of the electrodes and in the electrolyte has to be balanced. The separate transport phenomena in the electrolyte and in the active particle as well as their coupling due to the electrochemical reactions at the interfaces between the electrode particles and the electrolyte will inuence the performance and the lifetime of a battery. Any modeling of the complex phenomena during the usage of a battery has therefore to be based on sound physical and chemical principles in order to allow for reliable predictions for the response of the battery to changing load conditions. We will present a modeling approach for the transport processes in the electrolyte and the electrodesbased on non-equilibrium thermodynamics and transport theory. The assumption of local charge neutrality, which is known to be valid in concentrated electrolytes, is explicitly used to identify the independent thermodynamic variables and uxes. The theory guarantees strictly positive entropy production. Dierences to other theories will be discussed.

- Multigrid – adaptive local refinement solver for incompressible flows (2003)
- A non-linear multigrid solver for incompressible Navier-Stokes equations, exploiting finite volume discretization of the equations, is extended by adaptive local refinement. The multigrid is the outer iterative cycle, while the SIMPLE algorithm is used as a smoothing procedure. Error indicators are used to define the refinement subdomain. A special implementation approach is used, which allows to perform unstructured local refinement in conjunction with the finite volume discretization. The multigrid - adaptive local refinement algorithm is tested on 2D Poisson equation and further is applied to a lid-driven flows in a cavity (2D and 3D case), comparing the results with bench-mark data. The software design principles of the solver are also discussed.

- On upscaling heat conductivity for a class of industrial problems (2007)
- Calculating effective heat conductivity for a class of industrial problems is discussed. The considered composite materials are glass and metal foams, fibrous materials, and the like, used in isolation or in advanced heat exchangers. These materials are characterized by a very complex internal structure, by low volume fraction of the higher conductive material (glass or metal), and by a large volume fraction of the air. The homogenization theory (when applicable), allows to calculate the effective heat conductivity of composite media by postprocessing the solution of special cell problems for representative elementary volumes (REV). Different formulations of such cell problems are considered and compared here. Furthermore, the size of the REV is studied numerically for some typical materials. Fast algorithms for solving the cell problems for this class of problems, are presented and discussed.

- On numerical upscaling for flows in heterogeneous porous media (2007)
- A numerical upscaling approach, NU, for solving multiscale elliptic problems is discussed. The main components of this NU are: i) local solve of auxil- iary problems in grid blocks and formal upscaling of the obtained re sults to build a coarse scale equation; ii) global solve of the upscaled coarse scale equation; and iii) reconstruction of a fine scale solution by solving local block problems on a dual coarse grid. By its structure NU is similar to other methods for solving multiscale elliptic problems, such as the multiscale finite element method, the multiscale mixed finite element method, the numerical subgrid upscaling method, heterogeneous multiscale method, and the multiscale finite volume method. The difference with those methods is in the way the coarse scale equation is build and solved, and in the way the fine scale solution is reconstructed. Essential components of the presented here NU approach are the formal homogenization in the coarse blocks and the usage of so called multipoint flux approximation method, MPFA. Unlike the usual usage as MPFA as a discretiza- tion method for single scale elliptic problems with tensor discontinuous coefficients, we consider its usage as a part of a numerical upscaling approach. The main aim of this paper is to compare NU with the MsFEM. In particular, it is shown that the resonance effect, which limits the application of the Multiscale FEM, does not appear, or it is significantly relaxed, when the presented here numerical upscaling approach is applied.

- On approximation property of multipoint flux approximation method (2007)
- Approximation property of multipoint flux approximation (MPFA) approach for elliptic equations with discontinuous full tensor coefficients is discussed here. Finite volume discretization of the above problem is presented in the case of jump discontinuities for the permeability tensor. First order approximation for the fluxes is proved. Results from numerical experiments are presented and discussed.

- A one-dimensional model of the pressing section of a paper machine including dynamic capillary effects (2011)
- This work presents the dynamic capillary pressure model (Hassanizadeh, Gray, 1990, 1993a) adapted for the needs of paper manufacturing process simulations. The dynamic capillary pressure-saturation relation is included in a one-dimensional simulation model for the pressing section of a paper machine. The one-dimensional model is derived from a two-dimensional model by averaging with respect to the vertical direction. Then, the model is discretized by the finite volume method and solved by Newton’s method. The numerical experiments are carried out for parameters typical for the paper layer. The dynamic capillary pressure-saturation relation shows significant influence on the distribution of water pressure. The behaviour of the solution agrees with laboratory experiments (Beck, 1983).