On a numerical subgrid upscaling algorithm for Stokes-Brinkman ...

O. Iliev, Z. Lakdawala, V. Starikovicius

On a numerical subgrid upscaling algorithm for Stokes-Brinkman equations

Berichte des Fraunhofer ITWM, Nr. 189 (2010)

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Prof. Dr. Dieter Prätzel-Wolters Institutsleiter Kaiserslautern, im Juni 2001

On a numerical subgrid upscaling algorithm for Stokes–Brinkman equations O. Ilieva , Z. Lakdawalaa , V. Starikoviciusb a

Fraunhofer ITWM, Fraunhofer-Platz 1, D-67663 Kaiserslautern, Germany {iliev, lakdawala}@itwm.fhg.de b Vilnius Gediminas Technical University, Vilnius, Lithuania [email protected]

Abstract This paper discusses a numerical subgrid resolution approach for solving the Stokes-Brinkman system of equations, which is describing coupled flow in plain and in highly porous media. Various scientific and industrial problems are described by this system, and often the geometry and/or the permeability vary on several scales. A particular target is the process of oil filtration. In many complicated filters, the filter medium or the filter element geometry are too fine to be resolved by a feasible computational grid. The subgrid approach presented in the paper is aimed at describing how these fine details are accounted for by solving auxiliary problems in appropriately chosen grid cells on a relatively coarse computational grid. This is done via a systematic and a careful procedure of modifying and updating the coefficients of the Stokes-Brinkman system in chosen cells. This numerical subgrid approach is motivated from one side from homogenization theory, from which we borrow the formulations for the so called cell problem, and from the other side from the numerical upscaling approaches, such as Multiscale Finite Volume, Multiscale Finite Element, etc. Results on the algorithm’s efficiency, both in terms of computational time and memory usage, are presented. Comparison with solutions on full fine grid (when possible) are presented in order to evaluate the accuracy. Advantages and limitations of the considered subgrid approach are discussed. Keywords: Stokes-Brinkman equations, subgrid approach, multiscale problems, numerical upscaling.

Preprint submitted to Elsevier

July 4, 2010

1. Introduction The demands of the industry regularly pose new challenging problems to applied mathematics. In many cases the existing algorithms do not work, and new specialized algorithms for classes of industrial problems are demanded. Developing algorithms for a class of filtration problems (filtering solid particles out of liquid) is discussed in this paper. Numerical simulations, when they can be performed, allow significant reduction in time and costs for design of new filter elements with proper flow rate - pressure drop ratio. The CFD simulations assisting this design are characterized by three peculiarities: • The filtering porous layers are usually manufactured with certain variations in the weight (i.e., in porosity and permeability), and consistently with this, accuracy of 5 to 8 per cent for the pressure drop across a filter element is the target of the simulations. • High accuracy for the flow velocity within a filter element is not required, as long as the pressure drop over the complete filter element is properly computed. • 30-36 simulations with different flow rates and different viscosities have to be performed for each geometry to evaluate the performance of the filter element at different flow and temperature conditions. Our aim is to account for these peculiarities, in order to develop efficient algorithm for this particular class of problems. In the case of liquid filtration, the flow is usually laminar, and at low and moderate temperatures it is slow (due to higher viscosity), and it is described by the Stokes-Brinkman system of equations (see, e.g., [22] and references therein): Darcy

z }| { ∂~u −1 ˜ ~u + ∇p = f~˜ − ∇ · (˜ µ∇~u) + µ ˜K ρ | {z } ∂t | {z }

(1)

Stokes

∇ · ~u = 0, where ( µ µ ˜= µef f

in Ωf in Ωp

( f~N S ˜ f~ = f~B 2

in Ωf in Ωp

−1

˜ K

( 0 = K−1

in Ωf in Ωp

Here ~u, p stand for velocity and pressure respectively, ρ, µ ˜ and K denote the density, viscosity, and the permeability tensor of the porous medium, respectively. The Stokes equations (cf. [4, 19]) govern the flow in pure fluid regions. The Brinkman equations (cf. [7]) are introduced as an extension to the Darcy model for the flow in porous media for the case of highly porous media (note, that porosity of the non-woven filtering media, which is our primary interest, is often more than 0.9). Concerning the hierarchy of the models for the porous media flow, we refer to the recent review in [10]. The Stokes-Brinkman system is used to describe the coupled flow in plain and in porous media. For its justification, see [18]. Some details on modeling and simulation of flow through oil filters using the Stokes-Brinkman and Navier-Stokes-Brinkman systems of equations can be found in [30, 15, 22]. Most of the filters are characterized by complicated shape of the filtering medium/media (e.g. pleats and/or perforated porous layer/s), and/or by complicated shape of the filter element housing (e.g. ribs, perforated inner cylinder, etc.). Some examples are illustrated in Figure 1. The existing commercial (e.g. Gambit, http://www.fluent.de) and academic (e.g. Netgen, http://www.hpfem.jku.at/netgen/) grid generators, are often unable to generate grids in such complicated domains, even when an experienced researcher is the user. In many cases, the generated grid is either of very bad quality, or the grid cannot be generated at all. Most engineers working on the design of filter elements are not experienced in grid generation techniques, and an alternative approach is needed in this case. The grid generation procedure works robustly if one restricts to voxel or brick elements, and therefore we chose this approach. A pre-processor based on the level set method (c.f. [1]) is used to process given CAD data for attaining the assembly of a filter housing in the form of a computational domain (union of voxel or brick control volumes). Different conventions can be used to characterize a control volume by a certain material type. Most commonly, the material type of its center point is assigned to the whole CV, or the dominating material type in the control volume is assigned to it. This grid generation approach is stable, and a fluid flow solver using the aforementioned grid generator is successfully used for a variety of industrial applications (c.f. [16, 13, 14]). However, in general, such types of grids contain a large number of elements: in certain cases, a very fine grid has to be used to accurately resolve all geometrical features, what in turn results in a large number of elements. 3

Figure 1: Some examples of the complex geometries and complicated shapes of the filtering media.

For the cases where the geometrical features are at different scales, various approaches for multiscale problems can be adopted to provide an increase in the efficiency of the flow algorithms. Recall that the Darcy equation can be rigorously derived as a macroscopic model for flow in porous media, in the case of periodic or statistically homogeneous porous media. The Stokes system of equations at pore scale (c.f. [27]) is the basis of these derivations. Depending on the the porosity, Allaire [6] homogenizes the Stokes problem to the Darcy or the Brinkman system. Clearly, the homogenization approach is very restrictive, i.e. it can be used only for periodic or statistically homogeneous porous media, and for the Stokes (and not for Navier-Stokes) system at the pore scale. The homogenization theory works in the case of scale separation, and it allows for a drastic reduction of the computational costs: only one auxiliary problem is solved in a periodicity cell on the fine scale, its solution is post-processed to compute the coefficients of the macroscopic equation, and the macroscopic equation is further solved at the coarse scale. The coefficients of the macroscopic equation in this case are called upscaled, homogenized, or effective coefficients. For the cases when the homogenization theory does not work, its ideas can still be imported to the numerical upscaling approaches, such as the Multiscale Finite Element Method (MsFEM) [23], the Mixed MsFEM, [3], the Multiscale Finite Volume Method [21, 8, 5] and the Subgrid Method [26, 24, 9, 17]. Another approach which serves as a building block for numerical upscaling procedures, is the volume averaging approach combined with the Representative Elementary Volume (REV) concept (cf. [12]). Unlike the homogenization theory, this approach is not based on asymptotic expansions, but on the volumetric averaging of functions and their derivatives. For example, if a block of porous medium is large enough (i.e. representative), its Darcy permeability is defined from the 4

requirement that the macroscopic pressure drop is equal to the microscopic pressure drop computed by solving the Stokes problem at the pore level. In the case of saturated flow in porous media, homogenization is studied either in connection with the mesoscopic and macroscopic Darcy models (upscaling elliptic equation with oscillating coefficients to macroscopic elliptic equation, i.e. upscaling Darcy to Darcy (cf. [29])), or in connection with impermeable porous matrix and fluid flow in the (connected) pore space, i.e. upscaling Stokes to Darcy (cf. [27]). The Authors could not find articles on upscaling Stokes-Brinkman system at mesoscale to some macroscale system of equations, except a recent work concerning numerical study of homogenization of Brinkman equations [36]. In this paper, we discuss a subgrid upscaling algorithm for the Stokes-Brinkman system of equations. We also solve Navier-Stokes regimes in certain cases, working in the context of iterative upscaling, but this issue is not discussed here. The subgrid approach was recently used in some simpler applications, namely the Darcy problem (single phase flow in porous medium) (cf. [25], [32], [9]). In the subgrid approach, one solves a problem on a coarser grid, but accounts for the unresolved geometrical features by solving local auxiliary problems on the underlying finer grid in all (or some selected) coarse cells. This paper deals with applying a similar approach in solving the incompressible Stokes-Brinkman equations in highly complex domains. The numerical solution for such systems is computationally expensive in terms of memory usage and computational time, and the subgrid approach is developed to compute a reliable pressure drop at reasonable computational costs. The remainder of the paper is organized as follows. A one-scale model (it can be also called single grid model), i.e. the Stokes-Brinkman system of equations, is described in 2. It includes a short description of the Finite Volume discretization, and the Chorin projection method employed for solving the model numerically. Section 3 is devoted to introducing the concept of quasi-porous coarse cells and to the description of the subgrid algorithm for the Stokes-Brinkman system. Due to lacking theoretical results for the upscaling of the Stokes-Brinkman system, we perform a numerical study of this approach. In Section 4, we present validation results for the developed subgrid upscaling procedure. Results from numerical simulation of industrial filters are also presented in this Section. Finally, some conclusions are drawn.

5

Figure 2: Illustration of an example of quasi-porous cell/s where the fine scale resolves the geometrical details(the white cells represent pure fluid region; yellow cells on the left picture correspond to porous media, on the right pictures the yellow cells are of so called quasi-porous type).

2. Governing equations and the single grid algorithm The fine scale system includes the Stokes-Brinkman system of equations (1) in a complicated 3D domain Ω, consisting of porous, solid and fluid subdomains, i.e. Ω = Ωp ∪ Ωs ∪ Ωf . The flow domains considered in the current work are of different geometric characteristics, as shown in Figure 1. The governing equations are solved subject to the following boundary conditions which are typical in filtration problems. At the inlet of the free flow region, a velocity profile is specified. At the outlet, zero pressure is imposed. No slip boundary conditions are imposed elsewhere on walls. The coarse scale system also employs the Stokes-Brinkman system of equations but with an upscaled permeability for specified quasi-porous coarse cells, such that the Equation (1) is replaced with u0 ˜ −1 ~u0 + ∇p0 = f~˜0 − ∇ · (˜ µ∇~u0 ) + µ ˜K ρ ∂~ ef f ∂t

(2)

where ~u0 , p0 denote the coarse scale velocity and pressure respectively. Here ˜ ef f stands for the effective upscaled permeability, and the details of its K computations are given in Section 3. Single grid algorithm The computational domain is a connected union of control volumes (CVs), where each CV is a brick. The equations are discretized by the finite volume method. A collocated arrangement of the unknowns ~u and p is used, i.e. the unknowns are assigned to the centers of CVs. The Chorin method [2] for the Navier-Stokes equations [19, 4], along with a proper modification for 6

Stokes-Brinkman case [22], is used as a projection method decoupling velocity and continuity equations (note, the Chorin projection for unsteady Stokes problem is very close to the preconditioners for unsteady Stokes problem suggested in [35]) . The fractional time step discretization scheme can be written as



(ρ~u∗ − ρ~un ) + τ (−D + B)~u∗ = τ Gpn  n+1 ∗ ρ~u − ρ~u + τ (B~un+1 − B~u∗ ) = τ (Gpn+1 − Gpn ) GT ρ~un+1 = 0

(3) (4) (5)

where D~u denote the operator corresponding to the discretized viscous terms. The particular form of these operators depend on the spatial discretizations. G and GT denote the discretized gradient and divergence operator. B~u denotes the Darcy operator in the momentum equations whereas the superscript k+1 and k denotes the new and the old time level, respectively. The sum of Equations (3) and (4) results in an implicit discretization of the momentum equations. Equation (3) is solved with respect to the velocities using the old value of the pressure gradient, thus obtaining a prediction for the velocity. To solve the second equation for pressure correction, one takes the divergence from it and uses the continuity equation. The resulting equation is a Poisson type equation for the pressure correction which is discussed in detail in [22]. 3. Subgrid Algorithm In this section, a subgrid algorithm for the Stokes-Brinkman system is discussed. The goal is to develop an algorithm which computes not only the pressure drop across a filter element on a relatively coarse grid, but also preserves the pressure drop accuracy corresponding to a finer grid. The idea is to account for a subgrid resolution on the coarse grid by solving proper local auxiliary problems on the fine grid, and using their solution for calculating the permeability of quasi-porous coarse cells. In fact, solving local problems in order to account for the fine scale features is a common approach for numerical upscaling methods, such as MsFEM, MSFV, etc. For a given computational domain, Ω, we consider a fine grid and a coarse grid. The fine grid is assumed to contain all important geometrical details, 7

but solving the Stokes-Brinkman system of equations on this grid is memory and CPU intensive, or even impossible. As discussed in the Introduction, a material type (in our case fluid, solid, or porous) is assigned to each fine grid cell during the pre-processing stage. The coarse grid has a size, such that the problem is solvable at acceptable computational costs. Each coarse cell is a union of fine grid cells. Each coarse cell is considered, and the coarse grid cells containing mixture of solid & liquid, or liquid & porous, or solid & porous, or solid & liquid & porous fine grid cells, are defined as quasiporous cells, for which effective (upscaled) permeability tensors have to be computed. It is assumed that the Stokes-Brinkman equations describe the flow on the fine grid and on the coarse grid. In the latter case, the coarse scale ˜ ef f . Currently, permeability is the effective (upscaled) permeability tensor K ˜ ef f = {K11 , K22 , K33 } is considered, the diagonal permeability tensor i.e. K and the extension to the full tensor is ongoing work. The computation of upscaled coefficients requires solving the local fine scale problem in some coarse grid cells. Two approaches are considered: • The upscaled (effective) permeability is computed for each individual quasi-porous coarse grid cell. Similar to the block-permeability upscaling procedure for single phase flow in porous media (cf. [31]), the localization boundary conditions are specified on its boundary (see discussion below for more details; possible oversampling is also discussed below); • Alternatively, certain union of coarse cells is considered as a single block for which auxiliary problem is solved on the underlying fine grid. Thereafter, the upscaled (effective) permeability is assigned to each of the coarse cells forming the block. This approach is motivated by the Representative Elementary Volume concept, where a reasonably large heterogeneous (at a fine scale) volume has to be considered, in order to assign effective (upscaled) coarse scale properties to it. The upscaling approach considered here is similar to the one used by Durlofsky et al.[11] for the elliptic pressure equation, and to the approach considered and justified in the recent article [36]. The Darcy law is used there to compute the effective (upscaled) permeability of a coarse cell from the averaged fine grid pressure and velocity. In general, different types of (localization) boundary conditions can be specified for the auxiliary problems. Periodic boundary conditions suit very well for periodic geometries. The pressure drop in one 8

of the directions, and no flow (or symmetry) in the remaining directions, are often used. In the homogenization approach considered in [6], the constant velocity at infinity is prescribed as the boundary condition for the auxiliary problem. It is well known that the choice of the localization boundary conditions plays an important role in numerical upscaling methods, and we are currently working on comparing results for different (localization) boundary conditions. This work, however, will be reported elsewhere. Currently, the following local boundary conditions are considered for the auxiliary problems for the Stokes-Brinkman system: • inflow velocity Uin at the inlet face (for the current direction); • outflow b.c. at the outlet face: p = Pout ,

∂u ∂n

= 0;

• symmetry b.c. elsewhere. The auxiliary problems are solved for each direction d. Next, the average inlet pressure Pin is computed from the local fine scale solution and is used to obtain the effective permeability Kdd of the quasi-porous cell in direction d from the Darcy law [12, 36]: Kdd =

µUin Ld , Pin − Pout

(6)

where Ld is the distance between the inlet and outlet faces. The use of this approach is motivated by the fact that we are interested in computing the accurate pressure drop of the complete filter element. By calculating the effective permeability using Equation (6), we account for the resistance of the geometrical features which were otherwise unresolved on the coarse computational grid. Note, that in [36], this approach for computing the effective permeability was carefully and systematically compared to the standard formula from the homogenization literature (e.g., [6, 27]), and a very good agreement was found. Implementation of the subgrid method Here, we discuss the Subgrid method. The algorithm consists of four main components: 1. Selection of the quasi-porous coarse cells

9

2. Solution of the auxiliary problems with a fine grid resolution on selected coarse cells with boundary conditions, as specified above 3. Computation of the effective permeabilities for the quasi-porous cells 4. Solution of the full coarse scale problem, using the calculated effective permeabilities. Step (1) includes a mapping of each coarse cell onto the underlying union of fine grid cells. If the composition of fine grid cells contains a mixture of fluid and/or porous and/or solid, it is marked with a ’quasi-porous’ flag. Alternatively, at this stage, a block of coarse cell can be specified as quasiporous. Step (2) defines the local auxiliary problem on the quasi-porous coarse cells. The same space discretization, i.e. the finite-volume method, is considered on the fine scale and a solution is computed. Note, that additional layers can be added to a quasi-porous cell. In upscaling techniques, this is known as oversampling. Mostly, the domain is extended by a ’bordering ring’ with fine grid cells. We consider additional fine scale ’bordering layers’ of type ’fluid’ on a specified boundary. The number of additional layers is denoted by l in the next section. If l = 0, it is a purely local auxiliary solve. If l = 1, it means that the auxiliary problem is solved in coarse cell extended with one fluid layer in the specified direction, as shown on Figure 5(d). Step (3) assigns the effective permeability tensors to the coarse cells, postprocessing the auxiliary fine grid solution computed from step (2). Steps (2) and (3) are performed for every quasi-porous cell selected in step (1). Note, that in the case when the coarse cells/blocks are periodic (e.g., this is the case when each coarse quasi-porous block contains one pleat), the auxiliary problem needs to be solved once for each type of quasi-porous coarse cells. Furthermore, the once computed permeabilities are reused in the computations with the same geometry, but with a different flow rate or viscosity (recall that 30-36 simulations have to be performed for each filter element geometry). Step (4) includes the standard algorithm which solves the Stokes-Brinkman system of equations on the coarse grid. Remark 1 : (1) The same solver is used for solving global coarse scale problem and auxiliary problems on the fine grid. (2) Different stopping criteria can be specified for the local and for the global problems. 10

Remark 2: If the number of quasi-porous blocks is too large, Steps (2)-(3) become the most time consuming part of the solution procedure. Implementation is modular-based and the subgrid algorithm is parallelized, so that the auxiliary problems are distributed to different processes for faster computations. 4. Results and validation The results from numerical simulations are presented in this section. The performed simulations can be divided into three groups: • Validation for thin porous layer and for the upscaled permeability computation (Subsections 4.1 and 4.2). Stokes flow around a spherical obstacle is considered in the Subsection 4.1 to check the influence of the used voxel grid as well; • Numerical study of Brinkman to Brinkman upscaling. As mentioned earlier, the upscaling approaches for Stokes and for Darcy flows are theoretically considered in the literature, and due to the lack of theoretical results for the upscaling of Brinkman equation, we perform a numerical study here (we refer also to the recent paper [36]); • Simulations for the industrial filters with complicated porous media (perforated layer in one case, pleats in and complicated geometry (solid mesh supporting the in one case, and complicated filter element housing in

geometry of the the other case), perforated layer the other case).

Whenever possible, the results from the simulations with the upscaled equations are compared with the results obtained solving the fine grid problem for the same geometry. This is the methodology employed for the validation of the developed subgrid algorithm. It should be noted that the single grid simulations have already been validated against measurements(c.f. [30]). The computations and CPU time measurements were performed on dedicated node of Fraunhofer ITWM cluster ’Hercules’ with dual Intel Xeon 5148LV (’Woodcrest’) processor (2.33 GHz). 4.1. Permeability for a periodicity cell: flow around a sphere For better understanding of the subgrid algorithm, an example of an auxiliary problem for a quasi-porous coarse cell is considered. Consider a cube 11

occupied by the fluid with the sphere-shaped solid obstacle inside, as shown in Figure 3. Suppose that we consider a domain which is a periodic arrangement of such cells. Permeability for such a geometry is computed many times via homogenization and other upscaling approaches, and it is also measured (see,e.g., the discussion in [36]). Our goal for solving the Stokes flow in this geometry is twofold. i.) From one side, we want to check if the boundary conditions for the auxiliary cell problems and REV type approach for calculating the upscaled permeability, as described above, gives good results; ii) Additionally, we want to check the influence of the voxel-based discretization grid that is used to discretize the sphere. The auxiliary problem is solved on the fine grid resolution scale of 0.25mm with the following input data: cube size - L = 12 mm, with different sized spheres with radius r; inflow velocity Uin = 1.1574 mm/s, Pout = 0 kg/(mm s2 ), density(ρ) = 1.0 · 10−7 kg/mm3 , viscosity(µ) = 1.0 · 10−4 kg/(mm s). Table 4.1 shows the results obtained by our algorithm with permeability computed using Darcy law, denoted by KDarcy . The computed values are compared with the results on unit cell problem reported by Griebel and Klitz [36] and Sangani and Acrivos [34], denoted by KCell and KExperiment respectively. It is observed that the computed results are in good comparison with the cited results. A systematic study on grid resolution was performed, and it concluded that the error reduces with finer grid resolutions where the voxel-based disretization better approximates the solid sphere. Figure 3: Cube with solid sphere inside. Inlet, outlet and symmetry walls are marked with colors blue, green and red respectively.

r

KDarcy

KCell (Griebel, Klitz)

KExperiment (Sangani, Acrivos)

0.2 0.25 0.3 0.35

1.18 × 10−1 7.34 × 10−2 4.35 × 10−2 2.48 × 10−2

1.23 × 10−1 7.40 × 10−2 4.46 × 10−2 2.52 × 10−2

1.23 × 10−1 7.46 × 10−2 4.45 × 10−2 2.52 × 10−2

Table 1: Comparison of permeability in a 3D array of spheres (cell problem) for different radii

12

4.2. Channel filter with single porous layer This example is chosen to be a simplified filter element geometry (parallelepiped) with a thin layer of porous media. The computational domain is {(x, y, z) : 0 ≤ x ≤ 12, 0 ≤ y ≤ 24, 0 ≤ z ≤ 24 mm}, with a single porous layer {(x, y, z) : 4.5 ≤ x ≤ 5.5, 0 ≤ y ≤ 24, 0 ≤ z ≤ 24 mm}, as shown in Figure 4. On coarse grids, when the step size of the grid is bigger than the thickness of the layer, the latter can not be correctly resolved. The thickness of the filtering medium is correctly resolved on 0.5 mm grid. However, on grid 2 mm, the pre-processor provides a porous layer with 2 mm thickness (recall that the pre-processor assigns to a grid cell the material type of its center point). Solving problem (1) with Uin = 28.9 mm/s, ρ = 1.0 · 10−7 kg/mm3 , µ = 1.0 · 10−4 kg/(mm s), and isotropic permeability K = 1.0 · 10−4 mm2 , we obtain (correct) pressure drop of 28.95 kg/(mm s2 ) on 0.5 mm grid, and a twice bigger (wrong) value, i.e. 57.9 kg/(mm s2 ) on 2 mm grid. Note, that the considered flow is essentially one-dimensional one and one can easily find its analytical solution. Application of the subgrid algorithm for 2 mm coarse and 0.5 mm fine grids gives 144 (12 × 12) quasi-porous cells, which are all identical in this case, as shown in Figure 5 (a). Solution of the auxiliary problem in X direction gives the value of permeability K11 = 2.0 · 10−4 (the other components of the permeability tensor are not important in this case). Using the obtained permeability for the quasi-porous cells, we solve the problem on the 2 mm coarse grid, and get the correct pressure drop, i.e. 29.0. Including additional layers at inlet and outlet Using 1 mm size for the coarse grid, we get two different geometries of quasiporous cells in X direction, as illustrated in Figure 5 (b) and (c). Solving auxiliary problems for these geometries gives wrong values of the upscaled permeability in X direction: K11 = 1.60·10−4 for left cell and K11 = 2.67·10−4 for right cell. This is because the fine porous cells are touching the outlet of auxiliary problem in first case, and the inlet in the second, which does not allow full development of the flow. To deal with this issue, we have an option of adding ’border rings’, which are a specified number of layers of fine fluid cells before the inlet and/or outlet of the actual domain in any specified direction. Figure 5 (d) shows the auxiliary problem for the right 1 mm cell with 1 additional fluid layer in front of the inlet. Solving this auxiliary problem gives correct value of upscaled permeability: K11 = 2.0 · 10−4 . The 13

(a) 0.5 mm grid

(b) 1 mm grid

(c) 2 mm grid

Figure 4: Channel filter with the single porous layer.

same value is obtained for the left cell with additional fluid layer behind the outlet. Finally, solving the global problem on the coarse 1 mm grid with the upscaled permeabilities gives correct value of the global pressure drop, i.e. 28.95. 4.3. Channel filter with single porous layer with a hole As a next example, the same filter geometry is considered as in subsection 4.2 but with Uin = 2.8e − 03 and with a small hole in the porous layer {(x, y, z) : 4.5 ≤ x ≤ 5.5, 10.5 ≤ y ≤ 11.5, 10.5 ≤ z ≤ 11.5 mm}. This hole is correctly resolved on 0.5 mm grid. On 2 mm grid, this 1 × 1 × 1 mm hole is exactly in the center of 2 × 2 × 2 mm coarse cell, as illustrated in Figure 6 (a) and (b). Figure 6 (a) is a view along the channel, whereas Figure 6 (b) is a view across the channel. Therefore, as opposed to the previous case, this cell 14

(a) 2 mm

(b) 1 mm left

(c) 1 mm right

(d) right + 1 layer

Figure 5: Auxiliary problems for 2 and 1 mm quasi-porous cells in X direction for channel filter with the single porous layer.

is fluid on 2 mm grid, and the cross section of the hole as well as the thickness of the porous layer are wrongly depicted. On 1 mm grid, all 8 1 × 1 × 1 mm cells, covering the hole, have a porous center and therefore remain porous, as shown in Figure 6. Hence, 1 mm grid resolution remains the same as shown in Figure 4(b) in Section 4.2.

(a) 2 mm

(b) 2 mm

(c) 1 mm

(d) 1 mm

Figure 6: Auxiliary problems for 2 and 1 mm quasi-porous cells in X direction for channel filter with the single porous layer with a hole.

In Table 2, results for the different grid resolutions using the single grid algorithm are presented. On the 2 mm grid, the hole is too big, therefore we get relatively small pressure drop. The grids with step size 0.5 and 0.25 mm resolve the hole exactly (see Figure 6 (b)). Clearly it is expensive to compute accurate solution on a single fine grid, and a subgrid approach is the remedy for efficient simulations. Next, we discuss the results of the subgrid algorithm for this problem. In 15

Grid Number Memory CPU time resolution of CVs [MB] [s] 2 mm 0.5 mm 0.25 mm

864 55296 442368

0.80 33.73 247.1

Pressure drop [kg/(mm s2 )]

2 206 2489

0.00353 0.01486 0.01669

Table 2: Simulation results for channel filter with single porous layer with a hole.

Table 3, we present the results obtained for 2 mm coarse and several (0.5, 0.25 mm) fine scale combinations. Calculations were done with different number of additional fluid layers l used for the solution of the auxiliary problems. We show the computed values of the upscaled permeability K11 for the quasiporous cell with hole and the global pressure drop dP . Also, the CPU time TSG of preprocessing step only is given, because the subsequent solution of system on the coarse 2 mm grid was taking less than 10 seconds for all shown cases. The maximal memory used by the solver in all cases was 1.08 MB (coarse grid solver + upscaled permeability tensors). Scales

l

K11

TSG

dP

2/0.5 mm 2/0.5 mm 2/0.5 mm 2/0.25 mm 2/0.25 mm 2/0.25 mm

0 2 4 0 2 4

0.0226 0.0272 0.0273 0.0198 0.0217 0.0218

14 140 203 155 345 516

0.0168 0.0156 0.0156 0.0177 0.0172 0.0171

Table 3: Simulation results for channel filter with single porous layer with hole (obtained with Subgrid algorithm). l is the number of additional fluid layers. K11 is the upscaled permeability obtained for quasi-porous cell with hole. TSG is CPU time of preprocessing step. dP is the obtained global pressure drop.

4.4. Channel filter with periodic porous layer The next example is a channel/parallelepiped filter {(x, y, z) : 0 ≤ x ≤ 75, 0 ≤ y ≤ 15, 0 ≤ z ≤ 15 mm} with porous layer (30 ≤ x ≤ 45) consisting of two porous materials with a periodic structure, as shown in Figure 7. 16

(b) (3 × 3: 8 × 8)

(a)

Figure 7: Channel filter with two peridic porous layers (3 × 3) represented in yellow and red. (a) and (b) represent two different cross sections of x − z axis and y − z axis of the filter.

The first porous material(in yellow) in Figure 7 has permeability K = 1.0 · 10−4 mm2 . The second porous material (in red color) is more permeable with permeability K = 2.0 · 10−4 mm2 . The problem (1) is solved with the same parameters as before. For Figure 7, the results on the fine scale, 24 × 24 × 24, are obtained. The pressure drop dP = 80.005 is obtained in 9.42 seconds. The grid is further coarsened 8 times, such that 8 × 8 × 8 fine cells comprises of one coarse cell and the results using the coarse and the fine grid are obtained using the subgrid algorithm. The results obtained are dP = 80.008 in 4.55 seconds. Figure 8 (a) shows a geometry with a fine scale discretization containing 30 × 30 × 30 finite volumes. Compared to Figure 7(b), there is 1.5% more of the less permeable porous material and we expect a greater pressure drop across the filter element. The single grid algorithm results in dP = 86.21 in 21.65 seconds. The grid is then coarsened 3 times, such that 10 × 10 × 10 fine cells constitutes one coarse cell. Using both, fine and coarse scales, the subgrid algorithm results in dP = 86.21 in 8.39 seconds. Figure 8 (b) illustrates five, instead of three, periods of the second type of the

17

(a) (3x3: 10x10)

(b) (5x5: 8x8)

Figure 8: Channel filter with peridic porous layer.

porous material which has a higher permeability. The geometry compares with the geometry shown in Figure 7. We see that pressure drop dP = 80.012 is obtained in 52.67 seconds using the single grid algorithm with fine scale discretization of 40×40×40 cells. The coarse discretization contains 5×5×5 cells. Subgrid algorithm results in dP = 80.015 in 21.82 seconds. Lastly, instead of three, periods of the second type of the porous material which has a higher permeability for the geometry that compares with the geometry shown in Figure 8. We see that the pressure drop dP = 86.25 is obtained in 124.96 seconds using the single grid algorithm with fine scale discretization of 50×50×50 cells. The coarse discretization contains 5×5×5 cells. The subgrid algorithm results in dP = 86.22 in 41.23 seconds. 4.5. Channel filter with the combi layers The next example is chosen to serve the needs of the design of new generation of filters as shown in Figure 9 (a) . It should be noted that oil filters are often equipped with a so called bypass option. The bypass prevents the oil to filter through the porous media, especially at cold temperatures (i.e. high viscosity ). The idea behind such design was to prevent the destruction of 18

the filter due to the high pressure in such cases. Instead, the oil bypasses the filtering medium via some small pipe. Obviously, the bypassed oil is not filtered in this situation. Alternatively, the new league of filters are designed without the bypass option, but with an additional layer of a fine perforated filter layer that allows the oil to flow through the holes at cold temperatures. Additionally, a coarse filter layer in the form of a solid mesh is added to filter out the large particles. With the support of the numerical simulations, the size of the holes and the distance between the two porous layers are designed such that most oil flows through the porous part of the first layer at high temperature, and not through the holes. It is common understanding that the porous layers are the primary cause of the pressure drop across the filter. This is true only when the filter element is designed in a way that there is enough space between the inlet(bottom) and the porous layer, and between the porous layer and the outlet(top). Therefore, we study this filter in a simplified geometry, i.e. a simple channel geometry as shown in Figure 9 (b). The considered filter element is parallel piped {(x, y, z) : 2 ≤ x ≤ 67, −11 ≤ y ≤ −2, −67 ≤ z ≤ −2 mm} with two filtering porous layers and a supporting solid mesh between them, as shown in Figure 10 (b). Additionally, the first porous layer (−8 ≤ y ≤ −7) has a set of 6 × 6 holes, as shown in Figure 10 (b), where each hole is a cube: 1 × 1 × 1 mm, as illustrated in Figure 11 (a). When using the 2 mm resolution, the mesh layer is not captured at all and both porous layers have the wrong, doubled thickness, on this grid: −8 ≤ y ≤ −6 and −6 ≤ y ≤ −4, respectively. The 1 mm grid represents the channel and all three layers (including the solid mesh) correctly except for the holes in the first porous layer. As shown in Figure 11, all 4 1 × 1 × 1 mm cells, covering the hole, have a porous center and therefore are represented as porous in 1 mm grid. The 0.5 and 0.25 mm grids resolve the whole geometry of the filter exactly. System 1 is solved with the following flow parameters: Uin = 3.944773 mm/s, ρ = 1.0 · 10−7 kg/mm3 , µ = 1.0 · 10−4 kg/(mm s), isotropic permeability of first porous layer, K = 4.2 · 10−5 mm2 , and isotropic permeability of second porous layer, K = 7.5 · 10−4 mm2 . Note that in this test case the main flow is in Y direction. In Table 4, the results of single grid algorithm for different grid resolutions are presented. It is due to the aforementioned errors in the geometry resolution for 2 to 1 mm grids, that the resulting pressure drop is over estimated. As one 19

(a)

(b)

Figure 9: (a) A real industrial filter with multiple porous media layers. (b) Channel filter with the multiple porous layers. A simplified channel filter of the industrial filter to understand real processes.

goes to finer resolutions, i.e. 0.5 to 0.25 mm, the geometry is resolved more accurately resulting in a more accurate pressure drop. However, computation on the finer grids come with the cost of extensive CPU time consumption and memory usage. Next, we present the results from the subgrid algorithm for this problem. 2 mm resolution is too coarse to be used even as the coarse scale, where the 1 mm scale seems to be an appropriate choice. Alternatively, a bigger block of coarse cells can be used as an auxiliary problem for computing permeability. As discussed above, for 1 mm grid the only cells, which will be detected as ’quasi-porous’, are the cells covering the holes in first porous layer (see Figure 11 (a)). It means that the total of 4 × 6 × 6 = 144 quasi-porous cells will be detected. As can be seen from Figure 11 (c) and (b), solving the auxiliary problem for the main component of upscaled permeability tensor K22 , the flow will be almost aligned with the Y axis and we can expect good results even without the use of full permeability tensor. It is also clear that the additional fluid layers are needed for the auxiliary problems. In Table 5 we present the results obtained for two coarse/fine grids combi20

(a)

(b)

Figure 10: Cross sectional view of the complicated structure of the multiple porous layers in the channel filter.

nation: 1/0.5 and 1/0.25 mm. Calculations were done with different number of additional fluid layers l, Reynolds number Re, and accuracy  used for solution of auxiliary problems. We show the obtained values of upscaled permeability K22 for one of the quasi-porous cells, CPU time TSG of preprocessing step and the total solver time Ttotal (i.e. including the CPU time of subsequent solution on the coarse 1 mm grid), and finally the global pressure drop dP . The maximal memory used by the solver in all cases was 26.68 MB (coarse grid solver + upscaled permeability tensors). The results show that the additional number of layers (oversampling) is an essential parameter in our case. Also, the choice of the coarse grid is important for the selection of quasi-porous cells. 4.6. Simulation of Industrial Filters - Pleated filter Lastly, another example of a real filter with complicated filter media is considered. Figure 12 shows a pleated filter geometry on coarse and fine grids. The coarse grid does not resolve the shape of the filtering medium properly. Our aim is to solve the problem using equation 2 on the coarse grid, and ˜ ef f employ the subgrid algorithm described in Section 3 to compute the K 21

(a)

(b)

(c)

Figure 11: Resolution of one of the holes in first porous layer

Grid Number Memory CPU time resolution of CVs [MB] [s] 2 mm 1 mm 0.5 mm 0.25 mm

5445 37400 299200 2393600

4.42 26.49 182.70 1338.37

63.54 465.02 28271.47 518425.0

Pressure drop [kg/(mm s2 )] 16.762 11.627 1.579 1.762

Table 4: Simulation results for channel filter with multiple porous layers using the single grid algorithm.

for selected quasi-porous cells. Figure 12 (c) also illustrates the quasi-porous block on which fine scale auxiliary problem is solved. It should be noted that the number of cells in this block are fairly less compared to solving the fine scale problem on the complete filter element. In fact, computing permeability from simulation in a significantly smaller block gives the same results. In the case when all the pleats are of the same size, and the distances between them are equal, it is possible to consider only one smaller block, containing one pleat, to calculate the permeability for this block, and after that to assign this permeability to each coarse cell in the quasi-porous block from 12, which contains all the pleats. In Table 6, we present the results of solving this problem with the single grid numerical algorithm for two different grid resolutions. The coarse scale is of 1mm size and fine scale resolution is 0.5mm. The results illustrate the extent to which the computational time for solving a complete problem could increase with finer grid resolutions. Almost 100 times more computational

22

Scales

l

1/0.5 mm 1/0.5 mm 1/0.25 mm 1/0.25 mm 1/0.25 mm 1/0.25 mm

1 2 1 2 3 4

K22

TSG

0.01092 54.8 0.01177 93.4 0.00588 883.3 0.00863 787.7 0.00960 993.0 0.00972 1126.3

Ttotal

dP

781.5 838.3 1502.9 1464.8 1692.1 1829.4

1.6919 1.6289 2.3862 1.9173 1.8101 1.7984

Table 5: Simulation results for channel filter with combi layers (obtained with Subgrid algorithm). l is the number of additional fluid layers used for auxiliary problems. K22 is the upscaled permeability in flow direction obtained for one of the quasi-porous cells. TSG is CPU time of preprocessing step. Ttotal is the total CPU time. dP is obtained global pressure drop.

Figure 12: Pleats in the filter on different grids. Fine grid resolves the geometrical shape of the filter and the pleats, where as the coarse grid does not. Subgrid treats all pleats as a block of filtering medium with upscaled permeability computed using fine scale algorithm.

time is required for a problem that is twice finer in grid resolution. Note that different resolutions result in different geometries. For the coarse grid, not only are the shapes of the pleats not correctly accounted for, but also the inlet and outlet are not separated by the porous medium. In fact, the solution is sought for different geometries and for obvious reasons, the pressure drop on the coarse grid is an over estimation. Grid 1mm 0.5mm subgrid-1mm-0.5mm

dP (mbar)

TSG

Ttotal (s)

2700 1874 1760

63580

1421 149102 1420

Table 6: Results for geometries described in Figure 12.

Note that the subgrid algorithm is employed as a preprocessing step for the 23

single grid algorithm. Moreover, in industry, the subgrid method benefits from the fact that for each fixed geometry, many simulations are done using different physical parameters, such as density, viscosity, inflow velocities etc. for assessing the filter performance. In this case, the subgrid method can be used only once and precomputed upscaled permeabilities for the quasi-porous cells/blocks can be further used in subsequent simulations. 5. Summary A subgrid method is considered as a computational method for achieving desired accurate solutions for our problem. It is mainly employed for cases where the coarse computational grid is unable to accurately account for the complicated geometry and/or the filter media. The single grid algorithm is used as the building block for solving auxiliary problems on the quasiporous cells and for solving the global coarse scale problem, with special emphasis based on re-usability of solvers. The method emphasizes on the determination of upscaled quantities for use in subsequent coarse scale global simulations. Fine scale solution is sought only on some quasi-porous cell/s or collectively on a block (a collection of coarse cells). The results were presented for the computer simulation experiments using three dimensional models of oil filters. It is observed that the cell/block permeability is strongly influenced by the resolving grid. The CPU time and memory usage is reduced significantly using the subgrid method with the resulting desired accuracy of the pressure drop, which characterizes the flow through filters. [1] A. Kumar, Numerical Estimation of Surface Parameters by Level Set Methods, PhD Thesis, Technical University of Kaiserslautern, 2003. [2] A. J. Chorin, Numerical Solution of Navier-Stokes equation, Mathematics of Computation 22 (1968), 745-760. [3] A. F. Gulbransen, V.L. Hauge, K.A. Lie, A Multiscale Mixed Finite Element Method for Vuggy and Naturally Fractured Reservoirs, 21st Nordic Seminar on Computational Mechanics, Trondheim, 149-152, 2008. [4] C.A.J. Fletcher, Computational techniques for fluid dynamics. Springer, Berlin etc., 1991.

24

[5] G. Bonfigli, P. Jenny, An efficient multi-scale poisson solver for the incompressible navier-stokes equations with immersed boundaries. J. Comp. Phys., 228(12), 2009. [6] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. i: Abstract framework, a volume distribution of holes, Arch. Ration. Mech. Anal. 113 (1991), no. 3, 209-259. [7] H.C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Appli. Sci. Res., t. A1, (1947), 27-34. [8] H. Hajibeygi, G. Bonfigli, M. A. Hesse, and P. Jenny. Iterative multiscale finite-volume method. J. Comp. Phys., 227(19):8604-8621, Oct 2008. [9] J. Willems, Numerical Upscaling for Multiscale Flow Problems, PhD thesis, Technical University of Kaiserslautern, 2009. [10] K. R. Rajagopal, On a hierarchy of approximate models for flows of incompressible fluids through porous solids, Math. Models Meth. Appl. Sci., 17 (2007), 215-252. [11] L.J. Durlofsky, Numerical calculation og equivalent grid block permeability tensors for heterogeneous porous media, Water Resources Research 27, 669-708, 1991. [12] M. Kaviany, Principles of Heat Transfer in Porous Media. Springer, New York etc., 1991. [13] M. Dedering, W. Stausberg, O. Iliev, Z. Lakdawala, R. Ciegis, V. Starikovicius, On new Challenges for CFD Simulation in Filtration, Proceedings of World Filtration Congress, Leipzig, 2008. [14] O. Iliev, Z. Lakdawala, R. Ciegis, V. Starikovicius, M. Dedering, P.Popov, Advanced CFD simulation of filtration processes, Proceedings of Filtech, Wiesbaden, 2009. [15] O. Iliev, V. Laptev, On numerical simulation of flow through oil filters. Comput. Vis. Sci., 6 (2004) 139–146. [16] O. Iliev, V. Laptev, D. Vassileva, Algorithms and software for computer simulation of flow through oil filters. Proc. FILTECH Europa, 2003, D¨ usseldorf, pp.327–334. 25

[17] O. Iliev, R.D. Lazarov, and J. Willems, Fast numerical upscaling of heat equation for fibrous materials, J. Computing and Visualization in Science, 2009. [18] Ph. Angot, Analysis of singular perturbations on the Brinkman problem for fictitious domain models of viscous flows. Math. Meth. Appl. Sci., 22 (1999) 1395–1412. [19] P. Gresho, R. Sani, Incompressible flow and the finite element method. Volume 1: Advection-diffusion and isothermal laminar flow. In collaboration with M. S. Engelman. Chichester: Wiley. 1044 p. (1998). [36] M. Griebel and M. Klitz. Homogenisation and Numerical Simulation of Flow in Geometries with Textile Microstructures. SIAM MMS, 2009. Submitted. Also available as SFB611 preprint no 404 / 2008. [21] P. Jenny and I. Lunati. Multi-scale finite-volume method for elliptic problems with heterogeneous coefficients and source terms. Proc. Appl. Math. Mech., 6(1):485-486, 2006. [22] R. Ciegis, O. Iliev, Z. Lakdawala, On parallel numerical algorithms for simulating industrial filtration problems. Computational Methods in Applied Mathematics 7(2), 118-134(2007). [23] T.Y. Hou, Y. Efendiev, Multiscale Finite Element Methods. Surveys and Tutorials in the Applied Mathematical Sciences, Band 4, Springer, 2009. [24] T. Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems, SIAM J. Numer. Anal. 42 (2004), no. 2, 576598. [25] T. Arbogast, S. L. Bryant, A two-scale numerical subgrid technique for waterflood simulations, SPE Journal 7 (2002) 446-457. [26] T. Arbogast, Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow, Computational Geosciences 6, 453-481, 2002. [27] U. Hornung. Homogenization and porous media, Springer Verlag New York Inc., New York, USA, 1996. 26

[28] V. Starikovicius, R. Ciegis, O. Iliev, Z.Lakdawala, A parallel solver for the 3D Simulation of Flows through Oil Filters, Springer Optimization and its applications, Volume 27, 181-191(2008). [29] V.V. Jikov, S.M. Kozlov, Oleinik O.A., Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. [30] V. Laptev, Numerical solution of coupled flow in plain and porous media. PhD thesis, Technical University of Kaiserslautern, 2003. [31] X.H. Wu, Y. Efendiev and T.Y. Hou, Analysis of upscaling absolute permeability. Discrete Contin. Dyn. Syst., Series B 2, No. 2, 185-204 (2002). [32] Y. Chen, L.J. Durlofsky, M. Gerritsen, X.H. Wen, A Coupled LocalGlobal Upscaling Approach for Simulating Flow in Highly Heterogeneous Formations, Advances in Water Resources, 26, 1041-1060 2003. [33] Y. Jung, S. Torquato, Fluid permeabilities of triply periodic minimal surfaces, Physical Review Em 72, 056319, 2005. [34] A. Sangani, A.Acrivos, Int. J. Multiphase Flow, 8, 343, 1982. [35] James H. Bramble and Joseph E. Pasciak, 1988. A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comp., 50(181): 1-17. [36] M. Griebel, M. Klitz. Homogenisation and Numerical Simulation of Flow in Geometries with Textile Microstructures. SIAM MMS, 2009.

27

Published reports of the Fraunhofer ITWM The PDF-files of the ­following reports are available under: www.itwm.fraunhofer.de/de/ zentral__berichte/berichte

11. H. W. Hamacher, A. Schöbel On Center Cycles in Grid Graphs (15 pages, 1998)

(44 pages, 2001)

12. H. W. Hamacher, K.-H. Küfer Inverse radiation therapy planning a multiple objective optimisation approach (14 pages, 1999)

13. C. Lang, J. Ohser, R. Hilfer On the Analysis of Spatial Binary Images (20 pages, 1999)

1. D. Hietel, K. Steiner, J. Struckmeier A Finite - Volume Particle Method for Compressible Flows (19 pages, 1998)

14. M. Junk On the Construction of Discrete Equilibrium Distributions for Kinetic Schemes (24 pages, 1999)

2. M. Feldmann, S. Seibold Damage Diagnosis of Rotors: Application of ­Hilbert Transform and Multi-Hypothesis Testing Keywords: Hilbert transform, damage diagnosis, Kalman filtering, non-linear dynamics (23 pages, 1998)

3. Y. Ben-Haim, S. Seibold Robust Reliability of Diagnostic MultiHypothesis Algorithms: Application to Rotating Machinery Keywords: Robust reliability, convex models, Kalman filtering, multi-hypothesis diagnosis, rotating machinery, crack diagnosis (24 pages, 1998)

4. F.-Th. Lentes, N. Siedow Three-dimensional Radiative Heat Transfer in Glass Cooling Processes (23 pages, 1998)

5. A. Klar, R. Wegener A hierarchy of models for multilane vehicular traffic Part I: Modeling (23 pages, 1998)

Part II: Numerical and stochastic investigations (17 pages, 1998)

6. A. Klar, N. Siedow Boundary Layers and Domain Decomposition for Radiative Heat Transfer and Diffusion Equations: Applications to Glass Manufacturing Processes (24 pages, 1998)

7. I. Choquet Heterogeneous catalysis modelling and numerical simulation in rarified gas flows Part I: Coverage locally at equilibrium (24 pages, 1998)

8. J. Ohser, B. Steinbach, C. Lang Efficient Texture Analysis of Binary Images (17 pages, 1998)

9. J. Orlik Homogenization for viscoelasticity of the integral type with aging and shrinkage (20 pages, 1998)

10. J. Mohring Helmholtz Resonators with Large Aperture (21 pages, 1998)

24. H. W. Hamacher, S. A. Tjandra Mathematical Modelling of Evacuation Problems: A State of Art

15. M. Junk, S. V. Raghurame Rao A new discrete velocity method for NavierStokes equations (20 pages, 1999)

16. H. Neunzert Mathematics as a Key to Key Technologies (39 pages (4 PDF-Files), 1999)

17. J. Ohser, K. Sandau Considerations about the Estimation of the Size Distribution in Wicksell’s Corpuscle Problem (18 pages, 1999)

18. E. Carrizosa, H. W. Hamacher, R. Klein, S. Nickel Solving nonconvex planar location problems by finite dominating sets Keywords: Continuous Location, Polyhedral Gauges, Finite Dominating Sets, Approximation, Sandwich Algorithm, Greedy Algorithm (19 pages, 2000)

19. A. Becker A Review on Image Distortion Measures Keywords: Distortion measure, human visual system (26 pages, 2000)

20. H. W. Hamacher, M. Labbé, S. Nickel, T. Sonneborn Polyhedral Properties of the Uncapacitated Multiple Allocation Hub Location Problem Keywords: integer programming, hub location, facility location, valid inequalities, facets, branch and cut (21 pages, 2000)

21. H. W. Hamacher, A. Schöbel Design of Zone Tariff Systems in Public Transportation

25. J. Kuhnert, S. Tiwari Grid free method for solving the Poisson equation Keywords: Poisson equation, Least squares method, Grid free method (19 pages, 2001)

26. T. Götz, H. Rave, D. Reinel-Bitzer, K. Steiner, H. Tiemeier Simulation of the fiber spinning process Keywords: Melt spinning, fiber model, Lattice Boltzmann, CFD (19 pages, 2001)

27. A. Zemitis On interaction of a liquid film with an obstacle Keywords: impinging jets, liquid film, models, numerical solution, shape (22 pages, 2001)

28. I. Ginzburg, K. Steiner Free surface lattice-Boltzmann method to model the filling of expanding cavities by Bingham Fluids Keywords: Generalized LBE, free-surface phenomena, interface boundary conditions, filling processes, Bingham viscoplastic model, regularized models (22 pages, 2001)

29. H. Neunzert »Denn nichts ist für den Menschen als Menschen etwas wert, was er nicht mit Leidenschaft tun kann« Vortrag anlässlich der Verleihung des Akademie­preises des Landes RheinlandPfalz am 21.11.2001 Keywords: Lehre, Forschung, angewandte Mathematik, Mehrskalenanalyse, Strömungsmechanik (18 pages, 2001)

30. J. Kuhnert, S. Tiwari Finite pointset method based on the projection method for simulations of the incompressible Navier-Stokes equations Keywords: Incompressible Navier-Stokes equations, Meshfree method, Projection method, Particle scheme, Least squares approximation AMS subject classification: 76D05, 76M28 (25 pages, 2001)

31. R. Korn, M. Krekel Optimal Portfolios with Fixed Consumption or Income Streams

(30 pages, 2001)

Keywords: Portfolio optimisation, stochastic control, HJB equation, discretisation of control problems (23 pages, 2002)

22. D. Hietel, M. Junk, R. Keck, D. Teleaga The Finite-Volume-Particle Method for Conservation Laws

32. M. Krekel Optimal portfolios with a loan dependent credit spread

(16 pages, 2001)

Keywords: Portfolio optimisation, stochastic control, HJB equation, credit spread, log utility, power utility, non-linear wealth dynamics (25 pages, 2002)

23. T. Bender, H. Hennes, J. Kalcsics, M. T. Melo, S. Nickel Location Software and Interface with GIS and Supply Chain Management Keywords: facility location, software development, geographical information systems, supply chain management (48 pages, 2001)

33. J. Ohser, W. Nagel, K. Schladitz The Euler number of discretized sets – on the choice of adjacency in homogeneous lattices Keywords: image analysis, Euler number, neighborhod relationships, cuboidal lattice (32 pages, 2002)

34. I. Ginzburg, K. Steiner Lattice Boltzmann Model for Free-Surface flow and Its Application to Filling Process in Casting Keywords: Lattice Boltzmann models; free-surface phenomena; interface boundary conditions; filling processes; injection molding; volume of fluid method; interface boundary conditions; advection-schemes; upwind-schemes (54 pages, 2002)

35. M. Günther, A. Klar, T. Materne, R. Wegener Multivalued fundamental diagrams and stop and go waves for continuum traffic equations Keywords: traffic flow, macroscopic equations, kinetic derivation, multivalued fundamental diagram, stop and go waves, phase transitions (25 pages, 2002)

36. S. Feldmann, P. Lang, D. Prätzel-Wolters Parameter influence on the zeros of network determinants Keywords: Networks, Equicofactor matrix polynomials, Realization theory, Matrix perturbation theory (30 pages, 2002)

37. K. Koch, J. Ohser, K. Schladitz Spectral theory for random closed sets and es­timating the covariance via frequency space Keywords: Random set, Bartlett spectrum, fast Fourier transform, power spectrum (28 pages, 2002)

38. D. d’Humières, I. Ginzburg Multi-reflection boundary conditions for lattice Boltzmann models Keywords: lattice Boltzmann equation, boudary condistions, bounce-back rule, Navier-Stokes equation (72 pages, 2002)

39. R. Korn Elementare Finanzmathematik Keywords: Finanzmathematik, Aktien, Optionen, Port­ folio-Optimierung, Börse, Lehrerweiterbildung, Mathematikunterricht (98 pages, 2002)

40. J. Kallrath, M. C. Müller, S. Nickel Batch Presorting Problems: Models and Complexity Results Keywords: Complexity theory, Integer programming, Assigment, Logistics (19 pages, 2002)

41. J. Linn On the frame-invariant description of the phase space of the Folgar-Tucker equation Key words: fiber orientation, Folgar-Tucker equation, injection molding (5 pages, 2003)

42. T. Hanne, S. Nickel A Multi-Objective Evolutionary Algorithm for Scheduling and Inspection Planning in Software Development Projects Key words: multiple objective programming, project management and scheduling, software development, evolutionary algorithms, efficient set (29 pages, 2003)

43. T. Bortfeld , K.-H. Küfer, M. Monz, A. Scherrer, C. Thieke, H. Trinkaus Intensity-Modulated Radiotherapy - A Large Scale Multi-Criteria Programming Problem

Keywords: multiple criteria optimization, representative systems of Pareto solutions, adaptive triangulation, clustering and disaggregation techniques, visualization of Pareto solutions, medical physics, external beam radiotherapy planning, intensity modulated radiotherapy (31 pages, 2003)

44. T. Halfmann, T. Wichmann Overview of Symbolic Methods in Industrial Analog Circuit Design Keywords: CAD, automated analog circuit design, symbolic analysis, computer algebra, behavioral modeling, system simulation, circuit sizing, macro modeling, differential-algebraic equations, index (17 pages, 2003)

45. S. E. Mikhailov, J. Orlik Asymptotic Homogenisation in Strength and Fatigue Durability Analysis of Composites Keywords: multiscale structures, asymptotic homogenization, strength, fatigue, singularity, non-local conditions (14 pages, 2003)

46. P. Domínguez-Marín, P. Hansen, N. Mladenovi ´c , S. Nickel Heuristic Procedures for Solving the Discrete Ordered Median Problem Keywords: genetic algorithms, variable neighborhood search, discrete facility location (31 pages, 2003)

47. N. Boland, P. Domínguez-Marín, S. Nickel, J. Puerto Exact Procedures for Solving the Discrete Ordered Median Problem Keywords: discrete location, Integer programming (41 pages, 2003)

48. S. Feldmann, P. Lang Padé-like reduction of stable discrete linear systems preserving their stability Keywords: Discrete linear systems, model reduction, stability, Hankel matrix, Stein equation (16 pages, 2003)

49. J. Kallrath, S. Nickel A Polynomial Case of the Batch Presorting Problem Keywords: batch presorting problem, online optimization, competetive analysis, polynomial algorithms, logistics (17 pages, 2003)

50. T. Hanne, H. L. Trinkaus knowCube for MCDM – Visual and Interactive Support for Multicriteria Decision Making Key words: Multicriteria decision making, knowledge management, decision support systems, visual interfaces, interactive navigation, real-life applications. (26 pages, 2003)

51. O. Iliev, V. Laptev On Numerical Simulation of Flow Through Oil Filters Keywords: oil filters, coupled flow in plain and porous media, Navier-Stokes, Brinkman, numerical simulation (8 pages, 2003)

52. W. Dörfler, O. Iliev, D. Stoyanov, D. Vassileva On a Multigrid Adaptive Refinement Solver for Saturated Non-Newtonian Flow in Porous Media Keywords: Nonlinear multigrid, adaptive refinement, non-Newtonian flow in porous media (17 pages, 2003)

53. S. Kruse On the Pricing of Forward Starting Options under Stochastic Volatility Keywords: Option pricing, forward starting options, Heston model, stochastic volatility, cliquet options (11 pages, 2003)

54. O. Iliev, D. Stoyanov Multigrid – adaptive local refinement solver for incompressible flows Keywords: Navier-Stokes equations, incompressible flow, projection-type splitting, SIMPLE, multigrid methods, adaptive local refinement, lid-driven flow in a cavity (37 pages, 2003)

55. V. Starikovicius The multiphase flow and heat transfer in porous media Keywords: Two-phase flow in porous media, various formulations, global pressure, multiphase mixture model, numerical simulation (30 pages, 2003)

56. P. Lang, A. Sarishvili, A. Wirsen Blocked neural networks for knowledge extraction in the software development process Keywords: Blocked Neural Networks, Nonlinear Regression, Knowledge Extraction, Code Inspection (21 pages, 2003)

57. H. Knaf, P. Lang, S. Zeiser Diagnosis aiding in Regulation Thermography using Fuzzy Logic Keywords: fuzzy logic,knowledge representation, expert system (22 pages, 2003)

58. M. T. Melo, S. Nickel, F. Saldanha da Gama Large­scale models for dynamic multi­ commodity capacitated facility location Keywords: supply chain management, strategic planning, dynamic location, modeling (40 pages, 2003)

59. J. Orlik Homogenization for contact problems with periodically rough surfaces Keywords: asymptotic homogenization, contact ­problems (28 pages, 2004)

60. A. Scherrer, K.-H. Küfer, M. Monz, F. Alonso, T. Bortfeld IMRT planning on adaptive volume structures – a significant advance of computational complexity Keywords: Intensity-modulated radiation therapy (IMRT), inverse treatment planning, adaptive volume structures, hierarchical clustering, local refinement, adaptive clustering, convex programming, mesh generation, multi-grid methods (24 pages, 2004)

61. D. Kehrwald Parallel lattice Boltzmann simulation of complex flows Keywords: Lattice Boltzmann methods, parallel computing, microstructure simulation, virtual material design, pseudo-plastic fluids, liquid composite moulding (12 pages, 2004)

62. O. Iliev, J. Linn, M. Moog, D. Niedziela, V. Starikovicius On the Performance of Certain Iterative Solvers for Coupled Systems Arising in Discretization of Non-Newtonian Flow Equations

Keywords: Performance of iterative solvers, Preconditioners, Non-Newtonian flow (17 pages, 2004)

63. R. Ciegis, O. Iliev, S. Rief, K. Steiner On Modelling and Simulation of Different Regimes for Liquid Polymer Moulding Keywords: Liquid Polymer Moulding, Modelling, Simulation, Infiltration, Front Propagation, non-Newtonian flow in porous media (43 pages, 2004)

64. T. Hanne, H. Neu Simulating Human Resources in Software Development Processes Keywords: Human resource modeling, software process, productivity, human factors, learning curve (14 pages, 2004)

65. O. Iliev, A. Mikelic, P. Popov Fluid structure interaction problems in deformable porous media: Toward permeability of deformable porous media Keywords: fluid-structure interaction, deformable porous media, upscaling, linear elasticity, stokes, finite elements (28 pages, 2004)

66. F. Gaspar, O. Iliev, F. Lisbona, A. Naumovich, P. Vabishchevich On numerical solution of 1-D poroelasticity equations in a multilayered domain Keywords: poroelasticity, multilayered material, finite volume discretization, MAC type grid (41 pages, 2004)

67. J. Ohser, K. Schladitz, K. Koch, M. Nöthe Diffraction by image processing and its application in materials science Keywords: porous microstructure, image analysis, random set, fast Fourier transform, power spectrum, Bartlett spectrum (13 pages, 2004)

68. H. Neunzert Mathematics as a Technology: Challenges for the next 10 Years Keywords: applied mathematics, technology, modelling, simulation, visualization, optimization, glass processing, spinning processes, fiber-fluid interaction, trubulence effects, topological optimization, multicriteria optimization, Uncertainty and Risk, financial mathematics, Malliavin calculus, Monte-Carlo methods, virtual material design, filtration, bio-informatics, system biology (29 pages, 2004)

69. R. Ewing, O. Iliev, R. Lazarov, A. Naumovich On convergence of certain finite difference discretizations for 1­D poroelasticity interface problems Keywords: poroelasticity, multilayered material, finite volume discretizations, MAC type grid, error estimates (26 pages,2004)

70. W. Dörfler, O. Iliev, D. Stoyanov, D. Vassileva On Efficient Simulation of Non-Newtonian Flow in Saturated Porous Media with a Multigrid Adaptive Refinement Solver Keywords: Nonlinear multigrid, adaptive renement, non-Newtonian in porous media (25 pages, 2004)

71. J. Kalcsics, S. Nickel, M. Schröder Towards a Unified Territory Design Approach – Applications, Algorithms and GIS Integration Keywords: territory desgin, political districting, sales territory alignment, optimization algorithms, Geographical Information Systems (40 pages, 2005)

72. K. Schladitz, S. Peters, D. Reinel-Bitzer, A. Wiegmann, J. Ohser Design of acoustic trim based on ­geometric modeling and flow simulation for non-woven Keywords: random system of fibers, Poisson line process, flow resistivity, acoustic absorption, Lattice-Boltzmann method, non-woven (21 pages, 2005)

81. N. Marheineke, R. Wegener Fiber Dynamics in Turbulent Flows Part I: General Modeling Framework Keywords: fiber-fluid interaction; Cosserat rod; turbulence modeling; Kolmogorov’s energy spectrum; double-velocity correlations; differentiable Gaussian fields (20 pages, 2005)

Part II: Specific Taylor Drag

73. V. Rutka, A. Wiegmann Explicit Jump Immersed Interface Method for virtual material design of the effective elastic moduli of composite materials Keywords: virtual material design, explicit jump immersed interface method, effective elastic moduli, composite materials (22 pages, 2005)

74. T. Hanne Eine Übersicht zum Scheduling von Baustellen Keywords: Projektplanung, Scheduling, Bauplanung, Bauindustrie (32 pages, 2005)

75. J. Linn The Folgar-Tucker Model as a ­Differetial Algebraic System for Fiber Orientation ­Calculation Keywords: fiber orientation, Folgar–Tucker model, invariants, algebraic constraints, phase space, trace stability (15 pages, 2005)

76. M. Speckert, K. Dreßler, H. Mauch, A. Lion, G. J. Wierda Simulation eines neuartigen Prüf­systems für Achserprobungen durch MKS-Modellierung einschließlich ­Regelung Keywords: virtual test rig, suspension testing, multibody simulation, modeling hexapod test rig, optimization of test rig configuration (20 pages, 2005)

77. K.-H. Küfer, M. Monz, A. Scherrer, P. Süss, F. Alonso, A. S. A. Sultan, Th. Bortfeld, D. Craft, Chr. Thieke Multicriteria optimization in intensity modulated radiotherapy planning Keywords: multicriteria optimization, extreme solutions, real-time decision making, adaptive approximation schemes, clustering methods, IMRT planning, reverse engineering (51 pages, 2005)

78. S. Amstutz, H. Andrä A new algorithm for topology optimization using a level-set method Keywords: shape optimization, topology optimization, topological sensitivity, level-set (22 pages, 2005)

79. N. Ettrich Generation of surface elevation models for urban drainage simulation Keywords: Flooding, simulation, urban elevation models, laser scanning (22 pages, 2005)

80. H. Andrä, J. Linn, I. Matei, I. Shklyar, K. Steiner, E. Teichmann OPTCAST – Entwicklung adäquater Strukturoptimierungsverfahren für Gießereien Technischer Bericht (KURZFASSUNG) Keywords: Topologieoptimierung, Level-Set-Methode, Gießprozesssimulation, Gießtechnische Restriktionen, CAE-Kette zur Strukturoptimierung (77 pages, 2005)

Keywords: flexible fibers; k- e turbulence model; fiber-turbulence interaction scales; air drag; random ­Gaussian aerodynamic force; white noise; stochastic differential equations; ARMA process (18 pages, 2005)

82. C. H. Lampert, O. Wirjadi An Optimal Non-Orthogonal Separation of the Anisotropic Gaussian Convolution Filter Keywords: Anisotropic Gaussian filter, linear filtering, orientation space, nD image processing, separable filters (25 pages, 2005)

83. H. Andrä, D. Stoyanov Error indicators in the parallel finite element solver for linear elasticity DDFEM Keywords: linear elasticity, finite element method, hierarchical shape functions, domain decom-position, parallel implementation, a posteriori error estimates (21 pages, 2006)

84. M. Schröder, I. Solchenbach Optimization of Transfer Quality in Regional Public Transit Keywords: public transit, transfer quality, quadratic assignment problem (16 pages, 2006)

85. A. Naumovich, F. J. Gaspar On a multigrid solver for the three-dimensional Biot poroelasticity system in multilayered domains Keywords: poroelasticity, interface problem, multigrid, operator-dependent prolongation (11 pages, 2006)

86. S. Panda, R. Wegener, N. Marheineke Slender Body Theory for the Dynamics of Curved Viscous Fibers Keywords: curved viscous fibers; fluid dynamics; NavierStokes equations; free boundary value problem; asymptotic expansions; slender body theory (14 pages, 2006)

87. E. Ivanov, H. Andrä, A. Kudryavtsev Domain Decomposition Approach for Automatic Parallel Generation of Tetrahedral Grids Key words: Grid Generation, Unstructured Grid, Delaunay Triangulation, Parallel Programming, Domain Decomposition, Load Balancing (18 pages, 2006)

88. S. Tiwari, S. Antonov, D. Hietel, J. Kuhnert, R. Wegener A Meshfree Method for Simulations of Interactions between Fluids and Flexible Structures Key words: Meshfree Method, FPM, Fluid Structure Interaction, Sheet of Paper, Dynamical Coupling (16 pages, 2006)

89. R. Ciegis , O. Iliev, V. Starikovicius, K. Steiner Numerical Algorithms for Solving Problems of Multiphase Flows in Porous Media Keywords: nonlinear algorithms, finite-volume method, software tools, porous media, flows (16 pages, 2006)

90. D. Niedziela, O. Iliev, A. Latz On 3D Numerical Simulations of Viscoelastic Fluids Keywords: non-Newtonian fluids, anisotropic viscosity, integral constitutive equation (18 pages, 2006)

91. A. Winterfeld Application of general semi-infinite Programming to Lapidary Cutting Problems Keywords: large scale optimization, nonlinear programming, general semi-infinite optimization, design centering, clustering (26 pages, 2006)

92. J. Orlik, A. Ostrovska Space-Time Finite Element Approximation and Numerical Solution of Hereditary Linear Viscoelasticity Problems Keywords: hereditary viscoelasticity; kern approximation by interpolation; space-time finite element approximation, stability and a priori estimate (24 pages, 2006)

93. V. Rutka, A. Wiegmann, H. Andrä EJIIM for Calculation of effective Elastic Moduli in 3D Linear Elasticity Keywords: Elliptic PDE, linear elasticity, irregular domain, finite differences, fast solvers, effective elastic moduli (24 pages, 2006)

94. A. Wiegmann, A. Zemitis EJ-HEAT: A Fast Explicit Jump ­Harmonic ­Averaging Solver for the Effective Heat Conductivity of Composite Materials Keywords: Stationary heat equation, effective thermal conductivity, explicit jump, discontinuous coefficients, virtual material design, microstructure simulation, EJ-HEAT (21 pages, 2006)

95. A. Naumovich On a finite volume discretization of the three-dimensional Biot poroelasticity system in multilayered domains

Keywords: Elastic BVP, elastoplastic BVP, variational inequalities, rate-independency, hysteresis, linear kinematic hardening, stop- and play-operator (21 pages, 2006)

100. M. Speckert, K. Dreßler, H. Mauch MBS Simulation of a hexapod based suspension test rig Keywords: Test rig, MBS simulation, suspension, hydraulics, controlling, design optimization (12 pages, 2006)

101. S. Azizi Sultan, K.-H. Küfer A dynamic algorithm for beam orientations in multicriteria IMRT planning Keywords: radiotherapy planning, beam orientation optimization, dynamic approach, evolutionary algorithm, global optimization (14 pages, 2006)

102. T. Götz, A. Klar, N. Marheineke, R. Wegener A Stochastic Model for the Fiber Lay-down Process in the Nonwoven Production Keywords: fiber dynamics, stochastic Hamiltonian system, stochastic averaging (17 pages, 2006)

103. Ph. Süss, K.-H. Küfer Balancing control and simplicity: a variable aggregation method in intensity modulated radiation therapy planning Keywords: IMRT planning, variable aggregation, clustering methods (22 pages, 2006)

104. A. Beaudry, G. Laporte, T. Melo, S. Nickel Dynamic transportation of patients in hospitals Keywords: in-house hospital transportation, dial-a-ride, dynamic mode, tabu search (37 pages, 2006)

105. Th. Hanne Applying multiobjective evolutionary algorithms in industrial projects

Keywords: Biot poroelasticity system, interface problems, finite volume discretization, finite difference method (21 pages, 2006)

Keywords: multiobjective evolutionary algorithms, discrete optimization, continuous optimization, electronic circuit design, semi-infinite programming, scheduling (18 pages, 2006)

96. M. Krekel, J. Wenzel A unified approach to Credit Default Swap­ tion and Constant Maturity Credit Default Swap valuation

106. J. Franke, S. Halim Wild bootstrap tests for comparing signals and images

Keywords: LIBOR market model, credit risk, Credit Default Swaption, Constant Maturity Credit Default Swapmethod (43 pages, 2006)

97. A. Dreyer Interval Methods for Analog Circiuts Keywords: interval arithmetic, analog circuits, tolerance analysis, parametric linear systems, frequency response, symbolic analysis, CAD, computer algebra (36 pages, 2006)

98. N. Weigel, S. Weihe, G. Bitsch, K. Dreßler Usage of Simulation for Design and Optimization of Testing Keywords: Vehicle test rigs, MBS, control, hydraulics, testing philosophy (14 pages, 2006)

99. H. Lang, G. Bitsch, K. Dreßler, M. Speckert Comparison of the solutions of the elastic and elastoplastic boundary value problems

Keywords: wild bootstrap test, texture classification, textile quality control, defect detection, kernel estimate, nonparametric regression (13 pages, 2007)

107. Z. Drezner, S. Nickel Solving the ordered one-median problem in the plane Keywords: planar location, global optimization, ordered median, big triangle small triangle method, bounds, numerical experiments (21 pages, 2007)

108. Th. Götz, A. Klar, A. Unterreiter, R. Wegener Numerical evidance for the non-­existing of solutions of the equations desribing rotational fiber spinning Keywords: rotational fiber spinning, viscous fibers, boundary value problem, existence of solutions (11 pages, 2007)

109. Ph. Süss, K.-H. Küfer Smooth intensity maps and the BortfeldBoyer sequencer Keywords: probabilistic analysis, intensity modulated radiotherapy treatment (IMRT), IMRT plan application, step-and-shoot sequencing (8 pages, 2007)

110. E. Ivanov, O. Gluchshenko, H. Andrä, A. Kudryavtsev Parallel software tool for decomposing and meshing of 3d structures Keywords: a-priori domain decomposition, unstructured grid, Delaunay mesh generation (14 pages, 2007)

111. O. Iliev, R. Lazarov, J. Willems Numerical study of two-grid preconditioners for 1d elliptic problems with highly oscillating discontinuous coefficients Keywords: two-grid algorithm, oscillating coefficients, preconditioner (20 pages, 2007)

112. L. Bonilla, T. Götz, A. Klar, N. Marheineke, R. Wegener Hydrodynamic limit of the Fokker-Planckequation describing fiber lay-down processes Keywords: stochastic dierential equations, FokkerPlanck equation, asymptotic expansion, OrnsteinUhlenbeck process (17 pages, 2007)

113. S. Rief Modeling and simulation of the pressing section of a paper machine Keywords: paper machine, computational fluid dynamics, porous media (41 pages, 2007)

114. R. Ciegis, O. Iliev, Z. Lakdawala On parallel numerical algorithms for simulating industrial filtration problems Keywords: Navier-Stokes-Brinkmann equations, finite volume discretization method, SIMPLE, parallel computing, data decomposition method (24 pages, 2007)

115. N. Marheineke, R. Wegener Dynamics of curved viscous fibers with surface tension Keywords: Slender body theory, curved viscous bers with surface tension, free boundary value problem (25 pages, 2007)

116. S. Feth, J. Franke, M. Speckert Resampling-Methoden zur mse-Korrektur und Anwendungen in der Betriebsfestigkeit Keywords: Weibull, Bootstrap, Maximum-Likelihood, Betriebsfestigkeit (16 pages, 2007)

117. H. Knaf Kernel Fisher discriminant functions – a concise and rigorous introduction Keywords: wild bootstrap test, texture classification, textile quality control, defect detection, kernel estimate, nonparametric regression (30 pages, 2007)

118. O. Iliev, I. Rybak On numerical upscaling for flows in heterogeneous porous media

Keywords: numerical upscaling, heterogeneous porous media, single phase flow, Darcy‘s law, multiscale problem, effective permeability, multipoint flux approximation, anisotropy (17 pages, 2007)

119. O. Iliev, I. Rybak On approximation property of multipoint flux approximation method Keywords: Multipoint flux approximation, finite volume method, elliptic equation, discontinuous tensor coefficients, anisotropy (15 pages, 2007)

120. O. Iliev, I. Rybak, J. Willems On upscaling heat conductivity for a class of industrial problems Keywords: Multiscale problems, effective heat conductivity, numerical upscaling, domain decomposition (21 pages, 2007)

121. R. Ewing, O. Iliev, R. Lazarov, I. Rybak On two-level preconditioners for flow in porous media Keywords: Multiscale problem, Darcy‘s law, single phase flow, anisotropic heterogeneous porous media, numerical upscaling, multigrid, domain decomposition, efficient preconditioner (18 pages, 2007)

122. M. Brickenstein, A. Dreyer POLYBORI: A Gröbner basis framework for Boolean polynomials Keywords: Gröbner basis, formal verification, Boolean polynomials, algebraic cryptoanalysis, satisfiability (23 pages, 2007)

123. O. Wirjadi Survey of 3d image segmentation methods Keywords: image processing, 3d, image segmentation, binarization (20 pages, 2007)

124. S. Zeytun, A. Gupta A Comparative Study of the Vasicek and the CIR Model of the Short Rate Keywords: interest rates, Vasicek model, CIR-model, calibration, parameter estimation (17 pages, 2007)

125. G. Hanselmann, A. Sarishvili Heterogeneous redundancy in software quality prediction using a hybrid Bayesian approach Keywords: reliability prediction, fault prediction, nonhomogeneous poisson process, Bayesian model averaging (17 pages, 2007)

126. V. Maag, M. Berger, A. Winterfeld, K.-H. Küfer A novel non-linear approach to minimal area rectangular packing Keywords: rectangular packing, non-overlapping constraints, non-linear optimization, regularization, relaxation (18 pages, 2007)

127. M. Monz, K.-H. Küfer, T. Bortfeld, C. Thieke Pareto navigation – systematic multi-criteria-based IMRT treatment plan determination Keywords: convex, interactive multi-objective optimization, intensity modulated radiotherapy planning (15 pages, 2007)

128. M. Krause, A. Scherrer On the role of modeling parameters in IMRT plan optimization

137. E. Savenkov, H. Andrä, O. Iliev∗ An analysis of one regularization approach for solution of pure Neumann problem

Keywords: intensity-modulated radiotherapy (IMRT), inverse IMRT planning, convex optimization, sensitivity analysis, elasticity, modeling parameters, equivalent uniform dose (EUD) (18 pages, 2007)

Keywords: pure Neumann problem, elasticity, regularization, finite element method, condition number (27 pages, 2008)

129. A. Wiegmann Computation of the ­permeability of porous materials from their microstructure by FFFStokes Keywords: permeability, numerical homogenization, fast Stokes solver (24 pages, 2007)

130. T. Melo, S. Nickel, F. Saldanha da Gama Facility Location and Supply Chain Management – A comprehensive review Keywords: facility location, supply chain management, network design (54 pages, 2007)

131. T. Hanne, T. Melo, S. Nickel Bringing robustness to patient flow manage­ment through optimized patient transports in hospitals Keywords: Dial-a-Ride problem, online problem, case study, tabu search, hospital logistics (23 pages, 2007)

132. R. Ewing, O. Iliev, R. Lazarov, I. Rybak, J. Willems An efficient approach for upscaling properties of composite materials with high contrast of coefficients Keywords: effective heat conductivity, permeability of fractured porous media, numerical upscaling, fibrous insulation materials, metal foams (16 pages, 2008)

133. S. Gelareh, S. Nickel New approaches to hub location problems in public transport planning Keywords: integer programming, hub location, transportation, decomposition, heuristic (25 pages, 2008)

134. G. Thömmes, J. Becker, M. Junk, A. K. Vaikuntam, D. Kehrwald, A. Klar, K. Steiner, A. Wiegmann A Lattice Boltzmann Method for immiscible multiphase flow simulations using the ­Level Set Method Keywords: Lattice Boltzmann method, Level Set method, free surface, multiphase flow (28 pages, 2008)

135. J. Orlik Homogenization in elasto-plasticity Keywords: multiscale structures, asymptotic homogenization, nonlinear energy (40 pages, 2008)

136. J. Almquist, H. Schmidt, P. Lang, J. Deitmer, M. Jirstrand, D. Prätzel-Wolters, H. Becker Determination of interaction between MCT1 and CAII via a mathematical and physiological approach Keywords: mathematical modeling; model reduction; electrophysiology; pH-sensitive microelectrodes; proton antenna (20 pages, 2008)

138. O. Berman, J. Kalcsics, D. Krass, S. Nickel The ordered gradual covering location problem on a network Keywords: gradual covering, ordered median function, network location (32 pages, 2008)

139. S. Gelareh, S. Nickel Multi-period public transport ­design: A novel model and solution a ­ pproaches Keywords: Integer programming, hub location, public transport, multi-period planning, heuristics (31 pages, 2008)

140. T. Melo, S. Nickel, F. Saldanha-da-Gama Network design decisions in supply chain planning Keywords: supply chain design, integer programming models, location models, heuristics (20 pages, 2008)

141. C. Lautensack, A. Särkkä, J. Freitag, K. Schladitz Anisotropy analysis of pressed point processes Keywords: estimation of compression, isotropy test, nearest neighbour distance, orientation analysis, polar ice, Ripley’s K function (35 pages, 2008)

142. O. Iliev, R. Lazarov, J. Willems A Graph-Laplacian approach for calculating the effective thermal conductivity of complicated fiber geometries Keywords: graph laplacian, effective heat conductivity, numerical upscaling, fibrous materials (14 pages, 2008)

143. J. Linn, T. Stephan, J. Carlsson, R. Bohlin Fast simulation of quasistatic rod deformations for VR applications Keywords: quasistatic deformations, geometrically exact rod models, variational formulation, energy minimization, finite differences, nonlinear conjugate gradients (7 pages, 2008)

144. J. Linn, T. Stephan Simulation of quasistatic deformations using discrete rod models Keywords: quasistatic deformations, geometrically exact rod models, variational formulation, energy minimization, finite differences, nonlinear conjugate gradients (9 pages, 2008)

145. J. Marburger, N. Marheineke, R. Pinnau Adjoint based optimal control using meshless discretizations Keywords: Mesh-less methods, particle methods, Eulerian-Lagrangian formulation, optimization strategies, adjoint method, hyperbolic equations (14 pages, 2008

146. S. Desmettre, J. Gould, A. Szimayer Own-company stockholding and work effort preferences of an unconstrained executive Keywords: optimal portfolio choice, executive compensation (33 pages, 2008)

147. M. Berger, M. Schröder, K.-H. Küfer A constraint programming approach for the two-dimensional rectangular packing problem with orthogonal orientations Keywords: rectangular packing, orthogonal orientations non-overlapping constraints, constraint propagation (13 pages, 2008)

148. K. Schladitz, C. Redenbach, T. Sych, M. Godehardt Microstructural characterisation of open foams using 3d images Keywords: virtual material design, image analysis, open foams (30 pages, 2008)

149. E. Fernández, J. Kalcsics, S. Nickel, R. Ríos-Mercado A novel territory design model arising in the implementation of the WEEE-Directive Keywords: heuristics, optimization, logistics, recycling (28 pages, 2008)

150. H. Lang, J. Linn Lagrangian field theory in space-time for geometrically exact Cosserat rods Keywords: Cosserat rods, geometrically exact rods, small strain, large deformation, deformable bodies, Lagrangian field theory, variational calculus (19 pages, 2009)

151. K. Dreßler, M. Speckert, R. Müller, Ch. Weber Customer loads correlation in truck engineering Keywords: Customer distribution, safety critical components, quantile estimation, Monte-Carlo methods (11 pages, 2009)

152. H. Lang, K. Dreßler An improved multiaxial stress-strain correction model for elastic FE postprocessing Keywords: Jiang’s model of elastoplasticity, stress-strain correction, parameter identification, automatic differentiation, least-squares optimization, Coleman-Li algorithm (6 pages, 2009)

153. J. Kalcsics, S. Nickel, M. Schröder A generic geometric approach to territory design and districting Keywords: Territory design, districting, combinatorial optimization, heuristics, computational geometry (32 pages, 2009)

154. Th. Fütterer, A. Klar, R. Wegener An energy conserving numerical scheme for the dynamics of hyper­elastic rods Keywords: Cosserat rod, hyperealstic, energy conservation, finite differences (16 pages, 2009)

155. A. Wiegmann, L. Cheng, E. Glatt, O. Iliev, S. Rief Design of pleated filters by computer simulations Keywords: Solid-gas separation, solid-liquid separation, pleated filter, design, simulation (21 pages, 2009)

156. A. Klar, N. Marheineke, R. Wegener Hierarchy of mathematical models for production processes of technical textiles

Keywords: Fiber-fluid interaction, slender-body theory, turbulence modeling, model reduction, stochastic differential equations, Fokker-Planck equation, asymptotic expansions, parameter identification (21 pages, 2009)

157. E. Glatt, S. Rief, A. Wiegmann, M. Knefel, E. Wegenke Structure and pressure drop of real and virtual metal wire meshes Keywords: metal wire mesh, structure simulation, model calibration, CFD simulation, pressure loss (7 pages, 2009)

158. S. Kruse, M. Müller Pricing American call options under the assumption of stochastic dividends – An application of the Korn-Rogers model Keywords: option pricing, American options, dividends, dividend discount model, Black-Scholes model (22 pages, 2009)

159. H. Lang, J. Linn, M. Arnold Multibody dynamics simulation of geometrically exact Cosserat rods Keywords: flexible multibody dynamics, large deformations, finite rotations, constrained mechanical systems, structural dynamics (20 pages, 2009)

160. P. Jung, S. Leyendecker, J. Linn, M. Ortiz Discrete Lagrangian mechanics and geometrically exact Cosserat rods Keywords: special Cosserat rods, Lagrangian mechanics, Noether’s theorem, discrete mechanics, frame-indifference, holonomic constraints (14 pages, 2009)

161. M. Burger, K. Dreßler, A. Marquardt, M. Speckert Calculating invariant loads for system simulation in vehicle engineering Keywords: iterative learning control, optimal control theory, differential algebraic equations(DAEs) (18 pages, 2009)

162. M. Speckert, N. Ruf, K. Dreßler Undesired drift of multibody models excited by measured accelerations or forces Keywords: multibody simulation, full vehicle model, force-based simulation, drift due to noise (19 pages, 2009)

166. J. I. Serna, M. Monz, K.-H. Küfer, C. Thieke Trade-off bounds and their effect in multicriteria IMRT planning Keywords: trade-off bounds, multi-criteria optimization, IMRT, Pareto surface (15 pages, 2009)

167. W. Arne, N. Marheineke, A. Meister, R. Wegener Numerical analysis of Cosserat rod and string models for viscous jets in rotational spinning processes Keywords: Rotational spinning process, curved viscous fibers, asymptotic Cosserat models, boundary value problem, existence of numerical solutions (18 pages, 2009)

168. T. Melo, S. Nickel, F. Saldanha-da-Gama An LP-rounding heuristic to solve a multiperiod facility relocation problem Keywords: supply chain design, heuristic, linear programming, rounding (37 pages, 2009)

169. I. Correia, S. Nickel, F. Saldanha-da-Gama Single-allocation hub location problems with capacity choices Keywords: hub location, capacity decisions, MILP formulations (27 pages, 2009)

170. S. Acar, K. Natcheva-Acar A guide on the implementation of the Heath-Jarrow-Morton Two-Factor Gaussian Short Rate Model (HJM-G2++) Keywords: short rate model, two factor Gaussian, G2++, option pricing, calibration (30 pages, 2009)

171. A. Szimayer, G. Dimitroff, S. Lorenz A parsimonious multi-asset Heston model: calibration and derivative pricing Keywords: Heston model, multi-asset, option pricing, calibration, correlation (28 pages, 2009)

172. N. Marheineke, R. Wegener Modeling and validation of a stochastic drag for fibers in turbulent flows Keywords: fiber-fluid interactions, long slender fibers, turbulence modelling, aerodynamic drag, dimensional analysis, data interpolation, stochastic partial differential algebraic equation, numerical simulations, experimental validations (19 pages, 2009)

163. A. Streit, K. Dreßler, M. Speckert, J. Lichter, T. Zenner, P. Bach Anwendung statistischer Methoden zur Erstellung von Nutzungsprofilen für die Auslegung von Mobilbaggern

173. S. Nickel, M. Schröder, J. Steeg Planning for home health care services

Keywords: Nutzungsvielfalt, Kundenbeanspruchung, Bemessungsgrundlagen (13 pages, 2009)

Keywords: home health care, route planning, metaheuristics, constraint programming (23 pages, 2009)

164. I. Correia, S. Nickel, F. Saldanha-da-Gama Anwendung statistischer Methoden zur Erstellung von Nutzungsprofilen für die Auslegung von Mobilbaggern

174. G. Dimitroff, A. Szimayer, A. Wagner Quanto option pricing in the parsimonious Heston model

Keywords: Capacitated Hub Location, MIP formulations (10 pages, 2009)

165. F. Yaneva, T. Grebe, A. Scherrer An alternative view on global radiotherapy optimization problems Keywords: radiotherapy planning, path-connected sublevelsets, modified gradient projection method, improving and feasible directions (14 pages, 2009)

Keywords: Heston model, multi asset, quanto options, option pricing (14 pages, 2009) 174. G. Dimitroff, A. Szimayer, A. Wagner

175. S. Herkt, K. Dreßler, R. Pinnau Model reduction of nonlinear problems in structural mechanics Keywords: flexible bodies, FEM, nonlinear model reduction, POD (13 pages, 2009)

176. M. K. Ahmad, S. Didas, J. Iqbal Using the Sharp Operator for edge detection and nonlinear diffusion Keywords: maximal function, sharp function,image processing, edge detection, nonlinear diffusion (17 pages, 2009)

177. M. Speckert, N. Ruf, K. Dreßler, R. Müller, C. Weber, S. Weihe Ein neuer Ansatz zur Ermittlung von Erprobungslasten für sicherheitsrelevante Bauteile Keywords: sicherheitsrelevante Bauteile, Kundenbean­ spruchung, Festigkeitsverteilung, Ausfallwahrscheinlichkeit, Konfidenz, statistische Unsicherheit, Sicherheitsfaktoren (16 pages, 2009)

178. J. Jegorovs Wave based method: new applicability areas Keywords: Elliptic boundary value problems, inhomogeneous Helmholtz type differential equations in bounded domains, numerical methods, wave based method, uniform B-splines (10 pages, 2009)

179. H. Lang, M. Arnold Numerical aspects in the dynamic simulation of geometrically exact rods Keywords: Kirchhoff and Cosserat rods, geometri­ cally exact rods, deformable bodies, multibody dynamics,artial differential algebraic equations, method of lines, time integration (21 pages, 2009)

180. H. Lang Comparison of quaternionic and rotationfree null space formalisms for multibody dynamics Keywords: Parametrisation of rotations, differentialalgebraic equations, multibody dynamics, con­strained mechanical systems, Lagrangian mechanics (40 pages, 2010)

181. S. Nickel, F. Saldanha-da-Gama, H.-P. Ziegler Stochastic programming approaches for risk aware supply chain network design problems Keywords: Supply Chain Management, multi-stage stochastic programming, financial decisions, risk (37 pages, 2010)

182. P. Ruckdeschel, N. Horbenko Robustness properties of estimators in generalized Pareto Models Keywords: global robustness, local robustness, finite sample breakdown point, generalized Pareto distribution (58 pages, 2010)

183. P. Jung, S. Leyendecker, J. Linn, M. Ortiz A discrete mechanics approach to Cosserat rod theory – Part 1: static equilibria Keywords: Special Cosserat rods; Lagrangian mechanics; Noether’s theorem; discrete mechanics; frameindifference; holonomic constraints; variational formulation (35 pages, 2010)

184. R. Eymard, G. Printsypar A proof of convergence of a finite volume scheme for modified steady Richards’ equation describing transport processes in the pressing section of a paper machine Keywords: flow in porous media, steady Richards’ equation, finite volume methods, convergence of approximate solution (14 pages, 2010)

185. P. Ruckdeschel Optimally Robust Kalman Filtering Keywords: robustness, Kalman Filter, innovation outlier, additive outlier (42 pages, 2010)

186. S. Repke, N. Marheineke, R. Pinnau On adjoint-based optimization of a free surface Stokes flow Keywords: film casting process, thin films, free surface Stokes flow, optimal control, Lagrange formalism (13 pages, 2010)

187. O. Iliev, R. Lazarov, J. Willems Variational multiscale Finite Element ­Method for flows in highly porous media Keywords: numerical upscaling, flow in heterogeneous porous media, Brinkman equations, Darcy’s law, subgrid approximation, discontinuous Galerkin mixed FEM (21 pages, 2010)

188. S. Desmettre, A. Szimayer Work effort, consumption, and portfolio selection: When the occupational choice matters Keywords: portfolio choice, work effort, consumption, occupational choice (34 pages, 2010)

189. O. Iliev, Z. Lakdawala, V. Starikovicius On a numerical subgrid upscaling algorithm for Stokes-Brinkman equations Keywords: Stokes-Brinkman equations, subgrid approach, multiscale problems, numerical upscaling (27 pages, 2010)

Status quo: July 2010