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- 2008 (16) (remove)

#### Keywords

- energy minimization (2)
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- FETI (1)
- Gradual Covering (1)
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- optimal portfolio choice (1)
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#### Faculty / Organisational entity

- Fraunhofer (ITWM) (16) (remove)

We propose a constraint-based approach for the two-dimensional rectangular packing problem with orthogonal orientations. This problem is to arrange a set of rectangles that can be rotated by 90 degrees into a rectangle of minimal size such that no two rectangles overlap. It arises in the placement of electronic devices during the layout of 2.5D System-in-Package integrated electronic systems. Moffitt et al. [8] solve the packing without orientations with a branch and bound approach and use constraint propagation. We generalize their propagation techniques to allow orientations. Our approach is compared to a mixed-integer program and we provide results that outperform it.

Open cell foams are a promising and versatile class of porous materials. Open metal foams serve as crash absorbers and catalysts, metal and ceramic foams are used for filtering, and open polymer foams are hidden in every-day-life items like mattresses or chairs. Due to their high porosity, classical 2d quantitative analysis can give only very limited information about the microstructure of open foams. On the other hand, micro computed tomography (μCT) yields high quality 3d images of open foams. Thus 3d imaging is the method of choice for open cell foams. In this report we summarise a variety of methods for the analysis of the resulting volume images of open foam structures developed or refined and applied at the Fraunhofer ITWM over a course of nearly ten years: The model based determination of mean characteristics like the mean cell volume or the mean strut thickness demanding only a simple binarisation as well as the image analytic cell reconstruction yielding empirical distributions of cell characteristics.

The problem discussed in this paper is motivated by the new recycling directiveWEEE of the EC. The core of this law is, that each company which sells electrical or electronic equipment in a European country has the obligation to recollect and recycle an amount of returned items which is proportional to its market share. To assign collection stations to companies, in Germany for one product type a territory design approach is planned. However, in contrast to classical territory design, the territories should be geographically as dispersed as possible to avoid that a company, resp. its logistics provider responsible for the recollection, gains a monopoly in some region. First, we identify an appropriate measure for the dispersion of a territory. Afterwards, we present a first mathematical programming model for this new problem as well as a solution method based on the GRASP methodology. Extensive computational results illustrate the suitability of the model and assess the effectiveness of the heuristic.

We develop a framework for analyzing an executive’s own-company stockholding and work effort preferences. The executive, characterized by risk aversion and work effectiveness parameters, invests his personal wealth without constraint in the financial market, including the stock of his own company whose value he can directly influence with work effort. The executive’s utility-maximizing personal investment and work effort strategy is derived in closed-form, and an indifference utility rationale is demonstrated to determine his required compensation. Our results have implications for the practical and theoretical assessment of executive quality and the benefits of performance contracting. Assuming knowledge of the company’s non-systematic risk, our executive’s unconstrained own-company investment identifies his work effectiveness (i.e. quality), and also reflects work effort that establishes a base-level that performance contracting should seek to exceed.

An easy numerical handling of time-dependent problems with complicated geometries, free moving boundaries and interfaces, or oscillating solutions is of great importance for many applications, e.g., in fluid dynamics (free surface and multiphase flows, fluid-structure interactions [22, 18, 24]), failure mechanics (crack growth and propagation [4]), magnetohydrodynamics (accretion disks, jets and cloud simulation [6]), biophysics and -chemistry. Appropriate discretizations, so-called mesh-less methods, have been developed during the last decades to meet these challenging demands and to relieve the burden of remeshing and successive mesh generation being faced by the conventional mesh-based methods, [16, 10, 3]. The prearranged mesh is an artificial constraint to ensure compatibility of the mesh-based interpolant schemes, that often conflicts with the real physical conditions of the continuum model. Then, remeshing becomes inevitable, which is not only extremely time- and storage consuming but also the source for numerical errors and hence the gradual loss of computational accuracy. Apart from this advantage, mesh-less methods also lead to fundamentally better approximations regarding aspects, such as smoothness, nonlocal interpolation character, flexible connectivity, refinement and enrichment procedures, [16]. The common idea of mesh-less methods is the discretization of the domain of interest by a finite set of independent, randomly distributed particles moving with a characteristic velocity of the problem. Location and distribution of the particles then account for the time-dependent description of the geometry, data and solution. Thereby, the global solution is linearly superposed from the local information carried by the particles. In classical particle methods [20, 21], the respective weight functions are Dirac distributions which yield solutions in a distributional sense.

Recently we developed a discrete model of elastic rods with symmetric cross section suitable for a fast simulation of quasistatic deformations [33]. The model is based on Kirchhoff’s geometrically exact theory of rods. Unlike simple models of “mass & spring” type typically used in VR applications, our model provides a proper coupling of bending and torsion. The computational approach comprises a variational formulation combined with a finite difference discretization of the continuum model. Approximate solutions of the equilibrium equations for sequentially varying boundary conditions are obtained by means of energy minimization using a nonlinear CG method. As the computational performance of our model yields solution times within the range of milliseconds, our approach proves to be sufficient to simulate an interactive manipulation of such flexible rods in virtual reality applications in real time.

Summary. We present a model of exible rods | based on Kirchhoff\\\'s geometrically exact theory | which is suitable for the fast simulation of quasistatic deformations within VR or functional DMU applications. Unlike simple models of \\\"mass & spring\\\" type typically used in VR applications, our model provides a proper coupling of bending and torsion. The computational approach comprises a variational formulation combined with a nite dierence discretization of the continuum model. Approximate solutions of the equilibrium equations for sequentially varying boundary conditions are obtained by means of energy minimization using a nonlinear CG method. The computational performance of our model proves to be sucient for the interactive manipulation of exible cables in assembly simulation.

Abstract. An efficient approach to the numerical upscaling of thermal conductivities of fibrous media, e.g. insulation materials, is considered. First, standard cell problems for a second order elliptic equation are formulated for a proper piece of random fibrous structure, following homogenization theory. Next, a graph formed by the fibers is considered, and a second order elliptic equation with suitable boundary conditions is solved on this graph only. Replacing the boundary value problem for the full cell with an auxiliary problem with special boundary conditions on a connected subdomain of highly conductive material is justified in a previous work of the authors. A discretization on the graph is presented here, and error estimates are provided. The efficient implementation of the algorithm is discussed. A number of numerical experiments is presented in order to illustrate the performance of the proposed method.

This paper introduces methods for the detection of anisotropies which are caused by compression of regular three-dimensional point patterns. Isotropy tests based on directional summary statistics and estimators for the compression factor are developed. These allow not only for the detection of anisotropies but also for the estimation of their strength. Using simulated data the power of the methods and the dependence of the power on the intensity, the degree of regularity, and the compression strength are studied. The motivation of this paper is the investigation of anisotropies in the structure of polar ice. Therefore, our methods are applied to the point patterns of centres of air pores extracted from tomographic images of ice cores. This way the presence of anisotropies in the ice caused by the compression of the ice sheet as well as an increase of their strength with increasing depth are shown.

A Lattice Boltzmann Method for immiscible multiphase flow simulations using the Level Set Method
(2008)

We consider the lattice Boltzmann method for immiscible multiphase flow simulations. Classical lattice Boltzmann methods for this problem, e.g. the colour gradient method or the free energy approach, can only be applied when density and viscosity ratios are small. Moreover, they use additional fields defined on the whole domain to describe the different phases and model phase separation by special interactions at each node. In contrast, our approach simulates the flow using a single field and separates the fluid phases by a free moving interface. The scheme is based on the lattice Boltzmann method and uses the level set method to compute the evolution of the interface. To couple the fluid phases, we develop new boundary conditions which realise the macroscopic jump conditions at the interface and incorporate surface tension in the lattice Boltzmann framework. Various simulations are presented to validate the numerical scheme, e.g. two-phase channel flows, the Young-Laplace law for a bubble and viscous fingering in a Hele-Shaw cell. The results show that the method is feasible over a wide range of density and viscosity differences.

The theory of the two-scale convergence was applied to homogenization of elasto-plastic composites with a periodic structure and exponential hardening law. The theory is based on the fact that the elastic as well as the plastic part of the stress field two-scale converges to a limit, which is factorized by parts, depending only on macroscopic characteristics, represented in terms of corresponding part of the homogenised stress tensor and only on stress concentration tensor, related to the micro-geometry and elastic or plastic micro-properties of composite components. The theory was applied to metallic matrix material with Ludwik and Hocket-Sherby hardening law and pure elastic inclusions in two numerical examples. Results were compared with results of mechanical averaging based on the self-consistent methods.

Determination of interaction between MCT1 and CAII via a mathematical and physiological approach
(2008)

The enzyme carbonic anhydrase isoform II (CAII), catalysing the hydration and dehydration of CO2, enhances transport activity of the monocarboxylate transporter isoform I (MCT1, SLC16A1) expressed in Xenopus oocytes by a mechanism that does not require CAII catalytic activity (Becker et al. (2005) J. Biol. Chem., 280). In the present study, we have investigated the mechanism of the CAII induced increase in transport activity by using electrophysiological techniques and a mathematical model of the MCT1 transport cycle. The model consists of six states arranged in cyclic fashion and features an ordered, mirror-symmetric, binding mechanism were binding and unbinding of the proton to the transport protein is considered to be the rate limiting step under physiological conditions. An explicit rate expression for the substrate °ux is derived using model reduction techniques. By treating the pools of intra- and extracellular MCT1 substrates as dynamic states, the time dependent kinetics are obtained by integration using the derived expression for the substrate °ux. The simulations were compared with experimental data obtained from MCT1-expressing oocytes injected with di®erent amounts of CAII. The model suggests that CAII increases the e®ective rate constants of the proton reactions, possibly by working as a proton antenna.

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

In this paper we develop a network location model that combines the characteristics of ordered median and gradual cover models resulting in the Ordered Gradual Covering Location Problem (OGCLP). The Gradual Cover Location Problem (GCLP) was specifically designed to extend the basic cover objective to capture sensitivity with respect to absolute travel distance. Ordered Median Location problems are a generalization of most of the classical locations problems like p-median or p-center problems. They can be modeled by using so-called ordered median functions. These functions multiply a weight to the cost of fulfilling the demand of a customer which depends on the position of that cost relative to the costs of fulfilling the demand of the other customers. We derive Finite Dominating Sets (FDS) for the one facility case of the OGCLP. Moreover, we present efficient algorithms for determining the FDS and also discuss the conditional case where a certain number of facilities are already assumed to exist and one new facility is to be added. For the multi-facility case we are able to identify a finite set of potential facility locations a priori, which essentially converts the network location model into its discrete counterpart. For the multi-facility discrete OGCLP we discuss several Integer Programming formulations and give computational results.

In this paper, we are going to propose the first mathematical model for Multi- Period Hub Location Problems (MPHLP). We apply this mixed integer program- ming model on public transport planning and call it Multi-Period Hub Location Problem for Public Transport (MPHLPPT). In fact, HLPPT model proposed earlier by the authors is extended to include more facts and features of the real-life application. In order to solve instances of this problem where existing standard solvers fail, a solution approach based on a greedy neighborhood search is developed. The computational results substantiate the efficiency of our solution approach to solve instances of MPHLPPT.

Structuring global supply chain networks is a complex decision-making process. The typical inputs to such a process consist of a set of customer zones to serve, a set of products to be manufactured and distributed, demand projections for the different customer zones, and information about future conditions, costs (e.g. for production and transportation) and resources (e.g. capacities, available raw materials). Given the above inputs, companies have to decide where to locate new service facilities (e.g. plants, warehouses), how to allocate procurement and production activities to the variousmanufacturing facilities, and how to manage the transportation of products through the supply chain network in order to satisfy customer demands. We propose a mathematical modelling framework capturing many practical aspects of network design problems simultaneously. For problems of reasonable size we report on computational experience with standard mathematical programming software. The discussion is extended with other decisions required by many real-life applications in strategic supply chain planning. In particular, the multi-period nature of some decisions is addressed by a more comprehensivemodel, which is solved by a specially tailored heuristic approach. The numerical results suggest that the solution procedure can identify high quality solutions within reasonable computational time.