## Fraunhofer (ITWM)

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#### Keywords

- numerical upscaling (6)
- Darcy’s law (3)
- effective heat conductivity (3)
- facility location (3)
- non-Newtonian flow in porous media (3)
- poroelasticity (3)
- virtual material design (3)
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- Image based characterization and geometric modeling of 3d materials microstructures (2015)
- It is well known that the structure at a microscopic point of view strongly influences the macroscopic properties of materials. Moreover, the advancement in imaging technologies allows to capture the complexity of the structures at always decreasing scales. Therefore, more sophisticated image analysis techniques are needed. This thesis provides tools to geometrically characterize different types of three-dimensional structures with applications to industrial production and to materials science. Our goal is to enhance methods that allow the extraction of geometric features from images and the automatic processing of the information. In particular, we investigate which characteristics are sufficient and necessary to infer the desired information, such as particles classification for technical cleanliness and fitting of stochastic models in materials science. In the production line of automotive industry, dirt particles collect on the surface of mechanical components. Residual dirt might reduce the performance and durability of assembled products. Geometric characterization of these particles allows to identify their potential danger. While the current standards are based on 2d microscopic images, we extend the characterization to 3d. In particular, we provide a collection of parameters that exhaustively describe size and shape of three-dimensional objects and can be efficiently estimated from binary images. Furthermore, we show that only a few features are sufficient to classify particles according to the standards of technical cleanliness. In the context of materials science, we consider two types of microstructures: fiber systems and foams. Stochastic geometry grants the fundamentals for versatile models able to encompass the geometry observed in the samples. To allow automatic model fitting, we need rules stating which parameters of the model yield the best-fitting characteristics. However, the validity of such rules strongly depends on the properties of the structures and on the choice of the model. For instance, isotropic orientation distribution yields the best theoretical results for Boolean models and Poisson processes of cylinders with circular cross sections. Nevertheless, fiber systems in composites are often anisotropic. Starting from analytical results from the literature, we derive formulae for anisotropic Poisson processes of cylinders with polygonal cross sections that can be directly used in applications. We apply this procedure to a sample of medium density fiber board. Even if image resolution does not allow to estimate reliably characteristics of the singles fibers, we can fit Boolean models and Poisson cylinder processes. In particular, we show the complete model fitting and validation procedure with cylinders with circular and squared cross sections. Different problems arise when modeling cellular materials. Motivated by the physics of foams, random Laguerre tessellations are a good choice to model the pore system of foams. Considering tessellations generated by systems of non-overlapping spheres allows to control the cell size distribution, but yields the loss of an analytical description of the model. Nevertheless, automatic model fitting can still be obtained by approximating the characteristics of the tessellation depending on the parameters of the model. We investigate how to improve the choice of the model parameters. Angles between facets and between edges were never considered so far. We show that the distributions of angles in Laguerre tessellations depend on the model parameters. Thus, including the moments of the angles still allows automatic model fitting. Moreover, we propose an algorithm to estimate angles from images of real foams. We observe that angles are matched well in random Laguerre tessellations also when they are not employed to choose the model parameters. Then, we concentrate on the edge length distribution. In Laguerre tessellations occur many more short edges than in real foams. To deal with this problem, we consider relaxed models. Relaxation refers to topological and structural modifications of a tessellation in order to make it comply with Plateau's laws of mechanical equilibrium. We inspect samples of different types of foams, closed and open cell foams, polymeric and metallic. By comparing the geometric characteristics of the model and of the relaxed tessellations, we conclude that whether the relaxation improves the edge length distribution strongly depends on the type of foam.

- Test rig optimization (2014)
- Designing good test rigs for fatigue life tests is a common task in the auto- motive industry. The problem to find an optimal test rig configuration and actuator load signals can be formulated as a mathematical program. We in- troduce a new optimization model that includes multi-criteria, discrete and continuous aspects. At the same time we manage to avoid the necessity to deal with the rainflow-counting (RFC) method. RFC is an algorithm, which extracts load cycles from an irregular time signal. As a mathematical func- tion it is non-convex and non-differentiable and, hence, makes optimization of the test rig intractable. The block structure of the load signals is assumed from the beginning. It highly reduces complexity of the problem without decreasing the feasible set. Also, we optimize with respect to the actuators’ positions, which makes it possible to take torques into account and thus extend the feasible set. As a result, the new model gives significantly better results, compared with the other approaches in the test rig optimization. Under certain conditions, the non-convex test rig problem is a union of convex problems on cones. Numerical methods for optimization usually need constraints and a starting point. We describe an algorithm that detects each cone and its interior point in a polynomial time. The test rig problem belongs to the class of bilevel programs. For every instance of the state vector, the sum of functions has to be maximized. We propose a new branch and bound technique that uses local maxima of every summand.

- Construction of discrete shell models by geometric finite differences (2012)
- In the presented work, we make use of the strong reciprocity between kinematics and geometry to build a geometrically nonlinear, shearable low order discrete shell model of Cosserat type defined on triangular meshes, from which we deduce a rotation–free Kirchhoff type model with the triangle vertex positions as degrees of freedom. Both models behave physically plausible already on very coarse meshes, and show good convergence properties on regular meshes. Moreover, from the theoretical side, this deduction provides a common geometric framework for several existing models.

- Integration of nonlinear models of flexible body deformation in Multibody System Dynamics (2012)
- A simple transformation of the Equation of Motion (EoM) allows us to directly integrate nonlinear structural models into the recursive Multibody System (MBS) formalism of SIMPACK. This contribution describes how the integration is performed for a discrete Cosserat rod model which has been developed at the ITWM. As a practical example, the run-up of a simplified three-bladed wind turbine is studied where the dynamic deformations of the three blades are calculated by the Cosserat rod model.

- Geometrically exact Cosserat rods with Kelvin-Voigt type viscous damping (2012)
- We present the derivation of a simple viscous damping model of Kelvin–Voigt type for geometrically exact Cosserat rods from three–dimensional continuum theory. Assuming a homogeneous and isotropic material, we obtain explicit formulas for the damping parameters of the model in terms of the well known stiffness parameters of the rod and the retardation time constants defined as the ratios of bulk and shear viscosities to the respective elastic moduli. We briefly discuss the range of validity of our damping model and illustrate its behaviour with a numerical example.

- Multi-level Monte Carlo methods using ensemble level mixed MsFEM for two-phase flow and transport simulations (2012)
- In this paper, we propose multi-level Monte Carlo(MLMC) methods that use ensemble level mixed multiscale methods in the simulations of multi-phase flow and transport. The main idea of ensemble level multiscale methods is to construct local multiscale basis functions that can be used for any member of the ensemble. We consider two types of ensemble level mixed multiscale finite element methods, (1) the no-local-solve-online ensemble level method (NLSO) and (2) the local-solve-online ensemble level method (LSO). Both mixed multiscale methods use a number of snapshots of the permeability media to generate a multiscale basis. As a result, in the offline stage, we construct multiple basis functions for each coarse region where basis functions correspond to different realizations. In the no-local-solve-online ensemble level method one uses the whole set of pre-computed basis functions to approximate the solution for an arbitrary realization. In the local-solve-online ensemble level method one uses the pre-computed functions to construct a multiscale basis for a particular realization. With this basis the solution corresponding to this particular realization is approximated in LSO mixed MsFEM. In both approaches the accuracy of the method is related to the number of snapshots computed based on different realizations that one uses to pre-compute a multiscale basis. We note that LSO approaches share similarities with reduced basis methods [11, 21, 22]. In multi-level Monte Carlo methods ([14, 13]), more accurate (and expensive) forward simulations are run with fewer samples while less accurate(and inexpensive) forward simulations are run with a larger number of samples. Selecting the number of expensive and inexpensive simulations carefully, one can show that MLMC methods can provide better accuracy at the same cost as MC methods. In our simulations, our goal is twofold. First, we would like to compare NLSO and LSO mixed MsFEMs. In particular, we show that NLSO mixed MsFEM is more accurate compared to LSO mixed MsFEM. Further, we use both approaches in the context of MLMC to speed-up MC calculations. We present basic aspects of the algorithm and numerical results for coupled flow and transport in heterogeneous porous media.

- A direction splitting approach for incompressible Brinkmann flow (2012)
- The direction splitting approach proposed earlier in [6], aiming at the efficient solution of Navier-Stokes equations, is extended and adopted here to solve the Navier-Stokes-Brinkman equations describing incompressible flows in plain and in porous media. The resulting pressure equation is a perturbation of the incompressibility constrained using a direction-wise factorized operator as proposed in [6]. We prove that this approach is unconditionally stable for the unsteady Navier-Stokes-Brinkman problem. We also provide numerical illustrations of the method's accuracy and efficiency.

- Constitutive models for static granular systems and focus to the Jiang-Liu hyperelastic law (2012)
- Granular systems in solid-like state exhibit properties like stiffness dependence on stress, dilatancy, yield or incremental non-linearity that can be described within the continuum mechanical framework. Different constitutive models have been proposed in the literature either based on relations between some components of the stress tensor or on a quasi-elastic description. After a brief description of these models, the hyperelastic law recently proposed by Jiang and Liu [1] will be investigated. In this framework, the stress-strain relation is derived from an elastic strain energy density where the stable proper- ties are linked to a Drucker-Prager yield criteria. Further, a numerical method based on the finite element discretization and Newton- Raphson iterations is presented to solve the force balance equation. The 2D numerical examples presented in this work show that the stress distributions can be computed not only for triangular domains, as previoulsy done in the literature, but also for more complex geometries. If the slope of the heap is greater than a critical value, numerical instabilities appear and no elastic solution can be found, as predicted by the theory. As main result, the dependence of the material parameter Xi on the maximum angle of repose is established.

- An overview on the usage of some model reduction approaches for simulations of Li-ion transport in batteries (2012)
- In this work, some model reduction approaches for performing simulations with a pseudo-2D model of Li-ion battery are presented. A full pseudo-2D model of processes in Li-ion batteries is presented following [3], and three methods to reduce the order of the full model are considered. These are: i) directly reduce the model order using proper orthogonal decomposition, ii) using fractional time step discretization in order to solve the equations in decoupled way, and iii) reformulation approaches for the diffusion in the solid phase. Combinations of above methods are also considered. Results from numerical simulations are presented, and the efficiency and the accuracy of the model reduction approaches are discussed.

- Residual Demand Modeling and Application to Electricity Pricing (2012)
- Worldwide the installed capacity of renewable technologies for electricity production is rising tremendously. The German market is particularly progressive and its regulatory rules imply that production from renewables is decoupled from market prices and electricity demand. Conventional generation technologies are to cover the residual demand (defined as total demand minus production from renewables) but set the price at the exchange. Existing electricity price models do not account for the new risks introduced by the volatile production of renewables and their effects on the conventional demand curve. A model for residual demand is proposed, which is used as an extension of supply/demand electricity price models to account for renewable infeed in the market. Infeed from wind and solar (photovoltaics) is modeled explicitly and withdrawn from total demand. The methodology separates the impact of weather and capacity. Efficiency is transformed on the real line using the logit-transformation and modeled as a stochastic process. Installed capacity is assumed a deterministic function of time. In a case study the residual demand model is applied to the German day-ahead market using a supply/demand model with a deterministic supply-side representation. Price trajectories are simulated and the results are compared to market future and option prices. The trajectories show typical features seen in market prices in recent years and the model is able to closely reproduce the structure and magnitude of market prices. Using the simulated prices it is found that renewable infeed increases the volatility of forward prices in times of low demand, but can reduce volatility in peak hours. Prices for different scenarios of installed wind and solar capacity are compared and the meritorder effect of increased wind and solar capacity is calculated. It is found that wind has a stronger overall effect than solar, but both are even in peak hours.