## Fraunhofer (ITWM)

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#### Erscheinungsjahr

#### Dokumenttyp

- Bericht (198) (entfernen)

#### Schlagworte

- numerical upscaling (7)
- Integer programming (4)
- hub location (4)
- Darcy’s law (3)
- Heston model (3)
- Lagrangian mechanics (3)
- effective heat conductivity (3)
- facility location (3)
- non-Newtonian flow in porous media (3)
- poroelasticity (3)

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.

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.

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.

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.

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.

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.

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.

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.

In this work we extend the multiscale finite element method (MsFEM)
as formulated by Hou and Wu in [14] to the PDE system of linear elasticity.
The application, motivated from the multiscale analysis of highly heterogeneous
composite materials, is twofold. Resolving the heterogeneities on
the finest scale, we utilize the linear MsFEM basis for the construction of
robust coarse spaces in the context of two-level overlapping Domain Decomposition
preconditioners. We motivate and explain the construction
and present numerical results validating the approach. Under the assumption
that the material jumps are isolated, that is they occur only in the
interior of the coarse grid elements, our experiments show uniform convergence
rates independent of the contrast in the Young's modulus within the
heterogeneous material. Elsewise, if no restrictions on the position of the
high coefficient inclusions are imposed, robustness can not be guaranteed
any more. These results justify expectations to obtain coefficient-explicit
condition number bounds for the PDE system of linear elasticity similar to
existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner
and Scheichl [12]. Furthermore, we numerically observe the properties
of the MsFEM coarse space for linear elasticity in an upscaling framework.
Therefore, we present experimental results showing the approximation errors
of the multiscale coarse space w.r.t. the fine-scale solution.

The paper production is a problem with significant importance for the society
and it is a challenging topic for scientific investigations. This study is concerned
with the simulations of the pressing section of a paper machine. A two-dimensional
model is developed to account for the water flow within the pressing zone. Richards’
type equation is used to describe the flow in the unsaturated zone. The dynamic capillary
pressure–saturation relation proposed by Hassanizadeh and co-workers (Hassanizadeh
et al., 2002; Hassanizadeh, Gray, 1990, 1993a) is adopted for the paper
production process.
The mathematical model accounts for the co-existence of saturated and unsaturated
zones in a multilayer computational domain. The discretization is performed
by the MPFA-O method. The numerical experiments are carried out for parameters
which are typical for the production process. The static and dynamic capillary
pressure–saturation relations are tested to evaluate the influence of the dynamic
capillary effect.

In this paper, we present a viscoelastic rod model that is suitable for fast and accurate dynamic simulations. It is based on Cosserat’s geometrically exact theory of rods and is able to represent extension, shearing (‘stiff’ dof), bending and torsion (‘soft’ dof). For inner dissipation, a consistent damping potential proposed by Antman is chosen. We parametrise the rotational dof by unit quaternions and directly use the quaternionic evolution differential equation for the discretisation of the Cosserat rod curvature. The discrete version of our rod model is obtained via a finite difference discretisation on a staggered grid. After an index reduction from three to zero, the right-hand side function f and the Jacobian \(\partial f/\partial(q, v, t)\) of the dynamical system \(\dot{q} = v, \dot{v} = f(q, v, t)\) is free of higher algebraic (e. g. root) or transcendental (e. g. trigonometric or exponential) functions and therefore cheap to evaluate. A comparison with Abaqus finite element results demonstrates the correct mechanical behavior of our discrete rod model. For the time integration of the system, we use well established stiff solvers like RADAU5 or DASPK. As our model yields computational times within milliseconds, it is suitable for interactive applications in ‘virtual reality’ as well as for multibody dynamics simulation.

This work presents a proof of convergence of a discrete solution to a continuous one. At first, the continuous problem is stated as a system
of equations which describe filtration process in the pressing section of a
paper machine. Two flow regimes appear in the modeling of this problem.
The model for the saturated flow is presented by the Darcy’s law and the mass conservation. The second regime is described by the Richards approach together with a dynamic capillary pressure model. The finite
volume method is used to approximate the system of PDEs. Then the existence of a discrete solution to proposed finite difference scheme is proven.
Compactness of the set of all discrete solutions for different mesh sizes is
proven. The main Theorem shows that the discrete solution converges
to the solution of continuous problem. At the end we present numerical
studies for the rate of convergence.

An efficient mathematical model to virtually generate woven metal wire meshes is
presented. The accuracy of this model is verified by the comparison of virtual structures with three-dimensional
images of real meshes, which are produced via computer tomography. Virtual structures
are generated for three types of metal wire meshes using only easy to measure parameters. For these
geometries the velocity-dependent pressure drop is simulated and compared with measurements
performed by the GKD - Gebr. Kufferath AG. The simulation results lie within the tolerances of
the measurements. The generation of the structures and the numerical simulations were done at
GKD using the Fraunhofer GeoDict software.

Continuously improving imaging technologies allow to capture the complex spatial
geometry of particles. Consequently, methods to characterize their three
dimensional shapes must become more sophisticated, too. Our contribution to
the geometric analysis of particles based on 3d image data is to unambiguously
generalize size and shape descriptors used in 2d particle analysis to the spatial
setting.
While being defined and meaningful for arbitrary particles, the characteristics
were actually selected motivated by the application to technical cleanliness. Residual
dirt particles can seriously harm mechanical components in vehicles, machines,
or medical instruments. 3d geometric characterization based on micro-computed
tomography allows to detect dangerous particles reliably and with
high throughput. It thus enables intervention within the production line. Analogously
to the commonly agreed standards for the two dimensional case, we
show how to classify 3d particles as granules, chips and fibers on the basis of
the chosen characteristics. The application to 3d image data of dirt particles is
demonstrated.

Input loads are essential for the numerical simulation of vehicle multibody system
(MBS)- models. Such load data is called invariant, if it is independent of the specific system under consideration. A digital road profile, e.g., can be used to excite MBS models of different
vehicle variants. However, quantities efficiently obtained by measurement such as wheel forces
are typically not invariant in this sense. This leads to the general task to derive invariant loads
on the basis of measurable, but system-dependent quantities. We present an approach to derive
input data for full-vehicle simulation that can be used to simulate different variants of a vehicle
MBS model. An important ingredient of this input data is a virtual road profile computed by optimal control methods.

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

In this paper we study the possibilities of sharing profit in combinatorial procurement auctions and exchanges. Bundles of heterogeneous items are offered by the sellers, and the buyers can then place bundle bids on sets of these items. That way, both sellers and buyers can express synergies between items and avoid the well-known risk of exposure (see, e.g., [3]). The reassignment of items to participants is known as the Winner Determination Problem (WDP). We propose solving the WDP by using a Set Covering formulation, because profits are potentially higher than with the usual Set Partitioning formulation, and subsidies are unnecessary. The achieved benefit is then to be distributed amongst the participants of the auction, a process which is known as profit sharing. The literature on profit sharing provides various desirable criteria. We focus on three main properties we would like to guarantee: Budget balance, meaning that no more money is distributed than profit was generated, individual rationality, which guarantees to each player that participation does not lead to a loss, and the core property, which provides every subcoalition with enough money to keep them from separating. We characterize all profit sharing schemes that satisfy these three conditions by a monetary flow network and state necessary conditions on the solution of the WDP for the existence of such a profit sharing. Finally, we establish a connection to the famous VCG payment scheme [2, 8, 19], and the Shapley Value [17].

We introduce a refined tree method to compute option prices using the stochastic volatility model of Heston. In a first step, we model the stock and variance process as two separate trees and with transition probabilities obtained by matching tree moments up to order two against the Heston model ones. The correlation between the driving Brownian motions in the Heston model is then incorporated by the node-wise adjustment of the probabilities. This adjustment, leaving the marginals fixed, optimizes the match between tree and model correlation. In some nodes, we are even able to further match moments of higher order. Numerically this gives convergence orders faster than 1/N, where N is the number of dis- cretization steps. Accuracy of our method is checked for European option prices against a semi closed-form, and our prices for both European and American options are compared to alternative approaches.

In this paper we deal with dierent statistical modeling of real world accident data in order to quantify the eectiveness of a safety function or a safety conguration (meaning a specic combination of safety functions) in vehicles. It is shown that the eectiveness can be estimated along the so-called relative risk, even if the eectiveness does depend on a confounding variable which may be categorical or continuous. For doing so a concrete statistical modeling is not necessary, that is the resulting estimate is of nonparametric nature. In a second step the quite usual and from a statistical point of view classical logistic regression modeling is investigated. Main emphasis has been laid on the understanding of the model and the interpretation of the occurring parameters. It is shown that the eectiveness of the safety function also can be detected via such a logistic approach and that relevant confounding variables can and should be taken into account. The interpretation of the parameters related to the confounder and the quantication of the in uence of the confounder is shown to be rather problematic. All the theoretical results are illuminated by numerical data examples.

This report describes the calibration and completion of the volatility cube in the SABR model. The description is based on a project done for Assenagon GmbH in Munich. However, we use fictitious market data which resembles realistic market data. The problem posed by our client is formulated in section 1. Here we also motivate why this is a relevant problem. The SABR model is briefly reviewed in section 2. Section 3 discusses the calibration and completion of the volatility cube. An example is presented in section 4. We conclude by suggesting possible future research in section 5.