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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.