422 research outputs found
Numerical convergence for semilinear parabolic equations
We present a convergence result for finite element discretisations of semilinear parabolic equations, in which the evaluation of the nonlinearity requires some high order of regularity of the solution. For example a coefficient might depend on derivatives or pointevaluation of the solution. We do not rely on high regularity of the exact solution itself and as a payoff we can not deduce convergence rates. As an example the convergence result is applied to a nonlinear Fokker--Planck type battery model
Stationary solutions of liquid two-layer thin film models
We investigate stationary solutions of a thin-film model for liquid two-layer
flows in an energetic formulation that is motivated by its gradient flow
structure. The goal is to achieve a rigorous understanding of the contact-angle
conditions for such two-layer systems. We pursue this by investigating a
corresponding energy that favors the upper liquid to dewet from the lower
liquid substrate, leaving behind a layer of thickness . After proving
existence of stationary solutions for the resulting system of thin-film
equations we focus on the limit via matched asymptotic analysis.
This yields a corresponding sharp-interface model and a matched asymptotic
solution that includes logarithmic switch-back terms. We compare this with
results obtained using -convergence, where we establish existence and
uniqueness of energetic minimizers in that limit
Assessment of the potential impacts of plant traits across environments by combining global sensitivity analysis and dynamic modeling in wheat
A crop can be viewed as a complex system with outputs (e.g. yield) that are
affected by inputs of genetic, physiology, pedo-climatic and management
information. Application of numerical methods for model exploration assist in
evaluating the major most influential inputs, providing the simulation model is
a credible description of the biological system. A sensitivity analysis was
used to assess the simulated impact on yield of a suite of traits involved in
major processes of crop growth and development, and to evaluate how the
simulated value of such traits varies across environments and in relation to
other traits (which can be interpreted as a virtual change in genetic
background). The study focused on wheat in Australia, with an emphasis on
adaptation to low rainfall conditions. A large set of traits (90) was evaluated
in a wide target population of environments (4 sites x 125 years), management
practices (3 sowing dates x 2 N fertilization) and (2 levels). The
Morris sensitivity analysis method was used to sample the parameter space and
reduce computational requirements, while maintaining a realistic representation
of the targeted trait x environment x management landscape ( 82 million
individual simulations in total). The patterns of parameter x environment x
management interactions were investigated for the most influential parameters,
considering a potential genetic range of +/- 20% compared to a reference. Main
(i.e. linear) and interaction (i.e. non-linear and interaction) sensitivity
indices calculated for most of APSIM-Wheat parameters allowed the identifcation
of 42 parameters substantially impacting yield in most target environments.
Among these, a subset of parameters related to phenology, resource acquisition,
resource use efficiency and biomass allocation were identified as potential
candidates for crop (and model) improvement.Comment: 22 pages, 8 figures. This work has been submitted to PLoS On
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The behavior of a many particle cathode in a lithium-ion battery
We study the almost reversible storage process of charging and
discharging of lithium-ion batteries. That process is accompanied by a phase
transition and charging and discharging run along different paths, so that
hysteretic behavior is observed. We are interested in the storage problem of
the cathode of a lithium-ion battery consisting of a system of many iron
phosphate (FePO4) particles. There are mathematical models, see [DGJ08],
[DGGHJ09] and [DG09], that describe phase transitions and hysteresis
exclusively in a single storage particle and they can describe the observed
hysteretic voltage-charge plots with almost horizontal plateaus.
Interestingly the models predict that the coexistence of a 2-phase system in
an individual particle disappears, if its size is below a critical value. The
disappearance of the phase transition in the single particle model implies
the disappearance of the hysteresis. However, in the experiment hysteretic
behavior survives. In other words: The behavior of a storage system
consisting of many particles is qualitatively independent of the fact whether
the individual particles itself develop a 2-phase system or if they remain in
a single phase state. This apparent paradoxical observation will be resolved
in this article by a many particle model. It will be shown that if each of
the individual particles is in a homogeneous state, nevertheless the many
particle ensemble exhibits phase transition and hysteresis ..
Hysteresis in the context of hydrogen storage and lithium-ion batteries
The processes of reversible storage of hydrogen in a metal by
loading and unloading and of charging and discharging of
lithium-ion batteries have many things in common. The both
processes are accompanied by a phase transition and loading and
unloading run along different paths, so that hysteretic behavior
is observed.
For hydrogen storage we consider a fine powder of magnesium (Mg)
particles and lithium storage is studied for iron phosphate
(FePO) particles forming the cathode of a lithium-ion battery.
The mathematical models that are established in \cite{DGJ08} and
\cite{DGH09a}, describe phase transitions and hysteresis
exclusively in a single particle and on that basis they can
predict the observed hysteretic plots with almost horizontal
plateaus. Interestingly the models predict that the coexistence of
a 2-phase system in an individual particle disappears, if its size
is below a critical value. However, measurements reveal that this
is qualitatively not reflected by the mentioned hysteretic plots
of loading and unloading. In other words: The behavior of a
storage system consisting of many particles is qualitatively
independent of the fact whether the individual particles itself
develop a 2-phase system or if they remain in a single phase
state.
This apparent paradoxical observation will be resolved in this
article. It will be shown that if each of the individual particles
homogeneously distributes the supplied matter, nevertheless the
many particle ensemble exhibits phase transition and hysteresis,
because one of the two phases is realized in some part of the
particles while the remaining part is in the other phase
Gradient flow perspective of thin-film bilayer flows
We study gradient flow formulations of thin-film bilayer flows with triple-junctions between liquid/liquid/air. First we highlight the gradient structure in the Stokes free-boundary flow and identify its solutions with the well known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding thin-film model and formally identify solutions with those of the corresponding free-boundary problem. A robust numerical algorithm for the thin-film gradient flow structure is then provided. Using this algorithm we compare the sharp triple-junction model with precursor models. For their stationary solutions a rigorous connection is established using Gamma-convergence. For time-dependent solutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zero-contact angle case, where we compare numerical solutions with asymptotically exact source-type solutions
Gradient flow perspective of thin-film bilayer flows
We study gradient flow formulations of thin-film bilayer flows with
triple-junctions between liquid/liquid/air. First we highlight the gradient
structure in the Stokes free-boundary flow and identify its solutions with
the well-known PDE with boundary conditions. Next we propose a similar
gradient formulation for the corresponding thin-film model and formally
identify solutions with those of the corresponding free-boundary problem. A
robust numerical algorithm for the thin-film gradient flow structure is then
provided. Using this algorithm we compare the sharp triple-junction model
with precursor models. For their stationary solutions a rigorous connection
is established using [Gamma]-convergence. For time-dependentsolutions the
comparison of numerical solutions shows a good agreement for small and
moderate times. Finally we study spreading in the zero-contact angle case,
where we compare numerical solutions with asymptotically exact source-type
solutions
Stationary solutions for two-layer lubrication equations
We investigate stationary solutions of flows of thin liquid bilayers in an energetic formulation which is motivated by the gradient flow structure of its lubrication approximation. The corresponding energy favors the liquid substrate to be only partially covered by the upper liquid. This is expressed by a negative spreading coefficient which arises from an intermolecular potential combining attractive and repulsive forces and leads to an ultra-thin layer of thickness ε. For the corresponding lubrication models existence of stationary solutions is proven. In the limit ε to 0 matched asymptotic analysis is applied to derive sharp-interface models and the corresponding contact angles, i.e. the Neumann triangle. In addition we use Γ-convergence and derive the equivalent sharp-interface models rigorously in this limit. For the resulting model existence and uniqueness of energetic minimizers are proven. The minimizers agree with solutions obtained by matched asymptotics
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Blow-up versus boundedness in a nonlocal and nonlinear Fokker-Planck equation
Literaturverz
Blow-up versus boundedness in a nonlocal and nonlinear Fokker--Planck equation
We consider a Fokker-Planck equation on a compact interval where, as a constraint, the first moment is a prescribed function of time. Eliminating the associated Lagrange multiplier one obtains nonlinear and nonlocal terms. After establishing suitable local existence results, we use the relative entropy as an energy functional. However, the time-dependent constraint leads to a source term such that a delicate analysis is needed to show that the dissipation terms are strong enough to control the work done by the constraint. We obtain global existence of solutions as long as the prescribed first moment stays in the interior of an interval. If the prescribed moment converges to a constant value inside the interior of the interval, then the solution stabilises to the unique steady state
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