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A typical phase field approach for describing phase separation and
coarsening phenomena in alloys is the Cahn-Hilliard model. This
model has been generalized to the so-called Cahn-Larche system
by combining it with elasticity to capture non-neglecting deformation
phenomena,
which occur during phase separation and coarsening processes in the material.
In order to account for damage effects,
we extend the existing framework of Cahn-Hilliard and Cahn-Larche
systems by incorporating an internal damage variable of local character. This
damage variable allows to model the effect that damage of a material point is influenced by its local surrounding.
The damage process is described by a unidirectional rate-dependent evolution inclusion
for the internal variable.
For the introduced Cahn-Larche systems
coupled with rate-dependent
damage processes, we
establish a suitable notion of weak solutions and prove existence of weak solutions.
An electronic model for solar cells including active interfaces and energy resolved defect densities
(2011)
We introduce an electronic model for solar cells taking into account
heterostructures with active
interfaces and energy resolved volume and interface trap densities.
The model consists of continuity equations for electrons and holes with thermionic
emission transfer conditions at the interface and of ODEs for the trap
densities with energy level and spatial position as parameters,
where the right hand sides contain generation-recombination as well as
ionization reactions. This system is coupled with a Poisson
equation for the electrostatic potential.
We show the thermodynamic correctness of the model and prove a priori estimates
for the solutions to the evolution system. Moreover, existence and uniqueness
of weak solutions of the problem are proven. For this purpose we solve a
regularized problem and verify bounds of the corresponding solution
not depending on the regularization level.
Dissipativity is an important property of individual systems
that guarantees a stable interconnected system.
However, due to errors in the modeling process weakly non-dissipative
models may be constructed.
In this paper we introduce a method to perturb a non-dissipative LTI system in order to enforce dissipativity using spectral perturbation results for
para-Hermitian pencils.
Compared to earlier algorithms the new method
is applicable to a wider class of problems,
it utilizes a simpler framework, and
employs a larger class of allowable perturbations
resulting in smaller perturbations. Moreover, system stability can be enforced as well.
Numerical examples are provided to show the effectiveness of the new approach.
Time-stepping procedures for the solution of evolution equations can be performed on parallel architecture by parallelizing the space computation at each time step. This, however, requires heavy communication between processors and becomes inefficient when many time-steps are to be computed and many processors are available. In such
cases parallelization in time is advantageous.
In this paper we present a method for parallelization in time of linear multistep discretizations of linear evolution problems; we consider a model parabolic and a model hyperbolic problem, and their, respectively, A(theta)-stable and A-stable linear multistep discretizations. The method consists of a discrete decoupling procedure, whereby N+1 decoupled Helmholtz problems with complex frequencies are obtained; N being the number of time steps computed in parallel. The usefulness of the method rests on our ability to solve these Helmholtz problems efficiently. We discuss the theory and give numerical examples for multigrid preconditioned iterative solvers of relevant
complex frequency Helmholtz problems. The parallel
approach can easily be combined with a time-stepping procedure, thereby obtaining a block time-stepping method where each block of steps is computed in parallel. In this way we are able to optimize the algorithm with respect to the number of processors available, the difficulty of solving the Helmholtz problems, and the possibility of both time and space adaptivity. Extensions to other linear evolution problems and to Runge-Kutta time discretization
are briefly mentioned.
Affinely-Adjustable Robust Counterparts provide tractable alternatives to (two-stage) robust programs with arbitrary recourse. We apply them to robust network design with polyhedral demand uncertainty, introducing the affine routing principle. We compare the affine routing to the well-studied static and dynamic routing schemes for robust network design. All three schemes are embedded into the general framework of two-stage network design with recourse. It is shown that affine routing can be seen as a generalization of the widely used static routing still being tractable and providing cheaper solutions. We investigate properties on the demand polytope under which affine routings reduce to static routings and also develop conditions on the uncertainty set leading to dynamic routings being affine. We show however that affine routings suffer from the drawback that (even totally) dominated demand vectors are not necessarily supported by affine solutions. Uncertainty sets have to be designed accordingly. Finally, we present computational results on networks from SNDlib. We conclude that for these instances the optimal solutions based on affine routings tend to be as cheap as optimal network designs for dynamic routings. In this respect the affine routing principle can be used to approximate the cost for two-stage solutions with free recourse which are hard to compute.
This paper establishes the equivalence of conforming Courant finite element method and nonconforming Crouzeix-Raviart finite element method in the sense that the respective energy error norms are equivalent up to generic constants and higher-order data oscillations in a Poisson model problem. The Raviart-Thomas mixed finite element method is better than the previous two whereas the conjecture of the converse relation is proved to be false.
This paper completes the analysis of comparison initiated by Braess in Calcolo (2010). Two numerical benchmarks illustrate the comparison theorems and the possible strict superiority of the Raviart-Thomas mixed finite element method. Applications include least-squares finite element methods and equality of approximation classes for concepts of optimality for adaptive finite element methods.
This work concerns the approximation of the shape operator of smooth surfaces in $\mathbb{R}^{3}$ from polyhedral surfaces. We introduce two generalized shape operators that are vector-valued linear functionals on a Sobolev space of vector fields and can be rigorously defined on smooth and on polyhedral surfaces. We consider polyhedral surfaces that approximate smooth surfaces and prove two types of approximation estimates: one concerning the approximation of the generalized shape operators in the operator norm and one concerning the pointwise approximation of the (classic) shape operator, including mean and Gaussian curvature, principal curvatures, and principal curvature directions.
The estimates are confirmed by numerical experiments.
We propose a framework for deformation-based surface modeling that is interactive, robust and intuitive to use. The deformations are described by a non-linear optimization problem that models static states of elastic shapes under external forces which implement the user input. Interactive response is achieved by a combination of model reduction, a robust energy approximation, and an efficient quasi-Newton solver. Motivated by the observation that a typical modeling session requires only a fraction of the full shape
space of the underlying model, we use second and third derivatives of a deformation energy to construct a low-dimensional shape space that forms the feasible set for the optimization. Based on mesh coarsening, we propose an
energy approximation scheme with adjustable approximation quality. The quasi-Newton solver guarantees superlinear convergence without the need of costly Hessian evaluations during modeling. We demonstrate the effectiveness of the approach on different examples including the test suite introduced in [Botsch and Sorkine 2008].