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H. Gottschalk

Publications and source records attributed to H. Gottschalk.

At least 19 recordsLinked to original sources

Probabilistic Modeling of LCF Failure Times Using an Epidemiological Crack Percolation Model

The analysis of standardized low cycle fatigue (LCF) experiments shows that the failure times widely scatter. Furthermore, mechanical components often fail before the deterministic failure time is reached. A possibility to overcome these problems is to consider probabilistic failure times. Our approach for probabilistic life prediction is based on the microstructure of the metal. Since we focus on nickel-base alloys we consider a coarse grained microstructure, with random oriented FCC grains. This leads to random distributed Schmid factors and different anisotropic stress in each grain. To gain crack initiation times, we use Coffin-Manson- Basquin and Ramberg-Osgood equation on stresses corrected with probabilistic Schmid factors. Using these single grain crack initiation times, we have developed an epidemiological crack growth model over multiple grains. In this mesoscopic crack percolation model, cracked grains induce a stress increase in neighboring grains. This stress increase is realized using a machine learning model trained on data generated from finite element simulations. The resulting crack clusters are evaluated with a failure criterion based on a multimodal stress intensity factor. From the generated failure times, we calculate surface dependent hazard rates using a Monte Carlo framework. We compare the obtained failure time distributions to data from LCF experiments and find good coincidence of predicted and measured scatter bands.

cond-mat.mtrl-sci

Modeling, Minimizing and Managing the Risk of Fatigue for Mechanical Components

Mechanical components that are exposed to cyclic mechanical loading fail at loads that are well below the ultimate tensile strength. This process is known as fatigue. The failure time, that is the time when a first crack forms, is highly random. In this work we review some recent developments in the modelling of probabilistic failure times, understood as the time to the formation of a fatigue crack. We also discuss the how probabilistic models can be used in shape design with the design intent of optimizing the component's reliability. We give review a recent existence result for optimal shapes and we discuss continuous and discrete shape derivatives. Another application is optimal service scheduling. The mathematical fields involved range from reliability statistics over stochastic point processes, multiscale modeling, PDEs on variable geometries, shape optimization and numerical analysis to operations research.

math.OC

Assessment of Models for Pedestrian Dynamics with Functional Principal Component Analysis

Many agent based simulation approaches have been proposed for pedestrian flow. As such models are applied e.g.\ in evacuation studies, the quality and reliability of such models is of vital interest. Pedestrian trajectories are functional data and thus functional principal component analysis is a natural tool to asses the quality of pedestrian flow models beyond average properties. In this article we conduct functional PCA for the trajectories of pedestrians passing through a bottleneck. We benchmark two agent based models of pedestrian flow against the experimental data using PCA average and stochastic features. Functional PCA proves to be an efficient tool to detect deviation between simulation and experiment and to asses quality of pedestrian models.

physics.soc-ph

Probabilistic Schmid factors and scatter of LCF life

We investigate the scatter in the cycles to crack initiation for conventionally cast specimens of the superalloy RENE 80 in total strain controlled low cycle fatigue (LCF) tests at 850°C. The grains at the location of crack initiation are investigated using electron backscatter diffraction (EBSD) measurements with scanning electron microscopy (SEM). This results in determination of maximal Schmid factors for the slip system that lead to the initiation of the LCF surface crack. It is shown using stress - life and strain - life models, that taking into account the Schmid factor considerable reduces the scatter in LCF life prediction. This is used to propose a probabilistic model for LCF life based on random grain orientations. Using Mote Carlo simulation, we determine the probability distribution for the maximal Schmid factor in an uniaxial stress state under isotropic random orientations of the grain. This is used to calculate failure probabilities under the above mentioned model and to compare them with experimental data. Based on extreme value statistics, it is discussed whether a high maximal Schmid factor can be considered as causally related to an early initiation of an LCF crack and we present some rather surprising experimental evidence. In the last section conclusions are formulated and future directions of research are outlined.

cond-mat.mtrl-sci

On the unitary transformation between non-quasifree and quasifree state spaces and its application to quantum field theory on curved spacetimes

Using $\star$-calculus on the dual of the Borchers-Uhlmann algebra endowed with a combinatorial co-product, we develop a method to calculate a unitary transformation relating the GNS representations of a non-quasifree and a quasifree state of the free hermitian scalar field. The motivation for such an analysis and a further result is the fact that a unitary transformation of this kind arises naturally in scattering theory on non-stationary backgrounds. Indeed, employing the perturbation theory of the Yang-Feldman equations with a free CCR field in a quasifree state as an initial condition and making use of extended Feynman graphs, we are able to calculate the Wightman functions of the interacting and outgoing fields in a $ϕ^p$-theory on arbitrary curved spacetimes. A further examination then reveals two major features of the aforementioned theory: firstly, the interacting Wightman functions fulfil the basic axioms of hermiticity, invariance, spectrality (on stationary spacetimes), perturbative positivity, and locality. Secondly, the outgoing field is free and fulfils the CCR, but is in general not in a quasifree state in the case of a non-stationary spacetime. In order to obtain a sensible particle picture for the outgoing field and, hence, a description of the scattering process in terms of particles (in asymptotically flat spacetimes), it is thus necessary to compute a unitary transformation of the abovementioned type.

math-ph

AdS/CFT correspondence in the Euclidean context

We study two possible prescriptions for AdS/CFT correspondence by means of functional integrals. The considerations are non-perturbative and reveal certain divergencies which turn out to be harmless, in the sense that reflection-positivity and conformal invariance are not destroyed.

math-ph

Scattering theory for dipole quantum fields

In the present work a general frame for the scattering theory of local, relativistic dipole quantum fields is presented and some models of interacting dipole fields are considered, i.e. local, relativistic quantum fields with indefinite metric which asymptotically do not converge to free fields, but to free dipole fields. Also, we give explicit formulae for the (nontrivial) scattering matrix of dipole in- and out- fields for these models. Furthermore we show how related dipole degrees of freedom occur in the perturbation theory of certain two dimensional models, e.g. massive sine-Gordon or sinh-Gordon models.

math-ph

Feynman graphs for non-Gaussian measures

Partition- and moment functions for a general (not necessarily Gaussian) functional measure that is perturbed by a Gibbs factor are calculated using generalized Feynman graphs. From the graphical calculus, a new notion of Wick ordering arises, that coincides with orthogonal decompositions of Wiener-Itô type only if the measure is Gaussian. Proving a generalized linked cluster theorem, we show that the logarithm of the partition function can be expanded in terms of connected Feynman graphs ("linked cluster theorem").

math-ph

How to determine the law of the noise driving a SPDE

We consider a stochastic partial differential equation (SPDE) on a lattice \partial_t X=(Δ-m^2)X-λX^p+ηwhere $η$ is a space-time Lévy noise. A perturbative (in the sense of formal power series) strong solution is given by a tree expansion, whereas the correlation functions of the solution are given by a perturbative expansion with coefficients that are represented as sums over a certain class of graphs, called Parisi-Wu graphs. The perturbative expansion of the truncated (connected) correlation functions is obtained via a Linked Cluster Theorem as a sums over connected graphs only. The moments of the stationary solution can be calculated as well. In all these solutions the cumulants of the single site distribution of the noise enter as multiplicative constants. To determine them, e.g. by comparison with a empirical correlation function, one can fit these constants (e.g. by the methods of least squares) and thereby one (approximately) determines law of the noise.

math.PR

Systems of classical particles in the grand canonical ensemble, scaling limits and quantum field theory

Euclidean quantum fields obtained as solutions of stochastic partial pseudo differential equations driven by a Poisson white noise have paths given by locally integrable functions. This makes it possible to define a class of ultra-violet finite local interactions for these models (in any space-time dimension). The corresponding interacting Euclidean quantum fields can be identified with systems of classical "charged" particles in the grand canonical ensemble with an interaction given by a nonlinear energy density of the "static field" generated by the particles' charges via a "generalized Poisson equation". The infinite volume limit of such systems is discussed for models with trigonometric interactions using a representation of such models as Widom-Rowlinson models associated with a (formal) Potts models at imaginary temperature. The continuum limit of the particle systems under consideration is also investigated and the formal analogy with the scaling limit of renormalization group theory is pointed out. In some simple cases the question of (non-) triviality of the continuum limits is clarified.

math-ph

Nontrivial models with indefinite metric

The non perturbative construction of quantum field models with nontrivial scattering in arbitrary dimension $d$ of the underlying Minkowski space-time is much more simple in the framework of quantum field theory with indefinite metric than in the positive metric case. In particular, there exist a number of solutions in the physical dimension $d=4$, where up to now no positive metric solutions are known. Here we review, why this is so, and we discuss some examples obtained by analytic continuation from the solutions of Euclidean covariant stochastic partial differential equations (SPDEs) driven by non-Gaussian white noise.

math-ph

Indefinite metric

The quantization of gauge fields with the help of the Gupta-Bleuler formalism is reviewed.

math-ph

Partly Divisible Probability Distributions

Given a probability distribution $μ$ a set $Λ(μ)$ of positive real numbers is introduced, so that $Λ(μ)$ measures the "divisibility" of $μ$. The basic properties of $Λ(μ)$ are described and examples of probability distributions are given, which exhibit the existence of a continuum of situations interpolating the extreme cases of infinitely and minimally divisible probability distributions.

math.PR

Partly divisible probability measures on locally compact Abelian groups

A notion of admissible probability measures $μ$ on a locally compact Abelian group (LCA-group) $G$ with connected dual group $\hat G=\R^d\times \T^n$ is defined. To such a measure $μ$, a closed semigroup $Λ(μ)\subseteq (0,\infty)$ can be associated, such that, for $t\in Λ(μ)$, the Fourier transform to the power $t$, $(\hat μ)^t$, is a characteristic function. We prove that the existence of roots for non admissible probability measures underlies some restrictions, which do not hold in the admissible case. As we show for the example $\Z_2$, in the case of LCA-groups with non connected dual group, there is no canonical definition of the set $Λ(μ)$.

math.PR

Scattering theory for quantum fields with indefinite metric

In this work, we discuss the scattering theory of local, relativistic quantum fields with indefinite metric. Since the results of Haag--Ruelle theory do not carry over to the case of indefinite metric, we propose an axiomatic framework for the construction of in- and out- states, such that the LSZ asymptotic condition can be derived from the assumptions. The central mathematical object for this construction is the collection of mixed vacuum expectation values of local, in- and out- fields, called the ``form factor functional'', which is required to fulfill a Hilbert space structure condition. Given a scattering matrix with polynomial transfer functions, we then construct interpolating, local, relativistic quantum fields with indefinite metric, which fit into the given scattering framework.

math-ph

Remarks on some new models of interacting quantum fields with indefinite metric

We study quantum field models in indefinite metric. We introduce the modified Wightman axioms of Morchio and Strocchi as a general framework of indefinite metric quantum field theory (QFT) and present concrete interacting relativistic models obtained by analytical continuation from some stochastic processes with Euclidean invariance. As a first step towards scattering theory in indefinite metric QFT, we give a proof of the spectral condition on the translation group for the relativistic models.

math-ph