ESTIMATING UNCERTAINTIES FROM INACCURATE MEASUREMENT DATA USING MAXIMUM ENTROPY DISTRIBUTIONS

Modern engineering design often considers uncertainties in geometrical and material parameters and in the loading conditions. Based on initial assumptions on the stochastic properties as mean values, standard deviations and the distribution functions of these uncertain parameters a probabilistic analysis is carried out. In many application fields probabilities of the exceedance of failure criteria are computed. The out-coming failure probability is strongly dependent on the initial assumptions on the random variable properties. Measurements are always more or less inaccurate data due to varying environmental conditions during the measurement procedure. Furthermore the estimation of stochastic properties from a limited number of realisation also causes uncertainties in these quantities. Thus the assumption of exactly known stochastic properties by neglecting these uncertainties may not lead to very useful probabilistic measures in a design process. In this paper we assume the stochastic properties of a random variable as uncertain quantities caused by so-called epistemic uncertainties. Instead of predefined distribution types we use the maximum entropy distribution which enables the description of a wide range of distribution functions based on the first four stochastic moments. These moments are taken again as random variables to model the epistemic scatter in the stochastic assumptions. The main point of this paper is the discussion on the estimation of these uncertain stochastic properties based on inaccurate measurements. We investigate the bootstrap algorithm for its applicability to quantify the uncertainties in the stochastic properties considering imprecise measurement data. Based on the obtained estimates we apply standard stochastic analysis on a simple example to demonstrate the difference and the necessity of the proposed approach

Application of a hybrid parallelization technique to accelerate the numerical simulation of nonlinear mechanical problems

This paper presents the combination of two different parallelization environments, OpenMP and MPI, in one numerical simulation tool. The computation of the system matrices and vectors is parallelized with OpenMP and the solution of the system of equations is done with the MPIbased solver MUMPS. The efficiency of both algorithms is shown on several linear and nonlinear examples using the Finite Element Method and a meshless discretization technique

ADAPTIVE RESPONSE SURFACE APPROACH USING ARTIFICIAL NEURAL NETWORKS AND MOVING LEAST SQUARES

In engineering science the modeling and numerical analysis of complex systems and relations plays an important role. In order to realize such an investigation, for example a stochastic analysis, in a reasonable computational time, approximation procedure have been developed. A very famous approach is the response surface method, where the relation between input and output quantities is represented for example by global polynomials or local interpolation schemes as Moving Least Squares (MLS). In recent years artificial neural networks (ANN) have been applied as well for such purposes. Recently an adaptive response surface approach for reliability analyses was proposed, which is very efficient concerning the number of expensive limit state function evaluations. Due to the applied simplex interpolation the procedure is limited to small dimensions. In this paper this approach is extended for larger dimensions using combined ANN and MLS response surfaces for evaluating the adaptation criterion with only one set of joined limit state points. As adaptation criterion a combination by using the maximum difference in the conditional probabilities of failure and the maximum difference in the approximated radii is applied. Compared to response surfaces on directional samples or to plain directional sampling the failure probability can be estimated with a much smaller number of limit state points

Stochastic crack growth simulation in reinforced concrete structures by means of coupled finite element and meshless methods

The complex failure process of concrete structures can not be described in detail by standard engineering design formulas. The numerical analysis of crack development in concrete is essential for several problems. In the last decades a large number of research groups have dealt with this topic and several models and algorithms were developed. However, most of these methods show some difficulties and are limited to special cases. The goal of this study was to develop an automatic algorithm for the efficient simulation of multiple cracking in plain and reinforced concrete structures of medium size. For this purpose meshless methods were used to describe the growth of crack surfaces. Two meshless interpolation schemes were improved for a simple application. The cracking process of concrete has been modeled using a stable criterion for crack growth in combination with an improved cohesive crack model which can represent the failure process under combined crack opening and crack sliding very well. This crack growth algorithm was extended in order to represent the fluctuations of the concrete properties by enlarging the single-parameter random field concept for multiple correlated material parameters.Das komplexe Versagensverhalten von Betonstrukturen kann in der Regel nicht mit Standardbemessungsformeln beschrieben werden. Eine detaillierte numerische Analyse der Rissentwicklung in Beton ist für einige Problemstellungen unverzichtbar. In den letzten Jahrzehnten haben sich eine Vielzahl von Forschergruppen mit dieser Thematik aus-einandergesetzt. Dabei wurden verschiedene Modelle und Algorithmen entwickelt. Die meisten dieser Verfahren weisen jedoch verschiedene Probleme auf oder sind nur für Spezialfälle anwendbar. Das Ziel dieser Arbeit war die Entwicklung eines automatischen Algorithmus zur effizienten Simulation von mehrfacher Rissentwicklung in Beton- und Stahlbetonstrukturen mittlerer Größe. Dabei wurden netzfreie Verfahren angewendet, um die Änderung der Rissoberflächen abzubilden. Zwei netzfreie Interpolationstypen wurden im Hinblick auf eine unkomplizierte Anwendung angepaßt. Der Versagensprozess des Betons wurde mit Hilfe eines stabilen Risskriteriums in Kombination mit einem erwei-terten kohäsiven Rissmodell abgebildet. Dieses erweiterte Modell kann die Zusammenhänge bei kombinierter Rissöffnung und -gleitung sehr gut wiedergeben. Der entwickelte Algorithmus zur Risssimulation wurde in Hinblick auf eine stochastische Modellierung erweitert. Zu diesem Zweck wurde das Zufallsfeldkonzept für die Abbildung mehrerer untereinander korrelierter Materialparameter ergänzt

ESTIMATING UNCERTAINTIES FROM INACCURATE MEASUREMENT DATA USING MAXIMUM ENTROPY DISTRIBUTIONS

Modern engineering design often considers uncertainties in geometrical and material parameters and in the loading conditions. Based on initial assumptions on the stochastic properties as mean values, standard deviations and the distribution functions of these uncertain parameters a probabilistic analysis is carried out. In many application fields probabilities of the exceedance of failure criteria are computed. The out-coming failure probability is strongly dependent on the initial assumptions on the random variable properties. Measurements are always more or less inaccurate data due to varying environmental conditions during the measurement procedure. Furthermore the estimation of stochastic properties from a limited number of realisation also causes uncertainties in these quantities. Thus the assumption of exactly known stochastic properties by neglecting these uncertainties may not lead to very useful probabilistic measures in a design process. In this paper we assume the stochastic properties of a random variable as uncertain quantities caused by so-called epistemic uncertainties. Instead of predefined distribution types we use the maximum entropy distribution which enables the description of a wide range of distribution functions based on the first four stochastic moments. These moments are taken again as random variables to model the epistemic scatter in the stochastic assumptions. The main point of this paper is the discussion on the estimation of these uncertain stochastic properties based on inaccurate measurements. We investigate the bootstrap algorithm for its applicability to quantify the uncertainties in the stochastic properties considering imprecise measurement data. Based on the obtained estimates we apply standard stochastic analysis on a simple example to demonstrate the difference and the necessity of the proposed approach

Geodynamic evolution of the Eastern Pelagonian Zone in north-western Greece and the Republic of Macedonia. Implications from U/Pb, Rb/Sr, K/Ar, 40Ar/39Ar geochronology and fission track thermochronology

Die Pelagonische Zone ist eine ca. 420 km lange und 60 km breite NNW-SSE streichende Einheit der zentralen Helleniden. Sie ist in der Republik Mazedonien, in Albanien und auf dem griechischen Festland aufgeschlossen. Im W schließt sich die Pindos Zone, im Osten die Vardar Zone an. Das Arbeitsgebiet liegt in der kaum untersuchten nördlichen Pelagonischen Zone (Nordwest Griechenland und Republik Mazedonien). Aufgrund der Unterschiede in der lithologischen, strukturellen und metamorphen Entwicklung wird die nördliche Pelagonische Zone in Nordwest-Griechenland, der Republik Mazedonien und Albanien weiter unterteilt, in eine West und Ostpelagonische Zone. Das kristalline Basement der Westpelagonischen Zone besteht aus Graniten, Gneisen und Glimmerschiefern, welche von spätjurassischen Ophioliten und unmetamorphen paläozoischen Gesteinen überlagert werden. Die Ostpelagonische Zone zeigt eine Antiklinalstruktur und besteht aus einer unteren Einheit (hauptsächlich Granitoide und Gneise) und einer oberen Einheit (metaklastische und metakarbonatische Sequenz). Die vorliegende Arbeit konzentriert sich auf die Ostpelagonische Zone, wobei ergänzend Proben aus der Westpelagonischen Zone und der Vardar Zone untersucht worden. Die geochronologischen, strukturgeologischen und petrologischen Untersuchungen wurden mit dem Ziel durchgeführt, neue Daten für die Rekonstruktion der geodynamischen Entwicklung zur Verfügung zu stellen. U/Pb Zirkon Datierungen an ostpelagonischen Graniten zeigen zwei magmatische Ereignisse während des späten Paläozoikums an, welche mit der Schließung der Paleotethys im Zusammenhang stehen. In der West und Ostpelagonischen Zone, sowie in der Vardar Zone konnten mit Hilfe von K/Ar, Ar/Ar und Rb/Sr Datierungen fünf unterschiedliche tektonometamorphe Ereignisse während der alpidischen Orogenese datiert werden. Zirkon und Apatit Spaltspurendatierungen erlauben die zeitliche Rekonstruktion der späten bruchhaften Deformationsgeschichte.The Pelagonian Zone is a ca. 420 km long and 60 km broad NNW-SSE striking part of the central Hellenides, which is exposed in the Republic of Macedonia, Albania and the Greek mainland. It is flanked by the Pindos Zone in the W and the Vardar Zone in the E. The study area is located in the poorly studied northern Pelagonian Zone in north-western Greece and the Republic of Macedonia. According to differences in their lithology, structural and metamorphic evolution the northern Pelagonian Zone in north-western Greece, the Republic of Macedonia and Albania is subdivided into a West and East Pelagonian Zone. The crystalline basement of the West Pelagonian Zone consists of granites, gneisses and micaschists, which are superimposed by Late Jurassic ophiolites and unmetamorphosed Paleozoic rocks. The East Pelagonian Zone has an anticlinal shape and comprises a lower unit (mainly granitoids and gneisses) and an upper unit (metaclastic and metacarbonatic sequence). This study is concentrated on the East Pelagonian Zone, whereas additional samples derived from the West Pela-gonian Zone and Vardar Zone were also investigated. Geochronological, structural and petrological investigations were carried out to provide new data for the geodynamic reconstruction of this part of the Hellenic orogen. U/Pb zircon age determinations on East Pelagonian granites gave two different magmatic events during the Late Paleozoic, which were related to the closure of the Paleo-Tethyian ocean. In the West and East Pelagonian Zone and the Vardar Zone five different tectonometamorphic events, which are related to the Alpine orogeny, were dated by K/Ar, Ar/Ar and Rb/Sr. Zircon and apatite fission track thermochronology enable the reconstruction of the latest brittle deformation history

Application of the "fictious crack model" to meshless crack growth simulations

In this paper a meshless component is presented, which internally uses the common meshless interpolation technique >Moving Least SquaresFictitious Crack ModelFictitious Crack Model< finite interface elements are included between the crack surface nodes. The determination of the maximum principal strain at the crack tip is done by introducing an influence area around the singularity. On a practical example it is shown that the included elements improve the model by the transmission of the surface forces during monotonic loading and by the representation of the contact forces of closed cracks during reverse loading

A NATURAL NEIGHBOR BASED MOVING LEAST SQUARES APPROACH WITH INTERPOLATING WEIGHTING FUNCTION

The Element-free Galerkin Method has become a very popular tool for the simulation of mechanical problems with moving boundaries. The internally applied Moving Least Squares approximation uses in general Gaussian or cubic weighting functions and has compact support. Due to the approximative character of this method the obtained shape functions do not fulfill the interpolation condition, which causes additional numerical effort for the imposition of the essential boundary conditions. The application of a singular weighting function, which leads to singular coefficient matrices at the nodes, can solve this problem, but requires a very careful placement of the integration points. Special procedures for the handling of such singular matrices were proposed in literature, which require additional numerical effort. In this paper a non-singular weighting function is presented, which leads to an exact fulfillment of the interpolation condition. This weighting function leads to regular values of the weights and the coefficient matrices in the whole interpolation domain even at the nodes. Furthermore this function gives much more stable results for varying size of the influence radius and for strongly distorted nodal arrangements than classical weighting function types. Nevertheless, for practical applications the results are similar as these obtained with the regularized weighting type presented by the authors in previous publications. Finally a new concept will be presented, which enables an efficient analysis of systems with strongly varying node density. In this concept the nodal influence domains are adapted depending on the nodal configuration by interpolating the influence radius for each direction from the distances to the natural neighbor nodes. This approach requires a Voronoi diagram of the domain, which is available in this study since Delaunay triangles are used as integration background cells. In the numerical examples it will be shown, that this method leads to a more uniform and reduced number of influencing nodes for systems with varying node density than the classical circular influence domains, which means that the small additional numerical effort for interpolating the influence radius leads to remarkable reduction of the total numerical cost in a linear analysis while obtaining similar results. For nonlinear calculations this advantage would be even more significant

AN IMPROVED COHESIVE CRACK MODEL FOR COMBINED CRACK OPENING AND SLIDING UNDER CYCLIC LOADING

The modeling of crack propagation in plain and reinforced concrete structures is still a field for many researchers. If a macroscopic description of the cohesive cracking process of concrete is applied, generally the Fictitious Crack Model is utilized, where a force transmission over micro cracks is assumed. In the most applications of this concept the cohesive model represents the relation between the normal crack opening and the normal stress, which is mostly defined as an exponential softening function, independently from the shear stresses in tangential direction. The cohesive forces are then calculated only from the normal stresses. By Carol et al. 1997 an improved model was developed using a coupled relation between the normal and shear damage based on an elasto-plastic constitutive formulation. This model is based on a hyperbolic yield surface depending on the normal and the shear stresses and on the tensile and shear strength. This model also represents the effect of shear traction induced crack opening. Due to the elasto-plastic formulation, where the inelastic crack opening is represented by plastic strains, this model is limited for applications with monotonic loading. In order to enable the application for cases with un- and reloading the existing model is extended in this study using a combined plastic-damage formulation, which enables the modeling of crack opening and crack closure. Furthermore the corresponding algorithmic implementation using a return mapping approach is presented and the model is verified by means of several numerical examples. Finally an investigation concerning the identification of the model parameters by means of neural networks is presented. In this analysis an inverse approximation of the model parameters is performed by using a given set of points of the load displacement curves as input values and the model parameters as output terms. It will be shown, that the elasto-plastic model parameters could be identified well with this approach, but require a huge number of simulations