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Combination of model reduction and adaptive subset simulation for structural reliability problems
(2019)
A safe and robust design is a key criterion when building a structure or a component. Ensuring this criterion can either be performed by fullfilling prescribed safety margins, or by using a full probabilistic approach with a computation of the failure probability. The latter approach is particularly well suited for complex Problems with an interaction of different physical penomena that can be described in a numerical model. The bottleneck in this approach is the computational effort. Sampling methods such as Markov chain Monte Carlo methods are often used to evaluate the system reliability. Due to small failure probabilities (e.g. 10^6) and complex physical models with already and extensive computational effort for a single set of parameters, these methods a prohibitively expensive. The focus of this contribution is to demonstrate the advantages of combining model reduction techniques within the concept a variance reducing adaptive sampling procedures. In the developed method, a modification of the adaptive subset simulation based on Papaioannou et al. 2015 is used and coupled with a limit state function based on Proper Generalized Decomposition (PGD) (Chinesta et al. 2011). In the subset simulation the failure probability is expressed as a product of larger conditional failure probabilities. The intermediate failure events are chosen as a decreasing sequence. Instead of solving each conditional probability with a Markov chain approach, an importance sampling approach is used. It is be shown that the accuracy of the estimation depends mainly on the number of samples in the last sub-problem. For model reduction, the PGD approach is used to solve the structural problem a priori for a given Parameter space (physical space plus all random parameters). The PGD approach results in an approximation of the problem output within a prescribed range of all input Parameters (load factor, material properties, ..). The approximation of the solution by a separated form allows an evaluation of the limit state function within the sampling algorithm with almost no cost. This coupled PGD – adaptive subset Simulation approach is used to estimate the failure probability of examples with different complexity. The convergence, the error propagation as well as the reduction in computational time is discussed.
One of the most important goals in civil engineering is to guaranty the safety of constructions. National standards prescribe a required failure probability in the order of 10−6 (e.g. DIN EN 199:2010-12). The estimation of these failure probabilities is the key point of structural reliability analysis. Generally, it is not possible to compute the failure probability analytically.
Therefore, simulation-based methods as well as methods based on surrogate modeling or response surface methods have been developed. Nevertheless, these methods still require a few thousand evaluations of the structure, usually with finite element (FE) simulations, making reliability analysis computationally expensive for relevant applications.
The aim of this contribution is to increase the efficiency of structural reliability analysis by using the advantages of model reduction techniques. Model reduction is a popular concept to decrease the computational effort of complex numerical simulations while maintaining a reasonable accuracy. Coupling a reduced model with an efficient variance reducing sampling algorithm significantly reduces the computational cost of the reliability analysis without a relevant loss of accuracy.
One of the most important goals in civil engineering is to guaranty the safety of constructions. National standards prescribe a required failure probability in the order of 10-6 (e.g. DIN EN 199:2010-12). The estimation of these failure probabilities is the key point of structural reliability analysis. Generally, it is not possible to compute the failure probability analytically. Therefore, simulation-based methods as well as methods based on surrogate modelling or response surface methods have been developed. Nevertheless, these methods still require a few thousand evaluations of the structure, usually with finite element (FE) simulations, making reliability analysis computationally expensive for relevant applications.
The aim of this contribution is to increase the efficiency of structural reliability analysis by using the advantages of model reduction techniques. Model reduction is a popular concept to decrease the computational effort of complex numerical simulations while maintaining a reasonable accuracy. Coupling a reduced model with an efficient variance reducing sampling algorithm significantly reduces the computational cost of the reliability analysis without a relevant loss of accuracy.
In this paper, the impact problem and the subsequent wave Propagation are considered. For the contact discretization an intermediate non-uniform rational B-spline (NURBS) layer is added between the contacting finite element bodies, which allows a smooth contact formulation and efficient element-based integration.
The impact event is ill-posed and requires a regularization to avoid propagating stress oscillations. A nonlinear mesh-dependent penalty regularization is used, where the stiffness of the penalty regularization increases upon mesh refinement. Explicit time integration methods are well suited for wave propagation problems, but are efficient only for diagonal mass matrices. Using a spectral element discretization in combination with a NURBS contact layer the bulk part of the mass matrix is diagonal.
In this paper, the impact problem and the subsequent wave propagation are considered. For the contact discretization an intermediate NURBS layer is added between the contacting finite element bodies, which allows a smooth contact formulation and efficient element‐based integration. The impact event is ill‐posed and requires a regularization to avoid propagating stress oscillations. A nonlinear mesh dependent penalty regularization is used, where the stiffness of the penalty regularization increases upon mesh refinement. Explicit time integration methods are well suited for wave propagation problems, but are efficient only for diagonal mass matrices. Using a spectral element discretization and the coupled FE‐NURBS approach the bulk part of the mass matrix is diagonal.
Quasi-brittle materials exhibit strain softening. Their modeling requires regularized constitutive formulations to avoid instabilities on the material level. A commonly used model is the implicit gradient-enhanced damage model. For complex geometries, it still Shows structural instabilities when integrated with classical backward Euler schemes. An alternative is the implicit–explicit (IMPL-EX) Integration scheme. It consists of the extrapolation of internal variables followed by an implicit calculation of the solution fields. The solution procedure for the nonlinear gradient-enhanced damage model is thus transformed into a sequence of problems that are algorithmically linear in every time step. Therefore, they require one single Newton–Raphson iteration per time step to converge. This provides both additional robustness and computational acceleration. The introduced extrapolation error is controlled by adaptive time-stepping schemes. This paper introduced and assessed two novel classes of error control schemes that provide further Performance improvements. In a three-dimensional compression test for a mesoscale model of concrete, the presented scheme was about 40 times faster than an adaptive backward Euler time integration.
Die Brücken im Netz der Bundesverkehrswege sind überwiegend in einem ausreichenden bis guten Zustand. Allerdings steigt der Unterhalts- und Sanierungsaufwand aufgrund des inzwischen hohen Alters vieler Brücken sowie des ständig wachsenden Schwerlastverkehrs. Techniken zur Einschätzung der verbleibenden Lebensdauer von Brücken sowie zur dauerhaften Beobachtung des Tragverhaltens bzw. des Erfolges von Sanierungsmaßnahmen werden daher für den sicheren und wirtschaftlichen Betrieb dringend benötigt. Zur Evaluierung dafür geeigneter holistischer Ansätze wurde in der BAM das Projekt BLEIB - Bewertung, Lebensdauerprognose und Instandsetzung von Brückenbauwerken - ins Leben gerufen.
Ein zentrales Ergebnis des Projektes ist eine extern vorgespannte Stahlbetonbrücke als Zweifeldträger mit einer Gesamtlänge von 24 m, die für den Test verschiedenster Sensorsysteme, zur Validierung numerischer Modelle und zur Erprobung von Sanierungs- und Verstärkungsmaßnahmen entwickelt wurde. Für die Simulation unterschiedlicher Schädigungsgrade kann die Vorspannung der Brücke variiert werden. Die Brücke wird mit beweglichen Gewichten belastet und über einen Shaker zum Schwingen angeregt.
Das Brückenmodell wurde bewusst geschädigt, indem die Vorspannung der Struktur erstmalig schrittweise bis auf null reduziert wurde. Unter der Eigenlast verformte sich die Brücke, wodurch eine Rissbildung im Beton einsetzte. Die Zugspannung, die zuvor durch die Vorspannung aufgenommen wurde, übernahm Schritt für Schritt der Beton. Als die Zugspannungen die relativ geringe Zugfestigkeit des Betons überstiegen, begann dieser zu reißen und die schlaffe Bewehrung der Struktur nahm die Spannungen auf. Dieser Versuch wurde unter anderem von Schallemissionsmessungen begleitet. Der Rissbildungsprozess konnte damit, bei gleichzeitiger Aufzeichnung der Vorspannung, früh detektiert und die Risse geortet werden. Die Ergebnisse korrelieren gut mit den Ergebnissen der stereophotogrammetrischen Verformungsmessungen der Struktur.