Explicit Limit Research Articles

Inverse Response Surface Method in Reliability-Based Design

Abstract The paper introduces an inverse response surface method utilized when performing reliability-based design optimization of time-consuming problems. Proposed procedure is based on a coupling of the adaptive response surface method and the artificial neural network-based inverse reliability method. The validity and accuracy of the method is tested using examples with explicit nonlinear limit state functions. Obtained results as well as important aspects of the method are discussed.

Simplified reliability analysis of multi hazard risk in gravity dams via machine learning techniques

Deterministic analysis does not provide a comprehensive model for concrete dam response under multi-hazard risk. Thus, the use of probabilistic approach is usually recommended which is problematic due to high computational demand. This paper presents a simplified reliability analysis framework for gravity dams subjected to flooding, earthquakes, and aging. A group of time-variant degradation models are proposed for different random variables. Response of the dam is presented by explicit limit state functions. The probability of failure is directly computed by either classical Monte Carlo simulation or the refined importance sampling technique. Next, three machine learning techniques (i.e., K-nearest neighbor, support vector machine, and naive Bayes classifier) are adopted for binary classification of the structural results. These methods are then demonstrated in terms of accuracy, applicability and computational time for prediction of the failure probability. Results are then generalized for different dam classes (based on the height-to-width ratio), various water levels, earthquake intensity, degradation rate, and cross-correlation between the random variables. Finally, a sigmoid-type function is proposed for analytical calculation of the failure probability for different classes of gravity dams. This function is then specialized for the hydrological hazard and the failure surface is presented as a direct function of the dam's height and width.

Scaling Limits and Generic Bounds for Exploration Processes

We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become explored. Given an initial number of vertices $N$ growing to infinity, we study statistical properties of the proportion of explored nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the \emph{jamming constant}, through a diffusion approximation for the exploration process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e. random sequential adsorption. As opposed to homogeneous random graphs, these do not allow for a reduction in dimensionality. Instead we build on a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and obtain generic bounds: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, we give two trajectorial interpretations of our bounds by constructing two coupled processes that have the same fluid limits.

Limit state design and methodologies in ice class rules for ships and standards for Arctic offshore structures

Sea ice poses hazards to ships and offshore structures which create challenges for designers. Designers have to take into account the ice actions resulting from the various ice conditions, interactions and operational scenarios which may occur during the lifetime of structures. In this paper the methodologies used in ship and offshore structure standards, such as the Finnish Swedish Ice Class Rules, International Association of Classification Societies Polar Class rules and ISO 19906, are considered with respect to Limit State Design principles and design air temperature definitions. The paper presents an overview of the design approach used in these standards and especially the criteria for the structural limit used. Case study examples based on ship measurements in the Baltic, as well as design air temperature data from the Russian Arctic, are used to highlight the different approaches and their implementation. The analysis shows clearly that explicit design ice action and structural limit are required for the rational basis in the future development of Limit State definitions in standards accounting for ice actions.

New neural network-based response surface method for reliability analysis of structures

In this paper, a new algorithm is introduced for reliability analysis of structures using response surface method based on a group method of data handling-type neural networks with general structure (GS-GMDH-type NN). A multilayer network of quadratic neurons, GMDH, offers an effective solution to modeling nonlinear systems without an explicit limit state function. In the proposed method, the response surface function is determined using GMDH-type neural networks. This is then connected to a reliability method, such as first-order or second-order reliability methods (FORM or SORM) or Monte Carlo simulation method to predict the failure probability (Pf). In the proposed method, the use of the GMDH-type neural network with general structure, where all neurons from previous layers are used to produce neurons in the new layer, can improve the limit state function. In addition, the structure of the neural network and its weight are simultaneously optimized by genetic algorithm and singular value decomposition. As a result, the obtained model has no significant error, despite its simplicity. Moreover, the obtained limit state function is explicit and allows direct use of FORM and SORM methods. To determine the accuracy and efficiency of the proposed method, four numerical examples are solved and their results are compared to other conventional methods. The results show that the proposed method is simply applicable to analyzing the reliability of large complex and sophisticated structures without an explicit limit state function. The proposed approach is a high accurate method that can significantly reduce computing time compared with direct Monte Carlo method.

Cluster analysis of local convergent sequences of structures

AbstractThe cluster analysis of very large objects is an important problem, which spans several theoretical as well as applied branches of mathematics and computer science. Here we suggest a novel approach: under assumption of local convergence of a sequence of finite structures we derive an asymptotic clustering. This is achieved by a blend of analytic and geometric techniques, and particularly by a new interpretation of the authors' representation theorem for limits of local convergent sequences, which serves as a guidance for the whole process. Our study may be seen as an effort to describe connectivity structure at the limit (without having a defined explicit limit structure) and to pull this connectivity structure back to the finite structures in the sequence in a continuous way. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 674–728, 2017

Stability of cracked concrete hydraulic structures by nonlinear quasi-static explicit finite element and 3D limit equilibrium methods

Several hydraulic concrete structures suffer from severe tridimensional (3D) discrete cracking, producing an assembly of concrete blocks resting one on top of the other. It is important to consider the 3D particularities of the cracked surfaces in the nonlinear sliding safety evaluation of these structures. A methodology to assess a sliding safety factor (SSF) and a sliding direction D for any structure with an a priori known 3D discrete crack surface geometry using the quasi-static explicit nonlinear finite elements method (QSE-FEM) is presented herein in the context of the strength reduction approach. QSE-FEM is known for its efficiency in solving highly nonlinear problems compared to implicit FEM. However, in QSE-FEM, the determination of the incipient failure motion is challenging. In addition to the ratio of kinetic to internal strain energy, a new criterion is proposed to identify sliding initiation based on absolute displacements of a control point. As part of the proposed methodology, a complementary simple tool, 3D-LEM, has been developed as a 3D extension of the classical limit equilibrium method (LEM). It is useful in preliminary sliding analyses to estimate the critical friction coefficient inducing sliding and the corresponding direction D. Three benchmark examples, of increasing complexities, are presented to verify the performance of the proposed methodology. Finally, a case study is adapted from an existing cracked hydraulic structure to evaluate its sliding stability using the QSE-FEM and 3D-LEM approaches. In the three benchmark examples, the same results are found for SSF and D for the two approaches. In the hydraulic structure example, SSFLEM, computed with 3D-LEM, is a lower bound of SSFFEM from QSE-FEM, and a strong correlation is found between the sliding directions computed with the two approaches.

Limiting empirical distribution for eigenvalues of products of random rectangular matrices

We study the empirical spectral distribution of a product AN(m)=A1⋯Am of m random rectangular matrices with i.i.d. complex Gaussian entries. The product ensemble is of dimension N×N, and the rectangular matrix Aj is of size Nj×Nj+1 for j=1,…,m with Nm+1=N1=N. Explicit limit of empirical eigenvalue distribution of AN(m) is obtained in almost sure convergence as N goes to infinity. In particular, a rich feature of the limiting distributions is presented as the ratio Nj/N fluctuates for each j.

Hydrodynamic Limit Equation for a Lozenge Tiling Glauber Dynamics

We study a reversible continuous-time Markov dynamics on lozenge tilings of the plane, introduced by Luby et al. Single updates consist in concatenations of $n$ elementary lozenge rotations at adjacent vertices. The dynamics can also be seen as a reversible stochastic interface evolution. When the update rate is chosen proportional to $1/n$, the dynamics is known to enjoy especially nice features: a certain Hamming distance between configurations contracts with time on average and the relaxation time of the Markov chain is diffusive, growing like the square of the diameter of the system. Here, we present another remarkable feature of this dynamics, namely we derive, in the diffusive time scale, a fully explicit hydrodynamic limit equation for the height function (in the form of a non-linear parabolic PDE). While this equation cannot be written as a gradient flow w.r.t. a surface energy functional, it has nice analytic properties, for instance it contracts the $\mathbb L^2$ distance between solutions. The mobility coefficient $\mu$ in the equation has non-trivial but explicit dependence on the interface slope and, interestingly, is directly related to the system's surface free energy. The derivation of the hydrodynamic limit is not fully rigorous, in that it relies on an unproven assumption of local equilibrium.

A hybrid conjugate finite-step length method for robust and efficient reliability analysis

The robustness and efficiency of the first-order reliability method (FORM) are the important issues in the structural reliability analysis. In this paper, a hybrid conjugate search direction with finite-step length is proposed to improve the efficiency and robustness of FORM, namely hybrid conjugate finite-step length (CFSL-H). The conjugate scalar factor in CFSL-H is adaptively updated using two conjugate methods with a dynamic participation factor. The accuracy, efficiency and robustness of the CFSL-H are illustrated through the nonlinear explicit and structural implicit limit state functions with normal and non-normal random variables. The results illustrated that the proposed CFSL-H algorithm is more robust, efficient and accurate than the modified existing FORM algorithms for complex structural problems.

On curvature pinching of conic 2-spheres

We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff limit. This is a generalization of the previous results of Chen-Lin and Bartolucci for 2-spheres with one or two conic points.

Nesting in perception of affordances for stepping and leaping.

Perception of affordances for a given behavior typically reflects the task-specific action capabilities of the perceiver. However, many experiments have shown a discrepancy between the perceptual and behavioral boundaries for a given behavior. One possibility for such a discrepancy is that the context of many experimental tasks transformed what is typically a dynamic perception-action task into an analytical or reflective judgment. We investigated this hypothesis with respect to perception of maximum stepping and leaping distance. For both behaviors, perception of these affordances more closely reflected action capabilities when the perceptual task was nested within a superordinate task than when it was not (regardless of whether the behavior itself was performed). Additionally, verbal reports of perception of maximum leaping distance more closely reflected action capabilities when there was an explicit time limit on such reports. The results are discussed in the context of the ecological principle of nesting and in attentional focus during motor control tasks.

Stochastic six-vertex model

We study the asymmetric six-vertex model in the quadrant with parameters on the stochastic line. We show that the random height function of the model converges to an explicit deterministic limit shape as the mesh size tends to 0. We further prove that the one-point fluctuations around the limit shape are asymptotically governed by the GUE Tracy-Widom distribution. We also explain an equivalent formulation of our model as an interacting particle system, which can be viewed as a discrete time generalization of ASEP started from the step initial condition. Our results confirm an earlier prediction of Gwa and Spohn (1992) that this system belongs to the KPZ universality class.

Acoustic properties of pistonphones at low frequencies in the presence of pressure leakage and heat conduction

The wide concern on absolute pressure calibration of acoustic transducers at low frequencies prompts the development of the pistonphone method. At low frequencies, the acoustic properties of pistonphones are governed by the pressure leakage and the heat conduction effects. However, the traditional theory for these two effects applies a linear superposition of two independent correction models, which differs somewhat from their coupled effect at low frequencies. In this paper, acoustic properties of pistonphones at low frequencies in full consideration of the pressure leakage and heat conduction effects have been quantitatively studied, and the explicit expression for the generated sound pressure has been derived. With more practical significance, a coupled correction expression for these two effects of pistonphones has been derived. In allusion to two typical pistonphones, the NPL pistonphone and our developed infrasonic pistonphone, comparisons were done for the coupled correction expression and the traditional one, whose results reveal that the traditional one produces maximum insufficient errors of about 0.1dB above the lower limiting frequencies of two pistonphones, while at lower frequencies, excessive correction errors with an explicit limit of about 3dB are produced by the traditional expression. The coupled correction expression should be adopted in the absolute pressure calibration of acoustic transducers at low frequencies. Furthermore, it is found that the heat conduction effect takes a limiting deviation of about 3dB for the pressure amplitude and a small phase difference as frequency decreases, while the pressure leakage effect remarkably drives the pressure amplitude to attenuate and the phase difference tends to be 90° as the frequency decreases. The pressure leakage effect plays a more important role on the low frequency property of pistonphones.

Cumulative PSO-Kriging model for slope reliability analysis

The particle swarm optimization (PSO) algorithm is introduced in the Kriging modeling process to overcome the limits of pattern search method's single-point search scheme as well as its heavy dependence on the initial guess solution when obtaining the optimal correlation parameters. PSO-Kriging is proved to give better simulation by interpolating and extrapolating the unobserved points. In reliability analysis, cumulative formation of repeatedly using sampling points in previous iterations is introduced into PSO-Kriging and the classic response surface method (RSM). Cumulative formation can take full advantage of available sampling information and avoid reciprocating oscillation in the iterative process. One explicit nonlinear limit state function example demonstrated that cumulative scheme can make both PSO-Kriging and RSM much more effective, no matter latin hypercube sampling (LHS) or iteratively interpolating sampling (IIS) approach is utilized. Cumulative PSO-Kriging seems to be even more stable and efficient. Two slope reliability analysis examples including a practical nuclear plant breakwater's reliability analysis problem proved that the proposed cumulative PSO-Kriging is very suitable for the reliability analysis of real engineering structures.

Maximizing Probability Bounds Under Moment-Matching Restrictions

The problem of characterizing a distribution by its moments dates to work by Chebyshev in the mid-nineteenth century. There are clear (and close) connections with characteristic functions, moment spaces, quadrature, and other very classical mathematical pursuits. Lindsay and Basak posed the specific question of how far from normality could a distribution be if it matches k normal moments. They provided a bound on the maximal difference in cdfs, and implied that these bounds were attained. It will be shown here that in fact the bound is not attained if the number of even moments matched is odd. An explicit solution is developed as a symmetric distribution with a finite number of mass points when the number of even moments matched is even, and this bound for the even case is shown to hold as an explicit limit for the subsequent odd case. As Lindsay noted, the discrepancies can be sizable even for a moderate number of matched moments. Some comments on implications are proffered.

Asymptotic behavior of weighted cubic variation of sub-fractional brownian motion

ABSTRACTIn this article, we investigate the convergence of renormalized weighted cubic variation of a sub-fractional Brownian motion SH with Hurst index H. When , we prove by means of Malliavin calculus that the convergence holds in L2 toward an explicit limit which only depends on SH. We also numerically simulate the sample paths of such a type of sub-fractional Brownian motion regulated by different Hurst index H.

Limit distributions of generalized St. Petersburg games

The St. Petersburg game is known as a probability model with infinite expectation, and has some extensions. Gut and Martin-Löf (2013) studied convergence in distribution along a suitable specific subsequence under their generalization. In this article, we extend their results using residue analysis investigated by Vardi (1995). We give the explicit limit distributions via characteristic functions with gamma functions which become semi-stable or strictly semi-stable.

Limit shape of random convex polygonal lines: Even more universality

The paper concerns the limit shape (under some probability measure) of convex polygonal lines with vertices on Z+2, starting at the origin and with the right endpoint n=(n1,n2)→∞. In the case of the uniform measure, an explicit limit shape γ⁎:={(x1,x2)∈R+2:1−x1+x2=1} was found independently by Vershik (1994) [19], Bárány (1995) [3], and Sinaĭ (1994) [16]. Recently, Bogachev and Zarbaliev (1999) [5] proved that the limit shape γ⁎ is universal for a certain parametric family of multiplicative probability measures generalizing the uniform distribution. In the present work, the universality result is extended to a much wider class of multiplicative measures, including (but not limited to) analogs of the three meta-types of decomposable combinatorial structures — multisets, selections, and assemblies. This result is in sharp contrast with the one-dimensional case where the limit shape of Young diagrams associated with integer partitions heavily depends on the distributional type.

Limit your time, gamble responsibly: setting a time limit (via pop-up message) on an electronic gaming machine reduces time on device

In the current study, we assessed whether undergraduate electronic gaming machine (EGM) gamblers would be more apt to set a time limit and spend less time gambling when asked to consider setting an explicit time limit prior to their gambling session. To this end, participants (N = 43) were randomly assigned to a time limit pop-up condition or control condition, both of which involved gambling on an EGM in a virtual reality (VR) casino. In the time limit pop-up condition, participants were asked (via pop-up message) to consider setting a time limit on play and entering that limit in an available text box prior to commencing play. In the no time limit pop-up condition, participants engaged in play immediately upon accessing the EGM in the VR casino (i.e. they were not exposed to a time limit pop-up message). As predicted, participants who were explicitly asked to consider setting a time limit on their EGM play were significantly more likely to do so and spent less time gambling than those who were not given such instructions. The results provide preliminary support for the contention that setting a time limit on EGM play is an effective responsible gambling strategy.