Stochastic Deformation Research Articles

Effect of stochastic deformation on the vibration characteristics of a tube bundle in axial flow

Design evaluation of nuclear components using numerical methods has typically been focused on nominal geometries and conditions. In reality, the geometry of reactor components and operating conditions can differ from the design on the drawing table. Increasing understanding and safety of nuclear energy systems requires investigating these more realistic conditions as well. An example is the bow of fuel assemblies, a deformation that occurs after some residence time as a consequence of thermal and irradiation effects. A paradigm shift from purely deterministic simulations toward simulations involving stochastic processes can help to meet this demand.One of the aspects in the design assessment of nuclear reactors is flow-induced vibration (FIV). A lot of experimental effort has already been allocated to FIV phenomena in the past, and more recently also numerical approaches have been applied. However, it was not clear to what extent vibration characteristics would change for a geometry different from the design drawing. In this study the effect of a random bow deformation on the vibration behavior of a flexible tube in a rigid bundle is investigated using fluid–structure interaction simulations. As the (stressless) geometry presumably has a stochastic nature, the modal characteristics should be treated accordingly.For this uncertainty quantification of the modal characteristics, the generalized Polynomial Chaos method was applied. First, only the steady flow through the bundle was considered, without simulating the vibration. This sufficiently reduced the computational cost to apply the Monte Carlo method, which was used to validate the more efficient Polynomial Chaos method. This latter method was found adequate to predict the stochastic modal characteristics of the deformed, flexible tube.

On Two Non-Ergodic Reversible Cellular Automata, One Classical, the Other Quantum.

We propose and discuss two variants of kinetic particle models-cellular automata in 1 + 1 dimensions-that have some appeal due to their simplicity and intriguing properties, which could warrant further research and applications. The first model is a deterministic and reversible automaton describing two species of quasiparticles: stable massless matter particles moving with velocity ±1 and unstable standing (zero velocity) field particles. We discuss two distinct continuity equations for three conserved charges of the model. While the first two charges and the corresponding currents have support of three lattice sites and represent a lattice analogue of the conserved energy-momentum tensor, we find an additional conserved charge and current with support of nine sites, implying non-ergodic behaviour and potentially signalling integrability of the model with a highly nested R-matrix structure. The second model represents a quantum (or stochastic) deformation of a recently introduced and studied charged hardpoint lattice gas, where particles of different binary charge (±1) and binary velocity (±1) can nontrivially mix upon elastic collisional scattering. We show that while the unitary evolution rule of this model does not satisfy the full Yang-Baxter equation, it still satisfies an intriguing related identity which gives birth to an infinite set of local conserved operators, the so-called glider operators.

Moment Evolution Equations and Moment Matching for Stochastic Image EPDiff

Models of stochastic image deformation allow study of time-continuous stochastic effects transforming images by deforming the image domain. Applications include longitudinal medical image analysis with both population trends and random subject-specific variation. Focusing on a stochastic extension of the LDDMM models with evolutions governed by a stochastic EPDiff equation, we use moment approximations of the corresponding Itô diffusion to construct estimators for statistical inference in the full stochastic model. We show that this approach, when efficiently implemented with automatic differentiation tools, can successfully estimate parameters encoding the spatial correlation of the noise fields on the image.

The effects of couple stress lubricants and surface roughness on squeeze EHL motion between porous medium layer and elastic ball

This paper investigated the effects of porous media layer (PML) and couple stress (CS) lubricants at pure squeeze action in an isothermal elastohydrodynamic lubrication (EHL) point contact with surface roughness (SRN) under constant load condition. The modified transient stochastic Reynolds equation was derived in polar coordinates by means of the Christensen’s stochastic theory and the Stokes’s CS fluid theory. The Gauss-Seidel method (GSM) and the finite difference method (FDM) were both used to solve simultaneously modified transient stochastic Reynolds, force equilibrium, elasticity deformation, and rheology equations. The simulation results revealed that the greater permeability ( K) and porous layer thickness (Φ) are, the smaller central pressures ( Pc) are, and the smaller film thicknesses ( H) are. The times required for achieving maximum central pressures (Pcmax) decrease with increasing K and Φ. Due to squeeze and elastic deformation effects, the slope values of central velocity ( Vc) change from negative to positive. The magnitude of Vc and Vmin decreased with decreasing K and Φ. The Hcmin of circular type RN are greater than that of radial type RN. The Pcmax of radial type RN are slightly greater than that of circular type RN.

Исследование взаимосвязи продовольственных, энергетических и водных ресурсов с помощью трехсекторальной модели Лоренца

Работа посвящена проблеме создания новых методов для комплексного моделирования и управления риском, которые позволят исследовать синергетические взаимодействия между источниками рисков различного происхождения в условиях неопределенности. Предложен подход к исследованию взаимосвязи продовольственных, водных и энергетических ресурсов с помощью трехсекторальной модели Лоренца, которая объединяет в единой структуре однотипным образом описанные сектора экономики, каждый из которых рассматривается в сроках уровня производительности, количества рабочих мест. и уровня структурных нарушений В результате математического моделирования определены условия возникновения детерминированного хаоса в минимальной модели экономического развития и выявлены возможные причины возрастающей уязвимости глобальной экономики к малым изменениям параметров управления. Рассмотрена задача определения эффективных управлений с целью минимизации суммарных структурных нарушений за выбранный интервал времени. В результате модельных экспериментов обнаружены траектории изменения параметров управления, позволяющие уменьшить число структурных нарушений. Это достигается за счет изменений соотношения уровней пропозиции и спроса продукции, спроса и предложения на создание рабочих мест. Рассмотрено влияние случайных возмущений на стохастическую деформацию детерминированных аттракторов модели Лоренца. Показано, что при случайных возбуждениях траектории стохастической системы покидают детерминированный аттрактор и образуют вокруг него некоторый пучок с соответствующим вероятностным распределением. Рассмотрена возможность дальнейшего усложнения модели за счет учета других секторов экономики с помощью модели Лоренца в комплексной форме. Задача исследования n секторов экономик в этом случае сводится к рассмотрению поведения ансамбля n связанных осцилляторов, генерирующих колебания с частотами ωn соответственно. Коллективная синхронизация данных осцилляторов может быть исследована с помощью модели Курамото. Задача управления социально-экономическим развитием в условиях возникновения хаотических режимов сводится для комплексной модели к управлению частотой поля с ненулевым средним, генерируемым связанными осцилляторами.

A stochastic collocation approach for parabolic PDEs with random domain deformations

In this article we analyze the linear parabolic partial differential equation with a stochastic domain deformation. In particular, we concentrate on the problem of numerically approximating the statistical moments of a given Quantity of Interest (QoI). The geometry is assumed to be random. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and shown to admit an extension on a well defined region embedded in the complex hyperplane. The stochastic moments of the QoI are computed by employing a collocation method in conjunction with an isotropic Smolyak sparse grid. Theoretical sub-exponential convergence rates as a function to the number of collocation interpolation knots are derived. Numerical experiments are performed and they confirm the theoretical error estimates.

Evaluation of variability characteristics of mechanical parameters of artificially frozen clay in deep alluvium

The uncertain mechanical parameters of artificial frozen soil in deep alluvium are the key factors for computing the stochastic stress fields and stochastic deformation fields of artificial freezing engineering. In this study, a series of mechanical parameters tests for the artificially frozen clay are carried out, and the statistical data of elastic modulus, strength and Poisson ratio of frozen clay under three different low temperature conditions (−20 °C, −15 °C and −10 °C) are obtained. The experimental semi-variogram values of the mechanical parameters for artificially frozen clay under different temperature conditions are calculated and analyzed. The results show that the test data meets the requirements of the coefficient of variation analysis and the elastic modulus, strength and poisson ratio basically meet the normal distribution. The variability of the mechanical parameters for artificially frozen clay is basically in the degree of moderate to weak variability. The optimal theoretical variation function model of basic mechanical parameters of frozen clay is Gaussian model. The correlation of soil parameters calculated by Gauss model at smaller distances is greater than that of Exponential model and Spherical model. The mechanical parameters of artificially frozen clay are strongly dependent in spatial range. This study can improve our understanding of the variation characteristics of uncertain mechanical parameters for artificial frozen soil in deep alluvium and provide a theoretical basis and reference for the stochastic mechanical analysis of artificial freezing engineering.

Crystal structure and composition dependence of mechanical properties of single-crystalline NbCo2 Laves phase

Extended diffusion layers of the cubic C15 and hexagonal C14 and C36 NbCo2 Laves phases with concentration gradients covering their entire homogeneity ranges were produced by the diffusion couple technique. Single-phase and single-crystalline micropillars of the cubic and hexagonal NbCo2 Laves phases were prepared in the diffusion layers by focused ion beam (FIB) milling. The influence of chemical composition, structure type, orientation and pillar size on the deformation behavior and the critical resolved shear stress (CRSS) was studied by micropillar compression tests. The pillar orientation influences the activated slip systems, but the deformation behavior and the CRSS are independent of orientation. The deformation of the smallest NbCo2 micropillars (0.8 µm in top diameter) appears to be dislocation nucleation controlled and the CRSS approaches the theoretical shear stress for dislocation nucleation. The CRSS of the 0.8 µm-sized NbCo2 micropillars is nearly constant from 26 to 34 at.% Nb where the C15 structure is stable. It decreases as the composition approaches the Co-rich and Nb-rich boundaries of the homogeneity range where the C15 structure transforms to the C36 and the C14 structure, respectively. The decrease in the CRSS at these compositions is related to the reduction of shear modulus and stacking fault energy. As the pillar size increases, stochastic deformation behavior and large scatter in the CRSS values occur and obscure the composition effect on the CRSS.

Stochastic seismic lateral deformation of a multi-story subway station structure based on the probability density evolution method

Effective utilization of urban underground space has seen multi-story subway station structures become increasingly common. Two significant issues need to be solved in seismic evaluation of this kind of station structure. One is to set the appropriate seismic performance evaluation index; another is to choose a comprehensive method that can evaluate the structure performance with regard to the index. In this paper, a numerical model was established and validated by previously conducted shaking-table tests. Then stochastic seismic analyses were performed on the model using the probability density evolution method (PDEM). Inter-story drift and column drift ratio, which can reflect the lateral deformation of station structures induced by earthquake, were selected as seismic performance evaluation indexes and studied from the stochastic view. The stochastic analysis showed that the probability density distribution of the inter-story drift extremum of the middle-stories of the structure is larger than that of the top story and bottom story, similar to the lateral deformation mode of a ground frame structure with strong columns and weak beams. In addition, the probability density distribution of the column drift ratio extremum in each story is quite similar, and the probability density distribution of the column drift ratio extremum is larger than the inter-story drift extremum of the corresponding story. Compared with the inter-story drift, the column drift ratio is more suitable for evaluating the seismic performance of the multi-story subway station structure, because it reflects the influence of longitudinal beams with different height at the two ends of column on the lateral deformation of columns, which are key components and determine the safety of the structure to a great extent. By comparing the results of the PDEM with the shaking-table test, we found that stochastic analysis method can further obtain the probability of each numerical value of seismic performance evaluation indexes, thus making the evaluation results more comprehensive.

Stochastic deformation and shear transformation zones of the glassy matrix in CuZr-based metallic-glass composites

The incipient plasticity of an amorphous solid represents the onset of shear deformation, which is a stochastic progress closely related to microstructure evolution. It is well known that the cooling rate plays a crucial role in the microstructure evolution of a glass. Nevertheless, how the cooling rate affects the incipient plasticity is still unclear. In this work, the incipient plastic deformation of the amorphous phase in Cu47.5Zr48Al4Nb0.5 bulk metallic glass composites (BMGCs) cast with different cooling rates is systematically studied. The incipient plasticity of the glassy matrix of the BMGCs is found to occur via a first displacement burst, the occurrence of which is more stochastic as the cooling rate decreases. According to the auto-correlation functions and cooperative shear model, the stochastic behavior is attributed to the effects of the cooling rate on the initial free volume and subsequent activated shear transformation zones (STZs). A higher content of free volume is present in the glassy matrix of the studied samples cooled at a faster rate, which facilitates the operation of STZs. Moreover, the larger incipient burst size is correlated with the higher content of free volume and larger STZs in the glassy matrix. The larger STZs result in more preferable propagation of multiple shear bands, leading to less stochastic deformation and more obvious plastic deformation. This study provides further understanding on the incipient plasticity of glassy matrix in BMGCs.

Machine learning plastic deformation of crystals

Plastic deformation of micron-scale crystalline solids exhibits stress-strain curves with significant sample-to-sample variations. It is a pertinent question if this variability is purely random or to some extent predictable. Here we show, by employing machine learning techniques such as regression neural networks and support vector machines that deformation predictability evolves with strain and crystal size. Using data from discrete dislocations dynamics simulations, the machine learning models are trained to infer the mapping from features of the pre-existing dislocation configuration to the stress-strain curves. The predictability vs strain relation is non-monotonic and exhibits a system size effect: larger systems are more predictable. Stochastic deformation avalanches give rise to fundamental limits of deformation predictability for intermediate strains. However, the large-strain deformation dynamics of the samples can be predicted surprisingly well.

Fluid Deformation in Random Steady Three Dimensional Flow.

The deformation of elementary fluid volumes by velocity gradients is a key process for scalar mixing, chemical reactions and biological processes in flows. Whilst fluid deformation in unsteady, turbulent flow has gained much attention over the past half century, deformation in steady random flows with complex structure - such as flow through heterogeneous porous media - has received significantly less attention. In contrast to turbulent flow, the steady nature of these flows constrains fluid deformation to be anisotropic with respect to the fluid velocity, with significant implications for e.g. longitudinal and transverse mixing and dispersion. In this study we derive an ab initio coupled continuous time random walk (CTRW) model of fluid deformation in random steady three-dimensional flow that is based upon a streamline coordinate transform which renders the velocity gradient and fluid deformation tensors upper-triangular. We apply this coupled CTRW model to several model flows and find these exhibit a remarkably simple deformation structure in the streamline coordinate frame, facilitating solution of the stochastic deformation tensor components. These results show that the evolution of longitudinal and transverse fluid deformation for chaotic flows is governed by both the Lyapunov exponent and power-law exponent of the velocity PDF at small velocities, whereas algebraic deformation in non-chaotic flows arises from the intermittency of shear events following similar dynamics as that for steady two-dimensional flow.

Stochastic Generation and Deformation of Toroidal Oscillations in Neuron Model

The stochastic Hindmarsh–Rose neuron model in the parametric regions of complex nonlinear dynamics with quiescent and torus bursting regimes is studied. We show that in the zone where the deterministic system exhibits the quiescence, noise can lead to the generation of toroidal dynamical structure with the transition to the bursting regime. The studies of dispersion of random trajectories as well as power spectral density and interspike intervals distribution confirm these changes in system dynamics. Moreover, the stochastic emergence of torus is followed by the noise-induced chaotization. We show that the generation of the torus bursting oscillations is associated with the specificity of the arrangement of deterministic trajectories in the vicinity of the equilibrium as well as its stochastic sensitivity. A probabilistic mechanism of the stochastic generation of torus is investigated on the basis of the analysis of the stochastic sensitivity and geometry of confidence regions. We also show that in the parametric range of the torus bursting, noise can lead to the increase of number of bursts per time interval and to the reduction of spiking and quiescent phase duration.

Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters

• A general approach for the stochastic finite element analysis of soft tissue deformation is presented. • Sensitivity derivative Monte Carlo method to propagate uncertainty through hyperelastic models with stochastic parameters. • The proposed method is up to two orders of magnitude faster than the standard Monte Carlo approach. • The method is able to provide the user with statistical results on quantities of practical interest. • We applied the approach to a simple academic example and to the stochastic deformation of a brain. We present a simple open-source semi-intrusive computational method to propagate uncertainties through hyperelastic models of soft tissues. The proposed method is up to two orders of magnitude faster than the standard Monte Carlo method. The material model of interest can be altered by adjusting few lines of (FEniCS) code. The method is able to (1) provide the user with statistical confidence intervals on quantities of practical interest, such as the displacement of a tumour or target site in an organ; (2) quantify the sensitivity of the response of the organ to the associated parameters of the material model. We exercise the approach on the determination of a confidence interval on the motion of a target in the brain. We also show that for the boundary conditions under consideration five parameters of the Ogden–Holzapfel-like model have negligible influence on the displacement of the target zone compared to the three most influential parameters. The benchmark problems and all associated data are made available as supplementary material.

Stochastic analysis for uncertain deformation of foundations in permafrost regions

For foundations in permafrost regions, the displacement characteristics are uncertain because of the randomness of temperature characteristics and mechanical parameters, which make the structural system have an unexpected deviation and unpredictability. It will affect the safety of design and construction. In this paper, we consider the randomness of temperature characteristics and mechanical parameters. A stochastic analysis model for the uncertain displacement characteristic of foundations is presented, and the stochastic coupling program is compiled by Matrix Laboratory (MATLAB) software. The stochastic displacement fields of an embankment in a permafrost region are obtained and analyzed by Neumann stochastic finite element method (NSFEM). The results provide a new way to predict the deformation characteristics of foundations in permafrost regions, and it shows that the stochastic temperature has a different influence on the stochastic lateral displacement and vertical displacement. Construction disturbance and climate warming lead to three different stages for the mean settlement of characteristic points. For the stochastic settlement characteristic, the standard deviation increases with time, which imply that the results of conventional deterministic analysis may be far from the true value. These results can improve our understanding of the stochastic deformation fields of embankments and provide a theoretical basis for engineering reliability analysis and design in permafrost regions.

Disorder in the Sachdev–Ye–Kitaev model

We give qualitative arguments in support of the mesoscopic nature of the Sachdev–Ye–Kitaev (SYK) model in the regime with q2/N≪1 with N Majorana particles coupled by antisymmetric and random interactions of range q. Using a stochastic deformation of the SYK model, we show that its characteristic determinant obeys a viscid Burgers equation with a small spectral viscosity in the regime with q/N=1/2. In particular, the stochastic evolution of the SYK model can be mapped exactly onto that of random matrix theory, with universal Airy oscillations at the edges.

The Higgs boson in fractal quantum systems with active nanoelements

Stochastic deformation and stress fields within a fractal multilayer nanosystem are investigated theoretically and by numerical modeling. It is shown that the averaged displacement functions of lattice nodes are complex. Their behavior changes from regular to stochastic when the control parameters are altered. A set of ultracold 23Na atoms in an optical trap is chosen as the active nanoelement. It is demonstrated that certain physical properties (rate and quantization of the flow; hysteresis) of elementary excitations such as a vortex–antivortex pair are associated with the influence of a superfluid Bose–Einstein condensate (where a Higgs boson is the elementary excitation).

A weak approach to the stochastic deformation of classical mechanics

We establish a transfer principle, providing a canonical form of dynamics to stochastic models, inherited from their classical counterparts. The stochastic deformation of Euler$-$Lagrange conditions, and the associated Hamiltonian formulations, are given as conditions on laws of processes. This framework is shown to encompass classical models, and the so-called Schrödinger bridges. Other applications and perspectives are provided.

Fixture Capability Optimisation for Early-stage Design of Assembly System with Compliant Parts Using Nested Polynomial Chaos Expansion

This paper introduces the novel concept of fixture capability measure to determine fixture layout for the best assembly process yield by optimizing position of locators and reference clamps to compensate stochastic product variations and part deformation. This allows reducing the risk of product failures caused by product and process variation. The method is based on three main steps: (i) physics-based modelling of parts and fixtures, (ii) stochastic polynomial chaos expansion to calculate fixture capability, and (iii) fixture capability optimisation using surrogate modelling. The methodology is demonstrated and validated using the results of an aerospace wing sub-assembly joined by riveting technique.

Probing stochastic nano-scale inelastic events in stressed amorphous metal.

One fundamental yet longstanding issue in materials science is how local inelasticity arises within an amorphous structure before yielding occurs. Although many possible scenarios were postulated or predicted by theories and simulations,however, direct experimental evidence has been lacking today due to the lack of a sensitive way to detect nano-scale inelasticity. Through the carefully designed microcompression method as coupled with the state-of-art nano-scale electric resistance measurement, we here unfold a stochastic inelastic deformation process in a Zr-based metallic glass, which takes place via the recurrence of two types of short-lived inelastic events causing structural damage and recovery, respectively, prior to yielding. Our current findings reveal that these stochastic events not only self-organize into sub-critical events due to elastic coupling, but also compete with each other in a way that enables the whole amorphous structure to self-heal as well as to sustain local damage.