Smoothness Conditions Research Articles

Equilibrium problem for an inhomogeneous two-dimensional elastic body with two interacting thin rigid inclusions

A new nonlinear mathematical model is proposed that describes an equilibrium of a two-dimensional elastic body with two interacting thin rigid inclusions. Thin rigid inclusions are defined by straight line segments. In the initial state of the body, both inclusions are in contact at one breaking point. Both inclusions can move, subject to the conditions in the form of systems of inequalities describing a possible contact of the inclusions. These three systems of inequalities for infinitesimal rigid displacements correspond to three possible cases of mutual configurations of inclusions in equilibrium state. Inclusions exfoliate near the breaking point from the elastic matrix, in other words, there are cracks of a given length on both sides of thin inclusions. Nonpenetration conditions of the Signorini type are imposed on the curves defining the cracks. On a part of the outer boundary, clamping conditions are prescribed. The problem is formulated as a minimization of an energy functional over a non-convex set of possible displacements defined in an appropriate Sobolev space. The existence of a solution to the problem is proved. Optimality conditions of the solution have been obtained under the condition of sufficient smoothness of the solution.

Boundary Smoothness Conditions for Functions in $$R^p(X)$$

Boundary Smoothness Conditions for Functions in $$R^p(X)$$

Turbulent boundary layer over porous media with wall-normal permeability

Porous walls are a widely used passive flow control technique, which shows potential in reducing skin friction and mitigating flow-introduced noise. In the present study, porous media with wall-normal permeability is applied to a flat plate to investigate its interaction with the turbulent boundary layer at the Reynolds number based on friction velocity of Reτ=225. Time-resolved planar and tomographic particle image velocimetry were employed to identify the impact on mean statistics and coherent structures. An overall skin friction reduction of 22% is achieved. The porous wall induces counter-rotating streamwise vortex pairs at the spanwise sides of each pore, leading to momentum transport and the generation of alternative low- and high-speed regions close to the wall. Slip velocity is obtained, associated with the reduction in turbulent fluctuations and Reynolds shear stress. The streamwise velocity streaks and the hairpin vortices are significantly distorted compared with the smooth wall condition due to the downwash and upwash motion, featuring a notable reduction in the number and scale of the coherent structures, in which the skin friction reduction mechanism is related to. The proper orthogonal decomposition analysis returns the most energetic unsteady modes. Although the wall-coherent mode type remains to dominate the production of turbulent fluctuations, the scale and energy content of wall-incoherent modes increase, confirming the modification of the distribution and scale of near-wall turbulent structures.

C1-smooth isogeometric spline functions of general degree over planar mixed meshes: The case of two quadratic mesh elements

Splines over triangulations and splines over quadrangulations (tensor product splines) are two common ways to extend bivariate polynomials to splines. However, combination of both approaches leads to splines defined over mixed triangle and quadrilateral meshes using the isogeometric approach. Mixed meshes are especially useful for representing complicated geometries obtained e.g. from trimming. As (bi-)linearly parameterized mesh elements are not flexible enough to cover smooth domains, we focus in this work on the case of planar mixed meshes parameterized by (bi-)quadratic geometry mappings. In particular we study in detail the space of C1-smooth isogeometric spline functions of general polynomial degree over two such mixed mesh elements. We present the theoretical framework to analyze the smoothness conditions over the common interface for all possible configurations of mesh elements. This comprises the investigation of the dimension as well as the construction of a basis of the corresponding C1-smooth isogeometric spline space over the domain described by two elements. Several examples of interest are presented in detail.

Fatigue life evaluation of bridge stay cables subject to monitoring traffic and considering road roughness

Bridge stay cables are susceptible to fatigue damage caused by cyclic traffic loads. However, the effects of road roughness on fatigue assessment are often ignored, even though it is an essential factor that amplifies the response in vehicle–bridge interaction (VBI) systems. To address this gap, this study proposes a framework for cable fatigue assessment considering road roughness. The framework includes the VBI analysis method accounting for road roughness, the generation of random traffic flow, and the evaluation of cable fatigue. A case study based on a typical cable-stayed bridge is conducted. The results show that the proposed framework can effectively and feasibly evaluate cable fatigue. It is found that a poor road surface can reduce the fatigue life of cables, causing up to 4.47 times more damage compared to smooth road conditions. The decline of cable fatigue life caused by road roughness is primarily concentrated on the cables around the main towers, while the other cables are less affected by road roughness. Based on the measured traffic volume under poor road surface conditions, the estimated minimum fatigue life of the cable is 64 years. Moreover, the study indicates that cable fatigue damage rises with increasing traffic volume. These findings highlight the importance of considering road roughness and maintaining bridge decks to prevent the reduction of cable fatigue life.

The Novel Somatosensory Nose-Poke Adapted Paradigm (SNAP) Is an Effective Tool to Assess Differences in Tactile Sensory Preferences in Autistic-Like Mice.

One of the most prevalent deficits in autism spectrum disorder (ASD) are sensitivities to sensory stimuli. Despite the prevalence of sensory deficits in autism, there are few paradigms capable of easily assessing sensory behaviors in ASD-like mouse models. We addressed this need by creating the Somatosensory Nose-poke Adapted Paradigm (SNAP), which consists of an elevated platform with 6 holes in the center, half of which are lined with sandpaper and half are smooth, requiring mice to use their whiskers to sense the texture. The SNAP paradigm assesses tactile sensory preferences as well as stereotypy, anxiety, and locomotion. We used two wild-type (neurotypical) mouse strains, C57BL/6J (C57) inbred and CD-1 outbred mice, and two ASD mouse models, BTBR (a model of idiopathic ASD) and Cntnap2 -/- mice (a model of syndromic ASD). We found that both ASD models produced more nose pokes into the rough condition than the smooth condition, suggesting an increased preference for complex tactile stimulation when compared with the neurotypical groups, wherein no differences were observed. Furthermore, we found increased stereotypy and time spent in the center, suggestive of decreased anxiety, only for BTBR mice compared with the other mouse strains. Overall, SNAP is an easy to implement task to assess the degree of preference for complex tactile stimulation in ASD mouse models that can be further modified to exclude possible confounding effects of novelty or anxiety on the sensory preferences.

Case study of superelevation design at an expressway interchange by using ADSCivil Road software

Superelevation is known as a transverse slope made available to mitigate the impact of centrifugal force and minimize the vehicle’s tendency to overturn and skid. However, in an intersection or interchange that has horizontal curves with small radius consecutively, it is difficult to design superelevation. If the superelevation arrangement is not good enough, the alignment is not smooth, causing unfavourable and unsafe conditions. Moreover, for expressway interchange, it become more complicated to design by manual method because the expressway requirements in term of safety and aesthetics is higher than that of conventional intersection. On the other hand, ADSCivil Road, a software in the ADSCivil solution suite for infrastructure design developed by Baezeni Soft Co, Ltd. is a useful tool for geometric road design. This paper presents the results of ADSCivil Road software application in designing the superelevation at the expressway. ADSCivil Road has arranged superelevation for an interchange ramp (with a design speed of 60 km/h) that includes the small radius circular curves connecting together (R=125 m and R=100 m). The results show that using the superelevation diagram and corridor model in ADSCivil Road makes the superelevation design process more intuitive, make it easier to adjust the slope at each point, and controls the rate of superelevation difference. In addition, ADSCivil Road automatically updates the superelevation change for profile, cross-section and plan. Also, 3D simulation results show that the superelevation layout for this interchange meets technical, economic, and aesthetic requirements. Obviously, it can be seen that can become an effective tool to help engineers solve complex problems in road design.

Parametric modeling study for blown-dust secondary pollution and optimal ventilation velocity during tunnel construction

Tunnel construction often relies on drilling and blasting. High dust pollution is one of the primary problems of drilling and blasting construction. The level of secondary blown dust pollution caused by ventilation matches that of dust pollution caused by drilling construction. In this study, a critical flow model and blown dust rate model for deposited dust were established via force analysis, which was validated against the test data. The research results showed that the characteristic airflow velocity for blowing dust particles with a 100 μm diameter reached approximately 0.42 m/s for tunnel diameter is 10 m, and the ventilation Re values under smooth and rough conditions were 2.3 × 105 and 1.4 × 105, respectively. Furthermore, when ventilation Re reached 4 × 105, the blown dust pollution rate caused by ventilation under smooth conditions was approximately 1.8 × 10−2 kg/s. If dust particle size is more or less the critical dust particle size, the characteristic airflow velocity was increased. Moreover, the optimal velocity at which the deposited dust does not flow or move during tunnel construction was related to the tunnel size and roughness. For the smooth tunnel with a diameter of 10 m, the optimal ventilation velocity was 3.5 m/s. When the tunnel roughness was increased from 0.005 to 0.5 m, the optimal ventilation velocity decreased from 3.3 to 1.6 m/s. The deposited dust critical flow model and blown dust pollution rate model established in this study provide a sound theoretical basis for selecting the optimal velocity of tunnel ventilation and recognizing the risks of secondary blown dust pollution due to ventilation.

Background oriented schlieren image displacement estimation method based on global optical flow

Background oriented schlieren (BOS) technology is a recent non-contact optical diagnostic technology, which can be used for the visualization of variable density flow and the quantitative measurement of the refractive index field. The accuracy of BOS image displacement estimation directly determines the accuracy of refractive index field reconstruction. For BOS images, the optical path construction method, vibration and uneven illumination, blurring and defocusing effects, and the refocusing effect of volume refraction make it difficult for the BOS image sequence to satisfy the constant brightness assumption. In this study, the optical flow algorithm based on the assumption of a constant gradient in PIV is improved and applied to BOS image displacement estimation. A data item based on the assumption of constant brightness combined with the assumption of constant gradient is proposed. The first-order div-curl regularization constraint is used as the spatial smoothing condition, and the non-convex Charbonnier penalty function is used to determine the displacement field by the variational method to minimize the energy function. The proposed method is tested on the synthetic images obtained by two-dimensional turbulent direct numerical simulation (DNS), the BOS synthetic images with brightness changes, and the actual experimental BOS images of the flow field above a premixed butane/air combustion flame with different equivalence ratios. Compared with the current advanced optical flow algorithm, this method has higher accuracy and robustness. Overall, the proposed method can not only obtain more accurate results when the brightness of the BOS synthetic image and experimental image changes, but also helps in retaining the flow details such as small divergence and vorticity structure of the heat flow field while reducing the outliers.

Numerical investigation of a Mach 6 laminar shock-wave/boundary-layer interaction on a two-dimensional ramp with 3D controlled surface roughness

Direct Numerical Simulations (DNS) are performed to investigate roughness effects on a Mach 6 hypersonic compression-corner flow. Different variants of a roughness patch placed upstream of the ramp are considered in order to study their impact on the steady base flow characterized by a large separation bubble. The patches consist of sinusoidal elements, with modifications in position and spanwise extent of the roughness. Under adiabatic-wall conditions, compared to a smooth-wall case, the rough cases where the patch is upstream of the smooth-separation point cause a delayed boundary-layer separation and hence a smaller recirculation bubble. Moreover, the roughness-induced streamwise vorticity convected downstream by the detached shear layer generates mushroom-like structures once the flow reattaches on the ramp. The properties of such structures, although similar, reflect the character of the patch upstream of the bubble. The roughness inducing the largest local flow distortion is responsible for the widest structure on the ramp, characterized by the highest increase in wall-temperature. The patch occupying the entire width of the domain instead generates the weakest vortices on the ramp. No noticeable effects on the recirculation region can be noticed if the patch is placed downstream of the separation point for smooth conditions. Regarding the cold-wall case, the patch increases the heat-transfer rate on the ramp downstream of the roughness location, but with minimal effects on the bubble length. The analysis of the shear magnitude in the mushroom-shaped structures at reattachment suggests that, under adiabatic-wall conditions, the most upstream patch might have the highest transition potential.

Towards a finite-time singularity of the Navier–Stokes equations. Part 3. Maximal vorticity amplification

An exact analytical solution is obtained for the dynamical system derived in Part 1 of this series (Moffatt & Kimura, J. Fluid Mech, vol. 861, 2019a, pp. 930–967), which describes the approach of two initially circular vortices of finite but small cross-section symmetrically located on inclined planes. This exact solution, applicable in the inviscid limit, allows determination of the amplification $\mathcal {A}_{\omega }$ of the axial vorticity within the finite time $T$ during which the basic assumptions of the model continue to apply. It is first shown that, for arbitrarily prescribed $\mathcal {A}_{\omega }$ , it is possible to specify smooth initial conditions of finite energy such that, in the inviscid limit, this amplification is achieved within the time $T$ . When viscosity is included, an estimate is provided for the minimum vortex Reynolds number that is sufficient for the same result to hold. The predictions are broadly compatible with results from direct numerical simulations at moderate Reynolds numbers. Moreover, it is shown that one may come arbitrarily close to a finite-time singularity of the Navier–Stokes equation by appropriate choice of an initial, smooth, finite-energy velocity field; however, this approach to a singularity is ultimately thwarted through breach of the assumptions on which the dynamical system is based. Thus we make no claim here concerning realisation of a Navier–Stokes singularity. Moreover, we find that the conditions required to attain a large amplification $\mathcal {A}_{\omega }\gg 1$ during the time $T$ are far beyond those that can be realised in either experiment or direct numerical simulation.

Time-varying gravity field model of Sichuan-Yunnan region based on the equivalent mass source model

High-precision time-varying gravity field is an effective way to study the internal mass movement and understanding the spatio-temporal evolution process of the geodynamic system. Compared to the satellite gravity measurement, the repeated terrestrial gravity observation can provide a more high-order signal related to the shallow crust and subsurface. However, the suitable and unified method for gravity model estimation is a key problem for further applications. In this study, we introduce the spherical hexahedron element to simulate the field source mass and forward model the change of gravity field located at the Sichuan-Yunnan region (99–104°E, 23–29°N) in the four epochs from 2015 to 2017. Compared to the experimental results based on Slepian or spherical harmonics frequency domain method, this alternative approach is suitable for constructing the equivalent mass source model of regional-scale gravity data, by introducing the first-order smooth prior condition of gravity time-varying signal to suppress the high-frequency component of the signal. The results can provide a higher spatial resolution reference for regional gravity field modeling in the Sichuan-Yunnan region.

The Uncertainty Principle with Fourier Transform Derivatives

The classical uncertainty principle works for smooth signal functions. In our work, we apply the Fourier transform derivatives for the study of uncertainty principle, so that the smoothness condition for the signal functions is not required. At first, the amplitude and phase derivatives of vector-valued signal functions based on the Fourier transform are defined. Then we obtain a strong form of the uncertainty principle.

Convergence analysis of Inexact Newton–Landweber iteration with frozen derivative in Banach spaces

Abstract In this paper, we study the convergence analysis of the inexact Newton–Landweber iteration method (INLIM) with frozen derivative in Hilbert as well as Banach spaces. To study the convergence analysis, we incorporate the Hölder stability of the inverse mapping and Lipschitz continuity of the Fréchet derivative of the forward mapping. Moreover, we derive the convergence rates of INLIM in Hilbert as well as Banach spaces without using any extra smoothness condition. Finally, we compare our convergence rates results with that of several other frozen methods proposed in the literature to solve inverse problems.

Posterior Coreset Construction with Kernelized Stein Discrepancy for Model-Based Reinforcement Learning

Model-based approaches to reinforcement learning (MBRL) exhibit favorable performance in practice, but their theoretical guarantees in large spaces are mostly restricted to the setting when transition model is Gaussian or Lipschitz, and demands a posterior estimate whose representational complexity grows unbounded with time. In this work, we develop a novel MBRL method (i) which relaxes the assumptions on the target transition model to belong to a generic family of mixture models; (ii) is applicable to large-scale training by incorporating a compression step such that the posterior estimate consists of a Bayesian coreset of only statistically significant past state-action pairs; and (iii) exhibits a sublinear Bayesian regret. To achieve these results, we adopt an approach based upon Stein's method, which, under a smoothness condition on the constructed posterior and target, allows distributional distance to be evaluated in closed form as the kernelized Stein discrepancy (KSD). The aforementioned compression step is then computed in terms of greedily retaining only those samples which are more than a certain KSD away from the previous model estimate. Experimentally, we observe that this approach is competitive with several state-of-the-art RL methodologies, and can achieve up-to 50 percent reduction in wall clock time in some continuous control environments.

On Error and Compression Rates for Prototype Rules

We study the close interplay between error and compression in the non-parametric multiclass classification setting in terms of prototype learning rules. We focus in particular on a recently proposed compression-based learning rule termed OptiNet. Beyond its computational merits, this rule has been recently shown to be universally consistent in any metric instance space that admits a universally consistent rule---the first learning algorithm known to enjoy this property. However, its error and compression rates have been left open. Here we derive such rates in the case where instances reside in Euclidean space under commonly posed smoothness and tail conditions on the data distribution. We first show that OptiNet achieves non-trivial compression rates while enjoying near minimax-optimal error rates. We then proceed to study a novel general compression scheme for further compressing prototype rules that locally adapts to the noise level without sacrificing accuracy. Applying it to OptiNet, we show that under a geometric margin condition further gain in the compression rate is achieved. Experimental results comparing the performance of the various methods are presented.

Pre-filter analysis for retrieval of microphysical particle parameters: a quality-assurance method applied to 3 backscatter (β) +2 extinction (α) optical data taken with HSRL/Raman lidar.

We analyze the solution space of 3β+2α optical data inferred from lidar measurements, i.e., backscatter coefficients at three wavelengths and extinction coefficients at two wavelengths. These optical data are governed by microphysical parameters that can be expressed in terms of particle size distribution, effective radius, and complex refractive index (CRI). In our analysis, we consider two scenarios of the solution space. First, it can be expressed in terms of monomodal particle size distributions represented either by fine modes or by coarse modes. Secondly, the particle size distributions contain a fine mode as well as a coarse mode. Consideration of both scenarios and different values of the effective radius and CRI allows us to find synthetic 3β+2α optical data and corresponding intensive parameters (IPs) such as lidar ratios, backscatter- and extinction-related Ångström exponents at the available measurement wavelengths. Based on interdependencies between synthetic IPs and various microphysical properties, the qualitative and quantitative criteria for the optical data quality-assurance tool are developed. We derive the conditions of smoothness, closeness, convergence, and stability of the solution space for the quantitative criteria to test the quality of the 3β+2α optical data. Our novel methodology, to the best of our knowledge, can be used not only for particles of spherical shape, but also for cases in which particles are irregularly shaped. Another strength of our methodology is that it also works for the case of a size-dependent and wavelength-dependent CRI. We show the potential of this methodology for a measurement case from the ORACLES campaign. Data were taken with NASA Langley's airborne HSRL-2 instrument on September 24, 2016.

Distributed Projection-Free Online Learning for Smooth and Convex Losses

We investigate the problem of distributed online convex optimization with complicated constraints, in which the projection operation could be the computational bottleneck. To avoid projections, distributed online projection-free methods have been proposed and attain an O(T^{3/4}) regret bound for general convex losses. However, they cannot utilize the smoothness condition, which has been exploited in the centralized setting to improve the regret. In this paper, we propose a new distributed online projection-free method with a tighter regret bound of O(T^{2/3}) for smooth and convex losses. Specifically, we first provide a distributed extension of Follow-the-Perturbed-Leader so that the smoothness can be utilized in the distributed setting. Then, we reduce the computational cost via sampling and blocking techniques. In this way, our method only needs to solve one linear optimization per round on average. Finally, we conduct experiments on benchmark datasets to verify the effectiveness of our proposed method.

Primal-Dual First-Order Methods for Affinely Constrained Multi-block Saddle Point Problems

We consider the convex-concave saddle point problem , where the decision variables and/or are subject to certain multi-block structure and affine coupling constraints, and possesses certain separable structure. Although the minimization counterpart of this problem has been widely studied under the topics of ADMM, this minimax problem is rarely investigated. In this paper, a convenient notion of -saddle point is proposed, under which the convergence rate of several proposed algorithms are analyzed. When only one of and has multiple blocks and affine constraint, several natural extensions of ADMM are proposed to solve the problem. Depending on the number of blocks and the level of smoothness, or convergence rates are derived for our algorithms. When both and have multiple blocks and affine constraints, a new algorithm called xtra- radient Method of ultipliers (EGMM) is proposed. Under desirable smoothness conditions, an rate of convergence can be guaranteed regardless of the number of blocks in and . An in-depth comparison between EGMM (fully primal-dual method) and ADMM (approximate dual method) is made over the multi-block optimization problems to illustrate the advantage of the EGMM.

Invariant Measures for Uncountable Random Interval Homeomorphisms

A necessary and sufficient condition for the iterated function system {f(cdot ,omega ),|,omega in Omega } with probability P to have exactly one invariant measure mu _* with mu _*((0,1))=1 is given. The main novelty lies in the fact that we only require the transformations f(cdot ,omega ) to be increasing homeomorphims, without any smoothness condition, neither we impose conditions on the cardinality of Omega . In particular, positive Lyapunov exponents conditions are replaced with the existence of solutions to some functional inequalities. The stability and strong law of large numbers of the considered system are also proven.