Regions Of Large Gradients Research Articles

A Novel 2-D Phase Unwrapping Method Based on PUMA for Aerospace Topographic Mapping Systems

Two-dimensional phase unwrapping (PU) is the key step of interferometric synthetic aperture radar technology. The accuracy of the PU results directly affects the quality of the final product. Due to the limitation of the phase continuity assumption, the results obtained by the single-baseline PU in large gradient regions are not ideal. To solve this problem, this article proposes an improved PU max-flow/min-cut algorithm (PUMA) which can efficiently deal with the phase of large gradient regions. First, we construct a quasiconvex function which is the combination of logarithmic and quadratic functions as the potential function in the energy function model of the PUMA method. Then, we use the gradient information of the external digital elevation model to assist in setting the weight of the potential function. Finally, the unwrapped phase is obtained by minimizing the new energy function model. The experimental results on the TanDEM-X dataset show that the improved approach in this article can achieve higher accuracy PU results in the region of large gradient variation than the existing PU methods.

Attractor and fixed points in Bjorken flows

We consider a plasma of massless particles undergoing Bjorken expansion, mimicking the matter created in ultrarelativistic heavy ion collisions. We study the transition to hydrodynamics using kinetic theory in the relaxation time approximation. By allowing the relaxation time to depend on time, we can monitor the speed of the transition from the collisionless regime to hydrodynamics. By using a special set of moments of the momentum distribution, we reduce the kinetic equation to a coupled mode problem which encompasses all versions of second-order viscous hydrodynamics for Bjorken flows. This coupled mode problem is analyzed first using techniques of linear algebra. Then we transform this two-mode problem into a single nonlinear differential equation and proceed to a fixed point analysis. We identify an attractor solution as the particular solution of this nonlinear equation that joins two fixed points: one corresponding to the collisionless, early time regime, the other corresponding to late time hydrodynamics. We exploit the analytic solution of this equation in order to test several approximations and to identify generic features of the transition to hydrodynamics. We argue that extending the accuracy of hydrodynamics to early time, i.e., to the region of large gradients, amounts essentially to improve the accuracy of the location of the collisionless fixed point. This is demonstrated by showing that a simple renormalization of a second-order transport coefficient puts the free streaming fixed point at the right location, and allows us to reproduce accurately the full solution of the kinetic equation within second-order viscous hydrodynamics, even in regimes far from local equilibrium.

HR-Depth: High Resolution Self-Supervised Monocular Depth Estimation

Self-supervised learning shows great potential in monocular depth estimation, using image sequences as the only source of supervision. Although people try to use the high-resolution image for depth estimation, the accuracy of prediction has not been significantly improved. In this work, we find the core reason comes from the inaccurate depth estimation in large gradient regions, making the bilinear interpolation error gradually disappear as the resolution increases. To obtain more accurate depth estimation in large gradient regions, it is necessary to obtain high-resolution features with spatial and semantic information. Therefore, we present an improved DepthNet, HR-Depth, with two effective strategies: (1) re-design the skip-connection in DepthNet to get better high-resolution features and (2) propose feature fusion Squeeze-and-Excitation(fSE) module to fuse feature more efficiently. Using Resnet-18 as the encoder, HR-Depth surpasses all previous state-of-the-art(SoTA) methods with the least parameters at both high and low resolution. Moreover, previous SoTA methods are based on fairly complex and deep networks with a mass of parameters which limits their real applications. Thus we also construct a lightweight network which uses MobileNetV3 as encoder. Experiments show that the lightweight network can perform on par with many large models like Monodepth2 at high-resolution with only20%parameters. All codes and models will be available at https://github.com/shawLyu/HR-Depth.

The cubic interpolation spline for functions with boundary layer on a Bakhvalov mesh

The problem of cubic spline interpolation on the Bakhvalov mesh of functions with region of large gradients is considered. Asymptotically accurate error estimates O(N −4) are obtained for a class of functions with an exponential boundary layer in case 1/N ≤ ε, where N is number of nodes, ε is small parameter. In case ε ≤ 1/N we have experimentally shown that the error estimates of traditional spline interpolation are not uniform in a small parameter, and the error itself can increase indefinitely when the small parameter tends to zero at a fixed number of nodes N. A modified cubic spline is proposed for which uniform estimates of the order O(N −4) have been experimentally confirmed.

Large Gradient Micro-Structure Topography Measurement with Multi-Angle Stitching Digital Holographic Microscope

We present an effective method for the topography characterization of large gradient micro-structure based on digital holographic microscope (DHM). Due to the limitation of numerical aperture of DHM system, high frequency information corresponding to large gradient regions of specimen is prohibited from entering the imaging system and the complete collection of specimen features cannot be implemented. To solve this problem, we use a common configuration of off-axis DHM system to capture the holograms from multiple angles of the sample, and the various object waves coming from the sample of different tilt angle is corrected and spliced into a fully topography by multi-angle stitching process. In this way, the measurement of large gradient topography of specimen can be realized. Experimental results of large gradient microlens demonstrate the practicability and precision of the presented method.

A Characteristic-Featured Shock Wave Indicator for Conservation Laws Based on Training an Artificial Neuron

In this work, we use exact solutions of one-dimensional Burgers equation to train an artificial neuron as a shock wave detector. The expression of the artificial neuron detector is then modified into a practical form to reflect admissible jump of eigenvalues. We show the working mechanism of the practical form is consistent with compressing or intersecting of characteristic curves. In addition, we prove there is indeed a discontinuity inside the cell detected by the practical form, and smooth extrema and large gradient regions are never marked. As a result, we apply the practical form to numerical schemes as a shock wave indicator with its easy extension to multi-dimensional conservation laws. Numerical results are present to demonstrate the robustness of the present indicator under Runge–Kutta Discontinuous Galerkin framework, its performance is generally compared to TVB-based indicators more efficiently and accurately. To treat the initial inadmissible jumps, including linear contact discontinuities and those evolving into rarefaction waves, a preliminary strategy of combining a traditional indicator in the beginning with the present indicator is suggested. We believe the present indicator can be applied to unstructured mesh in the future.

Available Potential Energy in Density Coordinates

AbstractThe vertically integrated potential energy of an incompressible stratified fluid formulated in density coordinates can be simply written as a weighted vertical sum of the squares of the vertical displacements of density surfaces, a general expression valid for arbitrary displacements. The sum of this form of potential energy and kinetic energy is then a conserved quantity for the multilayer shallow water model. The formulation in density coordinates is a natural one to find the Lorenz reference state of available potential energy (APE). We describe the method to compute the APE of an ocean state and provide two applications. The first is the classical double-gyre, wind-driven circulation simulated by a shallow water model at high resolution. We show that the eddy kinetic and eddy potential energies are localized in regions of large gradients of mean APE. These large gradients surround an APE minimum found between the two gyres. The second is the time-mean World Ocean Circulation reconstructed from hydrography (World Ocean Atlas) and reference velocities at 1000 db from the Argo float program to obtain an absolute circulation. The total available potential energy exceeds the total mean kinetic energy of the World Ocean by three orders of magnitude, pointing out the very small Burger number of the circulation. The Gulf Stream, the Kuroshio, the Agulhas retroflection, and the confluence regions are four examples that confirm the shallow water model results that large gradients of mean available potential energy can be used as predictors for the presence of high eddy kinetic energy (obtained here from satellite altimetry).

On the Parameter-Uniform Convergence of Exponential Spline Interpolation in the Presence of a Boundary Layer

The paper is concerned with the problem of generalized spline interpolation of functions having large-gradient regions. Splines of the class C2, represented on each interval of the grid by the sum of a second-degree polynomial and a boundary layer function, are considered. The existence and uniqueness of the interpolation L-spline are proven, and asymptotically exact two-sided error estimates for the class of functions with an exponential boundary layer are obtained. It is established that the cubic and parabolic interpolation splines are limiting for the solution of the given problem. The results of numerical experiments are presented.

Structures of turbulent premixed flames in the high Karlovitz number regime – DNS analysis

Lean premixed turbulent methane-air flames have been investigated using direct numerical simulations (DNS) for different Karlovitz numbers (Ka), ranging from 65 to 3350. The flames are imposed to a high intensity small-scale turbulent environment, corresponding to high Ka conditions, and the effect on the flame structure is investigated during the transition from initial laminar flame to highly distorted turbulent flame. The focus is on the internal structure of different sub-layers of these flames. The preheat layer, fuel consumption layer and oxidation layer are characterized by the distribution of formaldehyde, the fuel consumption rate and the CO consumption rate, respectively. Different measures that quantify sub-layer thickness for turbulent flames have been defined and analyzed. The flame brush is broadened with time while the local thickness (excluding large scale wrinkling) of all three layers initially show thinning due to the interaction of the flame with the turbulent flow field. As time passes, the local thickness of the preheat layer and fuel consumption layer are restored while the oxidation layer remains thinned due to suppression of CO consuming reactions. As Ka increases there is an increasing probability of finding thinned, large gradient regions in each of these sub-layers. The contribution to the evolution of flame thickness from normal strain rate, chemical reaction and normal and tangential diffusion is analyzed in terms of a gradient transport equation. The relative size of the terms changes as Ka increase and, in particular, the term due to chemical reactions loses its relative significance. The observed thinning of the local flame structure is attributed to the preferential alignment of the flame normal with the compressive strain rate eigenvectors. Such alignment provides a mechanism for the flame thinning, consistent with the behavior of non-reacting scalars. A preferred angle of about 20 degrees is observed between the flame normal and the compressive strain rate eigenvector.

Corrections for lateral conduction error in steady state heat transfer measurements

Steady state measurements of convective heat transfer coefficient often use a constant wall heat flux condition along with measured temperature distribution to generate heat transfer coefficient maps. Large spatial gradients in the heat transfer coefficient may lead to lateral conduction within the heater foil, causing non-uniformity in the actual convective flux into the fluid, yielding errors in the calculated Nusselt numbers. This paper presents an analytical procedure for correcting such errors for a model system with a known heat transfer coefficient distribution and nominal 1-D applied wall heat flux. The resulting 2-D conduction problem is parametrized in terms of the Biot number Bi and the heat transfer coefficient distribution, expressed by the change in magnitude ρ and the peak gradient γ, as well as the proximity of the gradient region to a symmetry plane. Three model configurations are studied: a region of large gradients that is located far away from lateral boundaries, and two cases where the gradient region is located near a symmetry plane, viz. impingement heat transfer due to a slot jet and a round jet. It is shown that sharp spatial variations in heat transfer coefficient can lead to significant error in Nusselt number determination when the wall heat flux is assumed to be uniform. The error is shown to be amplified when the gradient region is located near a symmetry plane. Finally, the wall heat flux is correlated using of an expression that captures the results of the analytical calculations for the ranges of Bi, ρ and γ studied, which can be used to evaluate experimental designs for heat transfer measurement, and make corrections for the two-dimensional nature of heat transfer in the foil and insulation.

Molecular dynamics-based triangulation algorithm of free-form parametric surfaces for computer-aided engineering

A new computational approach for automated triangulation of Computer-Aided Design (CAD) surface models, applicable to various CFD (Computational Fluid Dynamics) problems of practical interest is proposed. The complex shaped product configurations are represented by a set of Non-Uniform Rational B-Splines (NURBS) surface patches. The suggested technique is based on the molecular dynamics method. The main idea of the approach is that the mesh nodes are considered as similarly charged interacting particles which move within the region to be meshed under the influence of internal (such as particle–particle interaction forces) and external forces as well as optional additional forces. Moreover, the particles experience a medium resistance due to which the system comes to equilibrium within a relatively short period of time. The proposed 3D surface mesh generation algorithm uses a parametric NURBS representation as initial definition of the domain boundary. This method first distributes the interacting nodes into optimal locations in the parametric domain of the NURBS surface patch using molecular dynamics simulation. Then, the well-shaped triangles can be created after connecting the nodes by Delaunay triangulation. Finally, the mapping from parametric space to 3D physical space is performed. Since the presented interactive algorithm allows to control the distance between a pair of nodes depending on the curvature of the NURBS surface, the method generates high quality triangular mesh. The algorithm enables to produce uniform mesh, as well as anisotropic adaptive mesh with refinement in the large gradient regions. The mesh generation approach has the abilities to preserve the representation accuracy of the input geometry model, create a close relationship between geometry modeling and grid generation process, be automated to a large degree. Some examples are considered in order to illustrate the method’s ability to generate a surface mesh for a complicated CAD model.

Adaptive Mesh Refinement for Immersed Boundary Methods

Immersed boundary methods coupled with adaptive mesh refinement (AMR) are a powerful tool to solve complex viscous incompressible flow problems, especially in the presence of moving and deforming boundaries. Immersed boundary methods have been traditionally used in the framework of fractional step formulations for temporal integration and are generally coupled to logically structured grids, where the elliptic problem for the pressure is solved using fast solution techniques. In many situations, especially at large Reynolds numbers, adaptive clustering of fluid grid points on large gradient regions is desirable. This article gives an overview of currently available AMR tools, with an emphasis on block structured grids that are a natural fit to immersed boundary methods, and discusses future trends.

Proton-radiography-based quality assurance of proton range compensator

The aim of this work was to study the feasibility of proton radiography (pRad) as a patient-specific range compensator (RC) quality assurance (QA) tool and to validate its clinical utility by performing QA on RCs having three kinds of possible defects. In order to achieve pRad for a single EBT film, proton beam currents were modulated with new weighting factors, maximizing the linearity of optical-density-to-thickness ratio. Two RCs, examined to be accurately manufactured as planned, were selected to estimate the feasibility of our pRad. The optical densities of the EBT film on which the RC was irradiated with the modulated proton beam were digitized to pixel values (pv) and then converted to thickness using a thickness-pv calibration curve. The thickness information on the pRad was compared with plan data that had been extracted from treatment planning system. The mean thickness difference (TD) over the flat RC regions was calculated as 0.39 mm, and the standard deviation as 0.22 mm, and the proton scattering effect was analyzed by step phantom measurement. Even proton scattering effected a TD of over 1 mm in the large gradient region, the percentage of pixels over the acceptance criterion was only within 1.11% and 3.49%, respectively, when a 1 mm distance to agreement tolerance limit was applied. The QA results for both precisely and imprecisely manufactured RCs demonstrated the high potential utility and clinical applicability of the pRad-based RC QA tool.

Verification of a Higher-Order Finite Difference Scheme for the One-Dimensional Two-Fluid Model

The one-dimensional two-fluid model is widely acknowledged as the most detailed and accurate macroscopic formulation model of the thermo-fluid dynamics in nuclear reactor safety analysis. Currently the prevailing one-dimensional thermal hydraulics codes are only first-order accurate. The benefit of first-order schemes is numerical viscosity, which serves as a regularization mechanism for many otherwise ill-posed two-fluid models. However, excessive diffusion in regions of large gradients leads to poor resolution of phenomena related to void wave propagation. In this work, a higher-order shock capturing method is applied to the basic equations for incompressible and isothermal flow of the one-dimensional two-fluid model. The higher-order accuracy is gained by a strong stability preserving multi-step scheme for the time discretization and a minmod flux limiter scheme for the convection terms. Additionally the use of a staggered grid allows for several second-order centered terms, when available. The continuity equations are first tested by manipulating the two-fluid model into a pair of linear wave equations and tested for smooth and discontinuous initial data. The two-fluid model is benchmarked with the water faucet problem. With the higher-order method, the ill-posed nature of the governing equations presents severe challenges due to a growing void fraction jump in the solution. Therefore the initial and boundary conditions of the problem are modified in order to eliminate a large counter-current flow pattern that develops. With the modified water faucet problem the numerical models behave well and allow a convergence study. Using the L1 norm of the liquid fraction, it is verified that the first and higher-order numerical schemes converge to the quasi-analytical solution at a rate of O(1/2) and O(2/3), respectively. It is also shown that the growing void jump is a contact discontinuity, i.e. it is a linearly degenerate wave. The sub-linear convergence rates are in exact agreement with the theory for a contact discontinuity.

Implementing high-order weighted compact nonlinear scheme on patched grids with a nonlinear interpolation

Weighted Compact Nonlinear Schemes (WCNSs) possess the merits of high-order accuracy, high resolution, and high endurance in low quality grids. In order to implement WCNS on patched grids, high-order interpolations are generally needed for the purpose of ensuring overall high-order accuracy. However, high-order linear interpolations are susceptible to numerical oscillations in the vicinity of discontinuities and large gradient regions. In order to prevent this kind of oscillations, a new weighted interpolation is developed following the idea of Deng and Zhang (J Comput Phys, 2000). The new interpolation contains three fourth-order sub-interpolations which are weighted together according their smoothness. The weights are designed in such a way that in smooth regions they could approach to the optimal weights to achieve sixth-order accuracy, whereas in regions near discontinuities, the weights of the stencils which contain discontinuities are assigned to be nearly zero. The fifth-order WCNS is combined with the new weighted interpolation to solve flow problems on patched grids. Several benchmark problems, including shock waves, vortex, and shock/vortex interaction, are simulated to test the method. The results indicate that the new interpolation has similar performance with the sixth-order Lagrange interpolation in smooth regions, and is superior to the Lagrange interpolation in preventing interpolation oscillations in the vicinity of discontinuities.

Weighting the Edge Stabilization

A modification of the edge stabilization technique is proposed to improve the behavior of the method when solving conservation equations with nonsmooth data or nonsmooth solutions. The key ingredient is tempering the edge stabilization in regions of large gradients through appropriate weights. The new method is shown to preserve the convergence properties of the original method on smooth solutions, and numerical tests indicate that it performs better on nonsmooth solutions.

Comparative study of 2-group [formula omitted] and diffusion theories for the calculation of the neutron noise in 1D 2-region systems

In this paper, the neutron noise, i.e. the stationary fluctuations of the neutron flux around its mean value, is calculated in 2-group P 1 and diffusion theories for a 2-region slab reactor using Green’s function technique. The applicability of diffusion theory for different types and locations of the perturbation, as well as different frequencies, is assessed. Material data, i.e. nuclear cross-sections and kinetic parameters, representative of a Light Water Reactor (LWR) and of a Heavy Water Reactor (HWR), respectively, are used in this work. It is demonstrated that for practical situations in LWRs and HWRs, there is no significant advantage to use P 1 theory since diffusion theory gives acceptable results. The largest deviations between the two formalisms are observed in regions of large gradients of the static neutron flux, such as close to the reflector interface and close to the perturbation. Such observations are in agreement with theoretical expectations. This study also indicates that neglecting the effect of cross-section perturbation on the diffusion coefficient gives a rather small impact on the solution. This allows drastically simplifying the determination of the neutron noise. When using numerical techniques for such a determination the memory requirements and computational effort can be significantly reduced.

On the kinetics of chemical reactions under detonation conditions

Physical mechanisms leading to the acceleration of chemical reactions in the regions of large gradients of parameters are considered, in particular, in detonation waves. It is shown that a kinetics is possible according to which the reaction rate depends on the temporal derivative of the specific volume.

Evaluation of a planetary boundary layer subgrid‐scale model that accounts for near‐surface turbulence anisotropy

In numerical modelling, far from regions of large gradients, the resolved scales contain most of the energy of turbulent motion whilst on subgrid scales (SGS), motions are less energetic. However, SGS contribution becomes larger than the resolved part near the surface. Recent studies have shown that near‐surface turbulence anisotropy has a dramatic effect on the mixing length to be used in SGS models which are generally derived for free‐stream isotropic turbulence or use the standard Prandtl mixing length κz. Using the flux measurements collected at the SIRTA observatory located near Paris (France), this paper shows that a SGS model suitable for both the surface layer and free‐stream turbulence must discard the Prandtl mixing length and account for near‐surface turbulence anisotropy. It also shows that such an SGS scheme, which is suited for all numerical models of the atmosphere, improves the representation of planetary boundary layer processes in most stability conditions.

Impulsive displacement of a quasi-linear viscoelastic material through accurate numerical inversion of the Laplace transform

An analytical model for the one-dimensional, impulsive displacement of a quasi-linear viscoelastic material has been developed. The quasi-linear model of Fung [1] has been used successfully for a wide range of soft biological tissues. Due to the integral definition of linear viscoelastic materials, solutions are conveniently performed in the Laplace transform plane. Complex kernels like the quasilinear model are challenging to invert back to the real plane. Here, the method of Gaver [2] and Stehfest [3] is used to numerically carry Laplace space solutions to the real plane. Parametric results for a basic impulsive disturbance problem are presented. Results indicate that stress wave propagation is weakly dependent on the fast time, slow time ratio and more strongly dependent on the logarithmic damping parameter. Limitations of the numerical inversion method in the face of discontinuities are discussed as well using asymptotic methods. As an alternative to the numerical/polynomial-based Gaver-Stehfest method, a semianalytical regularization function useful near large gradient regions method is developed. A composite method that utilizes both the fully numerical and semianalytical convolution-based method is also described. The composite model provides improved results in terms of reducing computational undershoot and overshoot (wiggles) which limit both the fully numerical and the semianalytical models alone.