Discrete Solver Research Articles

Zero Stability Well Predicts Performance of Convolutional Neural Networks

The question of what kind of convolutional neural network (CNN) structure performs well is fascinating. In this work, we move toward the answer with one more step by connecting zero stability and model performance. Specifically, we found that if a discrete solver of an ordinary differential equation is zero stable, the CNN corresponding to that solver performs well. We first give the interpretation of zero stability in the context of deep learning and then investigate the performance of existing first- and second-order CNNs under different zero-stable circumstances. Based on the preliminary observation, we provide a higher-order discretization to construct CNNs and then propose a zero-stable network (ZeroSNet). To guarantee zero stability of the ZeroSNet, we first deduce a structure that meets consistency conditions and then give a zero stable region of a training-free parameter. By analyzing the roots of a characteristic equation, we theoretically obtain the optimal coefficients of feature maps. Empirically, we present our results from three aspects: We provide extensive empirical evidence of different depth on different datasets to show that the moduli of the characteristic equation's roots are the keys for the performance of CNNs that require historical features; Our experiments show that ZeroSNet outperforms existing CNNs which is based on high-order discretization; ZeroSNets show better robustness against noises on the input. The source code is available at https://github.com/logichen/ZeroSNet.

Image recoloring for Red-Green dichromats with compensation range-based naturalness preservation and refined dichromacy gamut

People with color vision deficiency (CVD) have difficulty discriminating colors, which can cause loss of chromatic contrast in the perception of affected individuals. To compensate for contrast loss, image recoloring approaches have been proposed in the existing studies. In state-of-the-art studies, recoloring models were built based on an approximated gamut of CVD in the CIE L*a*b* (Lab) color space, which significantly deviates from the original gamut. In addition, luminance was not considered during recoloring. Moreover, existing methods also present problems, such as high computational costs and insufficient naturalness preservation . In this paper, we propose a novel recoloring method to compensate for CVD that enhances contrast through adopting a luminance channel-considered optimization model while preserving naturalness by imposing hard constraints on the amount of changes to the original colors. To obtain a better compensation effect, we fit a new curved surface for representing the gamut of dichromacy in the Lab color space more accurately. Furthermore, a discrete solver is implemented to solve the optimization problem efficiently. For effective assessment, qualitative, quantitative, and subjective experiments were conducted, and a new metric, called preference, is proposed to evaluate the contrast enhancement and naturalness preservation comprehensively.

Discrete Optimization for Shape Matching

AbstractWe propose a novel discrete solver for optimizing functional map‐based energies, including descriptor preservation and promoting structural properties such as area‐preservation, bijectivity and Laplacian commutativity among others. Unlike the commonly‐used continuous optimization methods, our approach enforces the functional map to be associated with a pointwise correspondence as a hard constraint, which provides a stronger link between optimized properties of functional and point‐to‐point maps. Under this hard constraint, our solver obtains functional maps with lower energy values compared to the standard continuous strategies. Perhaps more importantly, the recovered pointwise maps from our discrete solver preserve the optimized for functional properties and are thus of higher overall quality. We demonstrate the advantages of our discrete solver on a range of energies and shape categories, compared to existing techniques for promoting pointwise maps within the functional map framework. Finally, with this solver in hand, we introduce a novel Effective Functional Map Refinement (EFMR) method which achieves the state‐of‐the‐art accuracy on the SHREC'19 benchmark.

COMPARING THE DISCONTINUOUS GALERKIN AND HIGH-ORDER DIAMOND DIFFERENCING METHODS FOR THE TRANSPORT EQUATION ON A LOZENGE-BASED HEXAGONAL GEOMETRY

This paper presents an implementation and a comparison of two spatial discretisation schemes over a hexagonal geometry for the two-dimensional discrete ordinates transport equation. The methods are a high-order Discontinuous Galerkin (DG) finite element scheme and a high-order Diamond Differencing (DD) scheme. The DG method has been, and is being, studied on the hexagonal geometry, also called a honeycomb mesh – but not the DD method. In this research effort, it was chosen to divide the hexagons into (at least) three lozenges. An affine transformation is then applied onto said lozenges to cast them into the reference quadrilaterals usually studied in finite elements. In practice, this effectively means that the equations used in Cartesian geometry have their terms and operators altered using the Jacobian matrix of the transformation. This was implemented in the discrete ordinates solver of the code DRAGON5. Two 2D benchmark problems were then used for the verification and validation, including one based on the Monju 3D reactor benchmark. It was found that the diamond-differencing scheme seemed better. It converged much faster towards the solution at comparable mesh refinements for first-order expansion of the flux. Even if this difference was not present for second-order, DG was slower, about two to four times slower.

A constrained transport divergence-free finite element method for incompressible MHD equations

In this paper we study finite element method for three-dimensional incompressible resistive magnetohydrodynamic equations, in which the velocity, the current density, and the magnetic induction are divergence-free. It is desirable that the discrete solutions should also satisfy divergence-free conditions exactly especially for the momentum equations. Inspired by constrained transport method, we devise a new stable mixed finite element method that can achieve the goal. We also prove the well-posedness of the discrete solutions. To solve the resulting linear algebraic equations, we propose a GMRES solver with an augmented Lagrangian block preconditioner. By numerical experiments, we verify the theoretical results and demonstrate the quasi-optimality of the discrete solver with respect to the number of degrees of freedom. A comparison with other discretization using lid driven cavity is also given.

Scalable discrete adjoint for the transient Savage–Hutter equation on the primal consistent adaptive grid

Computing the adjoint for a transient non-linear hyperbolic PDE, like the Savage–Hutter equation for modeling dry granular mass flows, is a challenge with often unaffordable computing and memory costs. The challenges are further compounded for an adaptive mesh refinement (AMR) parallel code. For legacy codes, addressing these difficulties are constrained with the original design and data structures which refactoring those to be well-suited for the adjoint computation may end up to almost a new design of the code. In this paper, we develop a framework to compute the discrete adjoint for our parallel finite volume solver which supports AMR with a lower computational cost of the forward run. This is achieved using appropriate data structures and methods to reuse the computation of the forward AMR and repartitioning and eliminating any interpolation and prolongation errors in computing the adjoint. Furthermore, using analytical differentiation of the discrete forward solver, approximation error was avoided and the adjoint computed accurately. We illustrate our approach for the Savage–Hutter equation and use the adjoint for the gradient enhanced surrogate construction.

Suitability Analysis of Different Batteries with Off-Grid Photovoltaic System

The main objective of power generation and distribution companies is to ensure uninterrupted supply at reasonably low cost, minimum losses and high quality. Considering rural areas, where off-grid power systems using renewable resources are becoming more popular are generally equipped with storage elements like batteries. Hence energy storage and battery management system are a vital area of research nowadays. The study presents modelling of an off-grid solar PV model and testing it with different batteries (Lithium Ion (Li Ion), Nickel Cadmium (Ni Cd), Lead acid (Pb acid)) developed in Simulink on MATLAB 2017. Various aspects such as life cycle, energy density, SOC (State of Charge) and DOD (Depth of Discharge) are considered to study the technical performance of each battery. Comparisons among the three batteries have also been made considering environmental impact and safety measures. Pb acid and Ni Cd batteries pose as an environmental threat due to their toxic nature. This leads to an increase in usage of Li-ion batteries which are reasonably harmless and recyclable. The ultimate goal of the study is to choose the most economic and efficient battery with off-grid photovoltaic system as “the greenest power is the power you don’t have to produce”. From the comparisons performed in this paper, it is seen that Li ion batteries have superior overall performance when compared to Ni Cd and Pb acid batteries. Keywords: Performance comparison, battery life prediction, State of Charge (SOC), Depth of Discharge (DOD), result via discrete solver selection Cite this Article Kusum Tharani, Rishabh Mishra, Anurag Dobriyal et al. Suitability Analysis of Different Batteries with Off-Grid Photovoltaic System. Research & Reviews: Journal of Physics . 2018; 7(3): 11–19p

Large-Eddy Simulation of Soot Formation in a Model Gas Turbine Combustor

The computational modeling of soot in aircraft engines is a formidable challenge, not only due to the multiscale interactions with the turbulent combustion process but the equally complex physical and chemical processes that drive the conversion of gas-phase fuel molecules into solid-phase particles. In particular, soot formation is highly sensitive to the gas-phase composition and temporal fluctuations in a turbulent background flow. In this work, a large-eddy simulation (LES) framework is used to study the soot formation in a model aircraft combustor with swirl-based fuel and air injection. Two different configurations are simulated: one with and one without secondary oxidation jets. Specific attention is paid to the LES numerical implementation such that the discrete solver minimizes the dissipation of kinetic energy. Simulation of the model combustor shows that the LES approach captures the two recirculation zones necessary for flame stabilization very accurately. Further, the model reasonably predicts the temperature profiles inside the combustor. The model also captures variation in soot volume fraction with global equivalence ratio. The structure of the soot field suggests that when secondary oxidation jets are present, the inner recirculation region becomes fuel lean, and soot generation is completely suppressed. Further, the soot field is highly intermittent suggesting that a very restrictive set of gas-phase conditions promotes soot generation.

A Sparse Multiscale Algorithm for Dense Optimal Transport

Discrete optimal transport solvers do not scale well on dense large problems since they do not explicitly exploit the geometric structure of the cost function. In analogy to continuous optimal transport, we provide a framework to verify global optimality of a discrete transport plan locally. This allows the construction of an algorithm to solve large dense problems by considering a sequence of sparse problems instead. The algorithm lends itself to being combined with a hierarchical multiscale scheme. Any existing discrete solver can be used as internal black-box. We explicitly describe how to select the sparse sub-problems for several cost functions, including the noisy squared Euclidean distance. Significant reductions in run-time and memory requirements have been observed.

Register-My-Face: a tool to register three-dimensional face images

Efficient registration across pose is the most challenging research area for accurate recognition of human face images. Authors have discussed a tool that has been developed for registration of face images across poses using the nose tip of the face images. The nose tip has been considered here because of its stability in situations such as variations in pose, expression, etc. The aim of this investigation is to develop a face registration tool called “Register-My-Face” with a working methodology in all the three directions, namely yaw, pitch, and roll. This tool has been developed for three-dimensional (3-D) face registration, which is inspired by analyzing the “depth values” of face range images. The registration of the face is done using a geometrical technique which is based on computing the corresponding rotation in three orthogonal directions. The advantages of the designed tool are that it does not need any training phase for accurate detection of the nose tip, and this method can handle large pose variations, including 90 deg pose variations about the Y-axis in both the positive and negative directions. The method that has been followed to develop this tool is also independent of facial expression, occlusion and illumination variations. Moreover, it quickly detects the nose tip because it does not need to process the entire face surface, but only requires the isolated nose region. The tool has been integrated with three different databases; GavabDB, Bosphorus, and Frav3D, and the investigation highlights the robustness of the tool. Additionally, for exploring the performance of the tool, a SIMULINK model for hardware interface is also developed with a discrete solver and is tested on two different configuration setups and executed in two different execution modes with two simulation stop timings 10.0 and 1.0. This model can proceed according to the algorithm with a minimum of 6.640 s to register an unregistered raw 3-D face scan input image from the Frav3D database. Accuracies of the nose region of 98.87%, 94.44%, and 98.08% for the Frav3D, GavabDB, and Bosphorus databases, respectively, are observed. For nose tip detection, the success rates are 98.91% for the Frav3D database, 98.74% for the GavabDB database, and 96.03% for the Bosphorus database. Based on the success rate of nose tip detection, the registration process is implemented on three databases. Registration accuracy, computed between a neutral and the registered range face image for the Frav3D database is 87.5% for GavabDB and 89.87% for Bosphorus database, and the rate of success is 70.23%.

Coupling Function and Mechanism of the Bionic Coupling Functional Surface (BCFS) Caused by the Dual Factors of Form and Flexible Material

Some living creatures have special structures on their body surfaces, such as smooth and elastic epidermis with subcutaneous tissue having non-smooth structures under certain conditions. The elastic epidermis coupled with non-smooth structures has a special function called bio-coupling functional surface. Imitating this functional surface and applying it in engineering has a potential to solve some engineering problems. Based on the simulation method of fluid-structure interaction (FSI), simulation calculation of the bionic functional surface coupled by the two factors, form and flexible materials was conduct using ADINA software. A viscous and weakly compressible transient flow was selected as a working medium, a discrete solver was selected in numerical calculation and the basic model was chosen as a turbulence model. It is assumed that the coupling surface of the form/flexible materials results is large deformation and large strain. The boundary condition of fluid-structure interaction was set as the calculation surface. The simulation results showed that this coupling is a dynamic process, in which the two factors (form and flexible materials) are influenced by the flow field. As the pressure and velocity of the flow field increase, the coupling process changes from partial coupling to complete coupling, the pressure drag decreased due to the maximum effective stress of bionic coupling surface is very small and the smooth and flexible materials can redistribute pressure by absorbing and releasing energy, the pressure drag thus formed is decreased. Moreover, non-smooth structures (form factor) coupled with flexible materials reduced velocity of working face and minimise energy losses effectively, enabling the bionic coupling surface to reduce drag.

High quality solid texture synthesis using position and index histogram matching

The synthesis quality is one of the most important aspects in solid texture synthesis algorithms. In recent years several methods are proposed to generate high quality solid textures. However, these existing methods often suffer from the synthesis artifacts such as blurring, missing texture structures, introducing aberrant voxel colors, and so on. In this paper, we introduce a novel algorithm for synthesizing high quality solid textures from 2D exemplars. We first analyze the relevant factors for further improvements of the synthesis quality, and then adopt an optimization framework with the k-coherence search and the discrete solver for solid texture synthesis. The texture optimization approach is integrated with two new kinds of histogram matching methods, position and index histogram matching, which effectively cause the global statistics of the synthesized solid textures to match those of the exemplars. Experimental results show that our algorithm outperforms or at least is comparable to the previous solid texture synthesis algorithms in terms of the synthesis quality.

Drift-Wave Eigenmodes and Spectral Gaps in Tandem Mirrors

In recent years, breakthroughs in the GAMMA-10 tandem mirror device and new designs such as the Kinetically Stabilized Tandem Mirror have renewed enthusiasm for investigation of mirror plasmas. With a discrete eigenmode solver and shooting code we calculate kinetic Alfvén waves in a model GAMMA-10 experiment, and then examine the nature of the spectral gaps. We validate the results of these calculations by comparing with magnetic probe signals recorded in the LAPD as a function of frequency of the antenna and the axial position.

A methodology for the development of discrete adjoint solvers using automatic differentiation tools

A methodology for the rapid development of adjoint solvers for computational fluid dynamics (CFD) models is presented. The approach relies on the use of automatic differentiation (AD) tools to almost completely automate the process of development of discrete adjoint solvers. This methodology is used to produce the adjoint code for two distinct 3D CFD solvers: a cell-centred Euler solver running in single-block, single-processor mode and a multi-block, multi-processor, vertex-centred, magneto-hydrodynamics (MHD) solver. Instead of differentiating the entire source code of the CFD solvers using AD, we have applied it selectively to produce code that computes the transpose of the flux Jacobian matrix and the other partial derivatives that are necessary to compute sensitivities using an adjoint method. The discrete adjoint equations are then solved using the Portable, Extensible Toolkit for Scientific Computation (PETSc) library. The selective application of AD is the principal idea of this new methodology, which we call the AD adjoint (ADjoint). The ADjoint approach has the advantages that it is applicable to any set of governing equations and objective functions and that it is completely consistent with the gradients that would be computed by exact numerical differentiation of the original discrete solver. Furthermore, the approach does not require hand differentiation, thus avoiding the long development times typically required to develop discrete adjoint solvers for partial differential equations, as well as the errors that result from the necessary approximations used during the differentiation of complex systems of conservation laws. These advantages come at the cost of increased memory requirements for the discrete adjoint solver. However, given the amount of memory that is typically available in parallel computers and the trends toward larger numbers of multi-core processors, this disadvantage is rather small when compared with the very significant advantages that are demonstrated. The sensitivities of drag and lift coefficients with respect to different parameters obtained using the discrete adjoint solvers show excellent agreement with the benchmark results produced by the complex-step and finite-difference methods. Furthermore, the overall performance of the method is shown to be better than most conventional adjoint approaches for both CFD solvers used.

Fast example-based surface texture synthesis via discrete optimization

We synthesize and animate general texture patterns over arbitrary 3D mesh surfaces. The animation is controlled by flow fields over the target mesh, and the texture can be arbitrary user input as long it satisfies the Markov-Random-Field assumptions. We achieve this by extending the texture optimization framework over 3D mesh surfaces. We propose an efficient discrete solver inspired by k-coherence search, allowing interactive flow texture animation while avoiding the blurry blending problem for the least square solver in previous work. Our technique has potential applications ranging from simulation, visualization, and special effects.

Modelling of Switched Reluctance Machines: State of the Art

This article gives an overview of different modelling techniques to describe the nonlinear behaviour of both saturated switched reluctance machine (SRM) and power electronic converter. The current research contributes by comparing the decoupled single-phase superposition method (lookup-tables) with a coupled analysis between drive model in Matlab/Simulink® and finite element model of the SRM motor in Flux2D®. Experimental characterization techniques as well as finite element techniques are used to obtain the single-phase flux-linkage data. Transient electromagnetic analysis using the finite element method, coupled with a drive model and optimization strategy in Matlab/Simulink®, results in a more accurate SRM modelling. These results are more accurate because the mutual coupling between different phases is taken into account. Using this direct coupling, we can model a complete analysis of the motor behaviour (local saturation, iron losses, etc.) with the finite element software, keeping the complex drive and control strategy in Matlab/Simulink®. At every time-step of the discrete solver, data are exchanged between the drive model and the coupled electric circuit of the finite element analysis. Simulation results are compared with measurements of flux-linkage, torque, phase current, and iron losses. Advantages and disadvantages of the coupled and decoupled modelling technique are discussed.

Direct numerical calculation of acoustics: solution evaluation through energy analysis

The propagation of acoustic energy from a sound source to the far field is a fundamental problem of acoustics. In this paper the use of computational fluid dynamics (CFD) to directly calculate the acoustic field is investigated. The two-dimensional, compressible, inviscid flow about an accelerating circular cylinder is used as a model problem. The time evolution of the energy transfer from the cylinder surface to the fluid, as the cylinder is moved from rest to some non-negligible velocity, is shown. Energy is the quantity of interest in the calculations since various components of energy have physical meaning. By examining the temporal and spatial characteristics of the numerical solution, a distinction can be made between the propagating acoustic energy, the convecting energy associated with the entropy change in the fluid, and the energy following the body. In the calculations, entropy generation is due to a combination of physical mechanisms and numerical error. In the case of propagating acoustic waves, entropy generation seems to be a measure of numerical damping associated with the discrete flow solver.