Discrete Differential Operators Research Articles

An explicit fourth‐order hybrid‐variable method for Euler equations with a residual‐consistent viscosity

AbstractIn this article, we present a formally fourth‐order accurate hybrid‐variable (HV) method for the Euler equations in the context of method of lines. The HV method seeks numerical approximations to both cell averages and nodal solutions and evolves them in time simultaneously; and it is proved in previous work that these methods are supraconvergent, that is, the order of the method is higher than that of the local truncation error. Taking advantage of the supraconvergence, the method is built on a third‐order discrete differential operator, which approximates the first spatial derivative at each grid point using only the information in the two neighboring cells. Analyses of stability, accuracy, and pointwise convergence are conducted in the one‐dimensional case for the linear advection equation; whereas extension to nonlinear systems including the Euler equations is achieved using characteristic decomposition and the incorporation of a residual‐consistent viscosity to capture strong discontinuities. Extensive numerical tests are presented to assess the numerical performance of the method for both 1D and 2D problems.

Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier–Stokes equations

Abstract In this article we propose two finite-element schemes for the Navier–Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the point-wise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.

Lighter and faster simulations on domains with symmetries

A strategy to improve the performance and reduce the memory footprint of simulations on meshes with spatial reflection symmetries is presented in this work. By using an appropriate mirrored ordering of the unknowns, discrete partial differential operators are represented by matrices with a regular block structure that allows replacing the standard sparse matrix–vector product with a specialised version of the sparse matrix-matrix product, which has a significantly higher arithmetic intensity. Consequently, matrix multiplications are accelerated, whereas their memory footprint is reduced, making massive simulations more affordable. As an example of practical application, we consider the numerical simulation of turbulent incompressible flows using a low-dissipation discretisation on unstructured collocated grids. All the required matrices are classified into three sparsity patterns that correspond to the discrete Laplacian, gradient, and divergence operators. Therefore, the above-mentioned benefits of exploiting spatial reflection symmetries are tested for these three matrices on both CPU and GPU, showing up to 5.0x speed-ups and 8.0x memory savings. Finally, a roofline performance analysis of the symmetry-aware sparse matrix–vector product is presented.

Facial Expression Recognition using Discrete Differential Operator and CNN

Facial Expression Recognition using Discrete Differential Operator and CNN

Space division and adaptive selection strategy based differential evolution algorithm for multi-objective satellite range scheduling problem

Satellite range scheduling always plays a crucial role in tracking, telemetry, and control of the spacecraft. With the significant increase in the number and type of satellites in orbit, the demands of users for satellite range scheduling are becoming more and more diverse. However, existing studies for the satellite range scheduling problem (SRSP) rarely consider multiple optimization objectives, which is quite significant in practice. Hence, this paper investigates a multi-objective satellite range scheduling problem (MO-SRSP) that optimizes three objectives simultaneously: the overall profits of tasks, load-balance of antennas, and completion timeliness of tasks. To address MO-SRSP, this paper establishes a mathematical model on the basis of analysis of MO-SRSP and proposes a multi-objective evolutionary algorithm, which is called multi-objective differential evolution algorithm based on space division and adaptive selection strategy (MODE-SDAS). The space division strategy will uniformly divide the objective space into a set of subspaces and preserve a set of non-dominated solutions in each subspace during the environmental selection even if some of these solutions are dominated by other solutions in other subspaces, so as to maintain the diversity of the algorithm. The adaptive selection strategy will adaptively allocate computational resources to different subspaces to improve the convergence of the population. Besides, problem-specific designs such as coding and encoding methods, the discrete differential evolution operator, as well as objective-specific individual variation operators are incorporated into the algorithm to enhance the search capability. Finally, extensive experiments based on simulation test cases are carried out to verify the effectiveness and efficiency of MODE-SDAS. The comparison results show that MODE-SDAS significantly outperforms its competitors in terms of three metrics. Meanwhile, knee point analysis, sensitivity analysis, and application insights are presented.

Differential Operators on Sketches via Alpha Contours

A vector sketch is a popular and natural geometry representation depicting a 2D shape. When viewed from afar, the disconnected vector strokes of a sketch and the empty space around them visually merge into positive space and negative space , respectively. Positive and negative spaces are the key elements in the composition of a sketch and define what we perceive as the shape. Nevertheless, the notion of positive or negative space is mathematically ambiguous: While the strokes unambiguously indicate the interior or boundary of a 2D shape, the empty space may or may not belong to the shape's exterior. For standard discrete geometry representations, such as meshes or point clouds, some of the most robust pipelines rely on discretizations of differential operators, such as Laplace-Beltrami. Such discretizations are not available for vector sketches; defining them may enable numerous applications of classical methods on vector sketches. However, to do so, one needs to define the positive space of a vector sketch, or the sketch shape. Even though extracting this 2D sketch shape is mathematically ambiguous, we propose a robust algorithm, Alpha Contours , constructing its conservative estimate: a 2D shape containing all the input strokes, which lie in its interior or on its boundary, and aligning tightly to a sketch. This allows us to define popular differential operators on vector sketches, such as Laplacian and Steklov operators. We demonstrate that our construction enables robust tools for vector sketches, such as As-Rigid-As-Possible sketch deformation and functional maps between sketches, as well as solving partial differential equations on a vector sketch.

Direct Steady-State Calculation of Electromagnetic Devices Using Field-Circuit Models

Field-circuit models are very often used to model electromagnetic devices with conductive and non-linear magnetic materials. The numerical calculations of the field in the magnetic material must be combined with an equation of an external coil placed in the magnetic circuit. This means that the partial differential equations of the electromagnetic field in non-linear conductive materials and the non-linear ordinary differential equations must be solved together. Effective algorithms for solving such problems are still being developed. The article presents an algorithm directly providing the steady state solution without the simulation of transients. The basic assumption is that the solution can be predicted as a periodic time and space function, which is represented by appropriate Fourier series. The developed algorithm uses discrete partial differential operators for time and space derivatives. It allows us to create finite difference equations directly from the field and circuit equations, which take the form of algebraic equations, generally non-linear. This is a unique approach developed by us, which till now did not exist (and is not mentioned) in the literature. That algorithm is tested on a simple case of a solenoid coil with a ferromagnetic and conductive cylindrical core, in 2D space of radius and time. The calculation results confirm the effectiveness of the proposed approach both qualitatively, with regard to physical phenomena in ferromagnetic and conductive material, and quantitatively, in comparison with the results from specialized commercial software.

Consistency results for the dual-wind discontinuous Galerkin method

This paper further analyzes the dual-wind discontinuous Galerkin (DWDG) method for approximating Poisson’s problem by directly examining the relationship between the Laplacian and the underlying discrete Laplacian. DWDG methods are derived from the DG differential calculus framework that defines discrete differential operators to replace continuous differential operators. We establish a priori error estimates for the DWDG approximation of Δ. Since the DWDG method does not satisfy a Galerkin orthogonality condition, we also explore the relationship between the DWDG approximation and the Ritz projection defined to satisfy an exact Galerkin orthogonality condition linked to the DWDG method. Numerical experiments are provided to validate the theoretical results.

Combining Max pooling-Laplacian theory and k-means clustering for novel camouflage pattern design.

Camouflage is the main means of anti-optical reconnaissance, and camouflage pattern design is an extremely important step in camouflage. Many scholars have proposed many methods for generating camouflage patterns. k-means algorithm can solve the problem of generating camouflage patterns quickly and accurately, but k-means algorithm is prone to inaccurate convergence results when dealing with large data images leading to poor camouflage effects of the generated camouflage patterns. In this paper, we improve the k-means clustering algorithm based on the maximum pooling theory and Laplace's algorithm, and design a new camouflage pattern generation method independently. First, applying the maximum pooling theory combined with discrete Laplace differential operator, the maximum pooling-Laplace algorithm is proposed to compress and enhance the target background to improve the accuracy and speed of camouflage pattern generation; combined with the k-means clustering principle, the background pixel primitives are processed to iteratively calculate the sample data to obtain the camouflage pattern mixed with the background. Using color similarity and shape similarity for evaluation, the results show that the combination of maximum pooling theory with Laplace algorithm and k-means algorithm can effectively solve the problem of inaccurate results of k-means algorithm in processing large data images. The new camouflage pattern generation method realizes the design of camouflage patterns for different backgrounds and achieves good results. In order to verify the practical application value of the design method, this paper produced test pieces based on the designed camouflage pattern generation method and tested the camouflage effect of camouflage pattern in sunny and cloudy days respectively, and the final test results were good.

Penalty parameter and dual-wind discontinuous Galerkin approximation methods for elliptic second order PDEs

This article analyzes the effect of the penalty parameter used in symmetric dual-wind discontinuous Galerkin (DWDG) methods for approximating second order elliptic partial differential equations (PDE). DWDG methods follow from the DG differential calculus framework that defines discrete differential operators used to replace the continuous differential operators when discretizing a PDE. We establish the convergence of the DWDG approximation to a continuous Galerkin approximation as the penalty parameter tends towards infinity. We also test the influence of the regularity of the solution for elliptic second-order PDEs with regards to the relationship between the penalty parameter and the error for the DWDG approximation. Numerical experiments are provided to validate the theoretical results and to investigate the relationship between the penalty parameter and the L^2-error. For more information see https://ejde.math.txstate.edu/conf-proc/26/l1/abstr.html

Texture image classification based on a pseudo-parabolic diffusion model.

This work proposes a novel method based on a pseudo-parabolic diffusion process to be employed for texture recognition. The proposed operator is applied over a range of time scales giving rise to a family of images transformed by nonlinear filters. Therefore each of those images are encoded by a local descriptor (we use local binary patterns for that purpose) and they are summarized by a simple histogram, yielding in this way the image feature vector. Three main novelties are presented in this manuscript: (1) The introduction of a pseudo-parabolic model associated with the signal component of binary patterns to the process of texture recognition and a real-world application to the problem of identifying plant species based on the leaf surface image. (2) We also introduce a simple and efficient discrete pseudo-parabolic differential operator based on finite differences as texture descriptors. While the work in [26] uses complete local binary patterns, here we use the original version of the local binary pattern operator. (3) We also discuss, in more general terms, the possibilities of exploring pseudo-parabolic models for image analysis as they balance two types of processing that are fundamental for pattern recognition, i.e., they smooth undesirable details (possibly noise) at the same time that highlight relevant borders and discontinuities anisotropically. Besides the practical application, the proposed approach is also tested on the classification of well established benchmark texture databases. In both cases, it is compared with several state-of-the-art methodologies employed for texture recognition. Our proposal outperforms those methods in terms of classification accuracy, confirming its competitiveness. The good performance can be justified to a large extent by the ability of the pseudo-parabolic operator to smooth possibly noisy details inside homogeneous regions of the image at the same time that it preserves discontinuities that convey critical information for the object description. Such results also confirm that model-based approaches like the proposed one can still be competitive with the omnipresent learning-based approaches, especially when the user does not have access to a powerful computational structure and a large amount of labeled data for training.

An explicit weak Galerkin method for solving the first order hyperbolic systems

In this paper, we present and analyze a space–time weak Galerkin finite element (WG) method for solving the time-dependent symmetric hyperbolic systems. By introducing the discrete weakly differential operator, we construct a stable WG scheme which may be in the local or global form. The local scheme can be solved explicitly, element by element, under certain mesh condition. Then, we establish the stability and derive the optimal L2-error estimate of O(hk+12)-order for the WG solution when the k-order polynomials are used for k≥0. Numerical examples are provided to show the effectiveness of the proposed WG method.

Linear-scaling selected inversion based on hierarchical interpolative factorization for self Green's function for modified Poisson-Boltzmann equation in two dimensions

This paper studies an efficient numerical method for solving modified Poisson-Boltzmann (MPB) equations with the self Green's function as a state equation to describe electrostatic correlations in ionic systems. Previously, the most expensive point of the MPB solver is the evaluation of Green's function. The evaluation of Green's function requires solving high-dimensional partial differential equations, which is the computational bottleneck for solving MPB equations. Numerically, the MPB solver only requires the evaluation of Green's function as the diagonal part of the inverse of the discrete elliptic differential operator of the Debye-Hückel equation. Therefore, we develop a fast algorithm by a coupling of the selected inversion and hierarchical interpolative factorization. By the interpolative factorization, our new selected inverse algorithm achieves linear scaling to compute the diagonal of the inverse of this discrete operator. The accuracy and efficiency of the proposed algorithm will be demonstrated by extensive numerical results for solving MPB equations.

Discrete Differential Operators Immediately Applicable to Numerical Models of Solid Mechanics

The conventional gradient and related differential operators have been uniquely extended to a cluster of nodal points. Based on general algebraic grounds, such extensions are applicable to any discrete pattern while avoiding artificial shape functions or tessellations. Thus, various constitutive equations can be represented in a discrete form that enables the numerical modeling immediately in terms of nodal variables. Accuracy of this approach should ameliorate by the reduction of nodal spacing with the increasing computational power.

Solving 2D boundary-value problems using discrete partial differential operators

PurposeDiscrete differential operators of periodic base functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary linear and nonlinear differential equations with Dirichlet and Neumann boundary conditions.Design/methodology/approachThis paper presents a promising approach for solving two-dimensional (2D) boundary problems of elliptic differential equations. To create finite differential equations, specially developed discrete partial differential operators are used to replace the partial derivatives in the differential equations. These operators relate the value of the partial derivatives at each point to the value of the function at all points evenly distributed over the area where the solution is being sought. Exemplary 2D elliptic equations are solved for two types of boundary conditions: the Dirichlet and the Neumann.FindingsAn alternative method has been proposed to create finite-difference equations and an effective method to determine the leakage flux in the transformer window.Research limitations/implicationsThe proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This method can be extended to the 3D or time-periodic 2D cases.Practical implicationsThis paper presents a methodology for calculations of the self- and mutual-leakage inductances for windings arbitrarily located in the transformer window, which is needed for special transformers or in any case of the internal asymmetry of windings.Originality/valueThe presented methodology allows us to obtain the magnetic vector potential distribution in the transformer window only, for example, to omit the magnetic core of the transformer from calculations.

On approximation theory of nonlocal differential operators

AbstractRecently, several types of nonlocal discrete differential operators have emerged either from meshfree particle methods or from nonlocal continuum mechanics, such as peridynamics. In this article, we discuss the mathematical formulation as well as construction of the nonlocal discrete differential operators. Based on a least‐square minimization procedure and the associated Moore–Penrose inverse, we have found a general form of the shape tensor and a unified expression for the first type nonlocal differential operators. We then conduct a convergence study, which provides the interpolation error estimate for the first type discrete nonlocal different operators. We have shown that as the radius of the horizon approaches to zero, the first type nonlocal differential operators will converge to the local differential operators. Moreover, we have demonstrated the computational performance of the first type nonlocal differential operators in several numerical examples.

Transfer of conservative discrete differential operators between staggered grids: construction and duality relations

Abstract This paper is concerned with the construction of a discrete conservative operator on a mesh, knowing a conservative operator on a different mesh. We analyse the possibility of defining such a transfer based on the preservation of conservation properties and provide practical procedures, which can be effectively implemented for schemes that work on staggered grids. We show that such a transfer can be applied to obtain a discrete balance equation on a mesh starting from a discrete balance equation on a different mesh, and to prove local discrete duality relations for standard discrete operators defined on staggered grids.

Leakage Inductances of Transformers at Arbitrarily Located Windings

The article presents the calculation of the leakage inductance in power transformers. As a rule, the leakage flux in the transformer window is represented by the short-circuit inductance, which affects the short-circuit voltage, and this is a very important factor for power transformers. This inductance reflects the typical windings of power transformers well, but is insufficient for special transformers or in any case of the internal asymmetry of windings. This paper presents a methodology for calculations of the self- and mutual-leakage inductances for windings arbitrarily located in the air window. It is based on the 2D approach for analyzing the stray field in the air zone only, using discrete partial differential operators. That methodology is verified with the finite element method tested on real transformer data.

Learning to differentiate

Artificial neural networks together with associated computational libraries provide a powerful framework for constructing both classification and regression algorithms. In this paper we use neural networks to design linear and non-linear discrete differential operators. We show that neural network based operators can be used to construct stable discretizations of initial boundary-value problems by ensuring that the operators satisfy a discrete analogue of integration-by-parts known as summation-by-parts. Our neural network approach with linear activation functions is compared and contrasted with a more traditional linear algebra approach. An application to overlapping grids is explored. The strategy developed in this work opens the door for constructing stable differential operators on general meshes.

On Simplified Calculations of Leakage Inductances of Power Transformers

This paper deals with the problem of the leakage inductance calculations in power transformers. Commonly, the leakage flux in the air zone is represented by short-circuit inductance, which determines the short-circuit voltage, which is a very important factor for power transformers. That inductance is a good representation of the typical power transformer windings, but it is insufficient for multi-winding ones. This paper presents simple formulae for self- and mutual leakage inductance calculations for an arbitrary pair of windings. It follows from a simple 1D approach to analyzing the stray field using a discrete differential operator, and it was verified by the finite element method (FEM) calculation results.