Double Integral Research Articles

Transient processes in a gas/plate structure in the case of light loading

The problem of a pulse excitation in an acoustic half-space with a flexible wall described by a thin plate equation is studied. The solution is written as a double Fourier integral. A novel technique of estimation of this integral is developed. The surface of integration is deformed in such a way that the integrand is exponentially small everywhere except the neighbourhoods of several ‘special points’ that provide field components. Special attention is paid to the pulse associated with the coincidence point of the branches of the dispersion diagram of the acoustic medium and the plate. This pulse is shown to be a harmonic wave of a finite duration.

Fluid-structure interactions of internal pressure pipeline using the hierarchical finite element method

We study the influence of the fluid with the structure in vibration between fluid and structure of a cylinder of circular section granted by the phenomenon of the interaction fluid structure of a conditioned flow of laminar nature and incompressible in the form of the macrostructure. These two phenomena by the mechanical relations of stresses according to displacements, modelled by a cylinder. The analysis of the vibrations of cylinders filled with fluid is studied with limiting conditions of fluid and the solid with the coupling conditioned by its limits of action-reaction in forces. The problem of the cylindrical pipe is formulated by deriving the deformation and the kinetic energies of the vibrating cylinder with and its fluid to have different natural frequencies, we use the principle of Hamilton change the problem in the expression of the equation cylindrical differential which gives three displacement functions in a system of partial differential equations of the cylindrical coordinate of circular section which meet the limiting conditions imposed at both ends. Let us apply the Navier-Stocks equation in cylindrical coordinates, with the fluid continuity equation, for the solid equation of mechanical behaviour of stresses in terms of displacement by strain. To obtain the results of natural frequencies we use the Galerkin method for solid and for Galerkin-time fluid. Where the liquid influences the inner surface of the circular cylinder, depending on the condition of the coupling that the stresses of the solid are equal to the stresses fluid. The modelling is done by a computer language (MATLAB), the hierarchical finite element method is presented by a Legendre polynomial with double integral of Rodrigues, to arrive at the final formula of the mass-rigidity matrix, which dissects on three parts (fluid, coupling and structure). Based on a comparison with experimental results. We continue to study some geometrical and physical parameters which influence the natural frequencies, in a proportional or inversely proportional way.

Two Numerical Methods (RO (MSuM) and RO (SuMSu)) for Triple Integrals with for Continuous Functions

The main goal of this research is to calculate a triple integral included continuous integrands numerically by two composite rules. The first rule is the Mid-point method on the third dimension Z and the first dimension X with a suggested method on the second dimension Y, that is denoted by MSuM. The second rule is the suggested method on the third dimension Z and the first dimension X with a Mid-point method on the second dimension Y, that is denoted by SuMSu. The number of partial intervals is equals on the three dimensions. The study represented two theorems with the proofs to get such rules and the correction terms (the error terms) for each of rule. Moreover, to accelerate convergence and get better results, Romberg acceleration is used with both rules. These rules recalled by RO(MSuM) and RO(SuMSu) respectively such that the obtained results were high accuracy by relatively few partial intervals and shorter times.

Asymptotic expansion of the integral with two oscillations on an infinite interval

In this paper, we focus on constructing the asymptotic expansion for the highly oscillatory integral including of the product of exponential and Bessel oscillations with the stationary point. Based on the exact integral representation of Bessel function, the integral is transformed into a double oscillatory integral. For the resulting inner semi-infinite integral, we present a new way of a combination of the integration by parts and the Filon-type methods to produce the asymptotic expansion. Furthermore, the original oscillatory integral can be expanded in the sum of Gaussian hypergeometric function. The corresponding asymptotic property is also analysed. With increasing the oscillatory parameter, the error of the proposed asymptotic expansions decreases very fast. Numerical experiments are provided to illustrate the effectiveness of the expansion.

Approximate Calculation and Feature Analysis of Electric Field in Space by Thunderclouds

The calculation of electric field in space excited by thunderclouds is an important basis for lightning warning and protection. In numerical calculation of the electromagnetic field, it is often necessary to perform multiple loop nesting calculations on several triple integrals, which consume a lot of computing resources. In order to shorten the calculation time and improve the calculation efficiency, the electric field excited by the charged thunderclouds in space is theoretically derived with the analytical method by the thundercloud cylindrical charge pile model and based on the electrostatic field theory. The complex integrand function is approximated, so that the analytic expression of electric field in space is obtained in this paper. Through simulation and comparison, it is found that the approximate solution and the exact solution are similar in size, the change trend is the same, and the approximate analytical expression can be used for the approximate calculation of the electric field in a short range. Under certain conditions, the approximate solution can be converted into an accurate solution, which can be used for the accurate calculation of the electric field. Approximate calculation not only simplifies theoretical derivation but also improves calculation efficiency. The calculation time has been shortened from tens of hours to less than one second by using different calculation methods, which is a difference of 7 orders of magnitude. With approximate analytical expression, the electric field excited by charge pile with typical structures in thunderclouds in space is calculated and the characteristics of that are analyzed in this paper. For lightning protection of mobile targets, approximate calculation is of great significance in shortening the lightning warning time and enhancing the protection effect.

A revised contact stiffness model of rough curved surfaces based on the length scale

Based on the continuity of length scale for asperities, a complete model of normal stiffness between curved fractal surfaces considering friction factor is proposed. Through the coupling of critical base diameter and contact area, deformation modes of all asperities are determined, and contact stiffness of the whole rough surface is derived by double integral. Accordingly, effects of contact load, surface topography, friction factor, size parameters and material properties on the contact stiffness are investigated. Results show that contact stiffness depends sensitively on the fractal dimension, and exhibits nonmonotonic characteristics as fractal roughness varies across scales. Finally, predicted values of normal stiffness are compared with existing contact models as well as experimental data.

Dynamic response of a beam to the train of moving forces driven by an Erlang renewal process

Dynamic response of a beam to a random train of moving forces moving with the same velocity is considered. Unlike a widely used Poisson process model, a more adequate Erlang renewal process is used as a process driving the train of forces. Normal-mode approach is used to convert the problem into that of a renewal driven train of general pulses. Consequently the modal responses are the filtered renewal processes and are expressed as integrals with respect to the response to a single pulse (passage of a force) and to the increments of the counting renewal process. The expressions for the mean values and cross-correlations of modal responses are obtained as single and double integrals, respectively. The results are presented in terms of the renewal density of the underlying Erlang counting process. Mean value and variance of the mid-span deflection of the beam are determined by numerical evaluation of the pertinent integrals. Numerical analysis is carried out for the Erlang processes with integer parameter k=2, k=3 and 4 and, for comparison, also for a Poisson process. Different traffic conditions such as the velocity and the mean arrival rate of vehicles are taken into account.

Inversion formula for the windowed linear canonical transform

We study the inversion formula for recovering a signal from its windowed linear canonical transform. Different from the known inversion formula, where a double integral is invoked, we show that every signal can be recovered from its windowed linear canonical transform with a univariate integral. Moreover, we show that the integral involved is convergent almost everywhere on as well as in for all . Furthermore, we also obtain an inversion formula for functions in with the method of Cesàro summability.

Gromov\u2013Witten theory of K3 surfaces and a Kaneko\u2013Zagier equation for Jacobi forms

We prove the existence of quasi-Jacobi form solutions for an analogue of the Kaneko–Zagier differential equation for Jacobi forms. The transformation properties of the solutions under the Jacobi group are derived. A special feature of the solutions is the polynomial dependence of the index parameter. The results yield an explicit conjectural description for all double ramification cycle integrals in the Gromov–Witten theory of K3 surfaces.

The induced Aluthge transformations

The Aluthge transformation is generalized in the viewpoint of the axiom of operator means by using double operator integrals. It includes the mean transformation which is defined by S. H. Lee, W. Y. Lee and J. Yoon. Next we shall give some properties of it. Especially, we shall show that the n-th iteration of mean transformation of an invertible matrix converges to a normal matrix. Inclusion relations among numerical ranges of induced Aluthge transformations by some operator means are considered.

On reachable set estimation of nonlinear singular systems with distributed delay

SummaryThe estimation of the reachable set for singular systems with distributed delay and nonlinear term under zero initial condition is studied in this article. Based on the Lyapunov functional with triple integrals, a sufficient condition for bounding the reachable set is established by combining the free‐weighting matrix method, the integral mean value theorem, and the new inequality scaling method. Then the reachable set of nonlinear singular system is bounded by the prescribed ellipsoids. Finally, two numerical examples and a practical example verify the validity of the theoretical analysis.

A cookbook for approximating Euclidean balls and for quadrature rules in finite element methods for nonlocal problems

The implementation of finite element methods (FEMs) for nonlocal models with a finite range of interaction poses challenges not faced in the partial differential equations (PDEs) setting. For example, one has to deal with weak forms involving double integrals which lead to discrete systems having higher assembly and solving costs due to possibly much lower sparsity compared to that of FEMs for PDEs. In addition, one may encounter nonsmooth integrands. In many nonlocal models, nonlocal interactions are limited to bounded neighborhoods that are ubiquitously chosen to be Euclidean balls, resulting in the challenge of dealing with intersections of such balls with the finite elements. We focus on developing recipes for the efficient assembly of FEM stiffness matrices and on the choice of quadrature rules for the double integrals that contribute to the assembly efficiency and also posses sufficient accuracy. A major feature of our recipes is the use of approximate balls, e.g. several polygonal approximations of Euclidean balls, that, among other advantages, mitigate the challenge of dealing with ball-element intersections. We provide numerical illustrations of the relative accuracy and efficiency of the several approaches we develop.

Virtual Reality Design and Realization of Interactive Garden Landscape

It is very important to study and explore the application of virtual reality technology in landscape garden design, especially in the current environment of triple network integration and Internet of Things construction, to promote and facilitate the rapid development of digital landscape garden design in China. In this paper, we study the implementation method of virtual landscape gardening system and establish a virtual environment based on the ancient city of Yangcheng. On the computer platform, we study and realize a virtual roaming system of medium complexity with more complete roaming functions. Using the Quest3D software platform, a desktop-type virtual garden simulation system was developed, focusing on the virtual reality modeling technology method and virtual system implementation. The experimental results show that the GPU-accelerated drawing method based on GLSL can significantly improve the drawing frame rate of 3D garden landscape vegetation scenes with a small amount of scene data and has a certain feasibility. Based on the OpenSceneGraph (OSG) graphics rendering engine, the visualization of various types of 3D landscape models is realized, and the spatial layout of various types of landscape with parametric control is realized through digital vector layers, which flexibly manage and organize various garden elements and reasonably organize the spatial topological relationship between various types of landscape in 3D space. By integrating cross-platform ArcGISEngine components, the basic data of garden scenes including terrain data and vector data are managed. Through scene view cropping and hierarchical detail modeling technologies, the drawing efficiency and rendering real time of the garden landscape are improved. It realizes interactive 3D scene browsing and provides a six-degree-of-freedom all-round display of the overall landscape.

Influence functions for a 3D full-space under bilinear stationary loads

This manuscript brings the derivation of influence functions for a three-dimensional full-space under bilinearly-distributed time-harmonic loads. The differential equations describing the medium are decomposed in terms of uncoupled vector fields. A double Fourier transform allows the system of equations to be solved algebraically in the transformed space, where the bilinear-loading boundary conditions are imposed. The resulting displacement and stress solutions are presented in terms of double improper integrals to be evaluated numerically. The manuscript brings selected results from the evaluation of these solutions. These influence functions can be used in boundary element models of elastodynamic problems to yield computationally-efficient solutions and improved representation of sharply-varying contact traction fields.

Feynman Integrals and Scattering Amplitudes from Wilson Loops.

We study Feynman integrals and scattering amplitudes in N=4 super-Yang-Mills theory by exploiting the duality with null polygonal Wilson loops. As the main application, we compute for the first time the symbols of the general double pentagon integrals, which give the finite part of two-loop maximally helicity violating (MHV) amplitudes and finite components of next-to-MHV (NMHV) amplitudes to all multiplicities. The rational parts of the symbol consist of 164 letters, while the algebraic part contains 96 algebraic letters and cancel in MHV amplitudes and NMHV components which are free of square roots.

Some applications of the GRAN1 to analyze two-dimensional continuous probability distributions

The article considers the use of GRAN1 software in the analysis of two-dimensional continuous probability distributions. The learning process can be provided by freely distributable software, through the use of cloud technologies, in particular software from the GRAN package. However, the use of any technology in the educational process, including modern information and communication technologies, as well as the content of education, must be pedagogically balanced, which will avoid any negative consequences for the formation of the future member of society, his mental and physical development. Attention is drawn to the fact that children are increasingly looking for entertainment, using mobile devices, sharing new products with their peers and spending time together, playing a variety of online games, and parents cannot control this process. At the same time, teachers conduct lessons using mobile devices to draw attention to the subject, they are teaching. In the process of using computer technology, students can form a special type of thinking, the so-called “clip consciousness”, which leads to fragmentary thinking and the irreversibility of changes in consciousness. Therefore, the problem of excessive use of mobile devices by children requires constant analysis and coverage. The article discusses some examples of using the GRAN1 software to solve problems in probability theory. Examples of approximate calculation of the value of the double integral are given. Tables of values of the corresponding integrals are described. The use of modern information and communication technologies during mathematics lessons is considered.

A Conservative and Implicit Second-Order Nonlinear Numerical Scheme for the Rosenau-KdV Equation

In this paper, for solving the nonlinear Rosenau-KdV equation, a conservative implicit two-level nonlinear scheme is proposed by a new numerical method named the multiple integral finite volume method. According to the order of the original differential equation’s highest derivative, we can confirm the number of integration steps, which is just called multiple integration. By multiple integration, a partial differential equation can be converted into a pure integral equation. This is very important because we can effectively avoid the large errors caused by directly approximating the derivative of the original differential equation using the finite difference method. We use the multiple integral finite volume method in the spatial direction and use finite difference in the time direction to construct the numerical scheme. The precision of this scheme is O(τ2+h3). In addition, we verify that the scheme possesses the conservative property on the original equation. The solvability, uniqueness, convergence, and unconditional stability of this scheme are also demonstrated. The numerical results show that this method can obtain highly accurate solutions. Further, the tendency of the numerical results is consistent with the tendency of the analytical results. This shows that the discrete scheme is effective.

Second-order accurate BGK schemes for the special relativistic hydrodynamics with the Synge equation of state

This paper extends the second-order accurate BGK finite volume schemes for the ultra-relativistic flow simulations [5] to the 1D and 2D special relativistic hydrodynamics with the Synge equation of state. It is shown that such 2D schemes are very time-consuming due to the moment integrals (triple integrals) so that they are no longer practical. In view of this, the simplified BGK (sBGK) schemes are presented by removing some terms in the approximate nonequilibrium distribution at the cell interface for the BGK scheme without loss of accuracy. They are practical because the moment integrals of the approximate distribution can be reduced to the single integrals by some coordinate transformations. The relations between the left and right states of the shock wave, rarefaction wave, and contact discontinuity are also discussed, so that the exact solution of the 1D Riemann problem could be derived and used for the numerical comparisons. Several numerical experiments are conducted to demonstrate that the proposed gas-kinetic schemes are accurate and stable. A comparison of the sBGK schemes with the BGK scheme in one dimension shows that the former performs almost the same as the latter in terms of the accuracy and resolution, but is much more efficient.

A 64 × 64 SPAD Flash LIDAR Sensor Using a Triple Integration Timing Technique With 1.95 mm Depth Resolution

This paper presents a 64×64 dual mode flash LIDAR sensor that utilizes a triple integration timing technique. The sensor, fabricated in 130 nm HV CMOS, is capable of operating in both a direct time-of-flight (ToF) mode where the timestamp of the first arriving photon is recorded per-pixel, as well as a photon counting mode where the number of photons is recorded over a time interval. The timing technique utilizes both a time-to-digital converter (TDC) and a time-to-amplitude converter (TAC) with a counter measuring global clock cycles and the triple integration interpolator (TII) measuring between clock cycles. The TII uses an analog integration with an additional two reference integrations allowing the time measurement to be resistant to PVT variation and in turn, allowing the circuit to be miniaturized without causing a large timing non-uniformity across the array. Utilizing the TII, the sensor achieves a state-of-the-art timing performance, with a resolution of 13 ps (1.95 mm depth resolution), a maximum range of 220 μs (32 km), a single-shot jitter of 233 ps, and a differential non-linearity (DNL) of 6 ps (0.47 LSB). The sensor captures at a maximum frame rate of 8,300 fps and consumes 733 mW during operation. Experimental scenarios demonstrating the operation of the sensor are also provided.

Multiple Diamond-Alpha Integral in General Form and Their Properties, Applications

In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of alpha. Some of its properties are explored, and the relationship between it and the multiple mixed integral is provided. As an application, we establish some weighted Ostrowski type inequalities through the new integral. These new inequalities expand some known inequalities in the monographs and papers, and in addition, furnish some other interesting inequalities. Examples of Ostrowski type inequalities are posed in detail at the end of the paper.