Gauss Integral Research Articles

Derivative and partial integral methods and Gauss integral's indefinite integral solution and its use in wave function in quantum mechanics and exact solutions of the wave function depending on position and time

This study, , solution of Gaussian Integral with differential equation [18,19]; By using the partial integral method, the indefinite integral solution by taking x under the differential [29,30,31] and the indefinite integral solution by taking the Gaussian integral under the differential [33,38,39] both harmonic series and function solutions are found. Indefinite integral solution of Gaussian Integral [38] In Quantum Physics, the wave function solution f(x,α) in terms of α variable in x position and k space by substituting it in wave function [44,45] and in one-dimensional time dependent Schrödinger equation, wave function equation [66, 67] is found. When the wave function in direct space is differentiated by by the partial integral method without using the approximate value of w(k)[50] in space k, the general wave equation in k space by position [74] and approximately the wave function f(x,α) [80] is found. min in space k depending on location and time the exact solution of the wave equations [83,89] and the Taylor Series solution give the wave equation [87,90] in k space according to position and time.

Toward a simple and accurate Lagrangian-based error estimator for the BDF algorithms and adaptive time-stepping

PurposeThe purpose of this paper is to design a simple and accurate a-posteriori Lagrangian-based error estimator is developed for the class of backward differentiation formula (BDF) algorithms with variable time step size, and the adaptive time-stepping in BDF algorithms is demonstrated for efficient time-dependent simulations in fluid flow and heat transfer.Design/methodology/approachThe Lagrange interpolation polynomial is used to predict the time derivative, and then the accurate primary result is obtained by the Gauss integral, which is applied to evaluate the local error. Not only the generalized formula of the proposed error estimator is presented but also the specific expression for the widely applied BDF1/2/3 is illustrated. Two essential executable MATLAB functions to implement the proposed error estimator are appended for practical applications. Then, the adaptive time-stepping is demonstrated based on the newly proposed error estimator for BDF algorithms.FindingsThe validation tests show that the newly proposed error estimator is accurate such that the effectivity index is always close to unity for both linear and nonlinear problems, and it avoids under/overestimation of the exact local error. The applications for fluid dynamics and coupled fluid flow and heat transfer problems depict the advantage of adaptive time-stepping based on the proposed error estimator for time-dependent simulations.Originality/valueIn contrast to existing error estimators for BDF algorithms, the present work is more accurate for the local error estimation, and it can be readily extended to practical applications in engineering with a few changes to existing codes, contributing to efficient time-dependent simulations in fluid flow and heat transfer.

Concurrent topology optimization of multiscale structure under uncertain dynamic loads

This study proposes a robust concurrent topology optimization method with considering dynamic load uncertainty for the design of structures composed of periodic microstructures based on the bi-directional evolutionary structural optimization (BESO) method. The objective function is formulated as the summation of the mean and standard deviation of the structural dynamic compliance modulus. The constraints are imposed on the macrostructure and material microstructure volumes, respectively. The hybrid dimension reduction method and Gauss integral (HDRG) method is proposed to quantify and propagate load uncertainty to estimate the objective function. By the HDRG method, robust topology optimization with uncertainty modeled by probabilistic methods can be handed uniformly. To reduce the computational burden, a decoupled sensitivity analysis method is proposed to calculate the sensitivities of objective function with respect to the microstructure design variables. Five numerical examples are used to validate the effectiveness of the proposed robust concurrent topology optimization method and demonstrate the influence of load uncertainty on the design results. Results illustrate that the proposed methods can obtain the clear topologies of macro and micro structures, and the dynamic load uncertainty has a significant impact on the design results.

A FSDT meshfree method for free vibration analysis of arbitrary laminated composite shells and spatial structures

In this paper, a FSDT meshfree method based on the 3D continuous shell theory and moving-least squares approximation is proposed for the first time to investigate the free vibration behavior of arbitrary laminated composite shells and spatial structures. The novelty lies in that the resultant meshfree model has completely general geometric and kinematic descriptions, which are more readily applied to complex laminated shell geometries. In the formulation, the geometrical representation is based on the mapping technique without simplifying assumptions on the shell geometry, whereas the moving least-squares (MLS) approach and the first shear order theory (FSDT) are employed to build displacement fields. By using the mapping technique, a 3D arbitrary curved surface is expanded in a two-dimensional space, and the displacement approximation is established based on a set of nodes scattered in the Converted coordinate system. The simple Gauss integral are also formed in the Converted coordinate system. The essential boundary conditions are imposed by the full transformation method. According to Hamilton principle, the meshfree governing equations of free vibration of arbitrary laminated shells are obtained. The convergence, accuracy and applicability of the proposed method are demonstrated for the laminated shells of different geometrical shell shapes involving the square plates, shallow shells, open cylindrical shells, conical shells and corrugated plates with different boundary conditions and lamination schemes.

Performance Analysis of Non-uniform Sparse Segmentation Integral Method Based on Gauss Integral in EM Forward of Electrical Antenna Under Stratified Ocean

In order to solve the problem that it is difficult to take into account the computational efficiency and accuracy of electromagnetic field froward modeling of electrical antenna under the stratified ocean, evaluate the performance of two non-uniform sparse segmentation integral calculation methods based on Gauss integrals and improve the efficiency and accuracy of data interpretation in engineering applications, firstly two non-uniform sparse segmentation calculation methods based on Gauss integral nodes (Gauss-Legendre integral nodes and Gauss-Chebyshev integral nodes) are introduced. Then, two fast numerical integration methods corresponding of the time and frequency domain of the electrical antenna electromagnetic field under stratified ocean is established. Finally, by comparing and analyzing the results of the traditional uniform dense segmentation numerical integral calculation from the two aspects of computational accuracy and efficiency, the results show that two non-uniform sparse segmentation integral methods can improve the computational efficiency under the condition that the calculation accuracy is comparable, and the non-uniform sparse integral calculation method based on the Gauss-Chebyshev integral nodes is more significant than the calculation efficiency performance of the method based on the Gauss-Legendre integral node. At the same time, the general parameter setting of the non-uniform sparse segmentation integral method is given in this paper.

Thermal radiation calculation of complex surface based on Triangulation and Gauss Integral Method

To solve the problem of thermal radiation in complex geometric scenes, a finite element method based on Delaunay triangulation and three-point Gauss integral is proposed. The calculated value is compared with model existing theoretical solution of view factor, and the results show that the errors are within 3%. The method is applied to calculate the thermal radiation of combustion between adjacent liquefied petroleum gas tanks. The geometric model of conical flame and adjacent spherical tanks is also established. The result shows the maximum heat flux received by adjacent spherical tanks is 11.3 kW/m2 when the flame temperature is 1470 K without wind.

GISA: using Gauss Integrals to identify rare conformations in protein structures.

The native structure of a protein is important for its function, and therefore methods for exploring protein structures have attracted much research. However, rather few methods are sensitive to topologic-geometric features, the examples being knots, slipknots, lassos, links, and pokes, and with each method aimed only for a specific set of such configurations. We here propose a general method which transforms a structure into a ”fingerprint of topological-geometric values” consisting in a series of real-valued descriptors from mathematical Knot Theory. The extent to which a structure contains unusual configurations can then be judged from this fingerprint. The method is not confined to a particular pre-defined topology or geometry (like a knot or a poke), and so, unlike existing methods, it is general. To achieve this our new algorithm, GISA, as a key novelty produces the descriptors, so called Gauss integrals, not only for the full chains of a protein but for all its sub-chains. This allows fingerprinting on any scale from local to global. The Gauss integrals are known to be effective descriptors of global protein folds. Applying GISA to sets of several thousand high resolution structures, we first show how the most basic Gauss integral, the writhe, enables swift identification of pre-defined geometries such as pokes and links. We then apply GISA with no restrictions on geometry, to show how it allows identifying rare conformations by finding rare invariant values only. In this unrestricted search, pokes and links are still found, but also knotted conformations, as well as more highly entangled configurations not previously described. Thus, an application of the basic scan method in GISA’s tool-box revealed 10 known cases of knots as the top positive writhe cases, while placing at the top of the negative writhe 14 cases in cis-trans isomerases sharing a spatial motif of little secondary structure content, which possibly has gone unnoticed. Possible general applications of GISA are fold classification and structural alignment based on local Gauss integrals. Others include finding errors in protein models and identifying unusual conformations that might be important for protein folding and function. By its broad potential, we believe that GISA will be of general benefit to the structural bioinformatics community. GISA is coded in C and comes as a command line tool. Source and compiled code for GISA plus read-me and examples are publicly available at GitHub (https://github.com).

USING OPERATOR METHOD AND EFROS TRANSFORM FOR TAKING INTO ACCOUNT DISSIPATIVE PROPERTIES OF DEFORMABLE STRUCTURAL ELEMENTS

An approach for taking into account the dissipative properties of deformable structural elements (and in particular internal friction in the material) is proposed. This approach, based on the operator method and Efros transformation, is applied to analytical solutions obtained in the framework of elasticity theory, which can be represented as integrals of convolution type. The direct Laplace transform is applied to the operator equation. It is shown in the paper that changing to the Laplace variable enables one to pass from the relations taking into account dissipation energy to those for an elastic plate in the image space. The Efros theorem is used to perform the inverse Laplace transform of the complex function obtained. The posed problem is reduced to finding an unknown transforming function. Numerical calculations showed that in the problem considered (for small values of the time variable) the law of this function variation is very close to the law of normal distribution (Gaussian) with variable dispersion. Since the integral equations are discretized the kernel numerical modification is proposed by the Efros theorem. This modification is carried out by multiplying the discrete analogs of the original kernels by the special matrix obtained on the basis of the Gauss integral. The example of computing elastic and viscoelastic plate deflection is given, kernels of convolution integrals are presented for the "elastic" solution taking into account dissipation energy (internal friction in the material of the plate). Transversal impulse action on rectangular isotropic plates of medium thickness is considered. The plates are assumed to be hinged along the whole boundary. The plate deforming is simulated by the Timoshenko’s refined theory. It is possible to relatively quickly recalculate the deflections and deformations of the plate, taking into account internal friction for different values of the dissipation coefficients.

An efficient time adaptivity based on chemical potential for surface Cahn–Hilliard equation using finite element approximation

We present numerical simulations for the surface Cahn–Hilliard equation which describes phase separation phenomenon occurred on general surfaces. The spatial discretization is based on surface finite element method while the temporal discretization methods are first- and second-order stabilized semi-implicit schemes which guarantee the free energy decay. An efficient and parameter-free adaptive time-stepping strategy is proposed based on the numerical energy stability generated by stabilized semi-implicit scheme. The main idea is to use discrete chemical potential, the byproduct of stabilized semi-implicit scheme, to estimate the variation of numerical energy that is used as a indicator to update the time step, the operation avoid calculating numerical energy with Gauss integral in each time step and reduce the calculated cost. In addition, optimal error estimate of first-order stabilized semi-implicit scheme in the case of curved surface are provided. Finally, numerical experiments are presented to demonstrate the stability, accuracy and efficiency of the proposed algorithms.

A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics

This paper presents a step-by-step time integration method for transient solutions of nonlinear structural dynamic problems. Taking the second-order nonlinear dynamic equations as the model problem, this self-starting one-step algorithm is constructed using the Galerkin finite element method (FEM) and Newton–Raphson iteration, in which it is recommended to adopt time elements of degree m = 1,2,3. Based on the mathematical and numerical analysis, it is found that the method can gain a convergence order of 2m for both displacement and velocity results when an ordinary Gauss integral is implemented. Meanwhile, with reduced Gauss integration, the method achieves unconditional stability. Furthermore, a feasible integration scheme with controllable numerical damping has been established by modifying the test function and introducing a special integral rule. Representative numerical examples show that the proposed method performs well in stability with controllable numerical dissipation, and its computational efficiency is superior as well.

Extended layerwise method of laminated composite shells

This paper deals with the accurate description of the multiple delaminations and transverse cracks in double-curved laminated composite shells by using the extended finite element method (XFEM) and layerwise theory (LWT), and develops an extended layerwise method (XLWM) which had been applied to the laminated beams in our previous work (Li et al., 2015 [1]). In the displacement assumption of XLWM, the discontinuous function along the thickness direction is adopted to simulate the displacement discontinuous resulted from multiple delaminations. While the transverse cracks are modeled in in-plane displacements discretization by XFEM. The level set method (LSM) is employed in the present method to track the interfaces resulted from the transverse cracks. The interaction integral method and maximum circumferential tensile criterion are used to calculate the stress intensity factor (SIF) and crack growth angle, respectively. In order to ensure the accuracy of Gauss integral when certain nodes are very close to the crack surface as the crack grows, a local remeshing scheme is developed to shift these nodes without scarifying the mesh quality. In the numerical examples, spherical shells, cylindrical shells and plates with/without multiple delaminations and/or transverse crack are considered to the problems of static responses analysis, SIF calculation and transverse crack arbitrary growth.

Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows, part II

We describe a new approach to triple linking invariants and integrals, aiming for a simpler, wider and more natural applicability to the search for higher order helicities of fluid flows and magnetic fields. To each three-component link in Euclidean 3-space, we associate a geometrically natural generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers, but patterned after J.H.C. Whitehead's integral formula for the Hopf invariant. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of 3-space, while the integral itself can be viewed as the helicity of a related vector field on the 3-torus. In the first paper of this series [math.GT 1101.3374] we did this for three-component links in the 3-sphere. Komendarczyk has applied this approach in special cases to derive a higher order helicity for magnetic fields whose ordinary helicity is zero, and to obtain from this nonzero lower bounds for the field energy.

Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows

To each three-component link in the 3-sphere we associate a generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. We view this as a natural extension of the familiar situation for two-component links in 3-space, where the linking number is the degree of the classical Gauss map from the 2-torus to the 2-sphere. The generalized Gauss map, like its prototype, is geometrically natural in the sense that it is equivariant with respect to orientation-preserving isometries of the ambient space, thus positioning it for application to physical situations. When the pairwise linking numbers of a three-component link are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. This new integral is also geometrically natural, like its prototype, in the sense that the integrand is invariant under orientation-preserving isometries of the ambient space. Versions of this integral have been applied by Komendarczyk in special cases to problems of higher order helicity and derivation of lower bounds for the energy of magnetic fields. We have set this entire paper in the 3-sphere because our generalized Gauss map is easiest to present here, but in a subsequent paper we will give the corresponding maps and integral formulas in Euclidean 3-space.

Fast large-scale clustering of protein structures using Gauss integrals

Clustering protein structures is an important task in structural bioinformatics. De novo structure prediction, for example, often involves a clustering step for finding the best prediction. Other applications include assigning proteins to fold families and analyzing molecular dynamics trajectories. We present Pleiades, a novel approach to clustering protein structures with a rigorous mathematical underpinning. The method approximates clustering based on the root mean square deviation by first mapping structures to Gauss integral vectors--which were introduced by Røgen and co-workers--and subsequently performing K-means clustering. Compared to current methods, Pleiades dramatically improves on the time needed to perform clustering, and can cluster a significantly larger number of structures, while providing state-of-the-art results. The number of low energy structures generated in a typical folding study, which is in the order of 50,000 structures, can be clustered within seconds to minutes.

Calculation Algorithms for Automobiles’ Acceleration Performance

Acceleration time, as an important dynamic performance index of an automobile, is classified as acceleration time of shift and acceleration time at a given gear to evaluate an automobile’s acceleration performance. Compared with the traditional calculation method, this proposed algorithm calculates an automobile’s speed of optimal-power shift by using linear interpolation; calculates the acceleration time using Gauss integral, and calculates the acceleration time to a given distance by combining conversion method and interpolation method. A numerical example is given and the actual test results indicate that this algorithm is fit for automobile design.

Shape analysis for automated sulcal classification and parcellation of MRI data

We describe geometric invariants that characterize the shape of curves and surfaces in 3D space: curvature, Gauss integrals and moments. We apply these invariants to neuroimaging data to determine if they have application for automatically classifying and parcellating cortical data. The curves of sulci and gyri on the cortical surface can be obtained by reconstructing cortical surface representations of the human brain from magnetic resonance imaging (MRI) data. We reconstructed gray matter surfaces for 15 subjects, traced 10 sulcal curves on each surface and computed geometric invariants for each curve. These geometric features were used classify the curves into sulcal and hemispheric classes. The best classification results were obtained when moment-based features were computed on the sulcal curves in native space. Gauss integral measures showed that they were useful for differentiating the hemispheric location of a single sulcus. These promising results may indicate that moment invariants are useful for characterizing shape on a global scale. Gauss integral invariants are potentially useful measures for characterizing cortical shape on a local, rather than global scale. Gauss integrals have found biological significance in characterizing proteins so it is worthwhile to consider their possible application to neuroscientific data.

Evaluating protein structure descriptors and tuning Gauss integral based descriptors

The general development in the natural sciences from relative comparison to absolutedescription forces the current relative protein structure comparison, based on similaritymeasures, to be supplemented, or even replaced, by absolute description of each individualprotein structure. This paper addresses the question of what should be required from agood set of protein structure descriptors. As an example a Gauss integral based family ofprotein structure descriptors, that has been shown to successfully classify the geometry ofCATH2.4 connected protein domains, is examined. The CATH2.4 domains arehere observed to break a symmetry under reversal of the direction of traversal ofthe protein backbone that general folded tubes possess. It is thus a challengefor any large scale protein or polymer model to explain this broken symmetry.

The direct link model for polymer rings using topological field theory and the second topological moment in dense systems

Polymer rings in solution are either permanently entangled or not. Permanent topological restrictions give rise to additional entropic interactions apart from the ones arising due to mere chain flexibility or excluded volume. Conversely, entangled polymer rings systems may be formed by closing randomly entangled flexible linear chains. The dependance of linking numbers between randomly entangled rings on the chain length, more specifically the second topological moment $<n^2>$, i.e. the average squared linking number, may be determined. In this paper, an approach recently discussed in mathematical physics and called abelian BF theory, is presented which allows to express the linking constraint in its simplest form, the Gauss integral, in terms of two gauge fields. The model of Brereton and Shah for a single ring entangled with many other surrounding rings is rederived. The latter model is finally used to calculate the second topological moment, in agreement with a recent result by Ferrari, Kleinert, and Lazzizzera obtained by using ${\rm n}$-component $\phi^4$ theory in the limit ${\rm n}\to 0$.

Fourier - Gauss transforms of the Askey - Wilson polynomials

We discuss classical Fourier - Gauss integral transforms for a five-parameter family of the Askey - Wilson polynomials .

Writhing number of supercoiled DNA from its scanning force microscopy imaging

Scanning force microscopy can routinely image supercoiled DNA molecules under liquids or in air and the complete three-dimensional path of the chain can be identified. This makes it possible to evaluate the writhing number (Wr) of the coiled curve described by the chain axis in the imaged conformation of the molecule. Wr is a much better descriptor of supercoiling than the linking deficit (ΔLk) which has been commonly used so far. Wr of the DNA axis curve can be roughly estimated by the signed-intersection approach or calculated as the Gauss integral of the curve itself. We applied both methods to a model closed simple curve and to coiled DNA molecules exhibiting one or two turns of the chain about their superhelical axis.