Number Of Nodal Points Research Articles

Numerical solution of singular Poisson equations using sinc approximation

Abstract In this article, we use the Sinc approximation to solve the single Poisson problem (where the first derivative or higher order does not have an exact answer on the border of the boundary). We have examined the Sinc Galerkin approximation to solve the single Poisson problem, and finally, to solve the Poisson problem, we will use the sinc collocation method and in this method, we will reach a linear system. By carefully choosing the length of the steps and the number of nodal points, we will solve this system with two methods, with the orthogonalization technique, a numerical approximation will be obtained. Its accuracy can be exponential and of order where is a transaction parameter and c are a constant independent of N. In the final part, we will give some numerical examples of single Poisson problems.

Numerical solution of singular Poisson equations using sinc approximation

Abstract In this article, we use the Sinc approximation to solve the single Poisson problem (where the first derivative or higher order does not have an exact answer on the border of the boundary). We have examined the Sinc Galerkin approximation to solve the single Poisson problem, and finally, to solve the Poisson problem, we will use the sinc collocation method and in this method, we will reach a linear system. By carefully choosing the length of the steps and the number of nodal points, we will solve this system with two methods, with the orthogonalization technique, a numerical approximation will be obtained. Its accuracy can be exponential and of order where is a transaction parameter and c are a constant independent of N. In the final part, we will give some numerical examples of single Poisson problems.

AN METHOD FOR CREATING GEOINFORMATION 3D-MODEL OF AN AERODROME USING A DIGITAL TERRAIN MODEL

The creation of a three-dimensional geoinformation model of aerodrome is considered. An approach is proposed, consisting in supplementing a two-dimensional vector map with the values of the third coordinate, which are calculated using data from a digital terrain model. To implement this approach, the problem of interpolation of a function of two variables with a large number of nodal points is solved. The number of nodes where elevation values are set in a digital terrain model can be several thousand for each dimension. In this case, the use of global interpolation methods, for example, splines, in which function values at all nodal points are used to determine the values of a function at a given point falling into a grid cell, is computationally inefficient, since it requires a lot of machine time. It seems more promising to use a local approach to the interpolation problem. In this approach, the value of the function at an arbitrary point is determined by the values of the function at the nodal points of the grid closest to the specified point. Finite formulas are obtained that do not require solving equations, and allow interpolating functions for cases of linear and smooth approximation. An algorithm and a calculation program have been developed for a regular two-dimensional grid of nodes of a digital terrain model using the publicly available SRTM database. The results of numerical calculations of test cases with a comparison of linear and smooth approximations are presented. Using the proposed approach, a three-dimensional model of the Sochi aerodrome was created to simulate the movement of aircraft on its surface. The proposed approach and the interpolation method can be used to calculate the elevation of given points and to create three-dimensional cartographic models of the earth's surface.

Effective FE simulation methods for fillet welding process by continuous distributed heat input and element combination

This study proposed effective finite element methods for simulating the fillet welding process on T-shaped joints. The T-shaped fillet welded joints were fabricated, and the process was simulated by a Continuous Distributed heat input method (CD method) and combining shell and solid elements. This study compared the models by the Discontinuous Uniform heat input method (DU method) and the CD method. The results showed that the CD method not only simplified the modeling process but also reduced the calculation time compared to the DU method. Furthermore, combining solid and shell elements reduced the number of nodal points in the models. The calculation time of the models by the element combination was significantly reduced. By comparing with experiments, it was shown that the CD heat input method and the combination of solid and shell elements could save calculation time while keeping the analysis accuracy.

Sinc approximation for solving fourth-order pseudo-Poisson equations

In this article, we will investigate the Galerkin sinc approximation to solve the single pseudo-Poisson problem and in this method we will reach a linear system. We will solve this system by carefully choosing the length of the steps and the number of nodal points and with two methods; the first method, which is the usual method, is theoretically flawed and not practical and computationally efficient. Therefore, to solve the mentioned system, we introduce the orthogonalization technique, which solves both theoretical and computational problems. Numerical approximation is obtained whose accuracy is exponential and of order where N is a transaction parameter and c is a constant independent of N. In the final part, we give some numerical examples of individual pseudo-Poisson problems to demonstrate the efficiency of the method.

Order, Chaos and Born’s Distribution of Bohmian Particles

We study order, chaos and ergodicity in the Bohmian trajectories of a 2D quantum harmonic oscillator. We first present all the possible types (chaotic, ordered) of Bohmian trajectories in wavefunctions made of superpositions of two and three energy eigenstates of the oscillator. There is no chaos in the case of two terms and in some cases of three terms. Then, we show the different geometries of nodal points in bipartite Bohmian systems of entangled qubits. Finally, we study multinodal wavefunctions and find that a large number of nodal points does not always imply the dominance of chaos. We show that, in some cases, the Born distribution is dominated by ordered trajectories, something that has a significant impact on the accessibility of Born’s rule P=|Ψ|2 by initial distributions of Bohmian particles with P0≠|Ψ0|2.

Multi-domain multivariate spectral collocation method for (2+1) dimensional nonlinear partial differential equations

The novelty of this work rests upon the use of the domain partitioning technique in time variable when discretizing the domain of solution in spectral collocation algorithm. The single domain multivariate spectral collocation based methods have been proven to be effective in solving time-dependent partial differential equations (PDEs) defined over small time domains. However, there is a significant loss of accuracy as time computational domain proliferates and also when the number of grid nodes approaches a definite particular number. Therefore, the establishment of the new innovative multi-domain multivariate spectral quasilinearisation method (MDMV-SQLM) is described for the purpose of solving (2+1) dimensional nonlinear PDEs defined on large time intervals. The main output of this study is confirmation that minimizing the size of time computational domain at each subinterval assures sufficiently accurate results that are attained using minimal number of nodal points and less computational time. The solution algorithm involves partitioning the time domain into multiple non-overlapping sub-domains, simplification of the nonlinear PDEs using the quasilinearisation method and assumption of approximate solutions using triple Lagrange interpolating polynomials with Chebyshev–Gauss–Lobatto (CGL) points. The multi-domain spectral collocation procedure is executed on the linear systems of algebraic equations, where the subsequent matrix systems are solved separately in every time sub-interval with the continuity equation essentially used in obtaining initial conditions in the next subintervals. MATLAB software is used to implement the solution algorithm and numerical results are demonstrated graphically and in tabular form. To highlight the efficaciousness and accuracy of the MDMV-SQLM, error estimates, condition numbers and computational time are presented for well known (2+1) dimensional nonlinear initial-Dirichlet boundary value problems. The adoption of the domain decomposition technique is efficacious in suppressing the numerical challenges linked to large matrices and ill-conditioned nature of the resulting coefficient matrix. Also, the communicated results confirm that the numerical scheme is computationally cheap, fast and yield extremely accurate and stable results with the aid of fewer number of grid points for large time domains.

Predicting the effects of selected reservoir petrophysical properties on bottomhole pressure via three computational intelligence techniques

This study investigates the effects of selected petrophysical properties on predicting flowing well bottomhole pressure. To efficiently situate the essence of this investigation, genetic, imperialist competitive and whale optimization algorithms were used in predicting the bottomhole pressure of a reservoir using production data and some selected petrophysical properties as independent input variables. A total of 15,633 data sets were collected from Volvo field in Norway, and after screening the data, a total of 9161 data sets were used to develop apt computational intelligence models. The data were randomly divided into three different groups: training, validation, and testing data. Two case scenarios were considered in this study. The first scenario involved the prediction of flowing bottomhole pressure using only eleven independent variables, while the second scenario bothered on the prediction of the same flowing bottomhole pressure using the same independent variables and two selected petrophysical properties (porosity and permeability). Each of the two scenarios involved as implied in the first scenario, the use of three (3) heuristic search optimizers to determine optimal model architectures. The optimizers were allowed to choose the optimal number of layers (between 1 and 10), the optimal number of nodal points (between 10 and 100) for each layer and the optimal learning rate required per task/operation. the results, showed that the models were able to learn the problems well with the learning rate fixed from 0.001 to 0.0001, although this became successively slower as the leaning rate decreased. With the chosen model configuration, the results suggest that a moderate learning rate of 0.0001 results in good model performance on the trained and tested data sets. Comparing the three heuristic search optimizers based on minimum MSE, RMSE, MAE and highest coefficient of determination (R2) for the actual and predicted values, shows that the imperialist competitive algorithm optimizer predicted the flowing bottomhole pressure most accurately relative to the genetic and whale optimization algorithm optimizers.

Dynamic stiffness formulation for isotropic and orthotropic plates with point nodes

In this paper, the dynamic stiffness method for isotropic and orthotropic rectangular plates with point nodes is developed, making it possible to integrate the dynamic stiffness properties for plates with the dynamic stiffness properties of other elements such as bars and beams, but importantly, the advanced theory allows amalgamation of the dynamic stiffness method with the conventional finite element method for the first time. The derivation of the dynamic stiffness matrices for isotropic and orthotropic plates with point nodes has been accomplished by implementing the Fourier coefficients of the boundary values of the amplitudes of forces and displacements of the plate to form the force–displacement relationship at nodal points, including the corners. This innovative objective has been achieved by developing a new form of discrete Fourier transform technique for modified trigonometric functions. Using some carefully chosen illustrative examples, the convergence of results is ascertained by using different number of node points and their locations on the plate edges. The proposed theory has substantial advantages over conventional dynamic stiffness theories for plates, particularly when applying non-classical different boundary conditions on plate edges. The computed numerical results are discussed with significant conclusions drawn.

Structural deformation and metal-semiconductor transition in coupled carbon chains

The transition between gapped (semiconducting) and gapless (metallic) phases and tunability of bandgap in materials is a very lucrative yet considerably challenging goal for new-age device preparation. For bulk materials and for two-dimensional layered systems, this is a rapidly expanding field. We theoretically propose a one-dimensional pure carbon material with a tunable bandgap. We find that two parallel coupled polyyne chains show metallic behaviour with bands crossing on the Fermi level, unlike the single semiconducting chain. The number of nodal points (two) is robust under transverse and longitudinal strain, indicating the symmetry-protected nature of the metallic phase. Sliding one chain with respect to the other breaks reflection symmetry and a clear bandgap opens up at the nodes, leading to a gapped phase. By varying the slide parameter, the bandgap can be tuned efficiently. This work initiates and indicates possible topological phases of real one-dimensional materials without the involvement of edge modes.

Calculated Determination of the Seismic Resistance of Nuclear Power Plant Equipment

An algorithm to confirm the seismic resistance of equipment by a calculation method is proposed, and the limits of its application are determined. A mathematical model of the equipment is developed, and an example of the determination of natural frequencies and stresses for a three-dimensional structure is given. Two main types of calculation were used – static and dynamic. In the static calculation, the stress-strain state of a structure was determined. The values of the obtained stresses were compared with the allowable ones for the materials used, on the basis of which conclusions were made about the strength of the structure under seismic effects. The dynamic calculation resulted in the determination of the rigidity of the structure. The comparison of the stress values obtained for this equipment allowed us to make a conclusion regarding its resistance to seismic effects. The seismic resistance of the equipment was estimated on the example of the K-1000-60 / 1500 steam turbine condenser, and calculated at a seismic intensity of 6 points on the MSK-64 seismic intensity scale. In the course of solving this problem, results of the stress distribution in the housing and other structural elements of the condenser due to the action of combined normal operation and design-basis seismic loads were obtained. The seismic resistance of the equipment was calculated using the finite element method. This allowed us to present a solid body in the form of a set of individual finite elements that interact with each other in a finite number of nodal points. To these points are applied some interaction forces that characterize the influence of the distributed internal stresses applied along the real boundaries of adjacent elements. To perform such a calculation in CAD modeling software, a three-dimensional model was created. The obtained geometric model was imported into the software package, which significantly reduced complexity. The use of the calculation method allows us to significantly reduce the amount of testing when confirming the seismic resistance of equipment. Results of the assessment of the spatial complex stress state of the steam turbine condenser design due to the action of combined normal operation and design-basis seismic loads are obtained.

АЛГОРИТМИЧЕСКО-ПРОГРАММНОЕ ОБЕСПЕЧЕНИЕ ДЛЯ ИСПОЛЬЗОВАНИЯ КУБИЧЕСКИХ СПЛАЙНОВ ПРИ ВОССТАНОВЛЕНИИ ФУНКЦИОНАЛЬНОЙ ЗАВИСИМОСТИ В СЛУЧАЕ ГРАНИЧНЫХ УСЛОВИЙ ДЛЯ ПЕРВЫХ ПРОИЗВОДНЫХ

The work is devoted to the creation of a cubic splineS (x), which ensures the restoration of the functional dependence Y(x) when specifying conditions for the first derivative of a function on the boundaries of its definition. It is assumed that the function Y(x) is continuous, has continuous first and second derivatives on the interval [a,b], and at the nodal points x0=a, x1, x2, ...,xn=b takes known values equal, respectively, y0, y1, y2,... yn, k=0, 1, 2,…, n. A spline is represented as n polynomials of order no higher than three. The problem is reduced to finding 4n unknown spline coefficients. To determine them, a system of 4n-2 equations was formed: 2n equations taking into account the values of the function at the nodal points, n-1 equations reflecting the continuity of the first derivative and n-1 equations reflecting the continuity of the second derivative at the boundaries of neighboring polynomials, as well as two conditions, related to the values of the first derivative at the boundaries of the function. Solving the problem by the sweep method allowed us to develop algorithmic software for constructing the corresponding cubic spline S(x) in the case of a continuous function Y(x) with continuous first and second derivatives on the segment [a,b] with given values of the first derivative on its boundaries. A scheme for implementing a solution in an algorithmic language is proposed. The Scype program was developed in the Microsoft Visual Studio shell using the C++ language. The magnitude of the function recovery error using the proposed scheme was studied using the example of the function Y(x)=sin (x). The test results showed that the relative accuracy of interpolation with the number of nodal points n=10 on the segment [0,π/2] averaged about 1.4 • 10-3. The maximum accuracy was found in the first interval: (0.4-0.9)10-3. The minimum accuracy is recorded at the beginning of the second and last tenth interval: (4-6)10-2. An increase in the number of nodes to 102, 103 and 104 led to an increase in the average relative accuracy to 1•10-5, 1•10-7 and 2• 10-9, respectively. The data obtained can be used to interpolate functions and process experimental data. Key words: function recovery algorithm, interpolation polynomial, interpolation, cubic spline.

NODE-INDEPENDENT METHOD FOR GASTROENTEROLOGICAL SIGNAL PROCESSING BASED ON CUBIC SPLINES

This paper discusses a local cubic spline function built independently of node points using basic functions. the size of the calculations required to find the parameters to be determined during the construction of the spline function does not depend on the number of node points. Local-based splines are used to build such spline functions. Restoration of the gastroenterological signal was performed on the basis of the spline-function model discussed in the article. The result of a cubic spline-function error independent of the node points was compared with the result of the Lagrange classical polynomial error (Table 2).

Nodal intersections and geometric control

We prove that the number of nodal points on an $\mathcal{S}$-good real analytic curve $\mathcal{C}$ of a sequence $\mathcal{S}$ of Laplace eigenfunctions $\varphi_j$ of eigenvalue $-\lambda^2_j$ of a real analytic Riemannian manifold $(M, g)$ is bounded above by $A_{g , \mathcal{C}} \lambda_j$. Moreover, we prove that the codimension-two Hausdorff measure $\mathcal{H}^{m-2} (\mathcal{N}_{\varphi \lambda} \cap H)$ of nodal intersections with a connected, irreducible real analytic hypersurface $H \subset M$ is $\leq A_{g, H} \lambda_j$. The $\mathcal{S}$-goodness condition is that the sequence of normalized logarithms $\frac{1}{\lambda_j} \operatorname{log} {\lvert \varphi_j \rvert}^2$ does not tend to $-\infty$ uniformly on $\mathcal{C}$, resp. $H$. We further show that a hypersurface satisfying a geometric control condition is $\mathcal{S}$-good for a density one subsequence of eigenfunctions. This gives a partial answer to a question of Bourgain–Rudnick about hypersurfaces on which a sequence of eigenfunctions can vanish. The partial answer characterizes hypersurfaces on which a positive density sequence can vanish or just have $L^2$ norms tending to zero.

Determination of Rational Design and Geometric Parameters of a High-Performance Drill Based on a Mathematical Model of the Cutting Part

Processing of high-precision holes in one technological operation is an urgent problem of advanced manufacturing. Processing of precise holes in parts for aerospace and machine-building industries with a diameter of up to 30 mm is performed during countersinking, deployment or grinding operations. These operations are applied only if there already exists a pre-treated hole. Monolithic three-fluted drills have been becoming common for processing high-precision holes of 7-8 quality over the last few years. The processing of various types of materials such as stainless steels, cast iron and heat-resistant steels requires rational geometric and structural parameters of the cutting tool. The nature of the load distribution between all the teeth during drilling plays a huge role in the processing efficiency. Even load distribution between the three teeth and a positive geometry improves self-centering and reduces the deviation from the specified axis of the hole. The drill sharpening provides positive geometry along the entire main cutting edge. The influence of the geometric parameters of the cutting edge of the screw groove on the shape of the drill bit is equally important. Existing approaches to the design of the thinning do not account for the influence of the geometric parameters of the cutting edge on the section of the screw groove. Analytical approaches to modelling of the main cutting edges are typically married with difficulties associated with achieving a smooth change in the angle of inclination to the tangent of the cutting edge. The complexity of the aforementioned task is largely due to the presence of critical points at the interface of the spiral groove and thinning. Determining the rational shape of two sections of the main cutting edge performed in this study is a complicated task that includes several steps needed to find the number of nodal points. Achieving a positive rake angle in the normal section to the cutting edge at the gash area that was formed via a special sharpening is one of the most important results of this paper. The rational shape of the cutting edge and the front surface provides an increase in the strength of the cutting part by 1.3 times.

Numerical solutions of strongly non-linear generalized Burgers–Fisher equation via meshfree spectral technique

In this article, a meshfree spectral interpolation technique combined with Crank–Nicolson difference scheme is proposed to solve a class of strongly non-linear Burgers–Fisher type equation numerically. The proposed technique utilizes meshless shape functions for approximation of unknown spatial function and its derivatives. These shape functions are obtained by combining radial basis functions and point interpolation method in the spectral framework. The Crank–Nicolson finite difference scheme is employed for time integration. Stability of the proposed method is analysed theoretically and supported by numerical evidences for RBFs shape parameter , which is an equally important task. Measure of fitness quality is assessed via , and error norms. Efficiency and accuracy of the proposed technique is further examined via variation of time-step size and number of nodal points N. Comparison made with existing techniques in the literature confirms excellent performance of the proposed scheme.

Oscillations of offshore drilling platforms on layerd anisotropic base

The article presents the results of the studies, including the elements of a theoretical calculation, factors of internal strength and deformation field during volumetric vibrations of offshore drilling platforms on a layered basis. The influence of factors such as seismic dynamic impacts at the regional scale and dynamics of the geomorphology of the sea floor have been studied. Structures with spatial resilient links connected rigidly, hingedly or otherwise are subjected to dynamic force action. Depending on the size of the external forces and the forces of inertia, the flexible elements of the structure of the offshore drilling platform are in a complex three-dimensional stress-strain state. In dynamic calculation of the characteristics of the systems, the finite element method was used by replacing the platform with a system with a finite number of nodal points interconnected by inertia less rods. The masses are concentrated at the nodal points.

A New Numerical Method for an Asymptotic Coupled Model of Fractured Media Aquifer System

An asymptotic model coupling three-dimensional and two-dimensional equations is considered to demonstrate the flow in fractured media aquifer system in this paper. The flow is governed by Darcy’s law both in fractures and surrounding porous media. A new anisotropic and nonconforming finite element is constructed to solve the three-dimensional Darcy equation. The existence and uniqueness of the coupled solutions are deduced. Optimal error estimates are obtained in $$L^2$$ and $$H^1$$ norms. Numerical experiments show the accuracy and efficiency of the presented method. With the same number of nodal points and the same amount of computational costs, the results obtained by using the new element are much better than those by both $$Q_{1}$$ conforming element and Wilson nonconforming element on the same meshes.

A segregated spectral finite element method for the 2D transient incompressible Navier–Stokes equations

In this paper, the spectral element approximation and the velocity–pressuredecoupling method implementing the SIMPLE (semi-implicit method for pressure linked equations) algorithm are first combined to form a high-order segregated scheme for the solution of the two-dimensional transient incompressible Navier–Stokes equations. In contrast to previous segregated finite element methods based on the SIMPLE algorithm, the pressure equation is derived from the continuity equation using the element matrices to ensure convergence. High-order element basis functions are adopted, which greatly reduces the number of nodal points used in the calculation. The validation test that has an analytical solution demonstrates the high accuracy and convergence rate of the method. The flow in a lid-driven cavity with different inclination angles and the flow over a backward-facing step are investigated to further illustrate the performance of the scheme. The computed results are in excellent agreement with the benchmark solutions. The almost periodic solution for the flow of Re=10000 in a lid-driven square cavity is also captured by the present scheme.

Numerical solutions of time-fractional coupled viscous Burgers’ equations using meshfree spectral method

In this article, we compute numerical solutions of time-fractional coupled viscous Burgers’ equations using meshfree spectral method. Radial basis functions (RBFs) and spectral collocation approach are used for approximation of the spatial part. Temporal fractional part is approximated via finite differences and quadrature rule. Approximation quality and efficiency of the method are assessed using discrete $$E_{2}$$, $$E_{\infty }$$ and $$E_{\text {rms}}$$ error norms. Varying the number of nodal points M and time step-size $$\Delta t$$, convergence in space and time is numerically studied. The stability of the current method is also discussed, which is an important part of this paper.