Interpolation Nodes Research Articles

Superconvergence of the MINI mixed finite element discretization of the Stokes problem: An experimental study in 3D

Stokes flows are a type of fluid flow where convective forces are small in comparison with viscous forces, and momentum transport is entirely due to viscous diffusion. Besides being routinely used as benchmark test cases in numerical fluid dynamics, Stokes flows are relevant in several applications in science and engineering including porous media flow, biological flows, microfluidics, microrobotics, and hydrodynamic lubrication. The present study concerns the discretization of the equations of motion of Stokes flows in three dimensions utilizing the MINI mixed finite element, focusing on the superconvergence of the method which was investigated with numerical experiments using five purpose-made benchmark test cases with analytical solution. Despite the fact that the MINI element is only linearly convergent according to standard mixed finite element theory, a recent theoretical development proves that, for structured meshes in two dimensions, the pressure superconverges with order 1.5, as well as the linear part of the computed velocity with respect to the piecewise-linear nodal interpolation of the exact velocity. The numerical experiments documented herein suggest a more general validity of the superconvergence in pressure, possibly to unstructured tetrahedral meshes and even up to quadratic convergence which was observed with one test problem, thereby indicating that there is scope to further extend the available theoretical results on convergence.

Simulation of Heat Transfer in Single-Crystal Lithium Niobate in Interaction with Continuous-Wave Laser Radiation

The paper presents the simulation results of heat transfer in single-crystal lithium niobate (LiNbO3) in the form of cylinder of diameter mm and height mm in interaction with continuous-wave laser radiation with the output power of W and the wavelength of nm. The density of the LiNbO3 crystal is kg/m3; the thermal conductivity along the [001] direction is W/(m×K); the thermal conductivity in the (001) plane is W/(m×K); the specific heat at constant pressure is J/(kg×K); the absorption coefficient is %/cm @ 1064 nm. The laser beam propagates along the optical axis of the crystal. The laser beam intensity profile is represented as a Gaussian function, and the absorption of laser radiation of the single-crystal lithium niobate is described by Beer-Lambert’s law. The numerical solution of the non-stationary heat conduction problem is obtained by meshless scheme using anisotropic radial basis functions. The time interval of the non-stationary boundary-value problem is 2 h 30 min. The results of numerical calculations of the temperature distribution inside and on the surface of the single-crystal lithium niobate at times s are presented. The time required to achieve the steady-state heating mode of the LiNbO3 crystal, as well as its temperature range over the entire time interval, have been determined. The accuracy of the approximate solution of the boundary-value problem at the n-th iteration is estimated by the value of the norm of relative residual . The results of the numerical solution of the non-stationary heat conduction problem obtained by meshless method show its high efficiency even at a small number of interpolation nodes.

Fast simulation of fully non-stationary wind fields using a new matrix factorization assisted interpolation method

The classical spectral representation method and its improved versions have been widely used to generate non-stationary wind fields. However, they are not efficient for the simulation of non-stationary wind fields with time-varying coherence (i.e., fully non-stationary wind fields) due to the extensive computational demand regarding spectral matrix decomposition or harmonic superposition. To this end, this study develops a new matrix factorization assisted interpolation method, which is composed of an improved interpolation scheme and the monotone projected Barzilai-Borwein based non-negative matrix factorization (MPBB-NMF). Specifically, an improved interpolation scheme is firstly developed to quantify the desired distribution of time-frequency interpolation nodes, and only the decomposed spectra at these nodes are calculated by Cholesky decomposition. Then, MPBB-NMF with a fast global convergence is used to decouple these spectra into products of non-negative time- and frequency-dependent functions. Finally, one-dimensional interpolation is performed for these decoupled functions, which makes it possible to accelerate the harmonic superposition using a small amount of fast Fourier transforms (FFTs). Obviously, the proposed method realizes the decomposition of spectral matrix in the decoupling form. Besides reducing of the number of Cholesky decomposition and invoking FFT, its decoupling only aims at the low dimensional matrix and the interpolation merely involves a few decoupled items. Numerical simulation on the fully non-stationary wind field of a long-span bridge demonstrates the effectiveness and superiority of the proposed method.

SPH study of wave force on simplified superstructure of open-type sea access road

A numerical study is performed on the horizontal and uplift wave forces acting on open-type sea access roads. Numerical simulations are performed using the 2D δ-SPH model. In boundary condition, the static pressure difference between the fixed ghost particle and its interpolation point is calculated considering the distance between the interpolation node and the free surface. To avoid improper dynamic pressure measurements, an improved pressure measurement technique is incorporated with the numerical model. To increase the calculation accuracy for the upper region of the sea access road under wave conditions involving a large wave height with overtopping, Adaptive Particle Refinement (APR) is performed. Comparison of the numerical and experimental results demonstrates the high accuracy of the SPH scheme. Based on the numerical results, the characteristics of the wave force on the sea access road are investigated. Furthermore, the maximum horizontal and uplift wave forces on the sea access road are presented for cases involving different water depths, wave heights and wave periods. In this manner, the effects of the wave period on the wave force are clarified. Finally, the influence of the seaside slope on the wave force is examined.

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

Pricing of financial derivatives, in particular early exercisable options such as Bermudan options, is an important but heavy numerical task in financial institutions, and its speed-up will provide a large business impact. Recently, applications of quantum computing to financial problems have been started to be investigated. In this paper, we first propose a quantum algorithm for Bermudan option pricing. This method performs the approximation of the continuation value, which is a crucial part of Bermudan option pricing, by Chebyshev interpolation, using the values at interpolation nodes estimated by quantum amplitude estimation. In this method, the number of calls to the oracle to generate underlying asset price paths scales as widetilde{O}(epsilon ^{-1}), where ϵ is the error tolerance of the option price. This means the quadratic speed-up compared with classical Monte Carlo-based methods such as least-squares Monte Carlo, in which the oracle call number is widetilde{O}(epsilon ^{-2}).

On [formula omitted] matrix-based approximate inverse preconditioning technique for diagonal-plus-Toeplitz linear systems from spatial fractional diffusion equations

In this paper, we firstly explore the special structure of the discretized linear systems from the spatial fractional diffusion equations. The coefficient matrices of the resulting discretized systems have a diagonal-plus-Toeplitz structure. Because the resulting Toeplitz matrix is symmetric positive definite (SPD), then we can employ the τ matrix to approximate it. By making use of the piecewise interpolation polynomials, we propose a new approximate inverse preconditioner to handle the diagonal-plus-Toeplitz coefficient matrices. The τ matrix-based approximate inverse (TAI) preconditioning technique can be implemented very efficiently by using discrete sine transforms(DST). Theoretically, we have proved that the spectrum of the resulting preconditioned matrices are clustered around one. Thus, Krylov subspace methods with the proposed preconditioners converge very fast. To demonstrate the efficiency of the new preconditioners, numerical experiments are implemented. The numerical results show that with the proper interpolation node numbers, the performance of the τ-matrix based preconditioning technique is better than the other tested preconditioners.

INTERPOLATION OF TABULAR FUNCTIONS FROM ONE INDEPENDENT VARIABLE USING THE TAYLOR POLYNOMIAL

A method of local interpolation of tabular functions from one independent variable using the Taylor polynomial of the nth degree in arbitrarily located interpolation nodes has been developed. This makes it possible to calculate intermediate values of tabular functions between interpolation nodes. The conducted analysis of the latest research and publications in the field of interpolation of tabular functions showed that the main part of the research is a strict theory of interpolation, i.e. clarification of its fundamental mathematical provisions. Some features of the interpolation of tabular functions from one independent variable using the Taylor polynomial of the nth degree are considered, namely: the solution algorithm and mathematical formulation of the interpolation problem are given; its formalized notation is given, as well as the matrix notation of interpolation procedures for certain values of the argument. A scalar algorithm for solving the problem of interpolation of tabular functions from one independent variable using the Taylor polynomial of the 2nd, 3rd, and 4th degrees has been developed. The simplicity and clarity of this algorithm is one of its advantages, but the algorithm is inconvenient for software implementation. The mathematical formulation of the problem of interpolation of tabular functions in terms of matrix algebra is given. The interpolation task is reduced to performing the following actions: based on the values of nodal points known from the table, it is necessary to calculate the Taylor matrix of the nth degree; based on the function values specified in the table a column vector of interpolation nodes should be formed; solve a linear system of algebraic equations, the root of which is the numerical coefficients of the Taylor polynomial of the nth degree. A method of calculating the coefficients of the interpolant, given by the Taylor polynomial of the nth degree for one independent variable has been developed. The essence of the method reduces to the product of the matrix, inverse of the Taylor matrix, which is determined by the nodal points of the tabular function, by a column vector containing the values of the interpolation nodes. Specific examples demonstrate the peculiarities of calculating the interpolant coefficients of the 2nd, 3rd and 4th degrees for one independent variable, and for each of them the interpolated value of the function at a given point is calculated. Calculations were performed in the Excel environment, which by analogy can be successfully implemented in any other computing environment.

INTERPRETATION IN MAPLE SYSTEM THE RUNGE PHENOMENON AND ITS ELIMINATING

Questions on interpolation of functions with Lagrange polynomials are discussed. The Runge phenomenon is considered, when a sequence of Lagrange polynomials does not converge to an interpolated function uniformly. It is shown that the choice of roots of the Chebyshev polynomials as interpolation nodes provides more precise approximation than the interpolation by equidistant nodes. The Chebyshev polynomials are remarkable in sense that they admit least deviation from zero among all monic polynomials of the same degree. Сalculations and drawing graphs were performed using Maple program.

Towards a macroscopically consistent discrete method for granular materials: Delaunay strain-based formulation

We demonstrate that the Delaunay-based strain definition proposed by Bagi (Mech Mater 22:165–177, 1996) for granular media can be straightforwardly translated into a particle-based numerical method for continua. This method has a number of attractive features, including linear completeness and satisfaction of the patch test, exact conservation of linear and angular momenta in the absence of external forces and torques, and anti-symmetry of the gradient vectors for any two points not both on the boundary of the computational domain. The formulation in effect relies on nodal (particle) interpolation of the deformation gradient and is therefore inherently unstable. Drawing on the analogy with granular media, a pairwise interaction between particles is included to alleviate this issue. The underlying idea is to define a local, non-affine deformation of each bond or contact, and to introduce pairwise forces via a stored-energy functional expressed in terms of the corresponding local displacements. In this manner, a generalisation of the Ganzenmüller (Comput Methods Appl Mech Eng 286:87–106, 2015) hourglass stabilisation procedure to non-central forces is obtained. The performance of the method is demonstrated in a range of problems. This work can be considered a first step towards the development of a macroscopically consistent discrete method for granular materials.

Interpolation of Subdivision Features for Curved Geometry Modeling

We present a nodal interpolation method to approximate a subdivision model. The main application is to model and represent curved geometry without gaps and preserving the required simulation intent. Accordingly, we devise the technique to maintain the necessary sharp features and smooth the indicated ones. This sharp-to-smooth modeling capability handles unstructured configurations of the simulation points, curves, and surfaces. The surfaces correspond to initial linear triangulations that determine the sharp point and curve features. The method automatically suggests a subset of sharp features to smooth which the user modifies to obtain a limit model preserving the initial points. This model reconstructs the curvature by subdivision of the initial mesh, with no need of an underlying curved geometry model. Finally, given a polynomial degree and a nodal distribution, the method generates a piece-wise polynomial representation interpolating the limit model. We show numerical evidence that this approximation, naturally aligned to the subdivision features, converges to the model geometrically with the polynomial degree for nodal distributions with sub-optimal Lebesgue constant. We also apply the method to prescribe the curved boundary of a high-order volume mesh. We conclude that our sharp-to-smooth modeling capability leads to curved geometry representations with enhanced preservation of the simulation intent.

Another Hilbert inequality and critically separated interpolation nodes

AbstractAbstract: Estimates on the condition number of Vandermonde matrices have implications on several algorithms ranging from polynomial interpolation to sparse super resolution in fluorescence microscopy. Classically, the situation is studied for monomials on real intervals, the complex unit disk, and the complex unit circle. Except for roots of unity and well separated nodes on the unit circle, the condition number grows strongly with increasing polynomial degree. Here, we show that the condition number of the Vandermonde matrix for a particular instance of critically separated nodes on the complex unit circle grows logarithmically with the polynomial degree. The proof is based on a variant of Hilbert's inequality with remainder term.

PIES for 2D elastoplastic problems with singular plastic strain fields

The paper presents the approach for solving 2D elastoplastic problems with singular stress/strain fields by the parametric integral equation system (PIES) method. PIES is applied since it does not require the discretization of the plastic zone into elements, and instead, it uses globally declared parametric surface patches. The properties of such surfaces (e.g. the unit square is their domain) are used for automatically concentrating the interpolation nodes around the regions with singular stress/strain fields. This, in turn, limits the global number of nodes in favor of their accumulation in specific places. The proposed approach of node concentration requires only a few parameters to be defined. In order to take into account the complex nature of the plastic strain fields, a special method of their interpolation is selected. The Kriging approach effectively involves an interactive investigation of the spatial behavior of the strains to choose the best estimate. It is also the method that bases on irregularly spaced interpolation nodes. Some examples are solved, and obtained results are compared with the finite element method (FEM).

Variational integrators for forced Lagrangian systems based on the local path fitting technique

Variational integrators are particularly suitable for simulation of mechanical systems, where features such as symplecticity and momentum preservation are essential. They also exhibit excellent long-time energy behavior even if external forcing is involved. Motivated by this fact, we present a new approach, that is based on the local path fitting technique, to construct variational integrators for forced mechanical systems. The core technology exploited is to fit the local trajectory as the Lagrange interpolation polynomial by requiring that the forced Euler–Lagrange equations hold at the internal interpolation nodes. This operation also yields the essential terms of the discrete forced Euler–Lagrange equations and consequently formulates the final integrator. This new approach not only avoids numerical quadrature involved in the classical construction, but also significantly improves the precision of the resulting integrator, as illustrated by the given examples.

Interpolation method for evaluating weakly singular kernels

The solution of initial, boundary, and mixed value problems through the integral equation method yields certain boundary singular integral equations. In many scientific and industrial applications in artificial intelligence, biological systems, scattering, radiation, and image processing, there is a need to evaluate such types of singular integrals, especially the weakly singular kernels. This study presents a new numerical method for evaluating weakly singular kernels based on some advanced matrix-vector barycentric Lagrange interpolation formulas. We developed these formulas to be applied to numerically evaluating singular integrals. At the same time, we created three computational rules to determine the optimal locations for the distribution of the interpolation nodes to be within the integration domain and never be outside for any value of the interpolant degree. These rules are devised so that the equidistant nodes depend on the step-sizes, which are defined as functions of the interpolant degree by some small real number greater than or equal to zero. Thus, we overcame the singularity of the kernels on the whole integration domain and obtained uniform interpolation. Moreover, the presented method gives the kernel's values and the kernel's integral values at the singular points, whereas the numerical or exact values do not exist. The solutions to the illustrated four examples are shown in the given tables and figures. The interpolant solutions which we obtain by low-degree interpolants are faster to converge to the numerical or exact ones (if they exist). This confirms the originality of the presented method and its effectiveness in obtaining high-precision results.

Constructing Steklov-type cubature formulas for a finite element in the shape of a bipyramid

This paper reports the construction of cubature formulas for a finite element in the form of a bipyramid, which have a second algebraic order of accuracy. The proposed formulas explicitly take into consideration the parameter of bipyramid deformation, which is important when using irregular grids. The cubature formulas were constructed by applying two schemes for the location of interpolation nodes along the polyhedron axes: symmetrical and asymmetrical. The intervals of change in the elongation (compression) parameter of a bipyramid semi-axis have been determined, within which interpolation nodes of the constructed formulas belong to the integration region, while the weight coefficients are positive, which warrants the stability of calculations based on these cubature formulas. If the deformation parameter of the bipyramid is equal to unity, then both cubature formulas hold for the octahedron and have a third algebraic order of accuracy. The resulting formulas make it possible to find elements of the local stiffness matrix on a finite element in the form of a bipyramid. When calculating with a finite number of digits, a rounding error occurs, which has the same order for each of the two cubature formulas. The intervals of change in the elongation (compression) parameter of the bipyramid semi-axis have been determined, which meet the requirements, which are employed in the ANSYS software package, for deviations in the volume of the bipyramid from the volume of the octahedron. Among the constructed cubature formulas for a bipyramid, the optimal formula in terms of the accuracy of calculations has been chosen, derived from applying a symmetrical scheme of the arrangement of nodes relative to the center of the bipyramid. This formula is invariant in relation to any affinity transformations of the local bipyramid coordinate system. The constructed cubature formulas could be included in libraries of methods for approximate integration used by those software suites that implement the finite element method.

Interpolation by Series of Exponential Functions Whose Exponents Are Condensed in a Certain Direction

For complex interpolation nodes, we study the problem of interpolation by series of exponential functions whose exponents form a set, which is condensed at infinity in a certain direction. We obtain a criterion for all sets of nodes from a special class. For arbitrary sets of nodes, we obtain a necessary condition for the solvability of a more general problem of interpolation by functions that can be represented as Radon integrals of an exponential function over a set of exponents. The paper also contains well-known results on interpolation, which, in particular, allow studying the multipoint holomorphic Vallée Poussin problem for convolution operators.

The Algorithm of Determining an Anti-Collision Manoeuvre Trajectory Based on the Interpolation of Ship's State Vector.

The determination of a ship’s safe trajectory in collision situations at sea is one of the basic functions in autonomous navigation of ships. While planning a collision avoiding manoeuvre in open waters, the navigator has to take into account the ships manoeuvrability and hydrometeorological conditions. To this end, the ship’s state vector is predicted—position coordinates, speed, heading, and other movement parameters—at fixed time intervals for different steering scenarios. One possible way to solve this problem is a method using the interpolation of the ship’s state vector based on the data from measurements conducted during the sea trials of the ship. This article presents the interpolating function within any convex quadrilateral with the nodes being its vertices. The proposed function interpolates the parameters of the ship’s state vector for the specified point of a plane, where the values in the interpolation nodes are data obtained from measurements performed during a series of turning circle tests, conducted for different starting conditions and various rudder settings. The proposed method of interpolation was used in the process of determining the anti-collision manoeuvre trajectory. The mechanism is based on the principles of a modified Dijkstra algorithm, in which the graph takes the form of a regular network of points. The transition between the graph vertices depends on the safe passing level of other objects and the degree of departure from the planned route. The determined shortest path between the starting vertex and the target vertex is the optimal solution for the discrete space of solutions. The algorithm for determining the trajectory of the anti-collision manoeuvre was implemented in autonomous sea-going vessel technology. This article presents the results of laboratory tests and tests conducted under quasi-real conditions using physical ship models. The experiments confirmed the effective operation of the developed algorithm of the determination of the anti-collision manoeuvre trajectory in the technological framework of autonomous ship navigation.

On the stability and accuracy of the empirical interpolation method and gravitational wave surrogates

The combination of the reduced basis and the empirical interpolation method (EIM) approaches have produced outstanding results in many disciplines. In particular, in gravitational wave (GW) science these results range from building non-intrusive surrogate models for GWs to fast parameter estimation adding the use of reduced order quadratures. These surrogates have the salient feature of being essentially indistinguishable from or very close to supercomputer simulations of the Einstein equations, but can be evaluated in the order of a millisecond per multipole mode on a standard laptop. In this article we clarify a common misperception of the EIM as originally introduced and used in practice in GW science. Namely, we prove that the EIM at each iteration chooses the interpolation nodes so as to make the related Vandermonde-type matrix as invertible as possible; not necessarily optimizing its conditioning or accuracy of the interpolant as is sometimes thought. In fact, we introduce two new variations of the EIM, nested as well, which do optimize with respect to conditioning and the Lebesgue constant, respectively, and compare them through numerical experiments with the original EIM using GWs. Our analyses and numerical results suggest a subtle relationship between solving for the original EIM, conditioning, and the Lebesgue constant, in consonance with active research in rigorous approximation theory and related fields.

Solution of the 3D Helmholtz equation using barycentric Lagrange interpolation collocation method

PurposeThis meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.Design/methodology/approachIn this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.FindingsThe numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.Research limitations/implicationsCompared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.Practical implicationsThe Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.Originality/valueThis meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

On Marcinkiewicz–Zygmund Inequalities and $$A_p$$-Weights for $$L$$-Shape Arcs

Let $\Gamma$ be an $L$-shape arc consisting of 2 line segments that meet at an angle different from $\pi$ in the complex $z$-plane $\CC$. This paper is to investigate the behavior of the polynomial interpolants at the Fej\'er points, defined by $\{z_{n,k} = \psi(e^{i(2k\pi + \theta)/(n+1)})\}$ for any choice of $\theta$. In this regard, we recall that for the interval [-1, 1], the Fej\'er points $\{z_{n,k} = \psi^*(e^{i(2k+1)\pi/(n+1)})\}$ agree with the Chebyshev points and that the Chebyshev points are most commonly used as nodes for Lagrange polynomial interpolation. On the other hand, numerical experimentation demonstrates that for a typical open $L$-shape arc $\Gamma$, the Lebesgue constants tend to $\infty$ at the rate of $O((log(n))^2)$, as the polynomial degree $n$ increases, while the $A_{p}$-weight conditions for the Fej\'er points $\{z_{n,k}\}$ do not carry over from [-1, 1] to a truly $L$-shape arc. Further numerical experiments also demonstrate that the least upper bounds of the Marcinkiewicz-Zygmund inequalities for the canonical Lagrange interpolation polynomials at $\{z_{n,k}\}$ seem to grow at the rate of $n^{\beta}$, for some $\beta >0$ that depends on $p >1$.