Interpolation Nodes Research Articles

An enriched finite element method with interpolation cover functions for acoustic analysis in high frequencies

At present, the conventional finite element method (FEM) still has some defects for solving acoustic problems governed by the Helmholtz equation. Owing to the numerical dispersion error, the standard FEM always fails to provide sufficiently satisfactory results in high frequency range. To cure this drawback, an enriched finite element method (E-FEM) with interpolation cover functions is presented to deal with 2D and 3D acoustic problems in this work. In the present E-FEM, the nodal interpolation functions from the standard FEM are enriched by the additional interpolation cover functions over patches of elements to improve the performance (such as convergence rate and computation accuracy) of the standard FEM and the original mesh (such as the conventional triangular and tetrahedral mesh) in the standard FEM can still be used. In addition, the high gradient components of the considered field variable can be captured and the inter-element gradient jumps can be smoothed out by the used interpolation cover functions, so the present E-FEM is able to provide a smoother and more accurate acoustic field for acoustic analyses. Several numerical examples demonstrate that higher computational efficiency and accuracy can be obtained for analyzing acoustic problems compared to the conventional FEM. Moreover, the E-FEM can effectively suppress the dispersion error at high frequencies, so it can make an effective prediction for practical engineering problems.

Technology for restoring functional dependencies to determine reliability parameters

The problem of determining the properties of the object by analyzing the numerical and qualitative characteristics of a discrete sample is considered. A method has been developed to determine the probability of trouble-free operation of electronic systems for the case if the interpolation fields are different between several interpolation nodes. A method has been developed to determine the probability of trouble-free operation if the interpolation polynomial is the same for the entire interpolation domain. It is shown that local interpolation methods give more accurate results, in contrast to global interpolation methods. It is shown that in the case of global interpolation it is possible to determine the value of the function outside the given values by extrapolation methods, which makes it possible to predict the probability of failure. It is shown that the use of approximation methods to determine the probability of trouble-free operation reduces the error of the second kind. A method for analyzing the qualitative characteristics of functional dependences has been developed, which allows us to choose the optimal interpolation polynomial. With sufficient statistics, using the criteria of consent, it is possible to build mathematical models for the analysis of failure statistics of electronic equipment. Provided that the volume of statistics is not large, such statistics may not be sufficient and the application of consent criteria will lead to unsatisfactory results. Another approach is to use an approximation method that is applied to statistical material that was collected during testing or controlled operation. In this regard, it is extremely important to develop a method for determining the reliability of electronic systems in case of insufficiency of the collected statistics of failures of electronic equipment.

Computer Simulation System of Nonlinear Thermal Conductivity

The article discusses the computer simulation system of nonlinear processes that described by the one-dimensional nonstationary heat equation with power-law nonlinearity. The solution of the nonlinear heat equation is realized by meshless scheme, which is based on combination of the collocation method and radial basis functions. Radial basis functions are used to approximate the solution of a partial differential equation, while the collocation method is used to determine unknown coefficients. The computer simulation system allows is used the following radial basis functions: Gaussian, multiquadric, inverse quadratic, and inverse multiquadric. The computer simulation system allows setting the initial and boundary conditions of the boundary-value problem. In the computer simulation system, it is possible to set such parameters of the solution as the exponent in the nonlinear heat equation, the coefficient of thermal conductivity, the density, the specific heat at constant pressure, the size of the domain of the boundary-value problem, the distance between interpolation nodes, the time interval of the nonstationary boundary-value problem, the time step, and the shape parameter of the radial basis function. The visualization of an approximate solution of the one-dimensional nonstationary heat equation with power-law nonlinearity is realized in the form of the three-dimensional surface in the computer simulation system. The computer simulation system allows visualization of the solution at chosen time steps as three-dimensional plots. The results of numerical calculations are demonstrated by the benchmark problem that simulates the stopping of a heat wave front in a heat equation with power-law nonlinearity.

A Lagrange surrogate-based approach for uncertain nonlinear oscillators

Nonlinear oscillators are widely used in engineering equipment. Due to manufacturing errors, material defects, wear and external environments, some uncertainties inevitably exist in geometric, stiffness, and other parameters of nonlinear oscillators. Those parametric uncertainties may significantly impair response predictions of nonlinear oscillators. The present investigation proposes a Lagrange surrogate-based non-probabilistic method to evaluate reasonably the dynamic characteristics of uncertain nonlinear oscillators in the case of insufficient data information. The Lagrange interpolation polynomial is introduced to construct a Lagrange surrogate model for uncertain but bounded parameters. A matrix form of the least square method is applied to improve the efficiency of Lagrange polynomial coefficient calculation. Thus the uncertain problem is transformed into a nonlinear global optimization problem to locate maximum and minimum values of the surrogate model. An improved global search optimization strategy is designed by selecting initial values from the sorted interpolation nodes. The effectiveness and the accuracy of the proposed algorithm are examined by two numerical examples with nonlinearities, i.e. a double pendulum and a six degree-of-freedom chain type structure. Results demonstrate that the Lagrange surrogate-based approach could be a reasonable alternative to response bound estimation of uncertain nonlinear oscillators in the case without sufficient data information.

Interpolation sequences and nonspanning systems of exponentials on curves

Interpolation sequences of the form are investigated, and also the problem of when the system of exponentials is nonspanning on the family of arbitrary rectifiable curves in the uniform norm. In terms of the interpolation nodes (or equivalently, the exponents of the system of exponentials) a criterion for the interpolation problem to be solvable is established and the strong nonspanning property of is proved. This significantly improves some known results, in particular, results due to Korevaar, Dixon and Berndtsson. Bibliography: 23 titles.

The Synthesis of Dwell Mechanisms on the Basis of Straight-Line Linkages with Fivefold Interpolation Nodes

The Synthesis of Dwell Mechanisms on the Basis of Straight-Line Linkages with Fivefold Interpolation Nodes

Creating a bridge between cardinal B[formula omitted]-spline fundamental functions for interpolation and subdivision

This paper presents innovative contributions to the fields of cardinal spline interpolation and subdivision. In particular, it unifies cardinal Br-spline fundamental functions for interpolation that are made of r=ML+1 (L∈N∪{0}) distinct pieces between each pair of interpolation nodes and are featured by the properties of C2M−2 smoothness, approximation order 2M and support width 2M(r+1)r, with the basic limit functions of a special class of non-stationary subdivision schemes of arity M.After introducing a general result, we focus our attention on the subclass of fourth-order accurate, C2 smooth Br-splines with maximum width of the compact support 6. The binary subdivision scheme yielding these fundamental functions outperforms the existing interpolatory schemes and seems to be the most adequate starting point to obtain compactly supported fundamental (spline) functions for local interpolation over quadrilateral and triangular meshes.

Non-Polynomial Interpolation of Functions with Large Gradients and Its Application

Interpolation of a function of one variable with large gradients in the boundary layer region is studied. The problem is that the use of classical polynomial interpolation formulas on a uniform mesh to functions with large gradients can lead to errors of the order of $$O(1)$$ , despite a small mesh size. An interpolation formula based on fitting to the component that defines the boundary-layer growth of the function is investigated. An error estimate, which depends on the number of interpolation nodes and is uniform over the boundary layer component and its derivatives, is obtained. It is shown how the interpolation formula derived can be used to construct formulas for numerical differentiation and integration and in the two-dimensional case. The corresponding error estimates are obtained.

Inverse Identification of Temperature-Dependent Thermal Conductivity for Charring Ablators

Estimation of thermal conductivity of charring composite is a key issue for both design and optimization of an ablative thermal protection system. This paper presents a method for predicting temperature-dependent thermal conductivity of charring ablator using genetic algorithm and cubic spline interpolation, and a special ablative material composed of phenolic resin, glass fibers and quartz fibers is concerned. In this study, by considering the effects of pyrolysis, carbon–silica reaction, and the convective heat transfer between pyrolysis gas and charred layer, the ablative heat transfer model is developed to simulate the ablation process first. Then based on the measured temperature profiles and built model, the temperature-dependent thermal conductivity without prior information on the functional form is predicted which is parameterized by cubic spline. The predicted thermal conductivity curve is continuous and smooth everywhere, and the changes of the curve can reveal the ablation characteristic of the concerned material precisely. Consequently, the calculated temperature profiles using the predicted results show good agreement with experimental data. Also, the reliability of the prediction is discussed through analyzing the iterative process and the number of interpolation nodes. It is concluded that the improvement is limited through adding nodes when the thermal conductivity is already well defined by existing nodes.

An algorithm for best rational approximation based on barycentric rational interpolation

We present a novel algorithm for computing best uniform rational approximations to real scalar functions in the setting of zero defect. The method, dubbed BRASIL (best rational approximation by successive interval length adjustment), is based on the observation that the best rational approximation r to a function f must interpolate f at a certain number of interpolation nodes (xj). Furthermore, the sequence of local maximum errors per interval (xj− 1,xj) must equioscillate. The proposed algorithm iteratively rescales the lengths of the intervals with the goal of equilibrating the local errors. The required rational interpolants are computed in a stable way using the barycentric rational formula. The BRASIL algorithm may be viewed as a fixed-point iteration for the interpolation nodes and converges linearly. We demonstrate that a suitably designed rescaled and restarted Anderson acceleration (RAA) method significantly improves its convergence rate. The new algorithm exhibits excellent numerical stability and computes best rational approximations of high degree to many functions in a few seconds, using only standard IEEE double-precision arithmetic. A free and open-source implementation in Python is provided. We validate the algorithm by comparing to results from the literature. We also demonstrate that it converges quickly in some situations where the current state-of-the-art method, the minimax function from the Chebfun package which implements a barycentric variant of the Remez algorithm, fails.

Interpolation Method for Solving Weakly Singular Integral Equations of the Second Kind

We establish a new straightforward interpolation method for solving linear Volterra integral ‎‎equations with weakly singular kernels. The proposed method is fundamentally different from all other published methods for solving this type of equations. We have modified some vector-matrix barycentric Lagrange interpolation formulas to be convenient for interpolating the kernel twice concerning the two variables of the kernel and introducing new ideas for selecting interpolation nodes that ensure isolation of the singularity of the kernel. We create two rules for selecting the distribution nodes of ‎‎the two kernel variables that do not allow the ‎‎denominator of the kernel to contain an imaginary value. We interpolate the unknown and data functions ‎‎into the corresponding interpolant polynomial; each of the same degree via three matrices, one of ‎‎which is a monomial. By applying the presented method based on the two created rules, we transformed the ‎kernel into a double ‎interpolant polynomial with a degree equal to that of the unknown ‎function via five matrices, two of ‎which are monomials. We substitute the interpolate unknown ‎function twice; on the left side and on the ‎right side of the integral equation to get an ‎algebraic linear system without applying the ‎collocation method. The solution of this system yields ‎the unknown coefficients matrix that is necessary to find the interpolant solution. We ‎solve three ‎different examples for different values of the upper integration variable. The obtained ‎results as ‎shown in tables and figures prove that the obtained interpolate solutions are extraordinarily faster ‎‎to converge to the exact ones using interpolants of lowest degrees and give better results than those obtained by ‎other ‎methods. This confirms the originality and the potential of the presented method.‎

МОДЕЛІ КОНОЇДІВ ТА МЕТОД ПЕРЕРІЗІВ

Стаття присвячена дослідженню нових специфічних властивостей коноїдів – лінійчатих поверхонь Каталана (1843), які застосовуються в сучасному методі скінченних елементів (МСЕ). Коноїди з’явилися в МСЕ несподівано, коли у 1968 р. Ергатудіс, Айронс і Зенкевич сконструювали підбором перші серендипові скінченні елементи (СЕ): білінійний Q4, біквадратичний Q8 і бікубічний Q12. Коноїди застосовуються у якості базисних функцій (функцій впливу) у всіх (без винятку) моделях стандартних серендипових СЕ, незважаючи на неприродні спектри еквівалентних вузлових навантажень (фізична неадекватність). Саме коноїди, які асоціюються з проміжними вузлами інтерполяції, спричинили появу від’ємних навантажень у кутових вузлах СЕ. Найавторитетніший фахівець проф. О. Зінкевич радив змиритися з цим недоліком. Позбутися фізичної неадекватності в кутових вузлах можна, якщо відмовитись від коноїдів в проміжних вузлах. Але такі серендипові СЕ вже належать до альтернативних моделей. Варто зауважити, що коноїди використовують не тільки в МСЕ. Технологічні та естетичні якості коноїдів давно приваблюють архітекторів і будівельників. Потрібно знайти такі коноїди, які забезпечують фізичну адекватність моделей. Треба звернути увагу на тригонометричні коноїди, які недостатньо досліджені. Попередні дослідження свідчать, що тіло, яке утворюється коноїдом і носієм, може бути сімпсоновим. Поповнення модельного ряду сімпсонових тіл – цікава самостійна задача. Але на коноїдах правило трьох перерізів (кубатура Сімпсона) не завжди дає правильну відповідь. Головне – правильно обчислити площу середнього перерізу правильно вибраної трійки перерізів. Ця задача має самостійне значення. Підібрані приклади коноїдів дають можливість порівняти прості і наочні підходи з процедурою Монте-Карло. Когнітивно-графічний аналіз – найкраща інформаційна технологія, особливо у поєднанні з комп’ютерними експериментами.

Interpolation Problems of A. F. Leontiev Type

In this paper, we discuss free interpolation in the spaces of entire and analytic finite-order functions in the upper half-plane. A review of problems and basic results related to such problems is given. Solvability criteria are formulated in terms of canonical products of interpolation nodes and in terms of the measure determined by these nodes.

Penny-shaped crack simulation with a single high order smooth boundary element

Based on Lagrange interpolation polynomials, the penny-shaped cracks placed on but not limited to flat surface are simulated with a single high order smooth boundary element in the present work. By taking advantage of geometrical features of circular shape such as the symmetry and periodicity, the smoothness within the element is realized by repeated use of real nodes for interpolation in both the radial and circumferential directions of the element using high order shape functions, so that the end node/line effects existing in conventional low order elements have been removed. In the application of the boundary element method (BEM) for crack problems, the hyper-singular integrals are treated with the aid of the technique of shape function manipulations. The near hyper-singular integrals are treated with the tangential distance transformations together with the subdivision techniques. The crack-related parameters including the crack opening displacements (COD), the stresses ahead of crack tip, the stress intensity factors (SIF) and the T-stresses are computed under various loads and compared with the analytical solutions in the numerical examples, showing the accuracy, efficiency and effectiveness of the proposed high order smooth crack element.

A Meshless Collocation Method with Barycentric Lagrange Interpolation for Solving the Helmholtz Equation

In this paper, Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation. First of all, the interpolation basis function is applied to treat the spatial variables and their partial derivatives, and the collocation method for solving the second order differential equations is established. Secondly, the differential equations on a given test node. Finally, based on three kinds of test nodes, numerical experiments show that the present scheme can not only calculate the high wave numbers problems, but also calculate the variable wave numbers problems. In addition, the algorithm has the advantages of high calculation accuracy, good numerical stability and less time consuming.

Multiple Interpolation by the Functions of Finite Order in the Half-Plane

The aim of this paper is to study the multiple interpolation problem in the spaces of analytical functions of finite order $$\rho>1$$ in the half-plane. The necessary and sufficient conditions for solvability of interpolation problem are obtained. These conditions are obtained in terms of the Nevanlinna product of interpolation nodes. The solution of the interpolation problem is constructed in the form of the Jones interpolation series, which is a generalization of the Lagrange interpolation series.

A novel frequency-labeled adaptive sparse grid collocation method for uncertainty quantification of the frequency response of general viscoelastic damping structures

In this paper, a frequency-labeled adaptive sparse grid collocation (FL-ASGC) method is proposed to study the uncertainty quantification (UQ) of the frequency response of general viscoelastic damping structures. The advantage of the proposed method is that, on the one hand, it has a good approximation ability for both smooth and non-smooth responses so that it can robustly predict the variability of frequency responses under various damping levels. On the other hand, by evaluating the contribution of each interpolation node to the response at all frequencies, it can adaptively select samples and their accompanying analysis frequency domains, thereby minimizing the computational costs of probabilistic surrogate modeling. To obtain the deterministic solution of each sample, the layer-wise finite element method combined with the direct frequency response (DFR) method is employed. Two numerical examples with different damping materials are provided to test the proposed method. Uncertainties that obey normal distribution in material and geometry are considered. The results show the proposed method can accurately, efficiently, and robustly predict the variability of the frequency response of general viscoelastic damping structures.

Meshfree simulations of acoustic problems by a radial point interpolation method

The classical finite element approach cannot guarantee satisfactory accuracy for acoustic problems at large wavenumbers on account of the numerical pollution error effect. This negative effect stems from the fact that the approximate wavenumbers are usually in conflict with the real wavenumbers in many numerical methods. To suppress this effect, a radial point interpolation meshless technique with a modified scheme for selecting interpolation nodes is employed in this paper. One-dimensional dispersion analysis shows that this modified scheme can effectively reduce numerical errors compared with the original scheme. The results of several numerical examples have manifested that the present method can generate more accurate and reliable solutions than the standard finite element approach and the original radial point interpolation method in the acoustic analyses.

On one interpolating rational process of Fejer – Hermite

In this paper, a new approach to the definition of the interpolating rational process of Fejer – Hermite with first-type Chebyshev – Markov nodes on a segment is studied and some of its approximating properties are described. In the introduction a brief analysis of the results on the topic of the research is carried out. Herein, the methods of the construction of interpolating processes, in particular, Fejer – Hermite processes, in the polynomial and rational approximation are analysed. A new method to determine the interpolating rational Fejer – Hermite process is proposed. One of the main results of this paper is the proof of the uniform convergence of this process for an arbitrary function, which is continuous on the segment, under some restrictions for the poles of approximating functions. This result is preceded by some auxiliary statements describing the properties of special rational functions. The classic methods of mathematical analysis, approximation theory, and theory of functions of a complex variable are used to prove the results of the work. Moreover, we present the numerical analysis of the effectiveness of the application of the constructed interpolating Fejer – Hermite process for the approximation of a continuous function with singularities. The choice of parameters, on which the nodes of interpolation depend, is made in several standard ways. The obtained results can be applied to further study the approximating properties of interpolating processes.

On the numerical computation of bivariate Lagrange polynomials

In this paper we propose a simple procedure for numerically computing the Lagrange interpolation polynomial on a unisolvent set of points in the plane. We suggest the use of the canonical polynomial basis centered at the barycenter of the set of points and the PA=LU decomposition for solving the associated Vandermonde system to compute the coefficients of the Taylor polynomial. We show that the 1-norm condition number of the Vandermonde matrix is an upper bound for the Lebesgue constant of the interpolation node set in the unit disk. Therefore, the analysis of the condition number can be useful to select the unisolvent set of nodes in a set of scattered nodes. Numerical experiments show the efficiency and accuracy of the proposed method.