Hermite-type Interpolation Research Articles

The Representation of D-Invariant Polynomial Subspaces Based on Symmetric Cartesian Tensors

Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D-invariant polynomial subspace Pn in terms of Cartesian tensors, where Pn is a subspace with a maximal total degree equal to n,n≥1. For an arbitrary homogeneous polynomial p(k) of total degree k in Pn, p(k) can be rewritten as the inner products of a kth order symmetric Cartesian tensor and k column vectors of indeterminates. We show that p(k) can be determined by all polynomials of a total degree one in Pn. Namely, if we treat all linear polynomials on the basis of Pn as a column vector, then this vector can be written as a product of a coefficient matrix A(1) and a column vector of indeterminates; our main result shows that the kth order symmetric Cartesian tensor corresponds to p(k) is a product of some so-called relational matrices and A(1).

Wave propagation in meshless numerical wave tank by using Hermite-type RPIM

In this paper, the Hermite-type radial point interpolation method (RPIM) is proposed to develop the meshless numerical wave tank (NWT) for simulating the propagation of fully nonlinear water waves. The Hermite-type interpolation scheme, the radial basis functions (RBFs), and the polynomial functions are used to construct shape functions. These shape functions have better stability and can use any distribution nodes in the computational domain. The strong form of governing equation is transformed to a linear system of algebraic equations by the collocation approach. The material node approach is used to describe the fully nonlinear free surface boundary conditions, and the fourth-order Runge-Kutta (RK4) scheme is applied to update the free surface particles and the velocity potential. The numerical results are compared with other available solutions to confirm the accuracy and efficiency of the proposed numerical procedure. In addition, the accuracy and stability of Hermite-type RPIM compared with RPIM in this study. The numerical results demonstrate that good improvement in the accuracy of the numerical results is obtained after using the Hermite interpolation scheme.

New insight into meshless radial point Hermite interpolation through direct and inverse 2-D reaction–diffusion equation

In this paper, we propose an effective method to solve partial differential equations dependent on time with Neumann boundary condition, by examining its effectivity on direct and inverse reaction–diffusion equation. This method merges the radial point interpolation and the Hermite-type interpolation techniques to provide us suitable tools to impose the boundary condition. This technique is called meshless radial point Hermite interpolation “MRPHI” which utilizes the radial basis function and its derivative to prepare suitable shape functions that are the key for expanding the high-order derivative. This procedure is tested on some types of two-dimensional diffusion equations to show stability through the time in different arbitrary domains.

Extension of Some Generalized Hermite-Type Interpolation Operators to the Triangle with One Curved Side

We construct some generalized Hermite-type interpolation operators for the case of a triangle with one curved side, and we consider their product and Boolean sum operators. We study the interpolation properties of these operators, the orders of accuracy and the remainders of the interpolation formulas. Finally, we give some numerical examples.

Solving acoustic scattering problem by the meshless method based on radial basis function

In this paper, radial point collocation method (RPCM) is introduced to solve the acoustic scattering problem. This is a mathematically simple, easy-to-program and truly meshless method, which has been successfully applied to solve the solid mechanics and convection diffusion problems. However, application of this method to investigate acoustic problems, in particle the acoustic scattering problem is relatively new. The main advantage of this method is its mathematically simple, easy to program, and truly meshless. A Hermite-type interpolation method is employed to improve the solution accuracy while the Neumann boundary conditions exist. In addition, acoustic scattering problem is a typical unbounded domain problem, in order to solve it with RPCM, the domain is truncated to a finite region and an artificial boundary condition (ABC) is imposed. Finally, numerical example is presented to validate the accuracy and effectiveness of RPCM. In the future, the extension of RPCM to more complex and practical problems, especially the three-dimensional situations need to be investigated in more detail.

Fast systematic encoding of multiplicity codes

We present quasi-linear time systematic encoding algorithms for multiplicity codes. The algorithms have their origins in the fast multivariate interpolation and evaluation algorithms of van der Hoeven and Schost (2013), which we generalise to address certain Hermite-type interpolation and evaluation problems. By providing fast encoding algorithms for multiplicity codes, we remove an obstruction on the road to the practical application of the private information retrieval protocol of Augot et al. (2014).

Harmonic interpolation of Hermite type based on Radon projections in two directions

We study Hermite-type interpolation based on Radon projections. We give necessary and sufficient conditions for regularity in the cases in which chords lie in two directions. We also prove that the interpolation polynomials are continuous with respect to the distance from the origin to the chords.

Hermite type radial basis function-based differential quadrature method for higher order equations

In the paper, the radial basis function-based differential quadrature method (RBF-DQM) that uses Hermite type interpolation is developed. The method is an extension of the known RBF-DQM, which is a meshless numerical technique for solving differential equations. According to this technique, derivatives in a governing equation are approximated by a linear weighted sum of the sought function values defined at scattered nodes. To allow the method to be applied in higher order equations, where more than one boundary condition is imposed at an edge, Hermite type interpolation for the radial basis functions is used and appropriate weighting coefficients for differential quadrature method are determined in the paper. As a numerical test the method is used to discretize the governing equation for the free vibration of thin plates with various boundary conditions. Different shaped plates with various boundary conditions are analyzed. The convergence tests carried out in the work confirm usefulness of the method as a truly meshless technique.

An Improved Collocation Meshless Method Based on the Variable Shaped Radial Basis Function for the Solution of the Interior Acoustic Problems

As an efficient tool, radial basis function (RBF) has been widely used for the multivariate approximation, interpolating continuous, and the solution of the particle differential equations. However, ill‐conditioned interpolation matrix may be encountered when the interpolation points are very dense or irregularly arranged. To avert this problem, RBFs with variable shape parameters are introduced, and several new variation strategies are proposed. Comparison with the RBF with constant shape parameters are made, and the results show that the condition number of the interpolation matrix grows much slower with our strategies. As an application, an improved collocation meshless method is formulated by employing the new RBF. In addition, the Hermite‐type interpolation is implemented to handle the Neumann boundary conditions and an additional sine/cosine basis is introduced for the Helmlholtz equation. Then, two interior acoustic problems are solved with the presented method; the results demonstrate the robustness and effectiveness of the method.

Formula omitted] Hermite interpolation with circular precision

Recently some G 1 Hermite-type interpolation methods using a rational parametric cubic were proposed; the methods reproduce a circular arc when the input data come from it. A G 2 Hermite-type interpolation method is now proposed which reproduces a circular arc when the input data come from it.

Geometric Hermite interpolation with circular precision

We present several Hermite-type interpolation methods for rational cubics. In case the input data come from a circular arc, the rational cubic will reproduce it.

Interpolation and identification problems

The paper refines the relationship between the method of orthogonal moments for the identification of polynomial systems and the method of Lagrange-and Hermite-type operator interpolation in Hilbert space. The identification accuracy is estimated by the interpolation method and the minimum number of input signals that guarantee the prescribed accuracy is determined.

Hermite type moving-least-squares approximations

The moving-least-squares approach, first presented by McLain [1], is a method for approximating multivariate functions using scattered data information. The method is using local polynomial approximations, incorporating weight functions of different types. Some weights, with certain singularities, induce C ∞ interpolation approximation in ℝ n . In this work we present a way of generalizing the method to enable Hermite type interpolation, namely, interpolation to derivatives' data as well. The essence of the method is the use of an appropriate metric in the construction of the local polynomial approximations.

Parametric representation of fuzzy numbers and application to fuzzy calculus

We present several models to obtain simple parametric representations of the fuzzy numbers or intervals, based on the use of piecewise monotonic functions of different forms. The representations have the advantage of allowing flexible and easy to control shapes of the fuzzy numbers (we use the standard α - cuts setting, but also the membership functions are obtained immediately) and can be used directly to obtain error-controlled-approximations of the fuzzy calculus in terms of a finite set of parameters. The general setting is the Hermite-type interpolation, where the values and the slopes of the monotonic interpolators are given by appropriate parameters and the overall errors of the fuzzy computations can be controlled within a prefixed tolerance by eventually augmenting the total number of pieces (and of the parameters) by which the results are obtained. The representations are designed to model the lower and the upper extremal values of each α - cut (compact) intervals of the fuzzy numbers and are able to produce almost any possible configuration (differentiable, continuous or with a finite number of discontinuity points) by using parametric monotonic functions of different types. We show applications in the standard fuzzy calculus and we stress the generality and the applicability of the proposed representation to a large class of problems, including the numerical solution of fuzzy differential equations, the fuzzy linear regression and the stochastic extensions of the fuzzy mathematics. The proposed model is called the Lower–Upper representation and we denote the associated fuzzy numbers or intervals as LU-fuzzy.

Development of the meshless Hermite–Cloud method for structural mechanics applications

In this work, a novel true meshless numerical technique is proposed. It is termed the Hermite–Cloud method and is based on the classical reproducing kernel particle method except that a fixed reproducing kernel approximation is used instead. Another distinction is that the point collocation technique is used for the discretization of the governing partial differential equations. In this method, the Hermite theorem is employed for the construction of the interpolation functions. Through the constructed Hermite-type interpolation functions, we are able to generate the expressions of approximate solutions of both the unknown functions and the first-order derivatives, in a direct manner. A set of auxiliary conditions have also been developed so as to construct a complete set of PDEs with mixed Dirichlet and Neumann boundary conditions. Through several structural analysis examples, it is shown that the numerical results at the scattered discrete points generated by the Hermite–Cloud method are distinctly improved, for both the approximate solutions as well as the first-order derivatives.

One sided Hermite interpolation by piecewise different generalized polynomials

Piecewise generalized polynomials of different kinds of order n ( ECT-splines of order n) are constructed from different ECT-systems of order n via connection matrices which are nonsingular and totally positive. A well-known zero count for polynomial splines is extended to ECT-splines. It is used to construct ECT-B-splines and to show under which conditions ECT-splines will solve modified Hermite-type interpolation problems. Also conditions are specified such that piecewise generalized polynomials form rECT-systems and the interpolation problems associated with may be solved recursively.

Radial point interpolation collocation method (RPICM) for partial differential equations

This paper presents a truly meshfree method referred to as radial point interpolation collocation method (RPICM) for solving partial differential equations. This method is different from the existing point interpolation method (PIM) that is based on the Galerkin weak-form. Because it is based on the collocation scheme no background cells are required for numerical integration. Radial basis functions are used in the work to create shape functions. A series of test examples were numerically analysed using the present method, including 1-D and 2-D partial differential equations, in order to test the accuracy and efficiency of the proposed schemes. Several aspects have been numerically investigated, including the choice of shape parameter c with can greatly affect the accuracy of the approximation; the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on Neumann boundary for the solution of PDEs with Neumann boundary conditions. Particular emphasis was on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that good improvement on accuracy can be obtained after using Hermite-type interpolation. The h-convergence rates are also studied for RPICM with different forms of basis functions and different additional terms.

Radial point interpolation collocation method (RPICM) for the solution of nonlinear poisson problems

This paper applies radial point interpolation collocation method (RPICM) for solving nonlinear Poisson equations arising in computational chemistry and physics. Thin plate spline (TPS) Radial basis functions are used in the work. A series of test examples are numerically analysed using the present method, including 2D Liouville equation, Bratu problem and Poisson-Boltzmann equation, in order to test the accuracy and efficiency of the proposed schemes. Several aspects have been numerically investigated, namely the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on Neumann boundary for the solution of PDEs with Neumann boundary conditions. Particular emphasis was on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that a good accuracy can be obtained. The h-convergence rates are also studied for RPICM with coarse and fine discretization models.

A layerwise approach to piezo-electric plates accounting for adhesive flexibility and delaminated regions

Multilayer plates with piezoelectric layers are modeled considering the deformation of the adhesive layers. Special finite elements, that use linear Lagrange and conforming Hermite type interpolation functions, are developed. The results of the formulation are compared with the other results presented in the literature. Since flexible bonding formulation is also suitable for modeling delamination, case studies containing partially delaminated PZT patches are presented. It is shown for a cantilever plate that, while partial delamination between the piezoelectric and non-piezoelectric layers does not affect the maximum deformation significantly, it affects the local stresses near delaminated regions.

Hermite?Cloud: a novel true meshless method

In this paper, a novel true meshless numerical technique – the Hermite–Cloud method, is developed. This method uses the Hermite interpolation theorem for the construction of the interpolation functions, and the point collocation technique for discretization of the partial differential equations. This technique is based on the classical reproducing kernel particle method except that a fixed reproducing kernel approximation is employed instead. As a true meshless technique, the present method constructs the Hermite-type interpolation functions to directly compute the approximate solutions of both the unknown functions and the first-order derivatives. The necessary auxiliary conditions are also constructed to generate a complete set of partial differential equations with mixed Dirichlet and Neumann boundary conditions. The point collocation technique is then used for discretization of the governing partial differential equations. Numerical results show that the computational accuracy of the Hermite–Cloud method at scattered discrete points in the domain is much refined not only for approximate solutions, but also for the first-order derivative of these solutions.