Local Hermite Research Articles

Maximally smooth cubic spline quasi-interpolants on arbitrary triangulations

We investigate the construction of C2 cubic spline quasi-interpolants on a given arbitrary triangulation T to approximate a sufficiently smooth function f. The proposed quasi-interpolants are locally represented in terms of a simplex spline basis defined on the cubic Wang–Shi refinement of the triangulation. This basis behaves like a B-spline basis within each triangle of T and like a Bernstein basis for imposing smoothness across the edges of T. Any element of the cubic Wang–Shi spline space can be uniquely identified by considering a local Hermite interpolation problem on every triangle of T. Different C2 cubic spline quasi-interpolants are then obtained by feeding different sets of Hermite data to this Hermite interpolation problem, possibly reconstructed via local polynomial approximation. All the proposed quasi-interpolants reproduce cubic polynomials and their performance is illustrated with various numerical examples.

A novel local Hermite radial basis function‐based differential quadrature method for solving two‐dimensional variable‐order time fractional advection–diffusion equation with Neumann boundary conditions

AbstractA novel Hermite radial basis function‐based differential quadrature (H‐RBF‐DQ) method is presented in this paper based on 2D variable order time fractional advection–diffusion equations with Neumann boundary conditions. The proposed method is designed to treat accurately for derivative boundary conditions, which considerably improve the approximation results and extend the range of applicability for the method of RBF‐DQ. The advantage of the present method is that the Hermite interpolation coefficients are only dependent of the point distribution yielding a substantially better imposition of boundary conditions, even for time evolution. The proposed algorithm is thoroughly validated and is demonstrated to handle the fractional calculus problems with both Dirichlet and Neumann boundaries very well.

RBF collocation and hybrid-LHI methods for Stokes systems and its application to controllability problems

The purpose of this article is to introduce radial basis function (RBF) methods for solving both direct Stokes equations and controllability problems for the Stokes system with few internal scalar controls. In both cases, Dirichlet or Navier-slip boundary conditions are considered. We introduce two radial basis function solvers, one global and the other local, to solve Stokes equations. These methods are used to discretize the primal and adjoint systems related to the controllability problems. Both techniques are based on divergence-free global RBFs. A global colocation technique based on Div-free inverse multi-quadrics is formulated and analyzed. A generalization of scalar hybrid kernels to a vector divergence-free hybrid RBFs setting is defined. Based on these kernels, the local Hermite interpolation (LHI) method in vector form is introduced. Due to the properties of the hybrid kernel, we show that due to the properties of the hybrid kernel this local method, can reduce up to double precision, the value of the condition number of the local Gram matrices. Simultaneously, it is proved that the real components of the eigenvalues corresponding to the global LHI matrix are negative and that consequently backward difference formulas are stable for time integration. The conjugate gradient algorithm is adapted to the radial basis function setting to solve the controllability problems. Several benchmarks problems in two dimensions with a non-convex domain (a star shape) are numerically solved by these RBFs methods to display and compare their feasibility. The solutions to these problems are also implemented by finite element techniques to study their relative performance.

Argyris type quasi-interpolation of optimal approximation order

The paper investigates a quasi-interpolation framework for the construction of operators that provide approximations for bivariate functions from the space of Argyris macro-element splines on a general triangulation. An operator of such type is determined by local approximation operators that produce quintic polynomials and are naturally associated with vertices and triangles of the triangulation. The quasi-interpolant resembles the properties of the standard Argyris interpolant, especially in the determination of data corresponding to the vertices of the triangulation. The main difference is in the definition of the cross-boundary derivatives, where a novel approach with degree raising is used to eliminate the need for additional interpolation of derivatives at the edges. Despite reduction in degrees of freedom, a suitable choice of local approximation operators ensures that the quasi-interpolation operator has quintic precision and is of optimal approximation order. To demonstrate the usability of the presented approach, three approximation methods are derived based on local Hermite and Lagrange interpolation and least square fitting.

Radial basis function methods for optimal control of the convection–diffusion equation: A numerical study

In this paper, we perform a numerical study for the solution of optimal constrained optimization problems for linear convection–diffusion PDEs by local and global radial basis function techniques. To the best of our knowledge, these control problems have not been treated in the literature by RBFs methods. It is well-known that the algebraic system of RBFs methods presents a larger condition number and a higher numerical complexity as the number of nodes (or shape parameter), increases. In this work, and in the context of optimal constrained optimization problems, we explore a possible answer to both problems. Specifically, we introduce a local RBF method (denoted as LAM-DQ), based on the combination of an asymmetric RBFs local method (LAM), inspired in local Hermite interpolation (LHI), combined with the differential quadrature method (DQ). We also propose a preconditioning technique that in combination with extended arithmetic precision let us treat the ill-conditioning problem. We numerically prove that as the number of nodes increases, then for errors of the same order, the condition number remains tractable, in quad-precision, and the numerical complexity of the local method remains bounded.

A practical and effective regularized polynomial smoothing (RPS) method for high-gradient strain field measurement in digital image correlation

Digital image correlation (DIC) is an effective tool for studying the deformation of tested materials. However, high-gradient strain field (HSF) measurement remains a challenging issue in DIC, because multifold factors have a significant effect on the first correlation calculation step of the original displacement and the exact strain reconstruction in the second post-processing step. In this study, we focus on the second step. The formerly proposed local Hermite has been proven as an accurate and robust method, yet substantial programming work is involved in constructing a Hermite element. To solve this issue, a regularized polynomial smoothing (RPS) method is proposed, which replaces the Hermite function of the polynomial. The RPS applies the same kernel function as the point-wise least-squares (PLS) method for easy implementation. Tikhonov regularization and the generalized cross-validation (GCV) function are also used to decrease unavoidable noise and weaken the dependence on window size. The experiments demonstrate that, in the matched/overmatched case, Tikhonov regularization in RPS can greatly decrease the degree of morbidity caused by ill-posed problems in the PLS method, thereby ensuring smoothness and stability of the solution. Moreover, the influences of window size and polynomial order of both the RPS and PLS are also closely studied. In conclusion, the proposed RPS provides a practical, robust and accurate method for HSF measurement.

Low-temperature tunneling of CH3 quantum rotor in van der Waals solids

Motional quantum effects of tunneling methyl radical isolated in solid gases as they appear on experimental electron paramagnetic resonance (EPR) spectra are examined. Obtained analytical expressions of the tunneling frequency for methyl rotor/torsional-oscillator utilizing localized Hermite polynomials are compared to full numerical computations and tested against experimental EPR lineshape simulations. In particular, the X-band of methyl radical was displaying partial anisotropy averaging even at lowest temperatures. EPR lineshape simulations involving rotational dynamics were applied for the accurate determination of the potential barrier and the tunneling frequency. Tunneling frequency, as the splitting between the A and E torsional levels by the presence of a periodic C3 model potential with periodic boundary conditions, was computed and related to the EPR-lineshape alteration. The corresponding C2 rotary tunneling about the in-plane axes of methyl was also studied while both the C2 and C3 rotations were compared with the rotation of deuteriated methyl radical.

Three recipes for quasi-interpolation with cubic Powell–Sabin splines

We investigate the construction of bivariate quasi-interpolation methods based on C1 cubic Powell–Sabin B-spline representations. Rather than using a large set of functional data to specify all the parameters in such representations, we study how to reduce them by imposing different super-smoothness properties while retaining cubic precision. This results in three recipes, which are completely general in the sense that they can be implemented with any local cubic polynomial approximation scheme (or a mixture of them). More precisely, they embed C2 super-smoothness at the vertices and across the edges, C2 super-smoothness inside the macro-triangles, and smoothness of Clough–Tocher type, respectively. To demonstrate their usefulness, we derive four specific methods based on local Hermite and Lagrange interpolation. We conclude with a selection of numerical experiments.

Local Hermite (LH) Method: An accurate and robust smooth technique for high-gradient strain reconstruction in digital image correlation

High-gradient strain field (HSF) measurement is attracting more and more attention from engineering and scientific fields, where digital image correlation (DIC) is widely adopted to calculate the dramatically changing displacement fields on stressed samples. Accurate reconstruction of reliable strain field from noisy displacement field is one of the key issues, since strain calculation is highly sensitive to displacement noise. The formally proposed improved Hermite finite element smoothing method (IHFESM) and fast Hermite element method (FHEM) are effective global smoothing techniques based on Tikhonov regularization. However, they involve complex meshing and global regularization matrix assembly processes. In this work, we present a local version of FHEM, called local Hermite (LH) method, in which one Hermite element is taken as a smoothing window centered on one displacement data, and the whole field is handled point-wisely. Two types of LH method are obtained by employing Hermite element with two types of continuities, i.e. C1 and C2. The smoothed displacement and strain are consequently computed from the regularized local surface as done in FHEM. Two kinds of simulated noisy displacement fields and one high-gradient displacement field obtained from shape memory alloy (SMA) tensile test are used to compare the performance of the proposed LH methods with the widely used point-wise least-squares (PLS) method. Experiments show that LH method with C2 element is nearly insensitive to window size, which is a quite useful property for practical engineering test without much prior knowledge. More importantly, it generally produces lower error than PLS in resolving local HSF. Thus, LH is a robust and accurate method which is highly recommended for practical HSF measurement.

C1 and C2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames

The problem of constructing a rational adapted frame (f1(ξ),f2(ξ),f3(ξ)) that interpolates a discrete set of orientations at specified nodes along a given spatial Pythagorean-hodograph (PH) curve r(ξ) is addressed. PH curves are the only polynomial space curves that admit rational adapted frames, and the Euler–Rodrigues frame (ERF) is a fundamental instance of such frames. The ERF can be transformed into other rational adapted frame by applying a rationally-parametrized rotation to the normal-plane vectors. When orientation and angular velocity data at curve end points are given, a Hermite frame interpolant can be constructed using a complex quadratic polynomial that parametrizes the normal-plane rotation, by an extension of the method recently introduced to construct a rational minimal twist frame (MTF). To construct a rational adapted spline frame, a representation that resolves potential ambiguities in the orientation data is introduced. Based on this representation, a C1 rational adapted spline frame is constructed through local Hermite interpolation on each segment, using angular velocities estimated from a cubic spline that interpolates the frame phase angle relative to the ERF. To construct a C2 rational adapted spline frame, which ensures continuity of the angular acceleration, a complex-valued cubic spline is used to directly interpolate the complex exponentials of the phase angles at the nodal points.

A new approach to estimating a numerical solution in the error embedded correction framework

On the basis of the error correction method developed recently, an algorithm, so-called error embedded error correction method, is proposed for initial value problems. Two deferred equations are used to approximate the solution and the error, respectively, at each integration step. For the solution, the deferred equation, which is based on a modified Euler’s polygon including the information of both the solution and its estimated error at the previous integration step, is solved with the classical fourth-order Runge–Kutta method. For the error, the deferred equation, which is based on a local Hermite cubic polynomial with three pieces of information—the solution, its estimated error at the previous step, and the constructed solution—is solved by the seventh-order Runge–Kutta–Fehlberg method. The constructed algorithm controls the error and possesses a good behavior of error bound in a long time simulation. Numerical experiments are presented to validate the proposed algorithm.

Adaptive mesh refinement method for optimal control based on Hermite-Legendre-Gauss-Lobatto direct transcription

Direct transcription has been employed to transcribe the optimal control problem into a nonlinear programming problem. This paper presents a trajectory optimization method based on a combination of the direct transcription and mesh refinement algorithm. Hermite-Simpson method has the advantage of reasonable accuracy with highly sparse Hessian matrix and constraint Jacobians, and the pseudospectral method provides spectral accuracy for optimal control problems. The optimal control problem is discretized at a series of Legendre-Gauss-Lobatto points, then the trajectory states are approximated by using local Hermite interpolating polynomials. Thus, the method produces significantly smaller mesh size with a higher accuracy tolerance solution. The derived relative error estimation is then used to trade the number of mesh polynomials degree within each mesh interval with the number of mesh intervals. As a result, the suggested method can produce more small mesh size, requires less computation solution for the same optimal control problem. The simulation experiment results show that the suggested method has many advantages.

Efficient and Adaptive Methods for Computing Accurate Potential Surfaces for Quantum Nuclear Effects: Applications to Hydrogen-Transfer Reactions.

We present two sampling measures to gauge critical regions of potential energy surfaces. These sampling measures employ (a) the instantaneous quantum wavepacket density, an approximation to the (b) potential surface, its (c) gradients, and (d) a Shannon information theory based expression that estimates the local entropy associated with the quantum wavepacket. These four criteria together enable a directed sampling of potential surfaces that appears to correctly describe the local oscillation frequencies, or the local Nyquist frequency, of a potential surface. The sampling functions are then utilized to derive a tessellation scheme that discretizes the multidimensional space to enable efficient sampling of potential surfaces. The sampled potential surface is then combined with four different interpolation procedures, namely, (a) local Hermite curve interpolation, (b) low-pass filtered Lagrange interpolation, (c) the monomial symmetrization approximation (MSA) developed by Bowman and co-workers, and (d) a modified Shepard algorithm. The sampling procedure and the fitting schemes are used to compute (a) potential surfaces in highly anharmonic hydrogen-bonded systems and (b) study hydrogen-transfer reactions in biogenic volatile organic compounds (isoprene) where the transferring hydrogen atom is found to demonstrate critical quantum nuclear effects. In the case of isoprene, the algorithm discussed here is used to derive multidimensional potential surfaces along a hydrogen-transfer reaction path to gauge the effect of quantum-nuclear degrees of freedom on the hydrogen-transfer process. Based on the decreased computational effort, facilitated by the optimal sampling of the potential surfaces through the use of sampling functions discussed here, and the accuracy of the associated potential surfaces, we believe the method will find great utility in the study of quantum nuclear dynamics problems, of which application to hydrogen-transfer reactions and hydrogen-bonded systems is demonstrated here.

Estimating the temperature evolution of foodstuffs during freezing with a 3D meshless numerical method

Freezing processes are characterised by sharp changes in specific heat capacity and thermal conductivity for temperatures close to the freezing point. This leads to strong nonlinearities in the governing PDE that may be difficult to resolve using traditional numerical methods. In this work we present a meshless numerical method, based on a local Hermite radial basis function collocation approach in finite differencing mode, to allow the solution of freezing problems. By introducing a Kirchhoff transformation and solving the governing equations in Kirchhoff space, the strength of nonlinearity is reduced while preserving the structure of the heat equation. In combination with the high-resolution meshless numerical method, this allows efficient and stable numerical solutions to be obtained for freezing problems using 3D unstructured datasets. We demonstrate the proposed numerical formulation for the freezing of foodstuffs. By approximating the shape of the thermal conductivity and heat capacity curves in a piecewise linear fashion the temperature-dependent material curves may be described using eight independent parameters. We consider the optimisation of these parameters to match experimental data for the freezing of a hemispherical sample of mashed potato by using a simple manual procedure that requires a minimal number of simulations to be performed. Working in this way, a good approximation is obtained to the temperature profile throughout the sample without introducing instability into the numerical results.

Design of Robust Pulses to Insufficient Synchronization for OFDM/OQAM Systems in Doubly Dispersive Channels

This paper presents a pulse shaping method robust to insufficient synchronization in orthogonal frequency division multiplexing with offset quadrature amplitude modulation (OFDM/OQAM) systems over doubly dispersive (DD) channels. The proposed pulse is designed as a linear combination of several well localized Hermite functions. The coefficients optimization problem is modeled as a nonconvex constrained fractional programming problem based on the signal-to-interference ratio (SIR) maximization criterion. An efficient iterative algorithm is applied to simplify the problem to a series of quadratically constrained quadratic program (QCQP) problems which can be solved by semidefinite relaxation (SDR) method. Simulation results show that the proposed pulse is superior to traditional pulses with respect to SIR performance over DD channels in the presence of carrier frequency offset (CFO) and timing offset (TO).

Finite Element Analysis of Structures Using C1-Continuous Cubic B-Splines or Equivalent Hermite Elements

We compare contemporary practices of global approximation using cubic B-splines in conjunction with double multiplicity of inner knots (C1-continuous) with older ideas of utilizing local Hermite interpolation of third degree. The study is conducted within the context of the Galerkin-Ritz formulation, which forms the background of the finite element structural analysis. Numerical results, concerning static and eigenvalue analysis of rectangular elastic structures in plane stress conditions, show that both interpolations lead to identical results, a finding that supports the view that they are mathematically equivalent.

Efficient modeling of photonic crystals with local Hermite polynomials

Developing compact algorithms for accurate electrodynamic calculations with minimal computational cost is an active area of research given the increasing complexity in the design of electromagnetic composite structures such as photonic crystals, metamaterials, optical interconnects, and on-chip routing. We show that electric and magnetic (EM) fields can be calculated using scalar Hermite interpolation polynomials as the numerical basis functions without having to invoke edge-based vector finite elements to suppress spurious solutions or to satisfy boundary conditions. This approach offers several fundamental advantages as evidenced through band structure solutions for periodic systems and through waveguide analysis. Compared with reciprocal space (plane wave expansion) methods for periodic systems, advantages are shown in computational costs, the ability to capture spatial complexity in the dielectric distributions, the demonstration of numerical convergence with scaling, and variational eigenfunctions free of numerical artifacts that arise from mixed-order real space basis sets or the inherent aberrations from transforming reciprocal space solutions of finite expansions. The photonic band structure of a simple crystal is used as a benchmark comparison and the ability to capture the effects of spatially complex dielectric distributions is treated using a complex pattern with highly irregular features that would stress spatial transform limits. This general method is applicable to a broad class of physical systems, e.g., to semiconducting lasers which require simultaneous modeling of transitions in quantum wells or dots together with EM cavity calculations, to modeling plasmonic structures in the presence of EM field emissions, and to on-chip propagation within monolithic integrated circuits.

Shape-preserving interpolation of spatial data by Pythagorean-hodograph quintic spline curves

The interpolation of discrete spatial data-a sequence of points and unit tangents-by G(1) Pythagorean-hodograph (PH) quintic spline curves, under shape constraints, is addressed. To achieve this, a local Hermite scheme incorporating a tension parameter for each spline segment is employed, the imposed shape constraints being concerned with preservation of convexity at the knots and the sign of the discrete torsion over each spline segment. An asymptotic analysis in terms of the tension parameters is developed, and it is shown that satisfaction of the prescribed shape constraints can always be achieved for each spline segment by a suitable choice of the free angular parameters that characterize each PH quintic Hermite segment. In particular, it is proved that the cubic-cubic criterion (Farouki, R. T., Giannelli, C., Manni, C. & Sestini, A. (2008) Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures. Comput. Aided Geom. Design, 25, 274-297; Sestini, A., Landolfi, L. & Manni, C. (2013) On the approximation order of a space data dependent PH quintic Hermite interpolation scheme. Comput. Aided Geom. Design, 30, 148-158) for specifying these free parameters ensures satisfaction of the desired shape-preserving properties, requiring only mild application of the tension parameters that does not compromise the overall fairness of the interpolant. The performance of the method is illustrated through some computed examples.

A Local Pressure Boundary Condition Spectral Collocation Scheme for the Three-Dimensional Navier–Stokes Equations

A spectral collocation scheme for the three-dimensional incompressible $$({\varvec{u}},p)$$ formulation of the Navier–Stokes equations, in domains $$\varOmega $$ with a non-periodic boundary condition, is described. The key feature is the high order approximation, by means of a local Hermite interpolant, of a Neumann boundary condition for use in the numerical solution of the pressure Poisson system. The time updates of the velocity $${\varvec{u}}$$ and pressure $$p$$ are decoupled as a result of treating the pressure gradient in the momentum equation explicitly in time. The pressure update is computed from a pressure Poisson equation. Extension of the overall methodology to the Boussinesq system is also described. The uncoupling of the pressure and velocity time updates results in a highly efficient scheme that is simple to implement and well suited for simulating moderate to high Reynolds and Rayleigh number flows. Accuracy checks are presented, along with simulations of the lid-driven cavity flow and a differentially heated cavity flow, to demonstrate the scheme produces accurate three-dimensional results at a reasonable computational cost.

Vector field approximation using radial basis functions

In this article, we investigate the performance of RBF–PDE methods for approximating solenoidal fields. It is well known that global RBF collocation methods present a trade-off principle, which means that smoothness implies high convergence order plus ill-conditioning. On the other hand, local methods for solving this problem have recently appeared in the literature. In this paper, we perform a numerical investigation of the differences between RBF global and local methods, in order to investigate the possible advantage of using local methods for the approximation of vector fields. More precisely, we compare the local Hermite interpolation technique using inverse multiquadrics against the non-symmetric collocation method of Kansa.