Local Hermite Interpolation 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.

4-dimensional local radial basis function interpolation of large, uniformly spaced datasets

Background and ObjectiveLarge, uniformly spaced, complex and time varying datasets derived from high resolution medical image velocimetry can provide a wealth of information regarding small-scale transient physiological flow phenomena and pulsation of anatomical boundaries. However, there remains a need for interpolation techniques to effectively reconstruct a fully 4-dimensional functional relationship from this data. This paper presents a preliminary evaluation of a 4-dimensional local radial basis function (RBF) algorithm as a means of addressing this problem for laminar flows. MethodsA 4D interpolation algorithm is proposed based on a Local Hermitian Interpolation (LHI) using a combination of multi-quadric RBF with a partition of unity scheme. The domain is divided into uniform sub-systems with size restricted to immediately neighbouring points. The validity of the algorithm is first established on a known 4D analytical dataset and a CFD based laminar flow phantom. Application is then demonstrated through characterisation of a large 4D laminar flow dataset obtained from magnetic resonance imaging (MRI) measurements of cerebrospinal fluid velocities in the brain. ResultsPerformance of the algorithm is compared to that of a quad-linear interpolation, demonstrating favourable improvement in accuracy. The technique is shown to be robust, computationally efficient and capable of refined interpolation in Euclidean space and time. Application to MR velocimetry data is shown to produce promising results for the 4D reconstruction of the transient flow field and movement of the fluid boundaries at spatial and temporal locations intermediate to the original data. ConclusionThis study has demonstrated feasibility of an accurate, stable and efficient 4-dimensional local RBF interpolation method for large, transient laminar flow velocimetry datasets. The proposed approach does not suffer from ill-conditioning or high computational cost due to domain decomposition into local stencils where the RBF is only ever applied to a limited number of points. This work offers a potential tool to assist medical diagnoses and drug delivery through better understanding of physiological flow fields such as cerebrospinal fluid. Further work will evaluate the technique on a wider range of flow fields and against CFD simulation.

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.

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.

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.

An Upwind Scheme to Solve Unsteady Convection-Diffusion Equations Using Radial Basis Function Based Local Hermitian Interpolation Method with PDE Centres

Spurious oscillations occur in the numerical results of convective dominated problems with discontinuities when central interpo- lation schemes are used. Upwind interpolation is necessary to overcome these oscillations. In this work, an upwind scheme is proposed to solve the unsteady convection-diffusion equation using Radial Basis Function (RBF) based Local Hermitian Interpo- lation (LHI) with PDE centres. It is a meshless numerical method based on the interpolation of overlapping RBF systems and also satisfy the governing partial differential equations at certain specified locations. Local hermite interpolation gives symmetric non-singular interpolation matrix which can be inverted easily to solve the system of equations. The scheme is implemented to unsteady benchmark problems and validated against the corresponding analytical solutions. Comparisons with the benchmark solutions show that the proposed upwind scheme is stable and produces significantly accurate results especially for convection dominated problems.

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.

Two-dimensional meshless solution of the non-linear convection diffusion reaction equation by the Local Hermitian Interpolation method

A meshless numerical scheme is developed for solving a generic version of the non-linear convection-diffusion-reaction equation in two-dimensional do-mains. The Local Hermitian Interpolation (LHI) method is employed for thespatial discretization and several strategies are implemented for the solution of the resulting non-linear equation system, among them the Picard iteration, the Newton Raphson method and a truncated version of the Homotopy Analysis Method (HAM). The LHI method is a local collocation strategy in which Radial Basis Functions (RBFs) are employed to build the interpolation function. Unlike the original Kansa’s Method, the LHI is applied locally and the boundary and governing equation differential operators are used to obtain the interpolation function, giving a symmetric and non-singular collocation matrix. Analytical and Numerical Jacobian matrices are tested for the Newton-Raphson method and the derivatives of the governing equation with respect to the homotopy parameter are obtained analytically. The numerical scheme is verified by comparing the obtained results to the one-dimensional Burgers’ and two-dimensional Richards’ analytical solutions. The same resultsare obtained for all the non-linear solvers tested, but better convergence ratesare attained with the Newton Raphson method in a double iteration scheme.

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.

Interpolation with quintic Powell–Sabin splines

We discuss local Hermite interpolation by C2 quintic Powell–Sabin splines represented in a normalized B-spline basis. We derive explicit formulae for the spline coefficients in this B-spline representation to interpolate given Hermite data. As part of the analysis, we show how tensor algebra can be used to describe polynomials in Bernstein–Bézier form and to simplify their manipulation.

A scalable and implicit meshless RBF method for the 3D unsteady nonlinear Richards equation with single and multi‐zone domains

AbstractThe Local Hermitian Interpolation (LHI) method is a strong‐form meshless numerical technique which uses Radial Basis Function (RBF) interpolants to satisfy linear and nonlinear governing equations and boundary operators. Recent developments have shown that, for linear transport problems, applying the PDE governing equation directly to the basis functions can greatly improve the accuracy and stability of the resulting solutions. In this work, the LHI formulation with local PDE‐interpolation is extended to the nonlinear gravity‐driven Richards equation, in order to solve unsteady problems involving flow in unsaturated porous media.The application of the linearised PDE‐operator to the basis functions incorporates information, such as the effective velocity field, directly into the solution construction. This results in a form of ‘analytical upwinding’ which helps to stabilise the solution. In addition, the local interpolation itself satisfies the linearised governing equation, allowing for more accurate reconstruction of partial derivatives, and hence a more accurate solution. The procedure is tested using a 3D infiltration problem with a known analytical solution. The performance of the LHI method, both with and without local PDE interpolation, is compared to the Finite Element method via the FEMWATER software. The LHI formulation with PDE data centres shows consistent improvement over the FEMWATER solutions, reducing errors by several orders of magnitude.In addition, a procedure is introduced to model strongly heterogeneous and layered soils. The physically correct matching conditions are applied over layer interfaces, i.e. continuity of pressure and mass flux. The ‘double collocation’ property of the Hermitian RBF method is exploited to enforce both matching conditions at the same set of locations on the layer interface. This procedure allows the accurate capture of solutions across such interfaces, replicating the required discontinuities in the first derivatives of the pressure profile. The multi‐layer formulation is validated using a transient two‐layer infiltration problem, with the analytical solution replicated to a high precision in a variety of configurations. Copyright © 2010 John Wiley & Sons, Ltd.

A local hermitian RBF meshless numerical method for the solution of multi-zone problems

The local Hermitian interpolation (LHI) method is a strong-form meshless numerical technique in which the solution domain is covered by a series of small and heavily overlapping radial basis function (RBF) interpolation systems. Aside from its meshless nature and the ability to work on very large scattered datasets, the main strength of the LHI method lies in the formation of local interpolations, which themselves satisfy both boundary and governing PDE operators, leading to an accurate and stable reconstruction of partial derivatives without the need for artificial upwinding or adaptive stencil selection. In this work, an extension is proposed to the LHI formulation which allows the accurate capture of solution profiles across discontinuities in governing equation parameters. Continuity of solution value and mass flux is enforced between otherwise disconnected interpolation systems, at the location of the discontinuity. In contrast to other local meshless methods, due to the robustness of the Hermite RBF formulation, it is possible to impose both matching conditions simultaneously at the interface nodes. The procedure is demonstrated for 1D and 3D convection–diffusion problems, both steady and unsteady, with discontinuities in various PDE properties. The analytical solution profiles for these problems, which experience discontinuities in their first derivatives, are replicated to a high degree of accuracy. The technique has been developed as a tool for solving flow and transport problems around geological layers, as experienced in groundwater flow problems. The accuracy of the captured solution profiles, in scenarios where the local convective velocities exceed those typically encountered in such Darcy flow problems, suggests that the technique is indeed suitable for modeling discontinuities in porous media properties. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1201–1230, 2011

The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems

In this work an extension is proposed to the Local Hermitian Interpolation (LHI) method; a meshless numerical method based on interpolation with small and heavily overlapping radial basis function (RBF) systems. This extension to the LHI method uses interpolation functions which themselves satisfy the partial differential equation (PDE) to be solved. In this way, a much improved reconstruction of partial derivatives can be obtained, resulting in significantly improved accuracy in many cases. The implementation algorithm is described, and is validated via three convection-diffusion-reaction problems, for steady and transient situations. A Crank–Nicolson implicit time stepping technique is used for the time-dependent problems. In the proposed approach, a form of ‘analytical upwinding’ is implicitly implemented by the use of the partial differential operator of the governing equation in the interpolation function, which includes the desired information about the convective velocity field. The implicit upwinding scheme intrinsic to the proposed numerical approach is tested by solving a one-dimensional travelling front problem at Péclet numbers of 500, 1000, 2000, 5000 and infinity, which corresponds to a shock front in the case of infinity. In addition, the accuracy of the numerical scheme is validated against a one-dimensional steady state solution exhibiting strong boundary layer effects, and also against a steady and a transient three-dimensional convection-diffusion problem on irregular datasets. All the test cases are validated against the corresponding analytical solutions. Finally, the effect of various interpolation stencil configurations is investigated, and some important limitations on local data-centre distribution are identified.

An order-N complexity meshless algorithm for transport-type PDEs, based on local Hermitian interpolation

This work describes a new numerical method utilising radial basis function interpolants. Based on local Hermitian interpolation of function values and boundary operators, and using an explicit time advancement formulation, the method is of order-N complexity. Computational cost to advance the solution in time is minimal, and is largely dependent on local system support size. The explicit time advancement formulation allows a novel solution technique for many nonlinear partial differential equations. The performance of the method is examined for a variety of linear convection–diffusion–reaction problems, featuring both steady and unsteady solutions. The method is also demonstrated with a nonlinear Richards’ equation model, solving an unsaturated flow in porous media problem. The technique is named the local Hermitian interpolation (LHI) method.

A Meshless Solution Technique for the Solution of 3D Unsaturated Zone Problems, Based on Local Hermitian Interpolation with Radial Basis Functions

Recent developments in meshless numerical methods have led to algorithms that can be used to solve arbitrarily large problems without the support of a connected mesh, and without the computational cost and numerical ill-conditioning issues usually associated with such solution techniques. This work applies the Local Hermitian Interpolation (LHI) method, based on local interpolation with Radial Basis Functions (RBFs), to the solution of 3D unsaturated porous media problems. The proposed implementation is capable of handling real soil properties, provided either as an analytical function or as a series of pointwise measurements. The technique is implemented with implicit and explicit timestepping, and is validated against two transient Richards’ equation models, of which one has a known analytical solution. In addition, a real-world infiltration problem based on a saturated–unsaturated formulation is modelled, using a realistic variation of soil properties with water-pressure.

High-order approximation of conic sections by quadratic splines

Given a segment of a conic section in the form of a rational Bézier curve, a quadratic spline approximation is constructed and an explicit error bound is derived. The convergence order of the error bound is shown to be O( h 4) which is optimal, and the spline curve is both C 1 and G 2. The approximation method is very efficient as it is based on local Hermite interpolation and subdivision. The approximation method and error bound are also applied to an important subclass of rational biquadratic surfaces which includes the sphere, ellipsoid, torus, cone and cylinder.