Hermite Radial Basis Function Research Articles

Buckling analysis of functionally graded thin plate with in-plane material inhomogeneity

Buckling analysis of functionally graded material (FGM) thin plates with in-plane material inhomogeneity is investigated based on radial basis functions associated with collocation method. No background mesh is required in the discretization and solution which makes it a truly meshfree method. Two independent problems raised in the buckling analysis are studied according to the procedure. First, radial basis collocation method (RBCM) is employed to yield the non-uniform pre-buckling stresses by solving a 2D plane stress problem. Afterwards, based on Kirchhoff assumption and employing the predetermined non-uniform pre-buckling stresses, Hermite radial basis function collocation method (HRBCM) is proposed to study the buckling loads of FGM thin plates with in-plane material inhomogeneity. Compared to an over-determined system resulting from the conventional RBCM, HRBCM introducing more degrees of freedom on the boundary nodes can lead to a determined system for the eigenvalue problem. Convergence and comparisons studies with analytical solutions demonstrate that the proposed method possesses high accuracy and exponential convergence. Numerical examples illustrate that the material inhomogeneity has considerable effects on the buckling loads and mode shapes of thin plates. As a result, material inhomogeneity can be exploited to optimize the in-plane stress distribution and prevent the buckling of thin plates.

Composed Fluid–Structure Interaction Interface for Horizontal Axis Wind Turbine Rotor

In this paper, we propose a technique for high-fidelity fluid–structure interaction (FSI) spatial interface reconstruction of a horizontal axis wind turbine (HAWT) rotor model composed of an elastic blade mounted on a rigid hub. The technique is aimed at enabling re-usage of existing blade finite element method (FEM) models, now with high-fidelity fluid subdomain methods relying on boundary-fitted mesh. The technique is based on the partition of unity (PU) method and it enables fluid subdomain FSI interface mesh of different components to be smoothly connected. In this paper, we use it to connect a beam FEM model to a rigid body, but the proposed technique is by no means restricted to any specific choice of numerical models for the structure components or methods of their surface recoveries. To stress-test robustness of the connection technique, we recover elastic blade surface from collinear mesh and remark on repercussions of such a choice. For the HAWT blade recovery method itself, we use generalized Hermite radial basis function interpolation (GHRBFI) which utilizes the interpolation of small rotations in addition to displacement data. Finally, for the composed structure we discuss consistent and conservative approaches to FSI spatial interface formulations.

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.

The control volume radial basis function method CV-RBF with Richardson extrapolation in geochemical problems

The aim of this paper is to present how to implement a control volume approach improved by Hermite radial basis functions (CV-RBF) for geochemical problems. A multi-step strategy based on Richardson extrapolation is proposed as an alternative to the conventional dual step sequential non-iterative approach (SNIA) for coupling the transport equations with the chemical model. Additionally, this paper illustrates how to use PHREEQC to add geochemical reaction capabilities to CV-RBF transport methods. Several problems with different degrees of complexity were solved including cases of cation exchange, dissolution, dissociation, equilibrium and kinetics at different rates for mineral species. The results show that the solution and strategies presented here are effective and in good agreement with other methods presented in the literature for the same cases.

快速Hermite径向基函数曲面重构

In this paper, a fast method with compactly supported radial basis functions (CSRBFs) is presented for Hermite surface interpolation or approximation from scattered points. By constructing a hierarchy of the given points, a global surface reconstruction is achieved in a coarse-to- ne way which overcome the problems resulted from using CSRBFs. Moreover, we design a radial basis function center selection approach based on approximat- ing errors, to reduce the interpolating points on each level, which make the Hermite radial basis function implicits to be able to deal with point clouds more than one million of points. The experiments demonstrate our method is also suitable for handling extremely nonuniform or noisy point clouds.

Bilateral Hermite Radial Basis Functions for Contour‐based Volume Segmentation

AbstractIn this paper, we propose a novel contour‐based volume image segmentation technique. Our technique is based on an implicit surface reconstruction strategy, whereby a signed scalar field is generated from user‐specified contours. The key idea is to compute the scalar field in a joint spatial‐range domain (i.e., bilateral domain) and resample its values on an image manifold. We introduce a new formulation of Hermite radial basis function (HRBF) interpolation to obtain the scalar field in the bilateral domain. In contrast to previous implicit methods, bilateral HRBF (B‐HRBF) generates a segmentation boundary that passes through all contours, fits high‐contrast image edges if they exist, and has a smooth shape in blurred areas of images. We also propose an acceleration scheme for computing B‐HRBF to support a real‐time and intuitive segmentation interface. In our experiments, we achieved high‐quality segmentation results for regions of interest with high‐contrast edges and blurred boundaries.

Local mass conservative Hermite interpolation for the solution of flow problems by a multi-domain boundary element approach

This paper presents a local Hermite radial basis function interpolation scheme for the velocity and pressure fields. The interpolation for velocity satisfies the continuity equation (mass conservative interpolation) while the pressure interpolation obeys the pressure equation. Additionally, the Dual Reciprocity Boundary Element method (DRBEM) is applied to obtain an integral representation of the Navier–Stokes equations. Then, the proposed local interpolation is used to obtain the values of the field variables and their partial derivatives at the boundary of the sub-domains. This interpolation allows one to obtain the boundary values needed for the integral formulas for velocity and pressure at some nodes within the sub-domains. In the proposed approach the boundary elements are merely used to parameterize the geometry, but not for the evaluation of the integrals as it is usually done. The presented multi-domain approach is different from the traditional ones in boundary elements because the resulting integral equations are non singular and the boundary data needed for the boundary integrals are approximated using a local interpolation. Some accurate results for simple Stokes problems and for the Navier–Stokes equations at low Reynolds numbers up to Re = 400 were obtained.

Hermite Radial Basis Functions Implicits

Abstract The Hermite radial basis functions (HRBF) implicits reconstruct an implicit function which interpolates or approximates scattered multivariate Hermite data (i.e. unstructured points and their corresponding normals). Experiments suggest that HRBF implicits allow the reconstruction of surfaces rich in details and behave better than previous related methods under coarse and/or non‐uniform samplings, even in the presence of close sheets. HRBF implicits theory unifies a recently introduced class of surface reconstruction methods based on radial basis functions (RBF), which incorporate normals directly in their problem formulation. Such class has the advantage of not depending on manufactured offset‐points to ensure existence of a non‐trivial implicit surface RBF interpolant. In fact, we show that HRBF implicits constitute a particular case of Hermite–Birkhoff interpolation with radial basis functions, whose main results we present here. This framework not only allows us to show connections between the present method and others but also enable us to enhance the flexibility of our method by ensuring well‐posedness of an interesting combined interpolation/regularization approach.

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 Hermite radial basis function control volume method for multi-zones problems; A non-overlapping domain decomposition algorithm

In this paper a non-overlapping non-iterative multi-domain formulation for the control volume Hermite radial basis functions (CV-HRBF) method is proposed, where the local Hermitian RBF meshless collocation method is used to satisfy a physical matching condition at the sub-domain boundaries. In addition, the robustness of the Hermite interpolation is exploited even further to apply multiple flux continuities for those cases where more than two sub-domains converge in the same point. The algorithm is first validated in one-dimensional advection diffusion problems for which an analytical solution is known. Its accuracy is compared with a classic CV approach and a local radial basis function collocation method (LRBFCM). More general applications in two and three-dimensional domains are then considered. A heat transfer problem in strongly heterogeneous materials, and a groundwater flow problem in presence of geological layers characterised by different hydraulic conductivity, are taken as engineering applications to test the capabilities of the CV-HRBF method to handle multi-zone problems. Finally, the transport of a single species is simulated in a one-dimensional channel consisting of two adjacent zones that feature different Peclet numbers.

Hybrid evolutionary algorithm with Hermite radial basis function interpolants for computationally expensive adjoint solvers

In this paper, we present an evolutionary algorithm hybridized with a gradient-based optimization technique in the spirit of Lamarckian learning for efficient design optimization. In order to expedite gradient search, we employ local surrogate models that approximate the outputs of a computationally expensive Euler solver. Our focus is on the case when an adjoint Euler solver is available for efficiently computing the sensitivities of the outputs with respect to the design variables. We propose the idea of using Hermite interpolation to construct gradient-enhanced radial basis function networks that incorporate sensitivity data provided by the adjoint Euler solver. Further, we conduct local search using a trust-region framework that interleaves gradient-enhanced surrogate models with the computationally expensive adjoint Euler solver. This ensures that the present hybrid evolutionary algorithm inherits the convergence properties of the classical trust-region approach. We present numerical results for airfoil aerodynamic design optimization problems to show that the proposed algorithm converges to good designs on a limited computational budget.

Non-Overlapping Domain Decomposition Algorithm for the Hermite Radial Basis Function Meshless Collocation Approach: Applications to Convection Diffusion Problems

In the particular case of solving large-scale boundary value problems, the computational cost derived as a result of the application of any numerical scheme represent a determinant factor in the determination of its computational efficiency. The present work studies the influence of the non-overlapping domain decomposition and higher order interpolation functions on the Hermite radial basis collocation method, as a way to improve its efficiency under high demanding numerical conditions. A series of high Péclet convection diffusion and multi-zone problems are tested. Comparison between the numerical results and its analytical solution is carried out for each of them.

A Hermite radial basis function collocation approach for the numerical simulation of crystallization processes in a channel

A numerical model of crystal needles growing in a channel from an over-saturated solution is considered. The proposed numerical approach is based on the Hermite radial basis function meshless collocation technique. Numerically, the above phenomenon is very unstable and any small perturbation in the interface shape may give rise to the growth of an artificial (numerical) branch at the crystal interface. Our simulation does not show any evidences of such numerical instabilities and is able to predict the conditions under which real physical interface instabilities are expected. In the numerical algorithm, at each time step of the evolution, we only move the collocation points defining the crystal interface without changing the position of any of the other original collocation points. We are able to carry out this type of scheme due to robustness of the proposed Hermite interpolation approach. In this way, the proposed meshless collocation approach results in a very simple and efficient technique to deal with this type of moving boundary problems. Copyright © 2005 John Wiley & Sons, Ltd.

Radial basis function Hermite collocation approach for the solution of time dependent convection–diffusion problems

This work presents a meshless numerical approach for the solution of time dependent convection–diffusion problems in terms of a Hermite radial basis function interpolation numerical scheme. To test the proposed scheme several numerical examples are analysed including problems with variable convective velocity and reaction coefficient. Comparisons are made with available analytical solutions.

Symmetric boundary knot method

The boundary knot method (BKM) is a recent boundary-type radial basis function (RBF) collocation scheme for general partial differential equations. Like the method of fundamental solution (MFS), the RBF is employed to approximate the inhomogeneous terms via the dual reciprocity principle. Unlike the MFS, the method uses a non-singular general solution instead of a singular fundamental solution to evaluate the homogeneous solution so as to circumvent the controversial artificial boundary outside the physical domain. The BKM is meshfree, super-convergent, integration-free, very easy to learn and program. The original BKM, however, loses symmetricity in the presence of mixed boundary. In this study, by analogy with Fasshauer's Hermite RBF interpolation, we developed a symmetric BKM scheme. The accuracy and efficiency of the symmetric BKM are also numerically validated in some 2D and 3D Helmholtz and diffusion-reaction problems under complicated geometries.