Transfinite Interpolation Research Articles

A technical note on the geometric representation of a ship hull form

In a recent special issue on ship design of this Journal the applicability of NURBS surfaces for ship hull representation was commented upon, as part of a review of challenges. The authors rightfully point out that NURBS do have their problems when applied to ship hull modelling. The review mentions T-splines as a promising solution, and concludes that the development of applications that address the NURBS limitations remains a challenge. However, just lifting out T-splines as the only solution can be considered to be a bit meagre, because many more alternatives have been proposed in the literature over the years. In addition, applications for ship design that overcome the limitations of NURBS surfaces do exist and are being applied in the maritime industry.As an extension to the special issue paper, in this short technical note the NURBS deficiencies are put into a context, and other potential solutions besides T-splines are summarized. It is illustrated that a viable alternative is offered by a hybrid representation method, comprising elements of a solid model and transfinite interpolation of an irregular network of curves, combined with curve fairing functionality. Because no single method is superb, suggestions for further research are formulated at the end of this note.

Poisson Coordinates

Harmonic functions are the critical points of a Dirichlet energy functional, the linear projections of conformal maps. They play an important role in computer graphics, particularly for gradient-domain image processing and shape-preserving geometric computation. We propose Poisson coordinates, a novel transfinite interpolation scheme based on the Poisson integral formula, as a rapid way to estimate a harmonic function on a certain domain with desired boundary values. Poisson coordinates are an extension of the Mean Value coordinates (MVCs) which inherit their linear precision, smoothness, and kernel positivity. We give explicit formulas for Poisson coordinates in both continuous and 2D discrete forms. Superior to MVCs, Poisson coordinates are proved to be pseudoharmonic (i.e., they reproduce harmonic functions on n-dimensional balls). Our experimental results show that Poisson coordinates have lower Dirichlet energies than MVCs on a number of typical 2D domains (particularly convex domains). As well as presenting a formula, our approach provides useful insights for further studies on coordinates-based interpolation and fast estimation of harmonic functions.

Numerical analysis of dynamic temperature in response to different levels of reactive hyperaemia in a three-dimensional image-based hand model

Vascular reactivity (VR) is considered as an effective index to predict the risk of cardiovascular events. A cost-effective alternative technique used to evaluate VR called digital thermal monitoring (DTM) is based on the response of finger temperature to vessel occlusion and reperfusion. In this work, a simulation has been developed to investigate hand temperature in response to vessel occlusion and perfusion. The simulation consists of image-based mesh generation and finite element analysis of blood flow and heat transfer in tissues. In order to reconstruct a real geometric model of human hand, a computer programme including automatic image processing for sequential MR data and mesh generation based on the transfinite interpolation method is developed. In the finite element analysis part, blood flow perfused in solid tissues is considered as fluid phase through porous media. Heat transfer in tissues is described by Pennes bioheat equation and blood perfusion rate is obtained from Darcy velocities. Capillary pressure, blood perfusion and temperature distribution of hand are obtained. The results reveal that fingertip temperature is strongly dependent on larger arterial pressure. This simulation is of potential to quantify the indices used for evaluating the VR in DTM test if it is integrated with the haemodynamic model of blood circulation in upper limb.

Interpolation and [formula omitted]-approximation by trigonometric polynomials and blending functions

We present results on interpolation and L1-approximation of periodic functions by trigonometric polynomials and trigonometric blending functions. In Section 1, we obtain an error-representation formula for Hermite–Lagrange interpolation by trigonometric polynomials in terms of the differential operator D(2n+1):=D∏k=1n(D2+k2). In Sections 2 and 3, we establish canonical set characterization of the best and best one-sided trigonometric L1-approximants under some restrictions. In Section 4, we obtain an error-representation formula for multivariate Hermite–Lagrange transfinite interpolation by trigonometric blending functions that form the kernel of the differential operator Dθ(2m+1)Dη(2n+1). In Section 5, we give explicit constructions of the best trigonometric blending L1-approximants to multivariate periodic functions in terms of Hermite–Lagrange transfinite interpolation on canonical sets. Our results on best and best one-sided L1-approximation reveal the close relationship between interpolation and best L1-approximation (see e.g. Pinkus (1989) [15]). The non-linear problem of best L1-approximation becomes a linear interpolation problem on certain convexity functional cones. The interpolation point set of the interpolants that are best L1-approximants does not depend on the function to be approximated. For that reason, such an interpolation set is called canonical set of bestL1-approximation. In Section 6, we construct one-sided transfinite trigonometric blending interpolants to multivariate periodic functions. Then, we show that the best one-sided trigonometric blending L1-approximants to multivariate periodic functions are not transfinite trigonometric blending interpolants on interpolation sets consisting of vertical and horizontal line segments.

B-spline patches and transfinite interpolation method for PDE controlled simulation

This paper is to discuss an approach which combines B-spline patches and transfinite interpolation to establish a linear algebraic system for solving partial differential equations and modify the WEB-spline method developed by Klaus Hollig to derive this new idea. First of all, the authors replace the R-function method with transfinite interpolation to build a function which vanishes on boundaries. Secondly, the authors simulate the partial differential equation by directly applying differential operators to basis functions, which is similar to the RBF method rather than Hollig’s method. These new strategies then make the constructing of bases and the linear system much more straightforward. And as the interpolation is brought in, the design of schemes for solving practical PDEs can be more flexible. This new method is easy to carry out and suitable for simulations in the fields such as graphics to achieve rapid rendering. Especially when the specified precision is not very high, this method performs much faster than WEB-spline method.

Transfinite surface interpolation with interior control

There are various techniques to design complex free-form shapes with general topology. In contrast to the approaches based on trimmed surfaces and control polyhedra, in curve network-based design feature curves can be directly created and edited in 3D. Multi-sided patches interpolate this curve network with slopes given by associated tangent ribbons. The patches are smoothly connected and yield a natural and predictable surface model. This paper focuses on special design techniques to adjust the interior of transfinite patches when further shape control is needed. While the boundary constraints are retained, additional vertices, curves and even interior control surfaces are supplemented to gain more design freedom. The main idea is to apply different distance-based blending functions with special parameterizations over non-regular, n-sided domains. This concept can be naturally extended to create one- and two-sided patches as well. Shape variations will be demonstrated by a few simple examples.

CONSTRUCTION OF ANATOMICALLY ACCURATE FINITE ELEMENT MODELS OF THE HUMAN HAND AND A RAT KIDNEY

The aim of the paper is to develop a method for generating three-dimensional (3D) models of organs from medical images (computerized tomography (CT) images, magnetic resonance imaging (MRI), etc.). There were three main steps in the development of the model: the first step was image processing. Different image-processing operators including blurring, sharpening, edge detection, region segmentation, mathematical morphology transformation, rotation, and movement of the kidney slices were performed to automatically construct the accurate boundary information. The second step was mesh generation of each slice based on the boundary information by using the transfinite interpolation (TFI) technique. In this paper, the TFI method was improved to create grids from images directly. The last step was reconstructing the models by stacking the 2D grid models and visualizing the result in the Advanced Visual System (AVS) software. In order to verify the effectiveness of this method, the finite element (FE) models of a rat kidney, human hand, and blood vessels were reconstructed and good results were obtained.

Spectral Approximation of Partial Differential Equations in Highly Distorted Domains

In this paper we discuss spectral approximations of the Poisson equation in deformed quadrilateral domains. High order polynomial approximations are used for both the solution and the representation of the geometry. Following an isoparametric approach, the four edges of the computational domain are first parametrized using high order polynomial interpolation. Transfinite interpolation is then used to construct the mapping from the square reference domain to the physical domain. Through a series of numerical examples we show the importance of representing the boundary of the domain in a careful way; the choice of interpolation points along the edges of the physical domain may significantly effect the overall discretization error. One way to ensure good interpolation points along an edge is based on the following criteria: (i) the points should be on the exact curve; (ii) the derivative of the exact curve and the interpolant should coincide at the internal points along the edge. Following this approach, we demonstrate that the discretization error for the Poisson problem may decay exponentially fast even when the boundary has low regularity.

Online reconstruction of 3D objects from arbitrary cross-sections

We describe a simple algorithm to reconstruct the surface of smooth three-dimensional multilabeled objects from sampled planar cross-sections of arbitrary orientation. The algorithm has the unique ability to handle cross-sections in which regions are classified as being inside the object, outside the object, or unknown. This is achieved by constructing a scalar function on R 3 , whose zero set is the desired surface. The function is constructed independently inside every cell of the arrangement of the cross-section planes using transfinite interpolation techniques based on barycentric coordinates. These guarantee that the function is smooth, and its zero set interpolates the cross-sections. The algorithm is highly parallelizable and may be implemented as an incremental update as each new cross-section is introduced. This leads to an efficient online version, performed on a GPU, which is suitable for interactive medical applications.

Numerical analysis of microwave melting of ice-saturated porous medium filled in a rectangular waveguide with resonator using a combined transfinite interpolation and PDE methods

A numerical study is performed for the melting process of ice-saturated porous medium filled in a rectangular waveguide with a resonator subjected to electromagnetic energy. A microwave system supplies a monochromatic wave in a fundamental mode (TE 10 mode) with operating frequency of 2.45 GHz. Focus is placed on establishing a computationally efficient approach for solving moving boundary heat transfer problem in a two-dimensional structured grid. Numerically, preliminary grids are first generated by an algebraic method, based on a transfinite interpolation method, with subsequent refinement using a PDE mapping method. A preliminary case study indicates successful implementation of the numerical procedure. The predicted results from two-dimensional melting model are then validated against available experimental results and subsequently used as a tool for efficient computational prototyping. Based on the numerical results are performed illustrating the influence of resonator and layered configuration, in case of the installed resonator has strongly affected on the microwave power absorbed, temperature distribution, and the melting front during microwave melting process.

Thermo-Mechanical Analysis of Hot Shape Rolling of Steel by a Meshless Method

The numerical modeling of hot shape rolling of steel billet is represented by using a meshless Radial Basis function Collocation Method (RBFCM). The physical model consists of coupled thermal and mechanical models. Both models are solved through a strong formulation. The material in the present formulation is assumed to behave elastic or ideally plastic. The model decomposes the 3D geometry of the steel billet into a traveling 2D cross section which lets us analyze the large shape reductions by a sequence of small steps. Homogeneous compression and a uniform velocity over each of the cross-sections are assumed. A meshless method, based on collocation with radial basis functions, is proposed to solve the thermo-mechanical problem. The nodal distribution is adapted by Transfinite Interpolation (TFI) and Elliptic Node Generation (ENG) at each deformation step to the new form of the billet. The solution is calculated in terms of temperatures and displacements at each node. A simple benchmark solution is compared with FEM based commercial code DEFORM. A good agreement is found between the RBFCM and the commercial code DEFORM.

On transfinite interpolations with respect to convex domains

We propose an approach for constructing a transfinite interpolation where the domain of definition is a convex polytope. For multifaceted domains which are not of tensor product type, it is difficult to directly generalize the usual approach of transfinite interpolation which blends opposite faces and which then substracts that by a mixed term. Therefore, we suggest a short formula which uses topologic entities of the convex domain. Our formula uses some projection operator onto the faces of the polytope. Both representations ( V -setting and H -setting) of a polytope are used. We show also that the transfinite interpolation is stable under affine transformations. As a supplement to the theoretical demonstrations, we show some interesting practical illustrations.

VARIATIONAL PRINCIPLES FOR WATER WAVES FROM THE VIEWPOINT OF A TIME‐DEPENDENT MOVING MESH

The time-dependent motion of water waves with a parametrically defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic partial differential equations. The aim is to study the effect of transformation on variational principles for water waves such as Luke's Lagrangian formulation, Zakharov's Hamiltonian formulation, and the Benjamin–Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian. Also some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves.

DEVELOPMENT OF A THREE DIMENSIONAL MULTI-BLOCK STRUCTURED GRID DEFORMATION CODE FOR COMPLEX CONFIGURATIONS

In this study, a multi-block structured grid deformation code based on a hybrid of transfinite interpolation algorithm and spring analogy has been developed. The combination of spring analogy for block vertices and transfinite interpolation for interior grid points helps to increase the robustness and makes it suitable for distributed computing. Elliptic smoothing operator is applied to the block faces with sub-faces to maintain the grid’s smoothness and skewness. The capability of the developed code is demonstrated on a range of simple and complex configuration such as airfoil and wing body configuration.

Transfinite Thin Plate Spline Interpolation

Duchon’s method of thin plate splines defines a polyharmonic interpolant to scattered data values as the minimizer of a certain integral functional. For transfinite interpolation, i.e., interpolation of continuous data prescribed on curves or hypersurfaces, Kounchev has developed the method of polysplines, which are piecewise polyharmonic functions of fixed smoothness across the given hypersurfaces and satisfy some boundary conditions. Recently, Bejancu has introduced boundary conditions of Beppo–Levi type to construct a semicardinal model for polyspline interpolation to data on an infinite set of parallel hyperplanes. The present paper proves that, for periodic data on a finite set of parallel hyperplanes, the polyspline interpolant satisfying Beppo–Levi boundary conditions is in fact a thin plate spline, i.e., it minimizes a Duchon type functional. The construction and variational characterization of the Beppo–Levi polysplines are based on the analysis of a new class of univariate exponential ℒ-splines satisfying adjoint natural end conditions.

Automatic terrain modeling using transfinite element analysis

Abstract An automatic procedure for modeling terrain is developed based on L2 projection-based interpolation of discrete terrain data onto transfinite function spaces. The function space is refined automatically by the use of image processing techniques to detect regions of high error and the flexibility of the transfinite interpolation to add degrees of freedom to these areas. Examples are shown of a section of the Palo Duro Canyon in northern Texas.

Coupled analysis of multi-body motions in offshore connecting process

The coupled dynamic responses of two pontoons while connecting with each other in irregular waves are calculated by means of 3D potential flow theory. The computation is used to find the optimal status for connecting at certain sea state. On the basis of the relative motion of two pontoons in irregular waves, visual FORTRAN programming language, as well as open graphics library (OpenGL), is used to develop a set of virtual reality system, which is fully interactive and with realistic effect. The transfinite interpolation scheme is applied for the mesh generating of the wave surface, and the wave motion is simulated by surface elevation, which is calculated by 3D potential flow theory.

Construction of H1-conforming hierarchical shape functions for elements of all shapes and transfinite interpolation

The paper reviews the construction of hierarchical H 1-conforming shape functions for 1D (edge), 2D (triangle, quad) and 3D (hexahedron, tetrahedron, prism, pyramid) elements. The logic of implementation is based on identifying a set of core (kernel, bubble) 1D and 2D functions, and the use of specific edge-to-element and face-to-element extensions. The resulting implementation allows for an instantaneous modification (or redefinition) of core functions without any changes in the rest of the FE code. The 3D shape functions conform automatically to 1D edge and 2D face shape functions defined in local edge and face coordinates. In other words, the dependence of 3D shape functions on different edge and face orientations is taken into account at the level of element shape functions routine. This eliminates the use of sign factors during the assembly and simplifies dramatically the implementation of constrained approximation allowing for the presence of hanging nodes. The construction of shape functions is directly related to transfinite interpolation techniques used in mesh based geometry (MBG) descriptions.

Eigenanalysis of Two-Dimensional Acoustic Cavities Using Transfinite Interpolation

This paper discusses the efficient extraction of eigenfrequencies in two-dimensional acoustic cavities using higher order finite elements, called “Coons-patch macroelements”. The acoustic pressure is approximated within the entire domain using the bivariate transfinite (Coons-Gordon) interpolation formula. Basically, the proposed macroelements constitute a generalization of the well-known Serendipity and Lagrangian type elements. The paper investigates the performance of the proposed methodology in several examples and definitely correlates the necessity of using internal nodes or domain decomposition with the percentage of open boundaries over the entire boundary as well as their relative position. It was found that for simple shapes, the entire acoustic cavity can be considered as a single quadrilateral patch, i.e. as one Coons-patch macroelement, while for complex shapes the cavity should be divided into a small number of subregions.

Sweeping of unstructured meshes over generalized extruded volumes

An algorithm for the discretization of general extrudable volumes into semi-structured meshes is presented in this paper. The set of considered extrudable volumes is limited to volumes formed by two opposite topologically identical and geometrically similar cap surfaces, from which one is identified as the source and the other one as the target surface, and by the set of lateral quad-mappable surfaces. The source cap surface is discretized by an appropriate unstructured surface mesh which is projected onto the target cap surface using an affine mapping defined between the parametric spaces of the cap surfaces. This mapping is established by a least-squares fit of boundary nodes on the target surface. The structured quad meshes on lateral surfaces are obtained by transfinite interpolation. The volume interior nodes used to form inner layer of elements are defined by the mapping derived from a least-squares approximation of all boundary nodes. The obtained mesh is then subjected to a specific two-phase smoothing. The capabilities of the proposed algorithm are demonstrated by several examples.