On Wachspress quadrilateral patches

Wachspress initiated the study of rational basis functions for finite element construction over quadrilaterals and more general polygonal and curved elements. Later Apprato et al. (1979) and Gout (1979, 1985) studied the interpolatory and convergence properties of lower degree rational finite elements and their applications in solving second order boundary value problems. In the present paper we introduce higher degree Wachspress functions by an iterative technique and study their properties from the point of view of applications to surface fitting problems. It is indeed remarkable to note that these functions possess properties similar to tensor product Bernstein polynomials and hence could be effectively used to generate quadrilateral patches.     

Bridging art and engineering using Escher-based virtual elements

The geometric shape of an element plays a key role in computational methods. Triangular and quadrilateral shaped elements are utilized by standard finite element methods. The pioneering work of Wachspress laid the foundation for polygonal interpolants which introduced polygonal elements. Tessellations may be considered as the next stage of element shape evolution. In this work, we investigate the topology optimization of tessellations as a means to coalesce art and engineering. We mainly focus on M.C. Escher’s tessellations using recognizable figures. To solve the state equation, we utilize a Mimetic Finite Difference inspired approach, known as the Virtual Element Method. In this approach, the stiffness matrix is constructed in such a way that the displacement patch test is passed exactly in order to ensure optimum numerical convergence rates. Prior to exploring the artistic aspects of topology optimization designs, numerical verification studies such as the displacement patch test and shear loaded cantilever beam bending problem are conducted to demonstrate the accuracy of the present approach in two-dimensions.     

A unified, integral construction for coordinates over closed curves

We propose a simple generalization of Shephard's interpolation to piecewise smooth, convex closed curves that yields a family of boundary interpolants with linear precision. Two instances of this family reduce to previously known interpolants: one based on a generalization of Wachspress coordinates to smooth curves and the other an integral version of mean value coordinates for smooth curves. A third instance of this family yields a previously unknown generalization of discrete harmonic coordinates to smooth curves. For closed, piecewise linear curves, we prove that our interpolant reproduces a general family of barycentric coordinates considered by Floater, Hormann and Kós that includes Wachspress coordinates, mean value coordinates and discrete harmonic coordinates.     

A Finite Element Method on Convex Polyhedra

We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis functions guarantee convergence. Our method is a natural extension to linear interpolants on tetrahedra: for tetrahedral elements, the methods are identical. For fast and robust computations, we use an elasticity model based on Cauchy strain and stiffness warping. This more flexible discretization is particularly useful for simulations that involve topological changes, such as cutting or fracture. Since splitting convex elements along a plane produces convex elements, remeshing or subdivision schemes used in simulations based on tetrahedra are not necessary, leading to less elements after such operations. We propose various operators for cutting the polyhedral discretization. Our method can handle arbitrary cut trajectories, and there is no limit on how often elements can be split.     

Construction of a hermite rational “Wachspress type” finite element

This paper concerns the construction of a quadrilateral finite element whose interpolation space admits of rational fractions for basis functions of “Wachspress type” [1, 2]. The construction of this finite element, which is in a way the “rational” equivalent of the ADINI finite element[3, 4], is founded on a method analogous to the one used for Serendip degree-two finite element construction in[2]. The study of interpolation error is dealt with in a paper by Apprato, Arcangeli and Gout in this journal “Rational interpolation of Wachspress error estimates”.     

Honeycomb Wachspress finite elements for structural topology optimization

Traditionally, standard Lagrangian-type finite elements, such as linear quads and triangles, have been the elements of choice in the field of topology optimization. However, finite element meshes with these conventional elements exhibit the well-known “checkerboard” pathology in the iterative solution of topology optimization problems. A feasible alternative to eliminate such long-standing problem consists of using hexagonal (honeycomb) elements with Wachspress-type shape functions. The features of the hexagonal mesh include two-node connections (i.e. two elements are either not connected or connected by two nodes), and three edge-based symmetry lines per element. In contrast, quads can display one-node connections, which can lead to checkerboard; and only have two edge-based symmetry lines. In addition, Wachspress rational shape functions satisfy the partition of unity condition and lead to conforming finite element approximations. We explore the Wachspress-type hexagonal elements and present their implementation using three approaches for topology optimization: element-based, continuous approximation of material distribution, and minimum length-scale through projection functions. Examples are presented that demonstrate the advantages of the proposed element in achieving checkerboard-free solutions and avoiding spurious fine-scale patterns from the design optimization process.     

Interface element method: Treatment of non-matching nodes at the ends of interfaces between partitioned domains

A methodology for interface element method (IEM) to combine partitioned domains with non-matching nodes at the ends of interfaces is presented. The IEM is introduced to satisfy the continuity and the compatibility conditions on non-matching interfaces between partitioned finite element domains. Interface elements are defined on the finite elements bordering on the interfaces, and the moving least square (MLS) approximations are employed to construct the shape functions of the interface elements. By modifying the shape functions of the interface elements at the ends of non-matching interfaces, partitioned domains are glued such that all properties of the IEM are satisfied. The modifications are made to sub-domains and weight functions in the MLS approximations. The numerical examples show that the present IEM is very effective for the analysis of a partitioned system and for global-local analysis.     

A Complex View of Barycentric Mappings

Barycentric coordinates are very popular for interpolating data values on polyhedral domains. It has been recently shown that expressing them as complex functions has various advantages when interpolating two‐dimensional data in the plane, and in particular for holomorphic maps. We extend and generalize these results by investigating the complex representation of real‐valued barycentric coordinates, when applied to planar domains. We show how the construction for generating real‐valued barycentric coordinates from a given weight function can be applied to generating complex‐valued coordinates, thus deriving complex expressions for the classical barycentric coordinates: Wachspress, mean value, and discrete harmonic. Furthermore, we show that a complex barycentric map admits the intuitive interpretation as a complex‐weighted combination of edge‐to‐edge similarity transformations, allowing the design of “home‐made” barycentric maps with desirable properties. Thus, using the tools of complex analysis, we provide a methodology for analyzing existing barycentric mappings, as well as designing new ones.     

Barycentric coordinates for polytopes

In Wachspress (1975) [1], theory was developed for constructing rational basis functions for convex polygons and polyhedra. These barycentric coordinates were positive within the elements. Generalization to higher space dimensions is described here. The GADJ algorithm developed by Dasgupta (2003) [5] and in Dasgupta and Wachspress (2008) [6] is crucial for simple construction of rational barycentric basis functions.     

Overview and applications of the reproducing Kernel Particle methods

Summary Multiple-scale Kernel Particle methods are proposed as an alternative and/or enhancement to commonly used numerical methods such as finite element methods. The elimination of a mesh, combined with the properties of window functions, makes a particle method suitable for problems with large deformations, high gradients, and high modal density. The Reproducing Kernel Particle Method (RKPM) utilizes the fundamental notions of the convolution theorem, multiresolution analysis and window functions. The construction of a correction function to scaling functions, wavelets and Smooth Particle Hydrodynamics (SPH) is proposed. Completeness conditions, reproducing conditions and interpolant estimates are also derived. The current application areas of RKPM include structural acoustics, elastic-plastic deformation, computational fluid dynamics and hyperelasticity. The effectiveness of RKPM is extended through a new particle integration method. The Kronecker delta properties of finite element shape functions are incorporated into RKPM to develop a C ^m kernel particle finite element method. Multiresolution and hp-like adaptivity are illustrated via examples.     

The equation of the curved edge for isoparametric cubic finite elements

We derive the equation of the actual curve that results when a curved edge is approximated using an isopaiametric cubic finite element. This implied curve depends only on the parameters of the nodes associated with the curved side and does not depend on the shape of the element or on the basis functions used. By choosing a special set of basis functions for a nine-parameter cubic triangle we obtain an isoparametric transformation with a simple form that lends itself to elementary analysis. For triangular finite elements, both the Lagrange and Hermite interpolants are considered. A special case is described which permits an important class of cubic curves to be exactly matched very simply by an appropriate choice of mid-side nodes.     

Complex Transfinite Barycentric Mappings with Similarity Kernels

Transfinite barycentric kernels are the continuous version of traditional barycentric coordinates and are used to define interpolants of values given on a smooth planar contour. When the data is two‐dimensional, i.e. the boundary of a planar map, these kernels may be conveniently expressed using complex number algebra, simplifying much of the notation and results. In this paper we develop some of the basic complex‐valued algebra needed to describe these planar maps, and use it to define similarity kernels, a natural alternative to the usual barycentric kernels. We develop the theory behind similarity kernels, explore their properties, and show that the transfinite versions of the popular three‐point barycentric coordinates (Laplace, mean value and Wachspress) have surprisingly simple similarity kernels. We furthermore show how similarity kernels may be used to invert injective transfinite barycentric mappings using an iterative algorithm which converges quite rapidly. This is useful for rendering images deformed by planar barycentric mappings.     

Looking for the best constant in a Sobolev inequality: a numerical approach

A numerical method for the computation of the best constant in a Sobolev inequality involving the spaces H ²(Ω) and C⁰([`(W)])C^{0}(\overline{\Omega}) is presented. Green’s functions corresponding to the solution of Poisson problems are used to express the solution. This (kind of) non-smooth eigenvalue problem is then formulated as a constrained optimization problem and solved with two different strategies: an augmented Lagrangian method, together with finite element approximations, and a Green’s functions based approach. Numerical experiments show the ability of the methods in computing this best constant for various two-dimensional domains, and the remarkable convergence properties of the augmented Lagrangian based iterative method.     

A new front updating solution applied to some engineering problems

Summary An automatic mesh generation dealing with domains of an arbitrary shape could be realized by an advancing front method. The mesh generator based on this method creates triangle elements inside a domain starting with the polygonal (polyhedral in 3D) discretisation of its border. In this paper an original algorithm for the front updating procedures as a part of the mesh generator is presented. The proposed algorithm provides an efficient mesh generation procedure. It has been verified on the various domains with complex geometry and with nonuniform distribution of edge nodes such as the discretisation of the switched reluctance motor and power cable configuration, respectively. The related finite element calculations are carried out.     

Maximum-norm stability of the finite element Ritz projection under mixed boundary conditions

As a model of the second order elliptic equation with non-trivial boundary conditions, we consider the Laplace equation with mixed Dirichlet and Neumann boundary conditions on convex polygonal domains. Our goal is to establish that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon–Miranda) discrete maximum principle, and then prove the stability of the Ritz projection with mixed boundary conditions in \(L^\infty \) norm. Such results have a number of applications, but are not available in the literature. Our proof of the maximum-norm stability of the Ritz projection is based on converting the mixed boundary value problem to a pure Neumann problem, which is of independent interest.     

Analysis of a typical shaped tunnel lining by finite element method

In the present study, the diversion tunnel of Tehri Dam in India has been analyzed using finite element procedure. The tunnel is of circular shape from inside, from hydraulic considerations, and of horseshoe shape from outside, to facilitate excavation and construction of the tunnel. In the analysis, the 8-noded parabolic elements have been used and shape functions are chosen as to satisfy the continuity of displacements between the elements and the convergence criterion. A parametric study has been carried out and the results are presented in nondimensional form and are compared with results of conventional analysis of circular cross sections.     

The Approximation and Computation of a Basis of the Trace Space <Emphasis Type="Italic">H</Emphasis><Superscript>1/2</Superscript>

We present a method for construction of an approximate basis of the trace space H ^1/2 based on a combination of the Steklov spectral method and a finite element approximation. Specifically, we approximate the Steklov eigenfunctions with respect to a particular finite element basis. Then solutions of elliptic boundary value problems with Dirichlet boundary conditions can be efficiently and accurately expanded in the discrete Steklov basis. We provide a reformulation of the discrete Steklov eigenproblem as a generalized eigenproblem that we solve by the implicitly restarted Arnoldi method of ARPACK. We include examples highlighting the computational properties of the proposed method for the solution of elliptic problems on bounded domains using both a conforming bilinear finite element and a non-conforming harmonic finite element. In addition, we document the efficiency of the proposed method by solving a Dirichlet problem for the Laplace equation on a densely perforated domain.     

Multi-Level Shape Representation Using Global Deformations and Locally Adaptive Finite Elements

We present a model-based method for the multi-level shape, pose estimation and abstraction of an object's surface from range data. The surface shape is estimated based on the parameters of a superquadric that is subjected to global deformations (tapering and bending) and a varying number of levels of local deformations. Local deformations are implemented using locally adaptive finite elements whose shape functions are piecewise cubic functions with C ¹ continuity. The surface pose is estimated based on the model's translational and rotational degrees of freedom. The algorithm first does a coarse fit, solving for a first approximation to the translation, rotation and global deformation parameters and then does several passes of mesh refinement, by locally subdividing triangles based on the distance between the given datapoints and the model. The adaptive finite element algorithm ensures that during subdivision the desirable finite element mesh generation properties of conformity, non-degeneracy and smoothness are maintained. Each pass of the algorithm uses physics-based modeling techniques to iteratively adjust the global and local parameters of the model in response to forces that are computed from approximation errors between the model and the data. We present results demonstrating the multi-level shape representation for both sparse and dense range data.     

Interpolation error estimates on hermite rational “Wachspress type” third degree finite element

This paper is about the study of interpolation error for the Hermite rational “Wachspress type” third degree finite element that is constructed in[1]. We obtain results analogous with those of the “corresponding” ADINI (polynomial) finite element.     

Penalized approximation of the vibration frequencies of a fluid in a cavity

Here we point out some difficulties arising in the approximation of the vibration frequencies of a fluid in a cavity in the case of non convex polygonal domains. Since the eigensolutions must satisfy an irrotationality condition, a classical way to face the problem is to consider a penalized formulation. Unfortunately standard conforming finite elements fail to give good results. We intend to justify this failure and to suggest a finite element method based on a reduced integration strategy able to give reasonable results. Received: 29 April 1999 / Accepted: 29 September 1999