Recent advances in the construction of polygonal finite element interpolants

Summary This paper is an overview of recent developments in the construction of finite element interpolants, which areC ⁰-conforming on polygonal domains. In 1975, Wachspress proposed a general method for constructing finite element shape functions on convex polygons. Only recently has renewed interest in such interpolants surfaced in various disciplines including: geometric modeling, computer graphics, and finite element computations. This survey focuses specifically on polygonal shape functions that satisfy the properties of barycentric coordinates: (a) form a partition of unity, and are non-negative; (b) interpolate nodal data (Kronecker-delta property), (c) are linearly complete or satisfy linear precision, and (d) are smooth within the domain. We compare and contrast the construction and properties of various polygonal interpolants—Wachspress basis functions, mean value coordinates, metric coordinate method, natural neighbor-based coordinates, and maximum entropy shape functions. Numerical integration of the Galerkin weak form on polygonal domains is discussed, and the performance of these polygonal interpolants on the patch test is studied.     

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.     

Segmentation of discrete vector fields

In this paper, we propose an approach for 2D discrete vector field segmentation based on the Green function and normalized cut. The method is inspired by discrete Hodge decomposition such that a discrete vector field can be broken down into three simpler components, namely, curl-free, divergence-free, and harmonic components. We show that the Green function method (GFM) can be used to approximate the curl-free and the divergence-free components to achieve our goal of the vector field segmentation. The final segmentation curves that represent the boundaries of the influence region of singularities are obtained from the optimal vector field segmentations. These curves are composed of piecewise smooth contours or streamlines. Our method is applicable to both linear and nonlinear discrete vector fields. Experiments show that the segmentations obtained using our approach essentially agree with human perceptual judgement.     

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.     

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.     

Control of near‐grazing dynamics and discontinuity‐induced bifurcations in piecewise‐smooth dynamical systems

This paper develops a rigorous control paradigm for regulating the near‐grazing bifurcation behavior of limit cycles in piecewise‐smooth dynamical systems. In particular, it is shown that a discrete‐in‐time linear feedback correction to a parameter governing a state‐space discontinuity surface can suppress discontinuity‐induced fold bifurcations of limit cycles that achieve near‐tangential intersections with the discontinuity surface. The methodology ensures a persistent branch of limit cycles over an interval of parameter values near the critical condition of tangential contact that is an order of magnitude larger than that in the absence of control. The theoretical treatment is illustrated with a harmonically excited damped harmonic oscillator with a piecewise‐linear spring stiffness as well as with a piecewise‐nonlinear model of a capacitively excited mechanical oscillator. Copyright © 2009 John Wiley & Sons, Ltd.     

Necessary and sufficient optimality conditions for discrete time-invariant automaton-type systems

The optimal control of deterministic discrete time-invariant automaton-type systems is considered. Changes in the system’s state are governed by a recurrence equation. The switching times and their order are not specified in advance. They are found by optimizing a functional that takes into account the cost of each switching. This problem is a generalization of the classical optimal control problem for discrete time-invariant systems. It is proved that, in the time-invariant case, switchings of the optimal trajectory (may be multiple instantaneous switchings) are possible only at the initial and (or) terminal points in time. This fact is used in the derivation of equations for finding the value (Hamilton–Jacobi–Bellman) function and its generators. The necessary and sufficient optimality conditions are proved. It is shown that the generators of the value function in linear–quadratic problems are quadratic, and the value function itself is piecewise quadratic. Algorithms for the synthesis of the optimal closed-loop control are developed. The application of the optimality conditions is demonstrated by examples.     

基于Poisson方程的曲线形状渐变方法

以定义在分段线性曲线上的离散Poisson方程为理论基础,提出了一种同时适用于平面和空间曲线形状渐变的方法.通过在源曲线和目标曲线上定义局部标架,给出了一种非线性梯度场插值算法,使得源曲线的梯度场逐步过渡到目标曲线的梯度场,所得到的中间梯度场与用户指定的关键节点路径一起输入离散Poisson方程求解得到渐变序列 .该方法不直接插值顶点坐标,而是将源曲线与目标曲线视为定义在公共定义域上的标量场,并在梯度域进行梯度场操纵.对中间帧曲线周长以及平面曲线所包围的内部面积变化的统计表明:该算法尽可能地保持了几何形状的刚性,在中间帧求解的稳定性方面该算法优于同类其他方法.     

Biharmonic Coordinates

Barycentric coordinates are an established mathematical tool in computer graphics and geometry processing, providing a convenient way of interpolating scalar or vector data from the boundary of a planar domain to its interior. Many different recipes for barycentric coordinates exist, some offering the convenience of a closed‐form expression, some providing other desirable properties at the expense of longer computation times. For example, harmonic coordinates, which are solutions to the Laplace equation, provide a long list of desirable properties (making them suitable for a wide range of applications), but lack a closed‐form expression. We derive a new type of barycentric coordinates based on solutions to the biharmonic equation. These coordinates can be considered a natural generalization of harmonic coordinates, with the additional ability to interpolate boundary derivative data. We provide an efficient and accurate way to numerically compute the biharmonic coordinates and demonstrate their advantages over existing schemes. We show that biharmonic coordinates are especially appealing for (but not limited to) 2D shape and image deformation and have clear advantages over existing deformation methods.     

Stabilization of linear time varying systems over uncertain channels

In this paper, we study the problem of control of discrete‐time linear time varying systems over uncertain channels. The uncertainty in the channels is modeled as a stochastic random variable. We use exponential mean square stability of the closed‐loop system as a stability criterion. We show that fundamental limitations arise for the mean square exponential stabilization for the closed‐loop system expressed in terms of statistics of channel uncertainty and the positive Lyapunov exponent of the open‐loop uncontrolled system. Our results generalize the existing results known in the case of linear time invariant systems, where Lyapunov exponents are shown to emerge as the generalization of eigenvalues from linear time invariant systems to linear time varying systems. Simulation results are presented to verify the main results of this paper. Copyright © 2012 John Wiley & Sons, Ltd.     

Tools for computing tangent curves for linearly varying vectorfields over tetrahedral domains

We present some very efficient and accurate methods for computing tangent curves for three-dimensional flows. Our methods work directly in physical coordinates, eliminating the usual need to switch back and forth with computational coordinates. Unlike conventional methods, such as Runge-Kutta, for computing tangent curves which give only approximations, our methods produce exact values based upon piecewise linear variation over a tetrahedrization of the domain of interest. We use balycentric coordinates in order to efficiently track cell-to-cell movement of the tangent curves     

Link between Bézier and Lagrange curve and surface schemes

A class of curves that vary continuously between polynomial Lagrange interpolants and polynomial Bézier curves is discussed. An element in this class is specified by a real number which could be used as a shape parameter for Bézier curves. A geometric derivation of this scheme is given, and the connection to Pólya curves is pointed out. A generalization to the case of tensor product and triangular surface patches is also described.     

Projected Tikhonov Regularization of Large-Scale Discrete Ill-Posed Problems

The solution of linear discrete ill-posed problems is very sensitive to perturbations in the data. Confidence intervals for solution coordinates provide insight into the sensitivity. This paper presents an efficient method for computing confidence intervals for large-scale linear discrete ill-posed problems. The method is based on approximating the matrix in these problems by a partial singular value decomposition of low rank. We investigate how to choose the rank. Our analysis also yields novel approaches to the solution of linear discrete ill-posed problems with solution norm or residual norm constraints.     

Approaches to robust ℋ︁∞ static output feedback control of discrete‐time piecewise‐affine systems with norm‐bounded uncertainties

This paper investigates the problem of robust ??_∞ static output feedback controller design for a class of discrete‐time piecewise‐affine systems with norm‐bounded time‐varying parametric uncertainties. The objective is to design a piecewise‐linear static output feedback controller guaranteeing the asymptotic stability of the resulting closed‐loop system with a prescribed ??_∞ disturbance attenuation level. Based on a piecewise Lyapunov function combined with S‐procedure, Projection lemma, and some matrix inequality convexifying techniques, several novel approaches to the static output feedback controller analysis and synthesis are developed for the underlying piecewise‐affine systems. It is shown that the controller gains can be obtained by solving a set of strict linear matrix inequalities (LMIs) or a family of LMIs parameterized by one or two scalar variables, which are numerically efficient with commercially available software. Finally, three simulation examples are provided to illustrate the effectiveness of the proposed approaches. Copyright © 2010 John Wiley & Sons, Ltd.     

Enriched finite element subspaces for dual–dual mixed formulations in fluid mechanics and elasticity

In this paper we unify the derivation of finite element subspaces guaranteeing unique solvability and stability of the Galerkin schemes for a new class of dual-mixed variational formulations. The approach, which has been applied to several linear and nonlinear boundary value problems, is based on the introduction of additional unknowns given by the flux and the gradient of velocity, and by the stress and strain tensors and rotations, for fluid mechanics and elasticity problems, respectively. In this way, the procedure yields twofold saddle point operator equations as the resulting weak formulations (also named dual–dual ones), which are analyzed by means of a slight generalization of the well known Babuška–Brezzi theory. Then, in order to introduce well posed Galerkin schemes, we extend the arguments used in the continuous case to the discrete one, and show that some usual finite elements need to be suitably enriched, depending on the nature of the problem. This leads to piecewise constant functions, Raviart–Thomas of lowest order, PEERS elements, and the deviators of them, as the appropriate subspaces.     

Learning the stress function pattern of ordered weighted average aggregation using DBSCAN clustering

Ordered weighted average (OWA) operator provides a parameterized class of mean type operators between the minimum and the maximum. It is an important tool that can reflect the strategy of a decision maker for decision-making problems. In this study, the idea of obtaining the stress function from OWA weights has been put forward to generalize and characterize OWA weights. The main idea in this paper is mainly constructed on the basis that, generally, stress functions can be constructed using a mixture of constant and linear components. So, we can consider the stress function as a piecewise linear function. For obtaining stress functions as piecewise linear functions, we present a clustering-based approach for OWA weight generalization. This generalization is made using the DBSCAN algorithm as the learning method of a stress function associated with known OWA weights. In the learning process, the whole data set is divided into clusters, and then linear functions are obtained via a least squares estimator.     

Piecewise linear skeletonization using principal curves

Proposes an algorithm to find piecewise linear skeletons of handwritten characters by using principal curves. The development of the method was inspired by the apparent similarity between the definition of principal curves (smooth curves which pass through the "middle" of a cloud of points) and medial axes (smooth curves that run equidistantly from the contours of a character image). The central fitting-and-smoothing step of the algorithm is an extension of the polygonal line algorithm, which approximates principal curves of data sets by piecewise linear curves. The polygonal line algorithm is extended to find principal graphs and complemented with two steps specific to the task of skeletonization: an initialization method to capture the approximate topology of the character, and a collection of restructuring operations to improve the structural quality of the skeleton produced by the initialization method. An advantage of our approach over existing methods is that we optimize the skeleton graph by minimizing an intuitive and explicit objective function that captures the two competing criteria of smoothing the skeleton and fitting it closely to the pixels of the character image. We tested the algorithm on isolated handwritten digits and images of continuous handwriting. The results indicated that the proposed algorithm can find a smooth medial axis in the great majority of a wide variety of character templates and that it substantially improves the pixel-wise skeleton obtained by traditional thinning methods     

Surface Patches from Unorganized Space Curves

Recent 3D sketch tools produce networks of three‐space curves that suggest the contours of shapes. The shapes may be non‐manifold, closed three‐dimensional, open two‐dimensional, or mixed. We describe a system that automatically generates intuitively appealing piecewise‐smooth surfaces from such a curve network, and an intelligent user interface for modifying the automatically chosen surface patches. Both the automatic and the semi‐automatic parts of the system use a linear algebra representation of the set of surface patches to track the topology. On complicated inputs from ILoveSketch [ [BBS08] ], our system allows the user to build the desired surface with just a few mouse‐clicks.     

Harmonic Embeddings for Linear Shape Analysis

We present a novel representation of shape for closed contours in ℝ² or for compact surfaces in ℝ³ explicitly designed to possess a linear structure. This greatly simplifies linear operations such as averaging, principal component analysis or differentiation in the space of shapes when compared to more common embedding choices such as the signed distance representation linked to the nonlinear Eikonal equation. The specific choice of implicit linear representation explored in this article is the class of harmonic functions over an annulus containing the contour. The idea is to represent the contour as closely as possible by the zero level set of a harmonic function, thereby linking our representation to the linear Laplace equation. We note that this is a local represenation within the space of closed curves as such harmonic functions can generally be defined only over a neighborhood of the embedded curve. We also make no claim that this is the only choice or even the optimal choice within the class of possible linear implicit representations. Instead, our intent is to show how linear analysis of shape is greatly simplified (and sensible) when such a linear representation is employed in hopes to inspire new ideas and additional research into this type of linear implicit representations for curves. We conclude by showing an application for which our particular choice of harmonic representation is ideally suited.     

A method for the computation of self-sustained oscillations in systems with piecewise linear elements

In this paper a study of the self-sustained oscillations in systems which contain a nonlinear element of the piecewise linear type with or without memory is presented. A method is developed by which one may systematically detect possible symmetric limit cycles, and subsequently identify the oscillations completely. Vector-matrix techniques are employed to transform certain points between regions of the phase space. The boundaries of the regions of phase space correspond to the break points of the piecewise linear characteristic. A set of equations which determine points on a possible closed trajectory are derived using the transformations. Certain switching conditions which are necessary for the existence of an oscillation are applied to these equations. From the solution of the resulting equations, points on the closed trajectory are determined. Using the values of the coordinates of these points, one computes the waveform of the oscillation. Applications of the method to control systems with various configurations are discussed in theory and illustrative examples are presented.