Efficient numerical solution of Neumann problems on complicated domains

Abstract We consider elliptic partial differential equations with Neumann boundary conditions on complicated domains. The discretization is performed by composite finite elements. The a priori error analysis typically is based on precise knowledge of the regularity of the solution. However, the constants in the regularity estimates possibly depend critically on the geometric details of the domain and the analysis of their quantitative influence is rather involved. Here, we consider a polyhedral Lipschitz domain Ω with a possibly huge number of geometric details ranging from size O(ε) to O(1). We assume that Ω is a perturbation of a simpler Lipschitz domain Ω. We prove error estimates where only the regularity of the partial differential equation on Ω is needed along with bounds on the norm of extension operators which are explicit in appropriate geometric parameters. Since composite finite elements allow a multiscale discretization of problems on complicated domains, the linear system which arises can be solved by a simple multi-grid method. We show that this method converges at an optimal rate independent of the geometric structure of the problem.     

On the Topology and Geometry of Spaces of Affine Shapes

We define the space of affine shapes of k points in R ⁿ to be the topological quotient of (R ⁿ ) ^k modulo the natural action of the affine group of R ⁿ . These spaces arise naturally in many image-processing applications, and despite having poor separation properties, have some topological and geometric properties reminiscent of the more familiar Procrustes shape spaces Σ _n ^k in which one identifies configurations related by an orientation-preserving Euclidean similarity transformation. We examine the topology of the connected, non-Hausdorff spaces Sh _n ^k in detail. Each Sh _n ^k is a disjoint union of naturally ordered strata, each of which is homeomorphic in the relative topology to a Grassmannian, and we show how the strata are attached to each other. The top stratum carries a natural Riemannian metric, which we compute explicitly for k>n, expressing the metric purely in terms of “pre-shape” data, i.e. configurations of k points in R ⁿ .     

Global state and feedback equivalence of nonlinear systems

The problem of finding global state space transformations and global feedback of the form u(t)= α(x) + ν(t) to transform a given nonlinear system to a controllable linear system on Rⁿ or on an open subset of Rⁿ, is considered here. We give a complete set of differential geometric conditions which are equivalent to the existence of a solution to the above problem.     

Geometric Invariant Shape Representations Using Morphological Multiscale Analysis

In this paper, we present a new geometric invariant shape representation using morphological multiscale analysis. The geometric invariant is based on the area and perimeter evolution of the shape under the action of a morphological multiscale analysis. First, we present some theoretical results on the perimeter and area evolution across the scales of a shape. In the case of similarity transformations, the proposed geometric invariant is based on a scale-normalized evolution of the isoperimetric ratio of the shape. In the case of general affine geometric transformations the proposed geometric invariant is based on a scale-normalized evolution of the area. We present some numerical experiments to evaluate the performance of the proposed models. We present an application of this technique to the problem of shape classification on a real shape database and we study the well-posedness of the proposed models in the framework of viscosity solution theory.     

Best domain for an elliptic problem in cartesian coordinates by means of shape‐measure

In (ZAA J. Anal. Appl., Vol. 16, No. 1, pp. 143–155) we introduced a method to determine the optimal domains for elliptic optimal‐shape design problems in polar coordinates. However, the same problem in cartesian coordinates, which are more applicable, is found to be much harder, therefore we had to develop a new approach for these designs. Herein, the unknown domain is divided into a fixed and a variable part and the optimal pair of the domain and its optimal control, is characterized in two stages. Firstly, the optimal control for the each given domain is determined by changing the problem into a measure‐theoretical one, replacing this with an infinite dimensional linear programming problem and approximating schemes; then the nearly optimal control function is characterized. Therefore a function that offers the optimal value of the objective function for a given domain, is defined. In the second stage, by applying a standard optimization method, the global minimizer pair will be obtained. Some numerical examples are also given. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Skeleton-based Variational Mesh Deformations

In this paper, a new free-form shape deformation approach is proposed. We combine a skeleton-based mesh deformation technique with discrete differential coordinates in order to create natural-looking global shape deformations. Given a triangle mesh, we first extract a skeletal mesh, a two-sided Voronoibased approximation of the medial axis. Next the skeletal mesh is modified by free-form deformations. Then a desired global shape deformation is obtained by reconstructing the shape corresponding to the deformed skeletal mesh. The reconstruction is based on using discrete differential coordinates. Our method preserves fine geometric details and original shape thickness because of using discrete differential coordinates and skeleton-based deformations. We also develop a new mesh evolution technique which allow us to eliminate possible global and local self-intersections of the deformed mesh while preserving fine geometric details. Finally, we present a multi-resolution version of our approach in order to simplify and accelerate the deformation process. In addition, interesting links between the proposed free-form shape deformation technique and classical and modern results in the differential geometry of sphere congruences are established and discussed.     

A direct boundary integral equation method for the numerical construction of harmonic functions in three-dimensional layered domains containing a cavity

We consider boundary value problems for the Laplace equation in three-dimensional multilayer domains composed of an infinite strip layer of finite height and a half-space containing a bounded cavity. The unknown (harmonic) function satisfies the Neumann boundary condition on the exterior boundary of the strip layer (i.e. at the bottom of the first layer), the Dirichlet, Neumann or Robin boundary condition on the boundary surface of the cavity and the corresponding transmission (matching) conditions on the interface layer boundary. We reduce this boundary value problem to a boundary integral equation over the boundary surface of the cavity by constructing Green's matrix for the corresponding transmission problem in the domain consisting of the infinite layer and the half-space (not with the cavity). This direct integral equation approach leads, for any of the above boundary conditions, to boundary integral equations with a weak singularity on the cavity. The numerical solution of this equation is realized by Wienert's [Die Numerische approximation von Randintegraloperatoren für die Helmholtzgleichung im R ³, Ph.D. thesis, University of Göttingen, Germany, 1990] method. The reduction of the problem, originally set in an unbounded three-dimensional region, to a boundary integral equation over the boundary of a bounded domain, is computationally advantageous. Numerical results are included for various boundary conditions on the boundary of the cavity, and compared against a recent indirect approach [R. Chapko, B.T. Johansson, and O. Protsyuk, On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R ³ with cavities, J. Numer. Appl. Math. (Kyiv) 105 (2011), pp. 4–18], and the results obtained show the efficiency and accuracy of the proposed method. In particular, exponential convergence is obtained for smooth cavities.     

Nonlinear uncertain systems and robustness optimization

An approach to (adaptive) stabilization is developed for a class of uncertain nonlinearly perturbed linear dynamical systems modelled by a controlled differential inclusion on Rⁿ. The essential features are nonlinear feedback and optimization of robustness with respect to unmatched uncertainty. The approach is geometric: (i) a family ?? of proper subspaces V ? Rⁿ is introduced with the property that {0} ? V is an asymptotically stable equilibrium for the projection, onto V, of the unperturbed linear flow; (ii) for the perturbed system, it is then shown that each V ? ?? can be rendered attractive and (almost) invariant by some choice of feedback determined by the nature of the perturbation and other a priori system information; specifically, under differing hypotheses, we can render V ? ?? almost invariant and attractive, or invariant and finite-time attractive, or adaptively attractive via, respectively, linear high-gain feedback or discontinuous feedback, or adaptive feedback. Associated with each V ? ?? is a measure of robustness with respect to unmatched uncertainty (a stability radius), quantified by a real parameter η. The problem of optimizing this measure is formulated as an optimization problem over ??, solvable (in a supremal sense of η-maximizing sequences (V_j) ??) via solutions of particular algebraic Riccati equations of order k ≤ (n - m), where m ≤ n is the control dimension; typically k = n - m..     

Analysis and synthesis of robust H ∞ static output feedback control subject to actuator saturation

In this article, static output feedback (SOF) control analysis and synthesis are conducted for a linear continuous-time system subject to actuator saturation with the H ^∞ setting. Typically, the SOF problem is nonconvex; the existence of SOF control can be expressed in terms of the solvability of bilinear matrix inequalities. The actuator saturation problem is also considered since the driving capacity of an actuator is limited in practical applications. Using the singular value decomposition approach, the H ^∞ SOF controller design problem with actuator saturation can be expressed in terms of an eigenvalue problem (EVP) which can be efficiently solved using the LMI toolbox in Matlab. The balance between the minimization of the attenuation level of the H ^∞ performance and the maximisation of the estimation of the domain of attraction is considered in our approach via solving the corresponding EVP. To illustrate the proposed design procedure for two types of prescribing shape reference set, two numerical examples are included.     

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.     

GWCNN: A Metric Alignment Layer for Deep Shape Analysis

Deep neural networks provide a promising tool for incorporating semantic information in geometry processing applications. Unlike image and video processing, however, geometry processing requires handling unstructured geometric data, and thus data representation becomes an important challenge in this framework. Existing approaches tackle this challenge by converting point clouds, meshes, or polygon soups into regular representations using, e.g., multi‐view images, volumetric grids or planar parameterizations. In each of these cases, geometric data representation is treated as a fixed pre‐process that is largely disconnected from the machine learning tool. In contrast, we propose to optimize for the geometric representation during the network learning process using a novel metric alignment layer. Our approach maps unstructured geometric data to a regular domain by minimizing the metric distortion of the map using the regularized Gromov–Wasserstein objective. This objective is parameterized by the metric of the target domain and is differentiable; thus, it can be easily incorporated into a deep network framework. Furthermore, the objective aims to align the metrics of the input and output domains, promoting consistent output for similar shapes. We show the effectiveness of our layer within a deep network trained for shape classification, demonstrating state‐of‐the‐art performance for nonrigid shapes.     

Dynamic shape control for a flexible beam

In this paper we discuss the dynamic shape control for a flexible beam such that even under unknown step disturbances, the shape of the beam will track the desired shape after a certain regulation time. In order to treat the case of point controls and point observations, we formulate the flexible beam by a second-order partial differential equation on a Hilbert space H^?1(0, 1). We apply the theory of multivariable tuning regulators to the dynamic shape control problem. Two kinds of methods will be proposed for the determination of an integral feedback gain matrix M. Simulation results will show the efficiency of the theory developed in this paper.     

Möbius变换下四次有理抛物-PH曲线的C2 Hermite插值

目的 曲线插值问题在机器人设计、机械工业、航天工业等诸多现代工业领域都有广泛的应用，而已知端点数据的Hermite插值是计算机辅助几何设计中一种常用的曲线构造方法，本文讨论了一种偶数次有理等距曲线，即四次抛物-PH曲线的C² Hermite插值问题。方法 基于M bius变换引入参数，利用复分析的方法构造了四次有理抛物-PH曲线的C² Hermite插值，给出了具体插值算法及相应的Bézier曲线表示和控制顶点的表达式。结果 通过给出"合理"的端点插值数据，以数值实例表明了该算法的有效性，所得12条插值曲线中，结合最小绝对旋转数和弹性弯曲能量最小化两种准则给出了判定满足插值条件最优曲线的选择方法，并以具体实例说明了与其他插值方法的对比分析结果。结论 本文构造了M bius变换下的四次有理抛物-PH曲线的C² Hermite插值，在保证曲线次数较低的情况下，达到了连续性更高的插值条件，计算更为简单，插值效果明显，较之传统奇数次PH曲线具有更加自然的几何形状，对偶数次PH曲线的相关研究具有一定意义。     

Design and implementation of a new tuned hybrid intelligent model to predict the uniaxial compressive strength of the rock using SFS-ANFIS

This study proposes a novel design to systematically optimize the parameters for the adaptive neuro-fuzzy inference system (ANFIS) model using stochastic fractal search (SFS) algorithm. To affirm the efficiency of the proposed SFS-ANFIS model, the predicting results were compared with ANFIS and three hybrid methodologies based on ANFIS combined with genetic algorithm (GA), differential evolution (DE), and particle swarm optimization (PSO). Accurate prediction of uniaxial compressive strength (UCS) is of great significance for all geotechnical projects such as tunnels and dams. Hence, this study proposes the use of SFS-ANFIS, GA-ANFIS, DE-ANFIS, PSO-ANFIS, and ANFIS models to predict UCS. In this regard, the fresh water tunnel of Pahang–Selangor located in Malaysia was considered and the requirement data samples were collected. Different statistical metrics such as coefficient of determination (R²) and mean absolute error were used to evaluate the models. Referring to the efficiency results of SFS-ANFIS, it can be found that the SFS-ANFIS (with the R² of 0.981) has higher ability than PSO-ANFIS, DE-ANFIS, GA-ANFIS, and ANFIS models in predicting the UCS.

     

Medial Axis Isoperimetric Profiles

Recently proposed as a stable means of evaluating geometric compactness, the isoperimetric profile of a planar domain measures the minimum perimeter needed to inscribe a shape with prescribed area varying from 0 to the area of the domain. While this profile has proven valuable for evaluating properties of geographic partitions, existing algorithms for its computation rely on aggressive approximations and are still computationally expensive. In this paper, we propose a practical means of approximating the isoperimetric profile and show that for domains satisfying a “thick neck” condition, our approximation is exact. For more general domains, we show that our bound is still exact within a conservative regime and is otherwise an upper bound. Our method is based on a traversal of the medial axis which produces efficient and robust results. We compare our technique with the state-of-the-art approximation to the isoperimetric profile on a variety of domains and show significantly tighter bounds than were previously achievable.     

The indecomposability problem in binary morphology: An algorithmic approach

An indecomposable shape is like a prime number. It cannot be decomposed further as a Minkowski sum of two simpler shapes. With respect to Minkowski addition (dilation), therefore, the indecomposable shapes are the fundamental building blocks of all geometric shapes. However, just as it is difficult to identify whether a given number is a prime number or not, it is equally or more difficult to say whether a given shape is indecomposable or not. In this paper we take up a subdomain of binary images, called the weakly taxicab convex image domain, and show how the indecomposability problem in that shape domain can be approached in a manner closely analogous to the number theoretic way. Apart from our attempt to show that the indecomposability problem is an extremely interesting mathematical problem, our algorithmic treatment of the problem also leads to an efficient method of computing Minkowski addition and decomposition of binary images.     

GCT变换及几何图形形状相似性判定

目的 人类的视觉能力可以轻易地判定两个几何图形形状的相似性,但是,这对于计算机来说仍是一个开问题。在计算机视觉应用中,不仅需要对图形形状进行分类和相似性判定,还需要对图形形状的相似性度量给出与人类的视觉判断一致的结果,这是目前图形形状表示和分类算法没有较好解决的问题。方法 通过GCT变换,将图形形状从实数空间的坐标表示变换到复数空间的复数特征向量表示,进而将判定两个几何形状的相似性问题转化为判定它们的复数空间特征向量的相似性问题。GCT变换不仅可以判定图形形状的相似性,它还是可逆的,它可以近似重建原图形形状。结果 GCT变换具有位移、尺度和旋转不变性,它不仅可以判定几何图形形状的相似性,给出与人类视觉判断一致的相似性结果,而且在两个几何图形形状相似的情况下,还能计算出它们的角度旋转和尺度缩放。结论 对于封闭的几何形状,如果几何形状的中心点位于几何形状的内部且过中心点的任一直线与该几何形状有且只有两个交点,理论证明和实验验证,GCT变换可以高效准确地判定这类几何图形形状的相似性,并给出与人类视觉判定一致的结果。     

Signal space geometry

The purpose of this paper is to show the usefulness of the concepts of differential geometry for the problem of multiple frequency determination or multiple frequency demodulation. The exterior algebra of Grassmann provides the tool which is natural for analysing the problem. We describe a distance between k-dimensional subspaces in Rⁿ and a geometric “least squares” which allows us to uniquely associate with each r-dimensional subspace determined by the observations an r-dimensional subspace which determines the r unknown frequencies.     

Bézier curves and surfaces with shape parameters

Bézier curves with n shape parameters and triangular Bézier surfaces with 3n(n+1)/2 shape parameters are presented in this paper. The geometric significance of the shape parameters and the geometric properties of these curves and surfaces are discussed. The shapes of the curves and the surfaces can be modified intuitively, foreseeably and precisely by changing the values of the shape parameters.