Finite Circle Research Articles

An improved adaptive constraint aggregation for integrated layout and topology optimization

The purpose of this paper is to present a Kreisselmeier–Steinhauser (KS) function based adaptive constraint aggregation approach. It is implemented within the integrated layout and topology optimization of multi-component structure systems to avoid using large numbers of non-overlapping constraints defined by means of the previously proposed finite circle method (FCM). An improved adaptive approach is then put forward to obtain proper aggregation parameters for the KS function based constraint aggregation, contributing to less numerical difficulties while meeting the same aggregation precision compared with the existing adaptive approach. Furthermore, the complex step derivative approximation is utilized to yield better sensitivities for the aggregated constraint functions with high nonlinearity. Moreover, during the integrated layout and topology optimization, multi-point constraints (MPC) are applied to establish the interconnections between movable components and supporting structures, which can use fixed finite element meshes and analytical sensitivities. Finally, some numerical examples are tested to demonstrate the validity and effectiveness of the proposed formulation.

Deforming convex curves with fixed elastic energy

This paper presents a curve flow which preserves the elastic energy of the evolving curve. If the initial curve is a planar, simple and smooth curve with positive curvature then the local and global existence of the flow is proved. Under this flow, the evolving curve will converge to a finite circle in the C∞ metric as time goes to infinity.

AN AREA-PRESERVING FLOW FOR CLOSED CONVEX PLANE CURVES

Motivated by Gage [On an area-preserving evolution equation for plane curves, in Nonlinear Problems in Geometry, ed. D. M. DeTurck, Contemporary Mathematics, Vol. 51 (American Mathematical Society, Providence, RI, 1986), pp. 51–62] and Ma–Cheng [A non-local area preserving curve flow, preprint (2009), arXiv:0907.1430v2, [math.DG]], in this paper, an area-preserving flow for convex plane curves is presented. This flow will decrease the perimeter of the evolving curve and make the curve more and more circular during the evolution process. And finally, as t goes to infinity, the limiting curve will be a finite circle in the C∞ metric.

Integrated layout design of multi-component systems using XFEM and analytical sensitivity analysis

This study presents the integrated layout optimization of multi-component systems using a fixed mesh. The optimization formulation is established under the framework of the extended finite element method (XFEM). The level set method is used to represent components and is combined with the XFEM to describe material discontinuities across elements. Sensitivity analysis is proposed with respect to geometric variables of components and pseudo-densities of the basic structure. An analytical shape sensitivity analysis method with respect to positions and shapes of components is developed. Both solid and void components are considered to show the efficiency and accuracy of the proposed shape sensitivity analysis method. Furthermore, a revised finite circle method that adapts shape changes of elliptical components is proposed for the definition of non-overlapping constraints. Finally, numerical examples of maximizing the structural stiffness are tested to demonstrate the proposed method.

Some Recent Advances in the Integrated Layout Design of Multicomponent Systems

We provide an introduction and state of the art overview of integrated layout design of multicomponent systems. We review several packing optimization and overlap detection strategies, some tree-based methods, such as octrees and spheretrees, and a finite circle method (FCM) proposed to favor gradient-based optimization algorithms. Integrated layout design techniques for simultaneous packing and structure topology optimization of multicomponent systems are reviewed; two typical approaches for system stiffness maximization are reviewed and compared in detail. Design of multicomponent systems under inertia forces is presented using polynomial interpolation models; constraints to the centroid position, moment of inertia, and volume fraction are included. Applications to piezoelectric multi-actuated microtools and integrated layout design of bridge systems are presented. Finally, the effectiveness of the FCM, applications to 3D problems, and local optimum phenomena are discussed.

The asymptotic distribution of circles in the orbits of Kleinian groups

Let \(\mathcal{P}\) be a locally finite circle packing in the plane ℂ invariant under a non-elementary Kleinian group Γ and with finitely many Γ-orbits. When Γ is geometrically finite, we construct an explicit Borel measure on ℂ which describes the asymptotic distribution of small circles in \(\mathcal{P}\), assuming that either the critical exponent of Γ is strictly bigger than 1 or \(\mathcal{P}\) does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Γ. Our construction also works for \(\mathcal{P}\) invariant under a geometrically infinite group Γ, provided Γ admits a finite Bowen-Margulis-Sullivan measure and the Γ-skinning size of \(\mathcal{P}\) is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.

ON A CURVE EXPANDING FLOW WITH A NON-LOCAL TERM

This paper deals with a non-local curve evolution problem in the plane which will increase both the length of the evolving curve and the area it bounds and make the evolving curve more and more circular during the evolution process. And the limiting shape of the evolving curve will be a finite circle (i.e. a circle with finite radius) as the time t goes to infinity.

Gibbs phenomenon for Fourier series on finite sets

The Gibbs phenomenon is described for the Fourier series of a function at its jump, the function being defined along the finite circle ℤ/pℤ.

Integrated layout design of multi‐component system

AbstractA new integrated layout optimization method is proposed here for the design of multi‐component systems. By introducing movable components into the design domain, the components layout and the supporting structural topology are optimized simultaneously. The developed design procedure mainly consists of three parts: (i) Introduction of non‐overlap constraints between components. The finite circle method (FCM) is used to avoid the components overlaps and also overlaps between components and the design domain boundaries. (ii) Layout optimization of the components and supporting structure. Locations and orientations of the components are assumed as geometrical design variables for the optimal placement while topology design variables of the supporting structure are defined by the density points. Meanwhile, embedded meshing techniques are developed to take into account the finite element mesh change caused by the component movements. (iii) Consistent material interpolation scheme between element stiffness and inertial load. The commonly used solid isotropic material with penalization model is improved to avoid the singularity of localized deformation in the presence of design dependent loading when the element stiffness and the involved inertial load are weakened by the element material removal.Finally, to validate the proposed design procedure, a variety of multi‐component system layout design problems are tested and solved on account of inertia loads and gravity center position constraint. Solutions are compared with traditional topology designs without component. Copyright © 2008 John Wiley & Sons, Ltd.

Curvilinear polygons, finite circle packings, and normal grain growth

Curvilinear polygons, finite circle packings, and normal grain growth

Curvilinear polygons, finite circle packings, and normal grain growth

A key element of analytical models for curvature-driven grain growth is the formulation of a growth law for individual grains. Grain size distributions can be derived through the use of such growth laws. The present paper reviews some recent progress in developing a mean-field statistical theory of normal grain growth. A paradox related to curvilinear polygons is shown to support the expectation that the grain size distribution has a finite cutoff. The circumference of the circumscribing circle and the perimeter of the curvilinear polygon become exactly equal when the number of sides is 12. The boundary of an average 12-sided grain has an integral curvature that is equal in magnitude to the integral boundary of a completely circular grain. For this reason, grains with more than 12 sides are suggested to be energetically unfavourable, a result which is in excellent agreement with simulation data. In the concluding part, finite circle packings are employed to interpret a topological relationship found for Potts-model-type simulations of 2D grain growth. It is briefly indicated how spherical packings may help to develop some intuition about topological relationships found for normal grain growth in three dimensions.

Key Distribution Patterns Using Tangent Circle Structures

The problem of key management in a communications network is of primary importance. A key distribution pattern is an incidence structure which provides a secure method of distributing keys in a large network reducing storage requirements. It is of interest to find explicit constructions for key distribution patterns. In O’Keefe [5–7], examples are shown using the finite circle geometries (Minkowski, Laguerre and inversive planes); in Quinn [12], examples are constructed from conics in finite projective and affine planes. In this paper, we construct some examples using the finite tangent-circle structures, introduced in Quattrocchi and Rinaldi [10] and we give a comparison of the storage requirements.

Renormalization group analysis of tachyon condensation

Renormalization group analysis of boundary conformal field theory on bosonic D25-brane is used to study tachyon condensation. Placing the lump on a finite circle and triggering only the first three tachyon modes, the theory flows to nearby IR fixed point representing lumps that are extended object with definite profile. The boundary entropy corresponding to the D24-brane tension is calculated in the leading order in perturbative analysis which decreases under RG flow and agrees with the expected result to an accuracy of 8%. Multicritical behaviour of the IR theory suggests that the end point of the flow represents a configuration of two D24-branes. Analogy with Kondo physics is discussed.

Mechanical model of the pattern formation of lotus receptacles

A two-dimensional mechanical model for pattern formation is described. The lotus receptacle is modelled as a finite circle packing with an elastic boundary of minimum length. The deformation process is controlled by an increase in internal pressure. Equilibrium configurations are sought. During the process, the configuration is sometimes unchanged, sometimes changes continuously, but in some cases, by loss of stability, a dramatic rearrangement of the circles occurs. At the pressure under which the contact forces between circles vanish, the circles form a locally densest packing in a circle.

Fourier analysis on finite groups and applications

Introduction Cast of characters Part I: 1. Congruences and the quotient ring of the integers mod n 1.2 The discrete Fourier transform on the finite circle 1.3 Graphs of Z/nZ, adjacency operators, eigenvalues 1.4 Four questions about Cayley graphs 1.5 Finite Euclidean graphs and three questions about their spectra 1.6 Random walks on Cayley graphs 1.7 Applications in geometry and analysis 1.8 The quadratic reciprocity law 1.9 The fast Fourier transform 1.10 The DFT on finite Abelian groups - finite tori 1.11 Error-correcting codes 1.12 The Poisson sum formula on a finite Abelian group 1.13 Some applications in chemistry and physics 1.14 The uncertainty principle Part II. Introduction 2.1 Fourier transform and representations of finite groups 2.2 Induced representations 2.3 The finite ax + b group 2.4 Heisenberg group 2.5 Finite symmetric spaces - finite upper half planes Hq 2.6 Special functions on Hq - K-Bessel and spherical 2.7 The general linear group GL(2, Fq) 2.8. Selberg's trace formula and isospectral non-isomorphic graphs 2.9 The trace formula on finite upper half planes 2.10 The trace formula for a tree and Ihara's zeta function.

Quasicrystals and the Wulff-Shape

Infinite sphere packings give information about the structure but not about the shape of large dense sphere packings. For periodic sphere packings a new method was introduced in [W2], [W3], [Sc], and [BB], which gave a direct relation between dense periodic sphere packings and the Wulff-shape, which describes the shape of ideal crystals. In this paper we show for the classical Penrose tiling that dense finite quasiperiodic circle packings also lead to a Wulff-shape. This indicates that the shape of quasicrystals might be explained in terms of a finite packing density. Here we prove an isoperimetric inequality for unions of Penrose rhombs, which shows that the regular decagon is, in a sense, optimal among these sets. Motivated by the analysis of linear densities in the Penrose plane we introduce a surface energy for a class of polygons, which is analogous to the Gibbs—Curie surface energy for periodic crystals. This energy is minimized by the Wulff-shape, which is always a polygon and in certain cases it is the regular decagon, in accordance with the fivefold symmetry of quasicrystals.

Rotation and entropy

For a given mapf:X→Xf: X \to Xand an observableφ:X→Rd,\varphi : X \to \mathbb {R} ^{d},rotation vectors are the limits of ergodic averages ofφ.\varphi .We study which part of the topological entropy offfis associated to a given rotation vector and which part is associated with many rotation vectors. According to this distinction, we introduce directional and lost entropies. We discuss their properties in the general case and analyze them more closely for subshifts of finite type and circle maps.

Elimination of color in 1+1-dimensional QCD

We discuss QCD in one space and one time dimension on a finite circle. Boundary conditions force the retention of quantum mechanical (zero-mode) gauge degrees of freedom in a Hamiltonian formulation. We explicitely consider systems of static quarks and find that the dynamics of the zero modes lead to an elimination of fermionic non-singlet states from the spectrum in the continuum limit. Thus a mechanism is exhibited which justifies the quark model prescription of coupling the fermionic constituents alone already to a color singlet. As a further consequence, the effective interaction in larger clusters of quarks has a many-body structure of the flipflop type.

Fractons in the twisted multiflavor Schwinger model.

We consider two-dimensional QED with several fermion flavors on a finite spatial circle. A modified version of the model with {\em flavor-dependent} boundary conditions $\psi_p(L) = e^{2\pi ip/ N} \psi_p(0)$, $p = 1, \ldots , N$ is discussed ($N $ is the number of flavors). In this case a non-contactable contour in the space of the gauge fields is {\em not} determined by large gauge transformations. The Euclidean path integral acquires the contribution from the gauge field configurations with fractional topological charge. The configuration with $\nu = 1/N$ is responsible for the formation of the fermion condensate $\langle\bar{\psi}_p \psi_p\rangle_0$. The condensate dies out as a power of $L^{-1}$ when the length $L$ of the spatial box is sent to infinity. Implications of this result for non-abelian gauge field theories are discussed in brief.

On upper bounds for the logarithmic derivative of a meromorphic function

Upper bounds for the logarithmic derivative of a fast growing meromorphic function in the complex plane C as well as in a finite circle D = { z : |z| < R < ∞} are established.