Arbitrary Polygonal Domains Research Articles

The Harmonic Gbc Function Map is a Bijection If the Target Domain is Convex

Harmonic generalized barycentric coordinates (GBC) functions have been used for cartoon animation since an early work in 2006\cite{JMDGS06}. A computational procedure was further developed in \cite{SH15} for deformation between any two polygons. The bijectivity of the map based on harmonic GBC functions is still murky in the literature. In this paper, we present an elementary proof of the bijection of the harmonic GBC map transforming from one arbitrary polygonal domain $V$ to a convex polygonal domain $W$. This result is further extended to a more general harmonic map from one simply connected domain $V$ to a convex domain $W$ if the harmonic map preserves the orientation of the boundary of the domain $V$. In addition, we shall point out that the harmonic GBC map is also a diffeomorphism over the interior of $V$ to the interior of $W$. Finally, we remark on how to construct a harmonic GBC map from $V$ to $W$ when the number of vertices of $V$ is different from the number of vertices of $W$ and how to construct harmonic GBC functions over a polygonal domain with a hole or holes. We also point out that it is possible to use the harmonic GBC map to deform a nonconvex polygon $V$ to another nonconvex polygon $W$ by a good arrangement of the boundary map between $\partial V$ and $\partial W$. Several numerical deformations based on images are presented to show the effectiveness of the map based on bivariate spline approximation of the harmonic GBC functions.

An adaptive interpolation element free Galerkin method based on a posteriori error estimation of FEM for Poisson equation

In this paper, an adaptive element free Galerkin (EFG) method is presented to solve Poisson equation. In general, element free Galerkin method using moving least square (MLS) approximation needs a background mesh for integration. With the arbitrary polygonal influence domain technique, the shape function of MLS has almost interpolation property, and the Gaussian quadrature points in the background integration element only contribute to the vertices of that element, which enables us to compute the residual based on the background integration element just like the finite element method (FEM). The adaptive procedure based on triangular or tetrahedral background integration elements is then developed for EFG method, in which the residual-based a posteriori error estimation of FEM is used. Numerical examples are provided to illustrate the efficiency of the proposed adaptive EFG method.

C1 piecewise quadratic hierarchical bases

We present a construction of C 1 piecewise quadratic hierarchical bases of Lagrange type on arbitrary polygonal domains Ω ⊂ R 2 . Properly normalized, these bases are Riesz bases for Sobolev spaces H s ( Ω ) , with s ∈ ( 1 , 5 2 ) . The method is applicable to arbitrary initial triangulations of polygonal domains, and does not require a checkerboard quadrangulation needed for earlier C 1 cubic hierarchical Lagrange bases. Homogeneous boundary conditions can be taken into account in a natural way, and lead to Riesz bases for Sobolev spaces H 0 s ( Ω ) , s ∈ ( 1 , 5 / 2 ) ∖ { 3 / 2 } , and H 00 3 / 2 ( Ω ) .

Multiscale VEM for the Biot consolidation analysis of complex and highly heterogeneous domains

We introduce a novel heterogeneous multi-scale method for the consolidation analysis of two-dimensional porous domains with a complex micro-structure. A two-scale strategy is implemented wherein an arbitrary polygonal domain can be discretized into clusters of polygonal elements, each with its own set of fine scale discretization. The method harnesses the advantages of the Virtual Element Method into accurately capturing fine scale heterogeneities of arbitrary polygonal shapes. The upscaling is performed through a set of numerically evaluated multi-scale basis functions. The solution of the coupled governing equations is performed at the coarse-scale at a reduced computational cost. We discuss the computation of the multi-scale basis functions and corresponding virtual projection operators. The performance of the method in terms of accuracy and computational efficiency is evaluated through a set of numerical examples for poro-elastic materials with heterogeneities of various shapes.

Planar max flow maps and determination of lanes with clearance

One main challenge in multi-agent navigation is to generate trajectories minimizing bottlenecks in environments cluttered with obstacles. In this paper we approach this problem globally by taking into account the maximum flow capacity of a given polygonal environment. Given the difficulty in solving the continuous maximum flow of a planar environment, we present in this paper a GPU-based methodology which leads to practical methods for computing maximum flow maps in arbitrary two-dimensional polygonal domains. Once a flow map representation is obtained, lanes can be extracted and optimized in length while keeping constant the flow capacity achieved by the system of trajectories. This work extends our previous work on max flow maps by presenting a clearance-based flow generation method which takes into account the size of the agents at the flow generation phase. In this way we ensure that the maximum possible number of lanes with the needed clearance is always obtained, a property that was found to not be always obtained with our previous method. As a result we are able to generate trajectories of maximum flow from source to sink edges across a generic set of polygonal obstacles, enabling the deployment of large numbers of agents utilizing the maximum flow capacity of a continuous description of the environment and eliminating bottlenecks.

Optimal layout of ellipses and its application for additive manufacturing

The paper studies a layout problem of variable number of ellipses with variable sizes placed into an arbitrary disconnected polygonal domain with maximum packing factor. The ellipses can be continuously translated and rotated. Restrictions on the dimensions of the ellipses are taken into account. Tools for the mathematical modelling of placement constraints (distance constraints between ellipses and containment of ellipses into a polygonal domain) using the phi-function technique are introduced. The tools make it possible to formulate the layout problem in the form of MIP model that is equivalent to a sequence of nonlinear programming subproblems. We develop a new solution algorithm that involves the feasible starting point algorithm and optimisation procedure to search for efficient locally optimal solutions of the layout problem. This algorithm can be used in the design of parts for «support-free» additive manufacturing, taking into account the conditions for its static/ dynamic strength. Results of the algorithm implementation for a topologically optimised flat part with the analysis of a stress state are provided.

Optimal Packing in Additive Manufacturing

The paper studies a packing problem for ellipses placed into an arbitrary disconnected polygonal domain. Neither a number of ellipses nor their sizes are fixed. The objective is maximizing packing factor. The ellipses can be continuously translated and rotated. Restrictions on the dimensions of the ellipses are taken into account. Placement constraints (distance constraints between ellipses and containment of ellipses into a polygonal domain) are formulated using the phi-function technique. Resulting packing problem is stated in the form of MIP model that is equivalent to a sequence of NLP subproblems. A new solution approach is proposed combining a feasible starting point algorithm and an optimization procedure to search for efficient locally optimal solutions. The approach as an addition to existing topology optimization algorithms, can be used to automate design of parts for «support-free» additive manufacturing taking into account requirements of static and/or fatigue strength.

Efficient Spectral Methods for Some Singular Eigenvalue Problems

We propose and analyze some efficient spectral/spectral element methods to solve singular eigenvalue problems related to the Schrodinger operator with an inverse-power potential. For the Schrodinger eigenvalue problem $$-\Delta u +V(x)u=\lambda u$$ with a regular potential $$V(x)=c_1|x|^{-1}$$ , we first design an efficient spectral method on a ball of any dimension by adopting the Sobloev-orthogonal basis functions with respect to the Laplacian operator to overwhelm the homogeneous inverse potential and to eliminate the singularity of the eigenfunctions. Then we extend this spectral method to arbitrary polygonal domains by the mortar element method with each corner covered by a circular sector and origin covered by a circular disc. Furthermore, for the Schrodinger eigenvalue problem with a singular potential $$V(x)=c_3|x|^{-3}$$ , we devise a novel spectral method by modifying the former Sobloev-orthogonal bases to fit the stronger singularity. As in the case of $$|x|^{-1}$$ potential, this approach can be extended to arbitrary polygonal domains by the mortar element method as well. Finally, for the singular elliptic eigenvalue problem $$-\frac{\partial ^2}{\partial x^2}u-\frac{1}{x^2}\frac{\partial ^2}{\partial y^2}u =\lambda u$$ on rectangles, we propose a novel spectral method by using tensorial bases composed of the $$L^2$$ - and $$H^1$$ -simultaneously orthogonal functions in the y-direction and the Sobolev-orthogonal functions with respect to the Schrodinger operator with an inverse-square potential in the x-direction. Numerical experiments indicate that all our methods possess exponential orders of convergence, and are superior to the existing polynomial based spectral/spectral element methods and hp-adaptive methods.

Efficient Spectral and Spectral Element Methods for Eigenvalue Problems of Schrödinger Equations with an Inverse Square Potential

In this article, we study numerical approximation of eigenvalue problems of the Schrodinger operator $-\Delta u + \frac{c^2}{|x|^2}u$. There are three stages in our investigation: We start from a ball of any dimension, in which case the exact solution in the radial direction can be expressed by Bessel functions of fractional degrees. This knowledge helps us to design two novel spectral methods by modifying the polynomial basis to fit the singularities of the eigenfunctions. At the second stage, we move to circular sectors in the two-dimensional setting. Again the radial direction can be expressed by Bessel functions of fractional degrees. Only in the tangential direction some modifications are needed from stage one. At the final stage, we extend the idea to arbitrary polygonal domains. We propose a mortar spectral element approach: a polygonal domain is decomposed into several subdomains with each singular corner including the origin covered by a circular sector, in which origin and corner singularities a...

A family of triangular Hermite finite elements complementing the Bogner–Fox–Schmit rectangle

AbstractThe Bogner-Fox-Schmit rectangular element is one of the simplest elements that provide continuous differentiability of an approximate solution in the framework of the finite element method. However, it can be applied only on a simple domain composed of rectangles whose sides are parallel to two different straight lines.We propose a new family of triangular Hermite elements that involves straight-sided elements and elements with a curved side. Such an element can be used in combination with the Bogner-Fox-Schmit element near the boundary of an arbitrary polygonal domain or a domain with a curved part of the boundary and provides continuous differentiability of an approximate solution in thewhole domain up to the boundary.

Nonlinear graded Delaunay meshes

Starting with a boundary triangulation an algorithm automatically generates graded densities of triangular elements inside arbitrary polygonal domains. Interior nodes generated inside acute triangles receive spacing values from Bezier based nonlinear spacing functions defined on interior edges and a nonlinear interpolant defined over triangles. Nodal spacing values, adaptive breakpoints and a single G-value control densities of nodes and offer a range of graded meshes. Predetermined nodes with spacing values and internal boundaries provide additional initial control of nodal densities.

Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains

In this paper, a numerical verification method is presented for second-order semilinear elliptic boundary value problems on arbitrary polygonal domains. Based on the Newton-Kantorovich theorem, our method can prove the existence and local uniqueness of the solution in the neighborhood of its approximation. In the treatment of polygonal domains with an arbitrary shape, which gives a singularity of the solution around the re-entrant corner, the computable error estimate of a projection into the finite-dimensional function space plays an essential role. In particular, the lack of smoothness of the solution makes classical error estimates fail on nonconvex domains. By using the Hyper-circle equation, an alternative error estimate of the projection has been proposed. Additionally, a new residual evaluation method based on the mixed finite element method works well. It yields more accurate evaluation than the existing method. The efficiency of our method is shown through illustrative numerical results on several polygonal domains.

The Triangular Hermite Finite Element Complementing the Bogner-Fox-Schmit Rectangle

The Bogner-Fox-Schmit rectangular element is one of the simplest elements that provide continuous differentiability of an approximate solution in the framework of the finite element method. However, it can be applied only on a simple domain composed of rectangles or parallelograms whose sides are parallel to two different straight lines. We propose a new triangular Hermite element with 13 degrees of freedom. It is used in combination with the Bogner-Fox-Schmit element near the boundary of an arbitrary polygonal domain and provides continuous differentiability of an approximate solution in the whole domain up to the boundary.

Numerical integration over arbitrary polygonal domains based on Schwarz–Christoffel conformal mapping

AbstractThis paper presents a new numerical integration technique on arbitrary polygonal domains. The polygonal domain is mapped conformally to the unit disk using Schwarz–Christoffel mapping and a midpoint quadrature rule defined on this unit disk is used. This method eliminates the need for a two‐level isoparametric mapping usually required. Moreover, the positivity of the Jacobian is guaranteed. Numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results. Copyright © 2009 John Wiley & Sons, Ltd.

A Computable Fourier Condition Generating Alias-Free Sampling Lattices

We propose a Fourier analytical condition linking alias-free sampling with the Fourier transform of the indicator function defined on the given frequency support. Our discussions center around how to develop practical computation algorithms based on the proposed analytical condition. We address several issues along this line, including the derivation of simple closed-form expressions for the Fourier transforms of the indicator functions defined on arbitrary polygonal and polyhedral domains; a complete and nonredundant enumeration of all quantized sampling lattices via the Hermite normal forms of integer matrices; and a quantitative analysis of the approximation of the original infinite Fourier condition by using finite computations. Combining these results, we propose a computational testing procedure that can efficiently search for the optimal alias-free sampling lattices for a given polygonal or polyhedral shaped frequency domain. Several examples are presented to show the potential of the proposed algorithm in multidimensional filter bank design, as well as in applications involving the design of efficient sampling patterns for multidimensional band-limited signals.

Numerical conformal mapping method based Voronoi cell finite element model for analyzing microstructures with irregular heterogeneities

Numerical conformal mapping (NCM) is an effective method for transforming arbitrary polygonal domains into a regular domain such as a unit circle. This paper incorporates NCM-based stress functions in the Voronoi cell finite element model (VCFEM) for modeling microstructures with irregular shaped heterogeneities. Additional stress function components using the multi-resolution wavelet bases are also introduced to enrich stress functions in regions of sharp corners. The resulting enhanced model, termed as NCM–VCFEM, is able to effectively analyze real micrographs of heterogeneous materials with irregular shapes that have considerable effects on the evolution of stresses, strains and local damage. To optimize the use of the expensive NCM–VCFEM elements for the entire microstructure, a method of identification of heterogeneities that cannot be approximated as ellipses is first developed. Subsequently, a tessellation process is devised to accommodate arbitrary shapes in the VCFEM mesh. Validation studies comparing the results of NCM–VCFEM simulations with other numerical and analytical results demonstrate the effectiveness of NCM–VCFEM.

Formula omitted] hierarchical Riesz bases of Lagrange type on Powell–Sabin triangulations

In this paper we construct C 1 continuous piecewise quadratic hierarchical bases on Powell–Sabin triangulations of arbitrary polygonal domains in R 2 . Our bases are of Lagrange type instead of the usual Hermite type and under some weak regularity assumptions on the underlying triangulations we prove that they form strongly stable Riesz bases for the Sobolev spaces H s ( Ω ) with s ∈ ( 1 , 5 / 2 ) . Especially the case s = 2 is of interest, because we can use the corresponding hierarchical basis for preconditioning fourth-order elliptic equations leading to uniformly well-conditioned stiffness matrices. Compared to the hierarchical Riesz bases by Davydov and Stevenson (Hierarchical Riesz bases for H s ( Ω ) , 1 < s < 5 / 2 . Constructive Approximation, to appear) our construction is simpler.

Hierarchical Riesz Bases for Hs(Ω), 1 &lt; s &lt; 5/2

On arbitrary polygonal domains $\Omega \subset \RR^2$, we construct $C^1$ hierarchical Riesz bases for Sobolev spaces $H^s(\Omega)$. In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from $s \in (2,\frac{5}{2})$ to $s \in (1,\frac{5}{2})$. Since the latter range includes $s=2$, with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned.

On Schwarz's domain decomposition methods for elliptic boundary value problems

Summary. We study the additive and multiplicative Schwarz domain decomposition methods for elliptic boundary value problem of order 2 r based on an appropriate spline space of smoothness $r-1$ . The finite element method reduces an elliptic boundary value problem to a linear system of equations. It is well known that as the number of triangles in the underlying triangulation is increased, which is indispensable for increasing the accuracy of the approximate solution, the size and condition number of the linear system increases. The Schwarz domain decomposition methods will enable us to break the linear system into several linear subsystems of smaller size. We shall show in this paper that the approximate solutions from the multiplicative Schwarz domain decomposition method converge to the exact solution of the linear system geometrically. We also show that the additive Schwarz domain decomposition method yields a preconditioner for the preconditioned conjugate gradient method. We tested these methods for the biharmonic equation with Dirichlet boundary condition over an arbitrary polygonal domain using $C^1$ cubic spline functions over a quadrangulation of the given domain. The computer experiments agree with our theoretical results.

Adaptive mesh generation and transformation of triangular to quadrilateral meshes

AbstractIn this paper an algorithm for generating pure triangular or pure quadrilateral meshes for arbitrary polygonal domains is presented. The first part shows a triangulation procedure being as suitable for local mesh refinement as for local mesh coarsening. In the second part an algorithm for transformation of triangular grids to quadrilateral meshes of the same relative mesh density is introduced. Two simple strategies are shown which can easily be implemented into any existing triangulation algorithm.