Polygonal Domains Research Articles

Pattern formation on regular polygons and circles

We investigate the formation of Turing patterns on regular polygonal domains, as the number of edges grow, leading to the limiting case of the circle. Using linear and weakly nonlinear analysis, and evidence by simulations, we demonstrate how the domain shape can fundamentally change the expected bifurcation structure. Specifically, on the square domain we are able to derive pitchfork bifurcations for stripe and spot solutions, as well as show that both branches cannot bifurcate to produce stable patterns. This compares with the case of the equilateral triangle domain that causes the Turing bifurcation to be generically transcritical and, in some cases, none of the bifurcating branches are stable. Moreover, we find a monotonically increasing, but nonlinear relationship, between the minimal bifurcation area and the number of edges. Thus, patterns can occur on triangles with much smaller areas than circles. Overall, this work raises questions for researchers who are simulating applications on domains with simple shapes. Specifically, even small changes to domain geometry can have large impacts on the produced patterns; thus, domain perturbations should be considered in any sensitivity analyses.

A numerical study of the generalized Steklov problem in planar domains

Abstract We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for planar bounded domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate conjectures with relevance in spectral geometry.

Application of a finite element method variant in nonconvex domains to parabolic problems

In this paper we address one of the major difficulties which is the nonconvex behavior of the domains while finding the solution of the problems. The part of the domain where the nonsmoothness appears is where the challenge arises and the way that area is handled using different numerical methods reveals the effectiveness of these techniques. Here in this article, we study the semilinear parabolic problem in nonconvex polygonal domain. For the approximation of the solution we use the Composite Finite Element (CFE) method, which is a classification of the Finite Element Method. CFE discusses the two-scale discretization — the larger mesh also known as the coarse mesh with the size H and the smaller mesh, also known as the fine mesh with the size h. It helps in reducing the dimension of the domain space of consideration. The fine scale grid is used to resolve the nonconvexity of the boundary whereas the coarse scale grid is comprised of larger grids at an appropriate distance from the boundary. The degrees of freedom depends on the coarse grid. This is the precedence of CFE over other methods, i.e., it eases the task of reducing the domain complexity. In this article, we consider two approaches — the semi discrete analysis where only space discretization is carried out, and the fully discrete analysis where both the time and space discretization is done using both backward Euler and Crank–Nicolson method. We study the error analysis in the L∞(L2)-norm and in the L∞(H1)-norm for the semidiscrete case whereas for the fully discrete case, we study the error analysis in the L∞(L2)-norm. Also, we check for the optimal results. For the CFE technique in the L∞(L2)-norm, we derive the convergence having optimal order in time and almost optimal order in space even if the domain is nonconvex. We consider a T-shaped domain and another star shaped domain to carry out the theoretical findings. Thereafter, numerical computations are implemented to validate the theoretical results.

Exact Obstacle Avoidance for Autonomous Vehicles in Polygonal Domains

Exact Obstacle Avoidance for Autonomous Vehicles in Polygonal Domains

A note on the shift theorem for the Laplacian in polygonal domains

We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces.

Mechanical positional information guides the self-organized development of a polygonal network of creases in the skin of mammalian noses.

The glabrous skin of the rhinarium (naked nose) of many mammalian species exhibits a polygonal pattern of grooves that retain physiological fluid, thereby keeping their nose wet and, among other effects, facilitating the collection of chemosensory molecules. Here, we perform volumetric imaging of whole-mount rhinaria from sequences of embryonic and juvenile cows, dogs, and ferrets. We demonstrate that rhinarial polygonal domains are not placode-derived skin appendages but arise through a self-organized mechanical process consisting of the constrained growth and buckling of the epidermal basal layer, followed by the formation of sharp epidermal creases exactly facing an underlying network of stiff blood vessels. Our numerical simulations show that the mechanical stress generated by excessive epidermal growth concentrates at the positions of vessels that form rigid base points, causing the epidermal layers to move outward and shape domes-akin to arches rising against stiff pillars. Remarkably, this gives rise to a larger length scale (the distance between the vessels) in the surface folding pattern than would otherwise occur in the absence of vessels. These results hint at a concept of "mechanical positional information" by which material properties of anatomical elements can impose local constraints on an otherwise globally self-organized mechanical pattern. In addition, our analyses of the rhinarial patterns in cow clones highlight a substantial level of stochasticity in the pre-pattern of vessels, while our numerical simulations also recapitulate the disruption of the folding pattern in cows affected by a hereditary disorder that causes hyperextensibility of the skin.

SS-DNN: A hybrid strang splitting deep neural network approach for solving the Allen–Cahn equation

The Allen–Cahn equation is a fundamental partial differential equation that describes phase separation and interface motion in materials science, physics, and various other scientific domains. The presence of interfacial width (ϵ) between two stable phases, associated with a nonlinear term, is a small positive parameter which makes the problem more challenging to solve as ϵ approaches zero. This paper proposes a novel hybrid deep splitting method to efficiently and accurately solve the Allen–Cahn equation in a convex polygonal domain in Rd(d=1,2,3). The method combines the benefits of deep learning and splitting strategies, leveraging the strengths of both approaches. Essentially, a second-order splitting method is employed to split the Allen–Cahn equation into two simpler linear and non-linear sub-problems. While the nonlinear sub-problem can be solved analytically, the deep neural network is utilized to approximate the linear sub-problem. By integrating deep learning into the splitting strategy, we achieve a more efficient and accurate solution for the Allen–Cahn equation, demonstrating promising results. We also derive an error estimate for the proposed hybrid method. Modified space adaptivity and transform learning techniques are employed to enhance the efficiency of the neural network.

Necessary and sufficient conditions for avoiding Babuška’s paradox on simplicial meshes

Abstract It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to the failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown that this is optimal in the sense that a full realization of the boundary conditions leads to failure of convergence for conforming methods. The abstract conditions imply that standard nonconforming and discontinuous Galerkin methods converge correctly while conforming methods require a suitable relaxation of the boundary condition. The results are confirmed by numerical experiments.

Lowest-order nonstandard finite element methods for time-fractional biharmonic problem

In this work, we consider an initial-boundary value problem for a time-fractional biharmonic equation in a bounded polygonal domain with a Lipschitz continuous boundary in R2 with clamped boundary conditions. After stating the well-posedness, we focus on some regularity results of the solution with respect to the regularity of the problem data. The spatially semidiscrete scheme covers several popular lowest-order piecewise-quadratic finite element schemes, namely, Morley, discontinuous Galerkin, and C0 interior penalty methods, and includes both smooth and nonsmooth initial data. Optimal order error bounds with respect to the regularity assumptions on the data are proved for both homogeneous and nonhomogeneous problems. The numerical experiments validate the theoretical convergence rate results.

Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks

Solving Poisson Problems in Polygonal Domains with Singularity Enriched Physics Informed Neural Networks

Exponential Convergence and Computational Efficiency of BURA-SD Method for Fractional Diffusion Equations in Polygons

In this paper, we develop a new Best Uniform Rational Approximation-Semi-Discrete (BURA-SD) method taking into account the singularities of the solution of fractional diffusion problems in polygonal domains. The complementary capabilities of the exponential convergence rate of BURA-SD and the hp FEM are explored with the aim of maximizing the overall performance. A challenge here is the emerging singularly perturbed diffusion–reaction equations. The main contributions of this paper include asymptotically accurate error estimates, ending with sufficient conditions to balance errors of different origins, thereby guaranteeing the high computational efficiency of the method.

Very High-Order Accurate Discontinuous Galerkin Method for Curved Boundaries with Polygonal Meshes

Preserving the optimal convergence order of discontinuous Galerkin (DG) discretisations in curved domains is a critical and well-known issue. The proposed approach relies on the reconstruction for off-site data (ROD) method developed originally within the finite volume framework. The main advantages are simplicity, since the PDE solver only considers polygonal domains, and versatility, since any type of boundary condition can be imposed. The developed DG–ROD method consists in splitting the boundary conditions treatment and the leading discrete equations from a classical DG formulation into two independent solvers coupled in a simple and efficient iterative procedure. A numerical benchmark is provided to assess the capability of the method with Dirichlet and Neumann boundary conditions prescribed on curved boundaries, demonstrating that the optimal convergence order is effectively achieved.

Nonconforming virtual element methods for velocity-pressure Stokes eigenvalue problem

In this work, we investigate a divergence-free, nonconforming virtual element method for the numerical approximation of the Stokes eigenvalue problems using polygonal unstructured meshes on polygonal domain. By employing an elliptic projection operator, we “enhance” the definition of the virtual element space to make the L2-projection operator computable with optimal order. Furthermore, we prove the optimal order of convergence of the eigenfunction approximation and the double order of convergence of the eigenvalue approximation through compactness arguments for the solution operator (Babuška-Osborn Theory). Finally, we present a set of numerical experiments to assess the good approximation behavior of the method and show its computational performance.

Adaptive Deep Fourier Residual method via overlapping domain decomposition

The Deep Fourier Residual (DFR) method is a specific type of variational physics-informed neural network (VPINN). It provides a robust neural network-based solution to partial differential equations (PDEs). The DFR strategy is based on minimizing the dual norm of the weak residual of a PDE, which is equivalent to minimizing the energy norm of the error. To compute the dual norm of the weak residual, the DFR method employs an orthonormal spectral basis of the test space, known for rectangles or cuboids for multiple function spaces.In this work, we extend the DFR method with ideas of traditional domain decomposition (DD). This enables two improvements: (a) to solve problems in more general polygonal domains and (b) to develop an adaptive refinement technique in the test space using a Döfler marking algorithm. In the former case, we retain the desirable equivalence between the employed loss function and the H1 error under non-restrictive assumptions, numerically demonstrating adherence to explicit bounds in the case of the L-shaped domain problem. In the latter, we show how refinement strategies lead to potentially significant improvements against a reference, classical DFR implementation with a test function space of significantly lower dimensionality, allowing us to better approximate singular solutions at a more reasonable computational cost.

Adaptation of the composite finite element framework for semilinear parabolic problems

In this article, we discuss one type of finite element method (FEM), known as the composite finite element method (CFE). Dimensionality reduction is the primary benefit of CFE as it helps to reduce the complexity of the domain space. The number of degrees of freedom is greater in standard FEM compared to CFE. We consider the semilinear parabolic problem in a 2D convex polygonal domain. The analysis of the semidiscrete method for the problem is carried out initially in the CFE framework. Here, the discretization is carried out only in space. Then, the fully discrete problem is taken into account, where both the spatial and time components get discretized. In the fully discrete case, the backward Euler method and the Crank-Nicolson method in the CFE framework are adapted for the semilinear problem. The properties of convergence are derived and the error estimates are examined. It is verified that the order of convergence is preserved. The results obtained from the numerical computations are also provided.

Harmonic-measure distribution functions of simply connected and doubly connected polygonal domains

The h-function encodes aspects of the behaviour of Brownian particles released from a specified fixed point inside the region. In turn, this behaviour is influenced by the shape of the region's boundary and the location of the fixed point. We compute the h-functions of numerous simply connected and doubly connected polygonal regions. The Schottky–Klein prime function plays a key role in computing h-functions of these regions. Also, in the computation, we use the Schwarz–Christoffel mappings, the Cayley-type mappings, and their generalisations, which are expressed in terms of the prime function. We also cross-check the validity of all the h-functions we find by rederiving them via numerical simulation of Brownian motion.

Mass moments of functionally graded 2D domains and axisymmetric solids

We present a general methodology to evaluate the mass moments of two-dimensional domains and axisymmetric solids made of functionally graded materials. The approach developed in the paper is based on the sequence of two steps. First, the original domain integrals are converted to integrals extended to the relevant boundary by exploiting Gauss theorem. Second, for domains having a polygonal or circular shape, the boundary integrals are evaluated analytically by providing algebraic expressions that depend upon the parameters defining the density distribution, the position vectors of the vertices of the polygonal domain or the initial and ending points of an arbitrary circular sector, respectively. While the first step refers to moments of arbitrary order, the second step is limited to the most useful quantities for engineering applications, i.e. generalised mass, static moment and inertia tensor. The formulas derived in the paper are validated by means of examples retrieved from the specialised literature for which analytical results are available or have been specifically derived by the authors. Finally, in order to ascertain the computational savings entailed by the use of the proposed analytical formulas with respect to numerical techniques, the mass moments of a longitudinal section of a human femure, made of a functionally graded material and characterised by a linear density distribution, have been computed.

Limit shapes from harmonicity: dominos and the five vertex model

We discuss how to construct limit shapes for the domino tiling model (square lattice dimer model) and five-vertex model, in appropriate polygonal domains. Our methods are based on the harmonic extension method of Kenyon and Prause (2022 Duke Math J. 171 3003–22).

Efficient collision avoidance for autonomous vehicles in polygonal domains

Efficient collision avoidance for autonomous vehicles in polygonal domains

One Parameter Families of Conformal Mappings of Bounded Doubly Connected Polygonal Domains

One Parameter Families of Conformal Mappings of Bounded Doubly Connected Polygonal Domains