Nonsmooth Boundary Research Articles

Bursting oscillations and mechanism of sliding movement in piecewise Filippov system

Since the wide applications in science and engineering, the dynamics of non-smooth system has become one of the key research subjects. Furthermore, the interaction between different scales may result in special movement which can be usually described by the combination of large-amplitude oscillation and small-amplitude one. The influence of multiple scale on the dynamics of non-smooth system has received much attention recently. In this work, we try to explore the bursting oscillations and the mechanism of non-smooth Filippov system coupled by different scales in the frequency domain. Taking the typical periodically excited Duffing's oscillator for example a Filippov system coupled by two scales in the frequency domain is established when the difference in order between the excited frequency and the system natural frequency is obtained by introducing the piecewise control into the state variable and choosing suitable parameters. For the case in which the exciting frequency is far less than the natural frequency, the whole exciting term can be considered as a slow-varying parameter, also called slow subsystem, which leads to a generalized autonomous system, i.e., the fast subsystem. The equilibrium branches and the bifurcations of the fast subsystem along with the variation of the slow-varying parameter in different regions divided according to non-smooth boundary, can be derived. Two typical cases are taken into consideration, in which different distributions of the equilibrium branches and the relevant bifurcations of the fast subsystem may exist. It is pointed out that the variations of the parameters may influence not only the properties of the equilibrium branches, but also the structures of the bursting attractors. Furthermore, since the governing equation alternates between two subsystems located in different regions when the trajectory passes across the non-smooth boundary, the sliding movement along the non-smooth boundary of the trajectory can be observed under the condition of certain parameters. By employing the transformed phase portrait which describes the relationship between the state variable and the slow-varying parameter, the mechanisms of different bursting oscillations and sliding movements are investigated. The results show that bursting oscillations may exist in a non-smooth Filippov system coupled by two scales in the frequency domain. The alternations of the governing equation between different subsystems located in the two neighboring regions along the non-smooth boundary may result in a sliding movement of the trajectory along the non-smooth boundary.

Computing eigenvalues and eigenfunctions of the Laplacian for convex polygons

Recently a new transform method, called the Unified Transform or the Fokas method, for solving boundary value problems (BVPs) for linear and integrable nonlinear partial differential equations (PDEs) has received a lot of attention. For linear elliptic PDEs, this method yields two equations, known as the global relations, coupling the Dirichlet and Neumann boundary values. These equations can be used in a collocation method to determine the Dirichlet to Neumann map. This involves expanding the unknown functions in terms of a suitable basis, and choosing a set of collocation points at which to evaluate the global relations. Here, using these methods for the Helmholtz and modified Helmholtz equations and following the earlier results of [15], we determine eigenvalues of the Laplacian in a convex polygon. Eigenvalues are characterised by the points where the generalised Dirichlet to Neumann map becomes singular. We find that the method yields spectral convergence for eigenfunctions smooth on the boundary and for non-smooth boundary values, the rate of convergence is determined by the rate of convergence of expansions in the chosen Legendre basis. Extensions to the case of oblique derivative boundary conditions and constant coefficient elliptic PDEs are also discussed and demonstrated.

From path integrals to the Wheeler-DeWitt equation: Time evolution in spacetimes with a spatial boundary

We reexamine the relationship between the path integral and canonical formulation of quantum general relativity. In particular, we present a formal derivation of the Wheeler-DeWitt equation from the path integral for quantum general relativity by way of boundary variations. One feature of this approach is that it does not require an explicit $3+1$ splitting of spacetime in the bulk. For spacetimes with a spatial boundary, we show that the dependence of the transition amplitudes on spatial boundary conditions is determined by a Wheeler-DeWitt equation for the spatial boundary surface. We find that variations in the induced metric at the spatial boundary can be used to describe time evolution---time evolution in quantum general relativity is therefore governed by boundary conditions on the gravitational field at the spatial boundary. We then briefly describe a formalism for computing the dependence of transition amplitudes on spatial boundary conditions. Finally, we argue that for nonsmooth boundaries meaningful transition amplitudes must depend on boundary conditions at the joint surfaces.

Modeling of combustion-to-detonation transition in a cylindrical bubble with non-smooth boundary

The paper describes experimental modeling of combustion of a stoichiometric propane-oxygen mixture in quasi-cylindrical bubbles with non-smooth interface using polyvinyl chloride tubes with barriers inside (multiple or single water droplets; or the balls of polystyrene and steel). The transition from combustion to detonation occurs both before and after the barrier, for any kind of barrier material.

Solution of the Bitsadze–Samarskii problem for a parabolic system in a semibounded domain

We consider a nonlocal initial–boundary value Bitsadze–Samarskii problem for a spatially one-dimensional parabolic second-order system in a semibounded domain with nonsmooth lateral boundary. The boundary integral equation method is used to construct a classical solution of this problem under the condition that the vector function on the right-hand side in the nonlocal boundary condition only has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.

Statistics of fermions in a d-dimensional box near a hard wall

We study N noninteracting fermions in a domain bounded by a hard-wall potential in dimensions. We show that for large N, the correlations at the edge of the Fermi gas (near the wall) at zero temperature are described by a universal kernel, different from the universal edge kernel valid for smooth confining potentials. We compute this d-dimensional hard edge kernel exactly for a spherical domain and argue, using a generalized method of images, that it holds close to any sufficiently smooth boundary. As an application we compute the quantum statistics of the position of the fermion closest to the hard wall. Our results are then extended in several directions, including non-smooth boundaries such as a wedge, and also to finite temperature.

Clifford algebra valued boundary integral equations for three-dimensional elasticity

• Clifford analysis is applied to solve 3D problems of elasticity. • BIEs for Clifford algebra valued harmonic functions are derived. • The singularity-free BIEs are constructed without extra equations/variables/information. • The Clifford algebra valued singularity-free BEM is developed by a regular quadrature. • 3D problems of elasticity can be solved using the Clifford algebra valued BEM. Applications of Clifford analysis to three-dimensional elasticity are addressed in the present paper. The governing equation for the displacement field is formulated in terms of the Dirac operator and Clifford algebra valued functions so that a general solution is obtained analytically in terms of one monogenic function and one multiple-component spatial harmonic function together with its derivative. In order to solve numerically the three-dimensional problems of elasticity for an arbitrary domain with complicated boundary conditions, Clifford algebra valued boundary integral equations (BIEs) for multiple-component spatial harmonic functions at an observation point, either inside the domain, on the boundary, or outside the domain, are constructed. Both smooth and non-smooth boundaries are considered in the construction. Moreover, the singularities of the integrals are evaluated exactly so that in the end singularity-free BIEs for the observation point on the boundary taking values on Clifford numbers can be obtained. A Clifford algebra valued boundary element method (BEM) based on the singularity-free BIEs is then developed for solving three-dimensional problems of elasticity. The accuracy of the Clifford algebra valued BEM is demonstrated numerically.

Dirichlet problem for parabolic systems with Dini continuous coefficients on the plane

The Dirichlet problem for a one-dimensional (with respect to x) second-order parabolic system with Dini continuous coefficients is considered in an x-semibounded domain with a nonsmooth lateral boundary from the Dini–Holder class. The classical solvability of the problem is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.

A new accurate residual-based a posteriori error indicator for the BEM in 2D-acoustics

In this work we construct a new reliable, efficient and local a posteriori error estimate for the single layer and hyper-singular boundary integral equations associated to the Helmholtz equation in two dimensions. It uses a localization technique based on a generic operator Λ which is used to transport the residual into L2. Under appropriate conditions on the construction of Λ, we show that it is asymptotically exact with respect to the energy norm of the error. The single layer equation and the hyper-singular equation are treated separately. While the current analysis requires the boundary to be smooth, numerical experiments show that the new error estimators also perform well for non-smooth boundaries.

Identification of non-smooth boundary heat dissipation by partial boundary data

When electronic devices are in operation, the sharp change of the temperature on devices surface can be considered as an indicator of devices faults. Based on this engineering background, we consider an inverse problem for 1-dimensional heat conduction model, with the aim of detecting the nonsmooth heat dissipation coefficient from measurable temperature on the device surface. We establish the uniqueness for the nonsmooth dissipation coefficient and prove the convergence property of the minimizer of the regularizing cost functional for the inverse problem theoretically. Then a double-iteration scheme minimizing the data-match term and the regularization term alternatively is proposed to implement the reconstructions.

Bursting oscillations as well as the mechanism with codimension-1 non-smooth bifurcation

The coupling of different scales in nonlinear systems may lead to some special dynamical phenomena, which always behaves in the combination between large-amplitude oscillations and small-amplitude oscillations, namely bursting oscillations. Up to now, most of therelevant reports have focused on the smooth dynamical systems. However, the coupling of different scales in non-smooth systems may lead to more complicated forms of bursting oscillations because of the existences of different types of non-conventional bifurcations in non-smooth systems. The main purpose of the paper is to explore the coupling effects of multiple scales in non-smooth dynamical systems with non-conventional bifurcations which may occur at the non-smooth boundaries. According to the typical generalized Chua's electrical circuit which contains two non-smooth boundaries, we establish a four-dimensional piecewise-linear dynamical model with different scales in frequency domain. In the model, we introduce a periodically changed current source as well as a capacity for controlling. We select suitable parameter values such that an order gap exists between the exciting frequency and the natural frequency. The state space is divided into several regions in which different types of equilibrium points of the fast sub-system can be observed. By employing the generalized Clarke derivative, different forms of non-smooth bifurcations as well as the conditions are derived when the trajectory passes across the non-smooth boundaries. The case of codimension-1 non-conventional bifurcation is taken for example to investigate the effects of multiple scales on the dynamics of the system. Periodic bursting oscillations can be observed in which codimension-1 bifurcation causes the transitions between the quiescent states and the spiking states. The structure analysis of the attractor points out that the trajectory can be divided into three segments located in different regions. The theoretical period of the movement as well as the amplitudes of the spiking oscillations is derived accordingly, which agrees well with the numerical result. Based on the envelope analysis, the mechanism of the bursting oscillations is presented, which reveals the characteristics of the quiescent states and the repetitive spiking oscillations. Furthermore, unlike the fold bifurcations which may lead to jumping phenomena between two different equilibrium points of the system, the non-smooth fold bifurcation may cause the jumping phenomenon between two equilibrium points located in two regions divided by the non-smooth boundaries. When the trajectory of the system passes across the non-smooth boundaries, non-smooth fold bifurcations may cause the system to tend to different equilibrium points, corresponding to the transitions between quiescent states and spiking states, which may lead to the bursting oscillations.

A Sign-Definite Preconditioned High-Order FEM, Part I: Formulation and Simulation for Bounded Homogeneous Media Wave Propagation

In this article, we develop a novel computer model for the constant coefficient Helmholtz equation based on a sign-definite formulation [A. Moiola and E. A. Spence, SIAM Rev., 56 (2014), pp. 274--312], finite element methods (FEMs), and iterative solvers. This includes (i) a variant parameter version of the formulation for improved performance of the GMRES iterates compared to the standard formulation; (ii) development of a new frequency-robust sign-definite preconditioned high-order FEM model for the simulation of acoustic wave propagation in bounded homogeneous media; and (iii) efficient use of domain decomposition methods in conjunction with our preconditioned model. For simulations, we use a nonconvex star-shaped geometry comprising curved and nonsmooth boundaries and a non-star-shaped nonsmooth geometry and demonstrate our approach for medium- to high-frequency incident waves.

The heat equation source determination for the case of non-smooth boundary and initial conditions

An inverse problem of reconstructing the source of a special kind for parabolic equations in a bounded region with smooth boundary is considered. Solutions are sought in the Holder classes. We prove an uniqueness criterion for the solution and sufficient conditions of Fredholm property of the task at hand. As a consequence of the sufficient conditions for existence and uniqueness of solution of the inhomogeneous inverse problems are found.

Lagrange Multipliers and mixed formulations for some inequality constrained variational inequalities and some nonsmooth unilateral problems

This paper addresses a class of inequality constrained variational inequalities and nonsmooth unilateral variational problems. We present mixed formulations arising from Lagrange multipliers. First we treat in a reflexive Banach space setting the canonical case of a variational inequality that has as essential ingredients a bilinear form and a non-differentiable sublinear, hence convex functional and linear inequality constraints defined by a convex cone. We extend the famous Brezzi splitting theorem that originally covers saddle point problems with equality constraints, only, to these nonsmooth problems and obtain independent Lagrange multipliers in the subdifferential of the convex functional and in the ordering cone of the inequality constraints. For illustration of the theory we provide and investigate an example of a scalar nonsmooth boundary value problem that models frictional unilateral contact problems in linear elastostatics. Finally we discuss how this approach to mixed formulations can be further extended to variational problems with nonlinear operators and equilibrium problems, and moreover, to hemivariational inequalities.

Symmetry methods for option pricing

We obtain a solution of the Black–Scholes equation with a non-smooth boundary condition using symmetry methods. The Black–Scholes equation along with its boundary condition are first transformed into the one dimensional heat equation and an initial condition respectively. We then find an appropriate general symmetry generator of the heat equation using symmetries and the fundamental solution of the heat equation. The symmetry generator is chosen such that the boundary condition is left invariant; the symmetry can be used to solve the heat equation and hence the Black–Scholes equation.

Global error analysis of two-dimensional panel methods for dirichlet formulation

A rigorous analytical study of the global error of panel methods is presented. The analysis is performed for a wide variety of body shapes and different panel geometries to fully understand their effect on the convergence of the method. In particular, we study the global error associated with panel methods applied to thin or thick bodies with purely convex parts or with both convex and concave parts, and with smooth or non-smooth boundaries. Most previous studies focused on the analysis of local error, considering only the influence of the nearest panels and excluding the rest. The difference is shown to be appreciable in many configurations. Generally, there is a lack of consensus concerning the order of magnitude of the error for panel methods even in the simplest case with flat panels and a constant distribution of doublets along them. This paper clarifies apparently different or inconsistent results obtained by other authors.

Generation of Delaunay meshes in implicit domains with edge sharpening

A variational algorithm for the construction of 3D Delaunay meshes in implicit domains with a nonsmooth boundary is proposed. The algorithm is based on the self-organization of an elastic network in which each Delaunay edge is interpreted as an elastic strut. The elastic potential is constructed as a combination of the repulsion potential and the sharpening potential. The sharpening potential is applied only on the boundary and is used to minimize the deviation of the outward normals to the boundary faces from the direction of the gradient of the implicit function. Numerical experiments showed that in the case when the implicit function specifying the domain is considerably different from the signed distance function, the use of the sharpening potential proposed by Belyaev and Ohtake in 2002 leads to the mesh instability. A stable version of the sharpening potential is proposed. The numerical experiments showed that acceptable Delaunay meshes for complex shaped domains with sharp curved boundary edges can be constructed.

Bitsadze–Samarskii problem for a parabolic system on the plane

The solvability (in classical sense) of the Bitsadze–Samarskii nonlocal initial–boundary value problem for a one-dimensional (in x) second-order parabolic system in a semibounded domain with a nonsmooth lateral boundary is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the nonlocal boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.

Thin domains with non-smooth periodic oscillatory boundaries

In this work we study in detail how to adapt the unfolding operator method to thin domains with periodic oscillatory boundaries. We present the unfolding method as a general approach which allows us to analyze the behavior of the solutions of a Neumann problem for equation −Δu+u=f posed in two-dimensional thin domains with an oscillatory boundary. Assuming very mild hypothesis on the regularity of the oscillatory boundary we obtain the homogenized limit problem and corrector results for the three different cases depending on the order of the period of the oscillations.

Hypocycloidal Inclusions in Nonuniform Out-of-Plane Elasticity: Stress Singularity vs. Stress Reduction

Stress field solutions and Stress Intensity Factors (SIFs) are found for $n$-cusped hypocycloidal shaped voids and rigid inclusions in an infinite linear elastic plane subject to nonuniform remote antiplane loading, using complex potential and conformal mapping. It is shown that a void with hypocycloidal shape can lead to a higher SIF than that induced by a corresponding star-shaped crack; this is counter intuitive as the latter usually produces a more severe stress field in the material. Moreover, it is observed that when the order $m$ of the polynomial governing the remote loading grows, the stress fields generated by the hypocycloidal-shaped void and the star-shaped crack tend to coincide, so that they become equivalent from the point of view of a failure analysis. Finally, special geometries and loading conditions are discovered for which there is no stress singularity at the inclusion cusps and where the stress is even reduced with respect to the case of the absence of the inclusion. The concept of Stress Reduction Factor (SRF) in the presence of a sharp wedge is therefore introduced, contrasting with the well-known definition of Stress Concentration Factor (SCF) in the presence of inclusions with smooth boundary. The results presented in this paper provide criteria that will help in the design of ultra strong composite materials, where stress singularities always promote failure. Furthermore, they will facilitate finding the special conditions where resistance can be optimized in the presence of inclusions with non-smooth boundary.