Nonsmooth Boundary Research Articles

Surface term, corner term, and action growth in F(Rabcd) gravity theory

After reformulating $F($Riemann$)$ gravity theory as a second derivative theory by introducing two auxiliary fields to the bulk action, we derive the surface term as well as the corner term supplemented to the bulk action for a generic non-smooth boundary such that the variational principle is well posed. We also introduce the counter term to make the boundary term invariant under the reparametrization for the null segment. Then as a demonstration of the power of our formalism, not only do we apply our expression for the full action to evaluate the corresponding action growth rate of the Wheeler-DeWitt patch in the Schwarzchild anti-de Sitter black hole for the $F(R)$ gravity and critical gravity, where the corresponding late time behavior recovers the previous one derived by other approaches, but also in the asymptotically Anti-de Sitter black hole for the critical Einsteinian cubic gravity, where the late time growth rate vanishes but still saturates the Lloyd bound.

Hamilton–Jacobi–Bellman Equations with Time-Measurable Data and Infinite Horizon

In this paper we investigate the existence and uniqueness of weak solutions of the nonautonomous Hamilton–Jacobi–Bellman equation on the domain $$(0,\infty ) \times \Omega $$ . The Hamiltonian is assumed to be merely measurable in time variable and the open set $$\Omega $$ may be unbounded with nonsmooth boundary. The set $$\overline{\Omega }$$ is called here a state constraint. When state constraints arise, then classical analysis of Hamilton–Jacobi–Bellman equation lacks appropriate notion of solution because continuous solutions could not exist. In this work we propose a notion of weak solution for which, under a suitable controllability assumption, existence and uniqueness theorems are valid in the class of lower semicontinuous functions vanishing at infinity.

Torsion of elastic solids with sparse voids parallel to the twist axis

With the purpose of investigating a linear elastic solid containing a dilute distribution of cylindrical and prismatic holes parallel to the torsion axis, the full-field solution for an infinite elastic plane containing a single void and subject to torsion is derived. The obtained solution is exploited to derive the analytic expressions for the stress concentration factor related to the presence of an elliptical hole, the stress intensity factor for hypocycloidal holes and star-shaped cracks and the notch stress intensity factor for star-shaped polygons. Special sets of the void location are obtained for which peculiar mechanical behaviours are displayed, such as stress annihilation at some points along the boundary of elliptical voids and stress singularity removal at the cusps or points of hypocycloidal or isotoxal star-shaped polygonal voids. By means of finite-element simulations, it is finally shown that the presented closed-form expressions for the stress intensification provide reliable predictions, even for finite-domain realizations; in particular, the infinite-plane solution remains highly accurate when the the smooth and non-smooth external boundaries are larger than twice and five times the void dimension, respectively. Under these geometrical conditions, the derived analytical expressions represent a valid ‘guide tool’ in mechanical design.

Bornological Coderivative and Subdifferential Calculus in Smooth Banach Spaces

In this paper, we study bornological generalized differential properties of sets with nonsmooth boundaries, nonsmooth functions, and set-valued mappings in smooth Banach spaces. We establish a fuzzy intersection rule for bornological normal cones and develop fuzzy calculus for bornological generalized differential constructions as well as exact calculus for the limiting counterparts of these constructions.

Mixed Problem for a Parabolic System on a Plane and Boundary Integral Equations

We consider the mixed problem for a one-dimensional (with respect to the spatial variable) second-order parabolic system with Dini-continuous coefficients in a domain with nonsmooth lateral boundaries. Using the method of boundary integral equations, we find a classical solution of this problem. We investigate the smoothness of solution as well.

An adaptive method of fundamental solutions for solving the Laplace equation

In this paper, we propose a residual-type adaptive method of fundamental solutions (AMFS) for solving the two-dimensional Laplace equation. An error estimator is defined only on the boundary of the domain. Initial distributions of source points and collocation points are determined by using approaches proposed in Chen et al. (2006). The adding, removing, and stopping strategies are designed so that the required accuracy can be satisfied within finite steps. Numerical experiments reveal that AMFS improves the accuracy of the MFS approximation obtained from uniformly distributed sources and collocation points, which makes the MFS more practical for non-harmonic and non-smooth boundary conditions. Moreover, it is shown that the error estimator becomes equidistributed after an adaptive iteration. A detailed comparison between AMFS and MFS using uniformly distributed points is also presented for each numerical example.

Nonsmooth pseudodifferential boundary value problems on manifolds

We study pseudodifferential boundary value problems in the context of the Boutet de Monvel calculus or Green operators, with nonsmooth coefficients on smooth compact manifolds with boundary. In order to have a definition that is independent of the choice of (smooth) coordinates, we prove that nonsmooth Green operators are invariant under smooth coordinate transformations.

The Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow

A stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. The problem is motivated by modeling energy systems in rooms that possess an outlet where the fluid can freely flow, known as an “open boundary”. Numerical and experimental investigations available from the literature on heated cavities with open boundaries suggest that the heat transfer at the outlet depends on the temperature at the boundary, the velocity of the fluid, and the outside temperature. Aiming to include this feature in the model, we propose a novel non-smooth boundary condition which is deduced from physical assumptions. We show that this condition is compatible with two approaches at dealing with the “do-nothing” boundary condition for the fluid: (1) the “directional do-nothing” condition and (2) the “do-nothing” condition together with an integral bound for the backflow. Well-posedness of variational formulations is proved for each problem.

A robust asymmetrical contact algorithm for explicit solid dynamics

We describe a novel algorithm for the robust approximation of elastic, inelastic, and frictional contact problems in explicit computations. The method is based on a master-slave concept and a predictor/corrector split of the dynamic update. In the predictor step, the bodies move ignoring all contact interactions; in the correction, the nodes that have penetrated a body are pushed back while correcting their velocities to preserve linear momentum and balance the kinetic energy. In contrast with existing predictor/corrector contact algorithms, no iterations nor global computations are required in the correction step. Moreover, thanks to the geometrical basis of the method, the choice of an artificial penalty parameter is avoided. The contact algorithm does not require the computation of the normal vectors at the contacting surfaces, making it especially useful for simulations that employ finite element and certain meshfree discretizations, and for the simulation of contact among bodies with non-smooth boundaries.

Bursting Oscillations and the Mechanism with Sliding Bifurcations in a Filippov Dynamical System

The main purpose of the paper is to investigate the effect of multiple scales in frequency domain on the complicated oscillations of Filippov system with discontinuous right-hand side. A relatively simple model based on the Chua’s circuit with periodic excitation is introduced as an example. When the exciting frequency is far less than the natural frequency, implying that an order gap between the exciting frequency and the natural frequency exists, the whole exciting term can be considered as a slow-varying parameter, based on which the bifurcations of the two subsystems in different regions divided by the nonsmooth boundary are presented. Two typical cases are considered, which correspond to different distributions of equilibrium branches as well as the related bifurcations. In the first case, periodic symmetric Hopf/Hopf-fold-sliding bursting oscillations can be obtained, in which Hopf bifurcations may cause the alternations between the quiescent states and the spiking states, while fold bifurcations connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. While the second case is the periodic symmetric fold/fold-fold-sliding bursting, where the fold bifurcations not only lead to the alternations between the quiescent states and the spiking states, but also connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. It is pointed out that, different from the bursting oscillations in smooth dynamical systems in which the bifurcations may cause the alternations between quiescent states and spiking states, in the nonsmooth system, bifurcations may not only lead to the alternations, but also connect different forms of quiescent states. Furthermore, in the Filippov system, sliding movement along the nonsmooth boundary can be observed, the mechanism of which is presented based on the analysis of the two subsystems in different regions.

Uniqueness of Solution to the First Initial Boundary Value Problem for Parabolic Systems on the Plane in a Model Case

The first initial boundary value problem for a one-dimensional (in x) Petrovskii parabolic second-order system with constant coefficients in a semibounded (in x) domain with a nonsmooth lateral boundary is proved to have a unique classical solution in certain Holder classes.

Nonmonotone slip problem for miscible liquids

In this paper we prove the existence and uniqueness of a solution to the nonstationary two dimensional system of equations describing miscible liquids with nonsmooth, multivalued and nonmonotone boundary conditions of subdifferential type. We employ the regularized Galerkin method combined with results from the theory of hemivariational inequalities.

Smoothness in the Dini Space of a Single Layer Potential for a Parabolic System in the Plane

We prove estimates in the Dini space for the second order spatial derivative of a vector parabolic single layer potential. We obtain estimates for higher order derivatives of solutions to initial-boundary-value problems for one-dimensional parabolic systems with Dini-continuous coefficients in a curvilinear domain with nonsmooth lateral boundary.

A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

In axisymmetric fusion reactors, the equilibrium magnetic configuration can be expressed in terms of the solution to a semi-linear elliptic equation known as the Grad–Shafranov equation, the solution of which determines the poloidal component of the magnetic field. When the geometry of the confinement region is known, the problem becomes an interior Dirichlet boundary value problem. We propose a high order solver based on the Hybridizable Discontinuous Galerkin method. The resulting algorithm (1) provides high order of convergence for the flux function and its gradient, (2) incorporates a novel method for handling piecewise smooth geometries by extension from polygonal meshes, (3) can handle geometries with non-smooth boundaries and x-points, and (4) deals with the semi-linearity through an accelerated two-grid fixed-point iteration. The effectiveness of the algorithm is verified with computations for cases where analytic solutions are known on configurations similar to those of actual devices (ITER with single null and double null divertor, NSTX, ASDEX upgrade, and Field Reversed Configurations).

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

There are two main formulations for panel methods based on Green’s formula: Dirichlet and Neumann methods. In a previous work a rigorous analytical study of the global error of Dirichlet method was performed. In this work an analytical study of the global error is performed for the other formulation: Neumann method. 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. In order to validate the analytical results, both numerical and analytical solutions for different body geometries have been considered to compare the actual and predicted errors in each case. Finally, a comparison of the error order between both methods, Dirichlet and Neumann, has also been performed for different configurations.

Taking into Account the Influence of Relief in the Method of Microseismic Sounding

The traveling of surface waves in a model geophysical medium with a nonsmooth boundary is studied. It is shown that relief significantly distorts the microseismic noise field recorded at the day surface. This result should be taken into consideration when interpreting the data of passive surface-wave geophysical studies of deep crustal inhomogeneities.

Bitsadze–Samarskii problem for parabolic systems with Dini continuous coefficients

ABSTRACTWe consider a nonlocal initial-boundary value problem of the Bitsadze–Samarskii type for spatially one-dimensional parabolic second-order systems with Dini continuous coefficients in a domain with nonsmooth lateral boundaries. By the boundary integral equations method we construct a classical solution of this problem. The smoothness of the solution is studied.

Lovelock action with nonsmooth boundaries

We examine the variational problem in Lovelock gravity when the boundary contains timelike and spacelike segments nonsmoothly glued. We show that two kinds of contributions have to be added to the action. The first one is associated to the presence of a boundary in every segment and it depends on intrinsic and extrinsic curvatures. We can think of this contribution as adding a total derivative to the usual surface term of Lovelock gravity. The second one appears in every joint between two segments and it involves the integral along the joint of the Jacobson-Myers entropy density weighted by the Lorentz boost parameter which relates the orthonormal frames in each segment. We argue that this term can be straightforwardly extended to the case of joints involving null boundaries. As an application, we compute the contribution of these terms to the complexity of global AdS in Lovelock gravity by using the "complexity = action" proposal and we identify possible universal terms for arbitrary values of the Lovelock couplings. We find that they depend on the charge $a^*$ controlling the holographic entanglement entropy and on a new constant that we characterize.

Electric Field Computations Using Axial Green Function Method on Refined Axial Lines

The axial Green function method (AGM) uses so-called axial lines, which are parallel to axes and look like grids in the finite-difference method. These axial lines can easily be constructed in complicated domains compared to meshes in the FEM, because they are only parallel to axes no matter what the geometry of the problem domain is. The objective of this paper is laid on the computations of the electric potential problems with singularities, which can be caused by a nonsmooth boundary, discontinuous boundary values, and different types of boundary values. In the vicinity of the singularities, we select singular subdomains on which the axial lines can be independently refined. It is shown that the axial line refinements on subdomains near the singularities enable us to calculate the accurate and efficient electric field when using the AGM.

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in L P

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L2 spaces and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of L p spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator to prove that they have a bounded holomorphic functional calculus in those L p spaces. We also obtain functional calculus results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining L p results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and L p bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L2 extends to L p for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L2 proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces and about the relationship between conical and vertical square functions.