Boundary Value Formulation Research Articles

Role of interface tension in the thermoelastic analysis of inclusions: Unified formulation and closed-form results

We study the thermoelastic problem of an inclusion of general shape surrounded by a foreign elastic matrix under plane deformation and the influence of an in-plane far-field heat flux. The small-scale interface effects, related to not only interface conductivity, interface stretching rigidity and interface thermal expansion but also (residual) interface tension, are incorporated in the corresponding analysis. We establish general boundary value formulations in both heat conduction and thermoelasticity for an arbitrary inclusion shape, allowing for analytic or at least semi-analytic treatment of the problem. Specially, we derive closed-form solutions for the case of circular inclusion and obtain concise expressions for describing the remote heat flux-induced thermal stress near the interface (at both the inclusion side and matrix side). Surprisingly, it is shown that for the case of circular inclusion, interface tension plays the same role as interface stretching rigidity in determining the full thermal stress field, or in other words, the full thermal stress field depends on only the sum of interface tension and interface stretching rigidity but not on the specific value of either of them. Numerical examples are also given to illustrate the influence of interface tension on the von Mises stress distribution in the vicinity of the interface.

Asymptotic tracking for linear and nonlinear systems: a two-point boundary value formulation

In this paper, the output tracking problem for linear and nonlinear systems in normal form is revisited in the case of non-minimum phase dynamics and in the absence of any model for the exosystem generating the periodic reference signal. The asymptotic output tracking is obtained by replacing the classical stability condition for the zero dynamics driven by the reference signal with the existence of a solution to a two-point boundary value problem. The latter solution allows to generate the ideal steady-state evolution of the zero dynamics. This strategy allows to retain two interesting features of standard tracking and regulation approaches, namely, i) the use of a simple linear feedback structure and without the need of solving any partial differential equation (as in tracking) and ii) avoiding minimum phase requirements (as in output regulation). Since the same kind of two-point boundary value problem usually arises in the application of the Pontryagin Minimum Principle in optimal control, a natural question to ask is whether there is an underlying connection; hence, in the linear case, further insights are achieved by comparison with an alternative solution based on optimal control problem formulated for an auxiliary affine system.

Numerical initial boundary value problem for the generalized conformal field equations

In this paper we study a numerical implementation for the initial boundary value formulation for the generalized conformal field equations. We propose a formulation which is well suited for the study of the long-time behaviour of perturbed exact solutions such as a Schwarzschild or even a Kerr black hole. We describe the derivation of the implemented equations which we give in terms of the space-spinor formalism. We discuss the conformal Gauss gauge, and a slight generalization thereof which seems to be particularly useful in the presence of boundaries. We discuss the structure of the equations at the boundary and propose a method for imposing boundary conditions which allow the correct number of degrees of freedom to be freely specified while still preserving the constraints. We show that this implementation yields a numerically well-posed system by testing it on a simple case of gravitational perturbations of Minkowski space-time and subsequently with gravitational perturbations of Schwarzschild space-time.

Symmetry-breaking for a restricted n-body problem in the Maxwell-ring configuration

We investigate the motion of a massless body interacting with the Maxwell relative equilibrium, which consists of n bodies of equal mass at the vertices of a regular polygon that rotates around a central mass. The massless body has three equilibrium Zn-orbits from which families of Lyapunov orbits emerge. Numerical continuation of these families using a boundary value formulation is used to construct the bifurcation diagram for the case n=7, also including some secondary and tertiary bifurcating families. We observe symmetry-breaking bifurcations in this system, as well as certain period-doubling bifurcations.

Global error analysis and inertial manifold reduction

Four types of global error for initial value problems are considered in a common framework. They include classical forward error analysis and shadowing error analysis together with extensions of both to include rescaling of time. To determine the amplification of the local error that bounds the global error we present a linear analysis similar in spirit to condition number estimation for linear systems of equations. We combine these ideas with techniques for dimension reduction of differential equations via a boundary value formulation of numerical inertial manifold reduction. These global error concepts are exercised to illustrate their utility on the Lorenz equations and inertial manifold reductions of the Kuramoto–Sivashinsky equation.

Convergence Rate for the Ordered Upwind Method

The Ordered Upwind Method (OUM) is used to approximate the viscosity solution of the static Hamilton-Jacobi-Bellman (HJB) with direction-dependent weights on unstructured meshes. The method has been previously shown to provide a solution that converges to the exact solution, but no convergence rate has been theoretically proven. In this paper, it is shown that the solutions produced by the OUM in the boundary value formulation converge at a rate of at least the square root of the largest edge length in the mesh in terms of maximum error. An example with similar order of numerical convergence is provided.

A coherence-matched linear source mechanism for subsonic jet noise

We investigate source mechanisms for subsonic jet noise using experimentally obtained datasets of high-Reynolds-number Mach 0.4 and 0.6 turbulent jets. The focus is on the axisymmetric mode which dominates downstream sound radiation for low polar angles and the frequency range at which peak noise occurs. A linearized Euler equation (LEE) solver with an inflow boundary condition is used to generate single-frequency hydrodynamic instability waves, and the resulting near-field fluctuations and far-field acoustics are compared with those from experiments and linear parabolized stability equation (LPSE) computations. It is found that the near-field velocity fluctuations closely agree with experiments and LPSE computations up to the end of the potential core, downstream of which deviations occur, but the LEE results match experiments better than the LPSE results. Both the near-field wavepackets and the sound field are observed directly from LEE computations, but the far-field sound pressure levels (SPLs) obtained are more than an order of magnitude lower than experimental values despite close statistical agreement of the near hydrodynamic field up to the potential core region. We explore the possibility that this discrepancy is due to the mismatch between the decay of two-point coherence with increasing distance in experimental flow fluctuations and the perfect coherence in linear models. To match the near-field coherence, experimentally obtained coherence profiles are imposed on the two-point cross-spectral density (CSD) at cylindrical and conical surfaces that enclose near-field structures generated with LEEs. The surface pressure is propagated to the far field using boundary value formulations based on the linear wave equation. Coherence matching yields far-field SPLs which show improved agreement with experimental results, indicating that coherence decay is the main missing component in linear models. The CSD on the enclosing surfaces reveals that the application of a decaying coherence profile spreads the hydrodynamic component of the linear wavepacket source on to acoustic wavenumbers, resulting in a more efficient acoustic source.

Enrichment strategies and convergence properties of the XFEM for hydraulic fracture problems

In two recent papers Gordeliy and Peirce (2013) investigating the use of the Extended Finite Element Method (XFEM) for modeling hydraulic fractures (HF), two classes of boundary value problem and two distinct enrichment types were identified as being essential components in constructing successful XFEM HF algorithms. In this paper we explore the accuracy and convergence properties of these boundary value formulations and enrichment strategies. In addition, we derive a novel set of crack-tip enrichment functions that enable the XFEM to model HF with the full range of power law rλ behavior of the displacement field and the corresponding rλ−1 singularity in the stress field, for 12≤λ<1. This novel crack-tip enrichment enables the XFEM to achieve the optimal convergence rate, which is not achieved by existing enrichment functions used for this range of power law. The two XFEM boundary value problem classes are as follows: (i) a Neumann to Dirichlet map in which the pressure applied to the crack faces is the specified boundary condition and the XFEM is used to solve for the corresponding crack width (P→W); and (ii) a mixed hybrid formulation of the XFEM that makes it possible to incorporate the singular behavior of the crack width in the fracture tip and uses a pressure boundary condition away from it (P&W). The two enrichment schemes considered are: (i) the XFEM-t scheme with full singular crack-tip enrichment and (ii) a simpler, more efficient, XFEM-s scheme in which the singular tip behavior is only imposed in a weak sense. If enrichment is applied to all the nodes of tip-enriched elements, then the resulting XFEM stiffness matrix is singular due to a linear dependence among the set of enrichment shape functions, which is a situation that also holds for the classic set of square-root enrichment functions. For the novel set of enrichment functions we show how to remove this rank deficiency by eliminating those enrichment shape functions associated with the null space of the stiffness matrix. Numerical experiments indicate that the XFEM-t scheme, with the new tip enrichment, achieves the optimal O(h2) convergence rate we expect of the underlying piece-wise linear FEM discretization, which is superior to the enrichment functions currently available in the literature for 12<λ<1. The XFEM-s scheme, with only signum enrichment to represent the crack geometry, achieves an O(h) convergence rate. It is also demonstrated that the standard P→W formulation, based on the variational principle of minimum potential energy, is not suitable for modeling hydraulic fractures in which the fluid and the fracture fronts coalesce, while the mixed hybrid P&W formulation based on the Hellinger–Reissner variational principle does not have this disadvantage.

A two-point boundary value formulation of a mean-field crowd-averse game

We consider a population of “crowd-averse” dynamic agents controlling their states towards regions of low density. This represents a typical dissensus behavior in opinion dynamics. Assuming a quadratic density distribution, we first introduce a mean-field game formulation of the problem, and then we turn the game into a two-point boundary value problem. Such a result has a value in that it turns a set of coupled partial differential equations into ordinary differential equations.

Analysis of an arbitrary-profile, cylindrical, impedance reflector surface illuminated by an E-polarized complex line source beam

Electromagnetic scattering from a cylindrical reflector surface having an arbitrary conic section profile is studied. We assumed an electrically thin layer antenna illuminated by a complex line source in E-polarization mode. Our boundary value formulation, without loss of generality, involves an integral equation approach having impedance-type thin-layer boundary conditions. For simplicity, we also considered both faces of the reflector of the same uniform impedance value. Our computation employs the Method of Analytical Regularization (MAR) technique: the integral equations are converted into the discrete Fourier transform domain yielding two coupled dual series equations, which are then solved by the Fourier inversion and Riemann Hilbert Problem techniques. We demonstrate the accuracy and the convergence behaviors of our numerically solved MAR results that can serve as an accurate benchmark for comparison with widely used results obtained by approximate boundary conditions.

Boundary-value problem formulations for computing invariant manifolds and connecting orbits in the circular restricted three body problem

We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the circular restricted three-body problem (CR3BP), which models the motion of a satellite in an Earth–Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar vertical and halo orbits. We compute the unstable manifolds of selected vertical and halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits are of potential interest in space mission design.

О напряжённом состоянии кусочно-однородного упругого прямоугольника с двумя симметричными трещинами при антиплоской деформации.

This paper presents a boundary value problem and formulation for the analysis of linear elastic fracture mechanics problems involving piecewise homogeneous bimaterials. Investigation and mathematical works in this paper is focused on the stress- strain state of plates with two symmetric central cracks that are made of two bonded dissimilar materials which behave as a piecewise homogeneous elastic plate. Two cracks are on the bondness line of the two segments of plate that have two different shear modulus G1 and G2, and equal length and height. The antiplane distributed shear loading act on the edges of plate. Using sinusoidal Fourier transformation the equation governing this boundary value problem converts to a singular integral equation (S.I.E) of the Cauchy form, which can be solved with the aid of Gaussian numerical-analytical solution for singular integral equations. Consequently the dislocation field around the cracks boundaries and the tearing stresses of plate and the Stress Intensity Factor (S.I.F) equations at the tips of cracks are derived.

New stochastic model for dispersion in heterogeneous porous media: 2. Gaussian plume transmission across stepwise velocity fluctuations

The stochastic solute dispersion model studied in the previous article, can be applied to more realistic velocity variations by approximating them as piecewise constant. This requires treatment by a boundary value formulation, which raises problems connected with entropy considerations. A method is developed to deal with these by the introduction of a specially designed compensator function into the boundary value probability integral for calculating solute concentration. Applying this even for a single velocity step yields an intractable integration, but a suitable approximation is constructed that allows it to be evaluated in analytical form. The result is that a Gaussian solute plume impinging on a velocity step is transmitted as a modulated and compressed or dilated quasi-Gaussian. Plume dispersion is encapsulated in an enhancement factor F that multiplies the diffusive, linear time, dispersion. F is also time dependent; at the time of step penetration it equals kinematical dilation, but anneals away non-linearly so that a length scale can be established over which downstream effects of a velocity step on the dispersion extends. With multiple steps, the cumulative effect is expressed by a product of enhancement factors, each related to a single step. While kinematical compression and dilation effects cancel out across each stepped fluctuation, stochastic dispersion exceeds diffusive values for any plausible combination of fluctuation amplitude and length. A simple formula is obtained for the effective enhancement factor of a fluctuation in terms of its length and amplitude. Moreover, the algebra leads to the definition of a natural length scale Λ related to the Peclet number of the flow. Algebraic evaluation of the cumulative dispersion enhancement by a sequence of equally spaced identical fluctuations, leads to an expression for dispersivity as a function of the distance traversed by a solute plume. Key features of the model in agreement with observations are that the dispersivity behaves differently for traversal lengths above and below Λ, and that above this transition it is proportional to a fractional power of the traversal length. Simple stepwise models do not produce the full extent of the observed exponential increase below the transition, but modifying F by introduction of an adjustable parameter addresses this and leads to good agreement with experimental values over a range of 5 orders of magnitude in both traversal length and dispersivity.

Contour parallel milling tool path generation for arbitrary pocket shape using a fast marching method

Contour parallel tool paths are among the most widely used tool paths for planer milling operations. A number of exact as well as approximate methods are available for offsetting a closed boundary in order to generate a contour parallel tool path; however, the applicability of various offsetting methods is restricted because of limitations in dealing with pocket geometry with and without islands, the high computational costs, and numerical errors. Generation of cusps, segmentation of rarefied corners, and self-intersection during the offsetting operations and finding a unique offsetting solution for pocket with islands are among the associated problems in contour tool path generation. Most of methods are inherently incapable of dealing with such problems and use complex computational routines to identify and rectify these problems. Also, these rectifying techniques are heavily dependent on the type of geometry, and hence, the application of these techniques for arbitrary boundary conditions is limited and prone to errors. In this paper, a new mathematical method for generation of contour parallel tool paths is proposed which is inherently capable of dealing with the aforementioned problems. The method is based on a boundary value formulation of the offsetting problem and a fast marching method based solution for tool path generation. This method handles the topological changes during offsetting naturally and deals with the generation of discontinuities in the slopes by including an “entropy condition” in its numerical implementation. The appropriate modifications are carried out to achieve higher accuracy for milling operations. A number of examples are presented, and computational issues are discussed for tool path generation.

Sensitivity Analysis for Oscillating Dynamical Systems

Boundary value formulations are presented for exact and efficient sensitivity analysis, with respect to model parameters and initial conditions, of different classes of oscillating systems. Methods for the computation of sensitivities of derived quantities of oscillations such as period, amplitude and different types of phases are first developed for limit-cycle oscillators. In particular, a novel decomposition of the state sensitivities into three parts is proposed to provide an intuitive classification of the influence of parameter changes on period, amplitude and relative phase. The importance of the choice of time reference, i.e., the phase locking condition, is demonstrated and discussed, and its influence on the sensitivity solution is quantified. The methods are then extended to other classes of oscillatory systems in a general formulation. Numerical techniques are presented to facilitate the solution of the boundary value problem, and the computation of different types of sensitivities. Numerical results are verified by demonstrating consistency with finite difference approximations and are superior both in computational efficiency and in numerical precision to existing partial methods.

Alternative derivation of electromagnetic cloaks and concentrators

Beginning with a straightforward formulation of electromagnetic ‘cloaking’ that reduces to a boundary value problem involving a single Maxwell first-order differential equation, explicit formulae for the relative permittivity–permeability dyadic and fields of spherical and circular cylindrical annular cloaks are derived in terms of general compressed radial coordinate functions. The general formulation is based on the requirements that the cloaking occurs for all possible incident fields and that the cloaks with frequency ω>0 have continuous tangential components of E and H fields across their outer surfaces, and zero normal components of D and B fields at their inner material surfaces. The tangential-field boundary conditions at the outer surface of the cloak ensure zero scattered fields, and the normal-field boundary conditions at the inner surface of the cloak are compatible with zero total fields inside the interior cavity of the cloak. For spherical cloaks, unlike cylindrical cloaks, these boundary conditions lead to all the tangential components of the E and H fields being continuously zero across their inner surfaces—cylindrical cloaks having delta functions in polarization densities at their inner surfaces. For H-wave incident fields, a nonmagnetic circular cylindrical annulus is found that has nonzero scattered fields but zero total fields within its interior cavity. For bodies with no interior free-space cavities, the formulation is used to derive nonscattering spherical and cylindrical ‘concentrators’ that magnify the incident fields near their centers. For static fields (ω=0), the boundary value formulation is appropriately modified to obtain a relative permeability dyadic that will cloak magnetostatic fields. Causality-energy conditions imply that, unlike magnetostatic cloaking, electrostatic cloaking as well as low-frequency cloaking for ω>0 is not realizable. It is also confirmed that perfect cloaking over any nonzero bandwidth violates causality-energy conditions and thus the cloaking of realistic time-dependent fields must be approximate.

Influence of the future

Fundamental issues concerned with time, both the possible connection of the thermodynamic and cosmological arrows and the possibility of retrocausality, are analyzed using a two-time boundary value formulation. Ways in which the future fate of the universe could leave observable traces today are suggested. Finally, if the definiteness of quantum measurement arises from the occurrence of “special states” then the best existing candidate to justify the presence of those states is the demand for nonentanglement in the far future and far past.

Dynamic response of a viscously damped cantilever with a viscous end condition

This paper deals with the dynamical analysis of a cantilevered Bernoulli–Euler beam subjected to distributed external viscous damping in-span and with a viscous end condition by a single damper. In order to evaluate the vibration characteristics of the system, a procedure is presented where overdamped and underdamped ‘‘modes’’ are investigated simultaneously, via the dynamic stiffness matrix of the system. Further, the orthogonality conditions, which allow the decoupling of the equations of motion in terms of principal coordinates, are derived. Then, the complex frequency response function is obtained through a formula which was established for the receptance matrix previously. Finally, the dynamic impulse response function of the system is evaluated in the time domain. Comparison with the numerical results obtained via a boundary value formulation justifies the approaches used here.

Calculation of Point-to-Point Short-Time and Rare Trajectories with Boundary Value Formulation.

Sampling rare, short-time, and reactive trajectories is of considerable interest in molecular simulations. These trajectories, which are also called "activated", hop between stable states separated by energy or entropy barriers. Simulations of activated trajectories with random sampling of initial conditions are inefficient, since most initial conditions lead to trajectories that do not pass the barrier in short times. A boundary value formulation is proposed that selects these rare trajectories, making the sampling of point-to-point reactive trajectories more effective. Earlier boundary value formulations by one of us focused on computations of approximate trajectories. In the proposed method, trajectories are accurate even when we employ a relatively large integration step (by a factor of about 100 compared to initial value methods). The boundary value solutions to short-time reactive trajectories tend to be unique and have significant statistical weights compared to other reactive trajectories of the microcanonical ensemble. Three numerical examples are considered: a transition in the Mueller potential, a conformational change in alanine dipeptide, and an isomerization in a Lennard-Jones cluster.

Wellposedness of the boundary value formulation of a fixed strike Asian option

This work is the follow up to [J. Hugger, Numerical Mathematics and Advanced Applications—Enumath 2001, Springer, Italy, 2003] where a partial differential equation equivalent to the stochastic formulation for a fixed strike Asian option was derived. In the present work the differential equation is complemented with boundary value conditions that are derived from financial conditions. With the complete boundary value formulation thus recovered, wellposedness of the problem is adressed. It turns out that the problem takes the form of a degenerated parabolic boundary value problem with a second-order, linear, time-dependent PDE with non-negative characteristic form. Apart from the degeneracy in the PDE, also the boundary conditions (derived from the financial understanding) are “the wrong ones” or at least are non-standard. There are conditions on boundaries where none are expected to be needed bacause of the degeneracy and there are boundaries where conditions are expected to be needed but none can be found.