Admissible Boundary Conditions Research Articles

An analytical model for Love wave in a coated piezoelectric bar via nonlocal theory due to an impulsive source

The non-locality behavior of new advanced functionally graded (FG) and piezoelectric (PE) nanobar is presented in this study with an impulsive point source using green function derivatives. The primary aim of this study is to explore the impact of different physical parameters on the phase dispersion phenomena against wave number and depth of point source of Love wave propagation. The stress tensor is considered the linear variation of the shear stress distribution in the thickness direction, and the shear stress at the free surface of the piezoelectric bar is zero. Using the Fourier transform and Green’s function technique, the equations of motion and the dispersion relation is derived by applying admissible boundary conditions involving piezoelectricity, dielectricity, functional gradient, and nonlocal elastic parameters of the media. The obtained results are validated with existing results by considering special cases. The variation in the dispersion curves due to the impulsive point source and piezoelectricity, dielectricity, functional gradient, and non-locality parameters have been analyzed and illustrated using graphs along with a comparative study by considering four piezoelectric materials, PZT-4, PZT-5A, PZT-5H, and PZT-2. Furthermore, the relationship between nonlocal parameters in the layer and half-space is investigated to explore the influence of wave number, phase velocity, and depth of point source.

Thermophoresis and uniform flow in rarefied polyatomic gases: The role of constitutive relations and boundary conditions

Recently, Rana and Barve [“A second-order constitutive theory for polyatomic gases: Theory and applications,” J. Fluid Mech. 958, A23 (2023)] developed a second-order coupled constitutive relations (CCR) for polyatomic gases that include quadratic nonlinearities in the entropy flux and apply the second law. However, in that work, the boundary conditions were heuristically obtained to match the drag coefficient on a sphere and may not be accurate in situations where thermal transpiration and thermal stress are significant factors, as indicated by their asymptotic analysis. This article presents a systematic approach for deriving thermodynamically admissible boundary conditions for the CCR model. We also propose a set of higher-order boundary conditions based on an asymptotic analysis of the solutions for drag on flow past a sphere and thermophoretic drag. The goal of deriving these boundary conditions is to improve the accuracy of the CCR model when applied to external flows, such as slow flow past particles and thermophoretic motion of a spherical particle and doublet. The results of the study demonstrate that the combination of the newly derived boundary conditions in conjunction with the CCR equations shows excellent agreement with both theoretical predictions and experimental data over a wide range of Knudsen numbers. The study suggests that the approach presented in this article can be used to improve the accuracy of the CCR model in a variety of external flow applications.

Bouncing wave packets, Ehrenfest theorem, and uncertainty relation based upon a new concept for the momentum of a particle in a box

For a particle in a box, the operator −i∂x is not self-adjoint and thus does not qualify as the physical momentum. As a result, in general the Ehrenfest theorem is violated. Based upon a recently developed new concept for a self-adjoint momentum operator, we reconsider the theorem and find that it is now indeed satisfied for all physically admissible boundary conditions. We illustrate these results for bouncing wave packets which first spread, then shrink, and return to their original form after a certain revival time. We derive a very simple form of the general Heisenberg–Robertson–Schrödinger uncertainty relation and show that our construction also provides a physical interpretation for it.

Extended reacting boundary modeling of porous materials with thin coverings for time-domain room acoustic simulations

Modeling of acoustic boundary conditions has a significant impact on the accuracy of room acoustic simulations, which play an important role in the design phase of indoor built environments in order to improve the acoustical comfort. In this work, a numerical framework based on the discontinuous Galerkin (DG) method is presented for modeling extended reacting boundaries of porous absorbers covered by thin materials. The domain decomposition methodology is applied by treating the porous material as a subdomain. Equivalent fluid models are used to depict the acoustic properties of porous materials, whose effective density and compressibility as irrational functions are approximated by multipole rational functions in the frequency domain. By employing the auxiliary differential equation approach to calculate the time convolution, the augmented time-domain governing equations of porous materials can be expressed in the same unified hyperbolic form as the linear acoustic equations, which further enables a consistent upwind numerical flux formulation throughout the whole domain. The numerical coupling across the interface between propagation media is handled by solving the underlying Riemann problem. Compared to existing approaches with the DG method for extended reacting boundaries modeling for room acoustics, the derived upwind numerical flux formulation does not involve the computation of auxiliary variables. The presented framework yield a well-posed linear hyperbolic system with admissible boundary conditions as guided by the “uniform Kreiss condition” (Kreiss, 1970). Acoustic properties of the covering materials are illustrated by considering a limp permeable membrane model. A local time-stepping approach is utilized to improve computational efficiency. Numerical validations against analytical solutions in 1D are performed to verify the desired high-order convergence rate. A 3D case study on modeling spherical wave fronts demonstrates the broadband accuracy of the formulation.

Physical significance of generalized boundary conditions: An Unruh-DeWitt detector viewpoint on [formula omitted

On PAdS2×S2, we construct the two-point correlation functions for the ground and thermal states of a real Klein-Gordon field admitting generalized (γ,v)-boundary conditions. We follow the prescription recently outlined in [1] for two different choices of secondary solutions. For each of them, we obtain a family of admissible boundary conditions parametrized by γ∈[0,π2]. We study how they affect the response of a static Unruh-DeWitt detector. The latter not only perceives variations of γ, but also distinguishes between the two families of secondary solutions in a qualitatively different, and rather bizarre, fashion. Our results highlight once more the existence of a freedom in choosing boundary conditions at a timelike boundary which is often neglected, though it has a notable associated physical significance.

Extended water wave systems of Boussinesq equations on a finite interval: Theory and numerical analysis

Considered here is a class of Boussinesq systems of Nwogu type. Such systems describe propagation of nonlinear and dispersive water waves of significant interest such as solitary and tsunami waves. The initial-boundary value problem on a finite interval for this family of systems is studied both theoretically and numerically. First, the linearization of a certain generalized Nwogu system is solved analytically via the unified transform of Fokas. The corresponding analysis reveals two types of admissible boundary conditions, thereby suggesting appropriate boundary conditions for the nonlinear Nwogu system on a finite interval. Then, well-posedness is established, both in the weak and in the classical sense, for a regularized Nwogu system in the context of an initial-boundary value problem that describes the dynamics of water waves in a basin with wall-boundary conditions. In addition, a new modified Galerkin method is suggested for the numerical discretization of this regularized system in time, and its convergence is proved along with optimal error estimates. Finally, numerical experiments illustrating the effect of the boundary conditions on the reflection of solitary waves by a vertical wall are also provided.

On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundary

In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are shown. Furthermore, if the Friedrichs system is hyperbolic, the Cauchy problem is proved to be well-posed in the sense of Hadamard. Finally, examples of Friedrichs systems with admissible boundary conditions are provided.

Quantum magnetic billiards: boundary conditions and gauge transformations

We study admissible boundary conditions for a charged quantum particle in a two-dimensional region subjected to an external magnetic field, i.e. a quantum magnetic billiard. After reviewing some physically interesting classes of admissible boundary conditions (magnetic Robin and chiral boundary conditions), we turn our attention to the role of gauge transformations in a magnetic billiard: in particular, we introduce gauge covariant boundary conditions, and find a sufficient condition for gauge covariance which is satisfied by all the aforementioned examples.

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

The aim of this paper is to prove the existence of Hadamard states for Dirac fields coupled with MIT boundary conditions on any globally hyperbolic manifold with timelike boundary once a suitable propagation of singularities theorem is assumed. To this avail, we consider particular pairs of weakly-hyperbolic symmetric systems coupled with admissible boundary conditions. We then prove the existence of an isomorphism between the solution spaces to the Cauchy problems associated with these operators – this isomorphism is in fact unitary between the spaces of L^2 -initial data. In particular, we show that for Dirac fields with MIT boundary conditions, this isomorphism can be lifted to a * -isomorphism between the algebras of Dirac fields and that any Hadamard state can be pulled back along this * -isomorphism preserving the singular structure of its two-point distribution.

Classification of classical Friedrichs differential operators: One-dimensional scalar case

<p style='text-indent:20px;'>The theory of abstract Friedrichs operators, introduced by Ern, Guermond and Caplain (2007), proved to be a successful setting for studying positive symmetric systems of first order partial differential equations (Fried-richs, 1958), nowadays better known as Friedrichs systems. Recently, Antonić, Michelangeli and Erceg (2017) presented a purely operator-theoretic description of abstract Friedrichs operators, allowing for application of the universal operator extension theory (Grubb, 1968). In this paper we make a further theoretical step by developing a decomposition of the graph space (maximal domain) as a direct sum of the minimal domain and the kernels of corresponding adjoints. We then study one-dimensional scalar (classical) Friedrichs operators with variable coefficients and present a complete classification of admissible boundary conditions.</p>

Role of boundary conditions on Lifshitz spacetimes

On a class of four-dimensional Lifshitz spacetimes with critical exponent $z=2$, including a hyperbolic and a spherical Lifshitz topological black hole, we consider a real Klein-Gordon field. Using a mode decomposition, we split the equation of motion into a radial and into an angular component. As first step, we discuss under which conditions on the underlying parameters we can impose to the radial equation boundary conditions of Robin type and whether bound state solutions do occur. Subsequently, we show that, whenever bound states are absent, one can associate to each admissible boundary condition a ground and a KMS state whose associated two-point correlation function is of local Hadamard form.

Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

We consider the Klein-Gordon operator on an n-dimensional asymptotically anti-de Sitter spacetime (M,g) together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on ∂M of order up to 2. Using techniques from b-calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-solutions of the underlying equations of motion with a prescribed form of their wavefront set and whose anti-symmetric part is proportional to the difference between the advanced and the retarded fundamental solutions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary.

Mechanics of hyperelastic composites reinforced with nonlinear elastic fibrous materials in finite plane elastostatics

A model for the mechanics of a hyperelastic material reinforced with unidirectional and bidirectional fibers is presented in finite plane elastostatics. This includes the refinement/development of a series of continuum-based prediction models to accommodate the nonlinear responses of both the matrix material and the reinforcing fibers. The kinematics of the embedded fibers, including the torsional kinematics between two adjoining fibers, are formulated via the first and second gradient of continuum deformations. Within the framework of variational principles and a virtual work statement, the Euler equation and the admissible boundary conditions are derived. To this end, a set of inhouse experiments are performed for the purpose of cross-examination and model implementation. The obtained models successfully predict the strain-stiffening responses of the elastomer – polyester fiber composites together with other key design considerations such as, deformation profiles, shear strain distributions, and the deformed configurations of a local unit fiber mesh. The Euler – Almansi strain integrate model is also proposed through which the strain-softening behaviors of a certain type of polyurethane fiber composites are predicted, yet further implementation of the obtained Euler – Almansi model remains to be determined due to the paucity of available data. The practical utility of the proposed models may be expected in the design and analysis of hyperelastic composites exhibiting strain-stiffening/softening responses by providing instant estimations of the resultant properties of intended composites.

Surface phenomena of gradient materials

The behavior of third gradient materials is analyzed. They possess stress tensor fields of second, third and fourth order. Starting from the principle of virtual power, we derive the admissible boundary conditions. Those on free surfaces can only be obtained by the application of the divergence theorem of surfaces. On the other hand, such an application to fictitious internal cuts makes no sense although it is usually practiced. We prove that some of the boundary conditions on a free surface may be interpreted as the equilibrium conditions of a shell. So a crust shell exists on such a surface and a beam exists where patches of the surface meet. On the other hand, no such shells or beams can be found with fictitious surfaces in the interior of a continuum. Our finding does not depend on any specific constitutive assumption.

A model for the second strain gradient continua reinforced with extensible fibers in plane elastostatics

A second strain gradient theory-based continuum model is presented for the mechanics of an elastic solid reinforced with extensible fibers in plane elastostatics. The extension and bending kinematics of fibers are formulated via the second and the third gradient of the continuum deformation. The Euler equations arising in the third gradient of virtual displacement are then formulated by means of iterated integration by parts and variational principles. A rigorous derivation of the associated boundary conditions is also presented from which the expressions of triple forces and stresses are obtained. The obtained triple forces are found to be in conjugation with the Piola-type triple stress and are necessary to determine energy contributions on edges and points of Cauchy cuts. In particular, a complete linear model including admissible boundary conditions is derived within the description of superposed incremental deformations. The obtained analytical solution predicts smooth deformation profiles and, more importantly, assimilate gradual and dilatational shear angle distributions throughout the domain of interest.

Closed-form solutions for modelling the rotational stiffness of continuous and discontinuous compliant interfaces in two-layer Timoshenko beams

An analytical model for a two-layer Timoshenko beam with a compliant interface is presented. A family of closed-form solutions is given for several cases of contact conditions at the interface that can include or exclude interlayer slip and relative rotations of the layers’ cross sections (distortion), with particular attention devoted to the latter. Each kinematic field at the interface is related to a corresponding traction by means of a linear-elastic law. The novelty of the proposed model is the rotational stiffness at the interface, completely separate from the tangential stiffness, that can control the amount of interlayer distortion at the interface. The derived closed-form solutions allow an exact stiffness matrix for any case of admissible boundary conditions to be obtained, as well as for continuity conditions in the case where multiple elements with closed-form solutions are mutually connected. The application of the closed-form solutions for continuous interface (adhesive joints) or for discontinuous interface (shear connectors or adhesive defects) is presented. Accuracy and applicability of the closed-form solutions have been assessed in two representative numerical examples allowing to conclude that the rotational stiffness at the interface can strongly affect the behaviour of a two-layer beam.

Numerical homogenization of second gradient, linear elastic constitutive models for cubic 3D beam-lattice metamaterials

Generalized continuum mechanical theories such as second gradient elasticity can consider size and localization effects, which motivates their use for multiscale modeling of periodic lattice structures and metamaterials. For this purpose, a numerical homogenization method for computing the effective second gradient constitutive models of cubic lattice metamaterials in the infinitesimal deformation regime is introduced here. Based on the modeling of lattice unit cells as shear-deformable 3D beam structures, the relationship between effective macroscopic strain and stress measures and the microscopic boundary deformations and rotations is derived. From this Hill–Mandel condition, admissible kinematic boundary conditions for the homogenization are concluded. The method is numerically verified and applied to various lattice unit cell types, where the influence of cell type, cell size and aspect ratio on the effective constitutive parameters of the linear elastic second gradient model is investigated and discussed. To facilitate their use in multiscale simulations with second gradient linear elasticity, these effective constitutive coefficients are parameterized in terms of the aspect ratio of the lattices structures.

A rational beam-elastic substrate model with incorporation of beam-bulk nonlocality and surface-free energy

In this study, a rational beam-elastic substrate model with inclusion of nonlocal and surface-energy effects is developed for bending and buckling analyses of nanobeams lying on elastic substrate media. The beam-section kinematics follows Euler–Bernoulli beam hypothesis. The thermodynamics-based strain gradient theory is employed to represent the beam-bulk nonlocality while the Gurtin–Murdoch surface theory is utilized to account for the surface-free energy. Interaction between the beam and its underlying substrate medium is described by the Winkler foundation model. The governing equilibrium equation and admissible natural boundary conditions are consistently obtained using the virtual displacement principle. To characterize bending and buckling responses of the new beam-elastic substrate model, three numerical examples are employed: the first demonstrates the capability of the proposed model in eliminating the paradoxical behavior present in the Eringen nonlocal differential model; the second characterizes the bending response of the free–free nanobeam-elastic substrate system; and the third examines the influences of several model parameters as well as the size-dependent effect on the buckling response of the simply supported nanobeam-elastic substrate system.

Nitsche’s method for linear Kirchhoff–Love shells: Formulation, error analysis, and verification

Stable and accurate modeling of thin shells requires proper enforcement of all types of boundary conditions. Unfortunately, for Kirchhoff–Love shells, strong enforcement of Dirichlet boundary conditions is difficult because both displacement and normal rotation boundary conditions must be applied. A popular alternative is to employ Nitsche’s method to weakly enforce all boundary conditions. However, while many Nitsche-based formulations have been proposed in the literature, they lack comprehensive error analyses and verification. In fact, existing formulations are variationally inconsistent and yield sub-optimal convergence rates when used with common boundary condition specifications. In this paper, we present a novel Nitsche-based formulation for the linear Kirchhoff–Love shell that is provably stable and optimally convergent for general sets of admissible boundary conditions. To arrive at our formulation, we first present a framework for constructing Nitsche’s method for any abstract variationally constrained minimization problem. We then apply this framework to the linear Kirchhoff–Love shell and, for the particular case of NURBS-based isogeometric analysis, we prove that the resulting formulation yields optimal convergence rates in both the shell energy norm and the standard L2-norm. To arrive at this formulation, we derive the Euler–Lagrange equations for general sets of admissible boundary conditions and show that the Euler–Lagrange boundary conditions typically presented in the literature are incorrect. We verify our formulation by manufacturing solutions for a new shell obstacle course that encompasses flat, parabolic, hyperbolic, and elliptic geometric configurations with a variety of common boundary condition specifications. These manufactured solutions allow us to robustly measure the error across the entire shell in contrast with current best practices where displacement and stress errors are only measured at specific locations. We use NURBS discretizations to represent the shell geometry and show optimal convergence rates in both the shell energy norm and the standard L2-norm with varying polynomial degrees for all of the problems in the obstacle course.

An analysis of lipid membrane morphology in the presence of coordinate-dependent non-uniformity

A model for the mechanics of lipid membranes with non-uniform (coordinate-dependent) properties is discussed. The coordinate-dependent responses of the membranes are incorporated via the augmented non-uniform energy function and material parameters, which are dependent explicitly on the surface coordinates. We formulate the associated normal and tangential Euler equilibrium equations through which the coordinate-dependent responses of membranes are characterized. The admissible boundary conditions are taken from the existing non-linear model but reformulated and adopted to the present framework. Within the prescription of superposed incremental deformations, a compatible linear model is also formulated, from which a complete analytical solution describing the non-uniform responses of the membrane subjected to substrate–membrane interactions is obtained.