Choice Of Boundary Conditions Research Articles

A method to estimate the rectangular orthotropic plate elastic constants using least-squares and Chladni patterns

A method to retrieve the elastic constants of rectangular wooden plates is presented, relying on the measurement of a set of eigenfrequencies and the identification of the corresponding mode shapes, and belonging to the more general category of non-destructive inverse parameter estimation methods. Compared to previous work, the current method is effective with any choice of boundary conditions. Furthermore, the error function is linear in the elastic constants, which may be computed easily via a matrix inversion. This framework lends itself naturally to a physical interpretation of the results in terms of linear combinations of eigenmodes, yielding new sets of modes and associated combined mode shapes in which the elastic constants are completely uncoupled. Several numerical benchmark tests and experimental cases are treated in detail, highlighting the reliability of the proposed methodology in cases of interest in acoustics and musical acoustics.

Some effects of domain size and boundary conditions on the accuracy of airfoil simulations

This paper investigates a specific case of one of the most popular fluid dynamic simulations, the incompressible flow around an airfoil (NACA 0012 here) at a high Reynolds number (6×106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$6 \ imes 10^6$$\\end{document}). OpenFOAM software was used to study the effect of domain size and four common choices of boundary conditions on airfoil lift, drag, surface friction, and pressure. We also examine the relation between boundary conditions and the velocity, pressure, and vorticity distributions throughout the domain. In addition to the common boundary conditions, we implement the “point vortex” boundary condition that was introduced many years ago but is now rarely used. We also applied the point vortex condition for the outlet pressure instead of using the traditional Neumann condition. With the airfoil generating significant lift at incidence angles of 5∘,10∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$5^\\circ , 10^\\circ$$\\end{document}, and 14∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$14^\\circ$$\\end{document}, we confirm a previous finding that the boundary conditions combine with domain size to produce an induced (pressure) drag. The change in the pressure drag with domain size is significant for the commonly-used boundary conditions but is much smaller for the point vortex alternative. The point vortex boundary condition increases the execution time, but this is more than offset by the reduction in domain size needed to achieve a specified accuracy in the lift and drag. This study also estimates the error in total drag and lift due to domain size and shows it can be almost eliminated using the point vortex boundary condition. We also used the impulse form of the momentum equations to study the relation between drag and lift and spurious vorticity, which is generated as a result of using non-exact boundary conditions. These equations reveal that the spurious vorticity throughout the domain is associated with cancelling circulation around the domain boundaries.

Asymmetric membrane "sticky tape" enables simultaneous relaxation of area and curvature in simulation.

Biological lipid membranes are generally asymmetric, not only with respect to the composition of the two membrane leaflets but also with respect to the state of mechanical stress on the two sides. Computer simulations of such asymmetric membranes pose unique challenges with respect to the choice of boundary conditions and ensemble in which such simulations are to be carried out. Here, we demonstrate an alternative to the usual choice of fully periodic boundary conditions: The membrane is only periodic in one direction, with free edges running parallel to the single direction of periodicity. In order to maintain bilayer asymmetry under these conditions, nanoscale "sticky tapes" are adhered to the membrane edges in order to prevent lipid flip-flop across the otherwise open edge. In such semi-periodic simulations, the bilayer is free to choose both its area and mean curvature, allowing for minimization of the bilayer elastic free energy. We implement these principles in a highly coarse-grained model and show how even the simplest examples of such simulations can reveal useful membrane elastic properties, such as the location of the monolayer neutral surface.

De-Higgsing in eleven-dimensional supergravity on the squashed S 7

Abstract In this paper we construct the subset of modes on S 7 that are relevant in the compactification of eleven-dimensional supergravity on a squashed S 7 when restricted to the sector that comprises singlets under the S p ( 1 ) × S p ( 2 ) isometry of the squashed sphere. Some of the properties of these modes, connected to the transition from the round S 7 to the squashed S 7, are analysed in detail. Special features of the Rarita–Schwinger operator, described in earlier work by Buchdahl, are explained and related to properties of the squashed S 7 operator spectrum obtained in previous work by the authors. We then discuss how the singlet modes give rise to supermultiplets in the left-squashed case, the phenomenon of de-Higgsing, and what happens to the AdS4 fields in these supermultiplets under an orientation reversal (‘skew-whiffing’) of the squashed S 7. Finally, we consider the possible choices of boundary conditions that appear for some of these fields in AdS4 in the case of the right-squashed non-supersymmetric compactification, and how these choices may affect the stability of the gravity theory.

Математическое моделирование потокораспределения возмущенного состояния теплоэнергетической системы

The results of the development of a model for the analysis of the disturbed state of a thermal power hydraulic system on the example of a heat supply system are presented. The parameters of the operating mode of the thermal power hydraulic system after the influence of disturbing factors on it were evaluated. It is shown that the classification of stationary states of the studied fragment of the system with steady flow distribution is oriented to boundary conditions. The boundary conditions classified into four types are considered. It is noted that the choice of boundary conditions must be carried out taking into account the use of energy equivalence in the analysis of the perturbed state of the studied fragment of the hydraulic system. The developed model of steady-state flow distribution with a non-isothermal flow of a viscous medium is presented. It is concluded that the developed mathematical model represents a qualitatively new approach to formalizing the problems of flow distribution analysis in systems with adjustable parameters. The model can be considered as a generalized form of representation of particular flow distribution models when describing the object under study, which can be considered as a hydraulic circuit with adjustable parameters with a non-isothermal flow of a viscous medium. It is shown that this model can be used to analyze and describe the flows and properties of a viscous medium in such systems where it is possible to regulate the parameters, and non-isothermal flow is taken into account.

The interplay between bulk flow and boundary conditions on the distribution of microswimmers in channel flow

While previous experimental and numerical studies of dilute microswimmer suspensions have focused on the behaviours of swimmers in the bulk flow and near boundaries, models typically do not account for the interplay between bulk flow and the choice of boundary conditions imposed in continuum models. In our work, we highlight the effect of boundary conditions on the bulk flow distributions, such as through the development of boundary layers or secondary peaks of cell accumulation in bulk-flow swimmer dynamics. For the case of a dilute swimmer suspension in Poiseuille flow, we compare the distribution (in physical and orientation space) obtained from individual-based stochastic models with those from continuum models, and identify under what conditions it is mathematically sensible to use specific continuum boundary conditions to capture different physical scenarios (i.e. specular reflection, uniform random reflection and absorbing boundaries). We identify that the spread of preferred cell orientations is dependent on the interplay between rotation driven by the shear flow (Jeffery orbits) and rotational diffusion. We find that in the absence of hydrodynamic wall interactions, swimmers preferentially approach the walls perpendicular to the surface in the presence of high rotational diffusion, and that the preferential approach of swimmers to the walls is shape-dependent at low rotational diffusion (when suspensions tend towards a fully deterministic case). In the latter case, the preferred orientations are nearly parallel to the surface for elongated swimmers and nearly perpendicular to the surface for near-spherical swimmers. Furthermore, we highlight the effects of swimmer geometries and shear throughout the bulk-flow on swimmer trajectories and show how the full history of bulk-flow dynamics affects the orientation distributions of microswimmer wall incidence.

Semiclassical theory and the Koopman-van Hove equation * *Contribution to the Special Collection ‘Koopman Methods in Classical and Quantum-Classical Mechanics’.

The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton–Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton–Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys–Wentzel–Kramers–Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein–Brillouin–Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.

Effects of the ground surface thermal boundary condition on the long-term thermal performance of energy diaphragm walls

Energy diaphragm walls have attracted increasing attention due to their high heat transfer performance. Thus, the factors that affect the thermal performance of energy walls have been investigated to provide insights for further practical applications. Numerical simulation is widely used in energy wall analysis due to the complexity of conducting field tests. Therefore, the choice of boundary conditions in the simulations is crucial; it refers to whether the model reflects reality and has a significant impact on the results. In most cases, the assumed ground surface thermal boundary is not the same (e.g., thermal insulation, constant temperature), and the ground temperature is also assumed to be constant with depth, which may result in a large loss of thermal energy or inaccurate assessment of thermal performance for the energy wall. In this respect, this paper investigates the long-term differences in thermal performance under these boundaries and considers the ground temperature that varies with depth and time, which has not been comprehensively considered before in the analysis of energy walls. The results show that accounting for air temperature or ground temperature variations with depth and time can result in a great increase in thermal performance, e.g., a 34% increase in the cooling thermal load compared with the thermal insulation boundary. In the performed parametric analysis, the results highlight the significant influence of the thermal load and wall height on the thermal differences between these boundaries. These findings provide a reference for choosing ground surface thermal boundaries and better thermal design of energy diaphragm walls.

Dynamic density functional theory for sedimentation processes on complex domains: Modelling, spectral elements, and control problems.

Modelling of many real-world processes, such as drug delivery, wastewater treatment, and pharmaceutical production, requires accurate descriptions of the dynamics of hard particles confined in complicated domains. In particular, when modelling sedimentation processes or systems with driven flows, it is important to accurately capture volume exclusion effects. This work applies Dynamic Density Functional Theory to the evolution of a particle density under diffusion, external forces, particle-particle interaction, and volume exclusion. Using a spectral element framework, for the first time it is possible to include all of these effects in dynamic simulations on complex domains. Moreover, this allows one to apply complicated no-flux, and other non-local, non-linear, boundary conditions. The methodology is also extended to control problems, addressing questions of how to enhance production set-up in industrially-motivated processes. In this work the relevant models are introduced, numerical methods are discussed, and several example problems are solved to demonstrate the methods' versatility. It is shown that incorporating volume exclusion is crucial for simulation accuracy and we illustrate that the choice of boundary conditions significantly impacts the dynamics.

Hygrothermal simulation challenges: Assessing boundary condition choices in retrofitting historic European buildings

Hygrothermal simulation challenges: Assessing boundary condition choices in retrofitting historic European buildings

Non-supersymmetric vacua and self-adjoint extensions

Internal intervals spanned by finite ranges of a conformal coordinate z and terminating at a pair of singularities are a common feature of many string compactifications with broken supersymmetry. The squared masses emerging in lower-dimensional Minkowski spaces are then eigenvalues of Schrödinger-like operators, whose potentials have double poles at the ends of the intervals. For one-component systems, the possible self-adjoint extensions of Schrödinger operators are described by points in AdS3 × S1, and those corresponding to independent boundary conditions at the ends of the intervals by points on the boundary of AdS3. The perturbative stability of compactifications to Minkowski space time depends, in general, on these choices of self-adjoint extensions. We apply this setup to the orientifold vacua driven by the “tadpole potential” V=T {e}^{frac{3}{2}phi } and find, in nine dimensions, a massive scalar spectrum, a unique choice of boundary conditions with stable tensor modes and a massless graviton, and a wide range of choices leading to massless and/or massive vector modes.

Asymptotic dynamics of three dimensional supergravity and higher spin gravity revisited

We reconsider the Hamiltonian reduction of the action for three dimensional AdS supergravity and W3 higher spin AdS gravity in the Chern-Simons formulation under asymptotically anti-de Sitter boundary conditions. We show that the reduction gives two copies of chiral bosons on the boundary. In particular, we take into account the holonomy of the Chern-Simons connection which manifests itself as zero mode of the momentum of the boundary chiral boson. We provide an equivalent formulation of the boundary action which we claim to be the geometric action on symplectic leaves of a (super-)Virasoro or a higher spin WN Poisson manifold in the case of supergravity or higher spin gravity respectively, where the intersection of leaves (given in terms of leaves representatives) can be identified as the bulk holonomy. This concludes the extension to non-linear algebras where the notion of coadjoint representation is not well-defined. The boundary Hamiltonian depends on a choice of boundary conditions and is equivalent to the Schwarzian action for corresponding Brown-Henneaux boundary conditions. We make this connection explicit in the extended supersymmetric case. Moreover, we discuss the geometric action in the case of W3 AdS3 gravity in both mathfrak{sl} (3) highest weight representations based on principal and diagonal mathfrak{sl} (2) embeddings.

Scattering in black hole backgrounds and higher-spin amplitudes. Part II

We continue to investigate correspondences between, on the one hand, scattering amplitudes for massive higher-spin particles and gravitons in appropriate quantum-to-classical limits, and on the other hand, classical gravitational interactions of spinning black holes according to general relativity. We first construct an ansatz for a gravitational Compton amplitude, at tree level, constrained only by locality, crossing symmetry, unitarity and consistency with the linearized-Kerr 3-point amplitude, to all orders in the black hole’s spin. We then explore the extent to which a unique classical Compton amplitude can be identified by comparing with the results of the classical process of scattering long-wavelength gravitational waves off an exact Kerr black hole, determined by appropriate solutions of the Teukolsky equation. Up to fourth order in spin, we find complete agreement with a previously conjectured exponential form of the tree-level Compton amplitude. At higher orders, we extract tree-level contributions from the Teukolsky amplitude by an analytic continuation from a physical (a/GM < 1) to a particle-like (a/GM > 1) regime. Up to the sixth order in spin, we identify a unique conservative part of the amplitude which is insensitive both to the choice of boundary conditions at the black hole horizon and to branch choices in the analytic continuation. The remainder of the amplitude is determined modulo an overall sign from a branch choice, with the sign flipping under exchanging purely ingoing and purely outgoing boundary conditions at the horizon. Along the way, we make contact with novel applications of massive spinor-helicity variables pertaining to their relation to EFT operators and (spinning) partial amplitudes.

DeWitt Boundary Condition in One-Loop Quantum Cosmology

DeWitt’s suggestion that the wave function of the universe should vanish at the classical Big Bang singularity is considered here within the framework of one-loop quantum cosmology. For pure gravity at one loop about a flat four-dimensional background bounded by a 3-sphere, three choices of boundary conditions are considered: vanishing of the linearized magnetic curvature when only transverse-traceless gravitational modes are quantized; a one-parameter family of mixed boundary conditions for gravitational and ghost modes; and diffeomorphism-invariant boundary conditions for metric perturbations and ghost modes. A positive ζ(0) value in these cases ensures that, when the three-sphere boundary approaches zero, the resulting one-loop wave function approaches zero. This property may be interpreted by saying that, in the limit of small three-geometry, the resulting one-loop wave function describes a singularity-free universe. This property holds for one-loop functional integrals, which are not necessarily equivalent to solutions of the quantum constraint equations.

Inequalities between the lowest eigenvalues of Laplacians with mixed boundary conditions

The eigenvalue problem for the Laplacian on bounded, planar, convex domains with mixed boundary conditions is considered, where a Dirichlet boundary condition is imposed on a part of the boundary and a Neumann boundary condition on its complement. Given two different such choices of boundary conditions for the same domain, we prove inequalities between their lowest eigenvalues. As a special case, we prove parts of a conjecture on the order of mixed eigenvalues of triangles.

Vacuum polarization on three-dimensional anti-de Sitter space-time with Robin boundary conditions

We study a quantum scalar field, with general mass and coupling to the scalar curvature, propagating on three-dimensional global anti-de Sitter space-time. We determine the vacuum and thermal expectation values of the square of the field, also known as the vacuum polarisation (VP). We consider values of the scalar field mass and coupling for which there is a choice of boundary conditions giving well-posed classical dynamics. We apply Dirichlet, Neumann and Robin (mixed) boundary conditions to the field at the space-time boundary. We find finite values of the VP when the parameter governing the Robin boundary conditions is below a certain critical value. For all couplings, the vacuum expectation values of the VP with either Neumann or Dirichlet boundary conditions are constant and respect the maximal symmetry of the background space-time. However, this is not the case for Robin boundary conditions, when both the vacuum and thermal expectation values depend on the space-time location. At the space-time boundary, we find that both the vacuum and thermal expectation values of the VP with Robin boundary conditions converge to the result when Neumann boundary conditions are applied, except in the case of Dirichlet boundary conditions.

On the simulation of nematic liquid crystalline flows in a 4:1 planar contraction using the Leslie–Ericksen and Beris–Edwards models

Extrusion is an essential element in the production of nematic soft solids, ensuring that the final formulation is sufficiently refined and compacted. The interplay between complex microstructure and processing conditions affects the final product performance and may even lead to material degradation. This paper investigates the flow of nematic liquid crystals in a 4:1 planar contraction as a precursor for material extrusion. We compare two different frameworks for modelling the evolution of the nematic microstructure. The Leslie-Ericksen model represents the liquid crystal microstructure through a lower order (vector) field, where nematic elements are treated as headless rods. We find that the polar nature of the description leads to ambiguity in microstructure orientation on the boundary. As a result, boundary conditions with the same notional interpretation are found to produce significantly varying flow and microstructure distributions. The orientability issue is avoided in the higher-order Beris-Edwards model, where the local orientation and degree of molecular alignment are captured in a single tensor. We find that matching flow and microstructure predictions obtained from both approaches is possible, provided that the boundary orientation is correctly chosen in the Leslie-Ericksen theory. In the lower order framework, a poor choice of boundary condition is found to produce poorer pressure loss predictions in flows through the contraction. The effect is particularly pronounced at low Ericksen numbers, where significant over-prediction of pressure losses can arise. • An investigation of the flow of the Leslie-Ericksen and Beris-Edwards model through a 4:1 planar contraction. • The effect of elasticity and microstructure (director) boundary conditions on the flow is investigated. • Vectorial and tensorial frameworks of microstructure representation are compared. • Effects affecting the size of corner vortices are analysed.

S-folds and AdS3 flows from the D3-brane

We investigate supersymmetric flows in type IIB supergravity that preserve an SO(2, 2) space-time symmetry and asymptote to AdS5× S5 at both endpoints. The flows are constructed as Janus-type ℝ × AdS3 BPS domain-walls in the effective four-dimensional [SO(1, 1) × SO(6)] ⋉ ℝ12 gauged maximal supergravity describing the massless sector of type IIB supergravity compactified on S1 × S5. The compactification includes an S-duality hyperbolic twist along the S1 which, when combined with an appropriate choice of boundary conditions for the running scalars, generates special flows that develop an S-fold regime at their core, thus enhancing the space-time symmetry there to SO(2, 3). Via the AdS/CFT correspondence, the flows constructed here are conjectured to describe conformal interfaces in a circle compactification of mathcal{N} = 4 SYM4.

The holography of duality in mathcal{N} = 4 Super-Yang-Mills theory

The space of mathcal{N} = 4 supersymmetric Yang-Mills theories exhibits an intricate structure of global one-form symmetries and SL(2, ℤ) duality orbits. In this paper we study this structure from the point of view of the holographic dual Type IIB string theory. Generalizing work by Witten, we map the different theories based on the gauge algebras su(N), so(N), and sp(N) to a choice of boundary conditions on bulk gauge fields. We show how the one-form symmetries and their anomalies, as well as the duality properties of the gauge theories, arise in the holographic picture. Along the way we prove that the number of disjoint SL(2, ℤ) duality orbits for the su(N) theories is given by the number of square divisors of N.

Reconciling asymmetry observations in the permeability tensor of digital rocks with symmetry expectations

Modelling anisotropic flow in rocks requires their full permeability tensor. While theories derived from the upscaling of Stokes flow to Darcy’s law may justify the tensor symmetry, homogenisations from micro-scale rock samples often return a non-zero level of asymmetry. Since most studies dismiss these controversial observations as numerical errors, this contribution looks more closely at the physical possibility of such behaviour. Asymmetry of the permeability tensor, which induces a rotatory flow, is manifested at the micro-scale by tortuous streamlines. Conversely, when considering a larger scale – above the Representative Elementary Volume for permeability – these tortuous paths do not statistically affect the flow direction any longer. At this point, the homogenisation of Stokes flow to Darcy’s law reaches its domain of validity. We show that the asymmetry in the permeability tensor vanishes for this scale separation, regardless of the choice of boundary conditions unlike previously thought, if the boundary layer effect is disregarded.