Choice Of Boundary Conditions Research Articles

Asymptotically Justified Boundary Conditions for a Viscoelastic Two‐Phase Fiber Model

AbstractIn this work we present asymptotically justified boundary/interface conditions for a viscoelastic two‐phase fiber model with flow‐enhanced crystallization. While the model equations are well‐established in the literature, the choice of boundary conditions is still under discussion as they might cause artificial layering at the crystallization onset. We show that our conditions avoid such layering, yielding a regularly perturbed boundary value problem with either continuous or discontiunous interface.

Generalized multiscale discontinuous Galerkin method for convection–diffusion equation in perforated media

In this work, we present a Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for the convection–diffusion equation in perforated media. In order to numerically solve the problem on the fine grid and use the solution as a reference solution, we construct the grid that resolves perforation on the fine grid level and performs finite element approximation using the Interior Penalty Discontinuous Galerkin method (IPDG). To reduce the size of the fine grid system, we present the construction of the coarse grid approximation using a Generalized Multiscale Discontinuous Galerkin Method (GMsDGM). In GMsDGM, we construct the multiscale basis functions in the local domains by solutions of local convection–diffusion problems with different boundary conditions and performing dimension reduction by solutions of local spectral problems. We investigate several types of multiscale space constructions based on the various choices of boundary conditions of the local domains. Numerical results are presented for several test problems with different velocities fields and perforations distributions. We also investigate the influence of various constructions of multiscale basis functions with a different number of basis functions and different model parameters.

Brown-York charges with mixed boundary conditions

We compute the Hamiltonian surface charges of gravity for a family of conservative boundary conditions, that include Dirichlet, Neumann, and York’s mixed boundary conditions defined by holding fixed the conformal induced metric and the trace of the extrinsic curvature. We show that for all boundary conditions considered, canonical methods give the same answer as covariant phase space methods improved by a boundary Lagrangian, a prescription recently developed in the literature and thus supported by our results. The procedure also suggests a new integrable charge for the Einstein-Hilbert Lagrangian, different from the Komar charge for non-Killing and non-tangential diffeomorphisms. We study how the energy depends on the choice of boundary conditions, showing that both the quasi-local and the asymptotic expressions are affected. Finally, we generalize the analysis to non-orthogonal corners, confirm the matching between covariant and canonical results without any change in the prescription, and discuss the subtleties associated with this case.

P-form surface charges on AdS: renormalization and conservation

Surface charges of a p-form theory on the boundary of an AdSd+1 spacetime are computed. Counter-terms on the boundary produce divergent corner-terms which holographically renormalize the symplectic form. Different choices of boundary conditions lead to various expressions for the charges and the associated fluxes. With the usual standard AdS boundary conditions, there are conserved zero-mode charges. Moreover, we explore two leaky boundary conditions which admit an infinite number of charges forming an Abelian algebra and non-vanishing flux. Finally, we discuss magnetic p-form charges and electric/magnetic duality.

Higher-dimensional view on quantum cosmology

We argue that the choice of boundary condition for the wave function in quantum cosmology depends on the UV completion of general relativity. We provide an explicit example using a braneworld scenario in which a de Sitter cosmology is induced on the surface of a Coleman-de Luccia bubble in a 5-dimensional AdS space. The corresponding boundary conditions are unambigously fixed by demanding consistency with the known physics of bubble nucleation and this selects the Vilenkin weighting for the amplitude from a 4D viewpoint.

Hybrid Model of a Stationary Plasma Thruster

We propose a mathematical model of processes in a stationary plasma thruster (SPT). This model is based on hybrid equations of electromagnetic hydrodynamics (EMHD) of plasma that take into account ionisation of the working substance, viz. xenon. A thorough consideration is given to the 1D2V case of plane symmetry, in which an algorithm, based on the Particle-in-Cell method, is constructed for the numerical analysis of solutions of the hybrid EMHD equations. A number of key problems are solved: the calculation of average values, interpolation, the construction of the initial distribution of macroparticles, the choice of boundary conditions for the electric field, etc. The results of the calculations are given for the equations with and without the induction fields in the SPT being taken into account.

Pressure-driven micro-poro-mechanics: A variational framework for modeling the response of porous materials

Porous materials are highly relevant in engineering and medical applications due to their enhanced properties and lightweight nature. Current micromechanical models of porous materials can accurately predict the response under the assumptions of small deformations and drained conditions, typically driven by imposed deformations. However, the theoretical framework for the micromechanical modeling of porous material driven by pore pressure in the large-deformation range has been understudied. In this work, we develop a finite-deformation variational framework for pressure-driven foams, i.e., materials where the pore pressure in the cavities produces the deformation. We further consider different kinematical constraints in the formulation of boundary conditions: kinematic uniform displacements, periodic displacements and uniform traction. We apply the proposed model in the numerical simulation of lung porous tissue using a spherical alveolar geometry and an image-based geometry obtained from micro-computed-tomography images of rat lung. Our results show that the stress distributions in the spherical alveolar model are highly dependent on the kinematical constraints. In contrast, the stress distribution in the image-based alveolar model is not affected by the choice of boundary conditions. Further, when comparing the response of pressure-driven versus deformation-driven models, we conclude that hydrostatic stresses experience a marked shift in their distribution, whereas the deviatoric stresses remain unaffected. Our findings of how stresses are affected by the choice of boundary conditions and geometry take particular relevance in the simulation of the lungs, where the pressure-driven and deformation-driven cases are related to mechanical ventilation and spontaneous breathing.

Domain wall fermions on the brane

We study domain wall fermions and their condensation in the D3/probe D7 system. A spatially dependent mass term for the N=2 hypermultiplet can be arranged to isolate distinct two component fermions on two 2+1 dimensional domain walls. We argue that the system shows condensation/mass generation analogous to the D3/probe D5 anti-D5 system. The chiral condensate and pion mass can be directly computed on the domain wall. We provide evidence that these systems with the domains separated by a width w have a bare (current) quark mass that scales as 1/w when the spatial dependent mass is large. Adding a magnetic field does not induce chiral symmetry breaking between the separated domain wall fermions, but a similar phenomenological dilaton factor can be made strong enough to introduce spontaneous symmetry breaking. We show a Gell-Man-Oakes-Renner relation for the pions in that case and also for the case where the D7 probe is in a back-reacted dilaton flow geometry. The vacuum configurations can also be interpreted as having a spontaneously generated mass by a Nambu-Jona-Lasinio four fermion operator, depending on the choice of boundary conditions on fluctuations, according to Witten's multi-trace prescription.

ON STABILIZATION PROBLEM FOR A LOADED HEAT EQUATION:THE TWO-DIMENSIONAL CASE

One of the important properties that characterize the behavior of solutions of boundary value problems for differential equations is stabilization, which has a direct relationship with the problems of controllability. The problems of solvability of stabilization problems of twodimensional loaded equations of parabolic type with the help of feedback control given on the boundary of the region are investigated in the article. These equations have numerous applications in the study of inverse problems for differential equations. The problem consists in the choice of boundary conditions (controls), so that the solution of the boundary value problem tends to a given stationary solution at a certain speed at t → ∞. This requires that the control is feedback, i.e. that it responds to unintended fluctuations in the system, suppressing the results of their impact on the stabilized solution. The spectral properties of the loaded two-dimensional Laplace operator, which are used to solve the initial stabilization problem, are also studied. The paper presents an algorithm for solving the stabilization problem, which consists of constructively implemented stages. The idea of reducing the stabilization problem for a parabolic equation by means of boundary controls to the solution of an auxiliary boundary value problem in the extended domain of independent variables belongs to A.V. Fursikov. At the same time, recently, the so-called loaded differential equations are actively used in problems of mathematical modeling and control of nonlocal dynamical systems.

Mitigating Cutting-Induced Plasticity Errors in the Determination of Residual Stress at Cold Expanded Holes Using the Contour Method

BackgroundThe split sleeve cold expansion process is widely used to improve the fatigue life of fastener holes in the civil and military aircraft industry. The process introduces beneficial compressive residual stresses around the processed hole, but uncertainties remain about the character of the stress field immediately adjacent to the bore of the hole.ObjectiveThe primary objective of this study was to implement the contour method with minimising error associated with cutting-induced plasticity to provide detailed and reliable characterisation of the residual stress introduced by the split sleeve cold expansion process.MethodsA systematic FE study of plasticity effects by simulating different contour cutting strategies (a single cut, two sequential cuts and a 6-cut sequence) for a cold expanded hole in an aluminium alloy coupon was conducted. The identified ‘optimum’ cutting strategy was then applied experimentally on coupons containing a hole that had been processed to 3.16% applied expansion.ResultsThe FE study of different cutting simulations show that the locations of the stress error is consistent with the location where cutting-induced plasticity accumulated and that the magnitude and locations of stress re-distribution plasticity can be controlled by an optimised cutting strategy. In order to validate this hypothesis a high quality contour measurement was performed, showing that accurate near bore stress results can be achieved by the proposed 6-cut approach that controls cutting induced plasticity.ConclusionsThe present work has demonstrated that detailed FE simulation analysis can be a very effective tool in supporting the development of an optimum cutting sequence and in making correct choices of boundary conditions. Through optimizing these key aspects of the cutting sequence one is much more likely to have a successful, low error contour residual stress result.

Near and far zones in two-body dynamics: An effective field theory perspective

We revisit several aspects of the interaction of self-gravitating, slowly varying sources with their own emitted radiation within the context of post-Newtonian approximation to General Relativity. We discuss and clarify the choice of boundary conditions of Green's functions used to determine conservative potentials, and the interplay between the so-called near and far zones, as well as the relation between far zone ultra-violet divergences and emitted power. Both near and far zone contributions are required for the computation of the conservative dynamics. Within a field-theory approach we rederive far-zone self-energy processes, known as tail and memory effects, generalising the calculation of their divergent part to arbitrary order in the post-Newtonian expansion.

Twentieth century global glacier mass change: an ensemble-based model reconstruction

Abstract. Negative glacier mass balances in most of Earth's glacierized regions contribute roughly one-quarter to currently observed rates of sea-level rise and have likely contributed an even larger fraction during the 20th century. The distant past and future of glaciers' mass balances, and hence their contribution to sea-level rise, can only be estimated using numerical models. Since, independent of complexity, models always rely on some form of parameterizations and a choice of boundary conditions, a need for optimization arises. In this work, a model for computing monthly mass balances of glaciers on the global scale was forced with nine different data sets of near-surface air temperature and precipitation anomalies, as well as with their mean and median, leading to a total of 11 different forcing data sets. The goal is to better constrain the glaciers' 20th century sea-level budget contribution and its uncertainty. Therefore, five global parameters of the model's mass balance equations were varied systematically, within physically plausible ranges, for each forcing data set. We then identified optimal parameter combinations by cross-validating the model results against in situ annual specific mass balance observations, using three criteria: model bias, temporal correlation, and the ratio between the observed and modeled temporal standard deviation of specific mass balances. These criteria were chosen in order not to trade lower error estimates by means of the root mean squared error (RMSE) for an unrealistic interannual variability. We find that the disagreement between the different optimized model setups (i.e., ensemble members) is often larger than the uncertainties obtained via the leave-one-glacier-out cross-validation, particularly in times and places where few or no validation data are available, such as the first half of the 20th century. We show that the reason for this is that in regions where mass balance observations are abundant, the meteorological data are also better constrained, such that the cross-validation procedure only partly captures the uncertainty of the glacier model. For this reason, ensemble spread is introduced as an additional estimate of reconstruction uncertainty, increasing the total uncertainty compared to the model uncertainty merely obtained by the cross-validation. Our ensemble mean estimate indicates a sea-level contribution by global glaciers (outside of the ice sheets; including the Greenland periphery but excluding the Antarctic periphery) for 1901–2018 of 69.2 ± 24.3 mm sea-level equivalent (SLE), or 0.59 ± 0.21 mm SLE yr−1. While our estimates lie within the uncertainty range of most of the previously published global estimates, they agree less with those derived from GRACE data, which only cover the years 2002–2018.

The initial boundary value problem and quasi-local Hamiltonians in general relativity

We discuss relations between the initial boundary value problem (IBVP) and quasi-local Hamiltonians in GR. The latter have traditionally been based on Dirichlet boundary conditions, which however are shown here to be ill-posed for the IBVP. We present and analyze several other choices of boundary conditions which are better behaved with respect to the IBVP and carry out a corresponding Hamiltonian analysis, using the framework of the covariant phase space method.

A general solution procedure for the scaled boundary finite element method via shooting technique

The scaled boundary finite element method (SBFEM) is known for its inherent ability to simulate unbounded domains and singular fields, and its flexibility in the meshing procedure. Keeping the analytical form of the field variables along one coordinate intact, it transforms the governing partial differential equations of the problem into a system of one-dimensional (initial–)boundary value problems. However, closed-form solution of the said system is not available for most cases (e.g. transient heat transfer, acoustics, ultrasonics, etc.) since the system cannot be diagonalized in general. This paper aims to establish a numerical tool within the context of the shooting technique to evaluate the coefficient matrices of the subdomains without a priori knowledge of the analytical solution of the semi-discretized system. With proper choice of boundary conditions, the technique uses the strong form of the scaled boundary finite element equations to pass the required information and with the desired accuracy from one boundary to another. Due to generality of the technique, its procedure can be adjusted for any field equations. Since this technique is presented here for the first time, linear elastostatics, for which the closed-form solution is well-established, is formulated to provide valid comparisons. In addition, any direct solution method can be used for integrating the scaled boundary equations. Thus, without loss of generality, a Nyström extension of the classical fourth-order Runge–Kutta method is employed. A quantitative sensitivity analysis is also conducted, and efficiency of the classical and proposed solution techniques is compared in terms of computational time. Finally, some numerical examples, including bounded and unbounded domains, as well as singular stress fields are simulated based on the classical and proposed solution techniques.

Impact of Southern Ocean surface conditions on deep ocean circulation during the LGM: a model analysis

Abstract. Changes in water mass distribution are considered to be a significant contributor to the atmospheric CO2 concentration drop to around 186 ppm recorded during the Last Glacial Maximum (LGM). Yet simulating a glacial Atlantic Meridional Overturning Circulation (AMOC) in agreement with paleotracer data remains a challenge, with most models from previous Paleoclimate Modelling Intercomparison Project (PMIP) phases showing a tendency to simulate a strong and deep North Atlantic Deep Water (NADW) instead of the shoaling inferred from proxy records of water mass distribution. Conversely, the simulated Antarctic Bottom Water (AABW) is often reduced compared to its pre-industrial volume, and the Atlantic Ocean stratification is underestimated with respect to paleoproxy data. Inadequate representation of surface conditions, driving deep convection around Antarctica, may explain inaccurately simulated bottom water properties in the Southern Ocean. We investigate here the impact of a range of surface conditions in the Southern Ocean in the iLOVECLIM model using nine simulations obtained with different LGM boundary conditions associated with the ice sheet reconstruction (e.g., changes of elevation, bathymetry, and land–sea mask) and/or modeling choices related to sea-ice export, formation of salty brines, and freshwater input. Based on model–data comparison of sea-surface temperatures and sea ice, we find that only simulations with a cold Southern Ocean and a quite extensive sea-ice cover show an improved agreement with proxy records of sea ice, despite systematic model biases in the seasonal and regional patterns. We then show that the only simulation which does not display a much deeper NADW is obtained by parameterizing the sinking of brines along Antarctica, a modeling choice reducing the open-ocean convection in the Southern Ocean. These results highlight the importance of the representation of convection processes, which have a large impact on the water mass properties, while the choice of boundary conditions appears secondary for the model resolution and variables considered in this study.

Search efficiency of discrete fractional Brownian motion in a random distribution of targets

Efficiency of search for randomly distributed targets is a prominent problem in many branches of the sciences. For the stochastic process of L\'evy walks, a specific range of optimal efficiencies was suggested under variation of search intrinsic and extrinsic environmental parameters. In this article, we study fractional Brownian motion as a search process, which under parameter variation generates all three basic types of diffusion, from sub- to normal to superdiffusion. In contrast to L\'evy walks, fractional Brownian motion defines a Gaussian stochastic process with power law memory yielding anti-persistent, respectively persistent motion. Computer simulations of search by time-discrete fractional Brownian motion in a uniformly random distribution of targets show that maximising search efficiencies sensitively depends on the definition of efficiency, the variation of both intrinsic and extrinsic parameters, the perception of targets, the type of targets, whether to detect only one or many of them, and the choice of boundary conditions. In our simulations we find that different search scenarios favour different modes of motion for optimising search success, defying a universality across all search situations. Some of our numerical results are explained by a simple analytical model. Having demonstrated that search by fractional Brownian motion is a truly complex process, we propose an over-arching conceptual framework based on classifying different search scenarios. This approach incorporates search optimisation by L\'evy walks as a special case.

Boundary conditions at a thin membrane for the normal diffusion equation which generate subdiffusion.

We consider a particle transport process in a one-dimensional system with a thin membrane, described by the normal diffusion equation. We consider two boundary conditions at the membrane that are linear combinations of integral operators, with time-dependent kernels, which act on the functions and their spatial derivatives defined on both membrane surfaces. We show how boundary conditions at the membrane change the temporal evolution of the first and second moments of particle position distribution (the Green's function) which is a solution to the normal diffusion equation. As these moments define the kind of diffusion, an appropriate choice of boundary conditions generates the moments characteristic for subdiffusion. The interpretation of the process is based on a particle random walk model in which the subdiffusion effect is caused by anomalously long stays of the particle in the membrane.

Numerical examination of the effect of different boundary conditions on the method of approximate particular solutions for scalar and vector problems

This work presents a numerical study of the influence of different boundary conditions (BC) on the precision of the method of approximate particular solutions (MAPS) with radial basis functions in two dimensions. The state of the art merely mentions that the MAPS accuracy decreases near the boundaries. Therefore, the influence that the choice of boundary condition has on this method should be studied with more detail. The present analysis with numerical experiments was applied to the solution of homogeneous Stokes flow equations, backward facing step flow, Slip flow, convective flow between parallel vertical plates and to elliptic differential equations with variable coefficients. The results show that care must be taken in choosing the boundary conditions for this type of solution methods because this selection affects the precision of the results. It is shown that the error associated with Neumann boundary condition, can be reduced by using a refined nodal distribution towards the inlet and outlet zones. Conversely, mixed boundary conditions require careful combination of traction and velocity to achieve better accuracy with MAPS for some flow problems.

A Probabilistic Framework for the Fatigue Damage Assessment of Ships Navigating through Level Ice Fields

For ships navigating in ice-covered waters, fatigue damage due to ice loads is an important issue. The purpose of this paper is to provide ship researchers and engineers with a simulation-based procedure for a probabilistic fatigue damage assessment of ships moving through a level ice field. A novel procedure for fatigue damage assessment using numerical simulations is developed, and a hypothetical scenario with the Xuelong 2 icebreaker is used to illustrate the procedure. Long-duration time-domain simulations are performed to generate samples of the ice-load peaks due to ship-ice interactions. Based on the average conditional exceedance rate (ACER) method, the extreme value statistics of the line load peaks are predicted, and subsequently, the fatigue stresses caused by ice loads are estimated by means of beam theory. Several analytical models are applied to describe the probability distribution of the stress amplitudes, and the fatigue damage is calculated based on S-N curves and the Palmgren-Miner formula. Furthermore, a quantification of the effects of uncertainties related to discretization errors (mesh size, reference period), ship-model properties, choice of boundary conditions, and S-N curves is performed.

Modeling of Microstructure Evolution during Deformation Processes by Cellular Automata-Boundary Conditions and Space Reorganization Aspects.

Cellular automata (CA) are efficient and effective numerical tools for modeling various phenomena and processes, e.g., microstructure evolution in plastic working processes. In many cases, the analysis of phenomena can be carried out only in a limited space and on representative volume. This limitation determines the geometry of CA space hence boundary conditions are very important issues in modeling. The paper discusses different boundary conditions that can be applied to modeling. Taking into account the transformation of the modeling space, the model should allow the selection of boundary conditions. The modeling of certain phenomena and processes is directly related to changes in the geometry of a representative volume and therefore may require changes or reorganization of the modeled CA space. Four reorganization options are presented: halving, cutting and bonding, doubling, and straightening. A choice of boundary conditions may depend on particular space reorganization as used for the modeling of microstructure evolution. A set of decision rules for selecting space reorganization options taking into account the changes of CA shape and sizes is also presented. The modeling of flat and shape rolling processes utilizing some of the described techniques is shown.