Choice Of Boundary Conditions Research Articles

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.

Boundary effects on constituent quark masses and on chiral susceptibility in a four-fermion interaction model

In this work we investigate the finite-size effects on the phase structure of a two-flavor four-fermion interaction model with a flavor-mixing interaction and in the presence of a magnetic background, taking into account different boundary conditions. We employ mean-field approximation and Schwinger’s proper-time method in a toroidal topology with antiperiodic and periodic boundary conditions. The chiral susceptibility and constituent quark masses are studied under the change of the relevant parameters: size of compactified coordinates, temperature, chemical potential and magnetic field strength, within the different scenarios of boundary conditions and the value of flavor-mixing parameter. The findings suggest that the thermodynamic behavior of this system is strongly affected by the combined effects of relevant variables, depending on the range of their change, the value of flavor-mixing parameter and the choice of boundary conditions.

Discrete and higher-form symmetries in SCFTs from wrapped M5-branes

We analyze topological mass terms of BF type arising in supersymmetric M-theory compactifications to AdS5. These describe spontaneously broken higher-form gauge symmetries in the bulk. Different choices of boundary conditions for the BF terms yield dual field theories with distinct global discrete symmetries. We discuss in detail these symmetries and their ’t Hooft anomalies for 4d mathcal{N} = 1 SCFTs arising from M5-branes wrapped on a Riemann surface without punctures, including theories from M5-branes at a ℤ2 orbifold singularity. The anomaly polynomial is computed via inflow and contains background fields for discrete global 0-, 1-, and 2-form symmetries and continuous 0-form symmetries, as well as axionic background fields. The latter are properly interpreted in the context of anomalies in the space of coupling constants.

Intrinsically symmetric cosmological model in the presence of dissipative fluids

Based upon the intrinsic symmetries approach to inhomogeneous cosmologies, we propose an exact solution to Einstein’s field equations where the spatial sections are flat and the source is a nonperfect fluid such that the dissipative terms can be written in terms of spatial gradients of the energy density under a suitable choice of the coordinate system. It is shown through the calculation of the luminosity distance as a function of the redshift that the presence of such inhomogeneities may lead to an effective deceleration parameter compatible with either the standard [Formula: see text]CDM model or LTB models depending on the choice of boundary conditions with no exotic matter. This fact is another evidence that different inhomogeneous models should be carefully investigated in order to verify which model may be compatible with observations and still be as close as possible to the standard model regarding the underlying assumptions, without resorting necessarily to exotic matter components.

Bipartite fidelity for models with periodic boundary conditions

For a given statistical model, the bipartite fidelity is computed from the overlap between the groundstate of a system of size N and the tensor product of the groundstates of the same model defined on two subsystems A and B, of respective sizes N A and N B with N = N A + N B . In this paper, we study for critical lattice models in the case where the full system has periodic boundary conditions. We consider two possible choices of boundary conditions for the subsystems A and B, namely periodic and open. For these two cases, we derive the conformal field theory prediction for the leading terms in the 1/N expansion of , in a most general case that corresponds to the insertion of four and five fields, respectively. We provide lattice calculations of , both exact and numerical, for two free-fermionic lattice models: the XX spin chain and the model of critical dense polymers. We study the asymptotic behaviour of the lattice results for these two models and find an agreement with the predictions of conformal field theory.

A Momentum-Conserving Scheme for Flow Simulation in 1D Channel with Obstacle and Contraction

We consider the extension of the momentum conservative staggered-grid (MCS) scheme for flow simulation in channels with varying depth and width. The scheme is formulated using the conservative properties of the Saint-Venant equations. The proposed scheme was successful in handling various steady flows and achieved results that are in complete accordance with the analytical steady solutions. Different choices of boundary conditions have created steady solutions according to the mass and energy conservations. This assessment has served as a validation of the proposed numerical scheme. Further, in a channel with a contraction and a nonuniform bed, we simulate two cases of dam break. The simulation results show a good agreement with existing experimental data. Moreover, our scheme, that uses a quasi-1-dimensional approach, has shown some fair agreement with existing 2-dimensional numerical results. This evaluation demonstrates the merits of the MCS scheme for various flow simulations in channels of varying width and bathymetry, suitable for river flow modeling.

General solutions in Chern-Simons gravity and Toverline{T} -deformations

We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a Toverline{T} -deformation.

The effect of boundary roughness on dense granular flows

In the field of granular rheology, an important open question is to understand the influence of boundary conditions on granular flows. We perform experiments in a quasi-2D annular shear cell subject to 6 different boundaries with controlled roughness/compliance. We characterize the granular slip at the boundaries to investigate which aspects of a dense granular flow can be controlled by the choice of boundary condition. Photoelastic techniques are implemented to measure the stress fields P(r) and τ(r) throughout the material. A full inverse-analysis of the fringes within each disk provides the vector force at each contact. This allows us to measure the continuum stress field by coarse-graining internal forces. We have observed that boundary roughness and compliance strongly controls the flow profile v(r) and shear rate profile γ˙(r). We also observed that boundary roughness and compliance play a significant role in the pressure profile P(r) and shear stress profile τ(r).

Гибридная модель стационарного плазменного двигателя

A mathematical model is proposed for studying the processes in a stationary plasma thruster (SPT) taking into account the ionization of the working substance - Xenon, based on the hybrid equations of electromagnetic hydrodynamics (EMHD) of the plasma. The 1D2V case of plane symmetry is considered in detail, for which a numerical algorithm for investigating solutions of hybrid EMHD equations is constructed, based on the method of macroparticles. The solution of a number of fundamental issues is given: the calculation of average values, interpolation, construction of the initial distribution of macroparticles, the choice of boundary conditions for the electric field, etc. The results of calculations with and without taking into account the induction fields in the SPT are presented.

Quasiparticles as detector of topological quantum phase transitions

A number of tools have been developed to detect topological phase transitions in strongly correlated quantum systems. They apply under different conditions, but do not cover the full range of many-body models. It is hence desirable to further expand the toolbox. Here, we propose to use quasiparticle properties to detect quantum phase transitions. The approach is independent from the choice of boundary conditions, and it does not assume a particular lattice structure. The probe is hence suitable for, e.g., fractals and quasicrystals. The method requires that one can reliably create quasiparticles in the considered systems. In the simplest cases, this can be done by a pinning potential, while it is less straightforward in more complicated systems. We apply the method to several rather different examples, including one that cannot be handled by the commonly used probes, and in all the cases we find that the numerical costs are low. This is so, because a simple property, such as the charge of the anyons, is sufficient to detect the phase transition point. For some of the examples, this allows us to study larger systems and/or further parameter values compared to previous studies.

A finite-strain model for incomplete damage in elastoplastic materials

We address a three-dimensional model capable of describing coupled damage and plastic effects in solids at finite strains. Formulated within the variational setting of generalized standard materials, the constitutive model results from the balance of conservative and dissipative forces. Material response is rate-independent and associative and damage evolution is unidirectional. We assess the model features and performance on both uniaxial and non-proportional biaxial tests.The constitutive model is then complemented with the quasistatic equilibrium system and initial and boundary conditions. We produce numerical simulations with the help of the multiphysics finite element software NETGEN/NGSolve. We show the flexibility of the implementation and run simulations for various 2D and 3D settings under different choices of boundary conditions and possibly in presence of pre-damaged regions.

Quantum phase transition induced by topological frustration

In quantum many-body systems with local interactions, the effects of boundary conditions are considered to be negligible, at least for sufficiently large systems. Here we show an example of the opposite. We consider a spin chain with two competing interactions, set on a ring with an odd number of sites. When only the dominant interaction is antiferromagnetic, and thus induces topological frustration, the standard antiferromagnetic order (expressed by the magnetization) is destroyed. When also the second interaction turns from ferro to antiferro, an antiferromagnetic order characterized by a site-dependent magnetization which varies in space with an incommensurate pattern, emerges. This modulation results from a ground state degeneracy, which allows to break the translational invariance. The transition between the two cases is signaled by a discontinuity in the first derivative of the ground state energy and represents a quantum phase transition induced by a special choice of boundary conditions.

High-order finite element methods for a pressure Poisson equation reformulation of the Navier–Stokes equations with electric boundary conditions

Pressure Poisson equation (PPE) reformulations of the incompressible Navier–Stokes equations (NSE) replace the incompressibility constraint by a Poisson equation for the pressure and a suitable choice of boundary conditions. This yields a time-evolution equation for the velocity field only, with the pressure gradient acting as a nonlocal operator. Thus, numerical methods based on PPE reformulations are representatives of a class of methods that have no principal limitations in achieving high order. In this paper, it is studied to what extent high-order methods for the NSE can be obtained from a specific PPE reformulation with electric boundary conditions (EBC). To that end, implicit–explicit (IMEX) time-stepping is used to decouple the pressure solve from the velocity update, while avoiding a parabolic time-step restriction; and mixed finite elements are used in space, to capture the structure imposed by the EBC. Via numerical examples, it is demonstrated that the methodology can yield at least third order accuracy in space and time.

Constraining stratal architecture and pressure barriers in the subsalt Karachaganak Carboniferous carbonate platforms using forward stratigraphic modelling

Predicting reservoir quality and pressure barriers is a challenging task in carbonate systems. In this paper, we present new model results from the Lower Carboniferous Serpukhovian carbonates in the Karachaganak Field (Northern Kazakhstan), which is characterized by complex facies architectures and reservoir compartmentalization. The Serpukhovian sequence at Karachaganak is poorly imaged by seismic due to the presence of thick Permian salt canopies. The carbonate sequence comprises two carbonate build-ups: a larger Main build-up (MBU) in the eastern portion of the field, and a smaller isolated Western build-up (WBU). We used diffusion-based DionisosFlow forward stratigraphic modeling (FSM) coupled with a CougarFlow sensitivity analysis workflow to (1) review sub-salt seismic horizon picks based on process-modeling constraints, (2) understand the regional controls on the stratigraphic stacking patterns, (3) understand the distribution of reservoir facies, including potential lateral pressure barriers, and (4) constrain the impact and uncertainties of our choice of boundary conditions in the model. Results point out the need for revisions of existing reservoir-scale static models as FSM simulations indicate that the current seismic reflector pick for the base of the Serpukhovian Stage below the WBU is incompatible with the seismic geometries and facies distribution for this system. To reconcile modelling results with existing seismic geometries, the base of the Serpukhovian stage is expected to be about 180 m higher in the stratigraphy. This suggested revision is also supported by the drilling of a recent well that penetrated the top of the Permian 200 m higher in the stratigraphy on the WBU than anticipated. Modeling results also show that fast regional subsidence rates were the primary control for the Early Serpukhovian aggradational stacking pattern, but that slowing subsidence rate during the Late Serpukhovian due to the Uralian orogeny resulted in a switch to progradational sequence patterns. The forward stratigraphic model additionally demonstrates the major role of wind direction and short-term eustatic changes on the presence and distribution of reefs and breccia reservoir facies. Models show that the clinoforms’ angle of repose varies between the leeward and windward sides of the build-up, which leads to the deposition of mud draping on the leeward side that acted as fluid barriers. The presence and distribution of the reservoir and mud facies in the model can explain the observed reservoir compartmentalization at Karachaganak. Finally, our forward stratigraphic modeling workflow coupled with uncertainty analysis provides new quantitative constraints to understand the impact of diffusion coefficients and sediment production rates on the vertical and lateral reservoir heterogeneities and associated lateral pressure-barriers.

How Should We Choose the Boundary Conditions in a Simulation Which Could Detect Anyons in One and Two Dimensions?

We discuss the problem of anyonic statistics in one and two spatial dimensions from the point of view of statistical physics. In particular we want to understand how the choice of the Bornvon Karman or the twisted periodic boundary conditions necessary in a Monte Carlo simulation to mimic the thermodynamic limit of the many body system influences the statistical nature of the particles. The particles can either be just bosons, when the configuration space is simply connected as for example for particles on a line. They can be bosons and fermions, when the configuration space is doubly connected as for example for particles in the tridimensional space or in a Riemannian surface of genus greater or equal to one (on the torus, etc . . . ). They can be scalar anyons with arbitrary statistics, when the configuration space is infinitely connected as for particles on the plane or in the circle. They can be scalar anyons with fractional statistics, when the configuration space is the one of particles on a sphere. One can further have multi components anyons with fractional statistics when the configuration space is doubly connected as for particles on a Riemannian surface of genus greater or equal to one. We determine an expression for the canonical partition function of hard core particles (including anyons) on various geometries. We then show how the choice of boundary condition (periodic or open) in one and two dimensions determine which particles can exist on the considered surface. In the conclusion, we mention the Laughlin wavefunction and give a few comments about experiments.

Revisiting the Sun’s Strong Differential Rotation along Radial Lines

Abstract Current state-of-the-art models of the solar convection zone consist of solutions to the Navier–Stokes equations in rotating, 3D spherical shells. Such models are highly sensitive to the choice of boundary conditions. Here we present two suites of simulations differing only in their outer thermal boundary condition, which is either one of fixed entropy (FE) or fixed flux (FF; corresponding to a fixed gradient in the entropy). We find that the resulting differential rotation is markedly different between the two sets. The FF simulations have strong differential rotation contrast and isocontours tilted along radial lines (in good agreement with the Sun’s interior rotation revealed by helioseismology), whereas the FE simulations have weaker contrast and contours tilted in the opposite sense. We examine in detail the force balances in our models and find that the poleward transport of heat by Busse columns drives a thermal wind responsible for the different rotation profiles. We conclude that the Sun’s strong differential rotation along radial lines may result from the solar emissivity being invariant with latitude (which is similar to the FF condition in our models) and the poleward transport of heat by Busse columns. In future work on convection in the solar context, we strongly advise modelers to use an FF outer boundary condition.

Combining the Runge Approximation and the Whitney Embedding Theorem in Hybrid Imaging

Abstract This paper addresses enforcing non-vanishing constraints for solutions to a 2nd-order elliptic partial differential equation by appropriate choices of boundary conditions. We show that, in dimension $d\geq 2$, under suitable regularity assumptions, the family of $2d$ solutions such that their Jacobian has maximal rank in the domain is both open and dense. The case of less regular coefficients is also addressed, together with other constraints, which are relevant for applications to recent hybrid imaging modalities. Our approach is based on the combination of the Runge approximation property and the Whitney projection argument [ 44]. The method is very general and can be used in other settings.

Self-similar grooving solutions to the Mullins’ equation

In 1957, Mullins proposed surface diffusion motion as a model for thermal grooving. By adopting a small slope approximation, he reduced the model to the Mullins’ linear surface diffusion equation, ( M E ) y t + B y x x x x = 0 , \begin{equation} \nonumber ({\mathrm {ME}})\quad \quad y_t + B y_{xxxx}=0, \end{equation} known also more simply as the Mullins’ equation. Mullins sought self-similar solutions to (ME) for planar initial conditions, prescribing boundary conditions at the thermal groove, as well as far field decay. He found explicit series solutions which are routinely used in analyzing thermal grooving to this day. While (ME) and the small slope approximation are physically reasonable, Mullins’ choice of boundary conditions is not always appropriate. Here we present an in depth study of self-similar solutions to the Mullins’ equation for general self-similar boundary conditions, explicitly identifying four linearly independent solutions defined on R ∖ { 0 } \mathbb {R}\setminus \{0\} ; among these four solutions, two exhibit unbounded growth and two exhibit asymptotic decay, far from the origin. We indicate how the full set of solutions can be used in analyzing the effective boundary conditions from experimental profiles and in evaluating the governing physical parameters.