Extra Boundary Conditions Research Articles

An XFEM/CZM implementation for massively parallel simulations of composites fracture

Because of their widely generalized use in many industries, composites are the subject of many research campaigns. More particularly, the development of both accurate and flexible numerical models able to capture their intrinsically multiscale modes of failure is still a challenge. The standard finite element method typically requires intensive remeshing to adequately capture the geometry of the cracks and high accuracy is thus often sacrificed in favor of scalability, and vice versa. In an effort to preserve both properties, we present here an extended finite element method (XFEM) for large scale composite fracture simulations. In this formulation, the standard FEM formulation is partially enriched by use of shifted Heaviside functions with special attention paid to the scalability of the scheme. This enrichment technique offers several benefits since the interpolation property of the standard shape function still holds at the nodes. Those benefits include (i) no extra boundary condition for the enrichment degree of freedom, and (ii) no need for transition/blending regions; both of which contribute to maintaining the scalability of the code.Two different cohesive zone models (CZM) are then adopted to capture the physics of the crack propagation mechanisms. At the intralaminar level, an extrinsic CZM embedded in the XFEM formulation is used. At the interlaminar level, an intrinsic CZM is adopted for predicting the failure. The overall framework is implemented in ALYA, a mechanics code specifically developed for large scale, massively parallel simulations of coupled multi-physics problems. The implementation of both intrinsic and extrinsic CZM models within the code is such that it conserves the extremely efficient scalability of ALYA while providing accurate physical simulations of computationally expensive phenomena. The strong scalability provided by the proposed implementation is demonstrated. The model is ultimately validated against a full experimental campaign of loading tests and X-ray tomography analyzes.

AN ACCURATE CALIBRATION METHOD FOR THE MEASUREMENT OF ARTERIAL OXYGEN SATURATION USING PHOTOPLETHYSMOGRAPHY

The monitoring of arterial oxygen saturation (SaO2) is a common practice in both clinical and home environments, and the process of monitoring can be exercised invasively or non-invasively. In the past decades, the pulse oximeter is one of the most popular non-invasive devices that use photoplethysmography (PPG) to measure SaO 2. As the pulse oximeter requires calibration prior to application in clinical practice, a significant number of calibration methods have been proposed based on experimental data collected from human volunteers. Alternatively, models may be derived from the Lambert–Beer law or the photon diffusion equation (PDE). However, most of such calibrated oximeter can only provide accurate readings of SaO 2 at high versus the low levels. We propose to apply an extra-boundary condition to solve the PDE, and then to develop a model-based calibration method that relate optical measurements to the level of SaO 2 in this work. Then, we validate our method against previous model-based methods and an oximeter simulator with higher accuracy when the level of SaO 2 is greater than 90%. In practice, our model-based method can still maintain a good performance when the level of SaO 2 decreases to 60%, thereby demonstrating high potential in the accurate evaluation of the oxygen level of patients by PPG.

Realization of the lepton flavor structure from point interactions

We investigate a 5d gauge theory on S1 with point interactions. The point interactions describe extra boundary conditions and provide three generations, the charged lepton mass hierarchy, the lepton flavor mixing and tiny degenerated neutrino masses after choosing suitable boundary conditions and parameters. The existence of the restriction in the flavor mixing, which appears from the configuration of the extra dimension, is one of the features of this model. Tiny Yukawa couplings for the neutrinos also appears without the see-saw mechanism nor symmetries in our model. The magnitude of CP violation in the leptons can be a prediction and is consistent with the current experimental data.

Realization of the lepton flavor structure from point interactions

We investigate a 5d gauge theory on $S^1$ with point interactions. The point interactions describe extra boundary conditions and provide three generations, the charged lepton mass hierarchy, the lepton flavor mixing and tiny degenerated neutrino masses after choosing suitable boundary conditions and parameters. The existence of the restriction in the flavor mixing, which appears from the configuration of the extra dimension, is one of the features of this model. Tiny Yukawa couplings for the neutrinos also appears without the see-saw mechanism nor symmetries in our model. The magnitude of CP violation in the leptons can be a prediction and is consistent with the current experimental data.

A two-phase shallow debris flow model with energy balance

This paper proposes a thin layer depth-averaged two-phase model provided by a dissipative energy balance to describe avalanches of solid-fluid mixtures. This model is derived from a 3D two-phase model based on the equations proposed by Jackson [R. Jackson, The Dynamics of Fluidized Particles, 2000] which takes into account the force of buoyancy and the forces of interaction between the solid and fluid phases. Jackson's model is based on mass and momentum conservation within the two phases, i.e. two vector and two scalar equations. This system has five unknowns: the solid volume fraction, the solid and fluid pressures and the solid and fluid velocities, i.e. three scalars and two vectors. As a result, an additional equation is necessary to close the system. Surprisingly, this issue is inadequately accounted for in the models that have been developed on the basis of Jackson's work. In particular, Pitman and Le [E.B. Pitman, L. Le, Phil. Trans. R. Soc. A, 2005] replaced this closure simply by imposing an extra boundary condition. If the pressure is assumed to be hydrostatic, this condition can be considered as a closure condition. However, the corresponding model cannot account for a dissipative energy balance. We propose here a closure equation to complete Jackson's model, imposing incompressibility of the solid phase. We prove that the resulting whole 3D model is compatible with a dissipative energy balance. From this model, we deduce a 2D depth-averaged model and we also prove that the energy balance associated with this model is dissipative. Finally, we propose a numerical scheme to approximate the depth-averaged model. We present several numerical tests for the 1D case that are compared to the results of the model proposed by Pitman and Le.

An instrumental electrode model for solving EIT forward problems

An instrumental electrode model (IEM) capable of describing the performance of electrical impedance tomography (EIT) systems in the MHz frequency range has been proposed. Compared with the commonly used Complete Electrode Model (CEM), which assumes ideal front-end interfaces, the proposed model considers the effects of non-ideal components in the front-end circuits. This introduces an extra boundary condition in the forward model and offers a more accurate modelling for EIT systems. We have demonstrated its performance using simple geometry structures and compared the results with the CEM and full Maxwell methods. The IEM can provide a significantly more accurate approximation than the CEM in the MHz frequency range, where the full Maxwell methods are favoured over the quasi-static approximation. The improved electrode model will facilitate the future characterization and front-end design of real-world EIT systems.

Explaining Ground Effect Aerodynamics via a Real-Life Reference Frame

The principle of relative motion as the cause of forces on a body submersedin a uid is foundational in the study of uid mechanics. In aerodynamics the wind tunnelis used as a convenient and safe method by which to test the aerodynamic performance ofbodies. This body-stationary convention has carried over into the computational world withthe development of CFD, though there is no practical reason why the moving body/stationaryuid set-up that corresponds to reality cannot be used for computational modelling. This pointbecomes particularly important as the concept of ground e ect is introduced. With an extraboundary nearby it becomes harder to appropriatel y match the experimental set-up with reality,and the extra boundary condition also adds complexity to computational simulation. A studywas undertaken to compare the body-stationary and body-moving reference frames in grounde ect. The moving reference frame velocity elds allowed new insight into the aerodynamics ofground e ect.

Towards a Gradient Truss Model. Part I: Bar Element Displacement-Force Relations

Recently the author presented a note on gradient truss model by assuming that the bars can support strain gradients along their length. Hence, higher-order strain gradient terms and an internal length scale parameter (the gradient elastic coefficient) were introduced into the constitutive stress-strain relation of the truss bar element. However, in order to solve the governing higher order differential equation, extra non-classical boundary conditions are required and these need to be investigated in order to optimize proper combinations. In this paper, extra non-classical boundary conditions are studied by considering the support conditions of the bar based on imposed strain and strain gradient dominated deformation mechanisms. The quantitative response and applicability of the model under different combinations of extra non-classical boundary conditions are examined. Consequently, twelve non-homogeneous displacement-force relations are presented and form the basis for the gradient truss element stiffness matrices presented in Part II. With these, a relation between the macro-scale of the bar element and an internal length scale in the micro-scale is established thus micro-scale dependent behaviour in the material being modelled can be directly captured using this new truss bar elements. Moreover, the weakening and the strengthening effect of strain and strain gradients imposed at the bar nodes can also be revealed. As an illustration of the model a numerical example is presented

Higher derivative effects for 4d AdS gravity

Motivated by holography, we explore higher derivative corrections to four-dimensional Anti-de Sitter (AdS) gravity. We point out that in such a theory the variational problem is generically not well-posed given only a boundary condition for the metric. However, when one evaluates the higher derivative terms perturbatively on a leading order Einstein solution, the equations of motion are always second order and therefore the variational problem indeed requires only a boundary condition for the metric. The equations of motion required to compute the spectrum around the corrected background are still generically higher order, with the additional boundary conditions being associated with new operators in the dual conformal field theory. We discuss which higher derivative curvature invariants are expected to arise in the four-dimensional action from a top-down perspective and compute the corrections to planar AdS black holes and to the spectrum around AdS in various cases. Requiring that the dual theory is unitary strongly constrains the higher derivative terms in the action, as the operators associated with the extra boundary conditions generically have complex conformal dimensions and non-positive norms.

Natural Convection Boundary Layer Flow Past a Sphere with Constant Heat Flux in Viscoelastic Fluid

The steady natural convection boundary layer flow of a viscoelastic fluid over a solid sphere with constant heat flux is studied in this paper. The boundary layer equations of viscoelastic fluid are an order higher than those for the Newtonian (viscous) fluid. The adherence boundary conditions are insufficient to determine the solution of these equations completely. Thus, the augmentation an extra boundary condition is needed to perform the numerical computational. The governing boundary layer equations are first transformed into non–dimensional form by using special dimensionless variables and then solved by using an implicit finite difference scheme known as Keller box method. Numerical results for the velocity and temperature profiles, wall temperature, as well as skin friction are shown graphically for different values of viscoelastic parameters and Prandtl number. It is found that, when the viscoelastic parameter increased, the values of skin friction decreased while the values of wall temperature are increased.

Mixed Convection Boundary Layer Flow of Viscoelastic Fluids Past a Sphere

The mixed convection boundary layer of a viscoelastic fluid past a sphere with constant temperature is discussed. The boundary layer equations are an order higher than those for the Newtonian (viscous) fluid and the adherence boundary conditions are insufficient to determine the solution of these equations completely. The governing non-similar partial differential equations are first transformed into dimensionless forms and then solved numerically using the Keller-box method by augmenting an extra boundary condition at infinity. Numerical results are presented for different values of the viscoelastic and mixed convection parametersKand , respectively. It is found that for cases of cooling sphere and heating sphere, the boundary layer separates from the sphere. To the best of our knowledge, this important classical problem has not been studied before for the case of a viscoelastic fluid. Thus, the results are original and new for this type of fluids.

Constant heat flux solution for mixed convection boundary layer viscoelastic fluid

The mixed convection boundary layer of a viscoelastic fluid past a circular cylinder with constant heat flux is discussed. The boundary layer equations are an order higher than those for the Newtonian (viscous) fluid and the adherence boundary conditions are insufficient to determine the solution of these equations completely. The governing non-similar partial differential equations are transformed into dimensionless forms and then solved numerically using the Keller-box method by augmenting an extra boundary condition at infinity. Numerical results obtained in the form of velocity distributions and temperature profiles are presented for a range of values of the dimensionless viscoelastic fluid parameter. It is found that for some values of the viscoelastic parameter and some negative values of the mixed convection parameter (opposing flow) the momentum boundary layer separates from the cylinder. Heating the cylinder delays separation and can, if the cylinder is warm enough, suppress the separation completely. Similar to the case of a Newtonian fluid, cooling the cylinder brings the separation point nearer to the lower stagnation point.

Non-capacity or capacity model for fluvial sediment transport

The last several decades have witnessed the development of both capacity and non-capacity models for fluvial sediment transport. While recent investigations of the multiple timescales of fluvial processes have substantiated the understanding of conditional applicability of capacity models, the extent to which a capacity model differs from a non-capacity model remains to be unravelled. This paper presents a comparative investigation of one-dimensional capacity and non-capacity models. As a corollary to the theoretical analyses of the multiple timescales of fluvial processes, this study demonstrates that bed load transport can adapt to local flow sufficiently rapidly and, accordingly, a capacity model is applicable. However, as bed evolution modifies the flow considerably, a non-capacity model is needed if the flow is to be properly resolved in addition to bed load transport. Furthermore, it takes a long time and space for suspended sediment transport to adapt to capacity. Therefore non-capacity modelling is critical for suspended sediment transport, whereas a capacity model may result in considerable errors at best and become ill-posed at worst because of the requirement for extra boundary conditions. The findings of this work should facilitate the physically enhanced development and applications of mathematical river models.

Classical noncommutative electrodynamics with external source

In a $U(1)_{\star}$-noncommutative (NC) gauge field theory we extend the Seiberg-Witten (SW) map to include the (gauge-invariance-violating) external current and formulate - to the first order in the NC parameter - gauge-covariant classical field equations. We find solutions to these equations in the vacuum and in an external magnetic field, when the 4-current is a static electric charge of a finite size $a$, restricted from below by the elementary length. We impose extra boundary conditions, which we use to rule out all singularities, $1/r$ included, from the solutions. The static charge proves to be a magnetic dipole, with its magnetic moment being inversely proportional to its size $a$. The external magnetic field modifies the long-range Coulomb field and some electromagnetic form-factors. We also analyze the ambiguity in the SW map and show that at least to the order studied here it is equivalent to the ambiguity of adding a homogeneous solution to the current-conservation equation.

Chebyshev Spectral Methods and the Lane-Emden Problem

The three-dimensional spherical polytropic Lane-Emden problem is yrr + (2/r)yr + y m = 0, y(0) = 1, yr(0) = 0 where m∈ (0,5) is a constant parameter. The domain is r∈ (0,ξ) where ξ is the first root of y(r). We recast this as a non- linear eigenproblem, with three boundary conditions and ξ as the eigenvalue allow- ing imposition of the extra boundary condition, by making the change of coordinate x≡ r/ξ: yxx + (2/x)yx +ξ 2 y m = 0, y(0) = 1, yx(0) = 0, y(1) = 0. We find that a Newton-Kantorovich iteration always converges from an m-independent starting point y (0) (x) = cos((π/2)x),ξ (0) = 3. We apply a Chebyshev pseudospectral method to discretize x. The Lane-Emden equation has branch point singularities at the endpoint x = 1 whenever m is not an integer; we show that the Chebyshev coefficients are a n ∼ constant/n 2m+5 as n→∞. However, a Chebyshev truncation of N = 100 always gives at least ten decimal places of accuracy — much more accuracy when m is an inte- ger. The numerical algorithm is so simple that the complete code (in Maple) is given as a one page table. AMS subject classifications: 65L10, 65D05, 85A15

Solving two typical inverse Stefan problems by using the Lie-group shooting method

We consider two typical inverse Stefan problems: one is computing a heat flux boundary condition when the moving boundary ξ( t) is given, and another is recovering an unknown moving boundary ξ( t), by knowing an extra Dirichlet boundary condition on the accessible boundary. Through a domain embedding method, we can transform the inverse problem into a parameter identification problem of an advection–diffusion partial differential equation, where ξ( t) and ξ ˙ ( t ) play the role of unknown parameters for the second inverse problem. The ξ ˙ ( t ) appeared in the governing equation makes the identification of ξ( t) rather difficult. However, upon using the Lie-group shooting method (LGSM) we can derive a simple system of algebraic equations to iteratively calculate ξ ˙ ( t ) and then ξ( t) at some discretized times. It is demonstrated through numerical examples that the LGSM is accurate and stable, although under a large measurement noise.

Effects of slip, viscous dissipation and Joule heating on the MHD flow and heat transfer of a second grade fluid past a radially stretching sheet

The flow and heat transfer of an electrically conducting non-Newtonian second grade fluid due to a radially stretching surface with partial slip is considered. The partial slip is controlled by a dimensionless slip factor, which varies between zero (total adhesion) and infinity (full slip). Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective numerical scheme is adopted to solve the obtained differential equations even without augmenting any extra boundary conditions. The important findings in this communication are the combined effects of the partial slip, magnetic interaction parameter and the second grade fluid parameter on the velocity and temperature fields. It is interesting to find that the slip increases the momentum and thermal boundary layer thickness. As the slip increases in magnitude, permitting more fluid to slip past the sheet, the skin friction coefficient decreases in magnitude and approaches zero for higher values of the slip parameter, i.e., the fluid behaves as though it were inviscid. The presence of a magnetic field has also substantial effects on velocity and temperature fields.

An alternative treatment of phenomenological higher-order strain-gradient plasticity theory

Phenomenological higher-order strain-gradient plasticity is here presented through a formulation inspired by previous work for strain-gradient crystal plasticity. A physical interpretation of the phenomenological yield condition that involves an effect of second gradient of the equivalent plastic strain is discussed, applying a dislocation theory-based consideration. Then, a differential equation for the equivalent plastic strain-gradient is introduced as an additional governing equation. Its weak form makes it possible to deduce and impose extra boundary conditions for the equivalent plastic strain. A connection between the present treatment and strain-gradient theories based on an extended virtual work principle is discussed. Furthermore, a numerical implementation and analysis of constrained simple shear of a thin strip are presented.

Flow and heat transfer of a non-Newtonian fluid past a stretching sheet with partial slip

The entrained flow and heat transfer of a non-Newtonian third grade fluid due to a linearly stretching surface with partial slip is considered. The partial slip is controlled by a dimensionless slip factor, which varies between zero (total adhesion) and infinity (full slip). Suitable similarity transformations are used to reduce the resulting highly nonlinear partial differential equations into ordinary differential equations. The issue of paucity of boundary conditions is addressed and an effective second order numerical scheme has been adopted to solve the obtained differential equations even without augmenting any extra boundary conditions. The important finding in this communication is the combined effects of the partial slip and the third grade fluid parameter on the velocity, skin-friction coefficient and the temperature field. It is interesting to find that the slip and the third grade fluid parameter have opposite effects on the velocity and the thermal boundary layers.

Quantum singularities in the BTZ spacetime

The spinless Ba\~nados-Teiltelboim-Zanelli (BTZ) spacetime is considered in the quantum theory context. Specially, we study the case of negative mass parameter using quantum test particles obeying the Klein-Gordon and Dirac equations. We study if this classical singular spacetime, with a naked singularity at the origin, remains singular when tested with quantum particles. The need of additional information near the origin is confirmed for massive scalar particles and all the possible boundary conditions necessary to turn the spatial portion of the wave operator self-adjoint are found. When tested by massless scalar particles or fermions, the singularity is ``healed'' and no extra boundary condition are needed. Near infinity, no boundary conditions are necessary.