Impermeable Boundary Conditions Research Articles

Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law

<p style="text-indent:20px;">This paper is concerned with the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem and the initial-boundary value problem in the half space with impermeable wall boundary condition for a scalar conservation laws with an artificial heat flux satisfying Cattaneo's law. In our results, although the <inline-formula><tex-math id="M1">\begin{document}$ L^2\cap L^\infty- $\end{document}</tex-math></inline-formula>norm of the initial perturbation is assumed to be small, the <inline-formula><tex-math id="M2">\begin{document}$ H^1- $\end{document}</tex-math></inline-formula>norm of the first order derivative of the initial perturbation with respect to the spatial variable can indeed be large. Moreover the far fields of the artificial heat flux can be different. Our analysis is based on the <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> energy method.</p>

Nonlinear analysis of a crack in 2D piezoelectric semiconductors with exact electric boundary conditions

In this paper, taking the exact electric boundary conditions into account, we propose a double iteration method to analyze a crack problem in a two-dimensional piezoelectric semiconductor. The method consists of a nested loop process with internal and outside circulations. In the former, the electric field and electron density in governing equations are constantly modified with the fixed boundary conditions on crack face and the crack opening displacement; while in the latter, the boundary conditions on crack face and the crack opening displacement are modified. Such a method is verified by numerically analyzing a crack with an impermeable electric boundary condition. It is shown that the electric boundary condition on crack face largely affects the electric displacement intensity factor near a crack tip in piezoelectric semiconductors. Under exact crack boundary conditions, the variation tendency of the electric displacement intensity factor versus crack size is quite different from that under an impermeable boundary condition. Thus, exact crack boundary conditions should be adopted in analysis of crack problems in a piezoelectric semiconductor.

Antiplane Fracture Problem of Three Nanocracks Emanating from an Electrically Permeable Hexagonal Nanohole in One-Dimensional Hexagonal Piezoelectric Quasicrystals

Based on the Gurtin-Murdoch surface/interface model and complex potential theory, by constructing a new conformal mapping, the electrically permeable boundary condition with surface effect is established, and the antiplane fracture problem of three nanocracks emanating from a hexagonal nanohole in one-dimensional hexagonal piezoelectric quasicrystals with surface effect is studied. The exact solutions of the stress intensity factor of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate are obtained under the two electrically permeable and the electrically impermeable boundary conditions. The numerical examples show the influence of surface effect on the stress intensity factors of the phonon field and the phason field, the electric displacement intensity factor, and the energy release rate under the two boundary conditions. It can be seen that the surface effect leads to the coupling of the phonon field, phason field, and electric field, and with the decrease of cavity size, the influence of surface effect is more obvious.

Existence and uniqueness of a weak solution of an integro-differential aggregation equation on a Riemannian manifold

A class of integro-differential aggregation equations with nonlinear parabolic term is considered on a compact Riemannian manifold . The divergence term in the equations can degenerate with loss of coercivity and may contain nonlinearities of variable order. The impermeability boundary condition on the boundary of the cylinder is satisfied if there are no external sources of ‘mass’ conservation, . In a cylinder for a sufficiently small , the mixed problem for the aggregation equation is shown to have a bounded solution. The existence of a bounded solution of the problem in the cylinder is proved under additional conditions. For equations of the form with the Laplace-Beltrami operator and an integral operator , the mixed problem is shown to have a unique bounded solution. Bibliography: 26 titles.

Anti-plane fracture problem of four nano-cracks emanating from a regular 4<i>n</i>-polygon nano-hole in magnetoelectroelastic materials

According to the conformal mapping from the exterior region of the regular <i>n</i>-polygon hole to the exterior region of a unit circle and from the exterior region of four cracks emanating from a circle to the interior region of a unit circle, a new conformal mapping is constructed to map the exterior region of four cracks emanating from a regular 4<i>n</i>-polygon hole to the interior of a unit circle. Then, based on the Gurtin-Murdoch surface/interface model and complex method, the anti-plane fracture of four nano-cracks emanating from a regular 4<i>n</i>-polygon nano-hole in magnetoelectroelastic material is studied. The exact solutions of stress intensity factor, electric displacement intensity factor, magnetic induction intensity factor, and energy release rate are obtained under the boundary condition of magnetoelectrically impermeable with considering the surface effect. Without considering the effect of the surface effect, the exact solution of four cracks emanating from a regular 4<i>n</i>-polygon hole in a magnetoelectroelastic material can be obtained. The numerical results show the influences of surface effect and the size of defect on the stress intensity factor, electric displacement intensity factor, magnetic induction intensity factor and energy release rate under the magnetoelectrically impermeable boundary condition. It can be seen that the stress intensity factor, electric displacement intensity factor, and magnetic induction intensity factor are significantly size-dependent when considering the surface effects of the nanoscale defects. And when the size of defect increases to a certain extent, the influence of surface effect begins to decrease and finally tends to follow the classical elasticity theory. When the distance between the center and the vertex of the regular 4<i>n</i>-polygon nano-hole is constant, the dimensionless field intensity factor decreases gradually with the increase of the number of edges, and approaches to the conclusion of a circular hole with four cracks. With the increase of the relative size of the crack, the dimensionless field intensity factor increases gradually. The dimensionless energy release rate of the nanoscale cracked hole has a significant size effect. The increase of mechanical load will increase the normalized energy release rate. The normalized energy release rate first decreases and then increases with electrical load increasing. The normalized energy release rate decreases with magnetic load increasing.

Boundary effect on the hydraulic characteristics of gravelly soil slopes in physical model tests

The stability of a residual gravelly soil slope during heavy rainfall is closely related to the seepage field, which is directly affected by the seepage boundaries of the slope. To examine the boundary effect, a novel model flume that can implement permeable and impermeable boundary conditions was designed. With this flume, hydraulic model tests of a gravelly soil slope considering those two seepage boundaries were carried out. The results show that distinct spatial morphology of seepage fields is reflected by the hydraulic characteristics of the slope models with different seepage boundaries. The slope with permeable boundaries generally exhibits a larger average infiltration rate as well as a relatively smaller moving speed of wetting front in the lower part compared to the slope with impermeable boundaries. The residual moisture content in the deep-lower part of the slope with permeable boundaries is 6.6% cm3/cm3 smaller than that of the slope with impermeable boundaries. Nevertheless, the maximum pore water pressure in the middle part of the slope with permeable boundaries is 1.2 kPa higher than that in the corresponding part of the slope with impermeable boundaries. It is noted that regional pore water pressures in the shallow-lower and shallow-middle parts of the slope with impermeable boundaries exhibit large fluctuations with an amplitude over 0.5 kPa. The seepage field in the slope with permeable boundaries presents a multi-dimensional development, which may lead to a local failure of the slope. The results are of great significance in studying the similarity of seepage boundary condition in the field of similarity theory on landslide model tests and provide an experimental basis for establishing rainfall-induced landslide theory.

Multiple defects in a piezoelectric half-plane under electro-elastic in-plane loadings

This paper provides an analytical approach to calculate mixed-mode field intensity factors of multiple defects in a piezoelectric half-plane. Both the permeable and impermeable boundary conditions are adopted. In this study, the crack problem analysis caused by climb, glide and electric dislocation in the piezoelectric half-plane. The Fourier transform is employed and close-form solutions are obtained for the stress and electric displacements in the medium. Then, the solutions are used to derive singular integral equations for the dislocation density on the face of multiple cracks. These equations are solved numerically to determine the field intensity factors in a piezoelectric half-plane. Finally, several examples are solved and the effects of the cracks interactions, geometrical parameters and loading conditions are investigated.

Wave impacts on structures with rectangular geometries: Part 2 decks, baffles and seawalls with impermeable or porous surfaces

This paper considers wave impacts on baffles, on baffles or decks adjacent to a vertical wall, and on porous seawalls and/or sea beds. For seawalls and vertical baffles, impacts can occur in steep waves, whilst a deck can be struck from below by a rising wave crest either in open sea or in a tank with standing waves (sloshing). A simple analytical model for the pressure impulse, P, due to a wave of idealized geometry and dynamics is developed and applied to the following geometries with impermeable surfaces:•horizontal wave impact onto a vertical wall with a deck at the waterline,•vertical wave impact under a deck in the same configuration (equivalent to vertical water impact of a horizontal plate),•horizontal wave impact onto a surface-piercing vertical baffle in open sea,•as for 3. but with the baffle in front of a wall,•as for 4. but with a deck extending from the vertical wall to the baffle,•bottom-mounted baffle in front of a wall with impact occurring on the wall.We also consider cases that complement part 1 of this paper to include the effect on impacts on a seawall with a porous sea bed and/or sea wall with/without a berm. Finally we reconsider case 3) above but with a porous baffle.The method uses eigenfunction expansions in each of the rectangular regions that satisfy some of the impermeable or porous surface conditions, and a simplified free-surface condition. Their unknown coefficients are determined from the impact boundary condition, impermeable or porous boundary conditions and by matching the solutions, in any two neighbouring rectangles, along their common boundary. Although the fluid motion is treated rather crudely, the method yields the pressure impulse throughout the entire region. Impulses, I, and moment impulses, M, on all or parts of the structure are also presented.

The theory and application of eulerlets

Consider a fixed body in a uniform flow field in the limit as the Reynolds number approaches infinity and the flow field remains steady. Instead of using standard techniques and theory for describing the problem, a new method is employed based upon the concept of matching two different Green’s integral representations over a common boundary, one given by approximations valid in the near-field and the other by approximations in the far-field. Further novelty arises from the choice of a near-field, that is, the Euler flow matched to an Oseen flow far-field. This entails introducing and defining eulerlets that are Green’s functions of the Euler equation. One important consequence of the model is the presence of a new Euler wake velocity not captured in standard models. This has a constant unchanging downstream profile and arises from the matching to the far-field Oseen wake velocity. It is then shown how this representation reduces to classical inviscid ideal flow aerodynamics when applied to flow past aerofoils and wings. It is also shown how it reduces to slender body flow theory. Finally, the formulation is tested on uniform flow past a circular cylinder for mean-steady subcritical laminar flow and turbulent flow. The inviscid impermeability boundary condition is used, the drag coefficient is specified, and a constant distribution of drag eulerlets is modeled. The forward flow separation and pressure drop in the wake are captured and compare favorably with experiment. The future expectation is the modeling of multiple general shaped bodies.

一维六方压电准晶中正六边形孔边裂纹的反平面问题

The anti-plane problem of cracks near regular hexagonal holes in 1D hexagonal piezoelectric quasicrystals was studied. By means of the Cauchy integral formula in the complex variable functions and through construction of conformal mapping functions, the analytical solutions of stress distribution and field intensity factors at the crack tip near the hole were obtained under the electrically impermeable boundary condition. The effects of the edge length and the crack length of the regular hexagon as well as the shear stress on the field intensity factors were discussed with numerical examples.

Surface effects on an electrically permeable elliptical nano-hole or nano-crack in piezoelectric materials under anti-plane shear

An electrically permeable elliptical nano-hole or nano-crack embedded in an infinite piezoelectric material with surface effect is investigated based on the Gurtin–Murdoch surface/interface model, which is subjected to far-field anti-plane mechanical and in-plane electrical loads. The electric field inside the elliptical nano-hole has been taken into consideration, and exact electroelastic fields near the elliptical nano-hole are obtained by using the technique of conformal mapping. The size-dependent stress and electric displacement intensity factors at the crack tip are derived exactly for both electrically permeable and impermeable boundary conditions when the elliptical nano-hole reduces to the nano-crack. Numerical examples are illustrated to show the surface effects on the stress and electric displacement intensity factors of electrically permeable and impermeable cracks, and on the stress and electric field concentrations around the electrically permeable and impermeable holes. The results indicate that the size dependence of an electrically permeable nano-hole or nano-crack is entirely different from that of the electrically impermeable case.

Design safe distance between a buried pipe and a soil slope

Leakage from pipes buried in a soil slope can lead to failure and scour of the slope. Placing the pipe at a safe distance from the slope crest is a key concern in design. The objectives of the work reported here were to investigate the wetting front advancement in a soil slope due to leakage from buried pipes at variable distances from the slope crest considering different soil densities and boundary conditions of the slope, and to recommend a safe separation distance based on the relation between the factor of safety of the slope and the separation distance. Stability charts for slopes were established for different separation distances and infiltration times, which will serve as a guide to determining a safe separation distance. The safe distance depends on soil permeability, slope geometry and infiltration time. A safe distance, regardless of how long the leakage is expected to last, is recommended as twice the slope height for soil slopes 15 m in height and 35° in inclination compacted to 95% relative compaction under impermeable side and bottom boundary conditions. When a pipe is laid within this distance, the ability to detect and repair pipe leakage in a timely manner becomes an essential means of maintaining slope stability.

Prediction of shallow landslide by surficial stability analysis considering rainfall infiltration

In Korea, where shallow deposits of weathered residual soil exist on bedrock in hillslopes, the full saturation of the soil layer caused by reaching of rainwater from the slope surface to an impermeable bedrock is one of the important causes of landslides. Landslides can lead to a shallow failure surface that passes through the contact between soil and impervious bedrock. In this study, a stability analysis method for slopes with a shallow bedrock was developed to predict landslides. The method is based on a one-dimensional conceptual infiltration model that considers an initial inhomogeneous water content distribution. Constant intensity of rainfall was considered and shallow impermeable boundary conditions were imposed on the Green-Ampt model to simulate the impermeable bedrock underlying the weathered residual soil. To consider a hillslope with weathered residual soil when the rainfall intensity is less than the infiltration capacity, the Green-Ampt model was modified to simulate rainfall intensity that is smaller than the soil's saturated hydraulic conductivity. A series of stability analyses were performed for slopes with known hydraulic properties to evaluate the applicability of the proposed procedure. The prediction results were then compared with those obtained by the stability analysis based on the infiltration results from numerical analysis. The results showed that the proposed method can be used to predict landslides due to rainfall infiltration by efficiently considering the movements of the saturated region in hillslopes that have shallow impermeable bedrock.

Effects of thermal and electric boundary conditions on fracture of three-dimensional thermopiezoelectric media

An arbitrarily shaped planar crack under different thermal and electric boundary conditions on the crack surfaces is studied in three-dimensional transversely isotropic thermopiezoelectric media subjected to thermal–mechanical–electric coupling fields. Using Hankel transformations, Green functions are derived for unit point extended displacement discontinuities in three-dimensional transversely isotropic thermopiezoelectric media, where the extended displacement discontinuities include the conventional displacement discontinuities, electric potential discontinuity, as well as the temperature discontinuity. On the basis of these Green functions, the extended displacement discontinuity boundary integral equations for arbitrarily shaped planar cracks in the isotropic plane of three-dimensional transversely isotropic thermopiezoelectric media are established under different thermal and electric boundary conditions on the crack surfaces, namely, the thermally and electrically impermeable, permeable, and semi-permeable boundary conditions. The singularities of near-crack border fields are analyzed and the extended stress intensity factors are expressed in terms of the extended displacement discontinuities. The effect of different thermal and electric boundary conditions on the extended stress intensity factors is studied via the extended displacement discontinuity boundary element method. Subsequent numerical results of elliptical cracks subjected to combined thermal–mechanical–electric loadings are obtained.

Optimum configuration for accurate simulations of chaotic porous media with Lattice Boltzmann Methods considering boundary conditions, lattice spacing and domain size

Simulations of the flow field through chaotic porous media are powerful numerical challenges of special interest in science and technology. The simulations are usually done over representative samples which summarise the properties of the material. Several factors affect the accuracy of the results. Firstly the spatial resolution has to be fine enough to be able to capture the smallest geometrical details. Secondly the domain size has to be large enough to contain the large characteristic scale of the porous media. And finally the effects induced by the boundary conditions have to be diluted when more realistic options are not available. This is the case when the geometry is obtained by tomography and the periodic boundary conditions cannot be applied to delimit the sample because its geometry is not periodic. Impermeable boundary conditions are usually chosen to enclose the domain, forcing mass conservation. As a result, the flow field is over-restricted and the total pressure drop can be over-estimated. In this paper a new strategy is presented to optimise the computational resources consumption keeping the restrictions imposed by the accuracy criteria. The effects of the domain size, discretisation thickness and boundary condition disturbances are studied in detail. The study starts with the procedural generation of chaotic porous walls which mimics acicular mullite filters. An advantage of this process is the possibility to create periodic geometries. Periodicity permits the application of advanced techniques such as cyclic cross-correlations between the phase field and the velocity component fields without aliasing. From cross-correlation operations the large characteristic scale is obtained. The result is a lower threshold for the domain size. In second place a mesh independent study is done to find the upper threshold for the lattice spacing. The Minkowski–Bouligand fractal dimension of the fluid–solid interface corroborates the results. It has been demonstrated how the fractal dimension is a good candidate to replace the mesh independent study with lower computational cost for this type of problems. The last step is to compare the results obtained for a periodic geometry applying periodicity and symmetry as boundary conditions. Considering the periodic case as reference the resultant error is analysed. The explanation of the analysis includes how the intensity of the error changes in space and the limitations of symmetric boundary conditions.

Математическая модель взаимодействия ледового покрова и гидродинамического диполя в канале

In this paper, hydroelastic waves generated by a hydrodynamic dipole in a channel covered with ice are studied. The stationary dipole placed in a flow of a liquid with constant speed and the dipole that moves uniformly along the channel are described. A mathematical model is based on the Kelvin-Voigt differential equation of viscoelastic plate and the Laplace equation for a velocity potential under the ice cover. The velocity potential is equal to the sum of the dipole potential in the channel and a potential of the flow caused by the deflection of the plate. The equation for the dipole potential in the channel is obtained by a method of mirror images for four walls. These equations are supplemented by impermeability boundary conditions on the walls and bottom of the channel, clamped boundary conditions of ice on the walls of the channel, and kinematic and dynamic conditions at the ice-liquid interface. The dynamic condition is described by the linearized Cauchy-Lagrange integral. A formulation of the stationary problem is studied. The solution is sought in the form of a traveling wave, which is independent of time in a coordinate system moving with the dipole. Streamlines of the fluid motion and shapes of the flow are studied for the case of flowing around the dipole in the channel. Obtained results are compared with forms of flow in an unbounded fluid.DOI 10.14258/izvasu(2017)1-30

Natural consolidation characteristics of viscous debris flow deposition

: Pore water pressure and water content are important indicators to both deposition and consolidation of debris flows, enabling a direct assessment of consolidation degree. This article gained a more comprehensive understanding about the entire consolidation process and focused on exploring pore water pressure and volumetric water content variations of the deposit body during natural consolidation under different conditions taking the viscous debris flow mass as a study subject and by flume experiments. The results indicate that, as the color of the debris changed from initial dark green to grayish-white color, the initial deposit thickness declined by 3% and 2.8% over a permeable and impermeable sand bed, respectively. A positive correlation was observed between pore water pressure and depth in the deposit for both scenarios, with deeper depths being related to greater pore water pressure. For the permeable environment, the average dissipation rate of pore water pressure measured at depths of 0.10 m and 0.05 m were 0.0172 Pa/d and 0.0144 Pa/d, respectively, showing a positive changing trend with increasing depth. Under impermeable conditions, the average dissipation rates at different depths were similar, while the volumetric water content in the deposit had a positive correlation with depth. The reduction of water content in the deposit accelerated with depth under impermeable sand bed boundary conditions, but was not considerably correlated with depth under permeable sand bed boundary conditions. However, the amount of discharged water from the deposit was greater and consolidation occurred faster in permeable conditions. This indicates that the permeability of the boundary sand bed has a significant impact on the progress of consolidation. This research demonstrates that pore water and pressure dissipations are present during the entire viscous debris consolidation process. Contrasting with dilute flows, pore pressure dissipation in viscous flows cannot be completed in a matter of minutes or even hours, requiring longer completion time - 3 to 5 days and even more. Additionally, the dissipation of the pore water pressure lagged the reduction of the water content. During the experiment, the dissipation rate fluctuated substantially, indicating a close relationship between the dissipation process and the physical properties of broadly graded soils.

Method of Fundamental Solutions for Three-Dimensional Exterior Potential Flows

AbstractThe method of fundamental solutions for solving three-dimensional potential flow problems with an unbounded domain is developed in this paper. Because the present meshless method is free from treatments of singularities, meshes, and numerical integrations, the computational effort and memory storage required are minimal as compared with other numerical schemes. It is a suitable technique for the potential flow problems in an unbounded domain, because it only requires that the field points be located on the body surface to satisfy the impermeable boundary condition without defining an optimal finite-domain computation. For the flow passing an obstacle, the impermeable boundary condition on the body surface can be dealt with by the linear superposition scheme. With the validations for the uniform flow passing a two-dimensional wing section and some three-dimensional benchmark problems, an application of case study for the flow field in the multiconnected unbounded domain is carried out. From the com...

Complex variable-based analysis for two semi-permeable collinear cracks in a piezoelectromagnetic media

ABSTRACTThe problem of two equal collinear cracks weakening a poled transversely isotropic piezo-electro-magnetic plate is addressed under semi-permeable electro-magnetic boundary conditions on the crack faces. The problem is formulated employing Stroh formalism and solved using a complex variable technique. Closed form analytical expressions are derived for various fracture parameters. An illustrative numerical case study is presented for poled BaTiO3 − CoFe2O4 ceramic cracked plate to study the effect of prescribed electric load, magnetic load, and inter-crack distance on fracture parameters. Moreover, a comparative study is done of volume fraction, impermeable, and semi-permeable boundary conditions on fracture parameters.

Crack tip enrichment functions for extended finite element analysis of two-dimensional interface cracks in anisotropic magnetoelectroelastic bimaterials

In this paper, the extended finite element method (X-FEM) is employed to present a static fracture analysis of two-dimensional interfacial crack problems in linear magnetoelectroelastic (MEE) bimaterials. Magnetoelectrically impermeable crack-face boundary conditions are adopted and the multi-field coupled effect in MEE bodies is considered. In order to capture the oscillating singularity of the extended stresses near the interfacial crack tip, suitable crack tip enrichment functions for anisotropic and transversely isotropic MEE bimaterials are newly derived and further applied to perform X-FEM analysis. As the fracture parameter, the J-integral is evaluated using the domain form of the contour integral. By comparing yielded results with the analytical and numerical solutions of the corresponding interfacial crack problems, the validity of the proposed formulation is verified. Moreover, it is shown that the results obtained by way of the new enrichment functions are superior to those obtained by the fourfold enrichment functions and twelvefold enrichment functions, especially in the case of topological enrichment. If there is no special requirement for precision, the fourfold enrichment functions with less computational cost can also be used to conduct X-FEM analysis for the present problem.