Half-space Problem Research Articles

The Projection Games Conjecture and the hardness of approximation of super-SAT and related problems

The Super-SAT (SSAT) problem was introduced in [1,2] to prove the NP-hardness of approximation of two popular lattice problems - Shortest Vector Problem and Closest Vector Problem. SSAT is conjectured to be NP-hard to approximate to within a factor of nc (c is positive constant, n is the size of the SSAT instance). In this paper we prove this conjecture assuming the Projection Games Conjecture (PGC) [3]. This implies hardness of approximation of these lattice problems within polynomial factors, assuming PGC. We also reduce SSAT to the Nearest Codeword Problem and Learning Halfspace Problem [4]. This proves that both these problems are NP-hard to approximate within a factor of Nc′/log⁡log⁡n (c′ is positive constant, N is the size of the instances of the respective problems). Assuming PGC these problems are proved to be NP-hard to approximate within polynomial factors.

Aspects of modeling and numerical simulation of dry point contacts between viscoelastic solids

The virtual absence of pressure driven flow inside an elastohydrodynamic lubrication (EHL) contact leads to remarkable similarity between lubricated film thickness and separation by a viscoelastic layer as is suggested by experimental and theoretical results, see [1]. This offers interesting opportunities for mixed lubrication modeling without needing the flow continuity-based Reynolds equation. In this paper numerical simulation of viscoelastic concentrated contact modeling is revisited, first considering the problem of a viscoelastic half-space and a rigid sphere, both in static (squeeze) as well as (frictionless) rolling/sliding conditions. In particular modeling aspects are discussed so that the efficiency of computation of the viscoelastic deformation remains close to the efficiency of computation of an elastic deformation. This is achieved by rewriting Hunter’s [2] deformation equation from an integral form into a differential form by applying the Leibnitz integral theorem, even for complex materials with multiple relaxation times to represent complex viscoelastic properties of materials. The approach can be straightforwardly implemented in any (EHL) contact solver regardless of the numerical method. Detailed results are presented, such as the effect of the Deborah number, the relation between the extreme cases of the rolling problem and the static problem, and discussed in detail. In addition to the rigid sphere against a viscoelastic half-space, also the case of two viscoelastic bodies is considered.

Axisymmetric BEM analysis of layered elastic halfspace with volcano-shaped mantle and cavity under internal gas pressure

This paper presents an axisymmetric BEM analysis of layered elastic halfspace with volcano-shaped mantle and cavity under internal gas pressure. The problem is of interest to understand the behavior of volcanoes, tectonic earthquakes and other oil and gas reservoirs. The volcano-shaped mantle ground topography and the internal spherical or ellipsoidal cavity are added to the conventional model of layered halfspace. The analysis uses the axisymmetric boundary element method (BEM) associated with the fundamental solution of a multilayered elastic fullspace subject to the concentrated ring-body force. The BEM is further verified for its high efficiency and accuracy by comparing its result with the analytical and FEM solutions for the problem of a homogeneous elastic halfspace whose spherical cavity is under internal pressure. The BEM is used to examine the effect of the volcano-shaped mantle, the cavity shape, the layered material properties and the internal pressure on the elastic behavior of the halfspace under the internal pressure loading within the cavity. The displacements and stresses at the external and internal boundaries and within the layered halfspace are presented and analyzed in detail. The presence of the volcano-shaped mantle can reduce significantly the swelling ground movement induced by the expansive pressure in the cavity in the layered halfspace. The cavity shape can have significant effect to the location and magnitude of the maximum tensile principal hoop stress on the cavity surface induced by its internal pressure.

Critical boundary value problems in the upper half-space

We consider the critical problem −Δu−12x⋅∇u=0 in R+N,∂u∂ν=λ|u|p−2u+|u|2∗−2u on ∂R+N,where R+N=(x′,xN):x′∈RN−1,xN>0 is the upper half-space, N≥4, ν is the outward normal vector at the boundary and 2≤p<2∗:=2(N−1)/(N−2). Using a variational approach, we obtain nonnegative nonzero solutions according to the value of the parameter λ>0.

Stability properties of a crack inverse problem in half space

We show in this paper a Lipschitz stability result for a crack inverse problem in half space. The direct problem is a Laplace equation with zero Neumann condition on the top boundary. The forcing term is a discontinuity across the crack. This formulation can be related to geological faults in elastic media or to irrotational incompressible flows in a half space minus an inner wall. The direct problem is well posed in an appropriate functional space. We study the related inverse problem where the jump across the crack is unknown, and more importantly, the geometry and the location of the crack are unknown. The data for the inverse problem are of Dirichlet type over a portion of the top boundary. We prove that this inverse problem is uniquely solvable under some assumptions on the geometry of the crack. The highlight of this paper is showing a stability result for this inverse problem. Assuming that the crack is planar, we show that reconstructing the plane containing the crack is Lipschitz stable despite the fact that the forcing term for the underlying PDE is unknown. This uniform stability result holds under the assumption that the forcing term is bounded above and the Dirichlet data are bounded below away from zero in appropriate norms.

The periodic contact problem for spherical indenters and viscoelastic half-space

The interaction of a periodic system of rigid spherical indenters with a viscoelastic half-space is considered. The linear viscoelastic models (the standard viscoelastic solid model and the Maxwell model) are used to describe the mechanical behavior of the half-space. The case of a constant load applied to each indenter is investigated. Using the localization method and the solution of the corresponding elastic problem, the analytical expressions for the contact pressure under a single indenter, the radius of a single contact spot, and for the additional displacement function are derived. The influence of the contact density and the viscoelastic half-space mechanical properties on the variation of the contact characteristics in time is analyzed.

Trefftz Displacement Potential Function Method for Solving Elastic Half-Space Problems

The elastic half-space problem has been solved previously using Boussinesq, Papkovich, Love, and Green and Zerna, potential function methods. In this work, the Trefftz displacement potential function method is used to obtain the stress and displacement fields in an elastic half-space subjected to boundary loads. Point load and various distributed loads are considered. The problem is presented using displacement formulation as Navier–Lame equations. It is proved that the Trefftz functions are solutions of the Navier–Lame displacement equations. Strain fields are derived in terms of the Trefftz function using the strain-displacement relations. The stress fields are similarly derived. The Trefftz function for the case of a point load acting at the origin of the elastic half-space is derived using the double exponential Fourier transformation technique. Stress equilibrium boundary conditions are used to fully determine the Trefftz function. Stress and displacement fields for the point load are then determined. The solutions to stress and displacement fields for point load are then used as Green functions to obtain stress and displacement fields for uniformly distributed load over a finite line, circular area and rectangular area. It is found that the solutions obtained for the stress and displacement fields in the elastic half-space due to point and distributed loads are identical with previously obtained expressions, thus validating this work.

REALIZATION OF GROUND MOTION INPUT IN ABAQUS FOR LAYERED FOUNDATION UNDER SV WAVE OF OBLIQUE INCIDENCE OVER CRITICAL ANGLE

In the finite element numerical simulation, it is an urgent challenge to determine the ground motion input for the layered foundation under SV wave of oblique incidence over critical angle. Exact dynamic stiffness matrix in the frequency domain (i.e., frequency domain stiffness matrix method) of layered foundation is used to derive the formula for calculating the equivalent node force of ground motion input under arbitrary angular oblique incidence of SV wave, and the validity and accuracy of the ground motion input applied to ABAQUS numerical simulation are demonstrated by simulating seismic wave field of homogeneous half-space and layered foundation under oblique incidence of SV wave through ABAQUS software. It is shown that the inclined SV wave incidence with an arbitrary angle can be realized through ABAQUS simulation by means of stiffness matrix method, and the method is of high accuracy, especially in the case of SV wave incidence with an inclined angle greater than the critical one. The oval like motion trails of surface particles in uniform half space and the wave motion characteristics of layered half space are simulated commendably by finite element numerical modelling. On the above basis, the frequency domain stiffness matrix method is further combined with equivalent linearization method, which solves the ground motion input problem of two-dimensional layered half space considering soil nonlinearity for the inclined incidence with arbitrary angle.

Reflection of plane waves from the impedance boundary of a magneto-thermo-microstretch solid with diffusion in a fractional order theory of thermoelasticity

In this research article, the effect of the impedance boundary on the reflection problem in a magnetized thermo-microstretch diffusion solid half-space is analyzed. The governing equations in generalized Sherief’s Model of fractional-order magneto-thermo-microstretch elasticity with diffusion in specialized x-y plane are derived and are solved for a plane-wave solution. The plane-wave solution indicates the existence of four coupled longitudinal waves namely CLD-1, CLD-2, CLD-3, CLD-4 waves and two coupled transverse waves namely CTD and CTM waves propagating with distinct speeds in the half-space. The thermally insulated/isothermal plane surface of a half-space is subjected to impedance boundary conditions. The expressions for reflection coefficient and energy ratios for the reflected waves are obtained and the numerical results are illustrated graphically to observe the effect of the magnetic field, fractional-order and diffusion parameters against the angle of incidence on the reflection coefficient and energy ratios of reflected waves.

Analytical study of two-dimensional thermo-mechanical responses of viscoelastic skin tissue with temperature-dependent thermal conductivity and rheological properties

The generalized thermo-viscoelastic theory without energy dissipation is used in this work to investigate the transfer of bioheat and the mechanical heat-induced response in a human skin plane tissue with variable thermal and rheological properties. Integral transformations are used for Laplace and Fourier. In the scheme of two-dimensional equations, the exponential matrix procedure, which forms the basis of the state-space approach of modern control theory, is applied. The resulting formulation is applied to a thermal shock half-space problem. The inversion process for Fourier and Laplace transforms is carried out using a numerical method based on Fourier series expansions. Comparisons between computational measurements and Penne’s experimental verification indicate that the Green–Naghdi (II) bioheat mathematical model is an effective method for estimating the transition of bioheat to viscoelastic skin tissue. The influences of variable thermal conductivity and volume materials properties on all fields are examined.

Failure of Fredholm solvability for the Dirichlet problem corresponding to weakly elliptic systems

It is known (cf. Martell et al. in Rev Mat Iberoam 32(3):913–970, 2016) that the $$L^p$$ -Dirichlet problem for constant coefficient second-order systems satisfying the Legendre-Hadamard (strong) ellipticity condition is well posed in the upper half-space. However, this may fail if only weak ellipticity for the system in question is assumed. In this paper we shall show that the aforementioned failure is at a fundamental level, in the sense that there exist systems which are weakly elliptic (i.e., their characteristic matrix is invertible) for which the $$L^p$$ -Dirichlet problem in the upper half-space is not even Fredholm solvable.

Nonstationary Displacements in an Axisymmetric Problem for an Elastic Half-Space Under Mixed Boundary Conditions*

Nonstationary Displacements in an Axisymmetric Problem for an Elastic Half-Space Under Mixed Boundary Conditions*

Obliquely incident P-SV wave scattering by multiple structures in layered half space using combined zigzag-paraxial boundary condition

Seismic wave scattering by multiple structures in layered half space is a classically multiple scattering problem in infinite domain. The computationally efficient substructure method is commonly adopted to solve this problem, because the structures can be modelled by the finite element scheme with the combination of the high-accuracy boundary condition to provide the dynamic stiffness of the truncated infinite rock or soil media. However, it is difficult to establish a high-accuracy boundary condition for multiple scattering problem in layered half space. In this paper, a combined zigzag-paraxial boundary condition is presented, and it is coupled with the seismic wave input method, as well as with the finite element scheme, to form the substructure method for solving the multiple scattering problem in layered half space. To verify the proposed substructure method, it is firstly applied to obliquely incident P-SV wave scatterings by a canyon, a cavity and an alluvial basin in homogeneous half space. The results are validated by comparing with several existing analytical and numerical methods. To demonstrate the capability of the proposed substructure method, it is finally applied to obliquely incident P-SV wave scattering by multiple tunnels in layered half space, and the effects of twin tunnels, soft soil interlayer and a V-shaped canyon on the structural response are discussed.

The nonlinear Schrödinger equation in the half-space

The present paper is concerned with the half-space Dirichlet problem where mathbb {R}^{N}_{+}:= {,x in mathbb {R}^N: x_N > 0, } for some N ge 1 and p > 1, c > 0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to (P_c). We prove that the existence and multiplicity of bounded positive solutions to (P_c) depend in a striking way on the value of c > 0 and also on the dimension N. We find an explicit number {c_p}in (1,sqrt{e}), depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions N ge 2, we prove that, for 0< c < {c_p}, problem (P_c) admits infinitely many bounded positive solutions, whereas, for c > {c_p}, there are no bounded positive solutions to (P_c).

On a thermoelastic magnetized half-space problem considering presence and absence of rotation in the context of GN (II) model

In this paper, having or lacking a rotating, magnetic field, and a time-dependent source of heat on a two-dimensional homogeneous, isotropic, and thermoelastic half-space was illustrated. The governing equations were formulated in the considering Maxwell's stresses. The surface boundary settings were explored considering the thermoelasticity model of Green and Naghdi (type II). The techniques of the eigenvalue method and the normal mode analysis were utilized to resolve the non-dimensional coupled equations. We analyzed theoretically and computed numerically the rotation, time, wave number, frequency, and magnetic field. A comparison between having or lacking of the rotation was made with the results obtained. The displacement components, temperature, dilatation, and average of the normal stress were displayed in graphs to illustrate the physical meaning of the wavenumber, frequency, rotation, magnetic field, and time-dependent heat source. Normal stresses have been displayed graphically to show the physical meaning of the magnetic field, rotation, frequency, wave number, and time-dependent heat source. The results obtained show that having rotation, magnetic field, and time affects significantly eigenvalue issue, especially in astrophysics, aircraft, engineering, astronomy, petroleum extracting, and structures.

State Space Approach to Electro-Magneto-Thermoelasticity with Energy Dissipation

In this article a two-dimensional problem of generalized thermoelasticity has been formulated with state space approach. In this formulation, the governing equations are transformed into a matrix differential equation whose solution enables us to write the solution of any two-dimensional problem in terms of the boundary conditions. The resulting formulation is applied to an isotropic half space problem under Green-Naghdi type-III model i.e., with energy dissipation theory of thermoelasticity in the presence of a magnetic field. The bounding surface is traction free and subjected to a time dependent thermal shock. The solution for temperature distribution, displacements and stress components are obtained and presented graphically. The effect of magnetic field, electric field and phase velocity on the considered parameters is observed in the figures.

Resonant Frequency Derived from the Rayleigh-Wave Dispersion Image: The High-Impedance Boundary Problem

ABSTRACTWe present a simple and automated approach to estimate primary site-response resonance, layer thickness, and shear-wave velocity directly from a dispersion image for a layer over half-space problem. We demonstrate this for high-impedance boundary conditions that lie in the upper tens of meters. Our approach eliminates the need for time-consuming dispersion curve picking and 1D shear-wave velocity inversion for large data volumes that can capture velocity structure in profile. We highlight important relationships between dispersion characteristics and resonance parameters through synthetic modeling and field data acquired over Atlantic Coastal Plain sediments. In this environment, shallow soil conditions are critical to accurately estimate earthquake site response. We suggest that this image processing approach can be applied to a range of high-impedance conditions, at a range of scales, or can provide model constraints for more complex velocity structures.

Influence of frictional contact on indentation of elastic layer under surface energy effects

The paper examines the influence of frictional contact on an indentation problem of a finite-thickness elastic layer under a rigid flat-ended cylindrical indenter considering surface energy effects based on Coulomb friction model. The continuum-based approach corresponding to the complete version of Gurtin-Murdoch surface elasticity model is adopted to demonstrate the influence from surface energy. Normal and radial displacement boundary conditions in the contact region corresponding to the frictional indentation problem is formulated in the term of integral equations by employing the relevant displacement Green's function obtained by the use of Hankel integral transform. The contact tractions are solved by applying a collocation technique together with the assumption of a piecewise distribution for the tractions within each discretized annular ring element. The accuracy of proposed solution scheme is validated with existing solutions from classical continuum mechanics for frictional indentation problem of an elastic half-space. Selected numerical results are presented to demonstrate the influence of friction effects on the elastic fields of the layer under the influence of surface energy. It is found that the consideration of frictional contact makes the layer stiffer when compared to the frictionless indentation, and the influence of friction effect is smaller when the surface energy effects are accounted for. In addition, the elastic fields of the layer are clearly size-dependent and the elastic layer becomes stiffer with the presence of surface stresses.

A two-dimensional half-space problem in an initially stressed rotating medium with microtemperatures

PurposeThe purpose of this paper is to establish a model of two-dimensional half-space problem of linear, isotropic, homogeneous, initially stressed, rotating thermoelastic medium with microtemperatures. The expressions for different physical variables such as displacement distribution, stress distribution, temperature field and microtemperatures are obtained in the physical domain.Design/methodology/approachNormal mode analysis technique is adopted to procure the exact solution of the problem.FindingsNumerical computations have been carried out with the help of MATLAB programming, and the results are illustrated graphically. Comparisons are made to show the effects of rotation, time and microtemperatures on the resulting quantities. The graphical results indicate that the effects of rotation, microtemperatures and time are very pronounced on the field variables.Originality/valueIn the present work, we have investigated the effects of rotation, time and microtemperature in an initially stressed thermoelastic medium. Although various investigations do exist to observe the disturbances in a thermoelastic medium under the effects of different parameters, the work in its present form, i.e. the disturbances in a thermoelastic medium in the presence of angular velocity, initial stress and microtemperature have not been studied till now. The present work is useful and valuable for analysis of problems involving coupled thermal shock, rotation parameter, microtemperatures and elastic deformation.

Thermal shock fracture associated with a unified fractional heat conduction

Recently, a unified fractional heat conduction model is newly proposed, which is a further improvement of the fractional thermoelasticity theory. It would be interesting to know how the newly developed fractional derivative definitions affects the thermal fracture behavior when some defects such as cracks exist in the medium. This work is aimed at analyzing the thermal fracture problem of an elastic half-space and strip with an insulated Griffith crack based on the unified fractional heat conduction model. Fourier and Laplace transforms are employed to solve the mixed boundary problem associated with a time-fractional partial differential equation. Temperature and stress intensity factors are evaluated by solving a system of singular integral equations. Effects of different fractional derivatives and different parameters on the thermoelastic fields are analyzed. A comparison of the temperature and stress intensity factors between the present model and the Fourier model is made. Numerical results show that different fractional definitions have different memory effects on the transient responses, and the effects of fractional definitions on the responses will vary accordingly with the parameter values.