Half-space Problem Research Articles

Magneto-thermoelasticity Green–Naghdi theory with memory-dependent derivative in the presence of a moving heat source

In the present work, a mathematical model of the Green-Naghdi thermoelasticity theory of type III (GN-III) with memory-dependent derivative (MDD) heat transfer for a perfectly conducting isotropic media has been constructed. The state-space and Laplace transform techniques are adopted for the solution of a half-space problem in the presence of a moving heat source with constant velocity. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for all fields are given and illustrated graphically. Comparison is made with the results predicted by coupled thermoelasticity (DCT). The influences of MDD parameters and heat source speed on all fields are examined.

Half-Space Model Problem for Navier-Lamé Equations with Surface Tension

Recently, we have seen the phenomena in use of partial differential equations (PDEs) especially in fluid dynamic area. The classical approach of the analysis of PDEs were dominated in early nineteenth century. As we know that for PDEs the fundamental theoretical question is whether the model problem consists of equation and its associated side condition is well-posed. There are many ways to investigate that the model problems are well-posed. Because of that reason, in this paper we consider the <img src=image/13426244_01.gif>-boundedness of the solution operator families for Navier-Lamé equation by taking into account the surface tension in a bounded domain of <img src=image/13426244_02.gif>-dimensional Euclidean space (<img src=image/13426244_02.gif>≥ 2) as one way to study the well-posedess. We investigate the <img src=image/13426244_01.gif>-boundedness in half-space domain case. The <img src=image/13426244_01.gif>-boundedness implies not only the generation of analytic semigroup but also the maximal <img src=image/13426244_03.gif> regularity for the initial boundary value problem by using Weis's operator valued Fourier multiplier theorem for time dependent problem. It was known that the maximal <img src=image/13426244_03.gif> regularity class is the powerful tool to prove the well-posesness of the model problem. This result can be used for further research for example to analyze the boundedness of the solution operators of the model problem in bent-half space or general domain case.

Acoustic radiation from a sphere pulsating near an impedance plane using a boundary integral equation method

In this study, a boundary integral equation method is proposed for investigating acoustic pressure radiation from a sphere pulsating near a free surface or an impedance plane. The half-space and free-space problems are investigated for the acoustic radiation of pulsating sphere. The effects of free surface and impedance boundaries are introduced into the mathematical model by employing three different half-space Green’s functions, respectively. These Green’s functions are derived, respectively, using the single image-source method, multiple equivalent-source method, and complex equivalent-source method. Green’s functions are implemented into the boundary element (BE) formulation. The surface of the pulsating sphere is discretized with linear and quadratic BEs, and the Combined Helmholtz Integral Equation Formulation (CHIEF) is employed to overcome the non-uniqueness problem. Four different case studies are considered for the sphere pulsating near a free surface or an impedance plane. The first case study involves the sphere pulsating near a free surface (perfectly reflective) and the single image-source method is used in the boundary element method (BEM) formulation. In the second case study, the sphere is assumed as pulsating near a perfectly reflecting and perfectly absorbing impedance planes, respectively. The multiple equivalent-source method is employed for the perfectly reflecting plane, but the multiple equivalent-source method and complex equivalent-source methods for the perfectly absorbing plane. The third case study involves a general impedance plane, and all the methods are employed, respectively, in the BE formulation. The final case study assumes a general impedance plane forming a perpendicular incidence and the complex equivalent-source method is used in this particular case. It is observed that there is a very good comparison between the results obtained from all these methods.

Half-space albedo problem with modified FN method for linear and quadratic anisotropic scattering

Abstract One speed, time independent and homogeneous medium neutron transport equation can be solved with the anisotropic scattering which includes both the linear anisotropic and the quadratic anisotropic scattering properties. Having solved Case’s eigenfunctions and the orthogonality relations among these eigenfunctions, some neutron transport problems such as albedo problem can be calculated as numerically by using numerical or semi-analytic methods. In this study the half-space albedo problem is investigated by using the modified FN method.

Partial Fourier Transform Methods to Solve the Solution Formula of Stokes Equation in Half-Space

Fluids are a shape of a matter which have substance liquids, gases and plasmas. In our daily life, fluids become important part, such as part of our blood and also help our body getting nutrients. It is well known that fluid motion can be described in mathematical model in especially in form of partial differential equations (PDE) and called as Navier Stokes Equations (NSE). The Navier Stokes equation is derived from balance of conservation of mass and conservation of momentum. In this paper, we consider the solution formula of the linearized of the Navier Stokes Equation (NSE) with the initial boundary value (IBV) problem in half space without surface tension. The model problem under consideration covers of non-linear fluid type. We solve the solution formula of velocity and density of the model problem by using Fourier transform and partial Fourier transform method. The strategy geting the solution of the model problem is based on the analysis of some resolvent of the model problem which obtained by using Laplace transform of the Stokes equations. Therefore, In particular, the formula of velocity and density of the Stokes equation are obtained.

Partial Fourier Transform Methods to Solve the Solution Formula of Stokes Equation in Half-Space

Fluids are a shape of a matter which have substance liquids, gases and plasmas. In our daily life, fluids become important part, such as part of our blood and also help our body getting nutrients. It is well known that fluid motion can be described in mathematical model in especially in form of partial differential equations (PDE) and called as Navier Stokes Equations (NSE). The Navier Stokes equation is derived from balance of conservation of mass and conservation of momentum. In this paper, we consider the solution formula of the linearized of the Navier Stokes Equation (NSE) with the initial boundary value (IBV) problem in half space without surface tension. The model problem under consideration covers of non-linear fluid type. We solve the solution formula of velocity and density of the model problem by using Fourier transform and partial Fourier transform method. The strategy geting the solution of the model problem is based on the analysis of some resolvent of the model problem which obtained by using Laplace transform of the Stokes equations. Therefore, In particular, the formula of velocity and density of the Stokes equation are obtained.

Gurtin and Murdoch's surface effect on the elastic behavior of an elastic half space subjected to body forces

Surface effect have significant influence on mechanical behavior of material at nano scale. Classical solutions have limitations in predicting the micromechanics of materials due to the neglect of surface effect. This paper examines a generalized Mindlin's problem for an elastic half space with surface effect and subjected to arbitrarily orientated body forces. It employs the Gurtin and Murdoch's theory of surface mechanics to describe the surface effect. It fully considers both the residual stress and surface elastic constants. It utilizes the GKS based method to solve the boundary value problem. Analytical solutions of the full elastic field due to uniform circular loading and point loading are derived and expressed in terms of Hankel transform. These new solutions are general and can be reduced to several existing solutions for the surface loading problems and those derived based on simplified version of surface elasticity. Additionally, exact closed-form solutions for the half space with limiting boundary conditions, including traction free, rigidly fixed, inextensible and rolling surfaces can also be obtained from the present solutions. The new solutions have revealed some unique physical phenomena for incompressible solid, which are more general than existing findings. Numerical studies are performed using the solutions to investigate the elastic behavior of the half space with the presence of the surface effect. The numerical results show the substantial differences between the elastic fields predicted by present solutions and those by classical solutions without the consideration of surface effect. The new solutions for point loading can serve as Green's function to address related dislocation problems and other mixed boundary value problems in nano-scale materials, where the surface effect can play significant role on the mechanical behavior of material.

A singularity analysis in Keer’s elastic indentation problem

A non-axisymmetric three-dimensional frictionless contact problem for an elastic half-space and a cylindrical indenter with the face in a wedge form is considered under the assumption of a circular region of contact. Explicit formulas are derived for the critical indenter displacement and contact force that are needed to ensure full contact over the entire contact region. It is shown that the contact pressure beneath the indenter maintains a logarithmic singularity, which is produced by the wedge edge and is similar to the singularity of the contact pressure in the corresponding two-dimensional contact problem for a wedge rigid indenter.

A fast multipole boundary element method for half-space acoustic problems in a subsonic uniform flow

This paper presents a fast multipole boundary element method (FMBEM) for half-space acoustic problems in a subsonic uniform flow. First, the fundamental solution, i.e., Green's function for half-space acoustic wave problems in a subsonic uniform flow is derived. By employing the Burton-Miller method, a convected hybrid boundary integral formula is achieved to overcome the non-uniqueness difficulty. The developed formula does not need to consider the integral on the infinite plane and it is convenient for numerical implementation. The numerical evaluation of singular integrals in the convected boundary integral formulae is also introduced. Then, the fast multipole method (FMM) for convected BEM is derived and adopted to improve the calculation efficiency and increase the computational scale. Different from the previous FMM for full-space problems, the multipole expansion and translation of the mirror-image model are the key to the FMM for half-space problems. The impact of the infinite plane on the results can be computed by building the mirror-image model. And only the structural boundary in the real domain needs to be discretized and constructed for the tree structure. Numerical examples including a pulsating half-sphere problem, a sphere model and an aircraft model in moving flows are given to validate the accuracy and efficiency of the developed method.

Half-space albedo problem for the Anlı-Güngör scattering function

Abstract One speed, time-independent and homogeneous medium neutron transport equation is solved for second order scattering using the Anlı-Güngör scattering function which is a recently investigated scattering function. The scattering function depends on Legendre polynomials and the t parameter which is defined on the interval [−1, 1]. A half-space albedo problem is examined with the FN method and the recently developed SVD method. Albedo values are calculated with two methods and tabulated. Thus, the albedo values for the Anlı-Güngör scattering are compared with these methods. The behaviour of the scattering function is similar to İnönü’s scattering function according to calculated results.

On integral and differential formulations in nonlocal elasticity

The paper is concerned with comparative analysis of differential and integral formulations for boundary value problems in nonlocal elasticity. For the sake of simplicity, the focus is on an antiplane problem for a half-space with prescribed shear stress along the surface. In addition, 1D exponential kernel depending on the vertical coordinate is considered.First, a surface loading in the form of a travelling harmonic wave is studied. This provides a counter-example, revealing that within the framework of Eringen’s theory the solution to the differential model does not satisfy the equation of motion in nonlocal stresses underlying the related integral formulation.A more general differential setup, starting from singularly perturbed equations expressing the local stresses through the nonlocal ones, is also investigated. It is emphasised that the transformation of the original integral formulation to the differential one in question is only possible provided that two additional conditions on nonlocal stresses hold on the surface. As a result, the formulated problem subject to three boundary conditions appears to be ill-posed, in line with earlier observations for equilibrium of a nonlocal cantilever beam.Next, the asymptotic solution of the singularly perturbed problem, subject to a prescribed stress on the boundary, together with only one of the aforementioned extra conditions, is obtained at a small internal size. Such simplification may be justified when only one of the stresses demonstrates nonlocal behaviour; a similar assumption has been recently made within the so-called dilatational gradient elasticity. Three-term expansion is obtained, leading to a boundary value problem in local stresses over interior domain. The associated differential equations are identical to those proposed by Eringen, however, the derived effective boundary condition incorporates the effect of a nonlocal boundary layer which has previously been ignored. Moreover, the calculated nonlocal correction to the classical antiplane problem for an elastic half-space, coming from the boundary conditions is by order of magnitude greater than that appearing in the equations of motion. Finally, it is shown that the proposed effective condition supports an antiplane surface wave.

A Stochastic Algorithm for Fault Inverse Problems in Elastic Half Space with Proof of Convergence

A Stochastic Algorithm for Fault Inverse Problems in Elastic Half Space with Proof of Convergence

Эллиптические уравнения со сдвигами произвольных направлений в полупространстве

In this paper, we investigate the half-space Dirichlet problem for elliptic differential-difference equations with superpositions of differential operators and translation operators acting in arbitrary directions parallel to the boundary hyperplane. The summability assumption is imposed on the boundary-value function of the problem. The specified equations, substantially generalizing classical elliptic partial differential equations, arise in various models of mathematical physics with nonlocal and (or) heterogeneous properties or the process or medium: multi-layer plates and envelopes theory, theory of diffusion processes, biomathematical applications, models of nonlinear optics, etc. The theoretical interest to such equations is caused by their nonlocal nature: they connect values of the desired function (and its derivatives) at different points (instead of the same one), which makes many classical methods unapplicable. For the considered problem, we establish the solvability in the sense of generalized functions, construct Poisson-like integral representations of solutions, and prove the infinite smoothness of the solution outside the boundary hyperplane and its uniform convergence to zero (together with all its derivatives) as the timelike variable tends to infinity. We find a power estimate of the velocity of the specified extinction of the solution and each its derivative.

Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

&lt;p style='text-indent:20px;'&gt;In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ (1+t)^{-\frac{1}{2}} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ L^\infty $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; norm.&lt;/p&gt;

A NUMERICAL APPROACH TO THE CONTACT OF NOMINALLY FLAT SURFACES

Machined surfaces can be described by heights and wavelengths of the surface asperities that show a statistical variation. Considering that a regular wavy surface with a sinusoidal profile is the crudest model for a rough surface, studying the contact of regular wavy surfaces is a good approximation for the contact of nominally flat surfaces. Such contact problems exhibit periodicity that can be simulated with the aid of computational techniques derived for contact mechanics in the frequency domain. The displacement calculation, which is a necessary step in the resolution of the contact problem, is mathematically a convolution product that can be calculated in the frequency domain with increased computational efficiency. The displacement induced by a unit surface load can be expressed in the frequency domain by the frequency response functions, which are counterparts of the space domain solutions to half-space fundamental problems such as the Boussinesq problem. The displacement induced by a periodic pressure distribution can be computed by executing the convolution product between the frequency response function and pressure on a single period. It should be noted that the convolution calculation in the spectral domain implies that the contributions of all neighbouring pressure periods are accounted for. The need to treat numerically only a single period results in remarkable computational efficiency, allowing for high density meshes that can capture the essential features of any textured real surface. The displacement calculation promotes the solution of the contact problem by an iterative approach. The advanced method is benchmarked against existing analytical solutions for the 3D contact of surfaces possessing two-dimensional waviness. This essentially deterministic model, supported by a direct numerical solution that can be obtained for samples of real rough surfaces, presents itself as a worthy alternative to the existing statistical models for rough contact interaction.

An efficient ADER-DG local time stepping scheme for 3D HPC simulation of seismic waves in poroelastic media

Many applications from the fields of seismology and geoengineering require simulations of seismic waves in porous media. Biot's theory of poroelasticity describes the coupling between solid and fluid phases and introduces a stiff reactive source term (Darcy's Law) into the elastodynamic wave equations, thereby increasing computational cost of respective numerical solvers and motivating efficient methods utilising High-Performance Computing.We present a novel realisation of the discontinuous Galerkin scheme with Arbitrary High-Order DERivative time stepping (ADER-DG) that copes with stiff source terms. To integrate this source term with a reasonable time step size, we utilise an element-local space-time predictor, which needs to solve medium-sized linear systems – each with 1,000 to 10,000 unknowns – in each element update (i.e., billions of times). We present a novel block-wise back-substitution algorithm for solving these systems efficiently, thus enabling large-scale 3D simulations. In comparison to LU decomposition, we reduce the number of floating-point operations by a factor of up to 25, when using polynomials of degree 6. The block-wise back-substitution is mapped to a sequence of small matrix-matrix multiplications, for which code generators are available to generate highly optimised code.We verify the new solver thoroughly against analytical and semi-analytical reference solutions in problems of increasing complexity. We demonstrate high-order convergence of the scheme for 3D problems. We verify the correct treatment of point sources and boundary conditions, including homogeneous and heterogeneous full space problems as well as problems with traction-free boundary conditions. In addition, we compare against a finite difference solution for a newly defined 3D layer over half-space problem containing an internal material interface and free surface. We find that extremely high accuracy is required to accurately resolve the slow, diffusive P-wave at a or near a free surface, while we also demonstrate that solid particle velocities are not affected by coarser resolutions. We demonstrate that by using a clustered local time stepping scheme, time to solution is reduced by a factor of 6 to 10 compared to global time stepping. We conclude our study with a scaling and performance analysis on the SuperMUC-NG supercomputer, demonstrating our implementation's high computational efficiency and its potential for extreme-scale simulations.

A three-dimensional indirect boundary integral equation method for the scattering of seismic waves in a poroelastic layered half-space

A new indirect boundary integral equation method (IBIEM) is developed to solve the seismic wave scattering problem in a three-dimensional fluid-saturated layered half-space. Based on Biot's theory, the Green's functions of inclined circular loads in porous elastic layered half-space are firstly deduced. According to the single-layer potential theory, when establishing the scattered wave field, the uniform surface loads and fluid sources are located directly on the boundary surfaces of the scatterers, so as to avoid determining the optimal location of fictitious wave sources surface. At the same time, the radiation condition of wave in semi-infinite layered media can be accurately realized by using the dynamic Green's function of concentrated load, and the computational memory can be greatly reduced. The scattering of seismic waves by a canyon topography in saturated layered half-space is examined in detail. The numerical results indicate that with the increase of the porosity of the overlying soil, the amplitude of surface displacement can be amplified by 1∼6 times, and the amplification effect is more significant near the corner of the canyon, which can be attributed to the superposition of wave diffraction effect and resonance amplification effect of saturated soil layer.

Viscoelastic dissipation in repeated normal indentation of an Hertzian profile

Simple exact solutions are known for the indentation problem of a viscoelastic halfspace by a rigid sphere only as long as the contact area is growing. We consider instead a more general cyclic repeated indentation with a pulsating load with a period of zero load. We show that a combination of exact with empirical relaxation solutions coming from simple uniaxial cases is sufficiently accurate to estimate the energy dissipated per cycle, which we report for the standard ”3-elements” solid and periodic half-sine loading for various parameters. The theoretical predictions favourably compare with boundary element numerical simulations. We find more energy is dissipated during the first indentation cycle with respect to the subsequent ones, due to the residual indentation left in the viscoelastic half-space. In load controlled systems, the maximum dissipation is reached at an angular frequency that is close to the reciprocal of the relaxation time of the material both for the first and subsequent cycles, but this is in general not true when displacement controlled systems are considered, when dissipation is much lower for subsequent cycles.

The surface tension effect revealed via the indentation scaling index

The axisymmetric frictionless indentation problem for an isotropic elastic half-space with the uniform surface tension is considered. The first-order regular and singular asymptotic models are constructed for the limit situations of relatively small (in the membrane limit) or large (in the Hertzian limit) values of the contact radius. By using the method of collocation, a two-parameter analytical approximate solution is developed with coefficients evaluated in triple quadratures. The accuracy of the obtained analytical models is investigated in the case of a paraboloidal indenter by comparison with numerical solutions available in the literature. Based on the asymptotic behavior of the solution in the limit situations, the so-called indentation scaling index is introduced for the analysis of the depth-sensing indentation data. By making use of the novel contact parameter it is possible to extract the surface tension effect independently of determining the reduced elastic modulus of a tested semi-infinite sample. It is shown that by employing an appropriate normalization, the contact problem can be reduced to that parameterized by only one parameter (relative contact radius), which implies that irrespectively of the combination of the material parameters, the force–displacement curve drawn in the normalized variables will be universal.

New fractional model for 2 dimensional half space problem with in the theory of generalized thermoelastic diffusion

AbstractIn this manuscript, the generalized thermoelastic diffusion theory is modified by using fractional derivatives. The revised equation is employed to study a two‐dimensional problem of the axisymmetric temperature distribution during a half space with a permeating substance in touch with the bounding plane. The surface of the half space is taken into account to be traction‐free and subject to an axisymmetric temperature distribution. Laplace and Hankel transformation techniques are used. The analytical solution within the transform domain is obtained by an immediate method. By employing a numerical method supported the Fourier expansion technique, the inverse of the double transforms is often obtained. Numerical results for the temperature, displacement, stress, concentration, and chemical potential are administered and represented graphically. The figures show some comparisons to estimate the impact of fractional parameter on all research areas.