Nonhomogeneous Media Research Articles

The method of finite spheres in acoustic wave propagation through nonhomogeneous media: Inf-sup stability conditions

When the method of finite spheres is used for the solution of time-harmonic acoustic wave propagation problems in nonhomogeneous media, a mixed (or saddle-point) formulation is obtained in which the unknowns are the pressure fields and the Lagrange multiplier fields defined at the interfaces between the regions with distinct material properties. Then certain inf-sup conditions must be satisfied by the discretized spaces in order for the finite-dimensional problems to be well-posed. We discuss in this paper the analysis and use of these conditions. Since the conditions involve norms of functionals in fractional Sobolev spaces, we derive ‘stronger’ conditions that are simpler in form. These new conditions pave the way for the inf-sup testing, a tool for assessing the stability of the discretized problems.

Incompatible Graded Finite Elements for Orthotropic Nonhomogeneous Media

Orthotropic materials or composites are challenging to model due to different elastic properties in two in-plane directions, which requires sufficient mesh refinement to attain desired accuracy. In this regard, an accurate and efficient incompatible graded element is developed for modeling orthotropic functionally graded materials. Properties of Graded finite elements such as Young’s moduli (E11, E22), shear modulus (G12) and Poisson’s ratio (v12) vary. A new incompatible graded element is developed using user subroutines in Abaqus and are compared to available exact solutions. In this study, performance (QM6) graded element is compared with lower-order and higher-order compatible elements (Q4 and Q8) as well as linear triangular (T3) and quadratic triangular (T6) elements with aid of several numerical examples. Additionally, orthotropic graded plate with carbon fibers properties as well as a circular disc with orthotropic properties under internal pressure is included. Furthermore, a curved beam with radially graded properties under the bending moment is studied. Additionally, the performance of various graded elements is compared with analytical solutions using Russell error.

Homogenization of Stokes Equations in Perforated Domains: A Unified Approach

We consider the homogenization of the Stokes equations in a domain perforated with a large number of small holes which are periodically distributed. Allaire (Arch Ration Mech Anal 113(3):209–259, 1990, Arch Ration Mech Anal 113(3):261–298 1990), gave a systematic study on this problem. In this paper, we introduce a unified proof for different sizes of holes for the homogenization of the Stokes equations by employing a generalized cell problem inspired by Tartar (in: Sanchez-Palencia (ed) Nonhomogeneous media and vibration theory, 1980).

Experimental study of breathers and rogue waves generated by random waves over non-uniform bathymetry

We present experimental evidence of formation and persistence of localized waves, breathers, and solitons, occurring in a random sea state and uniformly traveling over non-uniform bathymetry. Recent studies suggest connections between breather dynamics and irregular sea states and between extreme wave formation and breathers, random sea states, or non-uniform bathymetry individually. In this paper, we investigate the joint connection between these phenomena, and we found that breathers and deep-water solitons can persist in more complex environments. Three different sets of significant heights have been generated within a Joint North Sea Wave Observation Project wave spectrum, and the wave heights were recorded with gauges in a wave tank. Statistical analysis was applied to the experimental data, including the space and time distribution of kurtosis, skewness, Benjamin–Feir index, moving Fourier spectra, and probability distribution of wave heights. Stable wave packages formed out of the random wave field and traveling over shoals, valleys, and slopes were compared with exact solutions of the nonlinear Schrödinger equation with a good match, demonstrating that these localized waves have the same structure as deep-water breathers. We identify the formation of rogue waves at moments and over regions where the kurtosis and skewness have local maxima. These results provide insights for understanding of the robustness of Peregrine and higher-order Akhmediev breathers, Kuznetsov–Ma solitons, and rogue waves, and their occurrence in realistic oceanic conditions, and may motivate analogous studies in other fields of physics to identify limitations of exact weakly nonlinear models in non-homogeneous media.

A new family of boundary‐domain integral equations for the Dirichlet problem of the diffusion equation in inhomogeneous media with H−1(Ω) source term on Lipschitz domains

The interior Dirichlet boundary value problem for the diffusion equation in nonhomogeneous media is reduced to a system of boundary‐domain integral equations (BDIEs) employing the parametrix obtained in Portillo (2019). We further extend the results obtained in Portillo (2019) for the mixed problem in a smooth domain with L2(Ω) right‐hand side to Lipschitz domains and partial differential equation (PDE) right‐hand side in the Sobolev space H−1(Ω), where neither the classical nor the canonical conormal derivatives are well‐defined. Equivalence between the system of BDIEs and the original BVP is proved along with their solvability and solution uniqueness in appropriate Sobolev spaces.

A machine learning based efficient and compact full-spectrum correlated k-distribution model

The full-spectrum correlated-k-distribution (FSCK) look-up table previously developed by Wang et al. (2018) provides an efficient means for accurate calculations of radiative properties in nonhomogeneous media, which was constructed for gas mixtures of three species, i.e., CO2, CO and H2O within a large range of thermodynamic states. However, the total size of the FSCK look-up table is about 3.2 GB. In order to obtain accurate k-values for arbitrary thermodynamic states from the tabulated values, a 6-D linear interpolation method has to be employed. In practical applications, transferring and loading such a huge database are inconvenient and time-consuming. In this study, we presented a machine learning approach to accurately fit the FSCK look-up table, in order to provide an efficient and compact FSCK model. Without interpolation on the fly, correct k-distributions for any arbitrary thermodynamic states can be obtained directly and the total size of the trained machine learning model is only about 35 MB.

Instantaneous and finite time blow-up of solutions to a reaction-diffusion equation with Hardy-type singular potential

We deal with radially symmetric solutions to the reaction-diffusion equation with Hardy-type singular potentialut=Δum+K|x|2um, posed in RN×(0,T), in dimension N≥3, where m>1 and 0<K<(N−2)2/4. We prove that, in dependence of the initial condition u0∈L∞(RN)∩C(RN), its solutions may either blow up instantaneously or blow up in finite time at the origin, thus developing a singularity at x=0, but they can be continued globally in weak sense. The instantaneous blow-up occurs for example for any data u0 such that u0(0)>0. The proofs are based on a transformation mapping solutions to our equation into solutions to a non-homogeneous porous medium equation.

Young bilateral supernova remnants evolving into a turbulent interstellar magnetic field

ABSTRACT We employ 3D magnetohydrodynamic simulations to study the morphology and synchrotron emission of young supernova remnants evolving in a turbulent interstellar magnetic field, seeking to shed new light on to the polarization structure of the emission and on the debate concerning the quasi-parallel and quasi-perpendicular acceleration mechanisms. In the simulations, we consider a non-homogeneous interstellar medium magnetic field by introducing small random perturbations in the direction and intensity of the field. In order to analyse the dependence of the radio morphology on the degree of magnetic field perturbation and the observer’s point of view, we compute synthetic maps of the polarized intensity, position-angle, polarization fraction, and the polar-reference angle. By comparing the distribution of this angle to the polarization intensity, we show that it is possible to identify what type of acceleration mechanism is taking place at the main shock front.

A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part II: Validation and discussion

In the first part of this two-paper series, a new computational approach is presented for analyzing transient heat conduction problems in anisotropic nonhomogeneous media. The approach consists of a truly meshless Fragile Points Method (FPM) being utilized for spatial discretization, and a Local Variational Iteration (LVI) scheme for time discretization. In the present article, extensive numerical results are provided as validations, followed by a discussion on the recommended computational parameters. The FPM + LVIM approach shows its capability in solving 2 D and 3 D transient heat transfer problems in complex geometries with mixed boundary conditions, including preexisting cracks. Both functionally graded materials and composite materials are considered. It is shown that, with appropriate computational parameters, the FPM + LVIM approach is not only accurate, but also efficient, and has reliable stability under relatively large time intervals.

A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part I: Theory and implementation

A new and effective computational approach is presented for analyzing transient heat conduction problems. The approach consists of a meshless Fragile Points Method (FPM) being utilized for spatial discretization, and a Local Variational Iteration (LVI) scheme for time discretization. Anisotropy and nonhomogeneity do not give rise to any difficulties in the present implementation. The meshless FPM is based on a Galerkin weak-form formulation and thus leads to symmetric matrices. Local, very simple, polynomial and discontinuous trial and test functions are employed. In the meshless FPM, Interior Penalty Numerical Fluxes are introduced to ensure the consistency of the method. The LVIM in the time domain is a combination of the Variational Iteration Method (VIM) and a collocation method in each finitely large time interval. The present methodology represents a considerable improvement to the current state of science in computational transient heat conduction in anisotropic nonhomogeneous media. In this first part of the two-paper series, we focus on the theoretical formulation and implementation of the proposed methodology. Numerical results and validation are then presented in Part II.

A Discontinuous Dual Reciprocity Method in Conjunction with a Regularized Levenberg–Marquardt Method for Source Term Recovery in Inhomogeneous Anisotropic Materials

This paper outlines a new approach to identify a source term of a [Formula: see text]D elliptic equation for anisotropic nonhomogenous media. The proposed methodology is based on the minimization of an objective function representing differences between the measured potential and those calculated by using the discontinuous dual reciprocity boundary element method, the measurements are required to render a unique solution and supposed to be pointwise in the problem domain. Since the additional data may be contaminated by measurement noises or the numerical computing errors, we adopt a regularizing Levenberg–Marquardt method to solve the nonlinear least-squares problem attained from the inverse source problem. The numerical performance of the proposed approach is studied at the end for both geometries: smooth and piecewise smooth one. The results show a very good agreement with the analytical solutions under exact and noisy data.

Determination of thermophysical properties and density volume fractions of Al2O3/Y-ZrO2 layered composite materials using transient thermography and two-stage inverse nonlinear heat conduction analysis

In the present study, transient thermography, a nondestructive imaging technique, is applied to evaluate the transient temperature response in a graded medium without the use of embedded thermocouples. A layered composite sample was fabricated from Al2O3 and Y-ZrO2 powders using powder metallurgy (PM). This sample was irradiated on one side with a direct current laser while the transient temperature was measured along its depth by a midinfrared camera. Also, a MATLAB code based on the truly meshless radial point interpolation method (t-RPIM) was developed and implemented to solve the problem of quasilinear transient heat transfer in PM solids. In the t-RPIM formulation, the Cartesian transformation method and the Crank-Nicolson scheme were used for the evaluation of domain integrals and time discretization, respectively, thereby yielding a truly mesh-free technique. In the conducted experiment, the thermophysical properties were assumed to be independent of temperature because of the small amount of temperature increase. These properties and the volume fractions of the constituent powders were determined using a combination of the t-RPIM and the damped Gauss-Newton method in an inverse analysis. Good agreement was found between the measured temperature and the reconstructed temperature profile using the identified thermal parameters and volume fractions, thus validating the accuracy and ability of the applied t-RPIM as a tool in an inverse scheme to solve the inverse transient heat conduction problem in nonhomogeneous media.

Dynamic analysis of an elastic plate on a cross-anisotropic and continuously nonhomogeneous viscoelastic half-plane under a moving load

The dynamic response of an elastic thin plate to a load moving on its surface with constant velocity is determined analytically. The plate rests on a cross-anisotropic and continuously nonhomogeneous viscoelastic half-plane soil medium. Soil nonhomogeneity is associated with elastic moduli increasing with depth. Viscoelastic effects are introduced via the hysteretic damping model. The solution of the problem is obtained with the aid of the complex Fourier series method involving the horizontal coordinate x and the time t. Thus, the governing partial differential equations of motion for the plate–soil system are reduced to an algebraic equation for the plate and a system of two ordinary differential equations with variable coefficients for the soil. This system is solved analytically by the method of Frobenius. Compatibility and dynamic equilibrium at the plate–soil interface as well as boundary conditions at the plate surface and at infinite soil depth enable one to finally obtain the response for both the plate and the soil analytically. Verification of the solution is done by means of comparisons with the known existing analytical solutions for the special cases of isotropy and homogeneity with or without the plate. Parametric studies are finally performed, and the effects of soil anisotropy, nonhomogeneity and damping on the plate and the supporting soil are assessed.

Visibility of Bioresorbable Vascular Scaffold in Intravascular Ultrasound Imaging.

Bioresorbable vascular scaffold (BVS) has recently been spotlighted for its unique characteristics of absorbing into blood vessels and eventually disappearing. Although intravascular ultrasound (IVUS) is the most common guiding tool for stent deployment, the echogenicity of BVS struts has changed as the center of stent lumen and scanning rotation is not concentric, which may cause a critical erroneous measurement in practice. This study investigated the physical conditions for dimming the stent brightness in IVUS images using a finite-difference method (FDM) to numerically solve acoustic wave propagation through nonhomogeneous medium. The dimmed brightness is caused by an angled rectangular cross section of a strut and its similar acoustic impedance with water. Imaging frequency is not a major cause. However, the angle between the acoustic beam and the BVS surface is the major cause of the dimmed brightness. As a solution, an approach using a frequency compounding method with signal polarity comparator was proposed to recover the reduced brightness without sacrificing spatial resolutions. Based on the simulation study, the signal level from BVS can be attenuated down by 17 dB when the angle between the acoustic beamline and the surface of BVS is more than 45°. With the proposed frequency compounding approach, the reduced signal can be recovered by 6 dB. In the experimental BVS IVUS imaging, strut brightness was reduced by 18 dB with an angled strut position and recovered by 5 dB with the proposed frequency compounding method. A pig coronary was imaged to demonstrate the performance of the proposed method.

Elastic Fields of a Nonhomogeneous Half‐Space Subject to an Inclined Circular Load

The paper examines the elastic fields of displacements and stresses for a nonhomogeneous elastic half‐space where the elastic parameters have a linear variation over a finite depth beyond which it is constant. The circular loading area is subjected to a uniform inclined load. The numerical method is developed by applying the fundamental solution of a layered elastic solid and integrating numerically it over the loading area. As a result, only the loading area needs to be discretized in using the proposed numerical method. Numerical examples of calculation of displacements are conducted, and excellent agreement with the existing closed‐form solutions is obtained. The results obtained are used to understand the elastic fields induced by different types of loads in a nonhomogeneous medium.

Effect of the rotation on the thermal stress wave propagation in non-homogeneous viscoelastic body

In this article, an analytical solution for the effect of the rotation on thermo-viscoelastic non-homogeneous medium with a spherical cavity subjected to periodic loading is studied. The distribution of displacements, temperature, redial stress, and hoop stress in non-homogeneous medium, in the context of generalized thermo-viscoelasticity using the GL theory, is discussed and obtained. The results are displayed graphically to illustrate the effect of the rotation. Comparisons with the previous work in the absence of rotation and viscosity are made.

Mathematical modeling of a hydraulic flume for carrying out numerical experiments on coastal waves and erosion of cohesive soil

Hydrowave flumes are applied in laboratory experiments in order to study coastal wave impact on shore facilities. The realization of such laboratory experiments is costly and time-consuming concerning its preparation. Thus, it is pressing to apply a mathematical model. The paper presents a material hydrowave flume mathematical model that significantly broadens the capability of the research on coastal wave propagation and its impact on sea floor cohesive soil. The paper considers a solitary wave climb to the shore in the areas of different configuration including sea floor irregularities and a layer of cohesive soil. The wave propagation on the surface and soil shift in the water are modelled by three-component viscous incompressible fluid where air, water and soil are considered as components of non-homogeneous medium. The solution is obtained through finite difference numerical algorithm based on splitting scheme by physical factors and prediction correction method. Results of numerical calculations are presented in the paper.

Continuity of weak solutions to nonlinear fourth-order equations with strengthened ellipticity via Wolff potentials

In the present article nonlinear fourth-order equations in the divergence form with L^1-right-hand sides and the strengthened ellipticity condition on the coefficients are analyzed. Such equations, but with sufficiently regular right-hand sides, first appeared in the works of Professor I.V. Skrypnik concerning the regularity of generalized solutions for multidimensional nonlinear elliptic equations of high order. This class of equations correctly generalizes the corresponding nonlinear second-order elliptic equations with non-standard growth conditions on the coefficients, which are models for numerous physical phenomena in non-homogeneous medium. The main result of the article is a theorem on an estimation of oscillations in a ball of solutions to the given equations via the Wolff potentials of their right-hand sides. To prove this, we use the improved Kilpelainen-Maly method and pointwise potential estimates of functions related to special subclasses of Sobolev spaces, akin to the well-known De Giorgi classes. A new point is the verification that these classes contain superpositions of solutions and Moser logarithmic functions that include the Wolf potential of the right-hand side of the equation. As a corollary, a new result is obtained on the interior continuity of solutions to the equations with right-hand sides from the Kato class, which is characterized by the uniform convergence to zero of the corresponding Wolff potentials. Some important cases of fulfilling this condition are considered: the right-hand side of the equation belongs to the Morrey space with an index exceeding a certain limiting value, then the solutions are locally Holder continuous; if the right-hand side belongs to the borderline Lorentz-Zygmund classes, then the solutions are only locally continuous, but they are not Holder continuous in the domain. In the case when the summability exponents of the right-hand sides of the equations under consideration are less than the borderline values, there are examples of unbounded discontinuous solutions. These facts are exact analogues of the corresponding results in the theory of second-order elliptic equations.

NDSHA: A new paradigm for reliable seismic hazard assessment

A New Paradigm is needed for Reliable Seismic Hazard Assessment RSHA, not only from consideration of (a) the huge human losses experienced in the many recently destructive earthquakes worldwide; but also from (b) theoretical considerations of seismic wave generation and propagation phenomena through often non-homogeneous media within the earth's crust, particularly when large and more complex fault ruptures occur. The Neo-Deterministic Seismic Hazard Assessment (NDSHA) method, proposed some twenty years ago, is found to reliably and realistically simulate the wide suite of earthquake ground motions that may impact civil populations as well as their heritage buildings. The scenario-based NDSHA modeling technique is developed from comprehensive physical knowledge of: (i) the seismic source process; (ii) the propagation of earthquake waves; and (iii) their combined interactions with site effects. Thus, NDSHA effectively accounts for the tensor nature of earthquake ground motions: (a) formally described as the tensor product of the earthquake source functions and the Green's functions of the transmitting (pathway) medium; and (b) more informally described as mathematical arrays of numbers or functions (indices) “that transform according to certain rules under a change of coordinates.” Importantly, NDSHA therefore uses all available information about the spacial distribution of large magnitude earthquake phenomena, including: (a) Maximum Credible Earthquake (MCE) – which is based on seismic history and seismotectonics; and (b) geological and geophysical data. Thus it does not rely on scalar empirical ground motion attenuation models (GMPEs), as these are often both: (a) weakly constrained by available observations; and (b) fundamentally unable to account for the tensor nature of earthquake ground motions.

A boundary element and radial basis function approximation method for a second order elliptic partial differential equation with general variable coefficients

A numerical method based on boundary integral equation and radial basis function approximation is presented for solving boundary value problems governed by a second‐order elliptic partial differential equation with variable coefficients. The equation arises in the analysis of steady‐state anisotropic heat or mass diffusion in nonhomogeneous media with properties that vary according to general smoothly varying functions of space. The method requires only the boundary of the solution domain to be discretized into elements. To check the validity and accuracy of the numerical solution, some specific problems with known solutions are solved.