Anisotropic Equation Research Articles

Simulations of the interaction between fiber bundles/yarns and the energy absorption of the fabric under hypervelocity impact

In this work, the interaction between fiber bundles/yarns and the energy absorption of the multiple fabrics under hypervelocity impact (HVI) are studied by numerical simulation and experiments. In numerical model, an anisotropic elastic-plastic constitutive equation coupled with the Mie-Grüneisen equation of state (EOS) is adopted to describe deviatoric deformation and strongly nonlinear volumetric compression induced by HVI. Considering the internal interactions within fabrics that possess a hierarchical structure and the evolution of impact-induced debris clouds, a meso‑scale model for woven fabrics is developed using the finite element-smoothed particle hydrodynamics (FEM-SPH) adaptive method. The meso‑scale fabric model is validated through HVI experiments, and the simulation results demonstrate excellent agreement with the experimental data. Notably the model reveals an approximately linear increase relationship between the absorbed energy by the fabric and the impact energy of the projectile. Moreover, the meso‑scale fabric model exhibits enhanced predictive capability in modeling the movement and dispersion of debris clouds. Furthermore, the results of the sensitivity analysis of yarn partition indicate that the model is capable of capturing the interactions between fiber bundles/yarns within the fabric during HVI. With increasing impact velocity, the impact time decreases, which reduces the interaction between fabric components and subsequently leading to a decrease in the generated frictional energy.

The effect of microstructural barriers on transient crack growth in shape memory alloys

There are several issues to be solved in the fracture mechanics of shape memory alloys, one of them being the resistance to crack growth and therefore to fracture. This paper discusses the crack growth in a single crystal CoNiAl shape memory alloy under cyclic loading and the effect of micro-structural barriers. To observe the crack growth in detail, tests are conducted on edge-notched specimens. The displacement field is obtained using digital image correlation (DIC), and the fracture parameters are calculated by fitting anisotropic crack tip displacement equations to DIC data. Similar crack growth behaviors are observed in both superelastic and shape memory specimens, with a comparatively higher crack growth rate in the superelastic case: first a crack initiates at the notch and grows, then new cracks are observed to form near the tip of the main crack, or on the notch when the growth slows down. Then, further cyclic loading leads to the growth of the main crack and the new crack simultaneously with the two cracks merging at the end. Test specimens are examined post-failure with optical microscopy to better understand this complicated behavior. Results showed the presence of a non-transforming secondary (γ) phase around the regions where the propagating cracks slowed down, deviated, and/or stopped, improving the resistance of the shape memory alloy specimen to fracture.

On a class of weighted anisotropic p-Laplace equation with singular nonlinearity

We consider a class of singular weighted anisotropic p-Laplace equations. We provide sufficient condition on the weight function that may vanish or blow up near the origin to ensure the existence of at least one weak solution in the purely singular case and at least two different weak solutions in the purturbed singular case.

Effects of interior angle and inclination angle of slot defect on electromagnetic-stress behaviours of superconducting swing system

Currently, copper-oxygen high-temperature superconducting materials have strongly anisotropic electromagnetic properties, which are difficult to describe in the equations. And in the rotating machine containing the permanent magnet (PM) rotor and bulk high-temperature superconductor (HTS) stator, when the HTSs have defects, their electromagnetic-stress behaviours may affect the mechanical stability of the equipment and even cause it to not work properly. In this paper, we proposed an anisotropic electromagnetic equation based on the H-formulation and established a three-dimensional coupled model with the magnetic, thermal and stress fields to study the electromagnetic-stress behaviours of a HTS with a slot defect during the swing of a PM, discussing the effects of the interior angle and inclination angle of the defect on the behaviours. The results show that the interior and inclination angles of the defect have a large influence on the electromagnetic-stress behaviours. For the electromagnetic characteristics, the main influence is on the rotational losses. Especially the losses at an inclination angle of 60° are 16.5 times those without damage. This is related to the ‘thin wall’ structure near the upper surface of the HTS. The stress concentration point appears on the defect boundary. The novelty of this paper is the proposal of the anisotropic electromagnetic equations based on the H-formulation and the study of the electromagnetic-stress behaviours of a superconducting swing system containing a PM and HTS with a slot defect from a three-dimensional perspective. The research results of this paper can be the references for the design and structural protection of superconducting rotating machines.

On Coupled Two-Level Variational Problem in Sobolev-Orlicz Space

We study a coupled two-level variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its consistence and solvability issues. At the first level, we deal with the so-called temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection-diffusion equation with two types of control functions — distributed $L^2$-control and $BV$-bounded control in coefficients. At the second level, we have a constrained minimization problem with the nonstandard growth energy functional that lives in a variable Sobolev-Orlicz space. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with non-standard growth in the objective functional, is unknown a priori and it depends on the solution of the first-level optimal control problem.

A structure preserving hybrid finite volume scheme for semiconductor models with magnetic field on general meshes

We are interested in the discretisation of a drift–diffusion system in the framework of hybrid finite volume (HFV) methods on general polygonal/polyhedral meshes. The system under study is composed of two anisotropic and nonlinear convection–diffusion equations with nonsymmetric tensors, coupled with a Poisson equation and describes in particular semiconductor devices immersed in a magnetic field. We introduce a new scheme based on an entropy-dissipation relation and prove that the scheme admits solutions with values in admissible sets – especially, the computed densities remain positive. Moreover, we show that the discrete solutions to the scheme converge exponentially fast in time towards the associated discrete thermal equilibrium. Several numerical tests confirm our theoretical results. Up to our knowledge, this scheme is the first one able to discretise anisotropic drift–diffusion systems while preserving the bounds on the densities.

On the theory of entropy suband supersolutions of nonlinear degenerate parabolic equations

We consider a second-order nonlinear degenerate anisotropic parabolic equation in the case when the flux vector is only continuous and the nonnegative diffusion matrix is bounded and measurable. The concepts of entropy sub- and supersolution of the Cauchy problem are introduced, so that the entropy solution of this problem, understood in the sense of Chen-Perthame, is both an entropy sub- and supersolution. It is established that the maximum of entropy subsolutions of the Cauchy problem is also an entropy subsolution of this problem. This result is used to prove the existence of the largest entropy subsolution (and the smallest entropy supersolution). It is also shown that the largest entropy subsolution and the smallest entropy supersolution are also entropy solutions.

Entire weak solutions for an anisotropic equation in the Heisenberg group

Here, we consider an anisotropic equation \[ − Δ H n , p → u + a ( q ) | u | p – 2 u = λ w ( q ) | u | m − 2 u − h ( q ) | u | l − 2 u , -\Delta _{{\mathbb {H}^n},\overrightarrow {p}}u +a(q)|u|^{p^–2}u=\lambda w(q)|u|^{m-2}u-h(q)|u|^{l-2}u, \] in the Heisenberg group H n {\mathbb {H}^n} , where the operator Δ H n , p → \Delta _{{\mathbb {H}^n},\overrightarrow {p}} is the horizontal anisotropic p p -Laplacian on the Heisenberg group and is defined in the sequel. By the variational methods, we prove the existence of the entire weak solutions.

Gas kinetic schemes for solving the magnetohydrodynamic equations with pressure anisotropy

In many astrophysical plasmas, the Coulomb collision is insufficient to maintain an isotropic temperature, and the system is driven to the anisotropic regime. In this case, magnetohydrodynamic (MHD) models with anisotropic pressure are needed to describe such a plasma system. To solve the anisotropic MHD equation numerically, we develop a robust Gas-Kinetic flux scheme for non-linear MHD flows. Using anisotropic velocity distribution functions, the numerical flux functions are derived for updating the macroscopic plasma variables. The schemes are suitable for finite-volume solvers which utilize a conservative form of the mass, momentum and total energy equations, and can be easily applied to multi-fluid problems and extended to more generalized double polytropic plasma systems. Test results show that the numerical scheme is very robust and performs well for both linear wave and non-linear MHD problems.

Existence and multiplicity of solutions for a class of critical anisotropic elliptic equations of Schrödinger–Kirchhoff‐type

In this article, we obtain the existence and infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters, via combining the variational method, and the concentration‐compactness principle for anisotropic variable exponent under suitable assumptions on the nonlinearities.

Relativistic modeling of stellar objects in a Schwarzschild’s coordinates with embedded class one spacetime

In this study, we investigate a new relativistic anisotropic Einstein field equations solution for compact stars under embedding class 1 conditions. In order to do this, we apply the Karmarkar condition and the embedding class one technique. By taking Buchdahl type metric potential g_{rr} into consideration, the precise analytical solution has been investigated. We have studied physical characteristics of various compact star using this analytical solution. Central singularities are absent from the solution. We have explored thermodynamic observables inside the stellar models, such as radial and tangential pressures, matter density, anisotropic factor, energy conditions, TOV, red-shift, and the speed of sound, etc., after establishing this space-time geometry for the stellar models. From the graphical representation of various physical characteristics, it is demonstrated that our model meets all the specification for ultra-high density compact bodies.

Decay for the 3D anisotropic MHD-Boussinesq equations near the hydrostatic equilibrium

Decay for the 3D anisotropic MHD-Boussinesq equations near the hydrostatic equilibrium

The stationary AKPZ equation: Logarithmic superdiffusivity

AbstractWe study the two‐dimensional Anisotropic KPZ equation (AKPZ) formally given by where ξ is a space‐time white noise and λ is a strictly positive constant. While the classical two‐dimensional KPZ equation, whose nonlinearity is , can be linearised via the Cole‐Hopf transformation, this is not the case for AKPZ. We prove that the stationary solution to AKPZ (whose invariant measure is the Gaussian Free Field (GFF)) is superdiffusive: its diffusion coefficient diverges for large times as up to corrections, in a Tauberian sense. Morally, this says that the correlation length grows with time like . Moreover, we show that if the process is rescaled diffusively (), then it evolves non‐trivially already on time‐scales of order approximately . Both claims hold as soon as the coefficient λ of the nonlinearity is non‐zero. These results are in contrast with the belief, common in the mathematics community, that the AKPZ equation is diffusive at large scales and, under simple diffusive scaling, converges to the two‐dimensional Stochastic Heat Equation (2dSHE) with additive noise (i.e., the case ).

Efficient pure qP-wave simulation and reverse time migration imaging for vertical transverse isotropic (VTI) media

ABSTRACT Anisotropic pseudo-acoustic forward modeling and migration imaging are critical for high-precision seismic exploration. However, the wavefields simulated by the traditional coupled anisotropic acoustic wave equation have the problems of shear wave noise and numerical simulation instability for epsilon that is less than delta. Furthermore, although the pure anisotropic acoustic wave equation expressed by the differential operators can solve the aforementioned noise interference and numerical simulation instability issues, its numerical simulation calculation is large, particularly in 3D industrial applications, because it has to be solved by the spectral-based method. In this paper, a pure anisotropic acoustic wave equation in a vertical transverse isotropic (VTI) medium that can be numerically computed using the efficient finite difference method is derived. This equation not only eliminates noise interference and numerical simulation instabilities, but also allows for an efficient wavefield simulation. We also implement reverse time migration (RTM) using the proposed VTI pure anisotropic acoustic wave equation. Two synthetic tests and one field data test are performed to evaluate the accuracy and robustness of the developed VTI RTM. The imaging results show that the proposed VTI RTM can correct the anisotropy effect on seismic wave propagation and improve migration imaging precision.

Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation As a Port-Hamiltonian System

The anisotropic and heterogeneous N-dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. The recent structure-preserving Partitioned Finite Element Method is applied, leading directly to a finite-dimensional port-Hamiltonian system, and its numerical analysis is done in a general framework, under usual assumptions for finite element. Compatibility conditions are then exhibited to reach the best trade off between the convergence rate and the number of degrees of freedom for both the state error and the Hamiltonian error. Numerical simulations in 2D are performed to illustrate the optimality of the main theorems among several choices of classical finite element families.

Global logarithmic stability of a Cauchy problem for anisotropic wave equations

Global logarithmic stability of a Cauchy problem for anisotropic wave equations

Tight sandstone gas reservoir characterization via amplitude‐versus‐offset attributes

AbstractThe low porosity and low permeability of tight sandstone pose considerable challenges on characterizing the reservoir fluid. Moreover, the geophysical characteristics of the tight sandstone are usually similar to the surrounding rocks, leading to small impedance contrast, compounding the difficulty of reservoir parameters inversion. Amplitude‐versus‐offset technology is an important tool to invert the reservoir parameters from seismic data. The existing amplitude‐versus‐offset inversion methods are challenged to predict the tight sandstone reservoir parameters for the reasons above. In this paper, we propose a new workflow based on amplitude‐versus‐offset attributes to characterize the tight sandstone gas reservoirs in the Xihu Depression, East China Sea. The proposed workflow utilizes the Backus average method and Gassmann's equation to construct the rock physics model. Based on the analysis of the anisotropic model, we introduce Rüger's anisotropic equation for amplitude‐versus‐offset forward modelling. We compare the amplitude‐versus‐offset gathers with the real data and adjust the reservoir parameters. Through the sensitivity analysis, we select the gas saturation and porosity as the inversion parameters. The relationship between the reservoir parameters and amplitude‐versus‐offset attributes is established by fitting the fifth‐order polynomials. We use Shuey's equation and the least squares inversion method to calculate the amplitude‐versus‐offset attributes (intercept and gradient). For real data, we introduce the Toeplitz‐sparse matrix factorization method to calculate the reflectivity of the amplitude‐versus‐offset gathers. Finally, we proposed a multi‐well joint inversion approach to invert the gas saturation and porosity. The application results demonstrate the effectiveness of the proposed workflow for the characterization of the gas reservoir in tight sandstone for the describedcase.

Anisotropic fast diffusion equations

We prove the existence of self-similar fundamental solutions (SSF) of the anisotropic porous medium equation in the suitable fast diffusion range. Each of such SSF solutions is uniquely determined by its mass. We also obtain the asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. The combination of self-similarity and anisotropy is essential in our analysis and creates serious mathematical difficulties that are addressed by means of novel methods.

Spatially extended dislocations produced by the dispersive Swift-Hohenberg equation.

Motivated by previous results showing that the addition of a linear dispersive term to the two-dimensional Kuramoto-Sivashinsky equationhas a dramatic effect on the pattern formation, we study the Swift-Hohenberg equationwith an added linear dispersive term, the dispersive Swift-Hohenberg equation(DSHE). The DSHE produces stripe patterns with spatially extended defects that we call seams. A seam is defined to be a dislocation that is smeared out along a line segment that is obliquely oriented relative to an axis of reflectional symmetry. In contrast to the dispersive Kuramoto-Sivashinsky equation, the DSHE has a narrow band of unstable wavelengths close to an instability threshold. This allows for analytical progress to be made. We show that the amplitude equationfor the DSHE close to threshold is a special case of the anisotropic complex Ginzburg-Landau equation(ACGLE) and that seams in the DSHE correspond to spiral waves in the ACGLE. Seam defects and the corresponding spiral waves tend to organize themselves into chains, and we obtain formulas for the velocity of the spiral wave cores and for the spacing between them. In the limit of strong dispersion, a perturbative analysis yields a relationship between the amplitude and wavelength of a stripe pattern and its propagation velocity. Numerical integrations of the ACGLE and the DSHE confirm these analytical results.

Determination of fiber orientation model parameters for injection molding simulations via automated metamodel optimization

Injection molded short fiber reinforced components reveal a sound light weight potential with moderate costs and thus are widely used in many demanding engineering applications. The accurate determination of the fiber orientation (FO) is essential for predicting the overall mechanical behavior of discontinuous (short or long, with varying aspect ratio) fiber reinforced composites. The simulation of the FO requires a proper modeling of the hydrodynamics, the closure transformation of the FO tensor and optionally the application of specific correction functions. The determination of parameters for the fiber orientation models commonly used in injection molding simulations is a challenging task because they cannot be determined directly in experiments. Hence, a novel way is shown in our paper to derive these parameters faster, more efficiently and accurately by the usage of an automated metamodel optimization. For this, injection molding simulations were performed iteratively by an optimization program until a minimal deviation error of the simulated parameters was reached. The optimization was performed based on proper computed tomography FO data of selected regions of interest. The new approach was tested for a rotationally symmetric Venturi tube geometry made from short glass fiber reinforced polyamide (PA-GF). The fiber orientation distribution models chosen were the iARD-RPR equation with 3 parameters and the novel anisotropic IISO equation with 5 parameters. It was shown that the optimization method is feasible for the calibration of fiber orientation models. Furthermore, the IISO equation with its 2 additional parameters allowed a more accurate prediction of the fiber orientation distribution, especially of the core layer of the injection molded part.