Removability conditions for anisotropic parabolic equations in a computational validation

In this paper, the extension of the generalized finite difference method to the solution of anisotropic elliptical and parabolic partial differential equations is given in the case of considering two space dimensions. The explicit finite difference formulae and the criterion of stability for the anisotropic parabolic equations are given. The stability condition has been expressed in function of the coefficients of the star equation for irregular clouds of nodes. Various anisotropic cases of simulation of cardiac tissue, including elliptic and parabolic equations, have been solved, and the results show the accuracy of the method.

Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

The aim of this paper is to establish sufficient conditions of the finite time blow-up in solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equations with variable nonlinearity \( u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} } \) . Two different cases are studied. In the first case ai ≡ ai(x), pi ≡ 2, σi ≡ σi(x, t), and bi(x, t) ≥ 0. We show that in this case every solution corresponding to a “large” initial function blows up in finite time if there exists at least one j for which min σj(x, t) > 2 and either bj > 0, or bj(x, t) ≥ 0 and Σπbj−ρ(t)(x, t) dx 0 depending on σj. In the case of the quasilinear equation with the exponents pi and σi depending only on x, we show that the solutions may blow up if min σi ≥ max pi, bi ≥ 0, and there exists at least one j for which min σj > max pj and bj > 0. We extend these results to a semilinear equation with nonlocal forcing terms and quasilinear equations which combine the absorption (bi ≤ 0) and reaction terms.

Consider an anisotropic parabolic equation with the variable exponents vt=∑i=1n(bi(x,t)vxipi(x)-2vxi)xi+f(v,x,t), where bi(x,t)∈C1(QT¯), pi(x)∈C1(Ω¯), pi(x)>1, bi(x,t)≥0, f(v,x,t)≥0. If {bi(x,t)} is degenerate on Γ2⊂∂Ω, then the second boundary value condition is imposed on the remaining part ∂Ω∖Γ2. The uniqueness of weak solution can be proved without the boundary value condition on Γ2.

Consider the anisotropic parabolic equation with the variable exponent ut=∑i=1N(ai(x)|uxi|pi(x)−2uxi)xi,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {u_{t}}=\\sum_{i=1}^{N} \\bigl(a_{i}(x)|u_{x_{i}}|^{p_{i}(x)-2}u_{x_{i}} \\bigr)_{x _{i}}, $$\\end{document} with a_{i}(x), p_{i}(x)in C^{1}(overline{Omega}), p_{i}(x)>1, a_{i}(x)geq0. If some of {a_{i}(x)} are degenerate on the boundary, a partial boundary value condition is imposed, the stability of weak solutions can be proved based on the partial boundary value condition.

We consider the Cauchy problem for a certain class of anisotropic parabolic second-order equations with double non-power nonlinearities. The equation contains an "inhomogeneity" in the form of a non-divergent term depending on the sought function and spatial variables. Non-linearities are characterized by N -functions, for which 2 -condition is not imposed. The uniqueness of renormalized solutions in Sobolev-Orlich spases is proved by the S.N.Kruzhkov method of doubling the variables.

The pseudoviscoacoustic anisotropic wave equation is widely used in the oil and gas industry for modeling wavefields in attenuating anisotropic media. Compared to the full viscoelastic anisotropic wave equation, it can greatly reduce the computational cost of wavefield modeling while maintaining the visco-qP-wave kinematics very well. However, even if we place the source in a thin isotropic layer, there will be some unwanted S-wave artifacts in the qP wavefield simulated by the pseudoviscoacoustic anisotropic wave equation due to the stepped approximation of inclined layer interfaces. Furthermore, the wavefield simulated by the pseudoviscoacoustic anisotropic wave equation may suffer from numerical instabilities when the anisotropy parameter epsilon is less than delta. To overcome these problems, we derive a pure-viscoacoustic tilted transversely isotropic (TTI) wave equation in media with anisotropy in velocity and attenuation based on the exact complex-valued phase velocity formula. The pure-viscoacoustic TTI wave equation has decoupled amplitude dissipation and phase dispersion terms, which is suitable for further reverse time migration with Q compensation. For numerical simulations, we adopt the second-order Taylor series expansion to replace the mixed-domain spatially variable fractional Laplacian operator, which guarantees the decoupling of the wavenumber from the space-related fractional order. Then, we use an efficient and stable hybrid finite-difference and pseudospectral method (HFDPSM) to solve the pure-viscoacoustic TTI wave equation. Numerical tests indicate that the simulation results of the newly derived pure-viscoacoustic TTI wave equation are stable, free from S-wave artifacts, and accurate. We further demonstrate that HFDPSM outperforms the pseudospectral method in terms of numerical simulation stability and computing efficiency.

Anisotropic mesh adaptation for finite element solution of Anisotropic Porous Medium Equation

In this paper, our main goal is to present a non-classical MPFA method coupled with a high order finite volume method for the simulation of oil water displacements in heterogeneous and anisotropic petroleum reservoirs using general polygonal meshes. The governing equations are solved using the IMPES (IMplicit Pressure and Explicit Saturation) strategy, where the elliptic pressure equation is discretized by a linear Multipoint Flux Approximation method using Harmonic points (MPFA-H) capable to handle strongly heterogeneous and anisotropic media. Besides, to approximate the advective term that characterizes the hyperbolic saturation equation, we use the Multidimensional Optimal Order Detection (MOOD) method. This technique is based in an “a posteriori” limitation procedure, i.e., the limitation procedure is done after the calculation of “candidate solutions” only where necessary to guarantee certain physical properties such as monotonicity. To show the potential of our finite volume formulation, we solve some benchmark problems found in literature.

The accuracy and computational efficiency of full waveform forward modelling in poroelastic media are crucial for microseismic monitoring. It enables intuitive, precise and efficient simulation of subsurface responses, thereby improving the reliability of moment tensor inversion and seismic source mechanism interpretation. Additionally, it reflects the role of fluid effects in waveform evolution. In this paper, based on the Biot mechanism, we derived the first‐order velocity–stress equation of poroelastic media and discretized the model using a rotated staggered grid. The rotated staggered grid can well adapt to anisotropic media with high contrast parameters. We provide the finite difference formula based on graphic process unit–acceleration and moment tensor and also provide the graphic process unit workflow for forward modelling of anisotropic poroelastic media. First, two models with different grid sizes were run based on single graphic process unit, 1‐thread Central Processing Unit (CPU) and 16‐thread CPU. The results show that the speedup factors are approximately 14.3 and 3.7, respectively, compared with the running time of 1‐thread CPU and 16‐thread CPU. Then, we compare and evaluate the response of three basic source mechanisms (isotropic expansion, double couple and compensated linear vector dipole) in the model. The comparison of analytical and numerical results verifies the effectiveness of the method. Furthermore, wavefield snapshots of two typical anisotropic media (vertical transversely isotropic and horizontal transversely isotropic) are analysed to correspond to different moment tensor sources. The results showed that the source mechanism does not change the isotropic and anisotropic plane and the wave travel time, but it does change the polarization amplitude of the velocity component. The attenuation of slow qP‐wave increases along with the increase of the value of viscosity. The effect of permeability on wavefield appears with the opposite effect of viscosity. Finally, the seismic waveform differences between multi‐layer elastic media and poroelastic media are compared and analysed. The results showed that the seismic wavefield and waveform of poroelastic media are more complex, the propagation speed of seismic waves is faster, but the attenuation of seismic waves is stronger.

Energy harvesting from ambient environment has received increasing attention over the last decade due to the need for minimizing the dependence on conventional batteries in wireless applications. Among the methods of vibration-to-electricity conversion, piezoelectric transduction has been investigated by numerous research groups due to the ease of application and high power density offered by piezoelectric materials. Electromechanical modeling efforts of piezoelectric energy harvesters have been mostly focused on deterministic forms of excitation input, as in the typical case of harmonic excitation. In most practical applications, however, ambient vibrational energy is often stochastic with broad frequency content. This paper presents analytical and numerical modeling, simulations, and experimental validations of piezoelectric energy harvesting from broadband random vibrations. The models employed herein are based on distributed-parameter electroelastic solution to ensure that the effects of higher vibration modes are included. The goal is to predict the expected value of the power output in terms of the given power spectral density (PSD) or time history of the random vibration input. The analytical estimations are based on the PSD of broadband random base excitation and distributed-parameter frequency response functions (FRFs) of the coupled voltage and vibration response. The numerical simulations use the Fourier series representation of base acceleration history in an ordinary differential equation solver that employs first-order electroelastic equations. The simulations are compared against the experiments for a brass-reinforced PZT-5H bimorph under different random excitation levels. The analytical and numerical simulations exhibit very good agreement with the experimental measurements. Soft and hard ceramic and single crystal bimorphs (made of PZT-5H, PZT-8, PMN-PZT, and PMN-PZT-Mn) are compared for broadband random excitation through a theoretical case study.

Deformation behavior of fluid-filled porous elastomers: Analytical estimates and validation

The maximal regularity properties of a fractional anisotropic elliptic equation are investigated. The equation involves an abstract linear operator A in Fourier type Banach space E and convolution terms. We find the sufficient conditions that guarantee the separability of this problems in E-valued weighted Lp spaces. We prove that an anisotropic elliptic operator generated by this problem is sectorial and generates an analytic semigroup. In application, the maximal regularity properties of the Cauchy problem for degenerate abstract anisotropic parabolic equation and the boundary value problem for anisotropic elliptic convolution equation are obtained in mixed Lp norms.

Conservation laws and non-invariant solutions of anisotropic wave equations with a source

Seismic wave propagation shows anisotropic characteristics in many sedimentary rocks. Modern seismic exploration in mountainous areas makes it important to calculate P wave travel times in anisotropic media with irregular surfaces. The challenges in this context are mainly from two aspects. First is how to tackle the irregular surface in a Cartesian coordinate system, and the other lies in solving the anisotropic eikonal equation. Since for anisotropic media the ray (group) velocity direction is not the same as the direction of the travel-time gradient, the travel-time gradient no longer serves as an indicator of the group velocity direction in extrapolating the travel-time field. Recently, a topography-dependent eikonal equation formulated in a curvilinear coordinate system has been established, which is effective for calculating first-arrival travel times in an isotropic model with an irregular surface. Here, we extend the above equation from isotropy to transverse isotropy (TI) by formulating a topography-dependent eikonal equation in TI media in the curvilinear coordinate system, and then use a fast sweeping scheme to solve the topography-dependent anisotropic eikonal equation in the curvilinear coordinate system. Numerical experiments demonstrate the feasibility and accuracy of the scheme in calculating P wave travel times in TI models with an irregular surface.