Exact Equations Research Articles

A nonlinear inversion method for Young’s modulus and shear modulus based on the exact Zoeppritz equations

IntroductionYoung’s modulus and shear modulus are essential mechanical parameters for evaluating subsurface rocks, playing a pivotal role in the exploration and development of unconventional resources. Young’s modulus indicates the brittleness of the reservoir, while shear modulus determines the ease of fracturing rock layers.MethodsTraditional methods estimate these moduli through indirect calculations and approximate expressions, which are prone to cumulative errors and rely on multiple assumptions, reducing inversion accuracy. This paper presents a direct inversion method for acquiring of Young’s modulus and shear modulus using the exact Zoeppritz equations, integrated within a Bayesian framework for pre-stack inversion. The quantum particle swarm optimization (QPSO) algorithm is introduced to achieve a nonlinear solution to the objective function.ResultsTests on synthetic and actual field data demonstrate the feasibility and effectiveness of the proposed method, yielding more accurate inversion results compared to traditional methods.DiscussionThese findings provide valuable insights for predicting reservoir brittleness and characterizing reservoirs in unconventional shale gas exploration and development.

Theoretical analysis of water-droplet condensation and evaporation in prescribed environment with radiation

Abstract A quasi-explicit algebraic solution of the problem of quasi-steady water droplet heat- and mass-transfer with condensation or evaporation in a prescribed environment including radiation has been obtained. The solution is based on the mass-fraction form of Fick’s law of species diffusive mass transport, which allows accounting for variation in gas density through the diffusion layer surrounding the droplet. A new term, the average vapor supersaturation through the diffusion layer, has been included that allows the model to be extended to environmental conditions significantly outside those for which it is theoretically derived (dilute or near-saturation), even to conditions of relatively high temperature, low relative humidity, and low pressure. The latter two have atmospheric significance because evaporative cooling can induce significant decrease in droplet temperature and possibly induce freezing. Basic modeling assumptions are confirmed by comparison with time-dependent temperature and evaporation-rate measurements of a laboratory laser-heated droplet. Accuracy of the new model is assessed by comparison with exact equations. For temperatures between −30 and 0°C and negligible radiation, relative-error magnitudes in droplet temperature and evaporation rate are less than 5%, even for relative humidity down to 10% and pressure down to 60 kPa. This represents an improvement in accuracy over the conventional model, in which property variations of density and supersaturation through the diffusion layer are not taken into account. Differences in predicted evaporation rates between the mass-fraction and the more common partial-density Fick’s laws, which can be significant (>10% above 70% relative humidity) in coupled, implicit equations, are shown to be made negligibly small (<0.5%) by linearizing about the saturation state.

Quantum field theory of electrons and nuclei

Abstract We develop a non-relativistic quantum field theory of electrons and nuclei with Coulomb interactions. We derive the exact equations of motion and write these equations in the form of Hedin's equations for all species of identical particles involved. Theory derived allows the computation of exact observables and provides a rigorous starting point to derive approximations in a systematic way.

Approximate Solutions of the Boussinesq Equation for Horizontal Unconfined Aquifers During Pure Drainage Phase

In this work, conceptual approximations of the Boussinesq equation were introduced and analyzed, resulting into a very accurate and well-applicable model for horizontal unconfined aquifers during the pure drainage phase, without any recharge and zero-inflow conditions. The model was constructed by employing a variety of methods that included wave solution, variable separation, and series expansion, and its analysis and performance against the Boussinesq equation, at early and later times, providing fruitful insights enlightening the main mechanisms and physical characteristics of the drainage phase. The modeled non-linear forms were finally linearized, concluding with explicit analytical expressions that accurately incorporated most of the basic characteristics regarding the evolution of the water table and the outflow from the exact Boussinesq equation under different initial conditions. The endeavors of this work can be utilized for theoretical and modeling purposes related to this problem.

Seismic scattering inversion for multiple parameters of overburden-stressed isotropic media

Subsurface reservoirs are compressed by overburden stress resulting from the gravity of overlying masses, and the resulting stress changes significantly affect the seismic reflection responses generated at the reservoir interfaces. Several exact reflection coefficient equations have been well established to delineate the role that in situ stress plays in altering the energy transition and amplitude of seismic reflection responses. These exact equations, however, cannot be effectively used in practice due to their intricate formulations and the difficult geophysical estimation for the third-order elastic constants (3oECs) embedded in reflection coefficients. Based on the theories of nonlinear elasticity and elastic wave inverse scattering, we derive an approximate seismic reflection coefficient equation for overburden-stressed isotropic media in terms of the P-wave modulus, shear modulus, density, and a defined stress-related parameter (SRP). The SRP is a combined quantity of elastic moduli, 3oECs, and overburden stress, which can be naturally treated as a dimensionless stress-induced anisotropy parameter. Its inclusion effectively eliminates the need for 3oECs information when using our equation to estimate the desired reservoir properties from seismic observations. By comparing our equation to the exact one, we confirm its validity within the moderate-stress range. Then, we introduce a Bayesian inversion approach incorporating the new reflection coefficient equation to estimate four model parameters. In our approach, the Cauchy and Gaussian distribution functions are used for the a priori probability and the likelihood distributions, respectively. The synthetic tests from two well-log data sets and a field example demonstrate that four parameters can be reasonably inverted using our approach with rather smooth initial models, which illustrates the feasibility of our inversion approach.

Efficient modeling of atmospheric infrasound propagation with the Semi-Analytical-Finite-Element (SAFE) method

Downward refraction in the lower atmosphere, attributed to winds and temperature inversion, can enable infrasound to efficiently propagate over long distances inside acoustic waveguides. In this study, we use the semi-analytic finite element (SAFE) method to predict this effect in stratified, inhomogeneous, moving air for range-independent noise propagation over reflective half-planes. We present solutions for different temperature and velocity profiles and compare them to direct numerical solutions of the two-dimensional linearized Euler equations. The SAFE method separates the exact two-dimensional wave equation for a stratified, inhomogeneous, moving medium by expressing the pressure field as a sum of vertical eigenmodes propagating in the range direction. This simplifies the acoustic problem into a one-dimensional eigenvalue problem. As this problem is addressed with the finite-element method, the method can accommodate arbitrary wind and temperature profiles. For large, range-independent domains, the proposed procedure proves to be much more efficient than the tested direct numerical computations. Additionally, it converges against the exact solution of the acoustic problem if enough modes are included in the calculation. Therefore, the method offers a viable procedure for benchmarking other pressure field prediction techniques.

Analytical, semi-analytical and numerical solutions for global stability analysis in buildings: Double-Beam Systems Timoshenko

In this article, two analytical solutions, one semi-analytical solution, and one numerical solution are proposed to calculate the global critical buckling load of buildings employing a continuum model parallel coupling two Timoshenko beams. The parallel coupling of two Timoshenko beams represents the four types of building behavior: global bending, global shear, local bending, and local shear, the latter of which has been widely ignored in the literature. The model is associated with one translational kinematic field and two rotational fields. The equilibrium equations, constitutive laws, and boundary conditions are derived using a variational approach by applying Hamilton’s principle to the relevant Lagrangian function. Specifically, to solve the classical problem of buildings with uniform properties along their height, the first analytical solution proposes exact closed-form equations for a vertical load applied at the top and a closed-form analytical solution based on the decomposition of two subsystems for a polynomial vertical load uniformly applied along the height of the beam. Meanwhile, the semi-analytical solution exactly solves the differential equation associated with a polynomial vertical load profile and is based on coefficient iteration, converging with only two or three iterations. To address the challenge of buildings with variable properties along their height, the combined application of the continuous method and the transfer matrix method allows us to propose an efficient numerical method leading to a 6 × 6 transfer matrix, constant regardless of the number of discretizations and derived analytically, which results in a computational advantage by drastically reducing computational costs. The numerical results are encouraging and show good agreement with the results of the finite element method, suggesting that the proposed method can be used for engineering purposes in both the preliminary and final phases of the structural analysis of tall buildings.

Neural force functional for non-equilibrium many-body colloidal systems

Abstract We combine power functional theory and machine learning to study non-equilibrium overdamped many-body systems of colloidal particles at the level of one-body fields. We first sample in steady state the one-body fields relevant for the dynamics from computer simulations of Brownian particles under the influence of randomly generated external fields. A neural network is then trained with this data to represent locally in space the formally exact functional mapping from the one-body density and velocity profiles to the one-body internal force field. The trained network is used to analyse the non-equilibrium superadiabatic force field and the transport coefficients such as shear and bulk viscosities. Due to the local learning approach, the network can be applied to systems much larger than the original simulation box in which the one-body fields are sampled. Complemented with the exact non-equilibrium one-body force balance equation and a continuity equation, the network yields viable predictions of the dynamics in time-dependent situations. Even though training is based on steady states only, the predicted dynamics is in good agreement with simulation results. A neural dynamical density functional theory can be straightforwardly implemented as a limiting case in which the internal force field is that of an equilibrium system. The framework is general and directly applicable to other many-body systems of interacting particles following Brownian dynamics.

An investigation of anisotropy in the bubbly turbulent flow via direct numerical simulations

We investigated the effects of bubble count, flow direction, and Eötvös number on deformable bubbles in turbulent channel flow. For a given shear Reynolds number Re = 180 and fixed bubble volume fractions (1.263% and 2.525%), we conducted a series of direct numerical simulations using a coupled level-set and volume-of-fluid solver to evaluate their impact on bubble volume fraction distribution, velocity fields, and turbulence characteristics. Each aspect was studied based on the microscopic equations of two-phase flow, and the accuracy of the modeling terms used in current Reynolds-averaged Navier–Stokes equation (RANS) models was assessed. The influence on the anisotropic state was analyzed using the Lumley triangle, and the anisotropy of Reynolds stresses was captured through the exact balance equations. The results indicate that in upward flow, bubbles tend to accumulate near the wall, with smaller Eötvös numbers leading to closer proximity to the wall and greater attenuation of the liquid-phase velocity. This distribution enhances energy dissipation and turbulence isotropy. In downward flow, bubbles cluster in the channel center, generating additional pseudo-turbulence and attenuating energy in the buffer layer. Moreover, the interfacial transfer of turbulent energy, as currently modeled in RANS, is found to be inadequate for upward flows.

Coherent Ultrafast Stimulated X-Ray Raman Spectroscopy of Dissipative Conical Intersections.

The quantum coherent dynamics of a vibronic wave packet in a molecule passing through a conical intersection can be revealed using attosecond transient coherent Raman spectroscopy. In particular, the time evolution of the electronic coherence can be monitored in the presence of vibrational dynamics. So far, the technique has been investigated without including environmental quantum noise. Here, we employ the numerically exact hierarchy equation of motion approach to show that the transient coherent Raman signals are robust and accessible on times of up to a few hundred femtoseconds with respect to electonic and vibrational dephasing.

Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice

We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The analysis of fixed points reveals information of location of the Fisher zeros and we provide a conjecture about the location of zeros in terms of the boundary of the basins of attraction.

Two Fluids in [formula omitted] Gravity with Observational Constraints

A locally rotationally symmetric Bianchi type-I model has been studied with two fluids within the framework of the f(T) theory of gravity. Bianchi type-I is an immediate generalization of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric. We have derived the exact field equations in f(T) gravity by considering the Bianchi type-I metric and applying the action for f(T) theory of gravity. We have utilized the torsion scalar T and the Lagrangian for matter. The field equations are obtained by taking the variation of the action with respect to the vierbein, leading to a set of equations that includes the energy–momentum tensor for two fluid sources: matter and radiation. We fit the H(z) curve using 57 data points and the R2-test, achieving an R2 value of 0.9321, indicating a strong fit with the Observational Hubble Dataset (OHD). Cosmological parameters like energy density, pressure, and state finder diagnostics are also discussed.

DKP spin-one particles confined in a two-dimensional ring

We study the confinement of relativistic spin-one particles in two spatial dimensions by a quantum ring for systems described by the Duffin–Kemmer–Petiau (DKP) equation. The ring-like geometry is realized by introducing a pseudo-scalar funnel-shape potential into the DKP equation, leading to a generalization of the planar DKP oscillator. Having used a ten-dimensional representation of the DKP matrices, the DKP equation is then mapped into two uncoupled subproblems, the first is associated with states of zero spin-projection (natural parity states) and the second is associated with states having a spin-projection number equal to ±1 (unnatural parity states). For both subproblems the exact wave functions of the system are analytically derived and expressed through the generalized Laguerre polynomials. The corresponding energy eigenvalues are explicitly obtained for natural parity states, while in the unnatural parity case an exact energy equation from which they can be determined is derived.

Linear Logistic Scoring Equations for Latent Class and Latent Profile Models: A Simple Method for Classifying New Cases

Researchers are often interested in using latent class or latent profile parameter estimates to obtain posterior class membership probabilities for observations other than those of the original sample. In this paper, we demonstrate that these probabilities typically take on the form of linear logistic equations with coefficients which are functions of the original model parameters. In other words, the posterior class membership probabilities can be computed with a prediction formula similar to that of a multinomial logistic regression model. We derive the scoring equations for nominal, ordinal, count, and continuous indicators, as well as investigate models with missing values on class indicators, local dependencies, covariates, or multiple latent variables. In addition to the mathematical derivations of the scoring equations, we describe how either exact or approximate scoring equations can be obtained by estimating a multinomial regression model using a weighted data set.

Quantum speedup and non-Markovianity of an atom in structured reservoirs: pseudomodes as a good description of environmental memory

Following the recent paper (Teittinen et al 2019 New J. Phys. 21 123041), one can see that in general there is no simple relation between non-Markovianity and quantum speed limit. Here, we investigate the connection between quantum speed limit time and non-Markovianity of an atom in structured environments (reservoirs) whose dynamics is governed by an exact pseudomode master equation (Garraway 1997 Phys. Rev. A 55 2290). In particular, we find an inverse relation between them, which means that the non-Markovian feature of the quantum process leads to speedup of evolution. Thus, there is a link between quantum speedup and memory effects for specific cases of dynamical evolution. Our results might shed light on the relationship between the speedup of quantum evolution and the backflow of information from the environment to the system.

Boundary integrated neural networks for 2D elastostatic and piezoelectric problems

In this paper, we make the first attempt to adopt the boundary integrated neural networks (BINNs) for the numerical solution of two-dimensional (2D) elastostatic and piezoelectric problems. The proposed BINNs combine the artificial neural networks with the exact boundary integral equations (BIEs) to effectively solve the boundary value problems based on the corresponding partial differential equations (PDEs). The BIEs are utilized to localize all the unknown physical quantities on the boundary, which are approximated by using artificial neural networks and resolved via a training process. In contrast to many traditional neural network methods based on a domain discretization, the present BINNs offer several distinct advantages. Firstly, by embedding the analytical BIEs into the learning procedure, the present BINNs only need to discretize the boundary of the problem domain, which reduces the number of the unknowns and can lead to a faster and more stable learning process. Secondly, the differential operators in the original PDEs are substituted by integral operators, which can effectively eliminate the need for additional differentiations of the neural networks (high-order derivatives of the neural networks may lead to instabilities in the learning process). Thirdly, the loss function of the present BINNs only contains the residuals of the BIEs, as all the boundary conditions have been inherently incorporated within the formulation. Therefore, there is no necessity for employing any weighting functions, which are commonly used in most traditional methods to balance the gradients among the different objective functions. Extensive numerical experiments show that the present BINNs are much easier to train and can usually provide more accurate solutions as compared to many traditional neural network methods.

Resonance theory of vibrational strong coupling enhanced polariton chemistry and the role of photonic mode lifetime

Recent experiments demonstrate polaritons under the vibrational strong coupling (VSC) regime can modify chemical reactivity. Here, we present a complete theory of VSC-modified rate constants when coupling a single molecule to an optical cavity, where the role of photonic mode lifetime is understood. The analytic expression exhibits a sharp resonance behavior, where the maximum rate constant is reached when the cavity frequency matches the vibration frequency. The theory explains why VSC rate constant modification closely resembles the optical spectra of the vibration outside the cavity. Further, we discussed the temperature dependence of the VSC-modified rate constants. The analytic theory agrees well with the numerically exact hierarchical equations of motion (HEOM) simulations for all explored regimes. Finally, we discussed the resonance condition at the normal incidence when considering in-plane momentum inside a Fabry-Pérot cavity.

Effect of interlayer dependence strength on continuous to discontinuous transitions in multi-layer networks

Studies on multilayer networks have aroused great interest in recent years because they allow a deeper description of the complexity present in many real systems. In this study, we introduce a dependency strength controller into multilayer interdependent networks, refining the traditional models that often inadequately represent cascading failures in real-world systems like urban transportation networks. This refined model enables a more realistic depiction of failure dynamics, where a node’s failure leads to its dependent node’s failure with a probability λ or survival with 1−λ. We derive exact equations that describe this process in a multilayer network and its structural evolution, and solve them numerically. We found the existence of a critical transition from continuous to discontinuous percolation as the strength of the dependence between layers increases. The critical percolation thresholds for these transitions are numerically derived for two-layer Erdős-Rényi networks and two-layer scale-free networks. This enhanced understanding of complex systems, incorporating initial disturbances and varying dependency strengths, contributes to designing more resilient multilayer complex systems.

Wavefield decomposition for viscoelastic anisotropic media

Separating wave modes on seismic records is an essential step in imaging of multicomponent seismic data. Viscoelastic anisotropic models provide a realistic description of subsurface formations that exhibit anisotropy of velocity and attenuation. However, mode separation has not been extended to viscoelastic anisotropic media yet. Here, we propose an efficient approach to wavefield decomposition that takes velocity and attenuation anisotropy into account. Our algorithm operates in the frequency-wavenumber domain and, therefore, is suitable for general dissipative models. We present exact equations for wavefield decomposition in arbitrarily anisotropic attenuative homogeneous media. Then the proposed approach is applied to viscoelastic constant-[Formula: see text] VTI (transversely isotropic with a vertical symmetry axis) models. Numerical examples demonstrate the accuracy and efficiency of our approach for piecewise-homogeneous media characterized by pronounced velocity and attenuation anisotropy.

Liouville models of particle-laden flow

Langevin (stochastic differential) equations are routinely used to describe particle-laden flows. They predict Gaussian probability density functions (PDFs) of a particle's trajectory and velocity, even though experimentally observed dynamics might be highly non-Gaussian. Our Liouville approach overcomes this dichotomy by replacing the Wiener process in the Langevin models with a (small) set of random variables, whose distributions are tuned to match the observed statistics. This strategy gives rise to an exact (deterministic, first-order, hyperbolic) Liouville equation that describes the evolution of a joint PDF in the augmented phase-space spanned by the random variables and the particle position and velocity. Analytical PDF solutions for canonical models of particle-laden flows serve to establish a relationship between the Langevin and Liouville approaches. Finally, our framework is used to derive a new analytical PDF model for fluidized homogeneous heating systems.