Multiscale Expansion Research Articles

Curved-pipe flow with boundary conditions involving Bernoulli pressure

In this article, we study the steady-state flow of the incompressible viscous fluid through a thin distorted pipe with an arbitrary central curve. We prescribe the inflow and outflow boundary conditions involving the Bernoulli pressure with a given pressure drop. Using the multiscale expansion technique with respect to the pipe's thickness, we construct the higher-order asymptotic approximation of the flow given by the explicit formulae for the velocity and pressure. We also perform a detailed error analysis justifying the usage of the proposed solution and indicating its order of accuracy. For more information see https://ejde.math.txstate.edu/Volumes/2024/63/abstr.html

A multiscale Gaussian expansion method for low-frequency multimodal damping analysis of variable stiffness acoustic black hole beams

To address the issues of slow convergence and poor accuracy in existing dynamics methods caused by non-uniform variable cross-sections and layered material stiffness variations in variable stiffness acoustic black hole structure (VSABH). A multiscale Gaussian expansion method (MSGEM) is proposed in this paper. The structural dimensions are taken into account, and the structural parameters of multiple shape functions are adaptively selected based on stiffness and cross-sectional variation parameters. This results in the formation of shape function groups of various scales, which in turn creates the matrix of multi-scale Gaussian function groups. The feature information of the displacement field is realized to be extracted from multiple scales, and the fitting error of MSGEM to the displacement field of VSABH beam is within 2.02%. The missing solution of eigenfrequency of Gaussian expansion method is avoided, and the modal vibration patterns are basically matched, which verifies the validity of MSGEM. Meanwhile, the advantages of low-frequency multimodal vibration reduction of VSABH beams are highlighted from several aspects, and the trend of the adjustment of the vibration reduction characteristics of VSABH beams by different parameters is clarified, which provides valuable design guidance for ABH metamaterials oriented to low-frequency vibration reduction.

Optimal Control of Underdamped Systems: An Analytic Approach

Optimal control theory deals with finding protocols to steer a system between assigned initial and final states, such that a trajectory-dependent cost function is minimized. The application of optimal control to stochastic systems is an open and challenging research frontier, with a spectrum of applications ranging from stochastic thermodynamics to biophysics and data science. Among these, the design of nanoscale electronic components motivates the study of underdamped dynamics, leading to practical and conceptual difficulties. In this work, we develop analytic techniques to determine protocols steering finite time transitions at a minimum thermodynamic cost for stochastic underdamped dynamics. As cost functions, we consider two paradigmatic thermodynamic indicators. The first is the Kullback–Leibler divergence between the probability measure of the controlled process and that of a reference process. The corresponding optimization problem is the underdamped version of the Schrödinger diffusion problem that has been widely studied in the overdamped regime. The second is the mean entropy production during the transition, corresponding to the second law of modern stochastic thermodynamics. For transitions between Gaussian states, we show that optimal protocols satisfy a Lyapunov equation, a central tool in stability analysis of dynamical systems. For transitions between states described by general Maxwell-Boltzmann distributions, we introduce an infinite-dimensional version of the Poincaré-Lindstedt multiscale perturbation theory around the overdamped limit. This technique fundamentally improves the standard multiscale expansion. Indeed, it enables the explicit computation of momentum cumulants, whose variation in time is a distinctive trait of underdamped dynamics and is directly accessible to experimental observation. Our results allow us to numerically study cost asymmetries in expansion and compression processes and make predictions for inertial corrections to optimal protocols in the Landauer erasure problem at the nanoscale.

Discrete-Ordinate-Lattice-Boltzmann Method for analyzing radiative heat transfer: Application to two-dimensional irregular enclosure

A new Discrete Ordinate-Lattice Boltzmann Method (DO-LBM) is developed by combining DOM and LBM to analyze radiative heat transfer in a two-dimensional irregular enclosure including absorbing, emitting, and scattering media. The accuracy of the proposed method was verified through Chapman-Enskog multi-scale expansion, and non-negativity analysis of the equilibrium distribution function was performed by introducing similarity numbers. Two-dimensional irregular enclosure including curved boundaries and obstacles were considered, and thn numerical results were compared with those obtained by other methods on standard problems. As a result, it was confirmed that the DO-LBM provides a valuable results in simulating radiative heat transfer in a complex boundary structure, which is simple, accurate and can reduce the computational cost compared to other methods.

Derivation and analysis of lattice Boltzmann form of the mild slope equation

The present study investigates the performance of the Lattice Boltzmann Method (LBM) as an alternative numerical method for simulating wave propagation in coastal zones. While formulations like the Mild-Slope Equation (MSE) remain efficient for coastal engineering applications due to their simplicity, LBM offers an explicit numerical solver for partial differential equations (PDEs). LBM utilizes local operations, making it well-suited for Graphics Processing Unit (GPU) acceleration and consequently efficient simulations. This study examines LBM's performance in several benchmark cases, including wave propagation behind a single breakwater, harbor resonance, refraction of long waves over a parabolic shoal, and wave propagation over an elliptic shoal. The Chapman-Enskog multi-scale expansion was used within a popular 2D LBM scheme to derive the relaxation time and equilibrium distribution function. The results showed a good agreement between LBM predictions and benchmark data. Furthermore, LBM demonstrated efficiency in reducing the computation time, achieving a notable 30–50% decrease in CPU time and an 85% reduction when utilizing a GPU.

Wavy approach for fluid–structure interaction with high Froude number and undamped structure

This paper addresses the fluid–structure interaction problem, with an interest on the interaction of a gravity wave with a flexible floating structure, anchored to a seabed of constant depth. To achieve this goal, we make use of the model equations, namely, the Navier–Stokes equations and the Navier–Lamé equation, as well as the associated the boundary conditions. Applying the multi-scale expansion method, these set of equations are reduced to a pair of nonlinearly coupled complex cubic Ginzburg–Landau equations (CCGLE). By applying the proposed modified expansion method, the group velocity dispersion and second-order dispersion relation are deduced. In the same vein, modulation instability (MI) is investigated as a mechanism of formation of pulse trains in fluid–structure system using a CCGLE. For the analytical analysis, we made use of the inverse scattering method to find analytical solutions to the coupled nonlinear equations. Through that method, the obtained solutions depict rogue-shaped waves. Our results suggest that uncontrolled MI within the interaction between a flexible body and gravity waves in viscous flow may be considered as the principal source of many structural ruptures, which are the first cause of critical damage due to the great energy and unpredictability of rogue waves. The present work aims to provide tools to model a wide range of physical problems regarding the interaction of surface gravity waves and an offshore-anchored structure, and it aims to be essential to our understanding of the nonlinear characteristics of offshore structures in real-sea states.

Innovative integration of multi-scale residual networks and MK-MMD for enhanced feature representation in fault diagnosis

This article proposes a method based on multi-scale expansion of residual neural networks (ResNets) to address challenges in the operation of rotating components, such as bearings and gears, under complex conditions where they are often affected by environmental noise. This interference leads to weaker fault characteristics, making feature selection difficult and increasing the presence of extraneous information features. To tackle this issue, the proposed method first employs a multi-scale feature ResNet to extract features from vibration signals of rotating machinery. The method decomposes the signal into multiple sub-signals of different scales, extracting local features at each scale. It then uses residual connections to combine these local features to obtain a global feature representation. Furthermore, the article introduces a construction of the maximum mean discrepancy (MMD) and minimization of entropy boundaries to adapt to the differences between two domains. The method utilizes multiple kernel functions to calculate distances between data at different scales and combines these distances to obtain a comprehensive measure. By employing the MMD and minimization of entropy boundary approach, the method can more accurately determine whether signals at different scales belong to the same category, thereby improving diagnostic accuracy and robustness. Experimental results demonstrate the effectiveness of the proposed method in unsupervised cross-domain fault diagnosis tasks. Future work will focus on further optimizing the architecture of ResNets, enhancing feature extraction capabilities, and exploring advanced data augmentation methods to further improve the model’s generalization performance.

Semi-Supervised Multiscale Dynamic Graph Convolution Network for Hyperspectral Image Classification.

In recent years, convolutional neural networks (CNNs)-based methods achieve cracking performance on hyperspectral image (HSI) classification tasks, due to its hierarchical structure and strong nonlinear fitting capacity. Most of them, however, are supervised approaches that need a large number of labeled data to train them. Conventional convolution kernels are fixed shape of rectangular with fixed sizes, which are good at capturing short-range relations between pixels within HSIs but ignore the long-range context within HSIs, limiting their performance. To overcome the limitations mentioned above, we present a dynamic multiscale graph convolutional network (GCN) classifier (DMSGer). DMSGer first constructs a relatively small graph at region-level based on a superpixel segmentation algorithm and metric-learning. A dynamic pixel-level feature update strategy is then applied to the region-level adjacency matrix, which can help DMSGer learn the pixel representation dynamically. Finally, to deeply understand the complex contents within HSIs, our model is expanded into a multiscale version. On the one hand, by introducing graph learning theory, DMSGer accomplishes HSI classification tasks in a semi-supervised manner, relieving the pressure of collecting abundant labeled samples. Superpixels are generally in irregular shapes and sizes which can group only similar pixels in a neighborhood. On the other hand, based on the proposed dynamic-GCN, the pixel-level and region-level information can be captured simultaneously in one graph convolution layer such that the classification results can be improved. Also, due to the proper multiscale expansion, more helpful information can be captured from HSIs. Extensive experiments were conducted on four public HSIs, and the promising results illustrate that our DMSGer is robust in classifying HSIs. Our source codes are available at https://github.com/TangXu-Group/DMSGer.

Numerical Simulation for Solitary Waves of the Generalized Zakharov Equation Based on the Lattice Boltzmann Method

The generalized Zakharov equation is a widely used and crucial model in plasma physics, which helps to understand wave particle interactions and nonlinear wave propagation in plasma. The solitary wave solution of this equation provides insights into phenomena such as electron and ion acoustic waves, as well as magnetic field disturbances in plasma. The numerical simulation of solitary wave solutions to the generalized Zakharov equation is an interesting problem worth studying. This is crucial for plasma-based technology, as well as for understanding nonlinear wave propagation in plasma physics and other fields. In this study, a numerical investigation of the generalized Zakharov equation using the lattice Boltzmann method has been conducted. The lattice Boltzmann method is a new modeling and simulating method at the mesoscale. A lattice Boltzmann model was constructed by performing Taylor expansion, Chapman–Enskog expansion, and time multiscale expansion on the lattice Boltzmann equation. By defining the moments of the equilibrium distribution function appropriately, the macroscopic equation can be restored. Furthermore, the numerical experiments for the equation are carried out with the parameter lattice size m=100, time step Δt=0.001, and space step size Δx=0.4. The solitary wave solution of the equation is numerically simulated. Numerical results under different parameter values are compared, and the convergence and effectiveness of the model are numerically verified. It is obtained that the model is convergent in time and space, and the convergence orders are all 2.24881. The effectiveness of our model was also verified by comparing the numerical results of different numerical methods. The lattice Boltzmann method demonstrates advantages in both accuracy and CPU time. The results indicate that the lattice Boltzmann method is a good tool for computing the generalized Zakharov equation.

A (2+1) -dimensional evolution model of Rossby waves and its resonance Y-type soliton and interaction solutions

In this paper, we derive a Kadomtsev-Petviashvili equation by using the multi-scale expansion and perturbation method, which is a model from the potential vorticity equation in the traditional approximation and describes the Rossby wave propagation properties. The N-soliton solutions, resonance Y-type soliton solutions, resonance X-type soliton solutions and interaction solutions of the equation are obtained with the help of dependent-variable transformation. In addition, the composite graphs are given to view the resonance phenomenon of Rossby waves. The results better enrich the research of Rossby waves in ocean dynamics and atmospheric dynamics.

Nonlinear instability and solitons in a self‐gravitating fluid

We study a spherical, self‐gravitating fluid model, which finds applications in cosmic structure formation. We argue that since the system features nonlinearity and gravity‐induced dispersion, the emergence of solitons becomes possible. We thus employ a multiscale expansion method to study, in the weakly nonlinear regime, the evolution of small‐amplitude perturbations around the equilibrium state. This way, we derive a spherical nonlinear Schrödinger (NLS) equation that governs the envelope of the perturbations. The effective NLS description allows us to predict a “nonlinear instability” (occurring in the nonlinear regime of the system), namely, the modulational instability, which, in turn, may give rise to spherical soliton states. The latter feature a very slow (polynomial) curvature‐induced decay in time. The soliton profiles may be used to describe the shape of dark matter halos at the rims of the galaxies.

A multiscale asymptotic expansion for combustion system with composite materials

This paper discusses the multiscale asymptotic expansion for combustion system with composite materials or porous media in thermal protection system. The combustion system consists of two nonlinear equations with rapidly oscillating coefficients, which are posed on a time-varying domain and computationally expensive to solve with the finite element method. To efficiently solve this system, we propose the standard/novel multiscale asymptotic expansions by combining them with boundary layer solutions. Numerical simulation is performed to validate the accuracy and efficiency of the newly proposed methods, which show that the novel multiscale asymptotic expansion is more accurate than the standard one.

A calibration and uncertainty quantification analysis of classical, fractional and multiscale logistic models of tumour growth

Background and Objective: The validation of mathematical models of tumour growth is frequently hampered by the lack of sufficient experimental data, resulting in qualitative rather than quantitative studies. Recent approaches to this problem have attempted to extract information about tumour growth by integrating multiscale experimental measurements, such as longitudinal cell counts and gene expression data. In the present study, we investigated the performance of several mathematical models of tumour growth, including classical logistic, fractional and novel multiscale models, in terms of quantifying in-vitro tumour growth in the presence and absence of therapy. We further examined the effect of genes associated with changes in chemosensitivity in cell death rates. Methods: The multiscale expansion of logistic growth models was performed by coupling gene expression profiles to the cell death rates. State-of-the-art Bayesian inference, likelihood maximisation and uncertainty quantification techniques allowed a thorough evaluation of model performance. Results: The results suggest that the classical single-cell population model (SCPM) was the best fit for the untreated and low-dose treatment conditions, while the multiscale model with a cell death rate symmetric with the expression profile of OCT4 (Sym-SCPM) yielded the best fit for the high-dose treatment data. Further identifiability analysis showed that the multiscale model was both structurally and practically identifiable under the condition of known OCT4 expression profiles. Conclusions: Overall, the present study demonstrates that model performance can be improved by incorporating multiscale measurements of tumour growth when high-dose treatment is involved.

A novel model for the fifth-order mKP equation describing (2+1)-dimensional shallow ocean solitary waves and their properties

In this paper, we develop a (2+1)-dimensional fifth-order modified Kadomtsev-Petviashvili (mKP) equation model using multiscale analysis and perturbation expansion, which is a novel model description of the shallow ocean solitary waves phenomenon, featuring stronger nonlinear terms and a higher-order compared to traditional models. We examine the conservation law within the model to further investigate the properties of solitary waves. Subsequently, we obtain the exact solutions of the model by employing the expansion method. We simulate the generation and propagation of solitary waves through appropriate parameter adjustments, deriving the solitary wave solutions, kink wave solutions, and periodic wave solutions. Additionally, we analyze the effects of different parameters on the propagation of solitary waves. Finally, we explore the evolution of solitary waves before and after generation under a given initial perturbation conditions. This study contributes to a deeper understanding of the intricate dynamics of shallow ocean solitary waves, shedding light on their behaviour and implications.

Efficient asymptotic models for axisymmetric eddy current problems in linear ferromagnetic materials

The problem under consideration is that of time-harmonic eddy current problems in linear ferromagnetic materials surrounded by a dielectric medium with a smooth common interface. Assuming axisymmetric geometries and orthoradial axisymmetric data, we construct an efficient multiscale expansion for the orthoradial solution that provides reduced computational costs. We investigate numerically the accuracy of the approach using an analytical procedure and infinite cylinders as well. It results that the computation of two asymptotics is sufficient to ensure accurate solutions in the case of low frequencies.

Higher-Order Benjamin–Ono Model for Ocean Internal Solitary Waves and Its Related Properties

In this study, the propagation of internal solitary waves in oceans at great depths was analyzed. Using multi-scale analysis and perturbation expansion, the basic equation is simplified to the classical Benjamin–Ono equation with variable coefficients. To better describe the propagation characteristics of solitary waves, we derived a higher-order variable-coefficient integral differential (Benjamin–Ono) equation. Subsequently, the bilinear form of the model was derived using Hirota’s bilinear method, and a multi-soliton solution was obtained. Based on the multi-soliton solution of the model, we further studied the interaction of the soliton, which led to the discovery of Mach reflection. Some conclusions were drawn, which are of potential value for further study of solitary waves in the ocean.

Comprehensive assessment of the partially-saturated-cells framework for convective problems

We present a comprehensive investigation of Partially-Saturated-Cells method (PSC) for the simulations of incompressible thermal flows. The methodology comprises an additional collision operator introduced into the conventional simplified thermal lattice-Boltzmann (STLBM) method that offers a unified evolution equation for solid and fluid media present in the computational domain. It is a type of diffuse-interface immersed-boundary method that is dictated by the volume fraction of the solid medium and facilitates a smooth transition of boundary conditions at the solid-fluid interface. In this work, we devise a mechanism to evaluate local heat flux at the solid-fluid interface and present its congruity with several heat transfer problems. We also derive the recovery of the macroscopic energy equation from PSC's evolution equation through Chapman-Enskog multiscale expansion. Our investigation also reveals that the method is first-order accurate for moving body simulations irrespective of the straight or curved boundary. It is also Galilean invariant and demonstrates identical heat transfer characteristics in stationary and moving reference frames. A comprehensive analysis of the PSC algorithm is presented to enlighten its suitability for predicting thermo-hydrodynamic characteristics in free, forced, and mixed convection phenomena involving both stationary and moving boundary scenarios. To the best of our knowledge, these aspects pertaining to PSC methodology have not been discussed in the literature so far.

General fourth-order Chapman–Enskog expansion of lattice Boltzmann schemes

In order to derive the equivalent partial differential equations of a lattice Boltzmann scheme, the Chapman Enskog expansion is very popular in the lattice Boltzmann community. A main drawback of this approach is the fact that multiscale expansions are used without any clear mathematical signification of the various variables and operators. Independently of this framework, the Taylor expansion method allows to obtain formally the equivalent partial differential equations. The general equivalency of these two approaches remains an open question. In this contribution, we prove that both approaches give identical results with acoustic scaling for a very general family of lattice Boltzmann schemes and up to fourth-order accuracy. Examples with a single scalar conservation illustrate our purpose.

Multiscale expansion modeling of concrete with free lime-based expansive additives under isotropic conditions using poromechanics

Expansive additives have been used in concrete to compensate for shrinkage and to induce chemical pre-stress. The cement paste incurs internal expansive stress owing to the formation of confined portlandite and ettringite crystals in the pore network, resulting in expansion at the macroscale. In this study, a poromechanical model was designed to model the expansion of cement paste and concrete due to the formation of such expansive hydrates. The effective expansive stress on the cement paste was calculated considering the volume of the hydrates derived from the hydration model and tangent poroelastic properties estimated using a micromechanical scheme. Experiments were performed to measure the expansion of the cement paste and concrete with different amounts of expansive additives under stress-free and restraint conditions. The model predictions were compared with the experimentally measured expansion, which confirmed the rationality of the model for estimating free expansion. Finally, the extension of the current modeling framework to estimate the restrained expansion of concrete is deliberated. The proposed multiscale model can be employed by the engineers to estimate the expansion of cement paste and concrete with EA and provide insights to estimate the amount of EA required for a given application.

Statistical Effective Diffusivity Estimation in Porous Media Using an Integrated On-site Imaging Workflow for Synchrotron Users

Transport in porous media plays an essential role for many physical, engineering, biological and environmental processes. Novel synchrotron imaging techniques and image-based models have enabled more robust quantification of geometric structures that influence transport through the pore space. However, image-based modelling is computationally expensive, and end users often require, while conducting imaging campaign, fast and agile bulk-scale effective parameter estimates that account for the pore-scale details. In this manuscript we enhance a pre-existing image-based model solver known as OpenImpala to estimate bulk-scale effective transport parameters. In particular, the boundary conditions and equations in OpenImpala were modified in order to estimate the effective diffusivity in an imaged system/geometry via a formal multi-scale homogenisation expansion. Estimates of effective pore space diffusivity were generated for a range of elementary volume sizes to estimate when the effective diffusivity values begin to converge to a single value. Results from OpenImpala were validated against a commercial finite element method package COMSOL Multiphysics (abbreviated as COMSOL). Results showed that the effective diffusivity values determined with OpenImpala were similar to those estimated by COMSOL. Tests on larger domains comparing a full image-based model to a homogenised (geometrically uniform) domain that used the effective diffusivity parameters showed differences below 2 % error, thus verifying the accuracy of the effective diffusivity estimates. Finally, we compared OpenImpala’s parallel computing speeds to COMSOL. OpenImpala consistently ran simulations within fractions of minutes, which was two orders of magnitude faster than COMSOL providing identical supercomputing specifications. In conclusion, we demonstrated OpenImpala’s utility as part of an on-site tomography processing pipeline allowing for fast and agile assessment of porous media processes and to guide imaging campaigns while they are happening at synchrotron beamlines.