Homogeneous Problem Research Articles

Thermal modeling of porous medium integrated in PCM and its application in passive thermal management of electric vehicle battery pack

The integration of a composite of porous medium with phase change material (PCM) offers significant advantages in thermal management systems, enhancing heat transfer efficiency and addressing various thermal regulation challenges. This approach utilizes the PCM's latent heat absorption and the enhanced thermal conductivity provided by the porous medium, resulting in optimized system performance. Its applicability spans across electronics cooling and building insulation systems. However, predicting the thermal behavior of this composite material is challenging, necessitating computational tools to anticipate its response under different conditions and evaluate its influence on cooling strategies. The objective of this study is to create a computational tool specifically tailored to evaluate constitutive parameters of this composite material, thereby providing a comprehensive description of its thermal behavior. To achieve this goal, the multiscale homogenization principle is employed to assess the composite's effective thermophysical material properties using the representative volume element approach. The repeating unit cell of the aluminum lattice is incorporated into the PCM to define a representative volume element. The finite element method (FEM) is utilized to solve the three-dimensional homogenization problem, yielding an orthotropic effective thermal conductivity due to the inherent symmetry of the repeating material cell. Moreover, the study leverages the apparent heat capacity method to effectively manage the phase transitions within the PCM domain, utilizing smooth and temperature-dependent functions to accurately describe the thermophysical properties of the PCM. Integrating the composite into battery pack thermal management, this study thoroughly examines thermal dynamics by comparing outcomes with and without PCM integration. The transient thermal problem is accurately tackled using the FEM, employing the evaluated effective constitutive parameters of the homogenized composite to minimize computational effort. The results indicate a notable decline in the highest temperatures of the battery pack, leading to a reduction of about 14 °C at the specific moment when the phase change material fully transitions into its liquid form. The obtained results emphasize the effectiveness and practical feasibility of the proposed thermal management strategy. The modeling approach presented provides a robust tool with significant efficiency in reducing computational time for analyzing the thermal behavior of large models, as the utilization of the homogenization technique notably decreases the computational time.

Two-scale asymptotic homogenization analysis of piezoelectric composite materials in generalized curvilinear coordinates

This paper presents a comprehensive study on applying the two-scale asymptotic homogenization method to derive the effective properties of piezoelectric composite materials characterized by generalized periodicity in curvilinear coordinates. The methodology involves formulating the homogenization problem in a general curvilinear framework suitable for materials with complex geometric configurations. By systematically deriving the homogenized equations and effective coefficients, the paper provides a robust theoretical foundation for understanding the macroscopic behavior of piezoelectric composites under various loading conditions. The model’s versatility is demonstrated through its application to specific curvilinear coordinate systems, including the helix coordinate system and cylindrical coordinates with wavy geometry. These examples highlight the potential of the proposed approach in analyzing and designing piezoelectric materials with intricate geometries. The state-of-the-art numerical homogenization methods have also validated the proposed homogenization method, ensuring its accuracy and robustness.

Smoothing and flattening the universe through slow contraction versus inflation

In a systematic study, we use an equivalent pair of improved numerical relativity codes based on a tetrad-formulation of the classical Einstein-scalar field equations to examine whether slow contraction or inflation (or both) can resolve the homogeneity, isotropy and flatness problems. Our finding, based on a set of gauge/frame invariant diagnostics and the models considered, is that slow contraction robustly and rapidly smooths and flattens spacetimebeginning from initial conditions that are outside the perturbative regime of the flat Friedmann-Robertson-Walker metric, whereas inflation fails these tests. We present new numerical evidence supporting the conjecture that the combination of ultralocal evolution and an effective equation-of-state with pressure much greater than energy density is the key to having robust and rapid smoothing. The opposite of ultralocality occurs in expanding spacetimes, which is the leading obstruction to smoothing following a big bang.

Finite variation sensitivity analysis in the design of isotropic metamaterials through discrete topology optimization

AbstractThis article extends recently developed finite variation sensitivity analysis (FVSA) approaches to an inverse homogenization problem. The design of metamaterials with prescribed mechanical properties is stated as a discrete density‐based topology optimization problem, in which the design variables define the microstructure of the periodic base cell. The FVSA consists in estimating the finite variations of the objective and constraint functions after independently switching the state of each variable. It is used to properly linearize the functions of binary variables so the optimization problem can be solved through sequential integer linear programming. Novel sensitivity expressions were developed and it was shown that they are more accurate than the ones conventionally used in literature. Referred to as the conjugate gradient sensitivity (CGS) approach, the proposed strategy was quantitatively evaluated through numerical examples. In these examples, metamaterials with prescribed homogenized Poisson's ratios and minimal homogenized Young's moduli were obtained. A hexagonal base cell with dihedral symmetry was used to produce only metamaterials with isotropic properties. It was shown that, by using the CGS approach instead of the conventional sensitivity analysis, the sensitivity error was substantially reduced for the considered problem. The proposed developments effectively improved the stability and robustness of the discrete optimization procedures. In all the considered examples, when more accurate sensitivity analyses were performed, the parameters of the topology optimization method could be tuned more easily, yielding effective solutions even if the settings were not ideal.

A knowledge-driven memetic algorithm for the energy-efficient distributed homogeneous flow shop scheduling problem

The reduction of carbon emissions in the manufacturing industry holds significant importance in achieving the national ”double carbon” target. Ensuring energy efficiency is a crucial factor to be incorporated into future generation manufacturing systems. In this study, energy consumption is considered in the distributed homogeneous flow shop scheduling problem (DHFSSP). A knowledge-driven memetic algorithm (KDMA) is proposed to address the energy-efficient DHFSSP (EEDHFSSP). KDMA incorporates a collaborative initialization strategy to generate high-quality initial populations. Furthermore, several algorithmic improvements including update strategy, local search strategy, and carbon reduction strategy are employed to improve the search performance of the algorithm. The effectiveness of KDMA in solving EEDHFSSP is verified through extensive simulation experiments. KDMA outperforms many state-of-the-art algorithms across various evaluation aspects.

Uniform boundary estimates for Neumann problems in parabolic homogenization

For a family of second-order parabolic systems with periodic, oscillating and time-dependent coefficients, we establish the uniform boundary Hölder and Lipschitz estimates for Neumann problems in [Formula: see text] and [Formula: see text] cylinders, respectively, by using the convergence rate method. Moreover, we establish the uniform [Formula: see text] estimates for initial-Neumann problems in [Formula: see text] cylinders for [Formula: see text] by using the real-variable method. As a by-product, we also obtain the Gaussian estimates for Neumann function and its derivatives.

Imposing Dirichlet boundary conditions directly for FFT-based computational micromechanics

We discuss how Dirichlet boundary conditions can be directly imposed for the Moulinec–Suquet discretization on the boundary of rectangular domains in iterative schemes based on the fast Fourier transform (FFT) and computational homogenization problems in mechanics. Classically, computational homogenization methods based on the fast Fourier transform work with periodic boundary conditions. There are applications, however, when Dirichlet (or Neumann) boundary conditions are required. For thermal homogenization problems, it is straightforward to impose such boundary conditions by using discrete sine (and cosine) transforms instead of the FFT. This approach, however, is not readily extended to mechanical problems due to the appearance of mixed derivatives in the Lamé operator of elasticity. Thus, Dirichlet boundary conditions are typically imposed either by using Lagrange multipliers or a “buffer zone” with a high stiffness. Both strategies lead to formulations which do not share the computational advantages of the original FFT-based schemes. The work at hand introduces a technique for imposing Dirichlet boundary conditions directly without the need for indefinite systems. We use a formulation on the deformation gradient—also at small strains—and employ the Green’s operator associated to the vector Laplacian. Then, we develop the Moulinec–Suquet discretization for Dirichlet boundary conditions—requiring carefully selected weights at boundary points—and discuss the seamless integration into existing FFT-based computational homogenization codes based on dedicated discrete sine/cosine transforms. The article culminates with a series of well-chosen numerical examples demonstrating the capabilities of the introduced technology.

High-frequency homogenization of multidimensional hyperbolic equations

In L 2 ( R d ) , we consider an elliptic differential operator A ϵ , ϵ > 0 , of the form A ϵ = − div ⁡ g ( x / ϵ ) ∇ + ϵ − 2 V ( x / ϵ ) with periodic coefficients. For hyperbolic equations with the operator A ϵ , analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator A 1 are studied (the so-called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in L 2 ( R d ) -norm for small ε are obtained.

The long wavelength limit of periodic solutions of water wave models

AbstractThe present essay is concerned with providing rigorous justification of a long‐standing practice in numerical simulation of partial differential equations. Theory often sets initial‐value problems on all of or . If the initial data are localized in space, it has been usual practice to approximate the problem by an associated periodic problem or a homogeneous Dirichlet problem set on a finite interval. While these strategies are commonplace, rigorous justification of the practice is sparse. It is our purpose here to indicate justification of this practice in the concrete context of a surface water wave model. While the theory worked out here is specific to the particular partial differential equation, it will be apparent to the reader that more general results may be derived using the same approach.

Kinetics Parameters Estimation for the Coupled Neutron-Photon Point Kinetics Model in Monte Carlo Eigenvalue Calculations

The delayed photoneutron effect should be considered in the point kinetics equation (PKE) applied to D2O and beryllium-moderated or -reflected systems because of its importance on reactor kinetics. This paper proposes neutron-photon-coupled PKE with a clear definition of adjoint-weighted kinetics parameters after providing the mathematical derivation from the neutron and photon transport and precursor equations. The Monte Carlo algorithms to estimate the kinetics parameters are induced from their definition and implemented in the eigenvalue calculation module of McCARD. They are verified through homogeneous infinite-medium problems with few-group neutron and photon cross sections by comparing with analytic solutions. The proposed PKE and kinetics parameters estimation methods are applied for the kinetics analyses of the artificial control rod drop transient problem in the RB reactor. Photonuclear physics is also implemented in a time-dependent Monte Carlo simulation to calculate the reference data of neutron amplitude in the core to be compared with the PKE results. As good agreement is shown in the neutron amplitude, it is validated that the developed PKE is capable of predicting the reactor kinetics considering photoneutrons.

A micromechanical scheme with nonlinear concentration functions by physics-guided neural network

The mechanical behavior of heterogeneous materials has been reported to be significantly influenced by the nonlinear properties of both the matrix and interface. However, the micromechanical homogenization methods for predicting the effective properties are challenged in nonlinear problems due to the difficulties in solving the analytical form of the concentration tensors. In this study, we develop a nonlinear micromechanical scheme for heterogeneous materials with complex interfacial behaviors, where the key component, namely nonlinear concentration functions, is determined by the devised physics-guided neural network. In particular, the nonlinear Mori–Tanaka method (NMT) implemented within this new micromechanical scheme yields accurate solutions to axisymmetric nonlinear homogenization problems considering the effects of finite deformation, loading conditions, volume fraction, etc. Furthermore, the NMT is equivalent to the linear Mori–Tanaka method in the condition of the small deformation. Notably, this neural-network-based micromechanical scheme shows good generalization for different types of nonlinear interfaces, while the corresponding approach for generating training data via the finite element method (FEM) is cost-effective. This theoretical framework introduces a novel approach to nonlinear physical modeling, namely, not by the direct regression from the dataset but by deeply embedding neural networks in physical laws.

Fundamental solutions and critical Lane-Emden exponents for nonlinear integral operators in cones

In this article we study the fundamental solutions or “α-harmonic functions” for some nonlinear positive homogeneous nonlocal elliptic problems in conical domains, such asF(u)=0inCω,u=0inRN∖Cω, where ω is a proper C2 domain in SN−1 for N≥2, Cω:={x:x≠0,|x|−1x∈ω} is the cone-like domain related to ω, and F is an extremal fully nonlinear integral operator. We prove the existence of two fundamental solutions that are homogeneous and do not change signs in the cone; one is bounded at the origin and the other at infinity.As an application, we use the fundamental solutions obtained to prove Liouville type theorems in cones for supersolutions of the Lane-Emden-Fowler equation in the formF(u)+up=0inCω,u=0inRN∖Cω. We also prove a generalized Hopf type lemma in domains with corners. Most of our results are new even when F is the fractional Laplacian operator.

Challenges and Coping Strategies Brought by Digital Media to the Development of B&B Industry under the Perspective of Field Theory

Based on Bourdieu's field theory, the article discusses the challenges and coping strategies of digital media in the development of the B&B industry. In the Internet field, the power struggle between B&B operators and consumers becomes the main focus. By utilizing the Internet platform, B&B operators compete to increase brand awareness and influence and attract consumers' attention. Meanwhile, consumers also influence B&B brands through the Internet field, forming a complex, competitive network. In the competition for symbolic capital, B&Bs gain social recognition through the Netflix effect and celebrity recommendations to further enhance their image and competitiveness, and through self-media accounts and matrix content marketing, B&B content is widely disseminated to enhance their exposure in the Internet arena. However, the B&B industry is also facing the problem of homogenization and needs to maintain its personalized characteristics in competition. Combining local characteristics and innovative models, B&B operators can achieve sustainable development and promote rural revitalization.

Optimal convergence rate for homogenization of convex Hamilton–Jacobi equations in the periodic spatial-temporal environment

We study the optimal convergence rate for the homogenization problem of convex Hamilton–Jacobi equations when the Hamitonian is periodic with respect to spatial and time variables, and notably time-dependent. We prove a result similar to that of (Tran and Yu (2021)), which means the optimal convergence rate is also O ( ε ).

The free boundary for a semilinear non-homogeneous Bernoulli problem

In the classical homogeneous one-phase Bernoulli-type problem, the free boundary consists of a “regular” part and a “singular” part, as Alt and Caffarelli have shown in their pioneer work (Alt and Caffarelli, 1981 [1]) that regular points are C1,γ in two-dimensions. Later, Weiss (1999) [34] first realized that in higher dimensions a critical dimension d⁎ exists so that the singularities of the free boundary can only occur when d⩾d⁎.In this paper, we consider a non-homogeneous semilinear one-phase Bernoulli-type problem, and we show that the free boundary is a disjoint union of a regular and a singular set. Moreover, the regular set is locally the graph of a C1,γ function for some γ∈(0,1). In addition, there exists a critical dimension d⁎ so that the singular set is empty if d<d⁎, discrete if d=d⁎ and of locally finite Hd−d⁎ Hausdorff measure if d>d⁎. As a byproduct, we relate the existence of viscosity solutions of a non-homogeneous problem to the Weiss-boundary adjusted energy, which provides an alternative proof to existence of viscosity solutions for non-homogeneous problems.

Regularity results for an anisotropic nonlinear Dirichlet problem

In this paper we consider the homogeneous Dirichlet problem associated to the model equation − div ⁡ ( a ( x ) | ∇u | p − 2 ∇u ) − div ⁡ ( | u | ( r − 1 ) q + 1 | ∇u | q − 2 ∇u ) = f in Ω , where Ω is a bounded open subset of 2, $ ]]> R N , N > 2 , 1<q<p<N, \\frac {1}{q'} $ ]]> r > 1 q ′ and f belongs to 1 $ ]]> L m ( Ω ) , m > 1 . We give some regularity results according to the values of the exponents ( r − 1 ) q + 1 and m in suitable ranges depending on N , p and q .

A Novel Meshfree Method for Nonlinear Equations in Flow through Porous Media and Electrohydrodynamic Flows

In this study, an efficient meshfree numerical method is introduced for solving the nonlinear boundary value problems. The method of fundamental solutions (MFS) is one of the most popular among meshfree methods. While traditionally limited to linear and homogeneous problems, this study extends the applicability of the MFS to include nonhomogeneous and nonlinear equations. To achieve this, an extended MFS is combined with a fixed-point iteration scheme. This developed framework is benchmarked to address two different flow problems. The first involves fluid flow through porous media in a channel governed by the nonlinear Brinkman-Forchheimer equation. The second problem pertains to electrohydrodynamic (EHD) flow in a circular conduit. The obtained solutions are compared with the finite element method and the solutions available in the existing literature.

A Multistage Nonlinear Method for Aeroengine Health Parameter Estimation Based on Adjacent Operating Points

Abstract Health parameter estimation is the core of engine gas path analysis (GPA), which is widely adopted for engine safety improvement, as well as for operation and maintenance cost reduction. The major challenge of GPA lies in the contradiction between the high dimensions of parameters under estimation, e.g., health parameters, and the limited measurements obtainable from a small number of sensors. Existent GPA methods for health parameters commonly apply dimension reduction before estimation, leading to information loss and hence inaccurate estimation. To tackle the challenge of limited sensor measurements and to have more system outputs than parameters under estimation, we proposed to augment the output vector of the system model by combining the measurements from multiple adjacent operating points. But the engine model can face the problem of homogenization if using data from adjacent operating points. This can, in turn, lead to a low identifiability of parameters. We analyze the internal mechanism of such large deviation of the parameter estimation results based on linear models and argue for the need of nonlinear method. Hence, we propose a multistage nonlinear parameter estimation method for health parameters, combining biased and unbiased estimation. In our extensive simulations based on 10 output measurements of a JT9D engine, our method can estimate 130% more parameters than the widely used GPA method, while reducing the maximum estimation error of health parameters from 2.2% to 0.1%.

Inverse design of polymorphic reconfigurable metamaterial absorbers based on a dual-input neural network

In recent years, deep learning-based design methods for metamaterial absorbers have attracted much attention; however, the problem of structural homogeneity in inverse design constrains their further development. This paper, proposes a metamaterial absorber composed of the phase change material Ge2Sb2Se4Te1 and titanium. To give the metamaterial absorber a richer structure, we divide its Ge2Sb2Se4Te1 layer and top titanium layer into 36 small squares. In a dual-input neural network-based inverse design, this means that metamaterial absorbers with more types of absorption characteristics can be designed. We utilize this approach to design a reconfigurable metamaterial absorber that exhibits a large absorption bandwidth when the Ge2Sb2Se4Te1 layer is in both the crystalline and amorphous. This absorption bandwidth covers the range of solar wavelengths available to humans. Compared with previous research methods, our method eliminates the step of finding the optimal structure. In addition, we have designed metamaterial absorbers with structural diversity and reconfigurability.

Properties of the solutions to periodic conformable non-autonomous non-instantaneous impulsive differential equations

In this article, we study properties of the solutions to periodic non-autonomous conformable non-instantaneous impulsive differential equations. We use a conformable Cauchy matrix and obtain some basic properties of the periodic solution to the homogeneous and non=homogeneous problems. We consider the periodicity of solutions to nonlinear problem via a fixed theorem. For more information see https://ejde.math.txstate.edu/Volumes/2024/30/abstr.html