Enabling heterogeneous catalysis to achieve carbon neutrality: Directional catalytic conversion of CO_(2) into carboxylic acids

The increase in anthropogenic carbon dioxide(CO_(2))emissions has exacerbated the deterioration of the global environment,which should be controlled to achieve carbon neutrality.Central to the core goal of achieving carbon neutrality is the utilization of CO_(2) under economic and sustainable conditions.Recently,the strong need for carbon neutrality has led to a proliferation of studies on the direct conversion of CO_(2) into carboxylic acids,which can effectively alleviate CO_(2) emissions and create high-value chemicals.The purpose of this review is to present the application prospects of carboxylic acids and the basic principles of CO_(2) conversion into carboxylic acids through photo-,electric-,and thermal catalysis.Special attention is focused on the regulation strategy of the activity of abundant catalysts at the molecular level,inspiring the preparation of high-performance catalysts.In addition,theoretical calculations,advanced technologies,and numerous typical examples are introduced to elaborate on the corresponding process and influencing factors of catalytic activity.Finally,challenges and prospects are provided for the future development of this field.It is hoped that this review will contribute to a deeper understanding of the conversion of CO_(2) into carboxylic acids and inspire more innovative breakthroughs.

Synergy of heterogeneous Co/Ni dual atoms enabling selective C-O bond scission of lignin coupling with in-situ N-functionalization

Selective cleavage of Csp^(2)-OCH_(3)bond in lignin without breaking other types of C-O bonds followed by N-functionalization is fascinating for on-purpose valorization of biomass.Here,a Co/Ni-based dual-atom catalyst CoNiDA@NC prepared by in-situ evaporation and acid-etching of metal species from tailor-made metal–organic frameworks was efficient for reductive upgrading of various lignin-derived phenols to cyclohexanols(88.5%–99.9%yields),which had ca.4 times higher reaction rate than the single-atom catalyst and was superior to state-of-the-art heterogeneous catalysts.The synergistic catalysis of Co/Ni dual atoms facilitated both hydrogen dissociation and hydrogenolysis steps,and could optimize adsorption configuration of lignin-derived methoxylated phenols to further favor the Csp^(2)-OCH_(3)cleavage,as elaborated by theoretical calculations.Notably,the CoNi_(DA)@NC catalyst was highly recyclable,and exhibited excellent demethoxylation performance(77.1%yield)in real lignin monomer mixtures.Via in-situ cascade conversion processes assisted by dual-atom catalysis,various high-value N-containing chemicals,including caprolactams and cyclohexylamines,could be produced from lignin.

Multiple Tin Compounds Modified Carbon Fibers to Construct Heterogeneous Interfaces for Corrosion Prevention and Electromagnetic Wave Absorption

Currently,the demand for electromagnetic wave(EMW)absorbing materials with specific functions and capable of withstanding harsh environments is becoming increasingly urgent.Multi-component interface engineering is considered an effective means to achieve high-efficiency EMW absorption.However,interface modulation engineering has not been fully discussed and has great potential in the field of EMW absorption.In this study,multi-component tin compound fiber composites based on carbon fiber(CF)substrate were prepared by electrospinning,hydrothermal synthesis,and high-temperature thermal reduction.By utilizing the different properties of different substances,rich heterogeneous interfaces are constructed.This effectively promotes charge transfer and enhances interfacial polarization and conduction loss.The prepared SnS/SnS_(2)/SnO_(2)/CF composites with abundant heterogeneous interfaces have and exhibit excellent EMW absorption properties at a loading of 50 wt%in epoxy resin.The minimum reflection loss(RL)is−46.74 dB and the maximum effective absorption bandwidth is 5.28 GHz.Moreover,SnS/SnS_(2)/SnO_(2)/CF epoxy composite coatings exhibited long-term corrosion resistance on Q235 steel surfaces.Therefore,this study provides an effective strategy for the design of high-efficiency EMW absorbing materials in complex and harsh environments.

Development of a convolutional neural network based geomechanical upscaling technique for heterogeneous geological reservoir

Geomechanical assessment using coupled reservoir-geomechanical simulation is becoming increasingly important for analyzing the potential geomechanical risks in subsurface geological developments.However,a robust and efficient geomechanical upscaling technique for heterogeneous geological reservoirs is lacking to advance the applications of three-dimensional(3D)reservoir-scale geomechanical simulation considering detailed geological heterogeneities.Here,we develop convolutional neural network(CNN)proxies that reproduce the anisotropic nonlinear geomechanical response caused by lithological heterogeneity,and compute upscaled geomechanical properties from CNN proxies.The CNN proxies are trained using a large dataset of randomly generated spatially correlated sand-shale realizations as inputs and simulation results of their macroscopic geomechanical response as outputs.The trained CNN models can provide the upscaled shear strength(R^(2)>0.949),stress-strain behavior(R^(2)>0.925),and volumetric strain changes(R^(2)>0.958)that highly agree with the numerical simulation results while saving over two orders of magnitude of computational time.This is a major advantage in computing the upscaled geomechanical properties directly from geological realizations without the need to perform local numerical simulations to obtain the geomechanical response.The proposed CNN proxybased upscaling technique has the ability to(1)bridge the gap between the fine-scale geocellular models considering geological uncertainties and computationally efficient geomechanical models used to assess the geomechanical risks of large-scale subsurface development,and(2)improve the efficiency of numerical upscaling techniques that rely on local numerical simulations,leading to significantly increased computational time for uncertainty quantification using numerous geological realizations.

Catalytic effect in lithium metal batteries: From heterogeneous catalyst to homogenous catalyst

Lithium metal batteries are regarded as prominent contenders to address the pressing needs owing to the high theoretical capacity.Toward the broader implementation,the primary obstacle lies in the intricate multi-electron,multi-step redox reaction associated with sluggish conversion kinetics,subsequently giving rise to a cascade of parasitic issues.In order to smooth reaction kinetics,catalysts are widely introduced to accelerate reaction rate via modulating the energy barrier.Over past decades,a large amount of research has been devoted to the catalyst design and catalytic mechanism exploration,and thus the great progress in electrochemical performance has been realized.Therefore,it is necessary to make a comprehensive review toward key progress in catalyst design and future development pathway.In this review,the basic mechanism of lithium metal batteries is provided along with corresponding advantages and existing challenges detailly described.The main catalysts employed to accelerate cathode reaction with emphasis on their catalytic mechanism are summarized as well.Finally,the rational design and innovative direction toward efficient catalysts are suggested for future application in metal-sulfur/gas battery and beyond.This review is expected to drive and benefit future research on rational catalyst design with multi-parameter synergistic impacts on the activity and stability of next-generation metal battery,thus opening new avenue for sustainable solution to climate change,energy and environmental issues,and the potential industrial economy.

Silicon-based optoelectronic heterogeneous integration for optical interconnection

The performance of optical interconnection has improved dramatically in recent years.Silicon-based optoelectronic heterogeneous integration is the key enabler to achieve high performance optical interconnection,which not only provides the optical gain which is absent from native Si substrates and enables complete photonic functionalities on chip,but also improves the system performance through advanced heterogeneous integrated packaging.This paper reviews recent progress of silicon-based optoelectronic heterogeneous integration in high performance optical interconnection.The research status,development trend and application of ultra-low loss optical waveguides,high-speed detectors,high-speed modulators,lasers and 2D,2.5D,3D and monolithic integration are focused on.

Intelligent Dynamic Heterogeneous Redundancy Architecture for IoT Systems

The conventional dynamic heterogeneous redundancy(DHR)architecture suffers from the security threats caused by the stability differences and similar vulnerabilities among the executors.To overcome these challenges,we propose an intelligent DHR architecture,which is more feasible by intelligently combining the random distribution based dynamic scheduling algorithm(RD-DS)and information weight and heterogeneity based arbitrament(IWHA)algorithm.In the proposed architecture,the random distribution function and information weight are employed to achieve the optimal selection of executors in the process of RD-DS,which avoids the case that some executors fail to be selected due to their stability difference in the conventional DHR architecture.Then,through introducing the heterogeneity to restrict the information weights in the procedure of the IWHA,the proposed architecture solves the common mode escape issue caused by the existence of multiple identical error output results of similar vulnerabilities.The experimental results characterize that the proposed architecture outperforms in heterogeneity,scheduling times,security,and stability over the conventional DHR architecture under the same conditions.

Blockchain-Enabled Cybersecurity Provision for Scalable Heterogeneous Network:A Comprehensive Survey

Blockchain-enabled cybersecurity system to ensure and strengthen decentralized digital transaction is gradually gaining popularity in the digital era for various areas like finance,transportation,healthcare,education,and supply chain management.Blockchain interactions in the heterogeneous network have fascinated more attention due to the authentication of their digital application exchanges.However,the exponential development of storage space capabilities across the blockchain-based heterogeneous network has become an important issue in preventing blockchain distribution and the extension of blockchain nodes.There is the biggest challenge of data integrity and scalability,including significant computing complexity and inapplicable latency on regional network diversity,operating system diversity,bandwidth diversity,node diversity,etc.,for decision-making of data transactions across blockchain-based heterogeneous networks.Data security and privacy have also become the main concerns across the heterogeneous network to build smart IoT ecosystems.To address these issues,today’s researchers have explored the potential solutions of the capability of heterogeneous network devices to perform data transactions where the system stimulates their integration reliably and securely with blockchain.The key goal of this paper is to conduct a state-of-the-art and comprehensive survey on cybersecurity enhancement using blockchain in the heterogeneous network.This paper proposes a full-fledged taxonomy to identify the main obstacles,research gaps,future research directions,effective solutions,andmost relevant blockchain-enabled cybersecurity systems.In addition,Blockchain based heterogeneous network framework with cybersecurity is proposed in this paper tomeet the goal of maintaining optimal performance data transactions among organizations.Overall,this paper provides an in-depth description based on the critical analysis to overcome the existing work gaps for future research where it presents a potential cybersecurity design with key requirements of blockchain across a heterogeneous network.

Dynamics and synchronization in a memristor-coupled discrete heterogeneous neuron network considering noise

Research on discrete memristor-based neural networks has received much attention.However,current research mainly focuses on memristor–based discrete homogeneous neuron networks,while memristor-coupled discrete heterogeneous neuron networks are rarely reported.In this study,a new four-stable discrete locally active memristor is proposed and its nonvolatile and locally active properties are verified by its power-off plot and DC V–I diagram.Based on two-dimensional(2D)discrete Izhikevich neuron and 2D discrete Chialvo neuron,a heterogeneous discrete neuron network is constructed by using the proposed discrete memristor as a coupling synapse connecting the two heterogeneous neurons.Considering the coupling strength as the control parameter,chaotic firing,periodic firing,and hyperchaotic firing patterns are revealed.In particular,multiple coexisting firing patterns are observed,which are induced by different initial values of the memristor.Phase synchronization between the two heterogeneous neurons is discussed and it is found that they can achieve perfect synchronous at large coupling strength.Furthermore,the effect of Gaussian white noise on synchronization behaviors is also explored.We demonstrate that the presence of noise not only leads to the transition of firing patterns,but also achieves the phase synchronization between two heterogeneous neurons under low coupling strength.

Phase reconfiguration of heterogeneous CoFeS/CoNiS nanoparticles for superior battery-type supercapacitors

Developing advanced battery-type materials with abundant active sites,high conductivity,versatile morphologies,and hierarchically porous structures is crucial for realizing high-quality hybrid supercapacitors.Herein,heterogeneous FeS@NiS is synthesized by cationic Co doping via surface-structure engineering.The density functional theory(DFT)theoretical calculations are firstly performed to predict the advantages of Co dopant by improving the OH^(−)adsorption properties and adjusting electronic structure,benefiting ions/electron transfer.The dynamic surface evolution is further explored which demonstrates that CoFeS@CoNiS could be quickly reconstructed to Ni(Co)Fe_(2)O_(4)during the charging process,while the unstable structure of the amorphous Ni(Co)Fe_(2)O_(4)results in partial conversion to Ni/Co/FeOOH at high potentials,which contributes to the more reactive active site and good structural stability.Thus,the free-standing electrode reveals excellent electrochemical performance with a superior capacity(335.6 mA h g^(−1),2684 F g^(−1))at 3 A g^(−1).Furthermore,the as-fabricated device shows a quality energy density of 78.1 W h kg^(−1)at a power density of 750 W kg^(−1)and excellent cycle life of 92.1%capacitance retention after 5000 cycles.This work offers a facile strategy to construct versatile morphological structures using electrochemical activation and holds promising applications in energy-related fields.

Sulfur vacancies and heterogeneous interfaces promote high performance sodium storage of bimetallic chalcogenide hollow nanospheres

Transition metal sulfides have high theoretical capacities and are considered as potential anode materials for sodium-ion batteries.However,due to low inherent conductivity and significant volume expansion,the electrochemical performance is greatly limited.In this study,a nickel/manganese sulfide material(Ni_(0.96)S_(x)/MnS_(y)-NC)with adjustable sulfur vacancies and heterogeneous hollow spheres was prepared using a simple method.The introduction of a concentration-adjustable sulfur vacancy enables the generation of a heterogeneous interface between bimetallic sulfide and sulfur vacancies.This interface collectively creates an internal electric field,improving the mobility of electrons and ions,increasing the number of electrochemically active sites,and further optimizing the performance of Na~+storage.The direction of electron flow is confirmed by Density functional theory(DFT)calculations.The hollow nano-spherical material provides a buffer for expansion,facilitating rapid transfer kinetics.Our innovative discovery involves the interaction between the ether-based electrolyte and copper foil,leading to the formation of Cu_9S_5,which grafts the active material and copper current collector,reinforcing mechanical supporting.This results in a new heterostructure of Cu_9S_5 with Ni_(0.96)S_(x)/MnS_(y),contributing to the stabilization of structural integrity for long-cycle performance.Therefore,Ni_(0.96)S_(x)/MnS_(y)-NC exhibits excellent electrochemical properties following our modification route.Regarding stability performance,Ni0_(.96)S_(x)/MnS_(y)-NC demonstrates an average decay rate of 0.00944%after 10,000 cycles at an extremely high current density of 10000 mA g^(-1),A full cell with a high capacity of 304.2 mA h g^(-1)was also successfully assembled by using Na_(3)V_(2)(PO_(4))_(3)/C as the cathode.This study explores a novel strategy for interface/vacancy co-modification in the fabrication of high-performance sodium-ion batteries electrode.

Traffic Flow Prediction with Heterogeneous Spatiotemporal Data Based on a Hybrid Deep Learning Model Using Attention-Mechanism

A significant obstacle in intelligent transportation systems(ITS)is the capacity to predict traffic flow.Recent advancements in deep neural networks have enabled the development of models to represent traffic flow accurately.However,accurately predicting traffic flow at the individual road level is extremely difficult due to the complex interplay of spatial and temporal factors.This paper proposes a technique for predicting short-term traffic flow data using an architecture that utilizes convolutional bidirectional long short-term memory(Conv-BiLSTM)with attention mechanisms.Prior studies neglected to include data pertaining to factors such as holidays,weather conditions,and vehicle types,which are interconnected and significantly impact the accuracy of forecast outcomes.In addition,this research incorporates recurring monthly periodic pattern data that significantly enhances the accuracy of forecast outcomes.The experimental findings demonstrate a performance improvement of 21.68%when incorporating the vehicle type feature.

Effects of connected automated vehicle on stability and energy consumption of heterogeneous traffic flow system

With the development of intelligent and interconnected traffic system,a convergence of traffic stream is anticipated in the foreseeable future,where both connected automated vehicle(CAV)and human driven vehicle(HDV)will coexist.In order to examine the effect of CAV on the overall stability and energy consumption of such a heterogeneous traffic system,we first take into account the interrelated perception of distance and speed by CAV to establish a macroscopic dynamic model through utilizing the full velocity difference(FVD)model.Subsequently,adopting the linear stability theory,we propose the linear stability condition for the model through using the small perturbation method,and the validity of the heterogeneous model is verified by comparing with the FVD model.Through nonlinear theoretical analysis,we further derive the KdV-Burgers equation,which captures the propagation characteristics of traffic density waves.Finally,by numerical simulation experiments through utilizing a macroscopic model of heterogeneous traffic flow,the effect of CAV permeability on the stability of density wave in heterogeneous traffic flow and the energy consumption of the traffic system is investigated.Subsequent analysis reveals emergent traffic phenomena.The experimental findings demonstrate that as CAV permeability increases,the ability to dampen the propagation of fluctuations in heterogeneous traffic flow gradually intensifies when giving system perturbation,leading to enhanced stability of the traffic system.Furthermore,higher initial traffic density renders the traffic system more susceptible to congestion,resulting in local clustering effect and stop-and-go traffic phenomenon.Remarkably,the total energy consumption of the heterogeneous traffic system exhibits a gradual decline with CAV permeability increasing.Further evidence has demonstrated the positive influence of CAV on heterogeneous traffic flow.This research contributes to providing theoretical guidance for future CAV applications,aiming to enhance urban road traffic efficiency and alleviate congestion.

Mitigating Lattice Distortion of High‑Voltage LiCoO_(2)via Core‑Shell Structure Induced by Cationic Heterogeneous Co‑Doping for Lithium‑Ion Batteries

Inactive elemental doping is commonly used to improve the structural stability of high-voltage layered transition-metal oxide cathodes.However,the one-step co-doping strategy usually results in small grain size since the low diffusivity ions such as Ti^(4+)will be concentrated on grain boundaries,which hinders the grain growth.In order to synthesize large single-crystal layered oxide cathodes,considering the different diffusivities of different dopant ions,we propose a simple two-step multi-element co-doping strategy to fabricate core–shell structured LiCoO_(2)(CS-LCO).In the current work,the high-diffusivity Al^(3+)/Mg^(2+)ions occupy the core of single-crystal grain while the low diffusivity Ti^(4+)ions enrich the shell layer.The Ti^(4+)-enriched shell layer(~12 nm)with Co/Ti substitution and stronger Ti–O bond gives rise to less oxygen ligand holes.In-situ XRD demonstrates the constrained contraction of c-axis lattice parameter and mitigated structural distortion.Under a high upper cut-off voltage of 4.6 V,the single-crystal CS-LCO maintains a reversible capacity of 159.8 mAh g^(−1)with a good retention of~89%after 300 cycles,and reaches a high specific capacity of 163.8 mAh g^(−1)at 5C.The proposed strategy can be extended to other pairs of low-(Zr^(4+),Ta^(5+),and W6+,etc.)and high-diffusivity cations(Zn^(2+),Ni^(2+),and Fe^(3+),etc.)for rational design of advanced layered oxide core–shell structured cathodes for lithium-ion batteries.

Distributed Matching Theory-Based Task Re-Allocating for Heterogeneous Multi-UAV Edge Computing

Many efforts have been devoted to efficient task scheduling in Multi-Unmanned Aerial Vehicle(UAV)edge computing.However,the heterogeneity of UAV computation resource,and the task re-allocating between UAVs have not been fully considered yet.Moreover,most existing works neglect the fact that a task can only be executed on the UAV equipped with its desired service function(SF).In this backdrop,this paper formulates the task scheduling problem as a multi-objective task scheduling problem,which aims at maximizing the task execution success ratio while minimizing the average weighted sum of all tasks’completion time and energy consumption.Optimizing three coupled goals in a realtime manner with the dynamic arrival of tasks hinders us from adopting existing methods,like machine learning-based solutions that require a long training time and tremendous pre-knowledge about the task arrival process,or heuristic-based ones that usually incur a long decision-making time.To tackle this problem in a distributed manner,we establish a matching theory framework,in which three conflicting goals are treated as the preferences of tasks,SFs and UAVs.Then,a Distributed Matching Theory-based Re-allocating(DiMaToRe)algorithm is put forward.We formally proved that a stable matching can be achieved by our proposal.Extensive simulation results show that Di Ma To Re algorithm outperforms benchmark algorithms under diverse parameter settings and has good robustness.

Toward Next-Generation Heterogeneous Catalysts:Empowering Surface Reactivity Prediction with Machine Learning

Heterogeneous catalysis remains at the core of various bulk chemical manufacturing and energy conversion processes,and its revolution necessitates the hunt for new materials with ideal catalytic activities and economic feasibility.Computational high-throughput screening presents a viable solution to this challenge,as machine learning(ML)has demonstrated its great potential in accelerating such processes by providing satisfactory estimations of surface reactivity with relatively low-cost information.This review focuses on recent progress in applying ML in adsorption energy prediction,which predominantly quantifies the catalytic potential of a solid catalyst.ML models that leverage inputs from different categories and exhibit various levels of complexity are classified and discussed.At the end of the review,an outlook on the current challenges and future opportunities of ML-assisted catalyst screening is supplied.We believe that this review summarizes major achievements in accelerating catalyst discovery through ML and can inspire researchers to further devise novel strategies to accelerate materials design and,ultimately,reshape the chemical industry and energy landscape.

Heterogeneous information fusion recognition method based on belief rule structure

To solve the problem that the existing situation awareness research focuses on multi-sensor data fusion,but the expert knowledge is not fully utilized,a heterogeneous informa-tion fusion recognition method based on belief rule structure is proposed.By defining the continuous probabilistic hesitation fuzzy linguistic term sets(CPHFLTS)and establishing CPHFLTS distance measure,the belief rule base of the relationship between feature space and category space is constructed through information integration,and the evidence reasoning of the input samples is carried out.The experimental results show that the proposed method can make full use of sensor data and expert knowledge for recognition.Compared with the other methods,the proposed method has a higher correct recognition rate under different noise levels.

Oscillatory Dynamics of Heterogeneous Stem Cell Regeneration

Stem cell regeneration is an essential biological process in the maintenance of tissue homeostasis;dysregulation of stem cell regeneration may result in dynamic diseases that show oscillations in cell numbers.Cell heterogeneity and plasticity are necessary for the dynamic equilibrium of tissue homeostasis;however,how these features may affect the oscillatory dynamics of the stem cell regeneration process remains poorly understood.Here,based on a mathematical model of heterogeneous stem cell regeneration that includes cell heterogeneity and random transition of epigenetic states,we study the conditions to have oscillation solutions through bifurcation analysis and numerical simulations.Our results show various model system dynamics with changes in different parameters associated with kinetic rates,cellular heterogeneity,and plasticity.We show that introducing heterogeneity and plasticity to cells can result in oscillation dynamics,as we have seen in the homogeneous stem cell regeneration system.However,increasing the cell heterogeneity and plasticity intends to maintain tissue homeostasis under certain conditions.The current study is an initiatory investigation of how cell heterogeneity and plasticity may affect stem cell regeneration dynamics,and many questions remain to be further studied both biologically and mathematically.

Channel assignment and power allocation for throughput improvement with PPO in B5G heterogeneous edge networks

In Beyond the Fifth Generation(B5G)heterogeneous edge networks,numerous users are multiplexed on a channel or served on the same frequency resource block,in which case the transmitter applies coding and the receiver uses interference cancellation.Unfortunately,uncoordinated radio resource allocation can reduce system throughput and lead to user inequity,for this reason,in this paper,channel allocation and power allocation problems are formulated to maximize the system sum rate and minimum user achievable rate.Since the construction model is non-convex and the response variables are high-dimensional,a distributed Deep Reinforcement Learning(DRL)framework called distributed Proximal Policy Optimization(PPO)is proposed to allocate or assign resources.Specifically,several simulated agents are trained in a heterogeneous environment to find robust behaviors that perform well in channel assignment and power allocation.Moreover,agents in the collection stage slow down,which hinders the learning of other agents.Therefore,a preemption strategy is further proposed in this paper to optimize the distributed PPO,form DP-PPO and successfully mitigate the straggler problem.The experimental results show that our mechanism named DP-PPO improves the performance over other DRL methods.

Effect of boundary conditions on shakedown analysis of heterogeneous materials

The determination of the ultimate load-bearing capacity of structures made of elastoplastic heterogeneous materials under varying loads is of great importance for engineering analysis and design. Therefore, it is necessary to accurately predict the shakedown domains of these materials. The static shakedown theorem, also known as Melan's theorem, is a fundamental method used to predict the shakedown domains of structures and materials. Within this method, a key aspect lies in the construction and application of an appropriate self-equilibrium stress field(SSF). In the structural shakedown analysis, the SSF is typically constructed by governing equations that satisfy no external force(NEF) boundary conditions. However, we discover that directly applying these governing equations is not suitable for the shakedown analysis of heterogeneous materials. Researchers must consider the requirements imposed by the Hill-Mandel condition for boundary conditions and the physical significance of representative volume elements(RVEs). This paper addresses this issue and demonstrates that the sizes of SSFs vary under different boundary conditions, such as uniform displacement boundary conditions(DBCs), uniform traction boundary conditions(TBCs), and periodic boundary conditions(PBCs). As a result, significant discrepancies arise in the predicted shakedown domain sizes of heterogeneous materials. Built on the demonstrated relationship between SSFs under different boundary conditions, this study explores the conservative relationships among different shakedown domains, and provides proof of the relationship between the elastic limit(EL) factors and the shakedown loading factors under the loading domain of two load vertices. By utilizing numerical examples, we highlight the conservatism present in certain results reported in the existing literature. Among the investigated boundary conditions, the obtained shakedown domain is the most conservative under TBCs.Conversely, utilizing PBCs to construct an SSF for the shakedown analysis leads to less conservative lower bounds, indicating that PBCs should be employed as the preferred boundary conditions for the shakedown analysis of heterogeneous materials.