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Increasing the conductivity of gas diffusion layers (GDLs) is an important way to improve the output performance of polymer electrolyte membrane fuel cells (PEMFCs). However, the complex porous fiber structures of GDLs significantly enhances the difficulty of quantitatively altering their conductivity which is determined by the carbon fibers and the conduction characteristics between fibers. In addition, the microstructures of various types of GDLs are different. Thus, it is a considerable challenge to explore the conductive mechanisms of these porous materials and optimize their structures to reduce their bulk resistances. In this work, a mathematical graph theory model that applies to the through-plane (T-P) bulk resistance prediction of two types of commonly used GDLs, carbon paper and carbon felt, is established to explain their different micro conduction mechanisms in depth. In addition to the number of fiber contact points, their distribution, as well as the resistance of the carbon fibers, are all important factors affecting the T-P conductivity. Optimizing fiber density and fiber diameter can significantly improve the T-P conductivity of carbon paper. In comparison, making the structure of carbon felt more compact so that the distribution of its contact points in the T-P direction can be more uniform will be more effective for the reduction of its T-P bulk resistance. Meanwhile, the T-P bulk resistance of carbon paper can also be effectively improved by optimizing the content and distribution of the binders. A method to decline the bulk resistance of carbon paper by aggregating the binders in the in-plane (I P) direction is proposed. The simulation results show that it can reduce the T-P bulk resistance of carbon paper by about 19.9% at a compressive stress of 1.5 MPa. This study provides further guidance for optimizing the structural designs of GDLs to optimize their conduction performance. • T-P bulk resistance prediction model suitable for different GDLs is constructed. • Micro conductivity mechanisms of carbon paper and carbon felt are compared. • Increasing fiber density and diameter improves carbon paper's T-P conductivity. • Making carbon felt's structure more compact can optimize its T-P conductivity. • To promote carbon paper's T-P conductivity by optimizing the binders inside. [ABSTRACT FROM AUTHOR]

Assembly error, misalignment of components, is inevitable in the fabrication of proton exchange membrane (PEM) fuel cells. It will induce complex and contradictory effects, such as the higher contact resistance and better gas permeability, making it difficult to evaluate the error tolerance. These coupled impact on the performance of fuel cell needs to be clarified. This study establishes a performance prediction model considering the contact resistance between bipolar plate (BPP) and gas diffusion layer (GDL), deformation of membrane electrode assembly (MEA), and distribution of porosity and permeability in GDL to analyze the effect of assembly error. Assembly error brings better performance due to the higher porosity, permeability, and a more uniform distribution of oxygen. However, contact resistance also increases. Cases with narrow channels, thinner GDL, and lower compression ratios are more susceptible to assembly error, mainly manifested in the higher contact resistance, lower porosity and permeability growth, and less oxygen distribution improvement. Cases with wider channels, thicker GDL, and higher compression ratios have a higher tolerance for assembly error. Considering that contact resistance has a greater impact in most cases, assembly error needs to be avoided. The tolerance of assembly error is proposed to guide the fuel cell fabrication. • A method for analyzing the effects of assembly error on performance is proposed. • Results of the assembly model are input into the performance prediction model. • MEA deformation, the distribution of GDL porosity and permeability are considered. • Effects of channel width, GDL thickness, and compression ratio are analyzed. • Tolerance of assembly error under different conditions is proposed. [ABSTRACT FROM AUTHOR]

Air-cooled proton exchange membrane fuel cell (PEMFC) is a promising electrochemical device in fields of unmanned aerial and light-duty ground vehicles， with the virtues of simple system, low parasitic power and low cost. However, the significantly uneven distribution of internal reaction state of air-cooled PEMFC greatly limits the output performance and stability of the cells. In this study, a PCB test board for the in-situ observation of cell temperature and current density is designed and applied in fuel cells. The result shows that when the average current density is 500 mA/cm2，the difference of temperature and current density reach 20 °C and 400 mA/cm2, respectively. For the optimization of uniformity of reaction state, the effects of hydrogen pulsing interval, hydrogen supply mode, fan configuration and air inlet speed are experimentally explored. Based on the observed results of the PCB test board, optimization proposals of control strategy for air-cooled PEMFC are given: 1) A 30-s pulsing interval of hydrogen is preferred due to its little changes on current density distribution while significant increasement of hydrogen utilization rate. 2) The proposed bidirectional hydrogen flow improves the reaction uniformity, and reduces the fluctuation of current density during hydrogen pulsing discharge; 3) Compared to two paralleled fans, employing a single fan helps to improve the temperature uniformity, with the standard deviation of temperature from 8.9 °C to 6.7 °C; 4) Medium velocity of air inlet speed is preferred due to poor temperature uniformity at low velocity, and low water content at high velocity. • A PCB test board is designed to obtain the internal reaction state of fuel cell. • The uneven distributed temperature and current density in fuel cell is analyzed. • Optimization proposals of control strategy of air-cooled PEMFC are raised. [ABSTRACT FROM AUTHOR]

Abstract It is necessary to obtain internal heat and water distribution of proton exchange membrane fuel cells in order to perform water-heat management. Most of the existing works focus on the measurement in small experimental fuel cells, which cannot effectively guide the development of large-area fuel cells. In this study, an in-situ measurement method using micro-sensors is developed to observe the temperature and relative humidity in a commercial-size fuel cell with parallel flow fields and an active area of 250 cm2. A sensor array is incorporated into the cathode flow field, and well-designed waterproof and sealing structures protect sensors from liquid water. The sensors are verified with accuracy and response speed. The fuel cell performance and electrochemical impedance spectroscopy under co-flow and counter-flow configurations considering anode gas humidification are investigated. The results show that the temperature distribution is more uneven in the cases with lower output voltage, and the fuel cell performance is significantly affected by the humidity near the air inlet area. As current density increases, the relative humidity drops while dew point temperature keeps almost unchanged. The experimental method and analysis are beneficial to the understanding of the water-heat state and optimization of fuel cell stack design. Highlights • Measurement with micro-sensors for the commercial-size fuel cell is effective. • Uneven temperature and humidity distribution in the cathode flow field is obtained. • Performance and resistance are greatly related to temperature-humidity distribution. • Counter-flow without humidification gets good performance and water-heat balance. [ABSTRACT FROM AUTHOR]

Highlights • Gas diffusion layer accounts for as much as 20% of the resistance in fuel cells. • Parallel fiber microstructure leads to lower electrical resistance. • Resistance of carbon paper and carbon cloth declines as the load cycles increases. • Fiber crack and loss in resistance occur when exceeding "break stress" of 2.0 MPa. • Carbon felt has the highest stability in electric resistance and microstructure. Abstract Electrical resistance accounts for a significant part of the performance loss in proton exchange membrane fuel cells. To the best of the authors' knowledge, this work represents the first direct experimental investigation and comparison of the bulk resistance and microstructure of commercially available gas diffusion layers, carbon paper, carbon cloth and carbon felt, under cyclic and steady loads, which are typical compression conditions in the fuel cell. It was found that with the improvement of contact conductivity between gas diffusion layer and bipolar plate, the bulk resistance of gas diffusion layer accounts for as much as 20% of the resistance in the fuel cell, especially when the assembly pressure is high enough. Experimental results indicate that three kinds of gas diffusion layers show various electrical behaviors under compression due to their different fiber structures. For carbon paper, the resistance displays a gradual decline as the load cycles increases. A reduction in the resistance and obvious fiber cracks are observed when the compression pressure exceeds the "break stress" of 2.0 MPa. For woven carbon cloth, more uniform decline of the resistance is caused by the increasing fiber cracks, which are pulled and bent in the middle of a weave. Although felt gas diffusion layer features the lowest electrical conductivity, its tortuous and thick fibers lead to higher stability in electric resistance and microstructure than bonded carbon paper and woven carbon cloth. This study is helpful for enhancing our understanding of the relationship between electrical resistance and compression loads in the fuel cell with various gas diffusion layers. [ABSTRACT FROM AUTHOR]

Contact behavior between the gas diffusion layer (GDL) and bipolar plate (BPP) is of significant importance for proton exchange membrane fuel cells. Most current studies on contact behavior utilize experiments and finite element modelling and focus on fuel cells with graphite BPPs, which lead to high costs and huge computational requirements. The objective of this work is to build a more effective analytical method for contact behavior in fuel cells and investigate the size effect resulting from configuration alteration of channel and rib (channel/rib). Firstly, a mathematical description of channel/rib geometry is outlined in accordance with the fabrication of metallic BPP. Based on the interface deformation characteristic and Winkler surface model, contact pressure between BPP and GDL is then calculated to predict contact resistance and GDL porosity as evaluative parameters of contact behavior. Then, experiments on BPP fabrication and contact resistance measurement are conducted to validate the model. The measured results demonstrate an obvious dependence on channel/rib size. Feasibility of the model used in graphite fuel cells is also discussed. Finally, size factor is proposed for evaluating the rule of size effect. Significant increase occurs in contact resistance and porosity for higher size factor, in which channel/rib width decrease. [ABSTRACT FROM AUTHOR]

• Conventional springback model works in micro bending using the surface layer model. • The model underestimates the channel height in microforming of parallel channels. • The transverse stress is found to play an important role. • An improved model was developed considering the transverse stress. • The shift distance of the neutral plane and channel height are better captured. To satisfy the higher accuracy requirement in the microforming of thin-walled structures, the springback behavior under miniaturizing dimensions need to be accurately predicted and controlled. However, as the feature size approaches the thickness in microforming, many assumptions in conventional springback model for macroforming are not valid making these models can hardly provide effective springback analysis in microforming. In the present work, the applicability of a springback analytical framework of conventional stretch-bending was first explored via bending tests of 0.1 mm-thick SS316L sheets under different holding conditions with a punch radius from 3 to 5 mm which is much greater than the thickness. The analytical springback angle results agree with the experimental ones with the maximum error of 4.2% for different punch radii, grain sizes and stretching forces conditions, indicating the correctness of the model. The model was further verified in the microforming of parallel channels with a fillet radius of 0.15 mm. Finite element simulations were conducted to obtain the stretching loads for each channel. However, a significant underestimation of the channel height after springback for over 15% was found especially for the middle channel. The transverse stress is revealed to be no longer neglectable due to the strong stretching load and the small die radius during the microforming of parallel channels, which substantially contributes to yielding, hardening and thickness reduction in the stretch-bending area. In addition, the uneven stretching also affects the springback angle and channel height, which is caused by the nonuniform plastic deformation of the adjacent channels. By incorporating the effects of the transverse stress and the non-uniform stretching loads, an improved springback model was developed for microforming of parallel channels in this work. The shift distance of the neutral plane is better estimated by the improved model than by the conventional one. The predictive heights by the improved model are also revealed to agree with the experimental ones during the microforming of parallel channels with a maximum error of less than 5%. [Display omitted] [ABSTRACT FROM AUTHOR]

Exposed to severe loading condition in fuel cells, the perfluorinated sulfonic-acid (PFSA) membrane degrades over time, which becomes the primary cause of the increasing transmembrane gas permeation during the service. As a result, the hydrogen crossover rate increases with the damage propagation, leading to loss in power and life. However, the effect of mechanical degradation on gas permeability of membrane cannot be quantitatively estimated due to difficulties in damage observation and modeling. This paper first establishes a numerical technique to predict the gas permeation through membrane with consideration of mechanical degradation. Blister test as well as gas permeation test are conducted for the verification of the numerical model. The model shows great prediction accuracy, based on which the uneven distributed damage and its effect on gas permeability are analyzed. Result shows that the gas permeation rate grows with pressure and mechanical degradation has a remarkable contribution, accounting for about 7.3% of total permeation rate at testing pressure of 130 kPa. Finally, the varied membrane shape and gas permeability, which all accelerate the gas permeation, are further analyzed based on the numerical model. • Accumulated damage before fracture of PFSA membrane is investigated. • Quantitative relationship between mechanical damage and gas permeability is built. • The mechanical damage in PFSA membrane is modeled by Gurson damage model. • Numerical model of gas permeation is built and matches well with experiment. [ABSTRACT FROM AUTHOR]

Electrical contact resistance (ECR) at the interface is of significant importance in many fields of science and engineering. Current methods for contact resistance estimation are based on the typical nearly incompressible rough surfaces, which is not suitable for porous material with large deformation in the compression process. The objective of this work is to build an analytical model for ECR between solid material and porous material, for example, which could be used to predict power loss between carbon fiber paper and bipolar plate in the fuel cell. First, mathematical description of solid roughness surface is built by classic Greenwood and Williamson model. Considering the porous structure, carbon fiber paper is modeled by multi-layer construction based on random line network model. Effect of large compression of carbon paper on contact behavior is furtherly given necessary attention in this study. Contact pressure and resistance are calculated based on statistical methods with consideration of multi-deformation states. Then, experiments are carried out to validate the numerical model. The results show good agreements with the numerical model. Finally, influences of carbon paper compression and main parameters are systematically discussed based on the numerical model. The model developed will enhance our understanding regarding the relation between contact pressure and contact resistance at the interface for solid material and fiber-structure material. [ABSTRACT FROM AUTHOR]

• A type of gradient wettability cathode CL with three sub-layers is designed. • Gradient CL's performance (895 mW/cm2@0.6 V) is higher than normal (721 mW/cm2@0.6 V). • This CL can adapt high current density, cathode humidity and air stoichiometry. • Gradient CL with best performance is found through comparison of 18 structures. The design of cathode catalyst layer (CL) is essential to improve the mass transfer capacity of proton exchange membrane (PEM) fuel cell and increase power density. In this work, cathode CL is divided into three sub-layers, and each sub-layer is added nano-particles with different wettability. The gradient CL with hydrophilic SiO 2 particles at inner layer and hydrophobic polytetrafluoroethylene (PTFE) particles at outer layer significantly enhances the performance of membrane exchange assembly (MEA). Its performance is 895 mW/cm2@0.6 V, which is 24.1 %higher than CL without any particles (721 mW/cm2@ 0.6 V). Under operating conditions of high current density, high cathode humidity and high air stoichiometry, the gradient CL has only a little voltage loss. Through Electrochemical Impedance Spectroscopies (EIS) impedance analysis under high current density (1.8A/cm2), mass transfer resistance of gradient CL is 25.4 Ω, and is much smaller than the mass transfer resistance of the homogeneous CL of 35.1 Ω, which reflects the significant enhancement in mass transfer capacity of gradient CL. The gradient catalyst layer is suitable for a wider range of current density, humidity, and stoichiometry, but excessive cathode gas stoichiometry causes a decrease in performance, which is caused by excessive drainage capacity. In addition, 18 different gradient CLs are designed and manufactured, and the gradient CL with catalyst coated membrane (CCM) structure has the best performance. In gradient CL, increasing the capillary pressure difference between sublayers is the key to performance improvement. It is confirmed that the property of MEA with appropriate wettability gradient design can be significantly improved. [ABSTRACT FROM AUTHOR]

• Down-the-channel performance model and genetic algorithm are used to optimize flow channel. • The local current density of PEMFC is measured by experiments. • Optimal constant and variable optimal flow channel designs are analyzed. • Optimal variable flow channel designs benefit output performance, especially at low output voltages and large SRCWs. It is necessary to establish an effective method to guide the flow channel design of the bipolar plate (BPP) to obtain the optimal performance of the proton exchange membrane fuel cells (PEMFC). Most of the existing works focused on the optimization of constant channel dimensions and structures without considering achieving the flow channel's optimal local performance. In this study, an approach combining the down-the-channel performance model and genetic algorithm (GA) is developed to optimize BPP channel/rib patterns for fuel cells. A flow channel is divided into several segments, and the channel dimensions of each segment are brought into the down-the-channel performance model as variables to obtain the optimal parameter design by GA. The model and the optimization results are validated by the CFD and local current density experiment. The performance results of both CFD and local current density experiments are in good agreement with the results of the down-the-channel performance model. It is found that the cell performance of variable rib to channel width ratio (RCWR) design along the flow channel is better than the performance of constant RCWR design. Performance improvements are more significant when the sum of rib and channel width (SRCW) is higher, and the output voltage is lower. Smaller SRCW design can effectively reduce the ohmic losses and concentration polarization which is beneficial to cell performance, and as the SRCW is reduced, the performance improvement of variable RCWR design along the flow channel is less necessary. A guide for the design of the cathode flow channel RCWR from upstream to downstream is provided. [Display omitted] [ABSTRACT FROM AUTHOR]

• Methodology using composited equivalent is built to evaluate stack mechanical state. • The cells approaching to mid-stack have more uniform pressure. • Numerical results show good agreements with experimental results. • Nonlinear compression behavior of MEA is applied in high-power PEM fuel cell stack. • Equivalent model has higher computational efficiency than traditional finite element model. A high-power proton exchange membrane fuel cell (PEMFC) stack usually composes many cells, which induce high difficulty in evaluating its mechanical state of stack assembly. A methodology based on composite model and material property equivalent is developed to predict the mechanical state in PEMFC stack. In this methodology, the stack system is modeled based on a finite element model (FEM), in which bipolar plate (BPP) and membrane exchange assembly (MEA) are combined into a composite component. Experiments with stamped BPP are carried out to validate the FEM of the stack, and both the predicted clamping force and endplates deformation of FEM have a great agreement with the experimental results. Based on the methodology, it is found that more uniform pressure distribution can be generated when high-stiffness endplates are applied and cell number of the stack increases. The cells approaching to mid-stack have more uniform pressure. The pressure distribution of the stack is very sensitive to the compression ratio. High compression ratios leads to large endplate deformation, which increases the average pressure deviation between simulated value and design value, and also increases non-uniformity of pressure distribution. This methodology offers the possibility of evaluating the mechanical state of high-power fuel cell stack and greatly improves the computational efficiency. [ABSTRACT FROM AUTHOR]

[Display omitted] • In-situ monitoring technology for the fuel cells is proposed. • A method for decoupling the three kinds of voltage losses in fuel cells is built. • The internal states and voltage losses in large area fuel cells are analyzed. • The effect of operating conditions on the voltage losses is investigated. Understanding the local operating state and voltage loss distributions are of great importance to guide the performance improvement of proton exchange membrane fuel cells (PEMFCs). Especially when the active area is large, the uneven electrochemical reaction becomes more severe. However, the current research does not enable the simultaneous monitoring of the multiple operating states and decoupling of the voltage losses in the fuel cell. In this study, a method is established that can obtain the distribution of activation polarization, ohmic loss, and concentration polarization by analyzing the measured data of temperature, humidity, current density, and high-frequency resistance (HFR), which are obtained inside a 320 cm2 and 3 kW level fuel cell stack by using the multilayer printed circuit board (PCB) technology with designed isolation segments. The effects of humidity, operating temperature, and current density on the distributions of output voltage and voltage losses are discussed. Results show that the potential distribution on the surface of membrane exchange assembly (MEA) isn't uniform in the large active area. The activation polarization gradually increases in the direction of coolant flow and the direction of the cathode flow and the ohmic loss and concentration polarization have opposite distribution characteristics. In addition, when the humidity rises to full humidity, the inhomogeneity of ohmic loss distribution becomes higher with three local increased areas of 20 mV and one decreased area of −20 mV. Higher temperatures can significantly reduce the flooding near the outlet of the active area. When the current density increases, the inhomogeneity of the ohmic loss distribution increases, and the concentration polarization near the cathode outlet of the active area become more severe. [ABSTRACT FROM AUTHOR]

• A discrete dislocation dynamic model on electroplasticity was constructed. • The scattering process between electrons and dislocations was captured. • The momentum and the energy transferred during scattering was quantified. • Introducing electricity leads to fewer dislocation cores and better uniformity. • The TEM results of aluminum agree with the DDD simulation remarks. Introducing electric current into metals and alloys to improve their ductility and to reduce the hardening tendency has been widely adopted. Nevertheless, how the electricity affects the plastic deformation of those materials remains a critical issue with controversial opinions. To clarify the material plastic deformation subjected to electric current, or the so-called electroplastic behavior, an in-depth understanding of the dislocation evolution during that process is vital. From that motivation, three-dimensional dislocation dynamics simulations were carried out to explore the dislocation behavior during the uniaxial tensile deformation of pure aluminum with the introduction of electric current. The scattering process between electrons and dislocations was first captured by a physical model. The electron wind force and local heat effect were quantitatively analyzed by figuring out the momentum and the energy transferred during the interaction of electrons and dislocations respectively. The dislocation density evolution, the activation of different slip systems and the dislocation distribution were further analyzed based on the simulations. The dislocation density in the [111] direction is revealed to increase more significantly than [101] and [001] directions with the introduction of electric current. The results show that the current can reduce the flow stress by promoting the activation of the difficult-to-move dislocations. The activation effect reduces the dislocation tangling tendency and leads to more uniform dislocation distribution. Therefore, a reduction of the flow stress can be observed in EAT comparing to TAT, via the discrete dislocation dynamics simulations even though the dislocation densities are similar. The simulation results are also confirmed by the EAT and TAT experiment results and TEM observations. [ABSTRACT FROM AUTHOR]

Metallic bipolar plates (BPPs) are promising candidates for replacing conventional carbon-based BPPs. However, manufacturing errors are unavoidable on metallic BPPs due to the characteristics of fabrication processes. This study is conducted to further develop tolerance design method for dimensional error and assembly error based on the contact behavior model proposed in our previous work. Contact pressure distribution, contact resistance and gas diffusion layer (GDL) porosity are used as the evaluation indexes combined by desirability function method. Based on the contact behavior model, effect of dimensional error is substituted into the model by Monte Carlo Simulation, and misalignment between the upper plate and lower plate is included in the model to obtain the effect of assembly error. It is found that dimensional error and assembly error contribute to uneven pressure distribution, larger contact resistance and higher GDL porosity. The acceptable maximum dimensional error of 0.007, 0.012, 0.010 mm and assembly error of 0.080, 0.10, and 0.060 mm are recommended for the fuel cells with the channel widths of 1.0, 2.0 and 4.0 mm, respectively. This methodology developed is beneficial to the understanding of the effect of dimensional error and assembly error, and effective supplement to fabrication management of metallic BPPs. Image 1 • Tolerance design method for fabrication error of metallic BPPs is introduced. • Response variables are almost in linear rise with increasing the dimensional error. • Assembly error smaller than 0.05 mm contributes little to the response variables. • Dimensional error of 7, 12, 10 μm considered acceptable for channels of 1, 2, 4 mm. • Assembly error of 80, 100, 60 μm considered acceptable for channels of 1, 2, 4 mm. [ABSTRACT FROM AUTHOR]

It is necessary to obtain the current density distribution in order to understand local reactions inside the proton exchange membrane (PEM) fuel cells. Enlarging the active area to commercial-size makes it difficult to maintain uniform assembly pressure, gas supply and heat dissipation inside the fuel cell stack. Existing studies for laboratory-level fuel cells cannot effectively guide the development of large-area fuel cells. In this study, current density distribution inside a 250 cm2 and 3 kW level fuel cell stack is measured and analyzed using the printed circuit board (PCB) technology. A four-layer PCB sensor plate covering 144 current collecting segments and shunt resistors for current measurement is well-designed. The in-house developed measurement system is proved with good function, and the accuracy of current density measurement is within 2%. Results show that the flow field configuration and stack assembly condition greatly affects the uniformity of local current densities. Higher current passes through the area with larger contact pressure, and the current density is low in the poor contact area. The local current density has larger deviations for higher output current, but the normalized current is more uniform. Effects of operating conditions including stoichiometric ratio and flow direction are investigated. • A large-area segmented PCB current sensor plate is designed and tested. • The current distribution of a commercial-size fuel cell stack is investigated. • Nonuniformity contact pressure has a great impact on current density distribution. [ABSTRACT FROM AUTHOR]

Multi-stage forming of the ultra-thin metallic sheets is a promising process for manufacturing fine width and high aspect ratio bipolar plates in proton exchange membrane fuel cell. Correlations between the processing parameters and the channel quality evaluation indicators are investigated experimentally and numerically. A novel parameter optimization method and stage number design guidance are established with the consideration of the indicator priorities and material properties. The forming depth can be increased by larger Stage II punch stroke along with smaller clearance and radius. Increasing or decreasing the Stage II punch stroke and radius simultaneously contributes to larger draft angle. Small Stage II punch stroke and radius are beneficial to greater rib width. More uniform thickness distribution can be achieved by using smaller Stage II punch stroke and clearance. Two-stage forming is enough for soft materials and three-stage forming is suitable for relatively hard materials. Both the output power density and voltage distribution uniformity of the proton exchange membrane fuel cell are improved by employing multi-sage formed bipolar plates. The relationship between the parameters and indicators, optimization method, stage number design guidelines, all serve as the design principles of the micro channel's multi-stage forming for bipolar plates with ultra-thin metallic sheets. Image 1 • High performance BPPs are fabricated by multi-stage forming in ultra-thin sheets. • Influences of processing parameters on quality evaluation indicators are studied. • Novel parameter optimization method is established in a quantitative manner. • Guidance of the stage number design for different material properties is proposed. [ABSTRACT FROM AUTHOR]