A new model,called object model,for the simulation of cold roll-forming of tubes is presented.The model inherits the advantages of old models and is the embodiment of forming process that the strip is rolled step by s...A new model,called object model,for the simulation of cold roll-forming of tubes is presented.The model inherits the advantages of old models and is the embodiment of forming process that the strip is rolled step by step from feed rollers to last rolling pass.The elastic-plastic large deformation spline finite strip method based on updated Lagrangian method has been developed by improving the stiffness and transition matrix.Combined theory formulas and new analytical model,the forming process of a tube has been simulated successfully as an example.The analytical results are submitted and indicate that the proposed simulation method and new model are applicable.展开更多
The experimental results of the thermal conductivities of xonotlite-type calcium silicate insulation materials were presented at different temperatures and pressures. Two appropriative surroundings, i.e. an elevated t...The experimental results of the thermal conductivities of xonotlite-type calcium silicate insulation materials were presented at different temperatures and pressures. Two appropriative surroundings, i.e. an elevated temperature surrounding from ambient temperature to 1450 K and a vacuum surrounding from atmosphere pressure to 10-3 Pa, were designed for the transient hot-strip （THS） method. The thermal conductivities of xonotlite-type calcium silicate with four densities from ambient temperature to 1000 K and 0.045 Pa to atmospheric pressure were measured. The results show that the thermal conductivity of xonotlite-type calcium silicate decreases apparently with the fall of density, and decreases apparently with the drop of pressure, and reaches the least value at about 100 Pa. The thermal conductivity of xonotlite-type calcium silicate increases almost linearly with T0, and increases more abundantly with low density than with high density. The thermal conductivity measurement uncertainty is estimated to be approximately 3% at ambient temperature, and 6% at 800 K.展开更多
Dynamic response of beam-like structures to moving vehicles has been extensively studied. However, the study on dynamic response of plates to moving vehicles has so far received but scant attention. A plate-vehicle st...Dynamic response of beam-like structures to moving vehicles has been extensively studied. However, the study on dynamic response of plates to moving vehicles has so far received but scant attention. A plate-vehicle strip for simulating the interaction between a rectangular plate and moving vehicles was described. For the portion of strips that are in direct contact with the moving vehicles, the plate-vehicle strips were employed. Conventional plate finite strips were used to model the portion of strips that are not directly under the action of moving vehicles. In the analysis, each moving vehicle is idealized as a one-foot dynamic system with tire unsprung mass and sprund mass interconnected by a spring and a dashpot. The numerical results obtained from the proposed method agree well with available results.展开更多
Thermal conductivity is an important material parameter of silicon when studying the performance and reliability of devices or for guiding circuit design when considering heat dissipation, especially when the self-hea...Thermal conductivity is an important material parameter of silicon when studying the performance and reliability of devices or for guiding circuit design when considering heat dissipation, especially when the self-heating effect becomes prominent in ultra-scaled MOSFETs.The cross-plane thermal conductivity of a thin silicon film is lacking due to the difficulty in sensing high thermal conductivity in the vertical direction.In this paper, a feasible method that utilizes an ultra-fast electrical pulse within 20 μs combined with the hot strip technique is adopted.To the best of our knowledge, this is the first work that shows how to extract the cross-plane thermal conductivity of sub-50 nm(30 nm, 17 nm, and 10 nm)silicon films on buried oxide.The ratio of the extracted cross-plane thermal conductivity of the silicon films over the bulk value is only about 6.9%, 4.3%, and 3.8% at 300 K, respectively.As the thickness of the films is smaller than the phonon mean free path, the classical heat transport theory fails to predict the heat dissipation in nanoscale transistors.Thus, in this study, a ballistic model, derived from the heat transport equation based on extended-irreversible-hydrodynamics(EIT), is used for further investigation, and the simulation results exhibit good consistence with the experimental data.The extracted effective thermal data could provide a good reference for precise device simulations and thermoelectric applications.展开更多
The hot strip method,as one typical transient method,is widely used to measure the effective thermal conductivity of thermal insulation materials at various temperatures.Since the test theory is based on solving the e...The hot strip method,as one typical transient method,is widely used to measure the effective thermal conductivity of thermal insulation materials at various temperatures.Since the test theory is based on solving the energy equation via heat conduction,the test result will be questionable when measuring thermal insulation materials,such as silica aerogel and photovoltaic glazing,in which the participating thermal radiation is a dominant heat transfer mode at high temperature.In this study,numerical investigation is employed to reveal the measurement reliability of hot strip method when applied to translucent thermal insulation materials.By reproducing the dynamic conduction-radiation coupled heat transfer process within the translucent materials numerically,the effective thermal conductivity of translucent materials with varying extinction coefficients are obtained at various temperatures.Comparisons are made for the effective thermal conductivity of translucent materials determined by the hot strip method,one-dimensional steady state method,transient plane source method and Rosseland model.Large discrepancies are found among the effective thermal conductivity determined by different methods for translucent materials with low extinction coefficient.The thermal conductivity obtained from the hot strip method is overestimated at elevated temperature when compared with that from one-dimensional steady state method.In order to measure the effective thermal conductivity of translucent materials accurately,the effect of thermal radiation should be considered for different transient methods.展开更多
A new method,the stream surface strip element method,for simulating the three-dimensional deformation of plate and strip rolling process was proposed.The rolling deformation zone was divided into a number of stream su...A new method,the stream surface strip element method,for simulating the three-dimensional deformation of plate and strip rolling process was proposed.The rolling deformation zone was divided into a number of stream surface(curved surface)strip elements along metal flow traces,and the stream surface strip elements were mapped into the corresponding plane strip elements for analysis and computation.The longitudinal distributions of the lateral displacement and the altitudinal displacement of metal were respectively constructed to be a quartic curve and a quadratic curve,of which the lateral distributions were expressed as the third-power spline function,and the altitudinal distributions were fitted in the quadratic curve.From the flow theory of plastic mechanics,the mathematical models of the three-dimensional deformations and stresses of the deformation zone were constructed.Compared with the streamline strip element method proposed by the first author of this paper,the stream surface strip element method takes into account the uneven distributions of stresses and deformations along altitudinal direction,and realizes the precise three-dimensional analysis and computation.The simulation example of continuous hot rolled strip indicates that the method and the model accord with facts and provide a new reliable engineering-computation method for the three-dimensional mechanics simulation of plate and strip rolling process.展开更多
The calculation precision and convergence speed of streamline strip element method are increased by (using) the method whose initial value of the exit lateral displacement is determined with strip element variation me...The calculation precision and convergence speed of streamline strip element method are increased by (using) the method whose initial value of the exit lateral displacement is determined with strip element variation method, and the accurate tension lateral distribution model is adopted based on the original third power spline function streamline strip element method. The basic theory of the strip element method is developed. The calculated results by the improved streamline strip element method and the original streamline strip element method are compared with the measured results, showing that the calculated results of the improved method are in good agreement with the measured results.展开更多
Double-layered graphene sheets （DLGSs） can be applied to the development of a new generation of nanomechanical sensors due to their unique physical properties. A rectangular DLGS with a nanoparticle randomly located...Double-layered graphene sheets （DLGSs） can be applied to the development of a new generation of nanomechanical sensors due to their unique physical properties. A rectangular DLGS with a nanoparticle randomly located in the upper sheet is modeled as two nonlocal Kirchhoff plates connected by van der Waals forces. The Galerkin strip transfer function method which is a semi-analytical method is developed to compute the natural frequencies of the mass- plate vibrating system. It can give exact closed-form solutions along the longitudinal direction of the strip. The results obtained from the semi-analytical method are compared with the previous ones, and the differences between the single-layered graphene sheet （SLGS） and the DLGS mass sensors are also investigated. The results demonstrate the similarity of the in-phase mode between the SLGS and DLGS mass sensors. The sensitivity of the DLGS mass sensor can be increased by decreasing the nonlocal parameter, moving the attached nanoparticle closer to the DLGS center and making the DLGS smaller. These conclusions are helpful for the design and application of graphene-sheet-based resonators as nano-mass sensors.展开更多
As the traditional forging process has many problems such as low efficiency, high consumption of material and energy, large cylindrical shell rolling is introduced. Large cylindrical shell rolling is a typical rotary ...As the traditional forging process has many problems such as low efficiency, high consumption of material and energy, large cylindrical shell rolling is introduced. Large cylindrical shell rolling is a typical rotary forming technology, and the upper and lower rolls have different radii and speeds. To quickly predict the three-dimensional stresses and eliminate fishtail defect, an improved strip layer method is developed, in which the asymmetry of the upper and lower rolls, non-uniform deformation and stress, as well as the asymmetrical spread on the end surface are considered. The deformation zone is divided into a certain number of layers and strips along the thickness and width, respectively. The transverse displacement model is constructed by polynomial function, in order to increase the computation speed greatly. From the metal plastic mechanics principle, the three-dimensional stress models are established. The genetic algorithm is used for optimization calculation in an industrial experiment example. The results show that the rolling pressure, the normal stresses, the upper and lower friction stress distributions are not similar with those of a general plate rolling. There are two relative maximum values in rolling pressure distribution. The upper and lower longitudinal friction stresses change direction nearby the upper and lower neutral points, respectively. The fishtail profile of spread on the end surface is predicted satisfactorily. The reduction could be helpful to eliminate fishtail defect. The large cylindrical shell rolling example illustrates the calculation results acquired rapidly are good agreements with the finite element simulation and experimental values of previous study. A highly effective and reliable three-dimensional simulation method is proposed for large cylindrical shell rolling and other asymmetrical rolling.展开更多
Quantitative level evaluation of cold-rolled strip was significant for the shape control and the product structure optimization. Based on the probability statistics method and the shape pattern recognition method, the...Quantitative level evaluation of cold-rolled strip was significant for the shape control and the product structure optimization. Based on the probability statistics method and the shape pattern recognition method, the shape evaluation model of cold-rolled strip was established to evaluate real-time shape of the online strip. The model was embedded in the shape control system, and the online strip shape state was real-time observed whether to meet the technical indicators, so the shape level could be identified and evaluated. Based on the shape evaluation indicators, the shape component could be well controlled to improve the shape hit rate and the control stability, so that the rolled strip shape could meet the technical requirements. At a 1050 cold strip mill, the shape data of the scene typical specifications strip were online collected and real-time evaluated, and the hit of macro shape in less than 8 I was 68.44%, indicating that the mill shape control had considerable room for improvement, as well as the evaluation method was feasible and practical for quantitative evaluation of the strip shape. So the new evaluation model has important guiding significance to increase value-added and reduce product quality objection.展开更多
In this work, we applied two electromagnetic models for the characterization of a planar structure including a flat, thick copper conductor. Indeed the first model is consisted by modeling two metal ribbons without bu...In this work, we applied two electromagnetic models for the characterization of a planar structure including a flat, thick copper conductor. Indeed the first model is consisted by modeling two metal ribbons without bulkiness, placed one above the other at a distance of h<sub>2</sub> equal to the thickness of the thick conductor. This approach has been implemented and tested by the iterative method. The results of simulations have been compared with those calculated by the Ansoft HFSS software, and they are in good concordance, validating the method of analysis used. The second model is based on the calculation of the effective permittivity of the medium containing the thick conductor. This medium consists of a metallic region of complex relative permittivity , the rest of this medium is filled with air e<sub>r</sub><sub>2</sub> = 1. The effective permittivity e<sub>eff</sub> calculated from these two relative permittivity e<sub>r</sub><sub>2</sub> and . Comparing the simulation results of this new formulation of the iterative method with those calculated by the software Ansoft HFSS shows that they are in good matching which validates the second model.展开更多
Hot rolled strips usually have higher strength and lower plasticity at the ends, and the mechanical properties are distributed unevenly along the length direction. Such phenomena are caused by the different cooling ra...Hot rolled strips usually have higher strength and lower plasticity at the ends, and the mechanical properties are distributed unevenly along the length direction. Such phenomena are caused by the different cooling rates between the end and the center. The ends of the coiled strip cool down faster than the center, inducing finer grains in the sections. Furthermore, the center of the coil is kept at high temperature for longer time, which affects the precipitation of the carbides and creates the different mechanical properties from the ends. In this paper, the temperature field of the strip during cooling was simulated to discover the characteristics of the temperature change and the effect on mechanical properties. Based on the analysis, a concept of concave cooling control was introduced and implemented in the production. Results indicated that applying the concave cooling control method could significantly improve the uniformity of the properties and promote the quality of the products.展开更多
This paper formulates a two-dimensional strip packing problem as a non- linear programming （NLP） problem and establishes the first-order optimality conditions for the NLP problem. A numerical algorithm for solving t...This paper formulates a two-dimensional strip packing problem as a non- linear programming （NLP） problem and establishes the first-order optimality conditions for the NLP problem. A numerical algorithm for solving this NLP problem is given to find exact solutions to strip-packing problems involving up to 10 items. Approximate solutions can be found for big-sized problems by decomposing the set of items into small-sized blocks of which each block adopts the proposed numerical algorithm. Numerical results show that the approximate solutions to big-sized problems obtained by this method are superior to those by NFDH, FFDH and BFDH approaches.展开更多
基金the National Natural Science Foundation of China (No. 50375135)the Talent Foundation of Beijing Jiaotong University (No. 2003RC059)
文摘A new model,called object model,for the simulation of cold roll-forming of tubes is presented.The model inherits the advantages of old models and is the embodiment of forming process that the strip is rolled step by step from feed rollers to last rolling pass.The elastic-plastic large deformation spline finite strip method based on updated Lagrangian method has been developed by improving the stiffness and transition matrix.Combined theory formulas and new analytical model,the forming process of a tube has been simulated successfully as an example.The analytical results are submitted and indicate that the proposed simulation method and new model are applicable.
基金supported by the National Natural Science Foundation of China (No.50806021)
文摘The experimental results of the thermal conductivities of xonotlite-type calcium silicate insulation materials were presented at different temperatures and pressures. Two appropriative surroundings, i.e. an elevated temperature surrounding from ambient temperature to 1450 K and a vacuum surrounding from atmosphere pressure to 10-3 Pa, were designed for the transient hot-strip （THS） method. The thermal conductivities of xonotlite-type calcium silicate with four densities from ambient temperature to 1000 K and 0.045 Pa to atmospheric pressure were measured. The results show that the thermal conductivity of xonotlite-type calcium silicate decreases apparently with the fall of density, and decreases apparently with the drop of pressure, and reaches the least value at about 100 Pa. The thermal conductivity of xonotlite-type calcium silicate increases almost linearly with T0, and increases more abundantly with low density than with high density. The thermal conductivity measurement uncertainty is estimated to be approximately 3% at ambient temperature, and 6% at 800 K.
文摘Dynamic response of beam-like structures to moving vehicles has been extensively studied. However, the study on dynamic response of plates to moving vehicles has so far received but scant attention. A plate-vehicle strip for simulating the interaction between a rectangular plate and moving vehicles was described. For the portion of strips that are in direct contact with the moving vehicles, the plate-vehicle strips were employed. Conventional plate finite strips were used to model the portion of strips that are not directly under the action of moving vehicles. In the analysis, each moving vehicle is idealized as a one-foot dynamic system with tire unsprung mass and sprund mass interconnected by a spring and a dashpot. The numerical results obtained from the proposed method agree well with available results.
基金Project supported by the Zhejiang Provincial Natural Science Foundation of China(Grant No.LZ19F040001)the National Natural Science Foundation of China(Grant No.61473287)the NSFC–Zhejiang Joint Fund for the Integration of Industrialization Informatization,China(Grant No.U1609213)
文摘Thermal conductivity is an important material parameter of silicon when studying the performance and reliability of devices or for guiding circuit design when considering heat dissipation, especially when the self-heating effect becomes prominent in ultra-scaled MOSFETs.The cross-plane thermal conductivity of a thin silicon film is lacking due to the difficulty in sensing high thermal conductivity in the vertical direction.In this paper, a feasible method that utilizes an ultra-fast electrical pulse within 20 μs combined with the hot strip technique is adopted.To the best of our knowledge, this is the first work that shows how to extract the cross-plane thermal conductivity of sub-50 nm(30 nm, 17 nm, and 10 nm)silicon films on buried oxide.The ratio of the extracted cross-plane thermal conductivity of the silicon films over the bulk value is only about 6.9%, 4.3%, and 3.8% at 300 K, respectively.As the thickness of the films is smaller than the phonon mean free path, the classical heat transport theory fails to predict the heat dissipation in nanoscale transistors.Thus, in this study, a ballistic model, derived from the heat transport equation based on extended-irreversible-hydrodynamics(EIT), is used for further investigation, and the simulation results exhibit good consistence with the experimental data.The extracted effective thermal data could provide a good reference for precise device simulations and thermoelectric applications.
基金supported by the National Natural Science Foundation of China(No.52130604,No.51825604)Innovative Talents Support Plan of China Postdoctoral Foundation(No.BX20180244)。
文摘The hot strip method,as one typical transient method,is widely used to measure the effective thermal conductivity of thermal insulation materials at various temperatures.Since the test theory is based on solving the energy equation via heat conduction,the test result will be questionable when measuring thermal insulation materials,such as silica aerogel and photovoltaic glazing,in which the participating thermal radiation is a dominant heat transfer mode at high temperature.In this study,numerical investigation is employed to reveal the measurement reliability of hot strip method when applied to translucent thermal insulation materials.By reproducing the dynamic conduction-radiation coupled heat transfer process within the translucent materials numerically,the effective thermal conductivity of translucent materials with varying extinction coefficients are obtained at various temperatures.Comparisons are made for the effective thermal conductivity of translucent materials determined by the hot strip method,one-dimensional steady state method,transient plane source method and Rosseland model.Large discrepancies are found among the effective thermal conductivity determined by different methods for translucent materials with low extinction coefficient.The thermal conductivity obtained from the hot strip method is overestimated at elevated temperature when compared with that from one-dimensional steady state method.In order to measure the effective thermal conductivity of translucent materials accurately,the effect of thermal radiation should be considered for different transient methods.
基金Sponsored by National Natural Science Foundation of China(50175095)Provincial Natural Science Foundation of Hebei of China(502173)
文摘A new method,the stream surface strip element method,for simulating the three-dimensional deformation of plate and strip rolling process was proposed.The rolling deformation zone was divided into a number of stream surface(curved surface)strip elements along metal flow traces,and the stream surface strip elements were mapped into the corresponding plane strip elements for analysis and computation.The longitudinal distributions of the lateral displacement and the altitudinal displacement of metal were respectively constructed to be a quartic curve and a quadratic curve,of which the lateral distributions were expressed as the third-power spline function,and the altitudinal distributions were fitted in the quadratic curve.From the flow theory of plastic mechanics,the mathematical models of the three-dimensional deformations and stresses of the deformation zone were constructed.Compared with the streamline strip element method proposed by the first author of this paper,the stream surface strip element method takes into account the uneven distributions of stresses and deformations along altitudinal direction,and realizes the precise three-dimensional analysis and computation.The simulation example of continuous hot rolled strip indicates that the method and the model accord with facts and provide a new reliable engineering-computation method for the three-dimensional mechanics simulation of plate and strip rolling process.
文摘The calculation precision and convergence speed of streamline strip element method are increased by (using) the method whose initial value of the exit lateral displacement is determined with strip element variation method, and the accurate tension lateral distribution model is adopted based on the original third power spline function streamline strip element method. The basic theory of the strip element method is developed. The calculated results by the improved streamline strip element method and the original streamline strip element method are compared with the measured results, showing that the calculated results of the improved method are in good agreement with the measured results.
基金supported by the National Natural Science Foundation of China(Grant No.11302254)
文摘Double-layered graphene sheets （DLGSs） can be applied to the development of a new generation of nanomechanical sensors due to their unique physical properties. A rectangular DLGS with a nanoparticle randomly located in the upper sheet is modeled as two nonlocal Kirchhoff plates connected by van der Waals forces. The Galerkin strip transfer function method which is a semi-analytical method is developed to compute the natural frequencies of the mass- plate vibrating system. It can give exact closed-form solutions along the longitudinal direction of the strip. The results obtained from the semi-analytical method are compared with the previous ones, and the differences between the single-layered graphene sheet （SLGS） and the DLGS mass sensors are also investigated. The results demonstrate the similarity of the in-phase mode between the SLGS and DLGS mass sensors. The sensitivity of the DLGS mass sensor can be increased by decreasing the nonlocal parameter, moving the attached nanoparticle closer to the DLGS center and making the DLGS smaller. These conclusions are helpful for the design and application of graphene-sheet-based resonators as nano-mass sensors.
基金Supported by National Science and Technology Major Project of China(Grant No.2011ZX04002-101)National Science and Technology Support Plan of China(Grant No.2011BAF15B02)National Natural Science Foundation of China(Grant No.51305388)
文摘As the traditional forging process has many problems such as low efficiency, high consumption of material and energy, large cylindrical shell rolling is introduced. Large cylindrical shell rolling is a typical rotary forming technology, and the upper and lower rolls have different radii and speeds. To quickly predict the three-dimensional stresses and eliminate fishtail defect, an improved strip layer method is developed, in which the asymmetry of the upper and lower rolls, non-uniform deformation and stress, as well as the asymmetrical spread on the end surface are considered. The deformation zone is divided into a certain number of layers and strips along the thickness and width, respectively. The transverse displacement model is constructed by polynomial function, in order to increase the computation speed greatly. From the metal plastic mechanics principle, the three-dimensional stress models are established. The genetic algorithm is used for optimization calculation in an industrial experiment example. The results show that the rolling pressure, the normal stresses, the upper and lower friction stress distributions are not similar with those of a general plate rolling. There are two relative maximum values in rolling pressure distribution. The upper and lower longitudinal friction stresses change direction nearby the upper and lower neutral points, respectively. The fishtail profile of spread on the end surface is predicted satisfactorily. The reduction could be helpful to eliminate fishtail defect. The large cylindrical shell rolling example illustrates the calculation results acquired rapidly are good agreements with the finite element simulation and experimental values of previous study. A highly effective and reliable three-dimensional simulation method is proposed for large cylindrical shell rolling and other asymmetrical rolling.
基金Item Sponsored by the National Science and Technology Support Plan of China(2011BAF15B03)National Natural Science Foundation of China(51305387)Hebei Province Colleges and Universities Natural Science Research Youth Foundation of China(Q2012104)
文摘Quantitative level evaluation of cold-rolled strip was significant for the shape control and the product structure optimization. Based on the probability statistics method and the shape pattern recognition method, the shape evaluation model of cold-rolled strip was established to evaluate real-time shape of the online strip. The model was embedded in the shape control system, and the online strip shape state was real-time observed whether to meet the technical indicators, so the shape level could be identified and evaluated. Based on the shape evaluation indicators, the shape component could be well controlled to improve the shape hit rate and the control stability, so that the rolled strip shape could meet the technical requirements. At a 1050 cold strip mill, the shape data of the scene typical specifications strip were online collected and real-time evaluated, and the hit of macro shape in less than 8 I was 68.44%, indicating that the mill shape control had considerable room for improvement, as well as the evaluation method was feasible and practical for quantitative evaluation of the strip shape. So the new evaluation model has important guiding significance to increase value-added and reduce product quality objection.
文摘In this work, we applied two electromagnetic models for the characterization of a planar structure including a flat, thick copper conductor. Indeed the first model is consisted by modeling two metal ribbons without bulkiness, placed one above the other at a distance of h<sub>2</sub> equal to the thickness of the thick conductor. This approach has been implemented and tested by the iterative method. The results of simulations have been compared with those calculated by the Ansoft HFSS software, and they are in good concordance, validating the method of analysis used. The second model is based on the calculation of the effective permittivity of the medium containing the thick conductor. This medium consists of a metallic region of complex relative permittivity , the rest of this medium is filled with air e<sub>r</sub><sub>2</sub> = 1. The effective permittivity e<sub>eff</sub> calculated from these two relative permittivity e<sub>r</sub><sub>2</sub> and . Comparing the simulation results of this new formulation of the iterative method with those calculated by the software Ansoft HFSS shows that they are in good matching which validates the second model.
文摘Hot rolled strips usually have higher strength and lower plasticity at the ends, and the mechanical properties are distributed unevenly along the length direction. Such phenomena are caused by the different cooling rates between the end and the center. The ends of the coiled strip cool down faster than the center, inducing finer grains in the sections. Furthermore, the center of the coil is kept at high temperature for longer time, which affects the precipitation of the carbides and creates the different mechanical properties from the ends. In this paper, the temperature field of the strip during cooling was simulated to discover the characteristics of the temperature change and the effect on mechanical properties. Based on the analysis, a concept of concave cooling control was introduced and implemented in the production. Results indicated that applying the concave cooling control method could significantly improve the uniformity of the properties and promote the quality of the products.
基金State Foundstion of Ph.D Units of China(2003-05)under Grant 20020141013the NNSF(10471015)of Liaoning Province,China.
文摘This paper formulates a two-dimensional strip packing problem as a non- linear programming （NLP） problem and establishes the first-order optimality conditions for the NLP problem. A numerical algorithm for solving this NLP problem is given to find exact solutions to strip-packing problems involving up to 10 items. Approximate solutions can be found for big-sized problems by decomposing the set of items into small-sized blocks of which each block adopts the proposed numerical algorithm. Numerical results show that the approximate solutions to big-sized problems obtained by this method are superior to those by NFDH, FFDH and BFDH approaches.
