To investigate the application of meta-model for finite element( FE) model updating of structures,the performance of two popular meta-model,i. e.,Kriging model and response surface model( RSM),were compared in detail....To investigate the application of meta-model for finite element( FE) model updating of structures,the performance of two popular meta-model,i. e.,Kriging model and response surface model( RSM),were compared in detail. Firstly,above two kinds of meta-model were introduced briefly. Secondly,some key issues of the application of meta-model to FE model updating of structures were proposed and discussed,and then some advices were presented in order to select a reasonable meta-model for the purpose of updating the FE model of structures. Finally,the procedure of FE model updating based on meta-model was implemented by updating the FE model of a truss bridge model with the measured modal parameters. The results showed that the Kriging model was more proper for FE model updating of complex structures.展开更多
A dynamic finite element method combined with finite element mixed formula for contact problem is used to analyze the dynamic characteristics of gear system. Considering the stiffness excitation, error excitation and ...A dynamic finite element method combined with finite element mixed formula for contact problem is used to analyze the dynamic characteristics of gear system. Considering the stiffness excitation, error excitation and meshing shock excitation, the dynamic finite element model is established for the entire gear system which includes gears, shafts, bearings and gearbox housing. By the software of I-DEAS, the natural frequency, normal mode, dynamic time-domain response, frequency-domain response and one-third octave velocity grade structure borne noise of gear system are studied by the method of theoretical modal analysis and dynamic response analysis. The maximum values of vibration and structure borne noise are occurred at the mesh frequency of output grade gearing.展开更多
Using Moore-Penrose inverse theory, a set of formulations for calculating the static responses of a changed finite element structure are given in this paper. Using these formulations by structural analysis may elimina...Using Moore-Penrose inverse theory, a set of formulations for calculating the static responses of a changed finite element structure are given in this paper. Using these formulations by structural analysis may eliminate the need of assembling the stiffness matrix and solving a set of simultaneous equations.展开更多
In the present paper, we hare mtroduced the random materials. loads. geometricalshapes, force and displacement boundary condition directly. into the functionalvariational formula, by. use of a small parameter perturb...In the present paper, we hare mtroduced the random materials. loads. geometricalshapes, force and displacement boundary condition directly. into the functionalvariational formula, by. use of a small parameter perturbation method, a unifiedrandom variational principle in finite defomation of elastieity and nonlinear randomfinite element method are esiablished, and used.for reliability, analysis of structures.Numerical examples showed that the methods have the advontages of simple andconvenient program implementation and are effective for the probabilistic problems inmechanics.展开更多
In this paper, a kind of rationalism theory of shell is established which is of different mechanic characters in tension and in compression, and the finite element numerical analysis method is also described.
The central solenoid is an important part of the HT-7U device. In this paper, the computational analysis of the stress and the displacement on the pre-load structures of the central solenoid have been made by the fin...The central solenoid is an important part of the HT-7U device. In this paper, the computational analysis of the stress and the displacement on the pre-load structures of the central solenoid have been made by the finite element analysis system COSMOS/M2.0 under room and/or operating temperature. According to the analytical results, the clip aprons and compression plates are all satisfied with safety design criteria.展开更多
Nonlinear solution of reinforced concrete structures, particularly complete load-deflection response, requires tracing of the equilibrium path and proper treatment of the limit and bifurcation points. In this regard, ...Nonlinear solution of reinforced concrete structures, particularly complete load-deflection response, requires tracing of the equilibrium path and proper treatment of the limit and bifurcation points. In this regard, ordinary solution techniques lead to instability near the limit points and also have problems in case of snap-through and snap-back. Thus they fail to predict the complete load-displacement response. The arc-length method serves the purpose well in principle, received wide acceptance in finite element analysis, and has been used extensively. However modifications to the basic idea are vital to meet the particular needs of the analysis. This paper reviews some of the recent developments of the method in the last two decades, with particular emphasis on nonlinear finite element analysis of reinforced concrete structures.展开更多
In order to improve the strength and stiffness of shield cutterhead, the method of fuzzy mathematics theory in combination with the finite element analysis is adopted. An optimal design model of structural parameters ...In order to improve the strength and stiffness of shield cutterhead, the method of fuzzy mathematics theory in combination with the finite element analysis is adopted. An optimal design model of structural parameters for shield cutterhead is formulated,based on the complex engineering technical requirements. In the model, as the objective function of the model is a composite function of the strength and stiffness, the response surface method is applied to formulate the approximate function of objective function in order to reduce the solution scale of optimal problem. A multi-objective genetic algorithm is used to solve the cutterhead structure design problem and the change rule of the stress-strain with various structural parameters as well as their optimal values were researched under specific geological conditions. The results show that compared with original cutterhead structure scheme, the obtained optimal scheme of the cutterhead structure can greatly improve the strength and stiffness of the cutterhead, which can be seen from the reduction of its maximum equivalent stress by 21.2%, that of its maximum deformation by 0.75%, and that of its mass by 1.04%.展开更多
A specific computational program SAFEM was developed based on semi-analytical finite element (FE) method for analysis of asphalt pavement structural responses under static loads. The reliability and efficiency of th...A specific computational program SAFEM was developed based on semi-analytical finite element (FE) method for analysis of asphalt pavement structural responses under static loads. The reliability and efficiency of this FE program was proved by comparison with the general commercial FE software ABAQUS. In order to further reduce the computational time without decrease of the accuracy, the infinite element was added to this program. The results of the finite-infinite element coupling analysis were compared with those of finite element analysis derived from the verified FE program, The study shows that finite-infinite element coupling analysis has higher reliability and efficiency.展开更多
For the safety protection of passengers when train crashes occur, special structures are crucially needed as a kind of indispensable energy absorbing device. With the help of the structures, crash kinetic-energy can b...For the safety protection of passengers when train crashes occur, special structures are crucially needed as a kind of indispensable energy absorbing device. With the help of the structures, crash kinetic-energy can be completely absorbed or dissipated for the aim of safety. Two composite structures(circumscribed circle structure and inscribed circle structure) were constructed. In addition, comparison and optimization of the crashworthy characteristic of the two structures were carried out based on the method of explicit finite element analysis(FEA) and Kriging surrogate model. According to the result of Kriging surrogate model, conclusions can be safely drawn that the specific energy absorption(SEA) and ratio of specific energy absorption to initial peak force(REAF) of circumscribed circle structure are lager than those of inscribed circle structure under the same design parameters. In other words, circumscribed circle structure has better performances with higher energy-absorbing ability and lower initial peak force. Besides, error analysis was adopted and the result of which indicates that the Kriging surrogate model has high nonlinear fitting precision. What is more, the SEA and REAF optimum values of the two structures have been obtained through analysis, and the crushing results have been illustrated when the two structures reach optimum SEA and REAF.展开更多
A physical value mapping (PVM) algorithm based on finite element mesh from the stamped part in stamping process to the product is presented, In order to improve the efficiency of the PVM algorithm, a search way from...A physical value mapping (PVM) algorithm based on finite element mesh from the stamped part in stamping process to the product is presented, In order to improve the efficiency of the PVM algorithm, a search way from the mesh of the product to the mesh of the stamped part will be adopted. At the same time, the search process is divided into two steps: entire search (ES) and local search (LS), which improve the searching efficiency. The searching area is enlarged to avoid missing projection elements in ES process. An arc-length method is introduced in LS process. The validity is confirmed by the results of the complex industry-forming product.展开更多
This paper presents a novel parallel implementation technology for wave-based structural health monitoring (SHM) in laminated composite plates. The wavelet-based B-spline wavelet on he interval (BSWI) element is cons...This paper presents a novel parallel implementation technology for wave-based structural health monitoring (SHM) in laminated composite plates. The wavelet-based B-spline wavelet on he interval (BSWI) element is constructed according to Hamilton’s principle, and the element by element algorithm is parallelly executed on graphics processing unit (GPU) using compute unified device architecture (CUDA) to get the responses in full wave field accurately. By means of the Fourier spectral analysis method,the Mindlin plate theory is selected for wave modeling of laminated composite plates while the Kirchhoff plate theory predicts unreasonably phase and group velocities. Numerical examples involving wave propagation in laminated composite plates without and with crack are performed and discussed in detail. The parallel implementation on GPU is accelerated 146 times comparing with the same wave motion problem executed on central processing unit (CPU). The validity and accuracy of the proposed parallel implementation are also demonstrated by comparing with conventional finite element method (FEM) and the computation time has been reduced from hours to minutes. The damage size and location have been successfully determined according to wave propagation results based on delay-and-sum (DAS). The results show that the proposed parallel implementation of wavelet finite element method (WFEM) is very appropriate and efficient for wave-based SHM in laminated composite plates.展开更多
A new type of air bearing with flexure structure is introduced. The new bearing is designed for precision mechanical engineering devices such as mechanical watch movement. The new design uses the flexure structure to ...A new type of air bearing with flexure structure is introduced. The new bearing is designed for precision mechanical engineering devices such as mechanical watch movement. The new design uses the flexure structure to provide 3D damping to absorb shocks from all directions. Two designs are presented: one has 12 T-shape slots in the radian direction while the other has 8 spiral slots in the radian direction. Both designs have flexure mountings on the axial directions. Based on the finite element analysis (FEA), the new bearing can reduce the vibration (displacement) by as much as 8.37% and hence, can better protect the shafts.展开更多
Increasing size of wind turbine and deep water deployment have raised the issue of appropriate selection of the most suitable support structure to make offshore wind energy cost competitive.The paper presents an optim...Increasing size of wind turbine and deep water deployment have raised the issue of appropriate selection of the most suitable support structure to make offshore wind energy cost competitive.The paper presents an optimization methodology for decision making process of bottom mounted supports of offshore wind turbines (OWTs) through reasonable engineering attributes derivation.Mathematic models of support structures are reduced by the generalized single-degree-of-freedom theory with relatively fewer structural parameters.Soft-stiff design optimization based on dynamic properties of OWTs is performed for monopile and lattice supports with different wind turbines,water depth and hub height.Attributes of support structures,wind turbines and environment conditions are applied in the multi-criteria decision making method——TOPSIS for benchmarking of those options.The results illustrate the effectiveness of the proposed optimazation methodology combined with economical and environmental attributes together.展开更多
The wave-induced seepage force is investigated on marine structures resting on or buried in the seabed.The bed is modelled as a poroelastic medium containing a nearly saturated water.The governing equations are solved...The wave-induced seepage force is investigated on marine structures resting on or buried in the seabed.The bed is modelled as a poroelastic medium containing a nearly saturated water.The governing equations are solved with Finite Element Method.For a pipeline buried in the seabed,agreement between the present numerical results and that of Cheng H.D.(1986)is quite satisfactory.展开更多
In order to reduce the risk of sealing and improve the structural strength for a coal mine mobile refuge chamber,a new type of one-piece model was designed.Mechanical and mathematical calculation performed an importan...In order to reduce the risk of sealing and improve the structural strength for a coal mine mobile refuge chamber,a new type of one-piece model was designed.Mechanical and mathematical calculation performed an important role.Calculated according to statics and relevant contents,the structure had the same total volume as the traditional segmented structure,but had shorter length,wider width and greater height.Those prevented the structure from stress or deformation failure.Some reinforcing ribs with enough moments of inertia were welded in the external shell.Because of the one-piece structure,this refuge chamber reduced the risk of sealing which was a serious problem of segmented structure.Impact load with 300 ms duration and0.6 MPa over-pressure was settled.Explicit nonlinear dynamic analysis program was used to simulate the response of the refuge chamber.The maximum stress and the maximum displacement were obtained.The refuge chamber including blast airtight doors could meet the rigidity requirement.Weak parts of the chamber were the front and back end shell where bigger displacement values occurred than others.Thus,the calculation indicated that the refuge chamber could meet structural safety requirements.Based on the numerical analysis,suggestions were put forward for further resistance ability improvement.Only large inclined shaft with larger wellhead was suitable for this one-piece coal mine mobile refuge chamber.展开更多
Non-dominated sorting genetic algorithm II(NSGA-II)with multiple constraints handling is employed for multi-objective optimization of the topological structure of telescope skin,in which a bit-matrix is used as the ...Non-dominated sorting genetic algorithm II(NSGA-II)with multiple constraints handling is employed for multi-objective optimization of the topological structure of telescope skin,in which a bit-matrix is used as the representation of a chromosome,and genetic algorithm(GA)operators are introduced based on the matrix.Objectives including mass,in-plane performance,and out-of-plane load-bearing ability of the individuals are obtained by fnite element analysis(FEA)using ANSYS,and the matrix-based optimization algorithm is realized in MATLAB by handling multiple constraints such as structural connectivity and in-plane strain requirements.Feasible confgurations of the support structure are achieved.The results confrm that the matrix-based NSGA-II with multiple constraints handling provides an effective method for two-dimensional multi-objective topology optimization.展开更多
基金Sponsored by the National Key Technology Research and Development Program of China(Grant No.2011BAK02B02)
文摘To investigate the application of meta-model for finite element( FE) model updating of structures,the performance of two popular meta-model,i. e.,Kriging model and response surface model( RSM),were compared in detail. Firstly,above two kinds of meta-model were introduced briefly. Secondly,some key issues of the application of meta-model to FE model updating of structures were proposed and discussed,and then some advices were presented in order to select a reasonable meta-model for the purpose of updating the FE model of structures. Finally,the procedure of FE model updating based on meta-model was implemented by updating the FE model of a truss bridge model with the measured modal parameters. The results showed that the Kriging model was more proper for FE model updating of complex structures.
基金Funded by the Natural Science Foundation of China (No. 50675232)the Natural Science Foundation of CQ CSTC (2006BB3008)
文摘A dynamic finite element method combined with finite element mixed formula for contact problem is used to analyze the dynamic characteristics of gear system. Considering the stiffness excitation, error excitation and meshing shock excitation, the dynamic finite element model is established for the entire gear system which includes gears, shafts, bearings and gearbox housing. By the software of I-DEAS, the natural frequency, normal mode, dynamic time-domain response, frequency-domain response and one-third octave velocity grade structure borne noise of gear system are studied by the method of theoretical modal analysis and dynamic response analysis. The maximum values of vibration and structure borne noise are occurred at the mesh frequency of output grade gearing.
文摘Using Moore-Penrose inverse theory, a set of formulations for calculating the static responses of a changed finite element structure are given in this paper. Using these formulations by structural analysis may eliminate the need of assembling the stiffness matrix and solving a set of simultaneous equations.
文摘In the present paper, we hare mtroduced the random materials. loads. geometricalshapes, force and displacement boundary condition directly. into the functionalvariational formula, by. use of a small parameter perturbation method, a unifiedrandom variational principle in finite defomation of elastieity and nonlinear randomfinite element method are esiablished, and used.for reliability, analysis of structures.Numerical examples showed that the methods have the advontages of simple andconvenient program implementation and are effective for the probabilistic problems inmechanics.
文摘In this paper, a kind of rationalism theory of shell is established which is of different mechanic characters in tension and in compression, and the finite element numerical analysis method is also described.
文摘The central solenoid is an important part of the HT-7U device. In this paper, the computational analysis of the stress and the displacement on the pre-load structures of the central solenoid have been made by the finite element analysis system COSMOS/M2.0 under room and/or operating temperature. According to the analytical results, the clip aprons and compression plates are all satisfied with safety design criteria.
文摘Nonlinear solution of reinforced concrete structures, particularly complete load-deflection response, requires tracing of the equilibrium path and proper treatment of the limit and bifurcation points. In this regard, ordinary solution techniques lead to instability near the limit points and also have problems in case of snap-through and snap-back. Thus they fail to predict the complete load-displacement response. The arc-length method serves the purpose well in principle, received wide acceptance in finite element analysis, and has been used extensively. However modifications to the basic idea are vital to meet the particular needs of the analysis. This paper reviews some of the recent developments of the method in the last two decades, with particular emphasis on nonlinear finite element analysis of reinforced concrete structures.
基金Project(51074180) supported by the National Natural Science Foundation of ChinaProject(2012AA041801) supported by the National High Technology Research and Development Program of China+2 种基金Project(2007CB714002) supported by the National Basic Research Program of ChinaProject(2013GK3003) supported by the Technology Support Plan of Hunan Province,ChinaProject(2010FJ1002) supported by Hunan Science and Technology Major Program,China
文摘In order to improve the strength and stiffness of shield cutterhead, the method of fuzzy mathematics theory in combination with the finite element analysis is adopted. An optimal design model of structural parameters for shield cutterhead is formulated,based on the complex engineering technical requirements. In the model, as the objective function of the model is a composite function of the strength and stiffness, the response surface method is applied to formulate the approximate function of objective function in order to reduce the solution scale of optimal problem. A multi-objective genetic algorithm is used to solve the cutterhead structure design problem and the change rule of the stress-strain with various structural parameters as well as their optimal values were researched under specific geological conditions. The results show that compared with original cutterhead structure scheme, the obtained optimal scheme of the cutterhead structure can greatly improve the strength and stiffness of the cutterhead, which can be seen from the reduction of its maximum equivalent stress by 21.2%, that of its maximum deformation by 0.75%, and that of its mass by 1.04%.
基金represented by German Federal Highway Research Institute (BASt)financed by the Federal Minister of Transport and Digital Infrastructure (BMVI)conducted under FE 04.0259/2012/NGB
文摘A specific computational program SAFEM was developed based on semi-analytical finite element (FE) method for analysis of asphalt pavement structural responses under static loads. The reliability and efficiency of this FE program was proved by comparison with the general commercial FE software ABAQUS. In order to further reduce the computational time without decrease of the accuracy, the infinite element was added to this program. The results of the finite-infinite element coupling analysis were compared with those of finite element analysis derived from the verified FE program, The study shows that finite-infinite element coupling analysis has higher reliability and efficiency.
基金Projects(51405516,U1334208)supported by the National Natural Science Foundation of ChinaProject(2013GK2001)supported by the Science and Technology Program for Hunan Provincial Science and Technology Department,ChinaProject(2013zzts040)supported by the Graduate Degree Thesis Innovation Foundation of Central South University,China
文摘For the safety protection of passengers when train crashes occur, special structures are crucially needed as a kind of indispensable energy absorbing device. With the help of the structures, crash kinetic-energy can be completely absorbed or dissipated for the aim of safety. Two composite structures(circumscribed circle structure and inscribed circle structure) were constructed. In addition, comparison and optimization of the crashworthy characteristic of the two structures were carried out based on the method of explicit finite element analysis(FEA) and Kriging surrogate model. According to the result of Kriging surrogate model, conclusions can be safely drawn that the specific energy absorption(SEA) and ratio of specific energy absorption to initial peak force(REAF) of circumscribed circle structure are lager than those of inscribed circle structure under the same design parameters. In other words, circumscribed circle structure has better performances with higher energy-absorbing ability and lower initial peak force. Besides, error analysis was adopted and the result of which indicates that the Kriging surrogate model has high nonlinear fitting precision. What is more, the SEA and REAF optimum values of the two structures have been obtained through analysis, and the crushing results have been illustrated when the two structures reach optimum SEA and REAF.
基金This project is supported by National Natural Science Foundation ofChina(No.l9832020) and National Outstanding Youth Science Foundation ofChina(No.10125208).
文摘A physical value mapping (PVM) algorithm based on finite element mesh from the stamped part in stamping process to the product is presented, In order to improve the efficiency of the PVM algorithm, a search way from the mesh of the product to the mesh of the stamped part will be adopted. At the same time, the search process is divided into two steps: entire search (ES) and local search (LS), which improve the searching efficiency. The searching area is enlarged to avoid missing projection elements in ES process. An arc-length method is introduced in LS process. The validity is confirmed by the results of the complex industry-forming product.
基金supported by the National Natural Science Foundation of China (Grant Nos. 51421004 & 51405369)the National Key Basic Research Program of China (Grant No. 2015CB057400)+1 种基金the China Postdoctoral Science Foundation (Grant No. 2014M560766)the China Scholarship Council,and the Fundamental Research Funds for the Central Universities(Grant No. xjj2014107)
文摘This paper presents a novel parallel implementation technology for wave-based structural health monitoring (SHM) in laminated composite plates. The wavelet-based B-spline wavelet on he interval (BSWI) element is constructed according to Hamilton’s principle, and the element by element algorithm is parallelly executed on graphics processing unit (GPU) using compute unified device architecture (CUDA) to get the responses in full wave field accurately. By means of the Fourier spectral analysis method,the Mindlin plate theory is selected for wave modeling of laminated composite plates while the Kirchhoff plate theory predicts unreasonably phase and group velocities. Numerical examples involving wave propagation in laminated composite plates without and with crack are performed and discussed in detail. The parallel implementation on GPU is accelerated 146 times comparing with the same wave motion problem executed on central processing unit (CPU). The validity and accuracy of the proposed parallel implementation are also demonstrated by comparing with conventional finite element method (FEM) and the computation time has been reduced from hours to minutes. The damage size and location have been successfully determined according to wave propagation results based on delay-and-sum (DAS). The results show that the proposed parallel implementation of wavelet finite element method (WFEM) is very appropriate and efficient for wave-based SHM in laminated composite plates.
基金Selected from Proceedings of the 7th International Conference on Frontiers of DesignManufacturing(ICFDM'2006)This project is partially sup-ported by the Research Grant from the Hong Kong Watch Manufacturers Association the Federation of Hong Kong Watch TradeIndustry Technology Commission China(No.ITS/001/05).
文摘A new type of air bearing with flexure structure is introduced. The new bearing is designed for precision mechanical engineering devices such as mechanical watch movement. The new design uses the flexure structure to provide 3D damping to absorb shocks from all directions. Two designs are presented: one has 12 T-shape slots in the radian direction while the other has 8 spiral slots in the radian direction. Both designs have flexure mountings on the axial directions. Based on the finite element analysis (FEA), the new bearing can reduce the vibration (displacement) by as much as 8.37% and hence, can better protect the shafts.
基金Supported by the National Natural Science Foundation of China (No.51309209,51279186) and the National Basic Research Program of China (No.2011CB013704).
文摘Increasing size of wind turbine and deep water deployment have raised the issue of appropriate selection of the most suitable support structure to make offshore wind energy cost competitive.The paper presents an optimization methodology for decision making process of bottom mounted supports of offshore wind turbines (OWTs) through reasonable engineering attributes derivation.Mathematic models of support structures are reduced by the generalized single-degree-of-freedom theory with relatively fewer structural parameters.Soft-stiff design optimization based on dynamic properties of OWTs is performed for monopile and lattice supports with different wind turbines,water depth and hub height.Attributes of support structures,wind turbines and environment conditions are applied in the multi-criteria decision making method——TOPSIS for benchmarking of those options.The results illustrate the effectiveness of the proposed optimazation methodology combined with economical and environmental attributes together.
文摘The wave-induced seepage force is investigated on marine structures resting on or buried in the seabed.The bed is modelled as a poroelastic medium containing a nearly saturated water.The governing equations are solved with Finite Element Method.For a pipeline buried in the seabed,agreement between the present numerical results and that of Cheng H.D.(1986)is quite satisfactory.
基金Supported by the Major State Basic Research Development Program of China(973Project)(2011CB706900)
文摘In order to reduce the risk of sealing and improve the structural strength for a coal mine mobile refuge chamber,a new type of one-piece model was designed.Mechanical and mathematical calculation performed an important role.Calculated according to statics and relevant contents,the structure had the same total volume as the traditional segmented structure,but had shorter length,wider width and greater height.Those prevented the structure from stress or deformation failure.Some reinforcing ribs with enough moments of inertia were welded in the external shell.Because of the one-piece structure,this refuge chamber reduced the risk of sealing which was a serious problem of segmented structure.Impact load with 300 ms duration and0.6 MPa over-pressure was settled.Explicit nonlinear dynamic analysis program was used to simulate the response of the refuge chamber.The maximum stress and the maximum displacement were obtained.The refuge chamber including blast airtight doors could meet the rigidity requirement.Weak parts of the chamber were the front and back end shell where bigger displacement values occurred than others.Thus,the calculation indicated that the refuge chamber could meet structural safety requirements.Based on the numerical analysis,suggestions were put forward for further resistance ability improvement.Only large inclined shaft with larger wellhead was suitable for this one-piece coal mine mobile refuge chamber.
基金supported by the National Natural Science Foundation of China(Nos.50905085 and 91116020)the National Science Foundation for Post-doctoral Scientists of China(No.2012M511263)
文摘Non-dominated sorting genetic algorithm II(NSGA-II)with multiple constraints handling is employed for multi-objective optimization of the topological structure of telescope skin,in which a bit-matrix is used as the representation of a chromosome,and genetic algorithm(GA)operators are introduced based on the matrix.Objectives including mass,in-plane performance,and out-of-plane load-bearing ability of the individuals are obtained by fnite element analysis(FEA)using ANSYS,and the matrix-based optimization algorithm is realized in MATLAB by handling multiple constraints such as structural connectivity and in-plane strain requirements.Feasible confgurations of the support structure are achieved.The results confrm that the matrix-based NSGA-II with multiple constraints handling provides an effective method for two-dimensional multi-objective topology optimization.