CN111859482A - Muscle type thin-wall structure lightweight design method - Google Patents
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Abstract
The invention discloses a muscle type thin-wall structure lightweight design method, which belongs to the field of aerospace equipment and comprises the following steps of determining a geometric model, a boundary constraint condition and a load condition of an initial structure design of an internal pressure-stressed part, designing the geometric model based on the initial structure of the internal pressure-stressed part, and establishing a numerical analysis model of a variable thickness optimization method, wherein a thin-wall region with variable thickness in the numerical analysis model is a design domain, and other regions are non-design domains; the method comprises the steps of taking maximum structural rigidity as an optimization target, taking the material consumption equal to a set value as an optimization constraint condition, taking the thickness of a unit in a design domain as a design variable, obtaining the optimal thickness distribution of the part subjected to internal pressure in the design domain by adopting a variable thickness optimization method, and establishing a final muscle type thin-wall structure lightweight design geometric model based on the optimal thickness distribution of the design domain, wherein the method is favorable for more fully utilizing the mechanical property of the material; the thickness variation is gentle in the muscle-type design, which is beneficial to reducing the maximum stress in the structure during the heating/cooling process, thereby improving the life of the structure.
Description
Technical Field
The invention relates to the field of aerospace equipment, in particular to a muscle type thin-wall structure lightweight design method.
Background
Lightweight designs for thin-walled structures have received much attention in many engineering fields, such as aerospace, chemical machinery, etc. The reinforcement design is a common method for meeting this requirement. Compared with the structure design with uniform thickness, the reinforced structure design can more effectively utilize the mechanical property of the material due to better bending resistance, thereby reducing the material consumption of the structure. However, the design of the reinforced thin-wall structure has several disadvantages, such as high difficulty in processing and manufacturing, high time consumption, low material utilization efficiency of part of ribs, high stress easily occurring under thermal coupling load, and the like. Another structural configuration that can be used for thin-walled, lightweight designs is a variable thickness design, but the variable thickness design has relatively few practical applications. The reasons can be roughly classified into the following points: 1) the thickness distribution of the variable thickness design is difficult to determine by engineering experience, but needs to be determined by means of a structure optimization design technology, and the technical threshold is higher; 2) variable thickness design geometric model modeling based on optimization design results presents difficulties. Therefore, the application of the variable thickness design in engineering is not wide. However, the variable thickness design has the potential to overcome the deficiencies of the stiffened design: 1) because the thickness change of each part in the structure in the variable thickness design is relatively smooth and gentle, the machining difficulty of the variable thickness design is lower than that of the reinforcement design; 2) the distribution of materials in the structure is determined by the structure optimization design technology, and the materials are more reasonably and fully utilized; 3) because the thickness variation gradient in the structure is small, the temperature difference level in the structure is low in the temperature rising or cooling process, and further the thermal stress level is low.
Disclosure of Invention
According to the problems in the prior art, the invention discloses a muscle type thin-wall structure lightweight design method, which comprises the following steps:
s1: determining a geometric model designed by an initial structure of the internal pressure bearing component, boundary constraint conditions and load conditions;
s2: designing a geometric model based on the initial structure of the internal pressure-bearing part, and establishing a numerical analysis model of a variable thickness optimization method, wherein a thin-wall region with variable thickness in the numerical analysis model is a design region, and other regions are non-design regions;
s3: the method comprises the steps of obtaining the optimal thickness distribution of the part subjected to internal pressure in a design domain by using a variable thickness optimization method by taking the maximum structural rigidity as an optimization target, taking the material consumption equal to a set value as an optimization constraint condition and taking the unit thickness in the design domain as a design variable;
s4: and establishing a final muscle type thin-wall structure lightweight design geometric model based on the optimal thickness distribution of the design domain.
Further, the muscle-type thin-wall structure is characterized as follows: the inner surface of the pressed part remains unchanged and the outer surface has a smooth and continuous thickness variation which is visible.
Further, the design domain of the internal pressure-bearing component is a thin-wall region of the initial structure design geometric model, and the design domain of the internal pressure-bearing component needs to be subjected to meshing by adopting shell units based on the inner surface of the thin-wall region of the initial structure design geometric model.
Further, the steps of obtaining the optimal thickness distribution in the design domain of the internal pressure-bearing component by the variable thickness optimization method are as follows:
s3-1, setting initial parameters including initial thickness parameter, convergence parameter and maximum thickness h of design fieldmaxAnd a minimum thickness hminImporting the grid cell number, the section attribute number and the corresponding thickness information of the design domain, setting an iteration step number k, initializing an initial iteration step number k to be 1, and initializing a design variable vector hk；
S3-2 based on cell thickness hkOutputting a file for recording the unit thickness in the design domain;
s3-3, reading in the file of the cell thickness in the design domain, and updating the thickness of each cell in the design domain according to the input file information;
s3-4, carrying out structural response numerical analysis and outputting and recording each grid unit in the design domainM of (A)x，My，MxyA file of information;
s3-5: reading in and recording unit bending moment M in design domainx，My，MxyThe document of (4), calculating a p value of each cell based on the following formula (5);
wherein: p is a radical ofiFor designing the state parameter of the ith unit in the domain, v is Poisson's ratio, Mx，My，MxyRespectively representing bending moment around an X axis, bending moment around a Y axis and torque in an XY plane under a unit local coordinate system;
s3-6, calculating the updated h based on the formula (6)k+1Wherein lambda is Lagrange multiplier, the value of lambda is controlled by a volume constraint formula (3), and the value of lambda is iteratively calculated by adopting a dichotomy until the updated h k+1The volume constraint requirement is met;
s3-7, when the design variable vector of two adjacent iterative steps meets | | hk+1-hkWhen the absolute value is less than or equal to the absolute value, a variable thickness optimization result is obtained; if hk+1-hk| > k +1, return to step S3-2.
Further, the establishing of the final muscle type thin-wall structure lightweight design geometric model based on the optimal thickness distribution of the design domain comprises the following steps:
s4-1, determining the optimal thickness of each unit in the design domain based on the variable thickness optimization result;
s4-2, numbering grid nodes in all design domains, acquiring the number NA of all nodes in the design domains, acquiring the number and the serial number of units to which each node belongs, defining a serial number parameter N representing the current calculated node, and setting the serial number of the initial calculated node as N as 1;
s4-3, calculating the normal vector of the unit to which the node with the number of N belongs;
s4-4, calculating the space coordinate of the outer surface point corresponding to the node number N;
s4-5, if N is less than NA, executing N to be N +1, and returning to the step S4-3; if N is NA, proceed to S4-6;
s4-6, sequentially generating all points based on the calculated space coordinates of all the outer surface points to obtain an optimal design outer surface point cloud;
s4-7: obtaining a geometric model of the outer surface of the optimally designed thin-walled region based on the optimally designed outer surface point cloud;
And S4-8, replacing the outer surface of the design domain of the initial model of the design object with the outer surface of the optimized design, repairing the damaged surface generated by the outer surface replacement, and generating the final muscle type thin-wall light-weight design.
Further, the method for calculating the space coordinates of the outer surface points corresponding to the node N comprises the following steps:
s4-4-1, determining the number of the unit to which the N node belongs, generating a set NN, and calculating the number NE of elements in the NN;
s4-4-2, determining normal vectors n ═ n of all units in the NN set (n)1,n2,n3)；
S4-4-3: the spatial coordinates of the outer surface points corresponding to node N are calculated based on the following formula,
wherein (x)outer，youter，zouter) X-, Y-and Z-axis coordinate values representing outer surface points, (X)inner，yinner，zinner) Coordinate values of X, Y and Z axes representing points on the inner surface, hNN(i)The optimum thickness of the cell numbered nn (i).
Due to the adoption of the technical scheme, the muscle type thin-wall structure lightweight design method provided by the invention has the advantages that the material distribution is obtained based on variable thickness optimization, and the mechanical property of the material is more fully utilized; the optimization design process has good integrity, starts from the most common geometric model for engineering application and ends with the optimization design geometric model, thereby overcoming the difficulty that the geometric model is difficult to establish by the optimization design result of the variable-thickness structure; the thickness change in the muscle type design is smooth, which is beneficial to reducing the maximum stress in the structure in the process of temperature rise/temperature drop, thereby prolonging the service life of the structure; meanwhile, the muscle type design is beneficial to reducing the time consumption of numerical control machining and reducing the difficulty of numerical control machining.
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In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments described in the present application, and other drawings can be obtained by those skilled in the art without creative efforts.
FIG. 1 is a flow chart of a thin-wall structure lightweight design method of the present invention;
FIG. 2 is a geometric model of an initial structural design of an internally pressurized component;
FIG. 3 is a finite element model during a variable thickness optimization process;
FIG. 4 is a finite element model with boundary constraints and loading conditions added;
FIG. 5 is a schematic of the optimal thickness profile obtained by variable thickness optimization;
FIG. 6 is a cloud point view of the outer surface;
FIG. 7 is an outer surface generated based on a point cloud result;
FIG. 8 illustrates the problem of local face separation caused by the replacement of the outer surface of the master model with an outer surface;
FIG. 9 is a geometric solid model of the muscular lightening design obtained after repair of the separating surface.
Detailed Description
In order to make the technical solutions and advantages of the present invention clearer, the following describes the technical solutions in the embodiments of the present invention clearly and completely with reference to the drawings in the embodiments of the present invention:
FIG. 1 is a flow chart of a thin-wall structure lightweight design method of the present invention, wherein the flow of the thin-wall structure lightweight design method comprises the following steps;
s1, determining a geometric model and boundary constraint conditions of the initial structural design of the internal pressure bearing component and load conditions;
the thin-wall component bearing the internal pressure effect is widely applied to the fields of aerospace, chemical engineering and the like, for example, a pipeline for conveying substances or a high-pressure compressor casing of an aeroengine, the internal pressure load effect caused by internal high-pressure fluid or gas can be borne by the internal part of the thin-wall component due to functional requirements, the internal shape of the thin-wall component is often limited by the functional requirements and cannot be changed at will, and the external appearance of the thin-wall component can be changed.
The design process of the patent requires that a design object needs to have a given initial design, but a finite element model used in optimization is obtained by dividing a grid not directly from an initial design geometric model, but by carrying out model processing on the geometric model and then dividing the grid, the model processing process requires that the action position, the boundary constraint position and the like of a load, such as internal pressure, temperature and the like, are clear, wherein an internal pressure action surface is usually the inner surface of a thin-wall structure and can not be changed at will, and fig. 2 shows that the geometric model is designed for the initial structure of an internal pressure-bearing component.
S2: designing a geometric model based on the initial structure of the internal pressure-bearing part, and establishing a numerical analysis model of a variable thickness optimization method, wherein a thin-wall region with variable thickness in the numerical analysis model is a design region, and other regions are non-design regions;
s3: the method comprises the steps of obtaining the optimal thickness distribution of the part subjected to internal pressure in a design domain by using a variable thickness optimization method by taking the maximum structural rigidity as an optimization target, taking the material consumption equal to a set value as an optimization constraint condition and taking the unit thickness in the design domain as a design variable;
by adopting the variable thickness optimization method, the thickness parameters of all units in the design domain are used as design variables, and for this reason, the shell units are required to divide the grids in the design domain according to the requirement of the finite element model for variable thickness optimization. It is noted that only the design domain has to be meshed with shell cells, and non-design domains may be meshed with other types of cells. Herein, a design domain refers to a structural design changeable region, and a non-design domain is a structural design unchangeable region.
Fig. 3 is a finite element model in the variable thickness optimization process, and fig. 4 is a finite element model after adding boundary constraint conditions and load conditions, in this embodiment, the models all adopt shell elements to divide meshes, and all serve as design domains.
An optimized design criterion, which is specially aimed at the thickness distribution of a thin-wall structure, reduces the deformation of the structure by reducing the total bending strain energy of the structure, thereby improving the rigidity of the structure, and has the similar point to the reduced strain energy of the classical topological optimization method, but the difference is that the latter pursues the reduction is the total strain energy of the structure. For thin-walled structures, the overall strain energy of the structure includes both bending strain energy and membrane strain energy, and compared to thin-walled structures, the deformation caused by bending moment is larger, so that more effective stiffness optimization design can be obtained by considering the minimization of bending strain energy, and the optimization column of the related problems is as follows,
find h＝{h1,...,hi,...,hN} (1)
minΦ (2)
wherein h is a design variable vector, hiIs the thickness of the ith cell, NeTo design the total number of variables, AiIs the area of the ith cell, hiniPhi represents the strain energy of the element caused by bending moment for the set initial element thickness, and the calculation formula is,
wherein E is elastic modulus, v is Poisson's ratio, w is deflection deformation, x, y and z are respectively x-axis coordinate, y-axis coordinate and z-axis coordinate, and z-axis is thickness direction.
Although the above expression is very complex, an efficient and convenient optimization criterion can be derived based on the formula, as shown in the following formula,
Wherein M isx，My，MxyThe values of the lagrange multiplier lambda are respectively the bending moment around the X axis, the bending moment around the Y axis and the torque in the XY plane in the unit local coordinate system, so that the volume constraint is ensured to be met, the volume constraint can be solved according to the dichotomy, and the specific solving method can refer to the accessory program in the report.
This patent has established the optimization method based on above-mentioned method and has realized the procedure, and the basic thinking is: and compiling a main program of steps of data processing, variable updating, ANSYS calling and the like by taking MATLAB as a platform, carrying out numerical analysis on the current design by taking ANSYS as a black box, and outputting structural response information required by optimization, such as unit bending moment and the like.
The steps of obtaining the optimal thickness distribution of the thin-wall area by the variable thickness optimization method are as follows:
s3-1, setting an iteration step k of 0 by MATLAB, and setting initial parameters including initial thickness HiniThe convergence criterion is used for importing required information, designing domain unit numbers, number, section attribute numbers, corresponding thicknesses and the like, and initializing a design variable h;
s3-2 MATLAB setting k ═ k +1, based on cell thickness hk(thickness variable in the kth iteration step), outputting a file for recording the thickness of the unit in the design domain, and making an ANSYS input file;
s3-3, reading the file of the thickness of the unit in the design domain by ANSYS, and updating the thickness of each unit in the design domain according to the information of the input file;
S3-4 ANSYS carries out structural response numerical analysisOutputting M per grid cell in design Domainx，My，MxyA file of information;
s3-5, reading MATLAB into and recording the bending moment M of the unit in the design domainx，My，MxyThe p value of each cell is calculated according to the formula (5);
s3-6 MATLAB entering a criterion iteration subroutine, calculating updated h based on formula (2)k+1Wherein lambda is Lagrange multiplier, the value of lambda is controlled by a volume constraint formula (3), and the value of lambda is iteratively calculated by adopting a dichotomy until the updated hk+1The volume constraint requirement is met;
s3-7, MATLA judges that the design variable vector of two adjacent iterative steps meets | | hk+1-hkWhen the absolute value is less than or equal to the absolute value, a variable thickness optimization result is obtained; if hk+1-hk| > then return to step S4-2.
Further, the value of lambda is calculated based on the volume constraint (formula (3)) by adopting a classical criterion method, so that h is determinedk+1The calculation flow is as follows:
s3-6-1: setting a value of λ to an upper bound λUAnd lower bound λL，λUIt is necessary to have a sufficiently large value such as 1e6 (there is a problem that the value of p is generally high, and λ needs to be further increasedUValue of) lambda), lambdaLIt is necessary to take a sufficiently small value, e.g. 0 (in some cases, p is smaller overall, and λ needs to be further reducedLValue of) is set, iterative convergence parameters are set λ，λThe value is a positive integer which is small enough, but if the value is too small, the number of sub-iteration steps is excessive;
s3-6-2: calculating a value of lambda, wherein the lambda is calculated by adopting the following formula;
λ＝(λU+λL)/2； (7)
s3-6-3: based on the current lambda value, h is calculated according to the formula (6)k+1；
S3-6-4: computing
S3-6-5, if
Then calculate
Taking a value, if the value result is positive, passing through lambdaUUpdating λ ═ λUAnd returning to S3-6-2, otherwise, if the value is negative, passing through lambdaLUpdating λ ═ λLAnd returning to S3-6-2; if it is
If yes, ending iteration and current hk+1And (4) taking values of design variables of the optimized iteration step.
The process is a classical process for determining the value of the Lagrange multiplier lambda based on a dichotomy mode, and simultaneously, the design variable value h of the (k + 1) th iteration step is obtainedk+1；
And S4, obtaining the final muscle type thin-wall structure lightweight design based on the optimal thickness distribution of the design domain.
Typical results of the optimal thickness distribution obtained in the above manner are shown in fig. 5; it should be noted that the thickness distribution optimization result cannot be directly used for machining, and a geometric model required for machining needs to be established according to the thickness distribution optimization result.
Further, the final muscle-type thin-wall structure lightweight design is obtained based on the variable thickness optimization result in the following way:
S4-1, determining the optimal thickness of each unit in the design domain based on the variable thickness optimization result;
s4-2, numbering each node in all design domains, acquiring the number NA of all nodes in the design domains, acquiring the number and the serial number of units to which each node belongs, defining a serial number parameter N representing the current calculated node, and setting the serial number of the initial calculated node as N as 1;
s4-3, calculating the normal vector of the unit to which the node with the number of N belongs;
the cell normal vector n is calculated as in equation (8),
wherein xi, yi and zi are x, y and z coordinate values of the ith node of the unit respectively, I is I, II and III are global numbers of any three nodes of the unit; note that the calculated normal vector direction is associated with cell node selection, and if the obtained normal is directed to the inside of the structure, then n is-n.
S4-4, calculating the space coordinate of the outer surface point corresponding to the node number N;
s4-5, if N is less than NA, executing N to be N +1, and returning to the step S4-3; if N is NA, proceed to S4-6;
s4-6, sequentially generating all points based on the calculated space coordinates of all the outer surface points to obtain an optimal design outer surface point cloud;
s4-7, based on the point cloud of the optimized design outer surface obtained in the step, reverse modeling is implemented by adopting commercial CAD software to obtain a geometric model of the outer surface of the optimized design thin-wall area;
And S4-8, replacing the outer surface of the design domain of the initial model of the design object with the rest outer surfaces of the optimally designed thin wall, repairing the damaged surface generated by surface replacement, adding chamfers and the like, and generating the final muscle type thin wall lightweight design.
Further, the process of calculating the spatial position of the node N of the outer surface corresponding to the node N in the design domain is as follows:
s4-54-1, determining the number set of the unit to which the N node belongs, producing a set NN, and calculating the number NE of elements in the NN;
s4-4-2, determining normal vectors n ═ n of all units in the NN set (n)1,n2,n3)；
S4-4-3: the spatial coordinates of the outer surface points corresponding to node N are calculated based on the following formula (9),
wherein (x)outer，youter，zouter) X-, Y-and Z-axis coordinate values representing outer surface points, (X)inner，yinner，zinner) Coordinate values of X, Y and Z axes representing points on the inner surface, hNN(i)The optimum thickness of the cell numbered nn (i).
FIG. 6 is a cloud image of the outer surface points based on the optimization results obtained in the above steps; FIG. 7 is an outer surface generated based on a point cloud result; FIG. 8 is a problem of partial face separation caused by replacing the outer surface of the original model with a newly generated outer surface; FIG. 9 is a geometric solid model of the muscular lightening design obtained after repair of the separating surface.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art should be considered to be within the technical scope of the present invention, and the technical solutions and the inventive concepts thereof according to the present invention should be equivalent or changed within the scope of the present invention.
Claims (6)
1. A muscle type thin-wall structure lightweight design method is characterized by comprising the following steps:
s1: determining a geometric model designed by an initial structure of the internal pressure bearing component, boundary constraint conditions and load conditions;
s2: designing a geometric model based on the initial structure of the internal pressure-bearing part, and establishing a numerical analysis model of a variable thickness optimization method, wherein a thin-wall region with variable thickness in the numerical analysis model is a design region, and other regions are non-design regions;
s3: the method comprises the steps of obtaining the optimal thickness distribution of the part subjected to internal pressure in a design domain by using a variable thickness optimization method by taking the maximum structural rigidity as an optimization target, taking the material consumption equal to a set value as an optimization constraint condition and taking the unit thickness in the design domain as a design variable;
s4: and establishing a final muscle type thin-wall structure lightweight design geometric model based on the optimal thickness distribution of the design domain.
2. The muscle-type thin-walled structure lightweight design method according to claim 1, further characterized by: the muscle type thin-wall structure is characterized as follows: the inner surface of the pressed part remains unchanged and the outer surface has a smooth and continuous thickness variation which is visible.
3. The muscle-type thin-walled structure lightweight design method according to claim 1, further characterized by: the design domain of the internal pressure bearing component is a thin-wall region of an initial structure design geometric model, and the design domain of the internal pressure bearing component needs to adopt shell units to perform grid division on the basis of the inner surface of the thin-wall region of the initial structure design geometric model.
4. The muscle-type thin-wall structure lightweight design method according to claim 1, characterized in that: the steps of obtaining the optimal thickness distribution in the design domain of the internal pressure bearing part by the variable thickness optimization method are as follows:
s3-1, setting initial parameters including initial thickness parameter, convergence parameter and maximum thickness h of design fieldmaxAnd a minimum thickness hminImporting the grid cell number, the section attribute number and the corresponding thickness information of the design domain, setting an iteration step number k, initializing an initial iteration step number k to be 1, and initializing a design variable vector hk；
S3-2, based on the cell thickness vector hkOutputting a file for recording the thickness of each unit in the design domain;
s3-3, reading in the file of the cell thickness in the design domain, and updating the thickness of each cell in the design domain according to the input file information;
s3-4, carrying out structural response numerical analysis and outputting M of each grid cell in the record design domainx，My，MxyA file of information;
s3-5: reading in and recording unit bending moment M in design domainx，My，MxyThe document of (4), calculating a p value of each cell based on the following formula (5);
wherein: p is a radical ofiTo be provided withThe state parameter of the ith unit in the counting domain, v is Poisson's ratio, Mx，My，MxyRespectively representing bending moment around an X axis, bending moment around a Y axis and torque in an XY plane under a unit local coordinate system;
S3-6, calculating the updated h based on the formula (6)k+1Wherein lambda is Lagrange multiplier, the value of lambda is controlled by a volume constraint formula (3), and the value of lambda is iteratively calculated by adopting a dichotomy until the updated hk+1The volume constraint requirement is met;
s3-7, when the design variable vector of two adjacent iterative steps meets | | hk+1-hkWhen the absolute value is less than or equal to the absolute value, a variable thickness optimization result is obtained; if hk+1-hk| > k +1, return to step S3-2.
5. The muscle-type thin-wall structure lightweight design method according to claim 1, characterized in that: the method for establishing the final muscle type thin-wall structure lightweight design geometric model based on the optimal thickness distribution of the design domain comprises the following steps:
s4-1, determining the optimal thickness of each unit in the design domain based on the variable thickness optimization result;
s4-2, numbering grid nodes in all design domains, acquiring the number NA of all nodes in the design domains, acquiring the number and the serial number of units to which each node belongs, defining a serial number parameter N representing the current calculated node, and setting the serial number of the initial calculated node as N as 1;
s4-3, calculating the normal vector of the unit to which the node with the number of N belongs;
s4-4, calculating the space coordinate of the outer surface point corresponding to the node number N;
S4-5, if N is less than NA, executing N to be N +1, and returning to the step S4-3; if N is NA, proceed to S4-6;
s4-6, sequentially generating all points based on the calculated space coordinates of all the outer surface points to obtain an optimal design outer surface point cloud;
s4-7: obtaining a geometric model of the outer surface of the optimally designed thin-walled region based on the optimally designed outer surface point cloud;
and S4-8, replacing the outer surface of the design domain of the initial model of the design object with the outer surface of the optimized design, repairing the damaged surface generated by the outer surface replacement, and generating the final muscle type thin-wall light-weight design.
6. The muscle-type thin-wall structure lightweight design method according to claim 5, characterized in that: calculating the space coordinates of the outer surface points corresponding to the node N, and the method comprises the following steps:
s4-4-1, determining the number of the unit to which the N node belongs, generating a set NN, and calculating the number NE of elements in the NN;
s4-4-2, determining normal vectors n ═ n of all units in the NN set (n)1,n2,n3)；
S4-4-3: the spatial coordinates of the outer surface points corresponding to node N are calculated based on the following formula,
wherein (x)outer，youter，zouter) X-, Y-and Z-axis coordinate values representing outer surface points, (X)inner，yinner，zinner) Coordinate values of X, Y and Z axes representing points on the inner surface, hNN(i)The optimum thickness of the cell numbered nn (i).
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| US20190080040A1 (en) * | 2017-09-13 | 2019-03-14 | Dalian University Of Technology | Integration method for accurate modeling and analysis and reliability-based design optimization of variable stiffness composite plate and shell structures |
| CN110909496A (en) * | 2019-11-06 | 2020-03-24 | 上海理工大学 | Two-stage collaborative optimization design method for constrained damping structure |
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| US20190080040A1 (en) * | 2017-09-13 | 2019-03-14 | Dalian University Of Technology | Integration method for accurate modeling and analysis and reliability-based design optimization of variable stiffness composite plate and shell structures |
| CN110909496A (en) * | 2019-11-06 | 2020-03-24 | 上海理工大学 | Two-stage collaborative optimization design method for constrained damping structure |
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