CN112464531B - B-spline parameterization-based reinforcement modeling and optimizing method for thin-wall structure - Google Patents

B-spline parameterization-based reinforcement modeling and optimizing method for thin-wall structure Download PDF

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CN112464531B
CN112464531B CN202011342095.7A CN202011342095A CN112464531B CN 112464531 B CN112464531 B CN 112464531B CN 202011342095 A CN202011342095 A CN 202011342095A CN 112464531 B CN112464531 B CN 112464531B
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张卫红
丰圣起
孟亮
徐钊
陈亮
酒丽朋
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Northwestern Polytechnical University
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Abstract

The invention relates to a B-spline parameterization-based reinforcement modeling and optimizing method for a thin-wall structure, belonging to the technical field of reinforcement optimization of the thin-wall structure; firstly, a shell unit is used for simulating a thin-wall structure, and an entity unit simulated by taking the allowable height of reinforcement as the maximum is defined on the thin-wall structureHeight H max The design domain of (2) is that a shell and entity coupling unit is used for modeling the thin-wall structure reinforcement at the connecting part of the shell and the entity. A B-spline height field in the parameter domain is then constructed with the control points as design variables. And aiming at the three-dimensional curved surface, mapping the B-spline height field under the parameter domain onto the thin-wall structure by using a parameter mapping method so as to represent the reinforcement on the thin-wall structure. The B spline parameterization method can achieve a filtering effect, and can realize the independence of design variables and grids by taking the control points as the design variables, thereby greatly reducing the design variables and improving the calculation efficiency. More importantly, smooth and continuous reinforcement can be obtained by adopting the B-spline height field.

Description

B-spline parameterization-based reinforcement modeling and optimizing method for thin-wall structure
Technical Field
The invention belongs to the technical field of reinforcement optimization of thin-wall structures, and particularly relates to a B-spline parameterization-based reinforcement modeling and optimization method of a thin-wall structure.
Background
The thin-wall structure is widely applied to the fields of aviation, aerospace, automobiles, ships and the like due to the high rigidity-weight ratio. Due to the poor mechanical properties of the structure, such as rigidity, strength, buckling, vibration and the like, stiffening ribs are usually designed in practical application to reinforce the shell structure, and the shell structure, such as an integral wall plate of an aircraft wing, an engine box, a missile shell, an external fuel tank of a space plane and the like, belong to typical stiffened structures. The shape, size and layout of the reinforcement directly affect the weight and performance of the structure, so that how to design the reinforcement layout of the thin-wall structure and improve the bearing capacity of the structure is a popular research direction with high importance of domestic and foreign scholars and industry.
A reinforcement structure design method based on drawing constraints is provided in the documents H-DGTP-a gravity-function based direct reinforcement mapping for design Optimization of compensator layout and height of thin-walled structure. According to the method, on the basis of a Solid isotropic material penalty method (SIMP), a Heaviside function is utilized to constrain pseudo density, so that design variables can be changed from large to small along a certain direction, and therefore the layout and height of the reinforcement can be designed simultaneously by utilizing a topology optimization method. The design variables and the final optimization result of the method still have grid dependency, and the obtained reinforced structure has the problem of jagged boundaries. The method uses the solid units to simulate the thin-wall structure, so that the solid units need to be divided relatively densely, and more grids are needed in the process. In addition, the method also has the problems that the number of design variables is large, the obtained structure cannot be directly butted with computer aided design software, and the like.
A topological optimization method based on B-spline parameterization is provided in the literature, namely, Topology optimization in B-spline space [ J ]. Qian X. computer Methods in Applied Mechanics & Engineering,2013,265(oct.1):15-35. The method improves the traditional SIMP method by using a B-spline parameter field, uses B-spline control points to replace the traditional units as design variables, and uses the B-spline parameter field to represent an optimized topological structure. The method realizes that the design variables are independent from the grid, and the obtained result is smooth and continuous due to the high-order continuity of the B spline. However, the method is only applied to the traditional two-dimensional plane at present, and the expansion in the aspect of reinforcement needs to be expanded.
Disclosure of Invention
The technical problem to be solved is as follows:
in order to avoid the defects of the prior art, aiming at the problems that the thin-wall structure reinforcement obtained by the prior art has grid dependency and cannot obtain a topological configuration with clear and smooth boundary, the invention provides a B-spline parameterization-based reinforcement modeling and optimizing method for the thin-wall structure. Firstly, the method uses the shell unit to simulate the thin-wall structure, and defines a solid unit simulated reinforced license on the thin-wall structureCan be at a maximum height H max The design domain of (2) is that a shell and entity coupling unit is used for modeling the thin-wall structure reinforcement at the connecting part of the shell and the entity. Then constructing a B spline height field under a parameter domain, taking the control point as a design variable, wherein the value range of the design variable is 0-H max . And aiming at the three-dimensional curved surface, mapping the B-spline height field under the parameter domain onto the thin-wall structure by using a parameter mapping method so as to represent the reinforcement on the thin-wall structure. Aiming at the inherent problem that the closed surface is difficult to process by the B-spline method, the extended B-spline parameterization method is provided, so that the structure can be smoothly and continuously formed at the closed part of the surface. In topology optimization, a cell material parameter of 0-1 is given as the pseudo-density in order to distinguish whether a cell is to be deleted. In the B-spline height field, the pseudo density of the entity unit is judged according to the position relation of the entity unit and the reinforcement represented by the B-spline height field: the pseudo density of the entity unit in the reinforced rib is 1; the pseudo density outside the reinforcement is 0; the unit is cut by the reinforced surface, and the ratio of the volume of the cut part to the volume of the solid unit is used as a pseudo density value. In order to prevent the checkerboard phenomenon, the traditional topology optimization method needs to adopt a sensitivity filtering method or a density filtering method, wherein the filtering method is to modify the pseudo density or sensitivity of a target unit by utilizing information in a certain radius around the unit. This method causes the optimization result to appear in a gray area and increases the amount of calculation. The B-spline parameterization method can achieve a filtering effect, and can realize the independence of design variables and grids by taking the control points as the design variables, thereby greatly reducing the design variables and improving the calculation efficiency. More importantly, smooth and continuous reinforcement can be obtained by adopting the B-spline height field.
The technical scheme of the invention is as follows: the B-spline parameterization-based reinforcement modeling and optimizing method for the thin-wall structure is characterized by comprising the following specific steps of:
the method comprises the following steps: selecting n multiplied by m control points on a parameter domain coordinate system as design variables, selecting the order of a B spline as p, and establishing a B spline height field:
Figure BDA0002798844840000031
wherein the content of the first and second substances,
Figure BDA0002798844840000032
to design a variable, its value range is
Figure BDA0002798844840000033
(i ═ 0, …, n-1, j ═ 0, …, m-1) ξ, η respectively represent the coordinates of the parameter domain; n is a radical of i,p (ξ)、N j,p (η) is a B-spline function, and the expression equation is as follows:
Figure BDA0002798844840000034
step two: aiming at the thin-wall structure, a parameter mapping method is utilized to map a B spline height field on a parameter domain to a three-dimensional curved surface:
f:(ξ,η,h(ξ,η))→(x,y,z). (3)
the parameter mapping equation is as follows:
Figure BDA0002798844840000035
step three: carrying out finite element meshing on a shell unit for a thin-wall structure, and stretching in the normal direction of the thin-wall structure according to the reinforcement height requirement to obtain a design domain, wherein the height of the design domain is the reinforcement height; carrying out mesh division on the entity units for the design domain, and connecting the shell units and the entity units through a coupling unit to obtain a finite element model;
step four: calculating the pseudo density of the entity units in the design domain obtained by stretching in the third step according to the B spline height field mapped in the second step;
the solid elements in the design domain are divided into three types by the B-spline height field: cells within the height field, cells outside the height field, and cells traversed by the interface; wherein the cell pseudo densities in and outside the height field are 0 and 1, respectively, and the cell pseudo density rho penetrated by the interface k The volume of the cell in the B-spline height field is divided by the volume of the solid cell, and since the base areas of the solid cells are the same, the ratio of the areas is expressed as the ratio of the heights:
Figure BDA0002798844840000036
to facilitate derivation of the design variables, the relationship of cell pseudo-density to height field is approximated using the Heaviside function:
Figure BDA0002798844840000041
wherein the content of the first and second substances,
Figure BDA0002798844840000042
is a Heaviside parameter, and the delta H is the unit height delta H approximately equal to H k -H k-1 ，h c Is the value of the cell center point B-spline height field, H k And H k-1 Upper boundary height, H, of kth and k-1 units, respectively 0 ＝0；
Step five: applying load and boundary conditions on the finite element model established in the step three, taking the minimum compliance of the thin-wall structure as an optimization target, and taking the total ratio of the B-spline height field in the design domain as a constraint function; setting an initial value with B-spline control points as design variables, and establishing an optimization model:
Figure BDA0002798844840000043
in the formula, C s The flexibility of the thin-wall structure is ensured; k and U are respectively a stiffness matrix and a displacement vector of the integral structure of the finite element model, K s And U c Respectively, a rigidity matrix and a displacement vector of a thin-wall structure, F is a load vector,
Figure BDA0002798844840000044
representing a volume constraint function, U c T Of thin-walled constructionTransposition of displacement vector, U d Is a displacement vector of the design domain;
step six: carrying out optimization solution on the optimization model established in the step five to obtain an optimization result;
firstly, calculating the flexibility of the thin-wall structure flexibility target function and the flexibility of the constraint function in the fifth step; then substituting the calculated sensitivity and the optimized model into a global convergent mobility asymptotic algorithm GCMMA to carry out optimization design to obtain an optimized design variable;
step seven: and substituting the design variables optimized in the step six into the step one and the step two to obtain the optimized reinforcement layout.
The further technical scheme of the invention is as follows: the thin-wall structure is a cylindrical barrel closed curved surface structure, and an expanded B spline parameter method is adopted to increase the high-order continuity of the structure at a joint; mapping a rectangular parameter domain B spline height field onto the cylindrical barrel by a parameter mapping method; because heavy nodes exist at two ends of the B spline, the height step or the position dislocation of the reinforced structure represented by the B spline at the joint is caused;
the method comprises the steps of expanding a B spline height field of a parameter domain from n multiplied by m control points to n multiplied by (m +2 multiplied by p +2) grid scale, adding p +1 rows of control points at two ends of the B spline height field of the parameter domain corresponding to a joint, and giving the control points corresponding to the other side of the joint so as to avoid the influence of heavy nodes on reinforcement at the joint and ensure the continuity of reinforcement of a cross joint.
The further technical scheme of the invention is as follows: in the third step, the coupling method of the shell unit and the solid unit is a rigidity superposition method.
The further technical scheme of the invention is as follows: in the fifth step, the optimization method of the flexibility of the thin-wall structure is a global convergence moving asymptotic algorithm GCMMA.
The further technical scheme of the invention is as follows: and in the sixth step, the sensitivity is the derivative of the flexibility objective function and the constraint function of the thin-wall structure to the design variable.
Advantageous effects
The invention has the beneficial effects that:
1. the invention adopts a shell and entity coupling method, and uses the shell unit and the entity unit to respectively simulate a thin-wall structure and a reinforced structure. Compared with the traditional simulation of two structures by adopting a single shell unit or a single solid unit, the method can reflect two structures with different thicknesses more truly and accurately.
2. In the aspect of topology optimization, the B spline parameterization method takes the control points as design variables, and compared with the traditional topology optimization method, the design variables are reduced by more than 70%. Due to the characteristics of the B-spline parameters, the method has the sensitivity filtering effect, does not need additional sensitivity or density filtering calculation, and improves the calculation efficiency.
3. By utilizing the high-order continuity of the B-spline, the obtained reinforcement configuration is smooth and continuous, the problem of saw-tooth shape is avoided, and the reinforcement layout with gradually-changed height can be obtained.
4. In the prior art, a B spline parameterization method is applied to design and optimization of a plane structure or a three-dimensional main body structure, and solves the technical problem of topology and optimization of the main body structure; the invention adopts a B-spline parameterization method to solve the problem of optimization of the layout of the reinforcement structure on the main body, the height and the layout of the reinforcement can be truly expressed through the mapping of a B-spline height field and parameters, and the obtained optimization result can be seamlessly integrated with computer aided design software.
Drawings
FIG. 1 is a schematic diagram of a parameter domain B-spline height field to cylindrical shell roof parameter mapping employed in the method of the present invention.
FIG. 2 is a block diagram of a method for solving the continuity problem of a closed cylinder by using an extended B-spline parameterization method proposed by the method of the present invention.
FIG. 3 is a schematic diagram of the calculation of the unit pseudo density in the B-spline height field by the method of the present invention.
FIG. 4 shows the design domain, boundary conditions and operating conditions of the method of the present invention for solving the problem of cylindrical shell roof ribbing.
FIG. 5 is an optimized iteration curve of the method of the invention about a cylindrical shell roof and a reinforcement result in an iteration process.
FIG. 6 is a diagram showing the effect of the present invention on the reinforcement distribution after the optimization of the cylindrical shell roof.
FIG. 7 shows the design domain, boundary conditions and working conditions for solving the problem of reinforcement of the thin-wall bent pipe by the method of the present invention.
FIG. 8 is an optimized iteration curve of the method of the present invention for a bent pipe and a reinforcement result in the iteration process.
FIG. 9 is a diagram showing the effect of the present invention on the reinforcement distribution after the elbow pipe is optimized.
Detailed Description
The embodiments described below with reference to the drawings are illustrative and intended to be illustrative of the invention and are not to be construed as limiting the invention.
Example one: the method is used for modeling and optimizing design of the cylindrical shell roof structure reinforcement. Referring to fig. 4, a thin-wall cylindrical shell roof with a target curvature radius of 2.mm, a length of 40mm and a thickness of 0.1mm and a circular arc angle of 2rad is designed, and the maximum height of the reinforcement is 4 mm. Four corners of the cylindrical shell roof are fixed, and a normal point load is applied to the center. The point load size is 1000N. The Young's modulus and Poisson's ratio of the solid material are respectively 2.1 × 10 5 Pa,ν＝0.3。
Step 1, selecting 40 multiplied by 40B spline control points as design variables, taking p-q-3 as the B spline order, and establishing a proper parameter domain B spline height field
Figure BDA0002798844840000061
The design variable has a value range of
Figure BDA0002798844840000062
Step 2, establishing a mapping relation between a parameter domain B spline height field and a cylindrical shell roof:
Figure BDA0002798844840000063
in the formula, L is 40, R is 20, theta is formed by [0,2], h is a B-spline height field, and xi and eta respectively represent the coordinates of the parameter domain. And mapping the B-spline height field of the parameter domain to the cylindrical shell roof structure, wherein the maximum height of the B-spline height field is the maximum height of the design domain, and the B-spline surface can perfectly represent the layout and height of the reinforcement.
And 3, establishing a design domain in the normal direction of the curved surface according to the reinforcement height as shown in fig. 4, and respectively adopting 40 × 40 shell units and 40 × 40 × 4 solid units to perform grid division on the thin-wall structure and the design domain. The counter shell and the solid contact portion are connected by a coupling unit.
Step 4, obtaining the height H of the upper surface of each solid unit according to the solid units obtained in the step 3 k And calculating the pseudo density of the unit through a Heaviside function according to the position relation between the B-spline height field and the unit. The height of the solid unit is 1 according to
Figure BDA0002798844840000071
We can get β ═ 5.9, and substitute the height of each cell and the B-spline height field into the Heaviside equation:
Figure BDA0002798844840000072
the corresponding pseudo-density of each cell is obtained.
And 5, applying load and boundary conditions to the model obtained in the step 3, completely constraining the left end freedom degree, and applying four point loads to the right end. And selecting an optimization target with the minimum compliance of the thin-wall structure, and constraining the structural component ratio to be 0.1. Setting an initial value of a design variable to be 0.1, and establishing an optimization model:
Figure BDA0002798844840000073
Figure BDA0002798844840000074
Figure BDA0002798844840000075
and 6, substituting the GCMMA algorithm into the optimization model, calculating an objective function and constraint sensitivity, and carrying out optimization design, wherein an optimization iteration curve chart 5 shows.
TABLE 1
Structural compliance (kJ) Total ratio of structure
Before optimization 149.97 0.082
After optimization 47.76 0.1
And 7, substituting the design variables optimized in the sixth step into the first step and the second step to obtain the optimized reinforcement layout as shown in the figure 6.
Example two: the method is used for modeling and optimizing design of the reinforcement of the thin-wall bent pipe structure. Referring to fig. 7, the design target is a thin-wall bent pipe with the radius of 10mm and the thickness of 0.1mm, the arc angle of the bent pipe is pi/2, the radius from the center line of the bent pipe to the curvature center of the bent pipe is 60mm, and the maximum reinforcement height is 2 mm. The left side of the bent pipe is fixed, and the pipe orifice at the lower end is uniformly loaded rightwards. The sum of the uniformly distributed loads is F-1000N. The Young's modulus and Poisson's ratio of the solid material are respectively 2.1 × 10 5 Pa,ν＝0.3。
Step 1, establishing a proper parameter domain B spline height field
Figure BDA0002798844840000081
The design variable has a value range of
Figure BDA0002798844840000082
The B spline order is p-q-3. As the structure is a closed cylinder, according to the method in the first step, 63 x 73B-spline control points are selected, wherein 63 x 63 control points are used as design variables, the rest control points are uniformly distributed at two ends of a control point grid and are respectively endowed with variable values with the same opposite positions at two sides of the same joint, so that the influence of a B-spline heavy node on the reinforcement layout of the joint is eliminated, and the reinforcement is in high-order continuity at the joint.
Step 2, establishing a mapping relation between a parameter domain B spline height field and a bent pipe:
Figure BDA0002798844840000083
r in the formula 1 ＝60,R 2 ＝50,θ∈[0,2π]H is a B spline height field, and xi and eta respectively represent the coordinates of the parameter domain. And mapping the B-spline height field of the parameter domain to the bent pipe, so that the maximum height of the B-spline height field is the maximum height of the design domain, and the B-spline surface can perfectly represent the layout and height of the reinforcement.
And 3, selecting 60 × 60 shell units and 60 × 60 × 2 entity units to perform grid division on the thin-wall structure and the design domain for the optimization problem shown in fig. 7. The counter shell and the solid contact portion are connected by a coupling unit.
And 4, calculating the pseudo density of the unit through a Heaviside function according to the position relation between the B-spline height field and the unit. The height of the solid unit is 1 according to
Figure BDA0002798844840000084
The beta is 5.9, and the height of each cell and the height field of the B-spline are respectively substituted into the Heaviside formula
Figure BDA0002798844840000085
The corresponding pseudo-density of each cell is obtained.
And 5, applying load and boundary conditions to the model obtained in the step 3, completely constraining the left end freedom degree, and applying four point loads to the right end. And selecting an optimization target with the minimum compliance of the thin-wall structure, and constraining the structural component ratio to be 0.2. Setting an initial value of a design variable to be 0.2, and establishing an optimization model:
Figure BDA0002798844840000091
Figure BDA0002798844840000092
Figure BDA0002798844840000093
and 6, substituting the GCMMA algorithm into the optimization model, calculating an objective function and constraint sensitivity, and carrying out optimization design, wherein an optimization iteration curve chart 8 is shown.
TABLE 2
Structural flexibility (kJ) Total ratio of structure
Before optimization 39.97 0.192
After optimization 17.58 0.2
And 7, substituting the design variables optimized in the sixth step into the first step and the second step to obtain the optimized reinforcement layout as shown in the figure 9.
Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention, and that variations, modifications, substitutions and alterations can be made in the above embodiments by those of ordinary skill in the art without departing from the principle and spirit of the present invention.

Claims (5)

1. The B-spline parameterization-based reinforcement modeling and optimizing method for the thin-wall structure is characterized by comprising the following specific steps of:
the method comprises the following steps: selecting n multiplied by m control points on a parameter domain coordinate system as design variables, selecting the order of a B spline as p, and establishing a B spline height field:
Figure FDA0002798844830000011
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0002798844830000012
to design a variable, its value range is
Figure FDA0002798844830000013
Xi and eta respectively represent the coordinates of the parameter domain; n is a radical of i,p (ξ)、N j,p (η) is a B-spline function, and the expression equation is as follows:
Figure FDA0002798844830000014
step two: aiming at the thin-wall structure, a parameter mapping method is utilized to map a B spline height field on a parameter domain to a three-dimensional curved surface:
f:(ξ,η,h(ξ,η))→(x,y,z). (3)
the parameter mapping equation is as follows:
Figure FDA0002798844830000015
step three: carrying out finite element meshing on a shell unit for a thin-wall structure, and stretching in the normal direction of the thin-wall structure according to the reinforcement height requirement to obtain a design domain, wherein the height of the design domain is the reinforcement height; carrying out mesh division on the entity units for the design domain, and connecting the shell units and the entity units through a coupling unit to obtain a finite element model;
step four: calculating the pseudo density of the entity units in the design domain obtained by stretching in the third step according to the B spline height field mapped in the second step;
the solid elements in the design domain are divided into three types by the B-spline height field: cells within the height field, cells outside the height field, and cells traversed by the interface; wherein the cell pseudo densities in and outside the height field are 0 and 1, respectively, and the cell pseudo density rho penetrated by the interface k The volume of the unit in the B-spline height field is divided by the volume of the solid unit, and since the base areas of the solid units are the same, the ratio of the areas is expressed as the ratio of the heights:
Figure FDA0002798844830000021
to facilitate derivation of the design variables, the relationship of cell pseudo-density to height field is approximated using the Heaviside function:
Figure FDA0002798844830000022
wherein the content of the first and second substances,
Figure FDA0002798844830000023
is a Heaviside parameter, and the delta H is the unit height delta H approximately equal to H k -H k-1 ，h c Is the value of the cell center point B-spline height field, H k And H k-1 Upper boundary heights, H, of the kth and k-1 cells, respectively 0 ＝0；
Step five: applying load and boundary conditions on the finite element model established in the step three, taking the minimum compliance of the thin-wall structure as an optimization target, and taking the total ratio of the B spline height field in the design domain as a constraint function; setting an initial value with B-spline control points as design variables, and establishing an optimization model:
Figure FDA0002798844830000024
in the formula, C s The flexibility of the thin-wall structure is ensured; k and U are respectively a rigidity matrix and a displacement vector of the integral structure of the finite element model, K s And U c Respectively, a rigidity matrix and a displacement vector of a thin-wall structure, F is a load vector,
Figure FDA0002798844830000025
representing a volume constraint function, U c T Is the transposition of the displacement vector of the thin-walled structure, U d Is a displacement vector of the design domain;
step six: carrying out optimization solution on the optimization model established in the fifth step to obtain an optimization result;
firstly, calculating the flexibility of the thin-wall structure flexibility target function and the flexibility of the constraint function in the fifth step; then substituting the calculated sensitivity and the optimized model into a global convergent mobility asymptotic algorithm GCMMA to carry out optimization design to obtain an optimized design variable;
step seven: and substituting the design variables optimized in the step six into the step one and the step two to obtain the optimized reinforcement layout.
2. The method for the ribbing modeling and optimization of the thin-walled structure based on the B-spline parameterization of claim 1, wherein: the thin-wall structure is a cylindrical barrel closed curved surface structure, and an expanded B spline parameter method is adopted to increase the high-order continuity of the structure at a joint; mapping a rectangular parameter domain B spline height field onto the cylindrical barrel by a parameter mapping method; because heavy nodes exist at two ends of the B spline, the height step or the position dislocation of the reinforced structure represented by the B spline at the joint is caused;
expanding a B spline height field of a parameter domain from n multiplied by m control points to n multiplied by (m +2 multiplied by p +2) grid scale, adding p +1 rows of control points at two ends of the B spline height field of the parameter domain corresponding to the joint respectively, and giving the control points corresponding to the other side of the joint so as to avoid the influence of heavy nodes on the reinforcement of the joint and ensure the continuity of the reinforcement of the bridging joint.
3. The method for the ribbing modeling and optimization of the thin-walled structure based on the B-spline parameterization of claim 1, wherein: in the third step, the coupling method of the shell unit and the solid unit is a rigidity superposition method.
4. The method for the ribbing modeling and optimization of the thin-walled structure based on the B-spline parameterization of claim 1, wherein: in the fifth step, the optimization method of the flexibility of the thin-wall structure is a global convergence moving asymptotic algorithm GCMMA.
5. The method for the ribbing modeling and optimization of the thin-walled structure based on the B-spline parameterization of claim 1, wherein: and in the sixth step, the sensitivity is the derivative of the flexibility target function and the constraint function of the thin-wall structure to the design variable.
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