CN112364546B - Fiber reinforced composite material structure optimization design method based on bilateral filtering - Google Patents
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Abstract
The invention belongs to the technical field related to the structural optimization design of fiber reinforced composite materials, and discloses a structural optimization design method of a fiber reinforced composite material based on bilateral filtering, which comprises the following steps: (1) Dividing a design domain of the fiber reinforced composite material structure into N units, and taking a fiber angle theta e at the center point of each unit as a design variable; (2) Establishing a unit stiffness matrix K e, and obtaining an overall displacement vector U and a unit displacement vector U e according to finite element analysis and calculation; (3) Calculating an objective function value according to an objective function formula c=f T U, and calculating a sensitivity value of the objective function with respect to the design variable θ e (4) Performing bilateral filtering processing on the sensitivity value at the center point of the unit e in a circular area N e; (5) Updating the design variable theta e by using a sensitivity-based optimization algorithm; (6) And (3) repeating the steps (2) to (5) until the optimization termination condition is met, thereby completing the optimization design conforming to the composite material structure. The invention improves the optimization quality.
Description
Technical Field
The invention belongs to the technical field related to the structural optimization design of fiber reinforced composite materials, and particularly relates to a structural optimization design method of a fiber reinforced composite material based on bilateral filtering.
Background
Compared with the conventional metal material, the fiber reinforced composite material has the advantages of high specific strength, large specific modulus, good fatigue resistance, corrosion resistance, good shock absorption performance and the like, and is widely applied to the fields of aerospace, automobiles, buildings and the like. With the continuous development of automatic tape laying/wire laying technology, the manufacture of the variable-rigidity composite material is possible, and fibers can be laid along a designed curve path. Compared with the linear fiber reinforced constant-stiffness structural design, the variable-stiffness composite structural design has better mechanical properties, and a designer can obtain a structure with better properties by optimally designing the fiber laying angle or path.
The fiber laying angle or path is reasonable, the structural mechanical property is met, and the fiber laying angle or path is an important content of the structural optimization design of the composite material. Existing optimization techniques have proven effective in providing design solutions with significantly enhanced structural performance, but inherent problems are that the fiber paths are not parallel, i.e. spatial continuity of fiber angles cannot be guaranteed, and adjacent fiber angles may be abrupt (vary widely), thus becoming one of the main sources of gaps or overlaps. In the paths set in these design methods, the amount of gaps, overlaps, can affect the structural response, manufacturing time, surface quality of the finished product, and can also present difficulties in modeling in numerical analysis. It is particularly important to consider the manufacturability of the structure during the optimization design stage, so as to minimize manufacturing defects such as gaps or overlapping caused by non-parallel fiber paths, and to ensure the manufacturing quality of the composite material.
Disclosure of Invention
Aiming at the defects or improvement demands of the prior art, the invention provides a fiber reinforced composite material structure optimization design method based on bilateral filtering, which divides a design domain of a fiber reinforced composite material structure into a limited number of units, sets a specific circular area for each unit, takes a unit center point as a circle center, has a radius not smaller than 10 times of the unit size, carries out bilateral filtering treatment on a sensitivity value at a unit e center point in the area to obtain a filtered sensitivity value, takes the filtered sensitivity value as the sensitivity value at the unit e center point, and then utilizes an optimization algorithm based on sensitivity information to update design variables until an optimization termination condition is met, thereby obtaining the fiber reinforced composite material structure design meeting manufacturing requirements, and greatly improving the manufacturing quality of the composite material.
In order to achieve the above object, according to one aspect of the present invention, there is provided a method for optimizing a fiber reinforced composite structure based on bilateral filtering, the method comprising the steps of:
(1) Taking a fiber reinforced composite material structure as a design domain, dividing the design domain into N units, taking a fiber angle theta e at the center point of each unit as a design variable, and setting an initial value of the fiber angle at the center point of each unit, wherein e=1, 2, … and N;
(2) Establishing a unit stiffness matrix K e, and obtaining an overall displacement vector U and a unit displacement vector U e according to finite element analysis and calculation, wherein the unit stiffness matrix depends on a fiber angle theta e;
(3) Calculating an objective function value c according to an objective function formula c=f T U, and calculating a sensitivity value of the objective function with respect to the design variable θ e
(4) Setting a circular area N e with the center point of the unit e as the center, and carrying out bilateral filtering processing on the sensitivity value at the center point of the unit e in the circular area N e to obtain a filtered sensitivity value
(5) Updating the design variable theta e by using a sensitivity-based optimization algorithm;
(6) And (3) repeating the steps (2) to (5) until the optimization termination condition is met, thereby completing the optimization design conforming to the composite material structure.
Further, by the calculation formulaA cell stiffness matrix K e is established that depends on the fiber angle θ e, where dΩ is the area infinitesimal, Ω e is the area of the cell e, B is the displacement strain matrix, and D (θ e) is the cell elastic matrix that depends on the fiber angle θ e.
Further, the calculation formula of D (θ e) is: d (θ e)＝T(θe)D0T(θe)T, where D 0 is the original elastic matrix when the fiber is not rotating, and T (θ e) is the rotation matrix.
Further, in step (3), the sensitivity value of the objective function with respect to the design variable is derived using the relationship between the unit stiffness matrix K e and θ e The calculation formula is/>Where U e is a unit displacement vector corresponding to unit e, and the number of unit e is n e, U e is a column vector composed of elements in the 2n e th and 2n e +1 th rows of the overall displacement vector U.
Further, the filtered sensitivity valueThe calculation formula is as follows:
Wherein θ p is the fiber angle at any cell center point within circular region N e; h (e) is a weight function, and the calculation formula is W c (p-e) is a domain filter function, and p-e represents the Euclidean distance between two unit center points; w s(|θp-θe |) is a range filter function, |θ p-θe | represents the absolute value of the difference in fiber angle at the center point of the two units; the domain filter function W c (x) is calculated/>, using the following formulaWherein σd is the domain filter coefficient; the range filter function W s (x) is calculated/>, using the following formulaWhere σr is the range filter coefficient.
Further, the sensitivity value of the processed objective function with respect to the design variable is used to determine the sensitivity value of the design variable using the objective function value cAnd upper and lower bounds θ max and θ min of the design variables, and updating the design variables in combination with a moving asymptote algorithm.
Further, θ max and θ min are respectively:
θ max＝90°-ε,θmin = -90 °, where ε is a very small positive value.
Further, the mathematical expression of the optimal design model adopted by the method is as follows:
findθe(e＝1,2,...,1600)
min c＝FTU
s.t. KU＝F
θmin≤θe≤θmax
Where the fiber angle value θ e at the cell center point is a design variable, the objective function is compliance c, the design objective minimizes the compliance c of the structure, and the constraints include the equilibrium equation ku=f, the upper and lower bounds θ max and θ min of θ e.
In general, compared with the prior art, the fiber reinforced composite material structure optimization design method based on bilateral filtering mainly has the following beneficial effects:
1. Dividing the design domain of the fiber reinforced composite material structure into a limited number of units, taking the center point of each unit as the center of a circle, setting a specific circular area, carrying out bilateral filtering treatment on the sensitivity value at the center point of the unit e in the corresponding area to obtain a filtered sensitivity value as the sensitivity value at the center point of the unit e, and updating design variables by using an optimization algorithm based on sensitivity information until the optimization termination condition is met, so that the fiber reinforced composite material structure design meeting the manufacturing requirement is obtained, and the manufacturing quality of the composite material is greatly improved.
2. The fiber reinforced composite material obtained by adopting the method of the invention is optimally designed, so that the manufacturing defects of gaps and overlapping caused by unparallel fiber ribbons can be avoided as much as possible, and the manufacturing quality of the composite material is improved to a great extent.
3. The method provided by the invention is easy to implement, has strong applicability and is beneficial to popularization and application.
Drawings
FIG. 1 is a schematic flow chart of a fiber reinforced composite structure optimization design method based on bilateral filtering;
FIG. 2 is a schematic diagram of an example of an optimized design of a planar simply supported beam structure provided by an embodiment of the present invention;
FIG. 3 is a schematic view of an arrangement with respect to a circular region N e;
FIG. 4 is a schematic view of the fiber angle layout at the cell center point obtained using the method provided by the present invention for the example of FIG. 2;
FIG. 5 is a schematic view of the fiber angle layout at the center point of the cell calculated without the method provided by the present invention for the example of FIG. 2.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. In addition, the technical features of the embodiments of the present invention described below may be combined with each other as long as they do not collide with each other.
Referring to fig. 1, the method for optimizing the design of the fiber reinforced composite material structure based on bilateral filtering provided by the invention mainly comprises the following steps:
Step one, taking a fiber reinforced composite material structure as a design domain, dividing the design domain into N units, taking a fiber angle theta e at the center point of each unit as a design variable, and setting an initial value of the fiber angle at the center point of each unit, wherein e=1, 2, … and N.
Step two, a unit stiffness matrix K e is established, and then an overall displacement vector U and a unit displacement vector U e are obtained according to finite element analysis and calculation, wherein the unit stiffness matrix depends on a fiber angle theta e.
Step three, calculating an objective function value c according to an objective function formula c=f T U, and calculating a sensitivity value of the objective function with respect to the design variable θ e
Wherein the sensitivity valueThe calculation formula of (2) is as follows:
Setting a circular area N e with the center point of the unit e as the center, and carrying out bilateral filtering processing on the sensitivity value at the center point of the unit e in the circular area N e to obtain a filtered sensitivity value
Wherein the radius of the circular region N e is not less than 10 times the cell size; filtered sensitivity valuesThe following calculation formula is adopted:
Wherein θ p is the fiber angle at any cell center point within circular region N e; h (e) is a weight function, H -1 (e) is the reciprocal thereof to ensure that the output of the bilateral filtering is the average value of sensitivity near similar values, and the calculation formula is W c (p-e) is a domain filter function, and p-e represents the Euclidean distance between two unit center points; w s(|θp-θe |) is a range filter function, |θ p-θe | represents the absolute value of the difference in fiber angle at the center point of the two units.
The domain filter function W c (x) is calculated using the following formula:
Wherein σd is a domain filter coefficient, and the selection of this parameter directly affects the performance of bilateral filtering, and needs to be adjusted according to different calculation examples.
The range filter function W s (x) is calculated using the following formula:
Wherein σr is a range filter coefficient, and the selection of the parameter directly affects the performance of bilateral filtering and needs to be adjusted according to different calculation examples.
And fifthly, updating the design variable theta e by using a sensitivity-based optimization algorithm.
And step six, repeating the steps 2 to 5 until the optimization termination condition is met, thereby completing the optimization design conforming to the material structure.
The present invention will be described in further detail with reference to the following examples.
Examples
Referring to fig. 2 to 5, the present embodiment illustrates the present invention by taking an optimization problem of minimizing the flexibility of the simple beam structure with in-plane load as an example, and given a rectangular design domain with a size of 1m×4m, the lower left corner and the lower right corner are fixed supporting points, and the lower center receives a downward concentrated load F with a size of 1N.
And (3) performing fiber angle layout optimization on the fiber reinforced composite material simply supported beam structure to minimize the flexibility, wherein the method comprises the following specific steps:
Step one, dividing the design domain of the fiber reinforced composite structure into 20×80 square units, wherein the unit side length is 0.05m, and setting the initial value of the fiber angle theta e at the center point of each unit to be 0 °, namely the fiber level, wherein e=1, 2, …,1600.
Step two, through calculation formulaEstablishing a cell stiffness matrix K e depending on a fiber angle theta e, wherein dΩ is an area infinitesimal, Ω e is an area of a cell e, B is a displacement strain matrix, D (theta e) is a cell elastic matrix depending on the fiber angle theta e, a calculation formula is D (theta e)＝T(θe)D0T(θe)T,D0 is an original elastic matrix when the fiber is not rotated, and T (theta e) is a rotation matrix, and the calculation formula is respectively adopted as follows:
Where E x and E y are Young's modulus, G xy is shear modulus, v xy and v yx are Poisson's ratio, satisfying v xyEy＝νyxEx. The overall rigidity matrix K is obtained through assembling K e, belongs to basic operation in a finite element analysis method, and then the overall displacement field U is obtained through solving according to a formula KU=F, wherein F is an external vector.
Step three, calculating an objective function value c according to c=f T U, and simultaneously deducing a sensitivity value of the objective function on the design variable by utilizing the relation between the unit stiffness matrix K e and the theta e The calculation formula is/>In the formula, U e is a unit displacement vector corresponding to a unit e, and the number of the unit e is n e, and U e is a column vector formed by elements of the 2n e th row and the 2n e +1 th row of the overall displacement vector U.
Fourth, referring to fig. 3, a circular area N e is set with a radius of 0.5m (i.e. 10 times of the unit side length) around the center point of the unit e, and the sensitivity value at the center point of the unit e is filtered in the area to obtain a filtered sensitivity valueThe calculation formula is as follows:
Wherein θ p is the fiber angle at any cell center point within circular region N e; h (e) is a weight function, H -1 (e) is the reciprocal thereof to ensure that the output of the bilateral filtering is the average value of sensitivity near similar values, and the calculation formula is W c (p-e) is a domain filter function, and p-e represents the Euclidean distance between two unit center points; w s(|θp-θe |) is a range filter function, |θ p-θe | represents the absolute value of the difference in fiber angle at the center point of the two units; the domain filter function W c (x) is calculated using the following formula: /(I)Wherein σd is a domain filtering coefficient, and the selection of the parameter directly affects the performance of bilateral filtering and needs to be adjusted according to different calculation examples; the range filter function W s (x) is calculated using the following formula: /(I)Where σr is a range filter coefficient, and the selection of this parameter directly affects the performance of bilateral filtering, and needs to be adjusted according to different calculation examples.
Step five, using the objective function value c, the sensitivity value of the processed objective function with respect to the design variableAnd upper and lower bounds θ max and θ min of the design variables, θ max＝90°-ε,θmin = -90 ° where ε is a very small positive value, here 1×10 -8, and updating the design variables in combination with a moving asymptote algorithm (Method of Moving Asymptotes, abbreviated as MMA).
Step six, repeating the steps two to five, wherein each repetition is called an iteration process, until the optimization termination condition is met, completing the whole optimization design, and the optimization termination condition is simply set to be that the iteration times reach 50 times.
In summary, the fiber reinforced composite structure optimization design model based on bilateral filtering can be summarized as follows:
findθe(e＝1,2,...,1600)
min c＝FTU
s.t. KU＝F
θmin≤θe≤θmax
Wherein the fiber angle value θ e at the cell center point is a design variable, the objective function is compliance c, the design objective minimizes the compliance c of the structure, the constraint includes the equilibrium equation ku=f, and the upper and lower bounds of θ e are θ max and θ min, respectively.
The optimization result of the invention is as follows: the fiber angle layout obtained after filtering by the method is shown in figure 4, and the flexibility value of the objective function is 99.3; the fiber angle layout obtained without any filtering method is shown in fig. 5, the flexibility value of the objective function is 36.7, and the structural rigidity obtained by filtering by the method is smaller than that obtained without any filtering method, but the structural fiber angle layout obtained by filtering by the method is reasonable, is more convenient to manufacture, can meet the manufacturing requirement, and greatly improves the manufacturing quality of the composite material.
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.
Claims (6)
1. A fiber reinforced composite material structure optimization design method based on bilateral filtering is characterized in that:
(1) Taking a fiber reinforced composite material structure as a design domain, dividing the design domain into N units, taking a fiber angle theta e at the center point of each unit as a design variable, and setting an initial value of the fiber angle at the center point of each unit, wherein e=1, 2, … and N;
(2) Establishing a unit stiffness matrix K e, and obtaining an overall displacement vector U and a unit displacement vector U e according to finite element analysis and calculation, wherein the unit stiffness matrix depends on a fiber angle theta e;
(3) Calculating an objective function value c according to an objective function formula c=f T U, and calculating a sensitivity value of the objective function with respect to the design variable θ e
(4) Setting a circular area N e with the center point of the unit e as the center, and carrying out bilateral filtering processing on the sensitivity value at the center point of the unit e in the circular area N e to obtain a filtered sensitivity value
(5) Updating the design variable theta e by using a sensitivity-based optimization algorithm;
(6) Repeating the steps (2) to (5) until the optimization termination condition is met, thereby completing the optimization design conforming to the composite material structure;
balance equation ku=f; filtered sensitivity values The calculation formula is as follows:
Wherein θ p is the fiber angle at any cell center point within circular region N e; h (e) is a weight function, and the calculation formula is W c (p-e) is a domain filter function, and p-e represents the Euclidean distance between two unit center points; w s(|θp-θe |) is a range filter function, |θ p-θe | represents the absolute value of the difference in fiber angle at the center point of the two units; the domain filter function W c (x) is calculated/>, using the following formulaWherein σd is the domain filter coefficient; the range filter function W s (x) is calculated/>, using the following formulaWhere σr is the range filter coefficient;
Sensitivity values of the processed objective function with respect to the design variables using the objective function value c And upper and lower bounds θ max and θ min of the design variables, and updating the design variables in combination with a moving asymptote algorithm.
2. The fiber reinforced composite material structure optimization design method based on bilateral filtering as claimed in claim 1, wherein the method comprises the following steps: by calculationA cell stiffness matrix K e is established that depends on the fiber angle θ e, where dΩ is the area infinitesimal, Ω e is the area of the cell e, B is the displacement strain matrix, and D (θ e) is the cell elastic matrix that depends on the fiber angle θ e.
3. The fiber reinforced composite material structure optimization design method based on bilateral filtering as claimed in claim 2, wherein the method is characterized in that: d (θ e) has the formula: d (θ e)＝T(θe)D0T(θe)T, where D 0 is the original elastic matrix when the fiber is not rotating, and T (θ e) is the rotation matrix.
4. The fiber reinforced composite material structure optimization design method based on bilateral filtering as claimed in claim 1, wherein the method comprises the following steps: in step (3), the sensitivity value of the objective function with respect to the design variable is derived using the relationship between the unit stiffness matrix K e and θ e The calculation formula is/>Where U e is a unit displacement vector corresponding to unit e, and the number of unit e is n e, U e is a column vector composed of elements in the 2n e th and 2n e +1 th rows of the overall displacement vector U.
5. The fiber reinforced composite material structure optimization design method based on bilateral filtering as claimed in claim 1, wherein the method comprises the following steps: θ max and θ min are respectively:
θ max＝90°-ε,θmin = -90 °, where ε is a very small positive value.
6. The fiber reinforced composite structure optimization design method based on bilateral filtering as in any one of claims 1-4, wherein: the mathematical expression of the optimal design model adopted by the method is as follows:
findθe，e＝1,2,...,1600
min c＝FTU
s.t.KU＝F
θmin≤θe≤θmax
Where the fiber angle value θ e at the cell center point is a design variable, the objective function is compliance c, and the design objective minimizes the compliance c of the structure, the upper and lower bounds θ max and θ min of θ e.
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