CN106202699A - A kind of sensitivity method for solving under many displacement constraints - Google Patents

A kind of sensitivity method for solving under many displacement constraints Download PDF

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CN106202699A
CN106202699A CN201610532640.6A CN201610532640A CN106202699A CN 106202699 A CN106202699 A CN 106202699A CN 201610532640 A CN201610532640 A CN 201610532640A CN 106202699 A CN106202699 A CN 106202699A
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displacement
matrix
constraint
array
sensitivity
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CN106202699B (en
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罗利龙
常亮
王立凯
李倩
张生贵
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AVIC Aircraft Strength Research Institute
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/15—Vehicle, aircraft or watercraft design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/10—Numerical modelling
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/06—Power analysis or power optimisation

Abstract

Sensitivity method for solving under a kind of many displacement constraints, comprises the steps: to build aircaft configuration Optimized model, obtains variable, boundary condition and displacement constraint information under each operating mode of Optimized model；Static analysis, calculates real displacement；By boundary condition and displacement constraint, operating mode is grouped, obtains real transposed matrix U；According to boundary condition yojan K, form Ki；Dummy load array Pri is defined according to displacement constraint；Solving equation group Ki*Upsei=Pri；Computing unit differential matrix Kde, package assembly differential matrix Kd；Extract each row the transposition of real displacement array U, be multiplied with the respective column of structure differential stiffness matrix Kd and virtual displacement array Upse, obtain each displacement constraint sensitivity of current variable.The present invention solves the problem that under different operating mode, the sensitivity of many displacement constraints solves, need not in advance border to empty operating mode do and assume, data reusing is considered so that law of planning based on sensitivity information can apply to the structural optimization problems under the global restrictions such as displacement when the equation of static equilibrium solves.

Description

A kind of sensitivity method for solving under many displacement constraints
Technical field
The invention belongs to field of airplane structure, relate to the many displacements in a kind of aircaft configuration law of planning optimization design about Sensitivity method for solving under Shu.
Background technology
In aircaft configuration optimization designs, law of planning based on variable sensitivity information is to solve to consider that displacement, torsional angle etc. are complete The effective means of office's constrained optimization problems.Analytic method and semi analytical method is used to carry out needing each when displacement constraint sensitivity solves Displacement constraint calculates its corresponding virtual displacement, owing to Aircraft Structural Analysis needs to simulate multiple flight progress, causes analyzing moulder Condition and boundary condition are varied.And to when accounting for the successional optimization of malformation such as structures such as wings, usually need To apply displacement constraint in batches, multi-state multiple constraint makes displacement sensitivity solve and directly affects efficiency and the essence that law of planning optimizes Degree.Existing optimization tool the most only comprises a set of real operating mode when displacement sensitivity solves, and overlaps operating mode even if having more, the most only takes a certain The boundary condition of set operating mode calculates all virtual displacement.Additionally, need when virtual displacement solves to solve the equation of static equilibrium, this During need, according to boundary condition, structural stiffness matrix is carried out yojan, when multiple empty operating modes use same boundary conditions, Stiffness matrix after yojan is the most identical, discounting for the reusing of stiffness matrix after yojan, but each operating mode is carried out one Secondary single stiffness matrix yojan, along with scale of model and the increase of constraint scale, computer is unacceptable.Disadvantages mentioned above is serious Constrain the law of planning based on sensitivity information application in solving Large-scale Optimization Problems.
Summary of the invention
It is an object of the invention to provide the sensitivity method for solving under a kind of many displacement constraints, solve multidigit under different operating mode Move the problem that the sensitivity of constraint solves.
The purpose of the present invention is achieved through the following technical solutions:
Step one, builds aircaft configuration Optimized model, obtains the operating mode of described Optimized model, variable, boundary condition, displacement The information of the displacement constraint point that constraint and displacement constraint apply；
Step 2, carries out static analysis to described Optimized model, obtains displacement and the stiffness matrix K of structure of structure；
Step 3, finds out the described operating mode making each described displacement constraint point produce maximum displacement, makes find respectively meanwhile Described displacement constraint point produces the described operating mode of maximum displacement and classifies by described boundary condition, will have identical described border The described operating mode of condition is divided in same group, arranges according to the order of the described displacement constraint point chosen, and forms real displacement array U；
Step 4, carries out yojan to described stiffness matrix K, eliminates the singularity of described stiffness matrix K, forms Ki matrix；
Step 5, in described Ki matrix, chooses the described displacement constraint point identical with step 3, in described displacement about Apply specific loading in spot, form dummy load array Pri；
Step 6, is calculated the structure virtual displacement array under different described displacement constraint and different described boundary condition Upsei, and multiple described structure virtual displacement arrays Upsei are constituted virtual displacement matrix U pse；
Step 7, calculates the unit differential stiffness matrix Kde of described variable, and according to described unit differential stiffness matrix Kde assembles structure differential stiffness matrix Kd；
Step 8, extracts each row the transposition of described real displacement array U successively, with described structure differential stiffness matrix Kd and The respective column of described virtual displacement array Upse is multiplied, and obtains each displacement constraint sensitivity of current variable.
Preferably, forming described Ki matrix in described step 4 can be according to described border different types of in step 3 The number of condition, as the cycle-index of stiffness matrix K described in yojan, carries out yojan to described stiffness matrix K, forms described Ki Matrix.
Preferably, in described step 6, structure virtual displacement array Upsei is according to solving equilibrium equation Ki*Upsei=Pri Obtain.
Preferably, in described step 8, the solution formula of each displacement constraint sensitivity of current variable is: Wherein,For the sensitivity of r displacement constraint in i-th variable,The transposition arranged for r in real transposed matrix U, For structure differential stiffness matrix Kd,R for virtual displacement square array Upse arranges.
Having the beneficial effects that of sensitivity method for solving under many displacement constraints provided by the present invention, it is not necessary in advance to void The border of operating mode is done it is assumed that respect engineering reality completely, has taken into full account the reusability of data, carry when the equation of static equilibrium solves Computer efficiency and accuracy are risen so that law of planning based on sensitivity information can apply in large-scale structure optimization problem.
Accompanying drawing explanation
Fig. 1 is the detail flowchart that in the sensitivity method for solving under the many displacement constraints of the present invention, sensitivity calculates；
Detailed description of the invention
Clearer for the purpose making the present invention implement, technical scheme and advantage, below in conjunction with in the embodiment of the present invention Accompanying drawing, the technical scheme in the embodiment of the present invention is further described in more detail.In the accompanying drawings, the most identical or class As label represent same or similar element or there is the element of same or like function.Described embodiment is the present invention A part of embodiment rather than whole embodiments.The embodiment described below with reference to accompanying drawing is exemplary, it is intended to use In explaining the present invention, and it is not considered as limiting the invention.Based on the embodiment in the present invention, ordinary skill people The every other embodiment that member is obtained under not making creative work premise, broadly falls into the scope of protection of the invention.
Below in conjunction with the accompanying drawings the sensitivity method for solving under many displacement constraints of the present invention is described in further details.
A kind of undercarriage course stiffness simulation method of the present invention, comprises the steps:
Step one, builds aircaft configuration Optimized model, obtains the operating mode of described Optimized model, variable, boundary condition, displacement Constraint information (the displacement constraint point applied containing displacement constraint and displacement constraint).
Operating mode is practical working situation, i.e. residing various working environments.
Wherein, variable can be the area of section of the thickness of metal, bar and beam and the gross thickness of composite and layering Thickness etc.；
Structure has different boundary conditions, such as one bar in different operating environments, retrain the node of with Node in the middle of constraint will form two arbitrary boundary conditions；
Displacement constraint information refers to the definition of displacement constraint in optimization, including applying node serial number and the constraint of displacement constraint Degree of freedom (X, Y, Z), wherein, has 6 degree of freedom at one node of three dimensions, but to some specific nodal Displacement Constraint Consider 3 translational degree of freedom (X, Y, Z), therefore, to a node, at most can only apply 3 displacement constraints, but an optimization Problem can apply displacement constraint at different nodes, and the total number applying displacement constraint is not limited by the inventive method in theory System, i.e. displacement constraint number is multiplied by number of degrees of freedom, equal to node number.It should be noted that displacement constraint point and node implication phase With, simply statement difference, it is possible to replace.
Step 2, carries out static analysis to aircaft configuration Optimized model, obtains displacement and the stiffness matrix of structure of structure K, wherein, carries out the most complete static analysis according to the operating mode of actual definition to aircaft configuration Optimized model, obtains structure institute There is the displacement result under actual condition.
Assume that Optimized model comprises n operating mode, Sub1、Sub2、…、Subn, then Us can be expressed as:Then first row is Sub1Real displacement array U of operating mode1, the position of each node of Optimized model Moving and be respectively a11, a21 ..., ak1, secondary series is Sub2Real displacement array U of operating mode2, arrive Sub by that analogynOperating mode, by n Individual displacement array U constitutes real transposed matrix Us, the displacement result under the most all actual conditions.(note: step 2 is also not directed to position Move constraint information, the displacement result under the actual condition that the most complete static analysis obtains.)
Stiffness matrix K is square formation, and its line number and columns are all completely the same with the line number of Us, all represents all nodes of structure Number of degrees of freedom, i.e. displacement constraint number.
Step 3, finds out the operating mode making each displacement obligatory point produce maximum displacement, meanwhile, makes each displacement constraint by find Point produces the operating mode of maximum displacement and classifies by boundary condition, and the operating mode with same boundary conditions is divided in same group, Arrange according to the order of the displacement constraint point chosen, form real displacement array U.
Assume that Optimized model has n operating mode, k displacement constraint, pick out each displacement obligatory point maximum displacement place respectively Operating mode C1, C2..., Ck, plurality of operating mode number may repeat, and the operating mode therefore picked out must be initial all works The subset of condition.The each operating mode that will be singled out is divided into m group, S by boundary condition1, S2... Sm, often group comprises identical perimeter strip Part, extracts real displacement corresponding to each constraint and by row storage to real according to the displacement constraint order that each operating mode in this m group is corresponding Displacement array U, then the columns of real displacement array U is displacement constraint number, the displacement constraint that each row shift value is corresponding with array S Sequence must keep consistent.
The operating mode how picking out each displacement obligatory point maximum displacement place is described in detail.Assume that Optimized model has N operating mode, k displacement constraint, first displacement constraint point is No. 2 nodes, knows that its corresponding maximum displacement occurs in from Us Sub1, then the first row that displacement array is Us picked out, second displacement constraint point is No. 5 nodes, knows that it is corresponding from Us Maximum displacement also appears in Sub1, then the displacement array picked out is still the first row of Us, if second displacement constraint point Be No. 3 nodes, know that its corresponding maximum displacement occurs in Sub2 from Us, then the displacement array picked out will be the second of Us Row, selected first row and secondary series are all that each operating mode reality displacement that will be singled out arrives real displacement by row storage it is assumed that by that analogy Array U.
Step 4, carries out yojan to stiffness matrix K, forms Ki matrix.According to boundary condition different types of in step 3 Number, as the cycle-index of yojan stiffness matrix K, stiffness matrix K is carried out yojan, forms Ki matrix.
To how carrying out yojan stiffness matrix K illustrate.By boundary condition, stiffness matrix K is divided into m group, and it is right only to need K constraint circulates m time, can reject the row and column repeated in stiffness matrix K, eliminates the singularity of stiffness matrix K, forms Ki square Battle array.If not being grouped circulation, will screen whole stiffness matrix K, operand is big, based on same boundary conditions circulation is easy to Calculate.
Step 5, in matrix K i, chooses the displacement constraint point identical with step 3, applies single on displacement obligatory point Position load, forms dummy load array Pri.
Single displacement constraint is consequently exerted on certain single-degree-of-freedom of certain node, and dummy load is exactly the constraint at this node Applying specific loading on degree of freedom, specific loading is 1, and remaining panel load is 0, and its interior joint is ascended the throne shifting obligatory point.
To how carrying out dummy load array Pri illustrate.Matrix K i is divided into m group by boundary condition, it is assumed that a certain group In have 3 displacement constraints, be respectively applied to No. 1 node, No. 2 nodes, No. 3 nodes, then corresponding virtual displacement load column is respectively For:
Pr1=[1,0 ..., 0]T
Pr2=[0,1 ..., 0]T
Pr3=[0,0,1 ..., 0]T
Wherein, the cycle-index in each group depends on that this group intrinsic displacement retrains number.I in dummy load array Pri It is node serial number.
It should be noted that applying node and the reality displacement array U interior joint one_to_one corresponding in above step three of matrix K i.
Step 6, according to equilibrium equation group Ki*Upsei=Pri, respectively obtains different displacement constraint and various boundary Under structure virtual displacement array Upsei, and by multiple structure virtual displacement arrays Upsei constitute virtual displacement matrix U pse.
Calculate virtual displacement Upsei obtained under each displacement constraint correspondence dummy load one by one, and be stored into Upse by row, The columns of whole Upse is displacement constraint number, and order is consistent with the sequence of displacement constraint in array U and S, and (implication of U and S is shown in Second segment in bright book detailed description of the invention step 3).
Step 7, calculates the unit differential stiffness matrix Kde of variable, and according to unit differential stiffness matrix Kde assembling knot Structure differential stiffness matrix Kd.
The variable mentioned in variable described herein and above step one represents same things, and each variable is to each Individual displacement constraint has a sensitivity value, and stiffness matrix K is differentiated by this step about current variable, and each variable is required for weight This process multiple.It is to solve displacement to bring displacement sensitivity analytic expression into that this step stiffness matrix K differentiates about current variable Sensitivity.
The origin of displacement sensitivity analytic expression be described below:
Displacement constraint is widely used a kind of overall situation deflection constraint form in Engineering Structure Optimum, hereafter will put down from static(al) Weighing apparatus equation sets out, the analytical expression of the derivation displacement sensitivity of system.
Equation of static equilibrium general expression is:
KU=F ... ... ... ... ... ... (1)
Above formula (1) two ends are to i-th variable xiDerivation can obtain:
∂ K ∂ x i U + K ∂ U ∂ x i = ∂ F ∂ x i ... ( 2 )
Owing in most of Static Models, load does not change with the change of variable, i.e. think load and variable not phase Close, therefore formula (2) can be reduced to:
∂ U ∂ x i = - K - 1 ∂ K ∂ x i U ... ( 3 )
Above formula (3) is the analytic expression of displacement sensitivity, but in Practical Project problem, scale of model is the biggest, and structure is firm Degree matrix wan or quasi-morbid state, it is difficult to directly invert.Introduce dummy load method herein and solve this problem.Process is as follows:
If the numbered r of the degree of freedom of displacement constraint, then can generate a dummy load array, be expressed as formula (4):
Formula (4) is substituted into formula (1), can obtain after conversion:
U p s e r = K - 1 P r ... ( 5 )
Formula (3) makees transposition, can obtain:
∂ U r T ∂ x i = - U r T ∂ K ∂ x 1 K - 1 ... ( 6 )
Formula (6) two ends simultaneously take advantage of a weak point in opponent's defence load column obtain:
∂ U r ∂ x i = - U r T ∂ K ∂ x i K - 1 P r ... ( 7 )
Formula (5) is substituted into formula (7), and the displacement derivatives of i-th variable can be expressed as by the r displacement:
∂ U r ∂ x i = - U r T ∂ K ∂ x i U p s e r ... ( 8 )
Above formula (8) is and introduces the full analytic expression of displacement sensitivity that dummy load method is later.
Wherein, observation type (3) and (8), it is possible to find two formulas all have For structure differential stiffness matrix Kd, Kd is Existing conventional means.
Step 8, extracts each row the transposition of real displacement array U successively, arranges with structure differential stiffness matrix Kd and virtual displacement The respective column of battle array Upse is multiplied, and obtains each displacement constraint sensitivity of current variable.
Being different for different variant structural differential stiffness matrix Kd, fetch bit moves the right of matrix U and matrix U pse successively Should arrange and the Kd of current variable, calculate the sensitivity of all displacement constraints corresponding to current variable according to the formula (8) tried to achieve above.
∂ U r ∂ x i = - U r T ∂ K ∂ x i U p s e r
This formula once calculates the variable sensitivity about a displacement constraint,For the r in i-th variable The sensitivity of individual displacement constraint,The transposition arranged for r in real transposed matrix U,For structure differential stiffness matrix Kd,For The r row of virtual displacement square array Upse.
Each displacement constraint sensitivity corresponding for one variable is calculated successively and stored, more next variable has been counted Calculate, finally by all sensitivity results Formatting Outputs.
Give one example below and be described in detail, it is assumed that aircaft configuration Optimized model comprises 6 operating modes, 6 variablees, 4 Individual displacement constraint information.
6 variablees of storage of array defined in the first step, 4 displacement constraint information.
Second step obtains real displacement array Us, totally 6 row, the displacement structure under each column one operating mode of storage, it is assumed thatFirst row is real displacement array U of the first operating mode1, modal displacement is respectively 4,2,3, 5, secondary series is real displacement array U of the second operating mode2, modal displacement is respectively 5,7,6,3, and rear three arrange by that analogy, by five Individual displacement array U constitutes this reality transposed matrix Us.
Due to only 4 displacement constraint information, then stiffness matrix K is the square formation of 4 × 4.
3rd this optimization problem of step has 4 displacement constraints, and the 1st displacement constraint point is No. 3 nodes, knows that it is corresponding from Us Maximum displacement (numerical value 8 in the i.e. the 3rd row) occurs in the 4th row (i.e. Sub4), then the displacement array is Us the 4th row picked out, 2nd displacement constraint point is No. 2 nodes, knows that its corresponding maximum displacement (numerical value 6 the i.e. the 2nd row) also appears in the 3rd from Us Row (i.e. Sub3), then the displacement array is Us the 3rd row picked out, the 3rd displacement constraint point is No. 1 node, knows it from Us Corresponding maximum displacement (numerical value 8 in the i.e. the 1st row) still occurs in the 3rd row (i.e. Sub3), then the displacement array picked out is Us The 3rd row, the 4th displacement constraint point is No. 4 nodes, knows that its corresponding maximum displacement (numerical value 9 the i.e. the 4th row) occurs from Us (i.e. Sub5) is arranged, then the displacement array is Us the 5th row picked out the 5th.I.e. real displacement array U has 4 row, stores Us's respectively 4th, 3,3,5 row.
Assuming that will be singled out 4 operating modes are divided into 2 groups by boundary condition, same boundary conditions divides one group, it is assumed that the 1st group Comprise constraint the 1 and 4, the 2nd group and comprise constraint 2 and 3.
It is expressed as with form:
U needs the border number arrangement corresponding according to operating mode, first arranges the various boundary conditions of identical group, rearranges difference Group, wherein, arranges according to the order of the displacement constraint point chosen, and selected example i.e. comes according to the order of constraint number 1,4,2,3 Arrangement, obtains
Stiffness matrix K is divided into 2 groups by boundary condition by the 4th step, only need to can eliminate 4 constraint circulations 2 times just The singularity of degree matrix K, rejects the row and column repeated in stiffness matrix K, forms Ki matrix.
5th step matrix K i is divided into 2 groups according to boundary condition, has 2 displacement constraints in first group, is respectively applied to No. 1 joint Point, No. 4 nodes, then corresponding virtual displacement load column is respectively as follows: Pr1=[1,0,0,0]TWith Pr4=[0,0,0,1]T；Second Have 2 displacement constraints in group, be respectively applied to No. 2 nodes, No. 3 nodes, then corresponding virtual displacement load column is respectively as follows: Pr2 =[0,1,0,0]TWith Pr3=[0,0,1,0]T, it should be noted that the applying node of matrix K i is real displacement with the 3rd step above Array U interior joint one_to_one corresponding.
6th step, according to equilibrium equation Ki*Upsei=Pri, can calculate the structure virtual displacement under each dummy load Array Upsei, and it is assembled into Upse；Then Upse1 is the first row of Upse, Upse4 For the secondary series of Upse, Upse2 is the 3rd row of Upse, and Upse3 is the 4th row of Upse.
7th step, calculates the unit differential stiffness matrix Kde of variable, and according to unit differential stiffness matrix Kde assembling knot Structure differential stiffness matrix Kd, wherein, variable is 6.
8th step extracts each row the transposition of real displacement array U successively, arranges with structure differential stiffness matrix Kd and virtual displacement The respective column of battle array Upse is multiplied, and obtains each displacement constraint sensitivity of current variable.
Real displacement array U, virtual displacement array Upse, structure differential stiffness matrix Kd are 4 × 4 matrixes, it is known that a change Amount has 4 displacement constraint sensitivitys of correspondence, and 4 corresponding for variable displacement constraint sensitivitys have been calculated successively and stored, then 5 variablees of residue are calculated, finally by all sensitivity results Formatting Outputs.
The above, the only detailed description of the invention of the present invention, but protection scope of the present invention is not limited thereto, and any Those familiar with the art in the technical scope that the invention discloses, the change that can readily occur in or replacement, all answer Contain within protection scope of the present invention.Therefore, protection scope of the present invention should be with described scope of the claims Accurate.

Claims (4)

1. the sensitivity method for solving under displacement constraint more than a kind, it is characterised in that comprise the steps:
Step one, builds aircaft configuration Optimized model, obtains the operating mode of described Optimized model, variable, boundary condition, displacement constraint And the information of the displacement constraint point of displacement constraint applying；
Step 2, carries out static analysis to described Optimized model, obtains displacement and the stiffness matrix K of structure of structure；
Step 3, finds out the described operating mode making each described displacement constraint point produce maximum displacement, meanwhile, by each for making of finding described Displacement constraint point produces the described operating mode of maximum displacement and classifies by described boundary condition, will have identical described boundary condition Described operating mode be divided in same group, arrange according to the order of the described displacement constraint point chosen, form real displacement array U；
Step 4, carries out yojan to described stiffness matrix K, eliminates the singularity of described stiffness matrix K, forms Ki matrix；
Step 5, in described Ki matrix, chooses the described displacement constraint point identical with step 3, at described displacement constraint point Upper applying specific loading, forms dummy load array Pri；
Step 6, is calculated structure virtual displacement array Upsei under different described displacement constraint and different described boundary condition, And multiple described structure virtual displacement arrays Upsei are constituted virtual displacement matrix U pse；
Step 7, calculates the unit differential stiffness matrix Kde of described variable, and fills according to described unit differential stiffness matrix Kde Distribution structure differential stiffness matrix Kd；
Step 8, extracts each row the transposition of described real displacement array U successively, with described structure differential stiffness matrix Kd and described The respective column of virtual displacement array Upse is multiplied, and obtains each displacement constraint sensitivity of current variable.
Sensitivity method for solving under many displacement constraints the most according to claim 1, it is characterised in that shape in described step 4 Become the described Ki matrix can be according to the number of described boundary condition different types of in step 3, as stiffness matrix described in yojan The cycle-index of K, carries out yojan to described stiffness matrix K, forms described Ki matrix.
Sensitivity method for solving under many displacement constraints the most according to claim 1, it is characterised in that tie in described step 6 Structure virtual displacement array Upsei obtains according to solving equilibrium equation Ki*Upsei=Pri.
Sensitivity method for solving under many displacement constraints the most according to claim 1, it is characterised in that in described step 8 when The solution formula of each displacement constraint sensitivity of front variable is:Wherein,For in i-th variable The sensitivity of r displacement constraint,The transposition arranged for r in real transposed matrix U,For structure differential stiffness matrix Kd,R for virtual displacement square array Upse arranges.
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