CN107515963A - A kind of bi-material layers Continuum Structure Multidisciplinary systems Topology Optimization Method based on uncertain but bounded - Google Patents
A kind of bi-material layers Continuum Structure Multidisciplinary systems Topology Optimization Method based on uncertain but bounded Download PDFInfo
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Abstract
The invention discloses a kind of bi-material layers Continuum Structure Multidisciplinary systems Topology Optimization Method based on uncertain but bounded.This method is first according to the load character of Continuum Structure, consider structural loads, material property under Small Sample Size, the uncertainty of design object, by building bi-material layers interpolation model, and using the optimization characteristic distance d based on non-probabilistic set-based reliability model as Multidisciplinary systems index, establish bi-material layers OPTIMIZATION OF CONTINUUM STRUCTURES mathematical modeling；Finally using non-probability decision degree as constraint, using architecture quality as optimization aim, using the relative density of unit as design variable, using movement asymptote optimized algorithm, Continuum Structure is obtained by iteration and is giving outer carry and the preferred configuration under boundary condition.The present invention rationally characterizes uncertain and two kinds of material bearing capacitys the different influences to node configuration during topology optimization design is carried out, and the effective loss of weight of structure can be achieved, it is ensured that design compromise between security itself and economy.
Description
Technical field
It is more particularly to a kind of to be based on uncertain but bounded the present invention relates to Topology Optimization Design of Continuum Structures technical field
Bi-material layers Continuum Structure Multidisciplinary systems Topology Optimization Method, this method consider elasticity modulus of materials, structure external applied load
With the uncertainty of displacement safety value, based on bi-material layers interpolation model, optimizing this Multidisciplinary systems index of characteristic distance d
Constraint under, to Continuum Structure carry out topological optimization.
Background technology
As there is more wide design space in structure optimization field.With fields pair such as Aero-Space, automobile, ships
The economy of structure and the requirement more and more higher of performance, status of the structure optimization in structure design is more and more important, optimizes hand
Section also extend to shape optimum and the more topological optimization of challenge from the optimization of simple dimensional parameters.Topological optimization avoids
Artificial subjective factor in dimensionally-optimised and shape optimum, can provide more superior optimum results, especially in complicated work
Under condition, topological optimization is strong design meanses.And the development of the advanced structure manufacturing technology such as 3D printing, topological optimization is given again
Practicality bring bigger technical support, the limitation of manufacture view has no longer been the main limit of topology optimization design
System, therefore topological optimization has more wide application space, the topological optimization of bi-material layers structure also possesses Practical Project should
Ability.Bi-material layers structure can utilize the performance of material to the full extent, by the selection of material stiffness and specific stiffness, be
Structure optimization provides more excellent optimum results.
Structural Topology Optimization can be divided into discrete topology topological optimization and Continuum Structure topology according to the difference of research object
Optimize two major classes, wherein OPTIMIZATION OF CONTINUUM STRUCTURES turns into structure optimization field in recent years closer in engineering reality
One of popular research direction.OPTIMIZATION OF CONTINUUM STRUCTURES passes through the development of nearly 30 years, has formd a variety of optimization sides
Method, wherein representational method has level set method, Varying-thickness method, ESO methods (evolutionary structural
Optimization), SIMP methods (solid isotropic material with penalization) etc..But it is worth noting
, current Topology Optimization Method is largely based on discriminating hypotheses.
With the development of science and technology, the complexity of engineering structure system is increasingly increasing, the influence of uncertain factor
It is more and more prominent therewith, consider that the influence of uncertain factor has also obtained the attention of people in Optimal Structure Designing.Herein
Under environment, the reliability topological optimization optimization based on probability theory is proposed out, you can by property topological optimization
(Reliability-Based Top Optimization,RBTO).However, be widely present in engineering structure system it is random,
Fuzzy, unknown and a variety of unascertained informations such as bounded, and structure sample data is often a lack of, and is not enough to build probability
Reliability model and fuzzy Reliability Model.For Practical Project problem, although the distributed intelligence of uncertain factor is difficult to accurately
Obtain, but its uncertain boundary is but easier to determine, considers this case, the Multidisciplinary systems based on salient rate
Concept be proposed out.But the achievement currently in terms of Multidisciplinary systems Topology Optimization is less, in bi-material layers knot
The Multidisciplinary systems Topology Optimization Method aspect or blank of structure.
The content of the invention
The technical problem to be solved in the present invention is：Overcome the deficiencies in the prior art, there is provided one kind is based on uncertain but bounded
Bi-material layers Continuum Structure Multidisciplinary systems Topology Optimization Method.The present invention takes into full account in Practical Project problem and generally deposited
Uncertain factor, based on the interpolation model of bi-material layers, to optimize characteristic distance d this Multidisciplinary systems measurement
Constraints of the index as Optimized model, resulting design result can comprehensively utilize the characteristic of two kinds of materials, have more excellent
Performance, engineering adaptability is stronger.
The technical solution adopted by the present invention is：A kind of non-probability of bi-material layers Continuum Structure based on uncertain but bounded can
It is as follows by property Topology Optimization Method, implementation step：
Step 1：Based on classical SIMP (solid isotropic material with penalization) model,
Build bi-material layers interpolation model：
Wherein EiFor the elasticity modulus of materials of i-th of unit after interpolation, E1For the modulus of elasticity of material 1, E2For material 2
Modulus of elasticity, x1,iFor the design variable 1, x of i-th of unit2,iFor the design variable 2 of i-th of unit, p (p>1) for punishment because
Son；
Step 2：The uncertainty of elasticity modulus of materials, structural loads and safe displacement is considered, using interval variable KI、FI
WithRepresent respectively structure integral rigidity interval matrix, load section vector sum displacement section to
Amount, wherein subscript I represent that variable is interval variable, had according to finite element equilibrium equations：
KIuI=FI
Then, using interval parameter summit combined method, structure is obtained on modulus of elasticity and the monotonicity of load by displacement
Bound of the displacement under the influence of uncertain but bounded parameter：
Wherein For the actual bit of j-th of displacement constraint
Move section, subscript ki=1,2, work as kiRepresent that respective value removes boundary, works as k when=1iRepresent that respective value takes the upper bound when=2, i.e.,(K-1)2=K -1, Fi 1=F i,I=1,2,…,N；
Step 3：Using non-probabilistic set-based reliability model, actual displacement section and safe displacement section are standardized, makes it
Interval range is [- 1,1]：
Wherein δ uj,aFor the actual displacement of j-th of displacement constraint of standardization, δ uj,sWithThe jth respectively standardized
The safe displacement of individual displacement constraint and safe displacement section,WithRespectively actual displacement uj,aWith safe displacement uj,sIn
Value,For section radius, according to Structural functional equation：
S(uj,s,uj,a)=uj,s-uj,a
To judge whether structure is safe, works as S>Structural failure when 0, the structure safety as S≤0；
Step 4：The Multidisciplinary systems index whether to be failed as measurement structure using optimization characteristic distance d, optimization are special
Sign distance d definition is：For original failure plane to the displacement of targeted failure plane, wherein targeted failure plane is failed with original
The parallel plane of plane, and its reliability is a set-point, and the non-probability that optimization characteristic distance d can quantify current design can
By degree；
Step 5：Based in general topological optimization mathematical modeling, using Multidisciplinary systems index as constraint, establish double
The mathematical modeling of material Continuum Structure Multidisciplinary systems topological optimization：
Wherein, M is the architecture quality of design domain, ViFor the volume of i-th of unit, N is the unit sum of design domain division,
x1And x2The respectively lower bound of design variable 1 and design variable 2,WithThe respectively upper bound of design variable 1 and design variable 2
Boundary.djIt is the reliability of j-th of constraint, m is the number of constraint.
Step 6：The sensitivity of displacement bound is solved using adjoint vector method, further according to the chain type derivation of compound function
Rule obtains optimizing characteristic distance d sensitivity.
Step 7：Using mobile asymptote optimized algorithm (Method of Moving Asymptotes), to minimize knot
Structure quality is target, using non-probability decision degree d as constraint, is changed using the sensitivity for optimizing characteristic distance d and architecture quality
In generation, solves, in an iterative process, if current design is unsatisfactory for Reliability Constraint d<0, or the design change of former and later two iteration steps
When amount change absolute value sum is more than preset value ε, then iterative steps add one, and return to step two, otherwise, carry out step 8；
Step 8：If current design meets Reliability Constraint d<0, and the design variable change of former and later two iteration steps
When absolute value sum is less than preset value ε, then iteration terminates, and obtains bi-material layers Continuum Structure Multidisciplinary systems topological optimization
Preferred configuration.
The present invention compared with prior art the advantages of be：
The invention provides a kind of bi-material layers Continuum Structure based on uncertain but bounded under displacement Reliability Constraint
Multidisciplinary systems topology optimization design new approaches, in the office for solving traditional reliability design approach based on probability theory
While sex-limited, the design topological optimization result being combined by bi-material layers is more excellent.Constructed Multidisciplinary systems are opened up
Optimized model is flutterred, on the one hand can significantly reduce the shadow to topological structure configuration to the dependence of sample information and quantization uncertainty
Ring, on the other hand, embody the performances of two kinds of materials by bi-material layers interpolation model, and subregion is therefrom preferentially.To displacement
When bi-material layers Continuum Structure under reliability constraint carries out topology optimization design, it can take into full account under uncertain effect
Structural topology changing rule, while ensuring that displacement structure meets certain constraints, structural behaviour can be greatly promoted, reduced
Financial cost.
Brief description of the drawings
Fig. 1 is that the present invention is directed to the bi-material layers Continuum Structure Multidisciplinary systems topological optimization based on uncertain but bounded
Flow chart；
Fig. 2 is the one-dimensional Interference Model of the non-probabilistic set-based reliability model used in the present invention；
Fig. 3 is the non-probabilistic set-based reliability model two dimension interference schematic diagram used in the present invention；
Fig. 4 is the different interference situation schematic diagram of six kinds of the non-probabilistic set-based reliability model used in the present invention；
Fig. 5 is the schematic diagram of the Multidisciplinary systems index optimization characteristic distance d used in the present invention；
Fig. 6 is critical slope schematic diagrames of the index d when calculating used in the present invention；
Fig. 7 is the initial model schematic diagram that the present invention is directed to bi-material layers OPTIMIZATION OF CONTINUUM STRUCTURES；
Fig. 8 is the optimum results schematic diagram that the present invention is directed to bi-material layers OPTIMIZATION OF CONTINUUM STRUCTURES, wherein, Fig. 8 (a) is
Deterministic optimization schematic diagram, Fig. 8 (b) are that Multidisciplinary systems optimize (Rtarg=0.90) schematic diagram, Fig. 8 (c) are that non-probability can
Optimize (R by propertytarg=0.95) schematic diagram, Fig. 8 (d) are that Multidisciplinary systems optimize (Rtarg=0.999) schematic diagram；
Fig. 9 is that the present invention is directed to bi-material layers OPTIMIZATION OF CONTINUUM STRUCTURES iteration course curve synoptic diagram, wherein, Fig. 9 (a)
For deterministic optimization schematic diagram, Fig. 9 (b) is that Multidisciplinary systems optimize (Rtarg=0.90) schematic diagram, Fig. 9 (c) are non-probability
Reliability optimization (Rtarg=0.95) schematic diagram, Fig. 9 (d) are that Multidisciplinary systems optimize (Rtarg=0.999) schematic diagram.
Embodiment
Below in conjunction with the accompanying drawings and specific embodiment further illustrates the present invention.
As shown in figure 1, the present invention propose a kind of non-probability of bi-material layers Continuum Structure based on uncertain but bounded can
By property Topology Optimization Method, comprise the following steps：
(1) classical SIMP (solid isotropic material with penalization) model is based on, is built
Bi-material layers interpolation model：
Wherein EiFor the elasticity modulus of materials of i-th of unit after interpolation, E1For the modulus of elasticity of material 1, E2For material 2
Modulus of elasticity, x1,iFor the design variable 1, x of i-th of unit2,iFor the design variable 2 of i-th of unit, p (p>1) for punishment because
Son；
(2) uncertainty of elasticity modulus of materials, structural loads and safe displacement is considered, using interval variable KI、FIWithStructure integral rigidity interval matrix, load section vector sum displacement interval vector are represented respectively,
Wherein subscript I represents that variable is interval variable, is had according to finite element equilibrium equations：
KIuI=FI (2)
Because governing equation is linear, it is possible to solved using following interval parameter vertex schemeIn any componentBound.
Interval parameter vertex scheme：If f (x1,x2,…,xn) to independent variable xi(i=1,2 ..., n) is dull, incite somebody to action oneself
When variable is thought of as interval variable, i.e.,：
From function limit, f span is：
Wherein r is that summit (section two-end-point) combines ordinal number,ki
=1,2, i=1,2 ..., n；R=1,2 ..., 2n。
So according to interval parameter vertex scheme, displacement corresponding to j-th of constraint is obtainedInterval be：
Wherein For the actual bit of j-th of displacement constraint
Move section, subscript ki=1,2, work as kiRepresent that respective value removes boundary, works as k when=1iRepresent that respective value takes the upper bound when=2, i.e.,(K-1)2=K -1, Fi 1=F i,I=1,2 ..., N；
(3) after obtaining the bound in displacement section by step (2), based on Non-probabilistic Set-based Model For Structural Reliability,
The Multidisciplinary systems model established under following displacement constraint.
If uj,aFor the actual displacement of j-th of displacement constraint, uj,sFor the safe displacement of j-th of displacement constraint, do not knowing
In the presence of property, displacement is interval variable, i.e.,：
Two above section on same number axis when representing, it will has the possibility of interference, as shown in Figure 2.In figure twoWithRespectively actual displacement uj,aWith safe displacement uj,sIntermediate value.If structural limits function of state is：
S(uj,s,uj,a)=uj,s-uj,a (7)
Its failure plane or limiting condition plane are：
S(uj,s,uj,a)=uj,s-uj,a=0 (8)
Represent that structure meets constraints, works as S as S≤0>Represent that structure is unsatisfactory for constraints when 0.To actual displacement
With safe displacement interval variable uj,a∈uj,a I、uj,s∈uj,s IDo standardized transformation：
Wherein,For section radius.Then pass through standardized transformation,
There are δ uj,a∈ [- 1,1], δ uj,s∈[-1,1].Above formula is substituted into failure plane equation, had：
It is hereby achieved that δ uj,sWith δ uj,aBetween relational expression be：
Above formula is drawn in rectangular coordinate system, and indicates δ uj,sWith δ uj,aSpan, as shown in Figure 3.
The region area S of constraints will be metABFEDWith the gross area S in variable regionABCDThe ratio between be defined as the non-of structure
Probability decision degree R.Situation about intersecting below for the plane that failed shown in Fig. 3 with variable region solves R.It is flat that failure is solved first
Face and straight line δ uj,s=-1 intersection point, make δ u in formula (11)j,s=-1, δ u can be obtainedj,aFor：
OrderIt can solveThen obtain failure plane with it is straight
Line δ uj,a=1 intersection point, make δ u in (11)j,a=1, it can solve：
OrderIt can solveBy the coordinate of two above intersection point,
The expression formula that reliability R can be tried to achieve is：
Wushu (12) and formula (13) substitute into formula (14), obtain：
The expression formula that remaining five kinds failure planes intersect R under form with variable region can similarly be tried to achieve：
(4) because non-probability decision degree R is in some cases constant, will be difficult to look for for gradient optimal method
To correct optimization direction, therefore another non-probability decision degree index is introduced on the basis of non-probability decision degree R：Optimization is special
Distance d is levied, as shown in figure 5, d definition is：Distance of the original failure plane to targeted failure plane.Wherein targeted failure plane is
The plane parallel with former failure plane, and its reliability RtargFor a set-point.
Such as Fig. 6, because the non-probability decision degree of target is normally close to 1, therefore targeted failure plane is normally at variable region
The lower right corner, it is illustrated that intersect two kinds of critical conditions of form with variable region for targeted failure plane.Calculate first under critical condition
The slope for the plane that fails.For k1, there is (2 × 2/k1× 1/2)/4=1-Rtarg, solve k1=1/2 (1-Rtarg), it can similarly obtain k2
=2 (1-Rtarg).When the slope for the plane that fails takes different values, its magnitude relationship with critical slope is considered, according to
The expression formula that the range formula of parallel lines obtains optimization characteristic distance d is：
Work as d>When 0, failure plane with the non-probability decision degree R of targettargAbove corresponding targeted failure plane, now by
Area in safety zone is less than desired value, corresponding non-probability decision degree R<Rtarg, it is unsatisfactory for requiring.As d≤0, failure
Plane with the non-probability decision degree R of targettargBelow corresponding targeted failure plane, now because the area of safety zone is more than
Equal to desired value, corresponding non-probability decision degree R>Rtarg, meet design requirement.
(5) in general topological optimization mathematical modeling is based on, using Multidisciplinary systems index as constraint, establishes bi-material layers
The mathematical modeling of Continuum Structure Multidisciplinary systems topological optimization：
Wherein, M is the quality in structure design domain, ViFor the volume of i-th of unit, N is that the unit of design domain division is total
Number, ρ1And ρ2The respectively density of material 1 and material 2, x1,iFor the design variable 1, x of i-th of unit2,iFor setting for i-th unit
Count variable 2, p (p>1) it is penalty factor；x1And x2The respectively lower bound of design variable 1 and design variable 2,WithRespectively set
Count the upper bound of variable 1 and design variable 2.djIt is the reliability of j-th of constraint, m is the number of constraint.
(6) present invention is using mobile progressive this gradient optimal method solving-optimizing problem of line method (MMA), it is therefore desirable to
Obtain object function and constrain the partial derivative of function pair design variable, that is, carry out sensitivity analysis.Because the design of the present invention becomes
For amount far more than constraint function number, sensitivity is solved by the way of difference can bring huge amount of calculation.For this feature, sheet
Invention carries out sensitivity analysis of the constraint function to design variable using adjoint vector method.
By compound function chain type Rule for derivation, for jth (j=1,2 ..., m) individual constraint, it optimizes characteristic distance djIt is right
Single design variable xi(i=1,2 ..., N) (xiFor x1,iOr x2,i) partial derivative be：
Wherein：
WhereinWithCan directly it be calculated by formula and (20) and (21), andWithThen can not direct solution, it is necessary to which the Augmented Lagrangian Functions for constructing following constraint function are solved indirectly：
Wherein, λj(j=1,2 ..., m) is Lagrange multiplier vector, also referred to as adjoint vector.Due to F-Ku=0,
ThereforeAbove formula is to design variable xiDerivative of demanding perfection obtains：
Wherein：
Above formula is set up to any λ, therefore can be chosen appropriate λ and be caused du/dxiThe coefficient of place item is zero, even：
Above formula can be changed to using the symmetry of stiffness matrix：
Then by applying Virtual Load to FEM modelThe displacement tried to achieve is λ.After solving λ,
Sensitivity of the obligatory point displacement bound to design variable is then given by：
WhereinRespectively correspond toAdjoint vector, element stiffness matrix and motion vector,λ j、K j、u jRespectively correspond tou j,aAdjoint vector, global stiffness matrix and motion vector.In the Optimized model of the present invention, load F
Do not change with design variable, i.e. dF/dxi=0, then above formula can be rewritten as：
It can be obtained by formula (1)：
Wherein K1,jFor the stiffness matrix of j-th of unit corresponding to material 1, K2,jFor j-th of unit corresponding to material 2
Stiffness matrix, so having：
So formula (28) is finally written as：
In addition, object function M is to the local derviation of design variable：
(7) calculating is iterated using MMA optimized algorithms, according to current unit design variable, constraint function djPair set
The sensitivity of the sensitivity, object function M of variable to design variable is counted, solves new design variable.In an iterative process, such as
Fruit current design is unsatisfactory for Reliability Constraint d<0, or former and later two iteration steps design variable change absolute value sum be more than it is pre-
If during value ε, then iterative steps add one, and return to step (2), otherwise, step (8) is carried out；
(8) while Reliability Constraint and relative variation are considered, if current design meets Reliability Constraint d<0, and
When the design variable change absolute value sum of former and later two iteration steps is less than preset value ε, then iteration terminates, and present topology is optimized
Result as final optimum results.
Embodiment：
The characteristics of in order to more fully understand the invention and its applicability actual to engineering, the present invention are directed to such as Fig. 7 institutes
The flat board shown carries out topology optimization design.Design section size such as Fig. 7 is marked, thickness 1mm, is divided into the node of bilinearity four
Flat unit.Elasticity modulus of materials E1=210Gpa, E2=70Gpa, Poisson's ratio μ=0.3.The left side of flat board is clamped, and upper end one is small
Section constraint vertical direction displacement, lower right applies power F=1500N straight up, constrains the displacement of load(ing) point so that u1,a≤
u1,s=1.5mm, choose penalty factor p=3.If elastic modulus E1And E2There is 5% fluctuation with respect to nominal value, load F is relative
Nominal value has 15% fluctuation, i.e. E1=[199.5,220.5] Gpa, E2=[66.5,73.5] Gpa, F1=[1275,1725] N；
If displacement constraint u1,sWith respect to the fluctuation that nominal value has 5%, i.e. u1,s=[1.425,1.575] mm.
Fig. 8 (a), Fig. 8 (b), Fig. 8 (c), Fig. 8 (d) are respectively certainty topological optimization result and Multidisciplinary systems difference
For R=0.90, R=0.95 and R=0.999 when topological optimization result, wherein black portions are the larger material 1 of modulus of elasticity,
Grey parts are the less material 2 of modulus of elasticity.It can be seen that certainty topological optimization and different Multidisciplinary systems topologys are excellent
There is larger difference in the configuration for dissolving the structure come, excellent compared to certainty topological optimization result, Multidisciplinary systems topology
The configuration of change employs the material of more high elastic modulus in the larger region of stress, to ensure to tie in the presence of uncertainty
The security of structure.When using the Uncertainty as Multidisciplinary systems, three restrained positions of deterministic optimization result
The Multidisciplinary systems of shifting are only R1=0.4504.That is the result of deterministic optimization is not enough to the shadow of coping with uncertainty variable
Ring.Shown in iteration history such as Fig. 9 (a), Fig. 9 (b), Fig. 9 (c), Fig. 9 (d) in process of topology optimization, compared to initial designs,
Weight loss effect is obvious；As reliability allowable value increases, structure tends to safety, and weight increased.
In summary, the present invention proposes a kind of non-probability decision of bi-material layers Continuum Structure based on uncertain but bounded
Property Topology Optimization Method.First, topological optimization mathematical modeling of the Continuum Structure under Multidisciplinary systems constraint is established, then
Consider that material, external applied load and displacement allowable value are uncertain, the bound of displacement is calculated using interval parameter vertex scheme；Connect down
To be based on non-probabilistic set-based reliability model, this new Multidisciplinary systems index of optimization characteristic distance d is established；Secondly, make
With adjoint vector method, and the Rule for derivation of compound function is combined, obtain optimization characteristic distance d sensitivity；It is finally excellent using MMA
Change algorithm, using reliability as constraint, using architecture quality as target, iterative calculation is optimized, so as to complete Continuum Structure
Multidisciplinary systems topology optimization design.
It the above is only the specific steps of the present invention, protection scope of the present invention be not limited in any way；Its it is expansible should
For the optimization design field of Defective structure, all technical schemes formed using equivalent transformation or equivalent replacement, fall
Within rights protection scope of the present invention.
Non-elaborated part of the present invention belongs to the known technology of those skilled in the art.
Claims (8)
1. a kind of bi-material layers Continuum Structure Multidisciplinary systems Topology Optimization Method based on uncertain but bounded, its feature exist
In realizing that step is as follows：
Step 1：Based on classical SIMP (solid isotropic material with penalization) model, structure
Bi-material layers interpolation model：
Wherein EiFor the elasticity modulus of materials of i-th of unit after interpolation, E1For the modulus of elasticity of material 1, E2For the elasticity of material 2
Modulus, x1,iFor the design variable 1, x of i-th of unit2,iFor the design variable 2 of i-th of unit, p (p>1) it is penalty factor；
Step 2：The uncertainty of elasticity modulus of materials, structural loads and safe displacement is considered, using interval variable KI、FIWithStructure integral rigidity interval matrix, load section vector sum displacement interval vector are represented respectively,
Wherein subscript I represents that variable is interval variable, is had according to finite element equilibrium equations：
KIuI=FI
Then, using interval parameter summit combined method, displacement structure is obtained on modulus of elasticity and the monotonicity of load by displacement
Bound under the influence of uncertain but bounded parameter：
Wherein For the actual displacement area of j-th of displacement constraint
Between, subscript ki=1,2, work as kiRepresent that respective value removes boundary, works as k when=1iRepresent that respective value takes the upper bound when=2, i.e.,(K-1)2=K -1, Fi 1=F i,
Step 3：Using non-probabilistic set-based reliability model, actual displacement section and safe displacement section are standardized, makes its section
Scope is [- 1,1]：
Wherein δ uj,aFor the actual displacement of j-th of displacement constraint of standardization, δ uj,sWithJ-th respectively standardized
Safe displacement and the safe displacement section of constraint are moved,WithRespectively actual displacement uj,aWith safe displacement uj,sIntermediate value,For section radius, according to Structural functional equation：
S(uj,s,uj,a)=uj,s-uj,a
To judge whether structure is safe, works as S>Structural failure when 0, the structure safety as S≤0；
Step 4：Using optimization characteristic distance d as the Multidisciplinary systems index that whether fails of structure is weighed, optimize feature away from
Definition from d is：Original failure plane arrives the displacement of targeted failure plane, and wherein targeted failure plane is and former failure plane
Parallel plane, and its reliability is a set-point, and optimization characteristic distance d can quantify the non-probability decision of current design
Degree；
Step 5：Based in general topological optimization mathematical modeling, using Multidisciplinary systems index as constraint, bi-material layers are established
The mathematical modeling of Continuum Structure Multidisciplinary systems topological optimization：
Wherein, M is the quality in structure design domain, ViFor the volume of i-th of unit, N is that the unit of design domain division is total, ρ1With
ρ2The respectively density of material 1 and material 2,x 1Withx 2The respectively lower bound of design variable 1 and design variable 2,WithRespectively
The upper bound circle of design variable 1 and design variable 2, djIt is the reliability of j-th of constraint, m is the number of constraint；
Step 6：The sensitivity of displacement bound is solved using adjoint vector method, further according to the chain type Rule for derivation of compound function
Obtain optimizing characteristic distance d sensitivity；
Step 7：Using mobile asymptote optimized algorithm (Method of Moving Asymptotes), to minimize structure matter
Measure as target, using non-probability decision degree d as constraint, be iterated and asked using the sensitivity for optimizing characteristic distance d and architecture quality
Solution, in an iterative process, if current design is unsatisfactory for Reliability Constraint d<0, or the design variable change of former and later two iteration steps
When change absolute value sum is more than preset value ε, then iterative steps add one, and return to step two, otherwise, carry out step 8；
Step 8：If current design meets Reliability Constraint d<0, and the design variable change of former and later two iteration steps is absolute
When value sum is less than preset value ε, then iteration terminates, and obtains the optimal of bi-material layers Continuum Structure Multidisciplinary systems topological optimization
Configuration.
2. a kind of bi-material layers Continuum Structure Multidisciplinary systems based on uncertain but bounded according to claim 1 are opened up
Flutter optimization method, it is characterised in that：Material interpolation model in the step 1 employs the interpolation model of bi-material layers.
3. a kind of bi-material layers Continuum Structure Multidisciplinary systems based on uncertain but bounded according to claim 1 are opened up
Flutter optimization method, it is characterised in that：The not true of elasticity modulus of materials, structural loads and safe displacement is considered in the step 2
It is qualitative, and showed in the form of interval matrix and interval vector.
4. a kind of bi-material layers Continuum Structure Multidisciplinary systems based on uncertain but bounded according to claim 1 are opened up
Flutter optimization method, it is characterised in that：Carry out the non-probability of quantizing structure in the step 3 using non-probabilistic set-based reliability model
Reliability, according to non-probabilistic set-based reliability model, the Variational Design domain area S of constraints will be metAEFWith Variational Design domain
Gross area SABCDThe ratio between be defined as the non-probability decision degree R of structure：
Substitute into δ uj,aWith δ uj,sExpression formula can obtain：
5. a kind of bi-material layers Continuum Structure Multidisciplinary systems based on uncertain but bounded according to claim 1 are opened up
Flutter optimization method, it is characterised in that：Employed in the step 4 optimization characteristic distance d as measurement structure whether fail it is non-
Probabilistic reliability index, optimization distance d expression formula are：
Wherein k1=1/2 (1-R), k2=2 (1-R) are two critical slopes.
6. a kind of bi-material layers Continuum Structure Multidisciplinary systems based on uncertain but bounded according to claim 1 are opened up
Flutter optimization method, it is characterised in that：The optimized mathematical model established in the step 5 considers the influence of bi-material layers.
7. a kind of bi-material layers Continuum Structure Multidisciplinary systems based on uncertain but bounded according to claim 1 are opened up
Flutter optimization method, it is characterised in that：Adjoint vector method is used in the step 6 and by considering bi-material layers interpolating function
The Rule for derivation of compound function has obtained optimization characteristic distance d sensitivity.
8. a kind of bi-material layers Continuum Structure Multidisciplinary systems based on uncertain but bounded according to claim 1 are opened up
Flutter optimization method, it is characterised in that：Optimization distance d has been concurrently set in the step 6<0, design variable change absolute value it
Constrained with two less than ε.
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