CN108009381A - A kind of Continuum Structure reliability Topology Optimization Method under displacement and global stress mixed constraints - Google Patents
A kind of Continuum Structure reliability Topology Optimization Method under displacement and global stress mixed constraints Download PDFInfo
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Abstract
The invention discloses the Continuum Structure Multidisciplinary systems Topology Optimization Method under a kind of displacement and global stress mixed constraints, this method obtains cell density using filter density method by design variable first, then displacement and the stress of structure are calculated with relaxation rule, and global stress is handled using stress resultant function constraint, obtain the bound of displacement and stress resultant function followed by vertex combined method；Convergence problem is solved instead of Multidisciplinary systems index using optimization characteristic displacement, and with the sensitivity of adjoint vector method and compound function derivation law solving-optimizing characteristic displacement；Calculating finally is iterated with mobile Asymptotical Method, until meeting corresponding convergence conditions, is met the optimization design scheme of Reliability Constraint.The present invention rationally characterizes the uncertain influence to Continuum Structure rigidity and strength character during topology optimization design is carried out, and can realize effective loss of weight, it is ensured that design compromise between security itself and economy.
Description
Technical field
The present invention relates to the topology optimization design field containing Continuum Structure, more particularly to a kind of displacement and global stress mix
Continuum Structure reliability Topology Optimization Method under contract beam, this method consider the not true of elasticity modulus of materials and magnitude of load
The influence of qualitative rigidity and intensity to structure and lower Continuum Structure based on displacement and global stress mixed constraints it is non-
The formulation of probabilistic reliability topological optimization scheme.
Background technology
With science and technology and productivity it is growing, Structural Optimization Design becomes more and more important.According to setting
Range of variables is counted, Optimal Structure Designing can be divided into three levels：Cross-sectional size optimization, geometry optimization and topological layout
Optimization.Compared with dimensionally-optimised and shape optimum, influence bigger of the structural Topology Optimization variable to optimization aim, has bigger
Economic benefit.Therefore, the Topology Optimization of Continuum Structure has important engineering practical value.
Currently, most of Topology Optimization is concentrated mainly on displacement or other global response constraints, for answering
Topological optimization under force constraint is then studied less.However, in Practical Project, stress constraint is highly important, without considering answering
The Topology Optimization of force constraint cannot be put to engineering practice.Stress constraint has following three property, i.e. singularity, office
Portion's property and strong nonlinearity, this causes the Topology Optimization under stress constraint to become very difficult.Some scholars are directed to stress
Singularity propose stress relaxation method, stress resultant functional based method is proposed for the locality of stress, for stress
Strong nonlinearity proposes correction factor method.The topological optimization that these methods can solve under stress constraint to a certain extent is asked
Topic.
However, increasingly accurate and complicated, the material properties caused by the manufacturing processing technic of material with engineering structure
Dispersiveness has an important influence on the performance of structure.Further, since the deterioration of structure Service Environment so that the military service safety of structure
Face more challenges.Therefore, consider that probabilistic influence is very necessary in Optimal Structure Designing.Topological optimization is made
For the conceptual phase of structure optimization, there is conclusive influence to final structure type, therefore, study displacement and the overall situation
Continuum Structure reliability Optimum Design method under stress mixed constraints is of great significance.
In Practical Project, structure sample experimental data is usually a lack of, thus probabilistic reliability model and it is fuzzy can
Tend not to be met by property Model Condition, but the uncertain border of uncertain information is become more readily available.In recent years,
Multidisciplinary systems theory is developed rapidly.Therefore, the non-probability under research structure displacement and global stress mixed constraints
Reliability Topology Optimization Method has significant realistic meaning.At present, relevant research is still insufficient, and existing method is calculated as
Ben Taigao, also or safety redundancy is excessive, causes time cost loss and the serious wasting of resources.
The content of the invention
The technical problem to be solved in the present invention is：Overcome the deficiencies of the prior art and provide a kind of displacement and global stress mixes
Continuum Structure Multidisciplinary systems Topology Optimization Method under contract beam.The present invention takes into full account universal in Practical Project problem
Existing uncertain factor, obtained design result are more in line with truth, and engineering adaptability is stronger.
The technical solution adopted by the present invention：A kind of Continuum Structure reliability under displacement and global stress mixed constraints is opened up
Optimization method is flutterred, realizes that step is as follows：
Step 1：Design variable is described using density variable method, comes description scheme material properties and load with interval model
The uncertainty of lotus, with the volume of structure target as an optimization, using displacement structure and global stress as constraining, establishes non-probability
Reliability topological optimization model is as follows：
Wherein, V be optimize region volume, ρiAnd ViThe relative density and volume of respectively i-th unit, and ρiIt is
The function of design variable r, N are the unit sum of optimization region division.It is the actual displacement section of j-th of displacement constraint point
Value,It is the Admissible displacement interval value of j-th of displacement constraint, m is the number of displacement constraint.It is i-th of stress constraint point
Actual stress interval value,It is the allowable stress interval value of stress constraint.RsIt is non-Making by Probability Sets reliability index,It is
The non-probability decision degree of the corresponding target of j-th of displacement constraint,It is the non-probability decision degree of the corresponding target of k-th of stress constraint,
R is the lower limit of design variable；R is the lower limit of design variable.
Step 2：Design variable is filtered using filter density method, obtains the density value of unit.Use section
Measure to describe the uncertainty of elasticity modulus of materials and load, using vertex combined method, and the bullet using relaxation rule to unit
Property modulus and Stress calculation relax.After obtaining the displacement of structure and the stress of unit, to the stress of all units
Integrated, obtain corresponding stress resultant functional value, be compared to obtain the bound of displacement structure and stress resultant function
Bound and its corresponding vertex combination.
Step 3：According to the bound of displacement and the bound of stress resultant function, displacement and stress resultant letter are obtained
Counting the non-Making by Probability Sets reliability constrained is：
Wherein, SIRepresent structure actual displacement or the stress resultant function region area of a room, RIRepresent structure displacement allowable or permitted
With the stressed zone area of a room,Represent that the upper bound S of structure actual displacement either stress resultant function represents structure actual displacement or should
The lower bound of power comprehensive function, R represent the upper bound of structure Admissible displacement and allowable stress, and R represents structure Admissible displacement and allows to answer
The lower bound of power.
Step 4：Convergence problem is improved instead of Multidisciplinary systems index using optimization characteristic displacement.Utilize optimization
Former Optimized model can be rewritten as by characteristic displacement：
Wherein, d (RI,SI) it is optimization characteristic displacement.
Step 5：According to displacement and the corresponding vertex combination of stress resultant function, structure bit is obtained with adjoint vector method
The sensitivity of bound and stress resultant function bound to cell density is moved, is then obtained using the Rule for derivation of compound function
Sensitivity to the optimization characteristic displacement of displacement and stress resultant function to design variable.First solve displacement (stress resultant function)
Sensitivity of the optimization characteristic displacement to displacement (stress resultant function) bound, then solving displacement bound again, (stress is comprehensive
Close function bound) sensitivity to cell density, sensitivity of the cell density to design variable is then solved, finally by three
Multiplication obtains sensitivity of the optimization characteristic displacement of displacement (stress resultant function) to design variable.
Step 6：By obtained displacement and stress resultant function constraint condition value and its sensitivity information to design variable
As the input condition of mobile Asymptotical Method (MMA), optimization problem is solved, is designed the renewal of variable.
Step 7：Repeat step two is designed the multiple renewal of variable, until current design meets reliably to step 6
Degree constraint, and when the Relative percent change of object function is less than preset value ξ, then stop optimization process.
The present invention compared with prior art the advantages of be：
(1) present invention just considers the influence of uncertain factor in the conceptual phase of structure, can be to greatest extent
The economic benefit of lift structure, and compromise between security；
(2) Multidisciplinary systems index of the present invention can reasonable consideration uncertain factor structural behaviour is brought
Influence, it is and smaller to sample size demand, be very suitable for engineer application；
(3) present invention optimizes calculating using MMA algorithms so that the method proposed can be suitable for the feelings of multiple constraint
Condition, the scope of application are more extensive.
Brief description of the drawings
Fig. 1 is that the present invention is excellent for the Continuum Structure Multidisciplinary systems topology under displacement and global stress mixed constraints
Change flow chart；
Fig. 2 is topology optimization design region and border and load-up condition schematic diagram in the embodiment of the present invention；
Fig. 3 is the optimum results schematic diagram that the present invention is directed to OPTIMIZATION OF CONTINUUM STRUCTURES, wherein, Fig. 3 (a) is certainty
Optimization, Fig. 3 (b) optimize (Rs=0.90) for Multidisciplinary systems, and Fig. 3 (c) optimizes (Rs=0.95) for Multidisciplinary systems, figure
3 (d) optimizes (Rs=0.999) for Multidisciplinary systems；
Fig. 4 is that the present invention is directed to OPTIMIZATION OF CONTINUUM STRUCTURES iteration course curve, wherein, Fig. 4 (a) is excellent for certainty
Change, Fig. 4 (b) optimizes (Rs=0.90) for Multidisciplinary systems, and Fig. 4 (c) optimizes (Rs=0.95) for Multidisciplinary systems, Fig. 4
(d) (Rs=0.999) is optimized for Multidisciplinary systems.
Embodiment
Below in conjunction with the accompanying drawings and embodiment further illustrates the present invention.
As shown in Figure 1, the present invention proposes the non-probability of Continuum Structure under a kind of displacement and global stress mixed constraints
Reliability Topology Optimization Method, comprises the following steps：
Step 1：Design variable is described using density variable method, comes description scheme material properties and load with interval model
The uncertainty of lotus, with the volume of structure target as an optimization, using displacement structure and global stress as constraining, establishes non-probability
Reliability topological optimization model is as follows：
Wherein, V be optimize region volume, ρiAnd ViThe relative density and volume of respectively i-th unit, and ρiIt is
The function of design variable r, N are the unit sum of optimization region division.It is the actual displacement section of j-th of displacement constraint point
Value,It is the Admissible displacement interval value of j-th of displacement constraint, m is the number of displacement constraint.It is i-th of stress constraint point
Actual stress interval value,It is the allowable stress interval value of stress constraint.RsIt is non-Making by Probability Sets reliability index,It is
The non-probability decision degree of the corresponding target of j-th of displacement constraintIt is the non-probability decision degree of the corresponding target of k-th of stress constraint, r
For the lower limit of design variable；R is the lower limit of design variable.
Step 2：Design variable is filtered using filter density method, obtains the density value of unit.Use section
Measure to describe the uncertainty of elasticity modulus of materials and load, using vertex combined method, and the bullet using relaxation rule to unit
Property modulus and Stress calculation relax.After obtaining the displacement of structure and the stress of unit, to the stress of all units
Integrated, obtain corresponding stress resultant functional value, be compared to obtain the bound of displacement structure and stress resultant function
Bound and its corresponding vertex combination.
Cell density can be obtained by filtration by the design variable of unit：
Wherein ρiIt is the density value of i-th of unit, djFor the corresponding design variable of j-th of unit.ΩiFor all and unit i
Distance is less than or equal to r0The set of the unit of (filtering radius), rjIt is the distance of the central point of unit j and unit i.
After obtaining the density of unit, following relaxation is carried out to the elasticity modulus of unit：
E (ρ)=ρ3E0
Wherein E (ρ) be some unit elasticity modulus, ρ be the unit density, E0For the elasticity modulus of solid material.
After obtaining the elasticity modulus of unit, FEM calculation can be carried out, obtains the displacement of cell node.
In order to preferably characterize the stress level of structure, using von mises stress come characterization unit stress.Its mathematical table
It is up to formula：
Wherein, σ1,σ2,σ3Refer to first, second and third principal stress respectively.
Displacement of elemental node and corresponding displacement shape function and strain matrix according to obtaining can obtain unit each side
To direct stress and shearing stress be：
σ=D ε=DBue=Sue
Calculated to simplify, and change in view of unit internal stress smaller, therefore chosen unit center point and answer masterpiece
For the characterization of element stress.
The von mises stress that central point can be obtained according to the direct stress of unit center point all directions and shearing stress is：
Wherein, σc=[σcx,σcy,σcz,τaxy,τayz,τazx]TFor the load column vector of unit center point.
In order to express easily, remember：
σcr=h (σc)
Wherein ueRepresent the displacement column vector of the displacement component composition of the node corresponding to stress constraint unit.If its with it is total
Position, which moves column vector, following relation：
ue=seu
S in formulaeMatrix is extracted for element displacement.For rectangle bi-linear elements, have：
(ue)8×1=(se)8×nun×1
Wherein matrix (se)8×nEach behavior some element be 1, remaining element is 0 row vector.
Using stress relaxation method, can calculate stress is：
After obtaining the stress of unit, the stress of all units is integrated, obtains corresponding stress resultant function
Value, element stress synthesis use following formula：
Wherein, veVolume when for unit being solid, for P, generally takes P=6 or P=8.
Since comprehensive stress function and global maximum stress have certain difference, it is therefore desirable to certain amendment is taken,
If：
σmax≈cσPN
The wherein c of kth stepkMeet following formula：
Using vertex combined method, each Uncertainty in elasticity modulus of materials and load column vector is taken respectively
Lower border value is combined, and calculates each combination displacement and the stress resultant functional value of structure according to the method described above, so
These values under all combinations are compared afterwards, obtain the bound of displacement structure and the bound of stress resultant function and its
Corresponding vertex combination.
Step 3：According to the bound of displacement and the bound of stress resultant function, displacement and stress resultant letter are obtained
Counting the non-Making by Probability Sets reliability constrained is：
Wherein, SIRepresent structure actual displacement or the stress resultant function region area of a room, RIRepresent structure displacement allowable or permitted
With the stressed zone area of a room,Represent that the upper bound S of structure actual displacement either stress resultant function represents structure actual displacement or should
The lower bound of power comprehensive function, R represent the upper bound of structure Admissible displacement and allowable stress, and R represents structure Admissible displacement and allows to answer
The lower bound of power.
Step 4：Convergence problem is improved instead of Multidisciplinary systems index using optimization characteristic displacement.Utilize optimization
Former Optimized model can be rewritten as by characteristic displacement：
Wherein, d (RI,SI) it is optimization characteristic displacement.
Optimization characteristic displacement d is defined as considered repealed plane to the moving displacement of targeted failure plane.Wherein targeted failure
Plane is the plane parallel with former failure plane, and its reliability is desired value.
Since target reliability degree is generally higher than 50%, so targeted failure plane is normally at the lower right in uncertain domain.
The slope for calculating the plane that fails under critical condition can be gone out, if η is target reliability degree.For k1, there is (2 × 2/k1×
1/2)/4=1- η, solve k1=1/2 (1- η), can similarly obtain k2=2 (1- η).
Can be calculated optimization characteristic displacement expression formula be：
As d ＞ 0, failure plane is corresponding non-above targeted failure plane corresponding with the non-probability decision degree η of target
Probability decision degree Rs＜ η, are unsatisfactory for requiring.As d≤0, failure plane is lost in target corresponding with the non-probability decision degree η of target
Imitate below plane, corresponding non-probability decision degree Rs>=η, meets design requirement.
Step 5：According to displacement and the corresponding vertex combination of stress resultant function, structure bit is obtained with adjoint vector method
The sensitivity of bound and stress resultant function bound to cell density is moved, is then obtained using the Rule for derivation of compound function
Sensitivity to the optimization characteristic displacement of displacement and stress resultant function to design variable.
Since sensitivity of the optimization characteristic displacement to design variable of displacement structure has derived in some documents,
The sensitivity of stress resultant function pair design variable is only derived below.First, consider that stress resultant function is：
Correction factor is taken as：
Correction factor is constant after taking calmly, is not involved in sensitivity analysis below.
The then sensitivity of stress resultant function pair cell density is：
Arrangement can obtain：
Right end Section 1 in formula：
It is not related to derivative of the unit displacement of joint to cell density, therefore can directly calculates, wherein
And right end Section 2：
It can be calculated using adjoint vector, if：
Then adjoint vector λ can be obtained by a FEM calculation.
Then right end Section 2 can be solved to：
In formulaCalculate as follows：
The form for being write as vector has：
So far, the sensitivity of stress resultant function pair cell density has been obtained.
The density p of unit iiTo the design variable d of unit jjSensitivity be given by：
Wherein ΩiIt is less than or equal to r with unit i distances to be all0The set of the unit of (filtering radius), rjUnit j with
The distance of the central point of unit i.
So far, sensitivity of the cell density on design variable has been obtained.
Stress resultant function bound can be obtained on the sensitive of design variable with the Rule for derivation of compound function
Degree.
Sensitivity analysis for the stress resultant function upper bound, then can combine generation by the vertex of its corresponding Uncertainty
Enter to be calculated, obtain the sensitivity of bound pair design variable on stress resultant function, can similarly solve under stress resultant function
The sensitivity of bound pair design variable.
The sensitivity of displacement structure or stress resultant function optimization characteristic displacement to design variable is solved to：
Wherein：
So far, sensitivity of the optimization characteristic displacement of displacement structure and stress resultant function to design variable has been obtained.
Step 6：By obtained displacement and stress resultant function constraint condition value and its sensitivity information to design variable
As the input condition of mobile Asymptotical Method (MMA), optimization problem is solved, is designed the renewal of variable.
Step 7：Repeat step two is designed the multiple renewal of variable, until current design meets reliably to step 6
Degree constraint, and when the Relative percent change of object function is less than preset value ξ, then stop optimization process.
Embodiment：
The characteristics of in order to more fully understand the invention and its applicability to engineering reality, the present invention are directed to such as Fig. 2 institutes
The rectangular flat shown carries out topology optimization design.Design section is the rectangular area of 25mm × 20mm, and thickness 0.25mm, divides
For 100 × 80 units.Elasticity modulus of materials E=201Mpa, Poisson's ratio μ=0.3.The left end of rectangular area is fixed, lower right
Apply the power of F=300N vertically downward, in order to avoid stress concentration effect, load F is to load on the node of design domain lower right 9
On, the load of 0.1 times of application on each node.Without considering the influence of gravity, the displacement of load(ing) point is constrained so that u ＜ 2cm, and
And cause the stress of structure to be no more than 250Mpa, choose penalty factor p=3.If elastic modulus E, load F have with respect to nominal value
10% fluctuation, i.e. E=[180.9,221.1] Mpa, F=[270,330] N；If displacement constraint u and allowable stress [σ] are with respect to name
Adopted value has 10% fluctuation, i.e. u=[1.8,2.2] cm, σ=[225,275] Mpa.
Fig. 3 is respectively certainty topological optimization result and Multidisciplinary systems are respectively Rs=0.90, Rs=0.95 and Rs
Topological optimization Comparative result when=0.999.It can be seen that certainty topological optimization and different Multidisciplinary systems topological optimizations go out
The configuration for the structure come is there are larger difference, compared to certainty topological optimization as a result, Multidisciplinary systems topological optimization knot
Fruit structure is more reasonable, and structure is more stable.When using the Uncertainty as Multidisciplinary systems, deterministic optimization knot
The Multidisciplinary systems of three restrained displacements of fruit are only respectively Rs=0.4504.That is the result of deterministic optimization is not enough to should
Influence to uncertain variable.Iteration history in process of topology optimization is as shown in figure 4, compared to initial designs, loss of weight effect
Fruit is obvious；As reliability allowable value increases, structure tends to safety, and weight increased.
The present invention proposes the Continuum Structure Multidisciplinary systems topology under a kind of displacement and global stress mixed constraints
Optimization method.Initially set up with construction weight target as an optimization, it is continuous as what is constrained using displacement structure and global stress
Body structure Multidisciplinary systems topological optimization model；Then it is close unit to be obtained by unit design variable using filter density method
Degree, then with relaxation rule calculate structure displacement and stress, using stress resultant function constraint to global stress constraint into
Row approximate processing, the bound of displacement and stress resultant function is obtained using vertex combined method, so as to obtain displacement and stress is comprehensive
Close the Multidisciplinary systems index of function；Then convergence is solved instead of Multidisciplinary systems index using optimization characteristic displacement
Problem, and with the sensitivity of adjoint vector method and compound function derivation law solving-optimizing characteristic displacement；Finally with movement
Asymptotical Method is iterated calculating, until meeting corresponding convergence conditions, is met the optimal design side of Reliability Constraint
Case.
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
It is all to use equivalents or wait for the Topology Optimization Design of Continuum Structures field under displacement and global stress mixed constraints
The technical solution that effect is replaced and formed, all falls 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 (7)
1. the Continuum Structure Multidisciplinary systems Topology Optimization Method under a kind of displacement and global stress mixed constraints, its feature
It is：Realize that step is as follows：
Step 1：Design variable is described using density variable method, comes description scheme material properties and load with interval model
Uncertainty, with the volume of structure target as an optimization, using displacement structure and global stress as constraining, establishes non-probability decision
Property topological optimization model is as follows：
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Wherein, V be optimize region volume, ρiAnd ViThe relative density and volume of respectively i-th unit, and ρiIt is design
The function of variable r, N are the unit sum of optimization region division,It is the actual displacement interval value of j-th of displacement constraint point,It is the Admissible displacement interval value of j-th of displacement constraint, m is the number of displacement constraint,It is the reality of i-th of stress constraint point
Border stress interval value,It is the allowable stress interval value of stress constraint, RsIt is non-Making by Probability Sets reliability index,It is jth
The corresponding non-probability decision degree of target of a displacement constraint,It is the non-probability decision degree of the corresponding target of k-th of stress constraint,rFor
The lower limit of design variable；
Step 2：Design variable is filtered using filter density method, obtains the density value of unit, with section amount come
The uncertainty of elasticity modulus of materials and load is described, using vertex combined method, and the springform using relaxation rule to unit
Amount and Stress calculation relax, and obtain the displacement of structure and the stress of unit and corresponding stress resultant function
Value, is compared to obtain the bound of the bound of displacement structure and stress resultant function and its combination of corresponding vertex；
Step 3：According to the bound of displacement and the bound of stress resultant function, with reference to Multidisciplinary systems model, obtain
Displacement and the non-Making by Probability Sets reliability of stress resultant function constraint；
Step 4：Convergence problem is improved instead of Multidisciplinary systems index using optimization characteristic displacement, using optimizing feature
Former Optimized model can be rewritten as by displacement：
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Wherein, d (RI,SI) it is optimization characteristic displacement；
Step 5：According to displacement and the corresponding vertex combination of stress resultant function, obtained with adjoint vector method on displacement structure
The sensitivity of lower bound and stress resultant function bound to cell density, is then obtained in place using the Rule for derivation of compound function
Move and sensitivity of the optimization characteristic displacement of stress resultant function to design variable；
Step 6：Using obtained displacement and stress resultant function constraint condition value and its to the sensitivity information of design variable as
The input condition of mobile Asymptotical Method (MMA), solves optimization problem, is designed the renewal of variable；
Step 7：Repeat step two is designed the multiple renewal of variable, until current design meets reliability about to step 6
Beam, and when the Relative percent change of object function is less than preset value ξ, then stop optimization process.
2. the Continuum Structure Multidisciplinary systems under a kind of displacement according to claim 1 and global stress mixed constraints
Topology Optimization Method, it is characterised in that：In the step 1 structural wood is characterized with the non-Making by Probability Sets reliability index of Qiu
Expect the uncertain influence to the rigidity of structure and strength character of elasticity modulus and magnitude of load, construct displacement and global stress
Non-probabilistic set-based reliability model under mixed constraints.
3. the Continuum Structure Multidisciplinary systems under a kind of displacement according to claim 1 and global stress mixed constraints
Topology Optimization Method, it is characterised in that：In the step 2 displacement structure and stress resultant letter are obtained with vertex combined method
Several bounds and its combination of corresponding vertex.
4. the Continuum Structure Multidisciplinary systems under a kind of displacement according to claim 1 and global stress mixed constraints
Topology Optimization Method, it is characterised in that：With displacement structure and the optimization characteristic displacement of stress resultant function in the step 4
To replace original reliability index, so as to improve the convergence of former optimization problem.
5. the Continuum Structure Multidisciplinary systems under a kind of displacement according to claim 1 and global stress mixed constraints
Topology Optimization Method, it is characterised in that：According to displacement and the corresponding set of vertices of stress resultant function bound in the step 5
Close, the sensitivity of displacement and stress resultant function bound to cell density is solved using adjoint vector method.
6. the Continuum Structure Multidisciplinary systems under a kind of displacement according to claim 1 and global stress mixed constraints
Topology Optimization Method, it is characterised in that：In the step 5 displacement and stress resultant letter are solved with compound function derivation law
Sensitivity of several optimization characteristic displacements to design variable, first solves displacement and the optimization characteristic displacement contraposition of stress resultant function
The sensitivity with stress resultant function bound is moved, then solves displacement bound and stress resultant function bound again to unit
The sensitivity of density, then solves sensitivity of the cell density to design variable, is finally multiplied three to obtain displacement and stress
Sensitivity of the optimization characteristic displacement of comprehensive function to design variable.
7. the Continuum Structure Multidisciplinary systems under a kind of displacement according to claim 1 and global stress mixed constraints
Topology Optimization Method, it is characterised in that：With MMA algorithms to non-under displacement and global stress mixed constraints in the step 6
Probabilistic reliability topology optimization problem is solved.
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