CN108875125A - A kind of non-individual body bi-material layers structural topological optimization method under displacement and global stress mixed constraints - Google Patents
A kind of non-individual body bi-material layers structural topological optimization method under displacement and global stress mixed constraints Download PDFInfo
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Abstract
The invention discloses the non-individual body bi-material layers structural topological optimization methods under a kind of displacement and global stress mixed constraints, this method is first depending on the design variable description that variable density method carries out bi-material layers topology optimization problem, two design variables are arranged to each unit, then filter density is carried out to two design variables with filter density method respectively and obtains the corresponding two pseudo- density of unit, on the basis of ε Method of Stress Relaxation, propose that bi-material layers stress relaxation rule carrys out the stress constraint criterion of construction unit；Stress is integrated using stress resultant function, obtains global stress constraint function, and solves the sensitivity of global stress constraint function with adjoint vector method and compound function derivation law；Finally optimization problem is solved with mobile Asymptotical Method, iterative calculation obtains the optimization design scheme for meeting constraint up to meeting corresponding convergence conditions.The present invention can ensure to obtain the bi-material layers topology configuration for meeting displacement and global stress mixed constraints condition, and can realize effective loss of weight.
Description
Technical field
The present invention relates to fields containing Topology Optimization Design of Continuum Structures, the in particular to structure of bi-material layers stress constraint function
It builds and is being displaced the formulation with the Continuum Structure bi-material layers topological optimization scheme under global stress mixed constraints.
Background technique
With the development of science and technology and society, people are higher and higher to the performance requirement of structure, so that structure optimization is ground
Studying carefully becomes particularly important.Optimal Structure Designing can be divided into three levels：Dimensionally-optimised, shape optimum and topological optimization.Wherein,
There is conclusive influence in the conceptual phase that topological optimization is designed as structure on final configuration, thus to structural behaviour
It influences also most significant.Therefore the topology optimization design of research research is of great significance.
Traditional Topology Optimization is concentrated mainly on using structure compliance as target, with structural volume be constraint or with
Structural volume is target, with structure compliance be constraint the problem of on, for stress constraint topology optimization problem research then
It is fewer.Stress constraint has three properties：Singularity, locality and strong nonlinearity.Cheng Gengdong etc. is in research stress constraint
Under topology optimization problem when, it was found that Their Exotic Properties of stress, and propose ε stress relaxation rule.Also some scholars propose
Q-p method solves the singularity problem of stress.For the isolated problem of stress constraint, some scholars propose p-shaped formula
Stress resultant functional based method, this method has good propinquity effect to global maximum stress.Also some scholars propose K-S and answer
Power comprehensive function method, K-S stress resultant function solve the sensitivity of design variable fairly simple.However, stress resultant letter
Counting method makes the non-linear reinforcement of stress constraint, so that the convergence that Optimized model solves is deteriorated, some scholars are directed to this
Problem proposes the method for subregion stress resultant, this reduces the non-linear of stress resultant function to a certain extent.
It should be noted, however, that the topology optimization problem research under current stress constraint is mainly for single material knot
Structure rarely has for the bi-material layers even research of poly-material structure.Some scholars have studied the miniature fortune of bi-material layers under stress constraint
The topology optimization problem of motivation structure.In existing document, there is not yet the non-individual body under displacement and global stress mixed constraints
Bi-material layers structural Topology Optimization problem, however in practical projects, the using of bi-material layers can make the designability of structure more
By force, to further increase the performance of structure.Therefore, the bi-material layers topological optimization under research displacement and global stress mixed constraints
Problem is of great significance.
Summary of the invention
The technical problem to be solved by the present invention is to：It overcomes the deficiencies of the prior art and provide a kind of displacement and global stress is mixed
Non-individual body bi-material layers structural topological optimization method under contract beam.The present invention considers non-individual body bi-material layers structure in displacement and the overall situation
Topology optimization problem under stress constraint breaches the limitation of traditional stress constraint topological optimization homogenous material, further increases
The designability of engineering structure.
The technical solution adopted by the present invention：A kind of non-individual body bi-material layers structure under displacement and global stress mixed constraints is opened up
Optimization method is flutterred, realizes that steps are as follows：
Step 1：Design variable is described using density variable method, using the volume of structure as optimization aim, with displacement structure
With global stress as constraining, it is as follows to establish topological optimization model：
Wherein, V is the volume for optimizing region, ρi(x, y) and ViThe relative density and volume of respectively i-th unit, and
ρiIt is the function of design variable x and y, N is the unit sum for optimizing region division, ujIt is the actual displacement of j-th of displacement constraint point
Value, uj,maxIt is the Admissible displacement value of j-th of displacement constraint, m is the number of displacement constraint, σiIt is the actual stress of i-th of unit
Value,For the allowable stress of material 1, i ∈ solid (1) indicates that unit i is material 1,For the allowable stress of material 2, i ∈
Solid (2) indicates that unit i is material 2.R be design variable lower limit, be in order to avoid stiffness matrix is unusual and be arranged one
In a small amount, 0.01 is generally taken；
Step 2：Design variable x and y are filtered respectively using filter density method, the unit i after being filtered
The pseudo- density value of corresponding two, then according to the density p of pseudo- density value computing unit ii(x, y), and then calculate the total of structure
The sensitivity of volume and structure total volume about unit puppet density；
Step 3：Form is respectively extended to micro-structure/material model (SIMP model) bi-material layers using the solid with penalty factor
The elasticity modulus of unit is calculated, the stiffness matrix of unit is obtained, and then obtains the global stiffness matrix of structure, then root
The displacement that structure is calculated according to the equation of static equilibrium establishes bi-material layers stress relaxation on the basis of single material ε Method of Stress Relaxation
Rule, the stress constraint function of construction unit calculate in stress path, using the von mises stress of unit center point as single
The characterization of first stress；
Step 4：All element stress constraint functions are integrated using the stress resultant function of p-shaped formula, obtain stress
Comprehensive function carries out stress comprehensive function further according to the deviation of stress resultant functional value and element stress constraint function maximum value
Amendment, obtains global stress constraint function；
Step 5：Displacement structure and global stress constraint function are solved to the sensitive of unit puppet density using adjoint vector method
Degree recycles compound function derivation law to obtain the sensitivity of displacement structure and global stress constraint function design variable, is asking
During solving sensitivity, by correction factor as constant；
Step 6：Using obtained displacement and global stress constraint functional 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 update of variable；
Step 7：Step 2 is repeated to step 6, the multiple update of variable is designed, until current design meets about
Beam, and the Relative percent change of objective function be less than preset value ξ when, then stop optimization process.
The advantages of the present invention over the prior art are that：
(1) present invention considers the bi-material layers structural Topology Optimization problem under displacement and global stress constraint, so that structure
Design domain it is bigger, compared to single material structure topological optimization under traditional displacement and overall situation stress constraint, the present invention can be with
Obtain more light-weighted structure；
(2) the invention proposes bi-material layers stress relaxation method, element stress constraint function is constructed, on this basis may be used
To propose more material stress relaxation methods, therefore the contents of the present invention scope of application is than wide.
Detailed description of the invention
Fig. 1 is the present invention for the non-individual body bi-material layers structural Topology Optimization process under displacement and global stress mixed constraints
Figure.
Specific embodiment
With reference to the accompanying drawing and specific embodiment further illustrates the present invention.
As shown in Figure 1, the invention proposes the non-individual body bi-material layers structures under a kind of displacement and global stress mixed constraints
Topology Optimization Method includes the following steps：
Step 1：Design variable is described using density variable method, using the volume of structure as optimization aim, with displacement structure
With global stress as constraining, it is as follows to establish topological optimization model：
Wherein, V is the volume for optimizing region, ρi(x, y) and ViThe relative density and volume of respectively i-th unit, and
ρiIt is the function of design variable x and y, N is the unit sum for optimizing region division, ujIt is the actual displacement of j-th of displacement constraint point
Value, uj,maxIt is the Admissible displacement value of j-th of displacement constraint, m is the number of displacement constraint, σiIt is the actual stress of i-th of unit
Value,For the allowable stress of material 1, i ∈ solid (1) indicates that unit i is material 1,For the allowable stress of material 2, i ∈
Solid (2) indicates that unit i is material 2.R be design variable lower limit, be in order to avoid stiffness matrix is unusual and be arranged one
In a small amount, 0.01 is generally taken；
Step 2：Design variable x and y are filtered respectively using filter density method, the unit i after being filtered
The pseudo- density value of corresponding two, then according to the density p of pseudo- density value computing unit ii(x, y), and then calculate the total of structure
Volume；
Filter density is carried out respectively to design variable x and y, filtered version is as follows：
Wherein x 'i,y′iIt is the corresponding two pseudo- density values of i-th of unit, xj,yjIt is set for j-th of unit corresponding two
Count variable.ΩiIt is less than or equal to r with unit i distance to be all0The set of the unit of (filtering radius), rjIt is unit j and unit
The distance of the central point of i.
The then density p of unit ii(x, y) can be solved with following formula：
ρi(x, y)=x 'i[y′iρ1+(1-y′i)ρ2]
Wherein, ρ1For the density of material 1, ρ2For the density of material 2.
Then the total volume of structure may be calculated：
Wherein, ViFor the unit volume of i-th of unit.
Particularly, pseudo- density x ' of the structure total volume to i-th of uniti,y′iDerivative may be calculated：
Step 3：Form is respectively extended to micro-structure/material model (SIMP model) bi-material layers using the solid with penalty factor
The elasticity modulus of unit is calculated, the stiffness matrix of unit is obtained, and then obtains the global stiffness matrix of structure, then root
The displacement that structure is calculated according to the equation of static equilibrium establishes bi-material layers stress relaxation on the basis of single material ε Method of Stress Relaxation
Rule, the stress constraint function of construction unit calculate in stress path, using the von mises stress of unit center point as single
The characterization of first stress；
Using the bi-material layers extension form of SIMP model come the elasticity modulus of computing unit i：
Wherein, E1For the elasticity modulus of material 1, E2For the elasticity modulus of material 2.
, can be with construction unit stiffness matrix after obtaining unitary elasticity modulus, and then the global stiffness matrix of structure is obtained, into
Row Static Calculation obtains the displacement of unit.
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,σ3Respectively refer to the first, second and third principal stress.
According to obtained Displacement of elemental node and corresponding displacement shape function and the available unit each side of strain matrix
To direct stress and shearing stress be：
σ=D ε=DBue=Sue
In formula, the elasticity modulus of element strain matrix is using the unitary elasticity modulus after punishment.
It is calculated to simplify, and in view of unit internal stress changes smaller, therefore selection unit central point answers masterpiece
For the characterization of element stress.
Von mises stress according to the direct stress of unit center point all directions and the available central point of 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)
After obtaining element stress, bi-material layers stress relaxation rule is constructed on the basis of single material ε Method of Stress Relaxation.
Topology optimization problem, for unit i, being constrained to originally are constrained for single material stress：
It is rewritten as：
This formula is equivalent to：
Constraint relaxation is carried out below：
It is rewritten as again：
On the basis of single material ε Method of Stress Relaxation, the bi-material layers stress constraint function of structural unit i is：
Wherein, (σi)crFor the von mises stress of i-th of unit center point,For the allowable stress of material 1,For material
The allowable stress of material 2, ε are that stress relaxation is a small amount of, there is ε2=r。
Step 4：All element stress constraint functions are integrated using the stress resultant function of p-shaped formula, obtain stress
Comprehensive function carries out stress comprehensive function further according to the deviation of stress resultant functional value and element stress constraint function maximum value
Amendment, obtains global stress constraint function；
All element stress constraint functions are integrated using the stress resultant function of p-shaped formula.In view of element stress
Constraint function is likely less than zero, therefore it is as follows to construct global stress constraint function：
Wherein, P is stress resultant index, and P is higher to represent stress resultant function to the propinquity effect of global maximum stress more
It is good, but the increase of P can also make the nonlinearity of stress resultant function increase, so that the convergence of problem is deteriorated.P desirable 6
~12, generally take P=8,10.
C is Stress Correction Coefficient, is the element stress constraint function according to maximum stress unit to global stress constraint letter
Number is modified.The element stress constraint function of maximum stress unit is as follows：
Therefore correction factor c expression formula is as follows：
When carrying out the sensitivity solution of global stress constraint function below, c is handled as constant.
Global stress constraint is：
Φ-1≤0
Step 5：Displacement structure and global stress constraint function are solved to the sensitive of unit puppet density using adjoint vector method
Degree recycles compound function derivation law to obtain the sensitivity of displacement structure and global stress constraint function design variable, is asking
During solving sensitivity, by correction factor as constant；
Since displacement structure has derived the sensitivity of design variable in some documents, only derive below global
Stress function is to unit puppet density x 'j,y′jSensitivity.
Global stress function is to x 'jSensitivity be solved to：
WhereinIt solves as follows：
Due to (σcr)iIt is about x 'i,y′iAnd the function of u, for convenience of explanation, if
ThenIt can solve as follows：
It enables Then have：
It is arranged, is had：
Wherein first itemIt can be straight
Result before connecting is calculated, Section 2
It needs to find out by adjoint vector method.If：
Wherein adjoint vectorIt can be obtained by a FEM calculation.ThenIt can be solved to：
In formula, KjFor the element stiffness matrix of unit j, u is the displacement structure under actual condition.So far, the overall situation has been obtained
Stress function is about unit design variable x 'jSensitivity.
Similarly, global stress function is to y 'jSensitivity be solved to：
WhereinIt solves as follows：
In formula,It solves as follows：
It enablesThen
Have：
It is arranged, is had：
Wherein first itemCan directly it pass through
Result before is calculated, Section 2It needs
It is found out by adjoint vector method.If：
Wherein adjoint vectorIt can be obtained by a FEM calculation.ThenIt can be solved to：
So far, the sensitivity for completing global stress function about unit puppet density solves.
Sensitivity of the unit puppet density about design variable is solved below.
The density x ' of unit iiTo the design variable x of unit jjSensitivity be given by：
Wherein ΩiIt is less than or equal to r with unit i distance to be all0The set of the unit of (filtering radius), rjUnit j with
The distance of the central point of unit i.
Similarly, the density y ' of unit iiTo the design variable y of unit jjSensitivity be given by：
Wherein ΩiIt is less than or equal to r with unit i distance 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 unit puppet density about unit design variable has been obtained.
With compound function derivation law, have：
Similarly, have：
So far, sensitivity of the global stress function about design variable has been obtained.
Step 6：Using obtained displacement and global stress constraint functional 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 update of variable；
Step 7：Step 2 is repeated to step 6, the multiple update of variable is designed, until current design meets about
Beam, and the Relative percent change of objective function be less than preset value ξ when, then stop optimization process.
The invention proposes the non-individual body bi-material layers structural Topology Optimization sides under a kind of displacement and global stress mixed constraints
Method.It has initially set up using construction weight minimum as optimization aim, using displacement structure and global stress as the non-individual body of constraint
Bi-material layers structural Topology Optimization model；Then unit design variable is filtered with filter density method respectively, obtains list
First puppet density calculates the elasticity modulus of unit using the bi-material layers extension form of SIMP model, and comes with finite element
The displacement of structure is calculated, then uses bi-material layers Method of Stress Relaxation construction unit stress constraint function, and then utilize stress resultant
Functional based method constructs global stress constraint function, and solves global stress about with adjoint vector method and compound function derivation law
The sensitivity of beam function；It finally optimizes with mobile Asymptotical Method, until meeting corresponding convergence conditions, obtains
Meet the topology optimization design scheme of displacement and global stress constraint.
The above is only specific steps of the invention, are not limited in any way to protection scope of the present invention；Its is expansible to answer
For be displaced and global stress mixed constraints under non-individual body bi-material layers Structural Topology Optimization Design field, it is all to use equivalents
Or equivalence replacement and the technical solution that is formed, it all falls within rights protection scope of the present invention.
Part of that present invention that are not described in detail belong to the well-known technology of those skilled in the art.
Claims (6)
1. the non-individual body bi-material layers structural topological optimization method under a kind of displacement and global stress mixed constraints, it is characterised in that：
Realize that steps are as follows：
Step 1：Design variable is described using density variable method, using the volume of structure as optimization aim, with displacement structure and entirely
It is as follows to establish topological optimization model as constraint for office's stress：
Wherein, V is the volume for optimizing region, ρi(x, y) and ViThe relative density and volume of respectively i-th unit, and ρiIt is
The function of design variable x and y, N are the unit sum for optimizing region division, ujIt is the actual displacement value of j-th of displacement constraint point,
uj,maxIt is the Admissible displacement value of j-th of displacement constraint, m is the number of displacement constraint, σiIt is the actual stress value of i-th of unit,For the allowable stress of material 1, i ∈ solid (1) indicates that unit i is material 1,For the allowable stress of material 2, i ∈ solid
(2) indicate that unit i is material 2,rFor the lower limit of design variable, be in order to avoid stiffness matrix is unusual and be arranged one is a small amount of,
Generally take 0.01；
Step 2：Design variable x and y are filtered respectively using filter density method, the unit i institute after being filtered is right
The pseudo- density value of two answered, then according to the density p of pseudo- density value computing unit ii(x, y), and then calculate the total volume of structure；
Step 3：Using the solid with penalty factor respectively to micro-structure/material model (SIMP model) bi-material layers extension form to list
The elasticity modulus of member is calculated, and the stiffness matrix of unit is obtained, and then obtain the global stiffness matrix of structure, then according to quiet
Equilibrium equation calculates the displacement of structure, then uses the stress constraint function of bi-material layers stress relaxation rule construction unit；
Step 4：All element stress constraint functions are integrated using the stress resultant function of p-shaped formula, obtain stress resultant
Function repairs stress comprehensive function further according to the deviation of stress resultant functional value and element stress constraint function maximum value
Just, global stress constraint function is obtained；
Step 5：Displacement structure and global stress constraint function are solved to setting using compound function derivation law and adjoint vector method
The sensitivity for counting variable, during solving sensitivity, by correction factor as constant；
Step 6：Using obtained displacement and global stress constraint functional value and its to the sensitivity information of design variable as moving
The input condition of Asymptotical Method (MMA), solves optimization problem, is designed the update of variable；
Step 7：Step 2 is repeated to step 6, is designed the multiple update of variable, until current design meets constraint, and
And the Relative percent change of objective function be less than preset value ξ when, then stop optimization process.
2. the non-individual body bi-material layers structural topology under a kind of displacement according to claim 1 and global stress mixed constraints is excellent
Change method, it is characterised in that：Two groups of design variables are filtered using filter density method respectively in the step 2, are obtained
Two pseudo- density values corresponding to each unit, and then obtain the density and volume of the unit.
3. the non-individual body bi-material layers structural topology under a kind of displacement according to claim 1 and global stress mixed constraints is excellent
Change method, it is characterised in that：Element stress constraint function is constructed with bi-material layers stress relaxation rule in the step 3.
4. the non-individual body bi-material layers structural topology under a kind of displacement according to claim 1 and global stress mixed constraints is excellent
Change method, it is characterised in that：All element stresses are integrated using the stress resultant function of p-shaped formula in the step 4,
Obtain bi-material layers overall situation stress constraint function.
5. the non-individual body bi-material layers structural topology under a kind of displacement according to claim 1 and global stress mixed constraints is excellent
Change method, it is characterised in that：Bi-material layers structure is solved according to compound function derivation law and adjoint vector method in the step 5
Sensitivity to design variable of displacement and global stress constraint function.
6. the non-individual body bi-material layers structural topology under a kind of displacement according to claim 1 and global stress mixed constraints is excellent
Change method, it is characterised in that：With MMA algorithm to Continuum Structure under displacement and global stress mixed constraints in the step 6
Bi-material layers topology optimization problem is solved.
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