CN112487684B - Non-probability reliability topological optimization method for laminate under force-heat coupling environment - Google Patents

Non-probability reliability topological optimization method for laminate under force-heat coupling environment Download PDF

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CN112487684B
CN112487684B CN202011380312.1A CN202011380312A CN112487684B CN 112487684 B CN112487684 B CN 112487684B CN 202011380312 A CN202011380312 A CN 202011380312A CN 112487684 B CN112487684 B CN 112487684B
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reliability
laminate
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CN112487684A (en
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王晓军
倪博文
王磊
李泽商
蒋晓航
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Beihang University
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Abstract

The invention discloses a topological optimization method for non-probabilistic reliability of a laminate under a force-heat coupling environment. The method comprises the steps of firstly, calculating equivalent elastic parameters and thermal property parameters of the composite material laminated plate according to a first-order shear deformation theory of a laminated structure. And then considering the influence of uncertain factors such as laminated plate material parameters, loads and the like, constructing a laminated structure reliability topological optimization framework by using an SIMP material interpolation model and taking the minimum structure weight as an optimization target, taking the relative density of units as a design variable and taking the allowable displacement response reliability of the concerned position under the action of the force-heat coupling load as a constraint, and obtaining the optimal configuration of the laminated structure under the force-heat coupling environment by iteration by adopting a moving asymptote optimization algorithm. The method takes the composite material laminated structure as an object, takes the coupling effect of a temperature field and a structural field into consideration in the process of topology optimization design, reasonably represents the influence of uncertainty on the structural configuration under the condition of a limited test sample, and gives consideration to safety and economy.

Description

Non-probability reliability topological optimization method for laminate under force-heat coupling environment
Technical Field
The invention relates to the technical field of structural topology optimization design, in particular to a non-probabilistic reliability topology optimization method for a laminate under a force-heat coupling environment.
Background
Looking at the aerospace blueprint in the 21 st century, great weight reduction is still an urgent need for aircraft design. The carbon fiber reinforced normal-stiffness laminated plate is an important structural composite material formed by laminating single layers of composite materials in a specific laminating mode, and is widely applied due to light weight and mature manufacturing technology. With the increasingly harsh flight service environment, the fine design of the aircraft structure gradually changes from a single bearing function to multiple functions of bearing, heat insulation, vibration reduction and the like, and the novel heat-conducting composite material taking the high-heat-conducting carbon fiber as a main reinforcement can effectively overcome the defects of metal heat-conducting materials, realize higher heat conductivity and show wide application prospects. The research of the existing composite material design mainly focuses on the size and geometric level, and the topological optimization design of the composite material laminated plate structure is still in the starting stage, so that the research forms a set of topological optimization design method considering the reliability of the bearing heat-proof multifunctional integrated laminated plate structure, and has important theoretical significance for improving the performance of the composite material laminated structure.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: aiming at a composite material laminated structure, the defects of the prior art are overcome, and a non-probability reliability topological optimization design method aiming at the composite material structure in a force-heat coupling environment is provided. The invention fully considers the influence of complex force-heat environment in the actual engineering, constructs the non-probability reliability analysis index considering various uncertain factors, obtains a result more conforming to the real situation, and has stronger engineering applicability.
The technical scheme adopted by the invention is as follows: a method for topological optimization of non-probability reliability of laminate under a force-heat coupling environment is realized by the following steps:
the method comprises the following steps: establishing a laminated plate force thermal coupling constitutive model according to given composite material unidirectional plate material parameters and a layering mode, calculating the equivalent thermal conductivity of the laminated plate by utilizing a first-order lamination theory, and defining by the following formula:
Figure BDA0002809157650000011
wherein Δ Q represents heat, Δ t represents time,
Figure BDA0002809157650000021
denotes heat flow rate, A denotes material verticalityIn the cross-sectional area of the heat flux, Δ T represents the temperature difference across the material of length x. Assuming that the number of layers in the laminated structure at an angle theta to the heat flow rate is nθThe total number of layers is NN, NN ═ Σ nθThe ply ratio is omegaθ，ωθ＝nθ/NN, the thermal conductivity of the laminate structure in the direction of heat flow is expressed as:
Mc,NN＝∑ωθ(Mc,pcos2θ+Mc,tsin2θ)
in the formula, Mc,pDenotes the thermal conductivity of the composite unidirectional sheet along the direction of the fibres, Mc,tIndicating the thermal conductivity of the composite unidirectional sheet perpendicular to the direction of the fibers. According to the classical lamination theory of composite materials, the temperature varies linearly along the thickness direction of the laminate, namely:
T(x,y,z)＝T0(x,y)+Tz'(x,y)·z
wherein T is0(x, y) represents the face temperature T 'in the laminated structure'z(x, y) represents a temperature gradient in the thickness z direction of the laminate.
Taking a laminate of symmetrical plies as an example, the stress of the k-th layer in the global coordinate system is obtained by the following formula:
Figure BDA0002809157650000022
wherein sigmax,σyAnd τxyRepresenting a stress component, ε, in a global coordinate systemx，εyAnd gammaxyRepresenting the strain component, α, in a global coordinate systemx，αyAnd alphaxyIndicating the coefficient of thermal expansion in the global coordinate system.
Figure BDA0002809157650000023
The stiffness matrix representing a single ply of the laminate under the global coordinate system can be calculated as follows:
Figure BDA0002809157650000024
in the formula E1，E2Young's modulus in the principal direction 1, Young's modulus in the principal direction 2, G, respectively, of the unidirectional sheet12Is the shear modulus, μ, of the principal directions 1 and 221And mu12Poisson's ratios of the main directions 1 and 2, respectively; t iskThe transformation matrix representing the k-th layer of the laminate, i.e.:
Figure BDA0002809157650000025
in the formula, thetakIndicating the fiber angle of the k-th layer.
Unit elastic matrix of composite laminated plate
Figure BDA0002809157650000026
Comprises the following steps:
Figure BDA0002809157650000027
wherein, tkIs the thickness of the k-th layer of the laminate and h is the total thickness of the laminate. In the structural finite element method, the element stiffness matrix KiCan be obtained by the following formula:
Figure BDA0002809157650000031
wherein B is a strain-displacement matrix. The overall stiffness matrix of the composite laminate may be obtained by summing the cell stiffness matrices:
Figure BDA0002809157650000032
step two: selecting an uncertainty parameter b for describing the structural characteristics of the laminated plate or the external force thermal load according to the service environment and the structural material properties of the laminated plate in the actual engineering, and according to a finite element balance equation:
KI(b)UI(b)＝FI(TI(b),b)
MI(b)TI(b)＝QI(b)
wherein the structure displacement field UIInvolving a temperature field TIThe resulting displacement. KIIs a matrix of overall stiffness intervals of the laminate, MIAs a matrix of integral heat-conducting zones, FIIs a force load interval vector including external load caused by temperature field and mechanical load, QIIs a heat load interval vector. And (3) solving upper and lower bounds of structural force thermal coupling response under the influence of uncertainty parameters by using an interval parameter vertex combination method, namely:
Figure BDA0002809157650000033
Figure BDA0002809157650000034
wherein the content of the first and second substances, iu(b)in order to displace the lower bound of the response,
Figure BDA0002809157650000035
is the displacement response upper bound;
Figure BDA0002809157650000036
indicates the section constituted by the uncertainty parameter b.
Step three: based on a structure non-probability set reliability model, limiting the actual displacement interval (u) of the laminated plate displacementj)IAnd allowable displacement interval
Figure BDA0002809157650000037
Normalizing to range between [ -1,1 [)]The method comprises the following steps:
Figure BDA0002809157650000038
Figure BDA0002809157650000039
wherein δ ujAnd
Figure BDA00028091576500000310
actual and allowable displacements for the normalized jth displacement constraint, (δ uj)IAnd
Figure BDA00028091576500000311
for the normalized actual displacement interval and the allowed displacement interval of the jth displacement constraint,
Figure BDA00028091576500000312
and
Figure BDA00028091576500000313
respectively the central value and the radius of the actual displacement interval,
Figure BDA00028091576500000314
and
Figure BDA0002809157650000041
respectively, the central value and the radius of the allowable displacement interval according to the structural function:
Φ(ρ,b)＝U*(b)-U(ρ,b)
and judging whether the structure is safe, wherein the structure is invalid when phi (rho, b) is less than 0, and the structure is safe when phi (rho, b) is more than or equal to 0.
Step four: the optimization characteristic distance d is used as a non-probability reliability index for measuring whether the structure is invalid or not, and is defined as follows: the moving distance from the original failure plane to the target failure plane, wherein the target failure plane is a plane parallel to the original failure plane and has a reliability of a given value RtargDistance d of optimization feature of jth constraintjThe non-probabilistic reliability of a laminate to satisfy this constraint can be quantified by the expression:
Figure BDA0002809157650000042
step five: adopting a variable density topological optimization model based on the SIMP method, and using the non-probability reliability index obtained in the fourth step, namely the characteristic distance djAs a constraint, establishing a non-probability reliability topological optimization formula of the composite material laminated structure:
find:ρ＝(ρ1,ρ2,…,ρN)
Figure BDA0002809157650000043
s.t.K(ρ,b)U(ρ,b)＝F(T(ρ,b),ρ,b)
M(ρ,b)T(ρ,b)＝Q(ρ,b)
dj(ρ,b)≤0,j＝1,2,…,m
0≤ρ≤ρi≤1
where ρ is (ρ)1,ρ2,…,ρN) The design variables for the cell pseudo-density, i.e., the topology optimization problem, N represents the number of design variables, W represents the structure weight,ρthe lower bound for the design variable is a small value preset to prevent singularity of the stiffness matrix.
Step six: solving the sensitivity of the upper and lower bounds of the response at the displacement constraint position of the laminated structure by adopting an adjoint vector method, and obtaining a non-probability reliability constraint d according to a chain type derivation rule of a composite functionjThe sensitivity of (c);
step seven: iterative solution is carried out on design variables by adopting a moving asymptote optimization algorithm (methods), and in the iterative process, if the current design does not meet the reliability constraint dj<0, or when the sum of the design variable change absolute values of the two iteration steps is greater than the tolerance epsilon, adding one to the iteration step number, and returning to the second step, or else, carrying out the next step;
step eight: if the current design satisfies the reliability optimization feature distance dj<0, when the sum of the design variable change absolute values of the front and back iteration steps is less than the tolerance epsilon, the iteration is finished to obtain the non-probability reliability of the laminated structureAnd (4) optimizing the configuration of the topological optimization design.
Compared with the prior art, the invention has the advantages that: the invention provides a novel non-probability reliability topological optimization method under the influence of force-heat coupling load, which aims at a composite material laminated structure. On the other hand, the non-probabilistic reliability topology optimization framework established by the invention provides an effective solution for the topology optimization design of the composite structure under the condition of small sample test data.
Drawings
FIG. 1 is a flow chart of a non-probabilistic reliability topology optimization method of the present invention for a composite laminate structure in a force-thermal coupling environment;
FIG. 2 is a schematic structural view of a composite material subjected to thermal coupling load used in the present invention
FIG. 3 is a schematic view of the calculation of the thermal conductivity of the unidirectional sheet used in the present invention;
FIG. 4 is a schematic view of a classical laminate used in the present invention;
FIG. 5 is a one-dimensional interference model of the actual displacement response interval and the allowable displacement interval at the laminate displacement constraints used in the present invention;
FIG. 6 is a two-dimensional interference diagram of a non-probabilistic reliability model after parameter normalization for use in the present invention;
FIG. 7 is a non-probabilistic reliability indicator, i.e., an optimized feature distance d, used in the present inventionjSchematic representation of (a).
Detailed Description
The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings.
As shown in FIG. 1, the invention provides a topological optimization method for non-probabilistic reliability of a laminate under a force-heat coupling environment, and the method is a composite material laminated structure bearing force-heat coupling loads as shown in FIG. 2, wherein Ω is a design domain, Q is a heat load such as a heat flow rate and a heat flow density, and F is a maximum valuemIs a mechanical carrierThe load, Δ T, is the difference between the internal temperature field of the structure and the initial temperature field due to the heat load. The invention comprises the following steps:
(1) establishing a laminated plate force thermal coupling constitutive model according to given composite material unidirectional plate material parameters and a layering mode, calculating the equivalent thermal conductivity of the laminated plate by utilizing a first-order lamination theory, and defining by the following formula:
Figure BDA0002809157650000051
wherein Δ Q represents heat, Δ t represents time,
Figure BDA0002809157650000061
indicating the heat flow rate, a the cross-sectional area of the material perpendicular to the heat flow, and Δ T the temperature difference across the material of length x. Assuming that the number of layers in the laminated structure at an angle theta to the heat flow rate is nθThe total number of layers is NN, NN ═ Σ nθThe ply ratio is omegaθ，ωθ＝nθ/NN, the thermal conductivity of the laminate structure in the direction of heat flow is expressed as:
Mc,NN＝∑ωθ(Mc,pcos2θ+Mc,tsin2θ) (2)
in the formula, Mc,pDenotes the thermal conductivity of the composite unidirectional sheet along the direction of the fibres, Mc,tIndicating the thermal conductivity of the composite unidirectional sheet perpendicular to the direction of the fibers. According to the classical lamination theory of composite materials, the temperature varies linearly along the thickness direction of the laminate, namely:
T(x,y,z)＝T0(x,y)+Tz'(x,y)·z (3)
wherein T is0(x, y) represents the face temperature in the laminate structure, T'z(x, y) represents a temperature gradient in the thickness z direction of the laminate.
Taking a laminate of symmetrical plies as an example, the stress of the k-th layer in the global coordinate system is obtained by the following formula:
Figure BDA0002809157650000062
wherein sigmax,σyAnd τxyRepresenting a stress component, ε, in a global coordinate systemx，εyAnd gammaxyRepresenting the strain component, α, in a global coordinate systemx，αyAnd alphaxyIndicating the coefficient of thermal expansion in the global coordinate system.
Figure BDA0002809157650000063
The stiffness matrix representing a single ply of the laminate under the global coordinate system can be calculated as follows:
Figure BDA0002809157650000064
in the formula E1,，E2，G12，μ21And mu12Respectively the young's modulus of the unidirectional plate in the principal direction 1, the young's modulus of the principal direction 2, the shear modulus of the principal directions 1 and 2, and the poisson's ratio of the principal directions 1 and 2. T iskThe transformation matrix representing the k-th layer of the laminate, i.e.:
Figure BDA0002809157650000065
in the formula, thetakIndicating the fiber angle of the k-th layer.
Unit elastic matrix of composite laminated plate
Figure BDA0002809157650000066
Comprises the following steps:
Figure BDA0002809157650000067
wherein, tkIs the thickness of the k-th layer of the laminate and h is the total thickness of the laminate. In the structural finite element method, the element stiffness matrix KiCan be obtained by the following formula:
Figure BDA0002809157650000071
wherein B is a strain-displacement matrix. The overall stiffness matrix of the composite laminate may be obtained by summing the cell stiffness matrices:
Figure BDA0002809157650000072
(2) selecting an uncertainty parameter b for describing the structural characteristics of the laminated plate or the thermal load of external force according to the complex service environment and the structural material properties of the laminated plate in the actual engineering, and according to a finite element balance equation
Figure BDA0002809157650000073
Wherein the structure displacement field UIInvolving a temperature field TIThe resulting displacement. KIIs a matrix of overall stiffness intervals of the laminate, MIAs a matrix of integral heat-conducting zones, FIIs a force load interval vector including external load caused by temperature field and mechanical load, QIIs a heat load interval vector. When calculating the structural force thermal coupling response, the unidirectional coupling of the temperature field and the displacement field of the laminated structure is considered, namely the temperature field distribution of the structure is calculated through steady-state heat conduction, and then the external load F caused by temperature is calculated through the temperature fieldthFinally F is addedthTemperature-independent mechanical load FthStacking, calculating the displacement field of the structure according to the thermoelastic theory of the laminated plate.
And (3) solving upper and lower bounds of structural force thermal coupling response under the influence of uncertainty parameters by using an interval parameter vertex combination method, namely:
Figure BDA0002809157650000074
wherein the content of the first and second substances, iu(b)in order to displace the lower bound of the response,
Figure BDA0002809157650000075
is the displacement response upper bound;
Figure BDA0002809157650000076
indicates the section constituted by the uncertainty parameter b.
(3) Actual displacement intervals (u) at the laminate displacement constraints shown in fig. 5 based on a structural non-probabilistic set reliability modelj)IAnd allowable displacement interval
Figure BDA0002809157650000077
Normalizing to range between [ -1,1 [)]The method comprises the following steps:
Figure BDA0002809157650000078
wherein δ ujAnd
Figure BDA0002809157650000079
actual and allowable displacements for the normalized jth displacement constraint, (δ uj)IAnd
Figure BDA00028091576500000710
for the normalized actual displacement interval and the allowed displacement interval of the jth displacement constraint,
Figure BDA00028091576500000711
and
Figure BDA0002809157650000081
respectively the central value and the radius of the actual displacement interval,
Figure BDA0002809157650000082
and
Figure BDA0002809157650000083
respectively, the center value and radius of the allowable displacement interval, according toStructural function:
Φ(ρ,b)＝U*(b)-U(ρ,b) (13)
to determine whether the structure is safe, as shown in fig. 6, the structure is disabled when Φ (ρ, b) <0, and the structure is safe when Φ (ρ, b) ≧ 0.
(4) The optimization characteristic distance d is used as a non-probability reliability index for measuring whether the structure is invalid or not, and is defined as follows: the moving distance from the original failure plane to the target failure plane, wherein the target failure plane is a plane parallel to the original failure plane and has a reliability RtargFor a given value, the optimization characteristic distance d of the jth constraintjThe non-probabilistic reliability of a laminate to satisfy this constraint can be quantified by the expression:
Figure BDA0002809157650000084
as shown in FIG. 7, when dj>At 0, the failure plane is at the target reliability RtargAnd the area of the safe area is smaller than the target value at the moment above the corresponding target failure plane, so that the requirement is not met. When d isjWhen the probability is less than or equal to 0, the failure plane has a non-probability reliability R with the targettargUnder the corresponding target failure plane, the area of the safe area is more than or equal to the target value, so that the design requirement is met;
(5) based on SIMP material interpolation model, the elastic modulus, shear modulus, thermal conductivity of the ith cell was expressed as a function of the relative density of the completely solid laminate material:
Figure BDA0002809157650000085
where p is a penalty factor for implementing a penalty on the intermediate density unit, typically taken as 3.
Using the non-probability reliability index obtained in the step (4), namely the characteristic distance djAs a constraint, establishing a non-probability reliability topological optimization column of a composite material laminated structure based on a variable density topological optimization model of SIMP (simple modeling and mapping) method：
Figure BDA0002809157650000091
Where ρ is (ρ)1,ρ2,…,ρN) The design variables for the cell pseudo-density, i.e., the topology optimization problem, N represents the number of design variables, W represents the structure weight,ρthe lower bound for the design variable is a small value preset to prevent singularity of the stiffness matrix.
(6) The invention adopts a gradient optimization algorithm of a mobile asymptote method (MMA) to solve an optimization problem, so that partial derivatives of a target function and a constraint function to design variables need to be obtained, namely sensitivity analysis is carried out. Because the number of design variables of the invention is much more than that of the constraint functions, the sensitivity is solved in a differential mode, and huge calculation amount is brought. Therefore, the sensitivity of the upper and lower bounds of the response of the laminated structure displacement constraint part is solved by adopting the adjoint vector method, and the non-probability reliability constraint d is obtained according to the chain derivation rule of the composite functionjThe sensitivity of (2).
Constructing an augmented Lagrangian function:
Figure BDA0002809157650000092
wherein λj1For the introduced Lagrangian multiplier, (F)m)ΙAnd (F)th)ΙForce interzone vectors due to mechanical and thermal loads respectively,
Figure BDA0002809157650000093
βi(ρi)＝Eiαidenotes the thermal stress coefficient, DeltaT, of the ith celliRepresents the average amount of change in temperature in the ith cell,
Figure BDA0002809157650000094
representing the unit load vector of the ith cell. According to equilibrium equation KIUI＝FIIt is obvious that
Figure BDA0002809157650000095
So the actual displacement interval u at the jth constraintj IWith respect to the design variable ρiThe partial derivatives of (a) are:
Figure BDA0002809157650000096
the above formula is for arbitrary lambdaj1All hold, in order to avoid calculations
Figure BDA0002809157650000097
The trouble of (2), order:
Figure BDA0002809157650000101
namely:
Figure BDA0002809157650000102
wherein the content of the first and second substances,
Figure BDA0002809157650000103
it can be seen that the formula formally and the equilibrium equation KIUI＝FIAgree, so to solve for the adjoint vector λj1The unit load can be applied at the jth node degree of freedom by one finite element calculation.
Will be lambdaj1Substitution formula:
Figure BDA0002809157650000104
in the formula (I), the compound is shown in the specification,
Figure BDA0002809157650000105
overall stiffness K between adjacent sections in any one topologyICan be written as:
Figure BDA0002809157650000106
therefore, there are:
Figure BDA0002809157650000107
chain-type derivation rule using complex function, reliability index djWith respect to the design variable ρiThe sensitivity of (d) can be expressed as:
Figure BDA0002809157650000108
in combination with, have
Figure BDA0002809157650000111
And
Figure BDA0002809157650000112
wherein the content of the first and second substances,
Figure BDA0002809157650000113
(uj)M,(uj *)Dand(uj)Dall can be obtained in the iterative process, and the reliability index djWith respect to the design variable ρiThe sensitivity of (a) can be solved completely and efficiently.
Further, the partial derivative of the objective function W to the design variable is:
Figure BDA0002809157650000114
so far, the sensitivities of the objective function and the constraint on the design variables are obtained, and a gradient algorithm can be called to update the design variables.
(7) Iterative solution is carried out on design variables by adopting a moving asymptote optimization algorithm (MMA), and in the iterative process, if the current design does not meet the reliability constraint dj<0, or when the sum of the design variable change absolute values of the two iteration steps is greater than the tolerance epsilon, adding one to the iteration step number, and returning to the step (2), otherwise, executing the step (8);
(8) considering both reliability constraint and relative variance, if the current design satisfies the reliability optimization feature distance dj<And 0, when the sum of the design variable change absolute values of the front and back iteration steps is less than the tolerance epsilon, the iteration is finished, and the optimal configuration of the non-probability reliability topological optimization design of the laminated structure is obtained.
In summary, the invention provides a non-probabilistic reliability topology optimization method for a laminate board under a force-heat coupling environment. The method comprises the steps of firstly, calculating equivalent elastic parameters and thermal property parameters of the composite material laminated plate according to a first-order shear deformation theory of a laminated structure. And then considering the influence of uncertain factors such as laminated plate material parameters, loads and the like, constructing a laminated structure reliability topological optimization framework by using an SIMP material interpolation model and taking the minimum structure weight as an optimization target, taking the relative density of units as a design variable and taking the allowable displacement response reliability of the concerned position under the action of the force-heat coupling load as a constraint, and obtaining the optimal configuration of the laminated structure under the force-heat coupling environment by iteration by adopting a moving asymptote optimization algorithm. The invention provides a novel non-probability reliability topological optimization method under the influence of force-heat coupling load, which aims at a composite material laminated structure. On the other hand, the non-probabilistic reliability topology optimization framework established by the invention provides an effective solution for the topology optimization design of the composite structure under the condition of small sample test data.
The above are only the specific steps of the present invention, and the protection scope of the present invention is not limited in any way; the method can be expanded and applied to the field of structural optimization design under the environment of force-heat coupling, and all technical schemes formed by adopting equivalent transformation or equivalent replacement fall within the protection scope of the invention.
The invention has not been described in detail and is part of the common general knowledge of a person skilled in the art.

Claims (9)

1. A method for topological optimization of non-probability reliability of laminate plates under a force-heat coupling environment is characterized by comprising the following steps:
the first step is as follows: establishing a laminated plate force-heat coupling constitutive model according to given composite material unidirectional plate material parameters and a layering mode, and calculating an equivalent thermal conductivity, an equivalent heat conduction matrix and an equivalent stiffness matrix of the laminated plate by utilizing a first-order layering theory;
the second step is that: selecting an uncertainty parameter b for describing the structural characteristics of the laminated plate or the external force thermal load according to the service environment and the structural material properties of the laminated plate in the actual engineering, and according to a finite element balance equation:
KI(b)UI(b)＝FI(TI(b),b)
MI(b)TI(b)＝QI(b)
wherein the structure displacement field UIInvolving a temperature field TIThe resulting displacement; kIIs a matrix of overall stiffness intervals of the laminate, MIAs a matrix of integral heat-conducting zones, FIAs force load interval vector, QIIs a heat load interval vector; and (3) solving an upper bound and a lower bound of the structural force thermal coupling response under the influence of uncertainty parameters by using an interval parameter vertex combination method, namely:
Figure FDA0003524170650000011
Figure FDA0003524170650000012
wherein the content of the first and second substances, iu(b)in order to displace the lower bound of the response,
Figure FDA0003524170650000013
is the displacement response upper bound;
Figure FDA0003524170650000014
representing an interval formed by uncertainty parameters b;
the third step: based on a structure non-probability set reliability model, limiting the actual displacement interval (u) of the laminated plate displacementj)IAnd allowable displacement interval
Figure FDA0003524170650000015
Normalizing to range between [ -1,1 [)]；
The fourth step: the optimization characteristic distance d is used as a non-probability reliability index for measuring whether the structure is invalid or not, and is defined as follows: the moving distance from the original failure plane to the target failure plane, wherein the target failure plane is a plane parallel to the original failure plane, the reliability of the target failure plane is a given value, and the optimization characteristic distance d of the jth constraintjA non-probabilistic reliability for quantifying that the laminate satisfies the constraint;
the fifth step: adopting a variable density topological optimization model based on the SIMP method, and using the non-probability reliability index obtained in the fourth step, namely the characteristic distance djAs a constraint, establishing a non-probability reliability topological optimization column of the composite material laminated structure;
and a sixth step: solving the sensitivity of the upper and lower bounds of the response at the displacement constraint position of the laminated structure by adopting an adjoint vector method, and obtaining a non-probability reliability constraint d according to a chain type derivation rule of a composite functionjThe sensitivity of (c);
the seventh step: iterative solution is carried out on the design variables by adopting a moving asymptote optimization algorithm, and the design variables are iteratedIn the process, if the current design does not satisfy the reliability constraint dj<0, or when the sum of the design variable change absolute values of the two iteration steps is greater than the tolerance epsilon, adding one to the iteration step number, and returning to the second step, or else, carrying out the next step;
eighth step: if the current design satisfies the reliability optimization feature distance dj<And 0, when the sum of the design variable change absolute values of the front and back iteration steps is less than the tolerance epsilon, the iteration is finished, and the optimal configuration of the non-probability reliability topological optimization design of the laminated structure is obtained.
2. The method for topological optimization of non-probabilistic reliability of laminate plates under force-heat coupling environment according to claim 1, wherein the method comprises the following steps: in the first step, the thermal conductivity M is defined by the formula:
Figure FDA0003524170650000021
wherein Δ Q represents heat, Δ t represents time,
Figure FDA0003524170650000022
representing the heat flow rate, a representing the cross-sectional area of the material perpendicular to the heat flow, and Δ T representing the temperature difference across the material of length x; assuming that the number of layers in the laminated structure at an angle theta to the heat flow rate is nθThe total number of layers is NN, NN ═ Σ nθThe ply ratio is omegaθ，ωθ＝nθ/NN, the thermal conductivity of the laminate structure in the direction of heat flow is expressed as:
Mc,NN＝∑ωθ(Mc,pcos2θ+Mc,tsin2θ)
in the formula, Mc,pDenotes the thermal conductivity of the composite unidirectional sheet along the direction of the fibres, Mc,tRepresents the thermal conductivity of the composite unidirectional plate perpendicular to the fiber direction; according to the classical lamination theory of composite materials, the temperature varies linearly along the thickness direction of the laminate, namely:
T(x,y,z)＝T0(x,y)+Tz'(x,y)·z
wherein T is0(x, y) represents the area temperature in the laminate structure, Tz' (x, y) denotes a temperature gradient in the thickness z direction of the laminate.
3. The method for topological optimization of non-probabilistic reliability of laminate plates under force-heat coupling environment according to claim 2, wherein the method comprises the following steps: in the first step, for a symmetrically laid up laminate, the stress of the k-th layer in the global coordinate system is obtained by:
Figure FDA0003524170650000023
wherein sigmax,σyAnd τxyRepresenting a stress component, ε, in a global coordinate systemx，εyAnd gammaxyRepresenting the strain component, α, in a global coordinate systemx，αyAnd alphaxyRepresenting the thermal expansion coefficient under the global coordinate system;
Figure FDA0003524170650000024
wherein i, j is 1,2,6, and represents a stiffness matrix of a laminate single layer under a global coordinate system, which is calculated by the following formula:
Figure FDA0003524170650000025
in the formula E1，E2Young's modulus in the principal direction 1, Young's modulus in the principal direction 2, G, respectively, of the unidirectional sheet12Is the shear modulus, μ, of the principal directions 1 and 221And mu12Poisson's ratios of the main directions 1 and 2, respectively; t iskThe transformation matrix representing the k-th layer of the laminate, i.e.:
Figure FDA0003524170650000031
in the formula (I), the compound is shown in the specification,θkrepresents the fiber angle of the k-th layer;
unit elastic matrix of composite laminated plate
Figure FDA0003524170650000032
Comprises the following steps:
Figure FDA0003524170650000033
wherein, tkIs the thickness of the k-th layer of the laminate, and h is the total thickness of the laminate; in the structural finite element method, the element stiffness matrix KiObtained by the following formula:
Figure FDA0003524170650000034
wherein B is a unit strain-displacement matrix, and the total stiffness matrix of the composite material laminated plate is obtained by summing the unit stiffness matrices:
Figure FDA0003524170650000035
。
4. the method for topological optimization of non-probabilistic reliability of laminate plates under force-heat coupling environment according to claim 1, wherein the method comprises the following steps: in the second step, the unidirectional coupling of the temperature field and the displacement field of the laminated structure is considered, namely, the temperature field distribution of the structure is calculated through steady-state heat conduction, and then the external load F caused by temperature is calculated according to the temperature fieldthFinally F is addedthTemperature-independent mechanical load FthStacking, calculating the displacement field of the structure according to the thermoelastic theory of the laminated plate.
5. The method of claim 3, wherein the method comprises the following steps: in the third step, the normalization method for normalizing the actual displacement interval and the allowable displacement interval is as follows:
Figure FDA0003524170650000036
Figure FDA0003524170650000037
wherein δ ujAnd
Figure FDA0003524170650000038
actual and allowable displacements for the normalized jth displacement constraint, (δ uj)IAnd
Figure FDA0003524170650000039
for the normalized actual displacement interval and the allowed displacement interval of the jth displacement constraint,
Figure FDA00035241706500000310
and
Figure FDA00035241706500000311
respectively the central value and the radius of the actual displacement interval,
Figure FDA00035241706500000312
and
Figure FDA0003524170650000041
respectively, the central value and the radius of the allowable displacement interval according to the structural function:
Φ(ρ,b)＝U*(b)-U(ρ,b)
and judging whether the structure is safe, wherein the structure is invalid when phi (rho, b) is less than 0, and the structure is safe when phi (rho, b) is more than or equal to 0, and rho is pseudo density.
6. A force and heat coupling ring according to claim 3The topological optimization method for the non-probability reliability of the environmental laminate is characterized by comprising the following steps: in the fourth step, the optimized characteristic distance d is adopted as a non-probability reliability index for judging whether the structure is invalid or not, and the optimized characteristic distance d corresponding to the jth constraintjThe expression is as follows:
Figure FDA0003524170650000042
in the formula RtargRepresenting a given reliability indicator.
7. The method of claim 5, wherein the method comprises the following steps: in the fifth step, SIMP is used, a solid anisotropic material with compensation model is used to avoid the generation of intermediate density units, and for the SIMP model, the elastic modulus parameter of the ith unit is a function of the relative density of the material:
Figure FDA0003524170650000043
wherein p is a penalty factor used for realizing the penalty to the intermediate density unit, and p is taken as 3;
establishing a non-probability reliability topological optimization formula of the composite material laminated structure:
find a set of p ═ p (p)1,ρ2,…,ρN) The requirements are met,
Figure FDA0003524170650000051
s.t.K(ρ,b)U(ρ,b)＝F(T(ρ,b),ρ,b)
M(ρ,b)T(ρ,b)＝Q(ρ,b)
dj(ρ,b)≤0,j＝1,2,…,m
0≤ρ≤ρi≤1
where ρ is (ρ)1,ρ2,…,ρN) The design variables for the cell pseudo-density, i.e., the topology optimization problem, N represents the number of design variables, W represents the structure weight,ρthe lower bound of the design variable is a predetermined small value preset to prevent singularity of the stiffness matrix.
8. The method of claim 7, wherein the method comprises the following steps: and the sixth step of solving the sensitivity of the upper and lower response boundaries at the laminated structure displacement constraint position by adopting a adjoint vector method specifically comprises the following steps: constructing an augmented Lagrangian function:
Figure FDA0003524170650000052
wherein
Figure FDA0003524170650000053
For the introduced Lagrangian multiplier, (F)m)ΙAnd (F)th)ΙForce interzone vectors due to mechanical and thermal loads respectively,
Figure FDA0003524170650000054
βi(ρi)＝Eiαidenotes the thermal stress coefficient, Δ T, of the i-th celliRepresents the average amount of change in temperature in the ith cell,
Figure FDA0003524170650000055
a unit load vector representing an ith cell;
Figure FDA0003524170650000056
is a representation of an augmentation function; according to equilibrium equation KIUI＝FIIt is obvious that
Figure FDA0003524170650000057
So the actual displacement interval u at the jth constraintj IWith respect to the design variable ρiThe partial derivatives of (a) are:
Figure FDA0003524170650000058
the above formula is arbitrary
Figure FDA0003524170650000059
All hold, in order to avoid calculations
Figure FDA00035241706500000510
The trouble of (2), order:
Figure FDA00035241706500000511
namely:
Figure FDA0003524170650000061
wherein
Figure FDA0003524170650000062
The above equation formally and balance equation KIUI＝FIAgree, so to solve for the accompanying vector
Figure FDA0003524170650000063
Only applying unit load at the jth node degree of freedom through finite element calculation;
will be provided with
Figure FDA0003524170650000064
Substituting into the actual displacement interval u at the jth constraintj IWith respect to the design variable ρiThe partial derivative of (c) is formulated as:
Figure FDA0003524170650000065
Figure FDA0003524170650000066
in the formula (I), the compound is shown in the specification,
Figure FDA0003524170650000067
overall stiffness K between adjacent sections in any one topologyIWrite as:
Figure FDA0003524170650000068
therefore, there are:
Figure FDA0003524170650000069
9. the method of claim 7, wherein the method comprises the following steps: in the sixth step, a non-probability reliability constraint d is obtained by using a chain type derivative rule of a composite functionjThe sensitivity of (2) is specifically processed as follows: reliability index djWith respect to the design variable ρiThe sensitivity of (d) is expressed as:
Figure FDA00035241706500000610
bonding reliability index djThe expression of (1) is:
Figure FDA0003524170650000071
and
Figure FDA0003524170650000072
wherein the content of the first and second substances,
Figure FDA0003524170650000073
(uj)M,
Figure FDA0003524170650000074
and(uj)Dcan be obtained in an iterative process and combined with the obtained
Figure FDA0003524170650000075
And
Figure FDA0003524170650000076
is solved to obtain the reliability index djWith respect to the design variable ρiThe sensitivity of (2).
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