CN111310377A - Non-probability reliability topological optimization design method for continuum structure under mixed constraint of fundamental frequency and frequency interval - Google Patents

Non-probability reliability topological optimization design method for continuum structure under mixed constraint of fundamental frequency and frequency interval Download PDF

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CN111310377A
CN111310377A CN202010110973.6A CN202010110973A CN111310377A CN 111310377 A CN111310377 A CN 111310377A CN 202010110973 A CN202010110973 A CN 202010110973A CN 111310377 A CN111310377 A CN 111310377A
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CN111310377B (en
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邱志平
夏海军
王磊
张泽晟
马铭
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Beihang University
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Abstract

The invention discloses a non-probability reliability topological optimization design method of a continuum structure under mixed constraint of fundamental frequency and frequency interval. Firstly, eliminating the problem of local mode by adopting a modified solid isotropic microstructure/material interpolation model with penalty factors; obtaining upper and lower limits of the natural frequency of the structure by adopting a vertex combination method, and overcoming the problem of modal exchange in the optimization process by adopting a boundary formula, thereby constructing a continuum structure non-probability reliability topological optimization model under mixed constraint of fundamental frequency and frequency interval; the original reliability index is converted by adopting a performance extreme value method so as to overcome the problem of convergence in optimization, and the sensitivity of a target performance extreme value to a design variable is solved by adopting a complex function derivation method; and adjusting parameters in a Mobile Marching Algorithm (MMA), and performing iterative optimization calculation until corresponding convergence conditions are met to obtain an optimization design scheme meeting fundamental frequency and frequency interval reliability constraints.

Description

Non-probability reliability topological optimization design method for continuum structure under mixed constraint of fundamental frequency and frequency interval
Technical Field
The invention relates to the field of topological optimization design of a continuum structure under frequency constraint, in particular to a topological optimization design method for non-probability reliability of the continuum structure under mixed constraint of fundamental frequency and frequency interval.
Background
In engineering practice, the structure is subject to not only static loads such as gravity, but also vibration loads. Such as aircraft, are subject to engine vibration and aerodynamic loads; when the machine tool works, the machine tool can be subjected to vibration caused by high-speed rotation of the motor and other factors; the automobile is subjected to vibration loads generated by engine operation and road bumps while running, and the like. The resonance phenomenon easily causes structural damage. Therefore, when the structure is designed, not only the structure needs to be subjected to static analysis, but also the structure needs to be subjected to dynamic characteristic analysis, and optimized design is carried out, so that the aims of avoiding resonance and ensuring the structure safety are fulfilled.
The structure optimization design can be mainly divided into three levels according to different design variables, namely size optimization, shape optimization and topology optimization. For the topology optimization of the continuum structure, the size optimization refers to the numerical optimization design of certain sizes when the structure topology configuration and the shapes of all parts are determined; shape optimization refers to the improved optimization of certain geometries due to manufacturing or other requirements, when the topology of the structure has been determined; the topological optimization is to find the optimal topological configuration for the structure and lay a foundation for subsequent design, and moreover, the topological optimization has higher design freedom, so that higher economic benefit can be obtained. Therefore, the research on the structural topology optimization under the frequency constraint is of great significance.
However, structural analysis and design is generally based on deterministic assumptions, i.e., various parameters are considered to be artificially defined as definite quantities, regardless of errors and uncertainties in the parameters. However, in engineering practice, uncertainties in structural systems are prevalent due to the effects of various factors. Results obtained by neglecting uncertainty analysis and design cannot be convinced, and the method is not consistent with the concept of scientific and technological refinement development. Therefore, while minimizing uncertainty, the uncertainty present in the engineered structure must be studied in order to obtain a highly reliable structure. Due to various uncertainties and insufficient information that can be obtained in engineering practice, probability density functions of uncertain parameters are difficult to obtain, and the variation ranges of the uncertain parameters are easy to obtain. In this case, it is very convenient to describe the uncertain parameters and the reliability of the system by using a non-probability set theory, and the dependency on the initial sample is weak. Therefore, it is necessary to research a non-probabilistic reliability topological optimization design method of a continuum structure under the frequency constraint.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method overcomes the defects of the prior art and provides a continuum structure non-probability reliability topological optimization design method under the mixed constraint of fundamental frequency and frequency interval. The invention fully considers the universal uncertain factors in the practical engineering problem, the obtained design result is more in line with the real situation, and the engineering applicability is stronger.
The technical scheme adopted by the invention is as follows: a non-probability reliability topological optimization design method for a continuum structure under mixed constraint of fundamental frequency and frequency interval comprises the following implementation steps:
the method comprises the following steps: aiming at the characteristics of a design structure, a finite element is used for discretizing a design domain, and a Modified Solid isotropic microstructure/Material interpolation model (Modified Solid isotropic microstructure/Material with Penalty, MSIMP) with a Penalty factor is used for describing the elastic modulus and the density of the Material, namely:
Figure BDA0002389372110000021
M(ρi)＝Mmin+ρi(M0-Mmin)
wherein E (ρ)i) And M (ρ)i) Respectively representing the material elastic modulus and the material density of the ith unit, p > 1 is a penalty factor and is set to be 3, E0And M0Respectively the modulus of elasticity and the density of the solid material, EminAnd MminRespectively representing the lower bounds of the modulus of elasticity and the density of the solid material and being set to Emin＝10-3E0,Mmin＝10-3M0To avoid local modal problems.
Step two: the uncertainty range of the structural material parameters (including the elastic modulus and the density of the material) is described by using an interval model, the result of the structural natural frequency under the influence of uncertainty is calculated by using a vertex combination method, the maximum value of the result is taken as the upper bound of the natural frequency, and the minimum value of the result is taken as the lower bound of the natural frequency, so that the limit range of the structural natural frequency is obtained as follows:
Figure BDA0002389372110000022
Figure BDA0002389372110000023
wherein the content of the first and second substances,
Figure BDA0002389372110000024
andω rrespectively an upper bound and a lower bound of the r-th order frequency, a is an uncertain parameter vector, m is the number of uncertain parameters,
Figure BDA0002389372110000025
the calculated order r frequency for the ith vertex combination.
Step three: a boundary formula is adopted to overcome the modal exchange problem in the optimization process, and a non-probability reliability index is used to measure the influence of uncertainty on the structure safety and the front sjAdopting reliability constraint for 1-order frequency, and constructing a continuum structure non-probability reliability topological optimization model under mixed constraint of fundamental frequency and frequency interval:
Figure BDA0002389372110000031
Figure BDA0002389372110000032
Figure BDA0002389372110000033
Figure BDA0002389372110000034
Figure BDA0002389372110000035
0≤ρi≤1,i＝1,2,…,N
where V denotes the total volume of the structure, and ρ ═ is (ρ1,ρ2,…,ρN)TIs a density design variable, ViIs the solid volume of the ith element, N is the number of elements of the divided finite element, R (-) is the non-probability reliability function,
Figure BDA0002389372110000036
in the interval of the natural frequency of the kth order structure,ω 1for fundamental frequency constraint value, η is target non-probability reliability, for preceding sjThe reliability constraint adopted for the-1 st order frequency is to overcome the mode in the optimization processA problem of state switching, and
Figure BDA0002389372110000037
denotes the s thj+1 st order and sjThe order frequency spacing constraint, J denotes the order of the mode under consideration.
Figure BDA0002389372110000038
And ω'rIs the natural frequency of the structure under a certain vertex combination, and the vertex combination is taken as the midpoint of the uncertain parameters. Also, constrain
Figure BDA0002389372110000039
And
Figure BDA00023893721100000310
is also added to overcome the problem of modality exchange during the optimization process;
step four: the performance extremum method is adopted to process the original non-probability reliability index to solve the convergence problem, the original target reliability constraint can be converted into the constraint of the target performance extremum point, and the converted optimization model can be expressed as follows:
Figure BDA00023893721100000311
Figure BDA00023893721100000312
Figure BDA00023893721100000313
Figure BDA00023893721100000314
Figure BDA00023893721100000315
0≤ρi≤1,i＝1,2,…,N
wherein g (-) is a target performance extremum function;
step five: solving the sensitivity of the upper and lower bounds of the natural frequency of the structure to the design variable, carrying out sensitivity calculation and judging the repetition frequency by adopting the peak combination of the elastic modulus and the density of the material corresponding to the upper and lower bounds of the frequency, considering the two-order frequency as the repetition frequency if the difference of the upper bound (or the lower bound) of the adjacent-order frequency is less than 0.005Hz, and adopting a repetition frequency sensitivity analysis method; otherwise, adopting a single frequency sensitivity analysis method. And further solving the sensitivity of the target performance extreme value to the upper and lower bounds of the natural frequency of the structure, and then solving the sensitivity of the target performance extreme value to the design variable according to a complex function derivation rule.
Step six: adjusting algorithm parameters in a Mobile Marching Algorithm (MMA) to enable the algorithm parameters to become a linear approximation optimization algorithm, further modifying a corresponding program to enable the movement of each step in the linear approximation optimization algorithm to be consistent with the original MMA algorithm, and solving an optimization problem by taking the obtained target performance value and the sensitivity of the target performance value to a design variable as input conditions of the algorithm so as to update the design variable;
and seventhly, repeating the second step to the sixth step, and updating the design variables for multiple times until the current design meets the reliability constraint and the relative change percentage of the objective function is less than a preset value ξ, and stopping the optimization process.
Compared with the prior art, the invention has the advantages that:
(1) the non-probability reliability index adopted by the invention can reasonably consider the influence of uncertain factors on the structural performance, can furthest improve the economic benefit of the structure, gives consideration to the safety, and is very suitable for engineering application;
(2) the MSIMP model provided by the invention can effectively overcome the problem of local modal in the frequency-based topology optimization problem, thereby well realizing the topology optimization design of the continuum structure under the frequency constraint;
(3) the improved mobile progressive optimization algorithm provided by the invention can effectively solve the topological optimization problem of the continuum non-probability reliability structure under the constraints of fundamental frequency and frequency interval, and provides a new algorithm for solving the topological optimization problem of the continuum structure under the constraints of fundamental frequency and frequency interval.
Drawings
FIG. 1 is a flow chart of a non-probabilistic reliability topology optimization design method of a continuum structure under fundamental frequency and frequency interval hybrid constraints according to the present invention;
FIG. 2 is a schematic diagram of the process of deriving the target performance extremum in the present invention, wherein FIG. 2(a) is a schematic diagram of four key slopes of the extreme state plane when deriving the target performance extremum, and FIGS. 2(b) - (e) are schematic diagrams of four cases when deriving the target performance extremum;
FIG. 3 is a schematic diagram of a topology optimization design area and boundary and load conditions in an embodiment of the invention;
fig. 4 is a schematic diagram of an optimization result of topology optimization for a continuum structure according to the present invention, where fig. 4(a) is deterministic optimization, fig. 4(b) is non-probabilistic reliability optimization (R ═ 0.90), fig. 4(c) is non-probabilistic reliability optimization (R ═ 0.95), and fig. 4(d) is non-probabilistic reliability optimization (R ═ 0.999).
Detailed Description
The invention is further described with reference to the following figures and detailed description.
As shown in FIG. 1, the invention provides a non-probabilistic reliability topological optimization design method for a continuum structure under mixed constraints of fundamental frequency and frequency interval, which comprises the following steps:
the method comprises the following steps: aiming at the characteristics of a design structure, a finite element is used for discretizing a design domain, and a Modified Solid isotropic microstructure/Material interpolation model (Modified Solid isotropic microstructure/Material with Penalty, MSIMP) with a Penalty factor is used for describing the elastic modulus and the density of the Material, namely:
Figure BDA0002389372110000051
M(ρi)＝Mmin+ρi(M0-Mmin)
wherein E: (A)ρi) And M (ρ)i) Respectively representing the material elastic modulus and the material density of the ith unit, p > 1 is a penalty factor and is set to be 3, E0And M0Respectively the modulus of elasticity and the density of the solid material, EminAnd MminRespectively representing the lower bounds of the modulus of elasticity and the density of the solid material and being set to Emin＝10-3E0,Mmin＝10-3M0To avoid local modal problems.
Step two: an interval model is used to describe the uncertainty range of the structural material parameters, including the elastic modulus and density of the material. The mathematical definition of the intervals is described below.
In general, a number of intervals can be defined as:
Figure BDA0002389372110000052
whereinaAnd
Figure BDA0002389372110000053
are respectively referred to as interval number aILower and upper bounds. When in use
Figure BDA0002389372110000054
The number of intervals aIThe degradation is a real number.
Based on the definition of the number of intervals, the interval matrix can be defined as follows:
Figure BDA0002389372110000055
interval matrix AICan also be expressed as:
Figure BDA0002389372110000056
wherein the content of the first and second substances,A＝(a ij)m×nand
Figure BDA0002389372110000057
are respectively provided withReferred to as interval matrix AILower and upper bounds of interval matrix AICan be expressed as:
Figure BDA0002389372110000058
calculating the result of the structure natural frequency under the influence of uncertainty by adopting a vertex combination method for the uncertain parameters, taking the maximum value as the upper bound of the structure natural frequency, and taking the minimum value as the lower bound of the structure natural frequency, thereby obtaining the limit range of the structure natural frequency as follows:
Figure BDA0002389372110000061
Figure BDA0002389372110000062
wherein the content of the first and second substances,
Figure BDA0002389372110000063
andω rrespectively an upper bound and a lower bound of the r-th order frequency, a is an uncertain parameter vector, m is the number of uncertain parameters,
Figure BDA0002389372110000064
the calculated order r frequency for the ith vertex combination.
Step three: and the influence of uncertainty on the structure safety is measured by using a non-probability reliability index, and the definition of the non-probability reliability is introduced below.
For the number of intervals xIAnd yIThe non-probabilistic reliability is defined as:
Figure BDA0002389372110000065
in particular, when only xIThe degradation is a real number x', and the corresponding reliability is calculated as:
Figure BDA0002389372110000066
when only yIThe degradation is a real number y', and the corresponding reliability is calculated as:
Figure BDA0002389372110000067
using boundary formula to overcome the problem of modal exchange in optimization process, for previous sjAdopting reliability constraint for 1-order frequency, and constructing a continuum structure non-probability reliability topological optimization model under mixed constraint of fundamental frequency and frequency interval:
Figure BDA0002389372110000071
Figure BDA0002389372110000072
Figure BDA0002389372110000073
Figure BDA0002389372110000074
Figure BDA0002389372110000075
0≤ρi≤1,i＝1,2,…,N
where V denotes the total volume of the structure, and ρ ═ is (ρ1,ρ2,…,ρN)TIs a density design variable, ViIs the solid volume of the ith element, N is the number of elements of the divided finite element, R (-) is the non-probability reliability function,
Figure BDA0002389372110000076
is an interval of the k-th order natural frequency,ω 1for fundamental frequency constraint value, η is target non-probabilityDegree of leaning, preceding sjThe reliability constraint adopted for the-1 st order frequency is to overcome the modal exchange problem during the optimization process, and
Figure BDA0002389372110000077
denotes the s thj+1 st order and sjThe order frequency spacing constraint, J denotes the order of the mode under consideration.
Figure BDA0002389372110000078
And ω'rIs the natural frequency of the structure under a certain vertex combination, and the vertex combination is taken as the midpoint of the uncertain parameters. Also, constrain
Figure BDA0002389372110000079
And
Figure BDA00023893721100000710
is also added to overcome the problem of modality exchange during the optimization process;
step four: the performance extremum method is adopted to process the original non-probability reliability index to solve the convergence problem. Firstly, constructing a reliability-performance extremum function as follows:
F(g)＝R(AI+g≥BI)
wherein A isIAnd BIIs the number of intervals, AICan be that
Figure BDA00023893721100000711
Or
Figure BDA00023893721100000712
Correspondingly, BICan be thatω 1Or
Figure BDA00023893721100000713
The original reliability constraint can be written as:
F(0)＝R(AI≥BI)≥η
taking inverse transform F for both sides of the above equation-1It is possible to obtain:
F-1(η)≤0
let g (A)I≥BI,η)＝F-1(η), the original target reliability constraint can be converted into a constraint of the target performance extreme point, and the converted optimization model can be expressed as:
Figure BDA0002389372110000081
Figure BDA0002389372110000082
Figure BDA0002389372110000083
Figure BDA0002389372110000084
Figure BDA0002389372110000085
0≤ρi≤1,i＝1,2,…,N
wherein g (-) is a target performance extremum function. How to solve the objective performance function is described below.
The extreme state plane is first constructed as:
M(A,B)＝A+g-B＝0
wherein A is ∈ AI,B∈BI. The above equation can also be written as:
Figure BDA00023893721100000810
and
Figure BDA0002389372110000086
wherein δ a ═ a-ac)/Ar(Ar≠0),δB＝(B-Bc)/Br(Br≠0)。
As shown in fig. 2(a), four special cases of the extreme state plane can be derived:
Figure BDA0002389372110000087
wherein k isiThe slope of the extreme state plane. Thus, g can be found in four cases:
case 1: if k is3＜Ar/Br＜k1η ≧ 0.5, the extreme state plane intersects the top and left sides of the feasible region, as shown in fig. 2 (B.) to obtain intersection points M1 and N1, substituting δ B-1 and δ a-1 into the extreme state plane yields:
Figure BDA0002389372110000088
and
Figure BDA0002389372110000089
based on the definition of the non-probabilistic reliability, one can obtain:
Figure BDA0002389372110000091
g can be solved from the above formula to give:
Figure BDA0002389372110000092
case 2: if A isr/Br≥k1η ≧ 0.5 or Ar/Br≥k2η is less than or equal to 0.5, the two formulas are integrated into Ar/Br≥max{k1,k2And then two intersection points are located at the upper side and the lower side, as shown in fig. 2 (c). Similarly, let δ B ± 1 obtain:
Figure BDA0002389372110000093
based on the definition of the non-probability reliability, there are:
Figure BDA0002389372110000094
solving for g from the above formula yields:
g＝-Ac+Bc+(2η-1)Ar
case 3: if k is4＜Ar/Br＜k2η < 0.5, the positional relationship between the extreme state plane and the feasible region is shown in FIG. 2(d), where g can be calculated as:
Figure BDA0002389372110000095
case 4: if A isr/Br≤k3η ≧ 0.5 or Ar/Br≤k3η is not more than 0.5, both formulae are equivalent to Ar/Br≤min{k3,k4Then g can be solved as:
g＝-Ac+Bc+(2η-1)Br
in summary, the target performance extremum function is a piecewise function, as follows:
Figure BDA0002389372110000096
particularly, when Ar＝0,BrNot equal to 0, g can be solved as:
Figure BDA0002389372110000097
similarly, when Ar≠0,BrWhen 0, we can get:
Figure BDA0002389372110000101
combining the above several cases, we can get:
Figure BDA0002389372110000102
where epsilon is a small positive number.
Step five: solving the sensitivity of the upper and lower bounds of the natural frequency of the structure to the design variable, carrying out sensitivity calculation and judging the repetition frequency by adopting the peak combination of the elastic modulus and the density of the material corresponding to the upper and lower bounds of the frequency, considering the two-order frequency as the repetition frequency if the difference of the upper bound (or the lower bound) of the adjacent-order frequency is less than 0.005Hz, and adopting a repetition frequency sensitivity analysis method; otherwise, adopting a single frequency sensitivity analysis method. And further solving the sensitivity of the target performance extreme value to the upper and lower bounds of the natural frequency of the structure, and then solving the sensitivity of the target performance extreme value to the design variable according to a complex function derivation rule. The method for solving the sensitivity of the upper bound of the natural frequency of the structure to the design variable under the condition of single frequency is described firstly. The structural natural frequency characteristic equation under the condition of no damping is considered as follows:
Figure BDA0002389372110000103
wherein the content of the first and second substances,
Figure BDA0002389372110000104
and
Figure BDA0002389372110000105
the total rigidity matrix and the mass matrix of the structure, omega, corresponding to the upper bound of the r-th order natural frequencyrAnd
Figure BDA0002389372110000106
respectively, an upper bound of the natural frequency of the r-th order and a corresponding modal vector. Design variable rho is simultaneously paired at two ends of the upper formulajTaking the derivative, one can get:
Figure BDA0002389372110000107
the formula is simplified and arranged to obtain:
Figure BDA0002389372110000108
both ends of the above formula are simultaneously multiplied by
Figure BDA0002389372110000109
It is possible to obtain:
Figure BDA00023893721100001010
since K and M are symmetric matrices, then:
Figure BDA0002389372110000111
by normalization conditions of the modal vector, i.e.
Figure BDA0002389372110000112
It is possible to obtain:
Figure BDA0002389372110000113
rigidity matrix of unit j corresponding to upper bound of structure natural frequency
Figure BDA0002389372110000114
And quality matrix
Figure BDA0002389372110000115
Only the density design variables of the present cell. Then substituting the MSIMP interpolation model into the above equation, there are:
Figure BDA0002389372110000116
wherein the content of the first and second substances,
Figure BDA0002389372110000117
features corresponding to element j for the upper bound of the r-th order natural frequencyAnd (5) vector quantity.
Similarly, the design variable rho is the lower bound of the natural frequency of the structurejThe sensitivity of (d) is solved as:
Figure BDA0002389372110000118
wherein the content of the first and second substances,ω rto the lower bound of the structure's order-r natural frequency,
Figure BDA0002389372110000119
the lower bound for the r-th order natural frequency corresponds to the eigenvector of cell j.
Figure BDA00023893721100001110
And
Figure BDA00023893721100001111
respectively a rigidity matrix and a mass matrix of the unit j corresponding to the upper bound of the natural frequency of the structure.
For the case of the repetition frequency, since the multiple characteristic frequencies do not have differentiability in the general sense, the corresponding sensitivity information cannot be directly given. For sensitivity analysis of multiple characteristic frequencies, at a repetition frequency
Figure BDA00023893721100001112
Selecting a group of characteristic modal vectors which are continuously changed along with design variables and are independent from each other from a modal space (upper bound of natural frequency)
Figure BDA00023893721100001113
The general form of the set of vectors can be expressed as a linear combination of basis vectors:
Figure BDA00023893721100001114
wherein, βrkTo determine the coefficients, [ β ] by appropriate processingrk]And r, k is n, …, and n + m-1 forms an m × m orthogonal matrix. To be provided with
Figure BDA00023893721100001115
And
Figure BDA00023893721100001116
substituting the characteristic value equation to obtain:
Figure BDA00023893721100001117
wherein the content of the first and second substances,
Figure BDA00023893721100001118
and
Figure BDA00023893721100001119
the derivation of the density design variables from the above equation yields a set of coefficients β for multiple frequency sensitivities and undetermined coefficientsrkThe equation of the sub-eigenvalues of (a) is specifically derived by using a perturbation method.
First, the case of a variation of a single design variable is considered, for which the design variable ρ is givenjA small perturbation Δ ρj. The changes in the frequency, eigenvectors, and stiffness and mass matrices in the eigenvalue equation due to perturbation of a single design variable are as follows (only first order approximations after perturbation are considered here, small terms of second order and above are ignored):
Figure BDA0002389372110000121
Figure BDA0002389372110000122
Figure BDA0002389372110000123
Figure BDA0002389372110000124
r＝n,…,n+m-1
wherein the content of the first and second substances,
Figure BDA0002389372110000125
and
Figure BDA0002389372110000126
respectively corresponding to multiple characteristic frequencies
Figure BDA0002389372110000127
And feature vectors
Figure BDA0002389372110000128
The sensitivity of (2). Substituting the above formula into generalized eigenvalue equation, and respectively multiplying the two ends of the equation by left
Figure BDA0002389372110000129
Merging of Δ ρjThe same terms are simplified to obtain a first-order approximation result (i.e. let Δ ρjCoefficients of terms are zero):
Figure BDA00023893721100001210
substituting the linear expression of the basis vector into the formula, and utilizing the orthonormality of the characteristic mode on the quality matrix to obtain:
Figure BDA00023893721100001211
wherein, deltaskIs a function of Kronecker the above equation gives a set of information about the coefficients βrkN, …, n + m-1, with the condition that the system determinant is zero, i.e.:
Figure BDA00023893721100001212
the above formula relates to multiple characteristic frequencies
Figure BDA00023893721100001213
Is sensitive toAlgebraic equation of degree, solution to obtain multiple eigenfrequencies
Figure BDA00023893721100001214
M sensitivity of
Figure BDA00023893721100001215
Likewise, the sensitivity of the lower bound of the characteristic frequency to design variables at a repetition frequency can be solved by the following algebraic equation:
Figure BDA00023893721100001216
wherein the content of the first and second substances,
Figure BDA00023893721100001217
is the lower bound of the characteristic frequency
Figure BDA00023893721100001218
The corresponding characteristics of the light-emitting diode are as follows,KandMrespectively a structural overall rigidity matrix and a quality matrix corresponding to the lower boundary of the characteristic frequency,
Figure BDA00023893721100001219
for multiple characteristic frequenciesω rThe sensitivity of (2).
The sensitivity of the target performance extrema function to the upper and lower bounds of the natural frequency of the structure is then introduced. The expression for the target performance limit point g is listed first, as follows:
Figure BDA0002389372110000131
according to
Figure BDA0002389372110000132
Comprises the following steps:
Figure BDA0002389372110000133
from the expression of g, one can obtain:
Figure BDA0002389372110000134
and
Figure BDA0002389372110000135
and
Figure BDA0002389372110000136
and
Figure BDA0002389372110000141
in summary, there are:
Figure BDA0002389372110000142
wherein the content of the first and second substances,
Figure BDA0002389372110000143
and substituting the sensitivity of the upper and lower boundaries of the prior structure natural frequency to the design variable into the formula to obtain the sensitivity of the target performance extreme value to the design variable. In the above equation, note the reliability constraint for the fundamental frequency, there is
Figure BDA0002389372110000144
This is because the right-hand term of the fundamental frequency constraint is a constant term. And for the reliability constraint of the frequency interval, then
Figure BDA0002389372110000145
This is because the right term of the frequency interval constraint is also the natural frequency of the structure, and therefore the right term of the frequency interval reliability constraint also needs to be derived.
Step six: adjusting algorithm parameters in a Mobile Advance Algorithm (MMA) with U set to 106Set L to-106The linear approximation optimization algorithm is formed, corresponding programs are further modified, the moving limit of each step in the linear approximation optimization algorithm is consistent with that of the original MMA algorithm, the obtained target performance value and the sensitivity of the target performance value to the design variable are used as input conditions of the algorithm, the optimization problem is solved, and therefore the design variable is updated;
and seventhly, repeating the second step to the sixth step, and updating the design variables for multiple times until the current design meets the reliability constraint and the relative change percentage of the objective function is less than a preset value ξ, and stopping the optimization process.
Examples
As shown in FIG. 3, the two-dimensional rectangular flat plate has a fixed support at both left and right ends, a sheet size of 50m × 40m, a plate thickness of 1m, a median value of elastic modulus E of 210GPa, and a median value of density ρ of 7800kg/m3The poisson ratio μ is 0.3. The structure is divided into 50 x 40 cells. Constraining the fundamental frequency of the structure to be greater than ω02Hz, i.e. first order eigenvalues λ0Greater than 4. Let the material parameters and λ0All have a fluctuation of 5%, i.e. EI＝[200,220]GPa，ρI＝[7410,8190]kg/m3，
Figure BDA0002389372110000151
In the calculation of the present example, the relative mass is taken as an objective function, and the constraint is that the fundamental frequency of the structure is larger than lambda0For reliability topology optimization, the constraint is that g is less than or equal to 0. And taking the initial value of the relative volume density of each unit as 0.7, and carrying out topological optimization on the structure. Fig. 4 and table 1 are the results of deterministic and reliable topology optimization. It can be seen that the configuration of the structure obtained by deterministic topological optimization and different non-probabilistic reliable topological optimizations has a larger difference, and compared with the deterministic topological optimization result, the reliable topological optimization result is more reasonable. Along with the gradual improvement of the reliability, the connecting parts of the middle area and the left side and the right side become thicker gradually, so that the structural rigidity is higher. Again, as the calculation process tends to reduce the mass, the connection starts to become porous again. As can be seen from table 1, with increasing non-probabilistic reliability,the relative volume fraction of the topology optimization results is also gradually increasing, i.e. the more reliable the structure occupies more volume, i.e. more material is needed to compose the structure.
TABLE 1 volume fraction table of the optimization results of the present invention for continuum structure topology optimization
Figure BDA0002389372110000152
The invention provides a non-probability reliability topological optimization design method of a continuum structure under mixed constraint of fundamental frequency and frequency interval. The method comprises the steps of firstly, describing the elastic modulus and the density of a Material by adopting a Modified Solid Isotropic Microstructure/Material interpolation model (Modified Solid Isotropic Microstructure/Material with Penalty, MSIMP) with Penalty factors, and selecting a proper elastic modulus and a proper lower density limit to avoid the problem of local mode; describing uncertainty of the elastic modulus and the density of the material by using an interval model, and obtaining upper and lower limits of the natural frequency of the structure by using a vertex combination method; the method comprises the steps of measuring the influence of uncertainty on structural safety by adopting a non-probability reliability index, and overcoming the problem of modal exchange in the optimization process by adopting a boundary formula, so as to construct a non-probability reliability topological optimization model of a continuum structure under mixed constraint of fundamental frequency and frequency interval; the original reliability index is converted by adopting a performance extreme value method so as to overcome the problem of convergence in optimization; solving the sensitivity of the upper and lower boundaries of the natural frequency of the structure to the design variable, and then solving the sensitivity of the target performance extreme value to the design variable according to a complex function derivation rule; and adjusting parameters in a Mobile Marching Algorithm (MMA), and performing iterative optimization calculation until corresponding convergence conditions are met to obtain an optimization design scheme meeting fundamental frequency and frequency interval reliability constraints.
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 non-probability reliability topological optimization design method of the continuum structure under the mixed constraint of fundamental frequency and frequency interval can be expanded and applied, 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 (4)

1. A non-probability reliability topological optimization design method for a continuum structure under mixed constraint of fundamental frequency and frequency interval is characterized by comprising the following implementation steps:
the method comprises the following steps: aiming at the characteristics of a design structure, a finite element is used for dispersing a design domain, a Modified Solid Isotropic Microstructure/Material interpolation model (Modified Solid Isotropic Microstructure/Material with Penalty factor, MSIMP) model is used for describing the elastic modulus and the density of the Material, and proper lower limits of the rigidity and the density of the Material are selected to avoid the problem of local modal;
step two: describing an uncertainty range of structural material parameters including the elastic modulus and the density of the material by using an interval model, calculating a result of structural natural frequency under the influence of uncertainty by using a vertex combination method, taking the maximum value as the upper bound of the natural frequency and taking the minimum value as the lower bound of the natural frequency, thereby obtaining a limit range of the structural natural frequency;
step three: the influence of uncertainty on the structure safety is measured by using a non-probability reliability index, and a boundary element formula is adopted to overcome the modal exchange problem in the optimization process, so that a continuum structure non-probability reliability topological optimization model under the mixed constraint of fundamental frequency and frequency interval is constructed:
Figure FDA0002389372100000011
Figure FDA0002389372100000012
Figure FDA0002389372100000013
Figure FDA0002389372100000014
Figure FDA0002389372100000015
0≤ρi≤1,i＝1,2,…,N
where V denotes the total volume of the structure, and ρ ═ is (ρ1,ρ2,…,ρN)TIs a density design variable, ViIs the solid volume of the ith element, N is the number of elements of the divided finite element, R (-) is the non-probability reliability function,
Figure FDA0002389372100000016
in the interval of the natural frequency of the kth order structure,ω 1for fundamental frequency constraint value, η target non-probability reliability, and for overcoming the modal exchange problem in the optimization process, front s is selectedjThe-1 order frequencies all adopt the reliability constraint, and
Figure FDA0002389372100000017
denotes the s thj+1 st order and sjThe order frequency spacing constraint, J denotes the order of the mode to be considered,
Figure FDA0002389372100000018
and ω'rIs the natural frequency of the structure under a certain combination of vertices, where the combination of vertices is taken as the midpoint of the uncertain parameter, and similarly, the constraint
Figure FDA0002389372100000019
And
Figure FDA00023893721000000110
is also added to overcome the problem of modality exchange during the optimization process;
step four: the performance extremum method is adopted to process the original non-probability reliability index to solve the convergence problem, and the original optimization model can be converted into the following model by utilizing the target performance extremum point:
Figure FDA0002389372100000021
Figure FDA0002389372100000022
Figure FDA0002389372100000023
Figure FDA0002389372100000024
Figure FDA0002389372100000025
0≤ρi≤1,i＝1,2,…,N
wherein g (-) is a target performance extremum function;
step five: solving the sensitivity of the target performance extreme value to the upper and lower bounds of the structure natural frequency and the sensitivity of the upper and lower bounds of the frequency to the design variable, then solving the sensitivity of the target performance extreme value to the design variable by using a complex function derivation method, judging the repetition frequency when solving the sensitivity of the frequency to the design variable, considering the two orders of frequency as the repetition frequency if the difference of the upper bound or the lower bound of the adjacent orders of frequency is less than 0.005Hz, and adopting a repetition frequency sensitivity analysis method; otherwise, adopting a single frequency sensitivity analysis method;
step six: adjusting algorithm parameters in a mobile incremental algorithm (MMA), modifying a corresponding program, taking the obtained target performance value and the sensitivity of the target performance value to a design variable as input conditions of the algorithm, and solving an optimization problem so as to update the design variable;
and seventhly, repeating the second step to the sixth step, and updating the design variables for multiple times until the current design meets the reliability constraint and the relative change percentage of the objective function is less than a preset value ξ, and stopping the optimization process.
2. The non-probabilistic reliability topological optimization design method of continuum structure under fundamental frequency and frequency interval hybrid constraint of claim 1, characterized in that: in the first step, an MSIMP model is adopted to describe the elastic modulus and the density of the material, and proper lower limits of the rigidity and the density of the material are selected to avoid the problem of local mode.
3. The non-probabilistic reliability topological optimization design method of continuum structure under fundamental frequency and frequency interval hybrid constraint of claim 1, characterized in that: in the fourth step, the original non-probability reliability index is processed by adopting a performance extremum method, so that the problem of convergence is well solved.
4. The non-probabilistic reliability topological optimization design method of continuum structure under fundamental frequency and frequency interval hybrid constraint of claim 1, characterized in that: and in the sixth step, the algorithm parameters in the MMA are adjusted, and the corresponding program is modified, so that the algorithm program suitable for solving the frequency constraint topology optimization problem is obtained.
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