CN111310377B - Non-probability reliability topological optimization design method for continuum structure - Google Patents

Non-probability reliability topological optimization design method for continuum structure Download PDF

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CN111310377B
CN111310377B CN202010110973.6A CN202010110973A CN111310377B CN 111310377 B CN111310377 B CN 111310377B CN 202010110973 A CN202010110973 A CN 202010110973A CN 111310377 B CN111310377 B CN 111310377B
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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
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: the design domain is discretized by using finite elements according to the characteristics of the design structure, and a Modified Solid Isotropic Microstructure/Material interpolation model (Modified Solid Isotropic Microstructure/Material with Penalty, MSIMP) model with Penalty factors is used to describe the elastic modulus and density of the Material, namely:
Figure GDA0003399511280000021
M(ρi)＝Mmin+ρi(M0-Mmin)
wherein E (ρ)i) And M (ρ)i) Respectively representing the modulus of elasticity and the density of the material, p, of the ith cell>1 is a penalty factor and is set to 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 GDA0003399511280000022
Figure GDA0003399511280000023
wherein the content of the first and second substances,
Figure GDA0003399511280000024
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 GDA0003399511280000025
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 GDA0003399511280000031
Figure GDA0003399511280000032
Figure GDA0003399511280000033
Figure GDA0003399511280000034
Figure GDA0003399511280000035
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 GDA0003399511280000036
in the interval of the natural frequency of the kth order structure,ω 1for fundamental frequency constraint value, eta is target non-probability reliability, for preceding sjThe reliability constraint adopted for the-1 st order frequency is to overcome the modal exchange problem during the optimization process, and
Figure GDA0003399511280000037
denotes the s thj+1 st order and sjThe order frequency spacing constraint, J denotes the order of the mode under consideration.
Figure GDA0003399511280000038
And ωr' is the natural frequency of the structure at a particular vertex combination, where the vertex combination is taken as the midpoint of the uncertainty parameter. Also, constrain
Figure GDA0003399511280000039
And
Figure GDA00033995112800000310
is also added to overcome the optimizationModality exchange problems in the 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 GDA00033995112800000311
Figure GDA00033995112800000312
Figure GDA00033995112800000313
Figure GDA00033995112800000314
Figure GDA00033995112800000315
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;
step seven: and 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 the preset value xi, 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: the design domain is discretized by using finite elements according to the characteristics of the design structure, and a Modified Solid Isotropic Microstructure/Material interpolation model (Modified Solid Isotropic Microstructure/Material with Penalty, MSIMP) model with Penalty factors is used to describe the elastic modulus and density of the Material, namely:
Figure GDA0003399511280000051
M(ρi)＝Mmin+ρi(M0-Mmin)
wherein E (ρ)i) And M (ρ)i) Respectively representing the modulus of elasticity and the density of the material, p, of the ith cell>1 is a penalty factor and is set to 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 GDA0003399511280000052
whereinaAnd
Figure GDA0003399511280000053
are respectively referred to as interval number aILower and upper bounds. When in use
Figure GDA0003399511280000054
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 GDA0003399511280000055
interval matrix AICan also be expressed as:
Figure GDA0003399511280000056
wherein the content of the first and second substances,A＝(a ij)m×nand
Figure GDA0003399511280000057
are respectively called interval matrix AILower and upper bounds of interval matrix AICan be expressed as:
Figure GDA0003399511280000058
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 GDA0003399511280000061
Figure GDA0003399511280000062
wherein the content of the first and second substances,
Figure GDA0003399511280000063
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 GDA0003399511280000064
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 GDA0003399511280000065
in particular, when only xIThe degradation is a real number x', and the corresponding reliability is calculated as:
Figure GDA0003399511280000066
when only yIThe degradation is a real number y', and the corresponding reliability is calculated as:
Figure GDA0003399511280000067
using boundary formula to overcome the problem of modal exchange in optimization process, for previous sjReliability of-1 order frequencyAnd (3) bundling to construct a continuum structure non-probability reliability topological optimization model under mixed constraint of fundamental frequency and frequency interval:
Figure GDA0003399511280000071
Figure GDA0003399511280000072
Figure GDA0003399511280000073
Figure GDA0003399511280000074
Figure GDA0003399511280000075
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 GDA0003399511280000076
is an interval of the k-th order natural frequency,ω 1for fundamental frequency constraint value, eta is target non-probability reliability, for preceding sjThe reliability constraint adopted for the-1 st order frequency is to overcome the modal exchange problem during the optimization process, and
Figure GDA0003399511280000077
denotes the s thj+1 st order and sjThe order frequency spacing constraint, J denotes the order of the mode under consideration.
Figure GDA0003399511280000078
And ωr' is the natural frequency of the structure at a particular vertex combination, where the vertex combination is taken as the midpoint of the uncertainty parameter. Also, constrain
Figure GDA0003399511280000079
And
Figure GDA00033995112800000710
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 GDA00033995112800000711
Or
Figure GDA00033995112800000712
Correspondingly, BICan be thatω 1Or
Figure GDA00033995112800000713
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 may be converted into a constraint of the target performance extremum, and the converted optimization model may be expressed as:
Figure GDA0003399511280000081
Figure GDA0003399511280000082
Figure GDA0003399511280000083
Figure GDA0003399511280000084
Figure GDA0003399511280000085
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 GDA0003399511280000086
and
Figure GDA0003399511280000087
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 GDA0003399511280000088
wherein k isiThe slope of the extreme state plane. Thus, g can be found in four cases:
case 1: if k is3<Ar/Br<k1Eta is equal to or greater than 0.5, and the extreme state plane intersects the upper and left edges 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 GDA0003399511280000089
and
Figure GDA00033995112800000810
based on the definition of the non-probabilistic reliability, one can obtain:
Figure GDA0003399511280000091
g can be solved from the above formula to give:
Figure GDA0003399511280000092
case 2: if A isr/Br≥k1Eta is not less than 0.5 or Ar/Br≥k2Eta is less than or equal to 0.5, and 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 GDA0003399511280000093
based on the definition of the non-probability reliability, there are:
Figure GDA0003399511280000094
solving for g from the above formula yields:
g＝-Ac+Bc+(2η-1)Ar
case 3: if k is4<Ar/Br<k2,η<0.5, the position relationship between the extreme state plane and the feasible region is shown in FIG. 2(d), and g can be calculated as:
Figure GDA0003399511280000095
case 4: if A isr/Br≤k3Eta is not less than 0.5 or Ar/Br≤k3Eta is less than or equal to 0.5, and the two formulas 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 GDA0003399511280000096
particularly, when Ar＝0,BrNot equal to 0, g can be solved as:
Figure GDA0003399511280000097
similarly, when Ar≠0,BrWhen 0, we can get:
Figure GDA0003399511280000101
combining the above several cases, we can get:
Figure GDA0003399511280000102
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 GDA0003399511280000103
wherein the content of the first and second substances,
Figure GDA0003399511280000104
and
Figure GDA0003399511280000105
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 GDA0003399511280000106
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 GDA0003399511280000107
the formula is simplified and arranged to obtain:
Figure GDA0003399511280000108
both ends of the above formula are simultaneously multiplied by
Figure GDA0003399511280000109
It is possible to obtain:
Figure GDA00033995112800001010
since K and M are symmetric matrices, then:
Figure GDA0003399511280000111
by normalization conditions of the modal vector, i.e.
Figure GDA0003399511280000112
It is possible to obtain:
Figure GDA0003399511280000113
rigidity matrix of unit j corresponding to upper bound of structure natural frequency
Figure GDA0003399511280000114
And quality matrix
Figure GDA0003399511280000115
Only the density design variables of the present cell. Then substituting the MSIMP interpolation model into the above equation, there are:
Figure GDA0003399511280000116
wherein the content of the first and second substances,
Figure GDA0003399511280000117
the feature vector corresponding to cell j is the upper bound of the r-th order natural frequency.
Similarly, the design variable rho is the lower bound of the natural frequency of the structurejThe sensitivity of (d) is solved as:
Figure GDA0003399511280000118
wherein the content of the first and second substances,ω rto the lower bound of the structure's order-r natural frequency,
Figure GDA0003399511280000119
the lower bound for the r-th order natural frequency corresponds to the eigenvector of cell j.
Figure GDA00033995112800001110
And
Figure GDA00033995112800001111
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 GDA00033995112800001112
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 GDA00033995112800001113
The general form of the set of vectors can be expressed as a linear combination of basis vectors:
Figure GDA00033995112800001114
wherein, betarkIs the undetermined coefficient. By appropriate treatment, [ beta ]rk]And r, k is n, …, and n + m-1 forms an m × m orthogonal matrix. To be provided with
Figure GDA00033995112800001115
And
Figure GDA00033995112800001116
substituting the characteristic value equation to obtain:
Figure GDA00033995112800001117
wherein the content of the first and second substances,
Figure GDA00033995112800001118
and
Figure GDA00033995112800001119
respectively a structural overall rigidity matrix and a quality matrix corresponding to the upper bound of the characteristic frequency. 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 GDA0003399511280000121
Figure GDA0003399511280000122
Figure GDA0003399511280000123
Figure GDA0003399511280000124
r＝n,…,n+m-1
wherein the content of the first and second substances,
Figure GDA0003399511280000125
and
Figure GDA0003399511280000126
respectively corresponding to multiple characteristic frequencies
Figure GDA0003399511280000127
And feature vectors
Figure GDA0003399511280000128
The sensitivity of (2). Substituting the above formula into generalized eigenvalue equation, and respectively multiplying the two ends of the equation by left
Figure GDA0003399511280000129
Merging Δ ρjSimplifying the same terms to obtain a first-order approximation result (i.e. making Δ ρ)jCoefficients of terms are zero):
Figure GDA00033995112800001210
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 GDA00033995112800001211
wherein, deltaskIs the Kronecker function. The above equation gives a set of equations for the coefficient βrkN, …, n + m-1, with the condition of a non-zero solution being a system matrixThe formula is zero, i.e.:
Figure GDA00033995112800001212
the above formula relates to multiple characteristic frequencies
Figure GDA00033995112800001213
The algebraic equation of sensitivity, the solution of which yields multiple eigenfrequencies
Figure GDA00033995112800001214
M sensitivity of
Figure GDA00033995112800001215
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 GDA00033995112800001216
wherein the content of the first and second substances,
Figure GDA00033995112800001217
is the lower bound of the characteristic frequency
Figure GDA00033995112800001218
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 GDA00033995112800001219
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 GDA0003399511280000131
according to
Figure GDA0003399511280000132
Comprises the following steps:
Figure GDA0003399511280000133
from the expression of g, one can obtain:
Figure GDA0003399511280000134
and
Figure GDA0003399511280000135
and
Figure GDA0003399511280000136
and
Figure GDA0003399511280000141
in summary, there are:
Figure GDA0003399511280000142
wherein the content of the first and second substances,
Figure GDA0003399511280000143
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 GDA0003399511280000144
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 GDA0003399511280000145
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;
step seven: and 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 the preset value xi, 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 GDA0003399511280000151
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 the reliability topology optimization, the constraint is that g is less than or equal to0. 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, as the non-probability reliability is gradually increased, the relative volume fraction of the topology optimization result is gradually increased, that is, the structure with higher reliability occupies larger volume, that is, more material is required to form the structure.
TABLE 1 volume fraction table of the optimization results of the present invention for continuum structure topology optimization
Figure GDA0003399511280000152
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 (2)

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 corrected MSIMP model with penalty factors is used for describing the elastic modulus and the density of a material, and a proper material rigidity and density lower limit are selected to avoid the problem of local mode;
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 FDA0003399511270000011
Figure FDA0003399511270000012
Figure FDA0003399511270000013
Figure FDA0003399511270000014
Figure FDA0003399511270000015
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 FDA0003399511270000016
in the interval of the natural frequency of the kth order structure,ω 1is a fundamental frequency constraint value, eta is a target non-probability reliability, and front s is matched in order to overcome the modal exchange problem in the optimization processjThe-1 order frequencies all adopt the reliability constraint, and
Figure FDA0003399511270000017
denotes the s thj+1 st order and sjThe order frequency spacing constraint, J denotes the order of the mode to be considered,
Figure FDA0003399511270000018
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 FDA0003399511270000019
And
Figure FDA00033995112700000110
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 FDA0003399511270000021
Figure FDA0003399511270000022
Figure FDA0003399511270000023
Figure FDA0003399511270000024
Figure FDA0003399511270000025
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 progressive 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;
step seven: and 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 the preset value xi, 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: 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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