CN108009381B - Continuum structure reliability topological optimization method under mixed constraint of displacement and global stress - Google Patents

Continuum structure reliability topological optimization method under mixed constraint of displacement and global stress Download PDF

Info

Publication number
CN108009381B
CN108009381B CN201711419513.6A CN201711419513A CN108009381B CN 108009381 B CN108009381 B CN 108009381B CN 201711419513 A CN201711419513 A CN 201711419513A CN 108009381 B CN108009381 B CN 108009381B
Authority
CN
China
Prior art keywords
stress
displacement
function
sensitivity
reliability
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Active
Application number
CN201711419513.6A
Other languages
Chinese (zh)
Other versions
CN108009381A (en
Inventor
王磊
夏海军
邱志平
刘东亮
梁金雄
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Beihang University
Original Assignee
Beihang University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Beihang University filed Critical Beihang University
Priority to CN201711419513.6A priority Critical patent/CN108009381B/en
Publication of CN108009381A publication Critical patent/CN108009381A/en
Application granted granted Critical
Publication of CN108009381B publication Critical patent/CN108009381B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation

Landscapes

  • Engineering & Computer Science (AREA)
  • Physics & Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • Computer Hardware Design (AREA)
  • Evolutionary Computation (AREA)
  • Geometry (AREA)
  • General Engineering & Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Management, Administration, Business Operations System, And Electronic Commerce (AREA)
  • Design And Manufacture Of Integrated Circuits (AREA)

Abstract

The invention discloses a topological optimization method for the non-probability reliability of a continuum structure under the mixed constraint of displacement and global stress, which comprises the steps of firstly adopting a density filtering method to obtain unit density from design variables, then calculating the displacement and stress of the structure by using a relaxation rule, processing the global stress by using stress comprehensive function constraint, and then obtaining the upper and lower bounds of the displacement and stress comprehensive function by using a vertex combination method; solving the convergence problem by adopting the optimized characteristic displacement to replace a non-probability reliability index, and solving the sensitivity of the optimized characteristic displacement by using an adjoint vector method and a complex function derivation method; and finally, carrying out iterative computation by using a mobile progressive method until corresponding convergence conditions are met, and obtaining an optimal design scheme meeting reliability constraint. The method reasonably represents the influence of uncertainty on the structural rigidity and strength performance of the continuum in the process of topology optimization design, can realize effective weight reduction, and ensures that the design gives consideration to safety and economy.

Description

Continuum structure reliability topological optimization method under mixed constraint of displacement and global stress
Technical Field
The invention relates to the field of topological optimization design of a continuum structure, in particular to a topological optimization method for the reliability of the continuum structure under mixed constraint of displacement and global stress.
Background
With the development of scientific technology and productivity, the research on structural optimization design becomes more and more important. According to the range of design variables, the structure optimization design can be divided into three levels: cross-sectional dimension optimization, geometric shape optimization and topological layout optimization. Compared with size optimization and shape optimization, the structural topological optimization variables have larger influence on the optimization target and have larger economic benefit. Therefore, the topological optimization research of the continuum structure has important engineering practical value.
Currently, most of the topology optimization studies are mainly focused on displacement or other global response constraints, and less are studied for topology optimization under stress constraints. However, in practical engineering, stress constraints are very important, and a topology optimization study without considering the stress constraints cannot be put into engineering practice. The stress constraint has three properties, namely singularity, locality and strong nonlinearity, which makes the topological optimization research under the stress constraint difficult. Some scholars propose a stress relaxation method aiming at the singularity of stress, a stress comprehensive function method aiming at the locality of stress, and a correction coefficient method aiming at the strong nonlinearity of stress. The methods can solve the topological optimization problem under the stress constraint to a certain extent.
However, as the engineering structure becomes more sophisticated and complex, the dispersion of material properties due to the manufacturing process of the material has a significant impact on the performance of the structure. In addition, due to the deterioration of the service environment of the structure, the service safety of the structure faces greater challenges. Therefore, it is necessary to consider the influence of uncertainty in the structure optimization design. The topological optimization is used as a conceptual design stage of the structural optimization and has a decisive influence on the final structural form, so that the method for researching the structural reliability optimization design of the continuum under the mixed constraint of displacement and global stress has great significance.
In actual engineering, structural sample experimental data are often lacked, so that the conditions of a probabilistic reliability model and a fuzzy reliability model cannot be met, but uncertain boundaries of uncertain information are easy to obtain. In recent years, the theory of non-probabilistic reliability has rapidly developed. Therefore, the method for researching the non-probability reliability topological optimization under the mixed constraint of the structural displacement and the global stress has obvious practical significance. At present, related research is not sufficient, and the calculation cost of the existing method is too high, or the safety redundancy is too large, so that the time cost loss and the serious resource waste are caused.
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 method under the mixed constraint of displacement and global stress. 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 method for topological optimization of the reliability of a continuum structure under mixed constraint of displacement and global stress comprises the following steps:
the method comprises the following steps: the design variables are described by adopting a variable density method, the uncertainty of the structural material attribute and the load is described by using an interval model, the volume of the structure is taken as an optimization target, the structural displacement and the global stress are taken as constraints, and a non-probability reliability topological optimization model is established as follows:
Figure BDA0001522749060000021
where V is the volume of the optimization region, ρiAnd ViRelative density and volume, respectively, of the ith cell, and piIs a function of a design variable r, N being an optimumThe total number of cells divided by the area is quantized.
Figure BDA0001522749060000022
Is the actual displacement interval value of the jth displacement constraint point,
Figure BDA0001522749060000023
is the allowable displacement interval value of the j-th displacement constraint, and m is the number of displacement constraints.
Figure BDA0001522749060000024
Is the actual stress interval value of the ith stress constraint point,
Figure BDA0001522749060000025
is the allowable stress interval value of the stress constraint. RsIs a non-probability set reliability indicator,
Figure BDA0001522749060000026
is the target non-probabilistic reliability corresponding to the jth displacement constraint,
Figure BDA0001522749060000027
the target non-probability reliability corresponding to the kth stress constraint is shown, and r is the lower limit of a design variable; r is the lower limit of the design variable.
Step two: and filtering the design variables by adopting a density filtering method to obtain the density value of each unit. And describing uncertainty of the elastic modulus and the load of the material by using interval quantity, and relaxing the elastic modulus and stress calculation of the unit by using a vertex combination method and a relaxation rule. And after the displacement of the structure and the stress of each unit are obtained, the stress of all the units is integrated to obtain corresponding stress integrated function values, and the upper and lower boundaries of the displacement of the structure, the upper and lower boundaries of the stress integrated function and corresponding vertex combinations are obtained by comparison.
Step three: according to the upper and lower bounds of the displacement and the upper and lower bounds of the stress comprehensive function, the reliability of the non-probability set constrained by the displacement and the stress comprehensive function is obtained as follows:
Figure BDA0001522749060000031
wherein S isIRepresenting the magnitude, R, of the integral function of the actual displacement or stress of the structureIIndicating the allowable displacement or allowable amount of stress in the structure,
Figure BDA0001522749060000032
the upper bound of the structure actual displacement or stress synthesis function is represented, S represents the lower bound of the structure actual displacement or stress synthesis function, R represents the upper bound of the structure allowable displacement and allowable stress, and R represents the lower bound of the structure allowable displacement and allowable stress.
Step four: and the convergence problem is improved by adopting the optimized characteristic displacement to replace a non-probability reliability index. The original optimization model can be rewritten as follows by utilizing the optimization characteristic displacement:
Figure BDA0001522749060000033
wherein d (R)I,SI) To optimize feature displacement.
Step five: according to the corresponding vertex combination of the displacement and stress comprehensive function, the sensitivity of the upper and lower boundaries of the structural displacement and the sensitivity of the upper and lower boundaries of the stress comprehensive function to the unit density are obtained by using an adjoint vector method, and then the sensitivity of the optimized characteristic displacement of the displacement and stress comprehensive function to the design variable is obtained by using a derivation method of a composite function. The sensitivity of the optimized characteristic displacement of the displacement (stress integrated function) to the upper and lower bounds of the displacement (stress integrated function) is solved, then the sensitivity of the upper and lower bounds of the displacement (the upper and lower bounds of the stress integrated function) to the unit density is solved, then the sensitivity of the unit density to the design variable is solved, and finally the sensitivity of the optimized characteristic displacement of the displacement (stress integrated function) to the design variable is obtained by multiplying the sensitivity of the upper and lower bounds of the displacement (the upper and lower bounds of the stress integrated function).
Step six: and (3) taking the obtained displacement and stress comprehensive function constraint condition values and the sensitivity information of the displacement and stress comprehensive function constraint condition values to the design variables as input conditions of a mobile evolutionary algorithm (MMA), solving the optimization problem, and updating the design variables.
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 influence of uncertainty factors is considered in the conceptual design stage of the structure, so that the economic benefit of the structure can be improved to the maximum extent, and the safety is also considered;
(2) the non-probability reliability index adopted by the method can reasonably consider the influence of uncertain factors on the structural performance, has small demand on the sample capacity, and is very suitable for engineering application;
(3) the invention adopts the MMA algorithm to carry out optimization calculation, so that the proposed method can be suitable for the condition of multiple constraints, and the application range is wider.
Drawings
FIG. 1 is a flow chart of the present invention for non-probabilistic reliability topology optimization of continuum structure under mixed constraints of displacement and global stress;
FIG. 2 is a schematic diagram of a topology optimization design area and boundary and load conditions in an embodiment of the present invention;
fig. 3 is a schematic diagram of an optimization result of the topology optimization of the continuum structure according to the present invention, where fig. 3(a) is deterministic optimization, fig. 3(b) is non-probabilistic reliability optimization (Rs 0.90), fig. 3(c) is non-probabilistic reliability optimization (Rs 0.95), and fig. 3(d) is non-probabilistic reliability optimization (Rs 0.999);
fig. 4 is a graph of the iterative process for topology optimization of continuum structure according to the present invention, where fig. 4(a) is deterministic optimization, fig. 4(b) is non-probabilistic reliability optimization (Rs 0.90), fig. 4(c) is non-probabilistic reliability optimization (Rs 0.95), and fig. 4(d) is non-probabilistic reliability optimization (Rs 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-probability reliability topological optimization method of a continuum structure under mixed constraint of displacement and global stress, which comprises the following steps:
the method comprises the following steps: the design variables are described by adopting a variable density method, the uncertainty of the structural material attribute and the load is described by using an interval model, the volume of the structure is taken as an optimization target, the structural displacement and the global stress are taken as constraints, and a non-probability reliability topological optimization model is established as follows:
Figure BDA0001522749060000041
where V is the volume of the optimization region, ρiAnd ViRelative density and volume, respectively, of the ith cell, and piIs a function of a design variable r, and N is the total number of cells divided by the optimization region.
Figure BDA0001522749060000042
Is the actual displacement interval value of the jth displacement constraint point,
Figure BDA0001522749060000051
is the allowable displacement interval value of the j-th displacement constraint, and m is the number of displacement constraints.
Figure BDA0001522749060000052
Is the actual stress interval value of the ith stress constraint point,
Figure BDA0001522749060000053
is the allowable stress interval value of the stress constraint. RsIs a non-probability set reliability indicator,
Figure BDA0001522749060000054
is the target non-probability reliability corresponding to the jth shift constraint
Figure BDA0001522749060000055
Is the target non-probability reliability corresponding to the kth stress constraint, r is the lower limit of the design variable(ii) a r is the lower limit of the design variable.
Step two: and filtering the design variables by adopting a density filtering method to obtain the density value of each unit. And describing uncertainty of the elastic modulus and the load of the material by using interval quantity, and relaxing the elastic modulus and stress calculation of the unit by using a vertex combination method and a relaxation rule. And after the displacement of the structure and the stress of each unit are obtained, the stress of all the units is integrated to obtain corresponding stress integrated function values, and the upper and lower boundaries of the displacement of the structure, the upper and lower boundaries of the stress integrated function and corresponding vertex combinations are obtained by comparison.
The cell density can be filtered through the design variables of the cell:
Figure BDA0001522749060000056
where ρ isiIs the density value of the i-th cell, djThe design variables corresponding to the jth cell. OmegaiAll distances from the unit i are less than or equal to r0(filter radius) set of units, rjIs the distance of cell j from the center point of cell i.
After the density of the cell is obtained, the elastic modulus of the cell is relaxed as follows:
E(ρ)＝ρ3E0
where E (ρ) is the modulus of elasticity of a cell, ρ is the density of the cell, E0Is the modulus of elasticity of a solid material.
After the elastic modulus of the element is obtained, finite element calculation can be performed to obtain the displacement of the element node.
To better characterize the stress level of the structure, von mises stress was used to characterize the cell stress. The mathematical expression is as follows:
Figure BDA0001522749060000057
wherein σ1,σ2,σ3Respectively mean the firstTwo, three principal stresses.
According to the obtained displacement of the unit node and the corresponding displacement shape function and the strain matrix, the normal stress and the shear stress of each direction of the unit can be obtained as follows:
σ＝Dε＝DBue＝Sue
in order to simplify the calculation and to take into account that the stress variation within the cell is small, the stress at the center point of the cell is taken as an indication of the cell stress.
The Von Mileiser stress at the central point can be obtained according to the normal stress and the shear stress of the central point of the unit in all directions as follows:
Figure BDA0001522749060000061
wherein σc＝[σcx,σcy,σcz,τaxy,τayz,τazx]TThe load column vector at the center point of the cell.
For convenience of description, note:
σcr＝h(σc)
wherein u iseAnd the displacement column vector is composed of displacement components of the nodes corresponding to the stress constraint units. Let it have the following relation with the overall displacement column vector:
ue＝seu
in the formula seA matrix is extracted for the element displacements. For a rectangular bilinear cell, there are:
(ue)8×1＝(se)8×nun×1
wherein the matrix(s)e)8×nEach row of (a) is a row vector with one element being 1 and the remaining elements being 0.
With the stress relaxation method, the stress can be calculated as:
Figure BDA0001522749060000062
after the stress of each unit is obtained, the stress of all the units is integrated to obtain a corresponding stress integrated function value, and the unit stress is comprehensively represented by the following formula:
Figure BDA0001522749060000063
wherein v iseThe volume of the solid unit is generally 6 or 8 for P.
Because the integrated stress function has a certain difference from the global maximum stress, a certain correction needs to be adopted, and the following steps are set:
σmax≈cσPN
wherein c of the k stepkSatisfies the following formula:
Figure BDA0001522749060000064
and respectively taking upper and lower boundary values for each uncertain quantity in the elastic modulus and the load column vector of the material to combine by adopting a vertex combination method, calculating the displacement and stress comprehensive function value of the structure for each combination according to the method, and comparing the values under all combinations to obtain the upper and lower boundaries of the structure displacement, the upper and lower boundaries of the stress comprehensive function and the corresponding vertex combinations thereof.
Step three: according to the upper and lower bounds of the displacement and the upper and lower bounds of the stress comprehensive function, the reliability of the non-probability set constrained by the displacement and the stress comprehensive function is obtained as follows:
Figure BDA0001522749060000071
wherein S isIRepresenting the magnitude, R, of the integral function of the actual displacement or stress of the structureIIndicating the allowable displacement or allowable amount of stress in the structure,
Figure BDA0001522749060000072
representing actual displacement or stress healds of a structureAnd the upper bound of the resultant function, S represents the lower bound of the actual displacement or stress comprehensive function of the structure, R represents the upper bound of the allowable displacement and the allowable stress of the structure, and R represents the lower bound of the allowable displacement and the allowable stress of the structure.
Step four: and the convergence problem is improved by adopting the optimized characteristic displacement to replace a non-probability reliability index. The original optimization model can be rewritten as follows by utilizing the optimization characteristic displacement:
Figure BDA0001522749060000073
wherein d (R)I,SI) To optimize feature displacement.
The optimized feature displacement d is defined as the movement displacement from the actual failure plane to the target failure plane. Wherein the target failure plane is a plane parallel to the original failure plane, and the reliability thereof is a target value.
Since the target reliability is typically greater than 50%, the target failure plane is typically located in the lower right of the uncertainty region.
The slope of the failure plane under the critical condition can be calculated, and eta is set as the target reliability. For k1Has (2X 2/k)1X 1/2)/4 ═ 1- η, and the solution is given by k11/2 (1-. eta.), similarly, k can be obtained2＝2(1-η)。
The expression of the optimized feature displacement can be calculated as:
Figure BDA0001522749060000074
when d is greater than 0, the failure plane is above the target failure plane corresponding to the target non-probabilistic reliability eta, corresponding to the non-probabilistic reliability Rs< eta, which does not satisfy the requirements. When d is less than or equal to 0, the failure plane is below the target failure plane corresponding to the target non-probability reliability eta, and the corresponding non-probability reliability RsNot less than eta, and meets the design requirement.
Step five: according to the corresponding vertex combination of the displacement and stress comprehensive function, the sensitivity of the upper and lower boundaries of the structural displacement and the sensitivity of the upper and lower boundaries of the stress comprehensive function to the unit density are obtained by using an adjoint vector method, and then the sensitivity of the optimized characteristic displacement of the displacement and stress comprehensive function to the design variable is obtained by using a derivation method of a composite function.
Since the sensitivity of the optimal characteristic displacement of the structural displacement to the design variable has already been derived in some literature, only the sensitivity of the stress synthesis function to the design variable is derived below. First, consider the stress integration function as:
Figure BDA0001522749060000081
the correction factor is taken as:
Figure BDA0001522749060000082
the correction coefficient is constant after being fixed, and does not participate in the sensitivity analysis later.
The sensitivity of the stress integration function to cell density is then:
Figure BDA0001522749060000083
finishing to obtain:
Figure BDA0001522749060000084
the first term at the right end of the formula:
Figure BDA0001522749060000085
does not involve the derivative of the cell node displacement with respect to the cell density and can therefore be calculated directly, wherein
Figure BDA0001522749060000086
And the second term on the right:
Figure BDA0001522749060000087
the adjoint vector can be used for calculation, and:
Figure BDA0001522749060000088
the accompanying vector lambda can be obtained by a finite element calculation.
Then the right second term can be solved as:
Figure BDA0001522749060000091
in the formula
Figure BDA0001522749060000092
The calculation is as follows:
Figure BDA0001522749060000093
Figure BDA0001522749060000094
Figure BDA0001522749060000095
Figure BDA0001522749060000096
Figure BDA0001522749060000097
Figure BDA0001522749060000098
written as a vector is:
Figure BDA0001522749060000099
thus, the sensitivity of the stress integration function to the cell density is obtained.
Density ρ of cell iiDesign variable d for cell jjThe sensitivity of (d) is given by:
Figure BDA00015227490600000910
wherein omegaiAll distances from the unit i are less than or equal to r0(filter radius) set of units, rjIs the distance of cell j from the center point of cell i.
Thus, sensitivity of cell density with respect to design variations is obtained.
The sensitivity of the upper and lower bounds of the stress synthesis function with respect to the design variable can be obtained by applying a derivative method of the complex function.
For the sensitivity analysis of the upper bound of the stress comprehensive function, the vertex combinations of the corresponding uncertain quantities can be substituted for calculation to obtain the sensitivity of the upper bound of the stress comprehensive function to the design variables, and the sensitivity of the lower bound of the stress comprehensive function to the design variables can be solved in the same way.
The sensitivity of the structural displacement or stress synthesis function optimization characteristic displacement to the design variable is solved as follows:
Figure BDA0001522749060000101
wherein:
Figure BDA0001522749060000102
Figure BDA0001522749060000103
therefore, the sensitivity of the optimization characteristic displacement of the structural displacement and stress comprehensive function to the design variable is obtained.
Step six: and (3) taking the obtained displacement and stress comprehensive function constraint condition values and the sensitivity information of the displacement and stress comprehensive function constraint condition values to the design variables as input conditions of a mobile evolutionary algorithm (MMA), solving the optimization problem, and updating the design variables.
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.
Example (b):
in order to more fully understand the characteristics of the invention and the practical applicability of the invention to engineering, the invention is designed for the topology optimization of the rectangular flat plate as shown in FIG. 2. The design area is a rectangular area of 25mm × 20mm, with a thickness of 0.25mm, divided into 100 × 80 cells. The elastic modulus E of the material is 201Mpa, and the Poisson ratio mu is 0.3. The left end of the rectangular area is fixed, and F is applied to the lower right side, wherein the force is 300N vertically downward, in order to avoid the stress concentration effect, the load F is applied to 9 nodes on the lower right side of the designed area, and 0.1 time of load is applied to each node. The displacement of the loading point is constrained, without taking into account the influence of gravity, so that u < 2cm and the stress of the structure does not exceed 250Mpa, with a penalty factor p of 3 being chosen. Setting the elastic modulus E and the load F to have 10% fluctuation relative to the nominal value, namely E is [180.9,221.1] Mpa, and F is [270,330] N; let the displacement constraint u and the allowable stress [ sigma ] fluctuate by 10% from the nominal values, i.e., u ═ 1.8,2.2] cm, sigma ═ 225,275 Mpa.
Fig. 3 shows a comparison between the deterministic topology optimization results and the non-probabilistic topology optimization results when Rs is 0.90, Rs is 0.95, and Rs is 0.999, respectively. 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 non-probabilistic reliable topological optimization result has a more reasonable structure and a more stable structure. When using the same amount of uncertainty as the non-probabilistic reliability, the non-probabilistic reliabilities of the three constrained displacements of the deterministic optimization result are only Rs 0.4504, respectively. I.e. the result of the deterministic optimization is not sufficient to cope with the influence of the uncertainty variables. The iteration history in the topology optimization process is shown in fig. 4, and compared with the initial design, the weight reduction effect is obvious; as the allowable value of reliability increases, the structure tends to be safe and the weight increases.
The invention provides a non-probability reliability topological optimization method of a continuum structure under mixed constraint of displacement and global stress. Firstly, establishing a continuum structure non-probability reliability topological optimization model which takes the structure weight as an optimization target and takes the structure displacement and the global stress as constraints; then, a density filtering method is adopted to obtain unit density from unit design variables, then the relaxation rule is used to calculate the displacement and stress of the structure, the stress comprehensive function constraint is used to carry out approximate processing on the global stress constraint, and the vertex combination method is used to obtain the upper and lower bounds of the displacement and stress comprehensive function, so that the non-probability reliability index of the displacement and stress comprehensive function is obtained; then, solving the convergence problem by adopting the optimized characteristic displacement to replace a non-probability reliability index, and solving the sensitivity of the optimized characteristic displacement by using an adjoint vector method and a complex function derivation method; and finally, carrying out iterative computation by using a mobile progressive method until corresponding convergence conditions are met, and obtaining an optimal design scheme meeting reliability constraint.
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 topological optimization design of the continuum structure under the mixed constraint of displacement and global stress, 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 (5)

1. A topological optimization method for the non-probability reliability of a continuum structure under the mixed constraint of displacement and global stress is used for carrying out topological optimization design on a rectangular flat plate, and is characterized in that: the method comprises the following steps:
the method comprises the following steps: the design variables are described by adopting a variable density method, the uncertainty of the structural material attribute and the load is described by using an interval model, the volume of the structure is taken as an optimization target, the structural displacement and the global stress are taken as constraints, and a non-probability reliability topological optimization model is established as follows:
Figure FDA0002962809200000011
where V is the volume of the optimization region, ρiIs the relative density of the i-th cell, ViIs the volume of the ith cell, and piIs a function of a design variable r, N is the total number of cells divided by the optimization region,
Figure FDA0002962809200000012
is the actual displacement interval value of the jth displacement constraint point,
Figure FDA0002962809200000013
is the allowable displacement interval value of the jth displacement constraint, m is the number of displacement constraints,
Figure FDA0002962809200000014
is the actual stress interval value of the ith stress constraint point,
Figure FDA0002962809200000015
is the allowable stress interval value of the stress constraint, RsIs a non-probability set reliability indicator,
Figure FDA0002962809200000016
is the target non-probability reliability corresponding to the jth shift constraint
Figure FDA0002962809200000017
Is the target non-probabilistic reliability corresponding to the kth stress constraint,rfor design changeA lower limit of the amount;
step two: adopting a density filtering method to filter design variables to obtain density values of each unit, describing uncertainty of elastic modulus and load of the material by using interval quantity, adopting a vertex combination method, adopting a relaxation rule to calculate and relax the elastic modulus and stress of the units to obtain displacement of the structure and stress of each unit, then integrating the stress of all the units to obtain corresponding stress integrated function values, and comparing to obtain upper and lower boundaries of the structure displacement and upper and lower boundaries of the stress integrated function and corresponding vertex combinations thereof;
the cell density can be filtered through the design variables of the cell:
Figure FDA0002962809200000018
where ρ isiIs the relative density of the i-th cell, djFor the design variable corresponding to the jth cell, ΩiAll distances from the unit i are less than or equal to the filter radius r0Set of units of rjIs the distance of cell j from the center point of cell i,
after the density of the cell is obtained, the elastic modulus of the cell is relaxed as follows:
E(ρ)＝ρ3E0
where E (ρ) is the modulus of elasticity of a cell, ρ is the density of the cell, E0The modulus of elasticity of a solid material;
after the elastic modulus of the unit is obtained, finite element calculation can be carried out to obtain the displacement of the unit node;
in order to better characterize the stress level of a structure, von mises stress is used to characterize the cell stress, the mathematical expression of which is:
Figure FDA0002962809200000021
wherein σ1,σ2,σ3Respectively indicating a first main stress, a second main stress and a third main stress;
according to the obtained displacement of the unit node and the corresponding displacement shape function and the strain matrix, the normal stress and the shear stress of each direction of the unit can be obtained as follows:
σ＝Dε＝DBue＝Sue
in order to simplify calculation and consider that the stress variation in the unit is small, the stress at the center point of the unit is selected as the representation of the unit stress;
the Von Mileiser stress at the central point can be obtained according to the normal stress and the shear stress of the central point of the unit in all directions as follows:
Figure FDA0002962809200000022
wherein σc＝[σcx,σcy,σcz,τaxy,τayz,τazx]TA load column vector that is the center point of the cell;
for convenience of description, note:
σcr＝h(σc)
wherein u iseAnd a displacement column vector consisting of the displacement components of the nodes corresponding to the stress constraint units is set to have the following relation with the overall displacement column vector:
ue＝seu
in the formula seFor a unit displacement extraction matrix, for a rectangular bilinear unit, there are:
(ue)8×1＝(se)8×nun×1
wherein the matrix(s)e)8×nEach row of (a) is a row vector with one element being 1 and the remaining elements being 0;
with the stress relaxation method, the stress can be calculated as:
Figure FDA0002962809200000031
after the stress of each unit is obtained, the stress of all the units is integrated to obtain a corresponding stress integrated function value, and the unit stress is comprehensively represented by the following formula:
Figure FDA0002962809200000032
wherein v iseThe volume of the solid unit is shown, and for P, P is 6 or 8;
because the integrated stress function has a certain difference from the global maximum stress, a certain correction needs to be adopted, and the following steps are set:
σmax≈cσPN
wherein c of the k stepkSatisfies the following formula:
Figure FDA0002962809200000033
respectively taking upper and lower boundary values for each uncertain quantity in the elastic modulus and the load column vector of the material to combine by adopting a vertex combination method, calculating the displacement and stress comprehensive function value of the structure for each combination according to the method, and then comparing the values under all combinations to obtain the upper and lower boundaries of the structure displacement, the upper and lower boundaries of the stress comprehensive function and the corresponding vertex combinations thereof;
step three: according to the upper and lower bounds of the displacement and the upper and lower bounds of the stress comprehensive function, the reliability of the non-probability set constrained by the displacement and the stress comprehensive function is obtained as follows:
Figure FDA0002962809200000034
wherein S isIRepresenting the magnitude, R, of the integral function of the actual displacement or stress of the structureIIndicating the allowable displacement or allowable amount of stress in the structure,
Figure FDA0002962809200000035
representing the upper bound of the structure's actual displacement or stress integration function,Srepresenting the lower bound of the structure's actual displacement or stress integration function,
Figure FDA0002962809200000036
representing an upper bound for the allowable displacement and allowable stress of the structure,Rrepresenting a lower bound for allowable displacement and allowable stress of the structure;
step four: the problem of convergence is improved by replacing a non-probability reliability index with optimized characteristic displacement, and an original optimized model can be rewritten into the following model by using the optimized characteristic displacement:
Figure FDA0002962809200000041
wherein d (R)I,SI) Shifting for optimizing features;
the optimization characteristic displacement d is defined as the movement displacement from an actual failure plane to a target failure plane, wherein the target failure plane is a plane parallel to the original failure plane, and the reliability of the target failure plane is a target value;
since the target reliability is generally greater than 50%, the target failure plane is generally located at the lower right of the uncertainty region;
the slope of the failure plane under critical conditions can be calculated, let η be the target reliability, for k1Has (2X 2/k)1X 1/2)4 ═ 1- η, and the solution obtained is k11/2 (1-. eta.), similarly, k can be obtained2＝2(1-η)；
The expression of the optimized feature displacement can be calculated as:
Figure FDA0002962809200000042
when d is greater than 0, the failure plane is above the target failure plane corresponding to the target non-probabilistic reliability eta, corresponding to the non-probabilistic reliability RsLess than eta, not meeting the requirement, when d is less than or equal to 0, the failure plane corresponds to the target non-probability reliability etaBelow the target failure plane, corresponding non-probabilistic reliability RsNot less than eta, and meets the design requirement;
step five: according to the corresponding vertex combination of the displacement and stress comprehensive function, the sensitivity of the upper and lower boundaries of the structural displacement and the sensitivity of the upper and lower boundaries of the stress comprehensive function to the unit density are obtained by using an adjoint vector method, and then the sensitivity of the optimized characteristic displacement of the displacement and stress comprehensive function to the design variable is obtained by using a derivation rule of a composite function;
the sensitivity of the stress integration function to design variables is derived below, first, considering the stress integration function as:
Figure FDA0002962809200000043
the correction factor is taken as:
Figure FDA0002962809200000044
the correction coefficient is constant after being fixed, and does not participate in the following sensitivity analysis;
the sensitivity of the stress integration function to cell density is then:
Figure FDA0002962809200000051
finishing to obtain:
Figure FDA0002962809200000052
the first term at the right end of the formula:
Figure FDA0002962809200000053
does not involve the derivative of the cell node displacement with respect to the cell density, becauseThis can be calculated directly, wherein
Figure FDA0002962809200000054
And the second term on the right:
Figure FDA0002962809200000055
the adjoint vector can be used for calculation, and:
Figure FDA0002962809200000056
the accompanying vector lambda can be obtained by a finite element calculation,
then the right second term can be solved as:
Figure FDA0002962809200000057
in the formula
Figure FDA0002962809200000058
The calculation is as follows:
Figure FDA0002962809200000061
Figure FDA0002962809200000062
Figure FDA0002962809200000063
Figure FDA0002962809200000064
Figure FDA0002962809200000065
Figure FDA0002962809200000066
written as a vector is:
Figure FDA0002962809200000067
thus, the sensitivity of the stress synthesis function to the unit density is obtained;
density ρ of cell iiDesign variable d for cell jjThe sensitivity of (d) is given by:
Figure FDA0002962809200000068
wherein omegaiAll distances from the unit i are less than or equal to the filter radius r0Set of units of rjIs the distance of cell j from the center point of cell i;
thus, the sensitivity of the cell density with respect to design variables is obtained;
the sensitivity of the upper and lower bounds of the stress comprehensive function on the design variable can be obtained by applying a derivative method of the composite function;
for the sensitivity analysis of the upper bound of the stress comprehensive function, the corresponding vertex combinations of the uncertain quantities can be substituted for calculation to obtain the sensitivity of the upper bound of the stress comprehensive function to the design variables, and the sensitivity of the lower bound of the stress comprehensive function to the design variables can be solved in the same way;
the sensitivity of the structural displacement or stress synthesis function optimization characteristic displacement to the design variable is solved as follows:
Figure FDA0002962809200000069
wherein:
Figure FDA00029628092000000610
Figure FDA0002962809200000071
thus, the sensitivity of the optimized characteristic displacement of the structural displacement and stress comprehensive function to the design variable is obtained;
step six: taking the obtained displacement and stress comprehensive function constraint condition value and the sensitivity information of the displacement and stress comprehensive function constraint condition value to the design variable as input conditions of a moving progressive method (MMA), setting extra moving step length limitation in the updating of each step of the design variable, solving the optimization problem, and updating 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 method of the continuum structure under the mixed constraint of displacement and global stress according to claim 1, wherein the topological optimization method comprises the following steps: in the first step, the influence of uncertainty of the elastic modulus and the load size of the structural material on the structural rigidity and strength performance is represented by using a non-probability set reliability index of Qiu, and a non-probability set reliability model under the mixed constraint of displacement and global stress is constructed.
3. The non-probabilistic reliability topological optimization method of the continuum structure under the mixed constraint of displacement and global stress according to claim 1, wherein the topological optimization method comprises the following steps: in the fourth step, the original reliability index is replaced by using the structure displacement and the optimized characteristic displacement of the stress comprehensive function, so that the convergence of the original optimization problem is improved.
4. The non-probabilistic reliability topological optimization method of the continuum structure under the mixed constraint of displacement and global stress according to claim 1, wherein the topological optimization method comprises the following steps: and in the fifth step, according to the vertex combination corresponding to the upper and lower boundaries of the displacement and stress comprehensive function, the sensitivity of the upper and lower boundaries of the displacement and stress comprehensive function to the unit density is solved by using an adjoint vector method.
5. The non-probabilistic reliability topological optimization method of the continuum structure under the mixed constraint of displacement and global stress according to claim 1, wherein the topological optimization method comprises the following steps: and fifthly, solving the sensitivity of the optimized characteristic displacement of the displacement and stress comprehensive function to the design variable by using a composite function derivation method, firstly solving the sensitivity of the optimized characteristic displacement of the displacement and stress comprehensive function to the displacement and the upper and lower bounds of the stress comprehensive function, then solving the sensitivity of the upper and lower bounds of the displacement and the upper and lower bounds of the stress comprehensive function to the unit density, then solving the sensitivity of the unit density to the design variable, and finally multiplying the sensitivity of the optimized characteristic displacement of the displacement and stress comprehensive function to the design variable.
CN201711419513.6A 2017-12-25 2017-12-25 Continuum structure reliability topological optimization method under mixed constraint of displacement and global stress Active CN108009381B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201711419513.6A CN108009381B (en) 2017-12-25 2017-12-25 Continuum structure reliability topological optimization method under mixed constraint of displacement and global stress

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201711419513.6A CN108009381B (en) 2017-12-25 2017-12-25 Continuum structure reliability topological optimization method under mixed constraint of displacement and global stress

Publications (2)

Publication Number Publication Date
CN108009381A CN108009381A (en) 2018-05-08
CN108009381B true CN108009381B (en) 2021-05-25

Family

ID=62061024

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201711419513.6A Active CN108009381B (en) 2017-12-25 2017-12-25 Continuum structure reliability topological optimization method under mixed constraint of displacement and global stress

Country Status (1)

Country Link
CN (1) CN108009381B (en)

Families Citing this family (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109063283B (en) * 2018-07-17 2021-02-02 北京航空航天大学 Continuum structure reliability topological optimization method under rigid-strength fusion constraint
CN109408939B (en) * 2018-10-18 2022-11-29 燕山大学 Improvement method for optimizing distribution of reinforcing ribs of sheet structure considering both stress and displacement constraints
CN109829209B (en) * 2019-01-15 2023-07-18 内蒙古工业大学 Fuzzy reliability analysis method based on perturbation principle
CN110704912B (en) * 2019-09-19 2021-05-25 北京航空航天大学 Method for topological optimization of bridge bracket arm structure under stress constraint
CN111027110B (en) * 2019-11-27 2023-06-30 中国科学院光电技术研究所 Comprehensive optimization method for topology and shape and size of continuum structure
WO2022188001A1 (en) 2021-03-08 2022-09-15 浙江大学 Reliability-based topology optimization design method for part structure by considering bounded hybrid uncertainty
CN113887098B (en) * 2021-09-28 2023-06-20 北京航空航天大学 Metal structure topology optimization method based on fatigue stress robustness and reliability

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102945307A (en) * 2012-11-27 2013-02-27 北京汽车股份有限公司 Automobile chassis key structural member structure optimization design method
CN105893661A (en) * 2016-03-30 2016-08-24 华南理工大学 Multiple-station-progressive mold base-body structure topology optimization method
CN106650147A (en) * 2016-12-30 2017-05-10 北京航空航天大学 Continuum structure non-probability topologicaloptimization method based on bounded uncertainty
CN106650148A (en) * 2016-12-30 2017-05-10 北京航空航天大学 Method of continuum structure non-probabilistic reliability topological optimization under mixed constraints of displacements and stresses

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN102945307A (en) * 2012-11-27 2013-02-27 北京汽车股份有限公司 Automobile chassis key structural member structure optimization design method
CN105893661A (en) * 2016-03-30 2016-08-24 华南理工大学 Multiple-station-progressive mold base-body structure topology optimization method
CN106650147A (en) * 2016-12-30 2017-05-10 北京航空航天大学 Continuum structure non-probability topologicaloptimization method based on bounded uncertainty
CN106650148A (en) * 2016-12-30 2017-05-10 北京航空航天大学 Method of continuum structure non-probabilistic reliability topological optimization under mixed constraints of displacements and stresses

Non-Patent Citations (3)

* Cited by examiner, † Cited by third party
Title
"Robust topology optimization for structures under interval uncertainty";Jinglai Wu 等;《Advances in Engineering Software》;20160509;第36–48页 *
"桁架结构概率-非概率混合可靠性拓扑优化";乔心州 等;《应用力学学报》;20110831;第28卷(第4期);第402-406页 *
"结构拓扑优化中棋盘格抑制方法的研究";豆麟龙 等;《应用数学和力学》;20140815;第35卷(第8期);第920-928页摘要，引言，第1-4节 *

Also Published As

Publication number Publication date
CN108009381A (en) 2018-05-08

Similar Documents

Publication Publication Date Title
CN108009381B (en) Continuum structure reliability topological optimization method under mixed constraint of displacement and global stress
CN109063283B (en) Continuum structure reliability topological optimization method under rigid-strength fusion constraint
CN106650148B (en) Non-probabilistic reliability topological optimization method for continuum steel structure under mixed constraint of displacement and stress
Wu et al. An interval uncertain optimization method for vehicle suspensions using Chebyshev metamodels
Kaveh et al. Performance-based multi-objective optimization of large steel structures
CN112733221A (en) Optimization design method and system for reinforcing side slope by using anti-slide pile
CN104573906B (en) System and method for analyzing oscillation stability in power transmission system
CN106022631B (en) Index weight analysis method
Hong et al. Parametric reduced-order models of battery pack vibration including structural variation and prestress effects
CN112214817B (en) Multilayer displacement response determination method considering interlayer condition and transverse isotropy
CN110991112B (en) Random defect mode superposition method based on response surface method
Blanchard et al. Modelling the different mechanical response and increased stresses exhibited by structures made from natural fibre composites
CN111274624B (en) Multi-working-condition special-shaped node topology optimization design method based on RBF proxy model
Shi et al. An interval updating model for composite structures optimization
CN113887098B (en) Metal structure topology optimization method based on fatigue stress robustness and reliability
Serafinska et al. A coupled approach of optimization, uncertainty analysis and configurational mechanics for a fail‐safe design of structures
CN103870614A (en) Structural probability optimized design method
KR20210112705A (en) A composite beams design apparatus and beam desing method of using the same
CN110704912B (en) Method for topological optimization of bridge bracket arm structure under stress constraint
Hu et al. Structural optimization and form-finding of cylindrical shells for targeted elastic postbuckling response
CN111159946A (en) Discontinuous problem partition solving method based on minimum potential energy principle
Lu Parametric reduced-order models for the structural dynamics of hybrid electric vehicle batteries
CN116484667B (en) Topology optimization and stability assessment method for support connector structure
Yang et al. Actuator Placement and Configuration Direction Optimation in Plate Structure Vibration Control System
Qiu et al. Evaluation of long-service-life cable-stayed bridge strengthening schemes using a two-phase finite element model updating technology

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant