CN111475976A - Robust topology optimization method for particle reinforced material member considering mixing uncertainty - Google Patents

Robust topology optimization method for particle reinforced material member considering mixing uncertainty Download PDF

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CN111475976A
CN111475976A CN202010239662.XA CN202010239662A CN111475976A CN 111475976 A CN111475976 A CN 111475976A CN 202010239662 A CN202010239662 A CN 202010239662A CN 111475976 A CN111475976 A CN 111475976A
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程锦
陆威
刘振宇
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Zhejiang University ZJU
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Abstract

The invention discloses a method for optimizing the robust topology of a particle reinforced material member by considering mixing uncertainty. The method comprises the following steps: considering the uncertainty of the member using the particle reinforced material in manufacturing and use, regarding the insufficient external loading of the sample as interval uncertainty, regarding the sufficient matrix and particle material properties of the sample, and regarding the volume distribution of the particle reinforced phase in the matrix as bounded probability uncertainty; discretizing the volume distribution of the design domain and the particle enhanced phase, setting physical and geometric constraints, and establishing a stable topological optimization design model; and iteratively solving the robust optimization design model of the mixed uncertainty of the considered interval and the bounded probability by using an optimal criterion method. The component robust topology optimization design model established by the invention truly reflects the distribution characteristics of multi-source uncertainty in actual engineering, and the proposed structural performance statistical moment estimation method and the optimal criterion method are used for solving, so that the optimization result can be efficiently obtained, and the method has good engineering application value.

Description

Robust topology optimization method for particle reinforced material member considering mixing uncertainty
Technical Field
The invention belongs to the field of equipment structure optimization design, and relates to a method for optimizing the robust topology of a particle reinforced material member by considering mixing uncertainty.
Background
Topology optimization, which is a method for adjusting the distribution of limited materials in a structural design domain to make a structure have an optimal target performance, has been widely applied in structural design, and further matured with the popularization of additive manufacturing technology in recent years. Because various uncertainties exist in the production, manufacturing and using processes, in order to enable the topological optimization design result not to be degraded after actual processing, steady topological optimization considering the uncertainties has also obtained abundant results. However, most of the current structure topology robust optimization methods only consider simple probability or simple interval uncertainty, and the structure robust topology optimization considering probability interval mixed uncertainty is not extensively and deeply researched, and mainly includes:
1) the modeling mode of probability uncertainty can influence the analysis precision of target performance, and further influence the stable topology optimization result. The existing means for describing probability uncertainty by adopting normal distribution has irrationality in engineering, namely: the normal distribution parameter can theoretically take a negative value and be positive and infinite, which is not in accordance with the fact that the uncertainty parameter fluctuates probabilistically only within a certain range in practical engineering. 2) The theoretical support of a target performance estimation mode is weak when probability interval mixing uncertainties exist simultaneously, the existing structure robust topology optimization method considering the probability interval mixing uncertainties can only give an upper bound (worst performance) of a target performance statistical moment, but cannot give a corresponding worst working condition (particularly when uncertainty external loads are used as interval uncertainty variables), namely the worst working condition has an important guiding effect in engineering practical analysis. 3) The existing structure robust topology optimization method considering probability interval mixing uncertainty only aims at the problem of using a single material, and lacks attention to structure robust topology optimization using generalized composite materials such as multiphase materials and functional gradient materials. In fact, selective laser sintering or melting and other means in additive manufacturing can be mature and applied to metal, ceramic materials or various composite reinforced materials with complex topological structures at present, and the composition of materials in the structure can be controlled accurately, so that the method has practical research value and practical requirements for structural robust topological optimization by using composite materials.
On the other hand, the existing structure robust topology optimization method aiming at the generalized composite material has certain defects in the aspect of practical application. The current method research trend is to utilize a microscopic variable lattice structure to realize gradient properties of the same material and different equivalent physical properties on a macroscopic structure. However, due to the state of the art additive manufacturing, there is often a degradation in the actual structural performance due to: 1) the tiny topological structure in the lattice may not be reproduced completely, especially the tiny rigid bar node with infinite theory in the five-mode metamaterial; 2) manufacturing errors cannot be avoided in the additive manufacturing process, and a working route for carrying out structure robust topology optimization by introducing geometric boundary uncertainty at a microscopic level of a lattice structure has certain difficulty, which is partially caused by the acquisition of calculation cost and objective function gradient information. Therefore, the single-scale composite material without taking the microstructure into account (such as carbon fiber reinforced plastics, particle reinforced metal/metal ceramic materials and the like which are widely applied at present) can still be a material form suitable for actual manufacturing and production in the future for a long time, and has particularly urgent research needs for the structural robustness and topology optimization of the material.
Disclosure of Invention
The method comprises the steps of considering the uncertainty of a particle reinforced material-based member in the manufacturing and using processes, regarding the external load with insufficient samples as interval uncertainty and regarding the volume distribution of the member matrix attribute and the particle reinforced phase with sufficient samples as bounded probability uncertainty, discretizing the volume distribution of a design domain and the particle reinforced phase in the member matrix, setting structural constraints and the external load, establishing a particle reinforced material member robust optimization design model considering the interval and the bounded probability mixed uncertainty, iteratively solving the model by using an optimal criterion method, namely decoupling the uncertainty of the probability interval mixed uncertainty, determining the worst working condition by using the gradient information of an optimization target, estimating the mean value of the optimization target under the worst working condition by using a univariate dimension reduction method and a Laguerre (L aguerre) integral format, constructing a target function, calculating the gradient of the target function and the constraint variable function respectively, and updating the variance of the design target function, thereby solving the problem of the efficient particle reinforced material under the uncertain conditions.
The invention is realized by the following technical scheme: a method of robust topology optimization of a particulate reinforced material structure taking into account mixing uncertainties, the method comprising the steps of:
1) the following uncertainties in the manufacture and use of the component based on the particulate reinforcing material are considered: the material property of the base material of the component, the material property of the particle reinforced phase, the volume fraction of the particle reinforced phase distributed in the base, and the amplitude and the direction of the external load applied to the component; wherein, because it is difficult to obtain sufficient sample information about the external load, the amplitude and direction uncertainty of the external load are treated as interval uncertainty; regarding the matrix material property, the particle reinforced phase material property and the volume fraction of the particle reinforced phase distributed in the matrix with sufficient sample information as bounded probability uncertainty processing, and describing each bounded probability uncertainty parameter by adopting a random variable obeying generalized beta distribution (GBeta distribution);
2) discretizing a component design domain, specifically:
the stress condition of the component is simplified to be a two-dimensional plane stress state, the holes (such as hinge holes) of the component are reserved, and meanwhile, detailed geometric shapes (such as chamfers, fillets and the like) are removed to improve the calculation efficiency; placing the simplified component in a regular rectangular design domain, and dividing the rectangular design domain into Nx×NyA square unit of Nx，NyThe division numbers along the directions of the x axis and the y axis are respectively; based on the classical isotropic material with penalty (SMIP) framework in topological optimization, each unit is assigned with a unique design variable rhoe∈[0,1](e＝1,2,…,NxNy)；
3) Discretizing the volume distribution of the particle reinforced phase in the component matrix, specifically:
3.1) according to a coordinate system specified by the discrete design domain, and taking the force transmission direction of the component as an x axis, each unit coordinate can be uniquely determined; in the particle reinforced material component, the volume fraction of the particle reinforced phase in the matrix is in layered gradual change, namely the volume fraction of the particle reinforced phase in the matrix is changed only along the direction perpendicular to the force transmission direction of the component (namely the y-axis direction), the theoretical change mode is Vol (y), the theoretical change mode is a continuous function of a coordinate y, the theoretical change mode is selected according to the actual use requirement of the component, and the volume fraction of the particle reinforced phase in the same thickness is constant; considering the uncertainty fluctuation of the volume fraction of the particle-reinforced phase in the actual production, the actual change mode is recorded as Vol*(y) with bounded probabilistic uncertainty;
3.2) calculating the ith (i-1, 2, …, N) according to the discrete component design domain and the volume fraction change mode of the particle reinforced phasey) Mean volume fraction Vol of intralamellar, particle-reinforced phase*(i) (ii) a The volume fractions of the particle-enhanced phases of all the units in the i layer are Vol*(i) Is a discrete function of layer number i;
3.3) calculation of ith (i ═ 1,2, …, N) using the hallin-Tsai microstructure modely) Young's modulus of each unit in layer
Figure BDA0002432132290000031
To poisson's ratio
Figure BDA0002432132290000032
3.4) introduce penalty factors based on the classical isotropic material with penalty (SMIP) framework in topology optimization, then the ith (i ═ 1,2, …, Ny) The young's modulus of each cell within a layer can ultimately be expressed as:
Figure BDA0002432132290000033
in the formula Eq.1, the compound,
Figure BDA0002432132290000034
Eminrespectively, 3.3) th calculated i (i ═ 1,2, …, Ny) Young's modulus of each unit in layerFirstly, the specified minimum Young modulus allowable value is a constant; p is a constant penalty factor specified in advance;
Figure BDA0002432132290000035
for the ith (i ═ 1,2, …, N) construction based on penalized isotropic material (SMIP) frameworksy) Young's modulus of each unit in the layer;
4) applying physical constraints and geometric constraints to the discretized structure, specifically:
4.1) physical constraints including the fixation or support of the structure, external loads, applied according to the classical finite element method;
4.2) geometric constraints including the specified holes in the structure and the areas where the material is forcibly retained by setting the design variable ρ for the cell covered by the holee≡ 0 and the design variable position ρ corresponding to the cell that requires forced retention of material position coveragee1 and does not change its value during the subsequent optimization;
5) the structural yield of the particle reinforced material member is an optimized design target, the structural yield under the joint influence of the interval and bounded probability mixing uncertainty is considered in a robust topological optimization framework, so the yield mean value and the standard deviation of the member under the worst working condition can be used as the representation of the optimized design target, and a robust optimized design model of the particle reinforced material member under the condition of considering the interval and bounded probability mixing uncertainty is established as shown in Eq.2:
Figure BDA0002432132290000036
in the formula Eq.2, the compound,
Figure BDA0002432132290000037
is (N)xNy) × 1 dimension design vector, ρminThe minimum allowable value of each design variable given in the design is a constant; x ═ X (X)1,X2,…,Xm)TIs a m × 1-dimensional bounded probability uncertainty vector whose elements include material properties of the component matrix material, material properties of the particle-enhanced phase, and the particle-enhanced phase in the matrix materialUncertainty of medium volume distribution; i ═ f1,f2,…,fn,α1,α2,…,αn)TIs a 2n × 1 dimensional interval uncertainty vector, where f1,f2,…,fnRespectively the amplitude of n uncertain external loads borne by the component, α1,α2,…,αnRespectively representing n direction angles of uncertain external load borne by the component;
Figure BDA0002432132290000041
is the volume of the structure corresponding to the current design vector ρ; v0＝NxNyIs the volume of the design domain;
Figure BDA0002432132290000042
is the volume fraction of the designed structure occupying the designed domain, which is a constant;
k (x) U ═ f (i) is the member balance equation, where k (x) is (2N)xNy)×(2NxNy) Maintaining an overall stiffness matrix, and constructing by using a classical finite element theory, wherein a specific numerical value of the overall stiffness matrix is influenced by a bounded probability uncertain vector X; f (I) is (2N)xNy) × 1 dimensional node force matrix is constructed by classical finite element theory, and its specific value is influenced by interval uncertainty vector I, U is (2N)xNy) × 1 dimension node displacement matrix;
Figure BDA0002432132290000048
the yield value of the worst working condition of the particle reinforced material component under the action of the interval uncertain vector I is calculated as follows:
A) the structural yield, which simultaneously considers the effects of interval and bounded probability uncertainty, can be written as eq.3 according to the classical finite element method:
c(ρ,X,I)＝UTK(X)U＝F(I)TK-1(X)F(I) Eq.3
B) definition of
Figure BDA0002432132290000043
To obtain a constant vector by taking the mean of each probability variable in the bounded probabilistic uncertainty vector X, we call μXIs a mean vector of a bounded probabilistic uncertainty vector X, where
Figure BDA0002432132290000044
Respectively, each uncertainty X1,X2,…,XmThe mean value of (a); let bounded probability uncertainty vector X ═ μ in structure yield c (ρ, X, I)XThen at this time, the structural yield only includes the influence of the interval uncertainty I, which can be written as c (ρ, μ)XI) ═ c (ρ, I); meanwhile, the overall rigidity matrix is also a constant matrix and can be written as K (mu)X)＝K；
C) Writing the node moment array into the form of the sum of the force vectors of all the externally-loaded nodes:
Figure BDA0002432132290000045
simultaneously, the method comprises the following steps:
Figure BDA0002432132290000046
in the formula Eq.5, fix＝ficosαi，fiy＝fisinαiRespectively an external load FiA magnitude component along the x, y directions; e.g. of the typeix，eiyRespectively corresponding to an external load FiA unit node force vector of the acting node along the x and y axis directions;
D) according to the linear elasticity assumption, the overall effect of n uncertain loads can be equivalent to the superposition of the effects of the individual effects of each load:
Figure BDA0002432132290000047
in eq.6, the amplitude and the direction angle of the uncertain load are differentiated, respectively, to give (i ═ 1,2, …, n):
Figure BDA0002432132290000051
respectively enabling the obtained derivative information according to Eq.7
Figure BDA0002432132290000052
Solve to obtain
Figure BDA0002432132290000053
(I-1, 2, …, n), i.e. the worst case of the particulate reinforcing material member under the influence of the interval uncertainty vector I
Figure BDA0002432132290000054
The yield value of the worst working condition can be written as
Figure BDA0002432132290000055
In the formula Eq.2, the compound,
Figure BDA0002432132290000056
respectively the yield value of the worst working condition under the influence of the bounded probability uncertainty vector X
Figure BDA0002432132290000057
The mean and standard deviation of (a) are calculated as follows:
a) reduction of
Figure BDA0002432132290000058
Mu inXFor the bounded probabilistic uncertainty vector X, the yield value of the worst case at this time may be written as
Figure BDA0002432132290000059
Including probability uncertainty;
b) yield value of worst working condition through single variable dimension reduction method of Rahman multivariable function
Figure BDA00024321322900000510
Can be developed by the following formula:
Figure BDA00024321322900000511
in equation Eq.8, m is the number of bounded probability uncertainty parameters contained in a bounded probability uncertainty vector X, X＜i＞(i ═ 1,2, …, m) defined by eq.9:
Figure BDA00024321322900000512
x in the formula Eq.9iIs the ith bounded probability uncertainty variable;
c) according to the expansion Eq.8, calculating the yield value of the worst working condition
Figure BDA00024321322900000513
The high-dimensional integral of the first-order and second-order origin moments can be converted into the operation of a plurality of one-dimensional integrals:
Figure BDA00024321322900000514
Figure BDA0002432132290000061
phi (X) in the formulae Eq.10, Eq.11i) Is the probability uncertainty Xi(i ═ 1,2, …, m) probability density function;
d) each one-dimensional integral in the equations eq.10, eq.11 is calculated using the laguerre (L aguerre) integral format:
Figure BDA0002432132290000062
in the equation Eq.12, t is the number of integration points of Laguerre (L aguerre) and is a predetermined constant, x(j),λ(j)(j ═ 1,2, …, t) are the standard integration points and their corresponding weights, given by the laguerre (L aguerre) integration rule, all constant;
Figure BDA0002432132290000063
by reaction at X＜i＞Intermediate order bounded probability uncertainty variable XiAre respectively as
Figure BDA0002432132290000064
Is obtained by
Figure BDA0002432132290000065
Wherein
Figure BDA0002432132290000066
Are also all constants, calculated by equiprobable transformation of each integration point, i.e.
Figure BDA0002432132290000067
e) Worst condition yield value
Figure BDA0002432132290000068
The mean and standard deviation of (d) can be obtained by eq.13:
Figure BDA0002432132290000069
6) iteratively solving a robust optimization design model of the particle reinforced material member with the mixture uncertainty of the considered interval and the bounded probability of Eq.2 by adopting a standard optimization criterion method (OC method) in topological optimization, wherein the calculation process in each iteration process specifically comprises the following steps:
6.1) defining a weighted objective function in Eq.14 for achieving
Figure BDA00024321322900000610
The double-target optimization:
Figure BDA00024321322900000611
in equation Eq.14, J (ρ) is a defined weighting objective function; w is a weight value which is specified in advance and is a constant;
6.2) respectively calculating the target function and the constraint function for each design variable rho according to Eq.15 and Eq.16e(e＝1,2,…,NxNy) Gradient:
Figure BDA0002432132290000071
Figure BDA0002432132290000072
6.3) updating the current design variable according to the gradient information of the objective function and the constraint function and the standard optimal criterion method;
6.4) checking the difference value between the objective function value in the current iteration and the objective function value in the previous iteration, wherein for the first iteration, the difference value is defined as the objective function value of the first generation, and if the difference value is smaller than a pre-specified convergence threshold value, the convergence condition is met and an updated design variable is output; otherwise, repeating the steps 6.1) to 6.4).
Further, in the step 1), regarding the volume fraction of the particle-reinforced phase distributed in the matrix with sufficient sample information as a bounded probability uncertainty process, specifically as follows:
the volume fraction of the particle-reinforced phase in the matrix is changed only along the thickness direction (i.e. the y-axis direction) of the component, and the change mode has a theoretical design expression Vol (y), but the volume fraction of the particle-reinforced phase has uncertainty due to factors such as the hysteresis of a reinforcing particle adding port control system and the like in the actual manufacturing process and can be represented by the product of a deterministic theoretical expression Vol (y) and an uncertain parameter, as shown in Eq.17:
Vol*(y)＝Vol(y)·θ Eq.17
in the formula Eq.17, Vol*(y) is the manner in which the volume fraction of the particulate reinforcing phase changes during actual manufacture; theta is an uncertain parameter reflecting fluctuation of the actual volume fraction of the enhanced phase relative to a theoretical design value, is called as the volume fraction uncertainty of the particle enhanced phase, and uncertain information of the uncertain parameter can be obtained by measuring relevant parameters (feeding port response time lag, adding mechanism friction resistance and the like) of the adding mechanism.
Further, in the step 1), random variables obeying a generalized beta distribution (GBeta distribution) are used to describe each bounded probability uncertainty parameter, which is as follows:
1.1) for bounded probability uncertainty parameter XiS samples of the parameter are obtained through experiments, and then a sample set is constructed
Figure BDA0002432132290000073
From this sample set, the parameter X is calculated in Eq.18iValue range of (1), calculating parameter X according to Eq.19iMean and variance of (a):
Figure BDA0002432132290000074
Figure BDA0002432132290000075
1.2) describing distribution in [ a ] by adopting generalized beta distributioni,bi]Inner and mean and variance are respectively
Figure BDA0002432132290000076
Parameter X ofiFirst, the mean and variance are normalized as shown in Eq.20:
Figure BDA0002432132290000081
then, the parameter X is calculated using Eq.21iDistribution parameter α of the generalized beta distributioni,βi：
Figure BDA0002432132290000082
Recording parameter XiObey ini,bi]Internal and distribution parameter αi,βiGeneralized beta distribution of (i.e. X)i～GBeta(ai,bi|αi,βi) And the probability density function is shown as Eq.22:
Figure BDA0002432132290000083
in the formula Eq.22, the compound,
Figure BDA0002432132290000089
is a parameter XiA probability density function of; (. cndot.) is a gamma function.
Further, in the step 3.3), the ith (i ═ 1,2, …, N) is calculated using a hallin-Tsai microstructure modely) Young's modulus of each unit in layer
Figure BDA0002432132290000084
To poisson's ratio
Figure BDA0002432132290000085
The method comprises the following specific steps:
3.3.1) the physical properties of the reinforcing particles are: average length of particles lGAverage width wGAnd an average thickness tGYoung's modulus EGEtc.;
3.3.2) define the following parameters:
Figure BDA0002432132290000086
in the formula Eq.23, EMIs the Young's modulus of the matrix material;
3.3.3) th i (i ═ 1,2, …, Ny) Young's modulus of each unit in layer
Figure BDA0002432132290000087
Calculated by eq.24:
Figure BDA0002432132290000088
in the formula Eq.24, Vol*(i) Is the ith (i ═ 1,2, …, Ny) The actual average volume fraction of the particle-enhanced phase within the layer, with probabilistic uncertainty;
3.3.4) th i (i ═ 1,2, …, Ny) The poisson ratio of each unit in the layer is calculated by Eq.25:
Figure BDA0002432132290000091
in the formula Eq.25, the compound,
Figure BDA0002432132290000092
νG、νMare respectively the ith (i ═ 1,2, …, Ny) The Poisson's ratio of each unit, reinforcing particle and matrix material in the layer.
Further, the step 6.2) is specifically as follows:
6.2.1) substituting Eq.10, Eq.11 into Eq.15:
Figure BDA0002432132290000093
gradient term in equation Eq.26
Figure BDA0002432132290000094
Worst case yield values calculated for Eq.10 and Eq.11, respectively
Figure BDA0002432132290000095
First and second order origin moments vs. design variables ρe(e＝1,2,…,Nx·Ny) A gradient of (a);
6.2.2) gradient term
Figure BDA0002432132290000096
The specific calculation formula is shown as Eq.27 and Eq.28;
Figure BDA0002432132290000097
Figure BDA0002432132290000098
gradient terms in the equations Eq.27 and Eq.28
Figure BDA0002432132290000099
Given by the classical topology optimization framework SIMP:
Figure BDA00024321322900000910
in the formula Eq.29
Figure BDA0002432132290000101
Is that
Figure BDA0002432132290000102
The reason for this is that the bounded probability uncertainty vectors in the aforementioned 4 terms are all constant values; p is a constant penalty factor specified in advance; k is a radical ofeThe element stiffness matrix of the element e can be calculated by a classical finite element theory; u. ofeThe element displacement matrix of the element e can be extracted from a node displacement matrix U obtained by solving finite element equations of each time through a classical finite element theory;
6.2.3) substituting the calculated result of Eq.29 into Eq.27 and Eq.28, and further substituting the calculated result of Eq.26 and Eq.27 into an expression Eq.26 to obtain a final objective function gradient result.
Further, the step 6.3) is specifically as follows:
6.3.1) for iteratively determining an appropriate constraint function Lagrangian multiplier lambda for updating the design variable by using binary search (Bisection algorithmm), firstly, specifying the upper and lower bounds lambda of the value of the Lagrangian multiplier lambda1、λ21E-3, 1E9, respectively;
6.3.2) taking the current Lagrange multiplier as an average value of the upper and lower bounds of the value is shown as Eq.30:
λ＝λmiddle＝(λ1+λ2)/2 Eq.30
6.3.3) update the design variables by eq.31 (e ═ 1,2, …, NxNy)：
Figure BDA0002432132290000103
In the formula Eq.31, m is 0.2 and is the search movement step length, η is 0.5 and is the search damping coefficient, BeDerived from objective function and constraint function gradient information, e.g. Eq32, and:
Figure BDA0002432132290000104
6.3.4) calculating the volume of the structure corresponding to the updated design variable
Figure BDA0002432132290000105
And adjusting the value boundary of the Lagrange multiplier according to the violation condition of the constraint function: if it is
Figure BDA0002432132290000106
Contraction value from lower bound to λ1＝λmiddle(ii) a If it is
Figure BDA0002432132290000107
Contraction values from upper bound to λ2＝λmiddle；
6.3.5) check the convergence criterion (λ) of the binary search2-λ1)/(λ2+λ1) If the value is less than 1E-3, returning to repeat 6.3.2) to 6.3.5) to search the Lagrangian for the next time; if so, indicating that the current Lagrangian operator meets the requirement, and outputting the current design variable
Figure BDA0002432132290000108
The invention has the beneficial effects that:
1) considering the multi-source uncertainty of the particle reinforced material component in the manufacturing and using processes, wherein the multi-source uncertainty comprises the material property of the component matrix material, the volume fraction of the particle reinforced phase distributed in the matrix, and the amplitude and the direction of the external load applied to the component; wherein, because it is difficult to obtain sufficient sample information about the external load, the amplitude and direction uncertainty thereof are treated as interval uncertainty; the method has the advantages that the matrix material property with sufficient sample information, the particle reinforced phase material property and the volume fraction of the particle reinforced phase distributed in the matrix are regarded as bounded probability uncertainty processing, and the generalized beta distribution (GBeta distribution) is adopted to describe each bounded probability uncertainty parameter, so that the defect that the existing product structure robust topology optimization design method only considers probability uncertainty or interval uncertainty is overcome, and the constructed component robust optimization model is more in line with engineering practice.
2) With the classical finite element framework, the target performance can be displayed and expressed by design variables, deterministic parameters and uncertainty parameters; the existing component robust topology optimization method considering probability interval mixed uncertainty usually considers probability uncertainty in a design model at first, then introduces interval uncertainty in a target performance statistical moment to estimate a value boundary of the statistical moment so as to represent the robust performance of a component, so that specific worst working conditions cannot be given; the existing component robust topology optimization research considering the worst working condition of the structure only considers single type of uncertainty, and the lack of precision and rationality in the aspect of uncertainty modeling may cause the unreliable optimization result; on the premise of simultaneously considering the mixing uncertainty, the invention introduces the linear elastic deformation hypothesis, namely the final deformation of the component under the action of the uncertainty external loads of a plurality of intervals can be obtained by superposing the deformation generated by the independent action of each external load; according to the method, gradient information of the target performance to the uncertainty external load is calculated to obtain the worst working condition corresponding to the worst target performance of the component, and the solving process does not need to repeatedly solve the structural response through a finite element method, so that the defect that the worst working condition cannot be given by the existing mixed uncertainty steady topology optimization method is overcome, and a theoretical basis is provided for further analysis reference of engineers.
3) The existing component robust topological optimization considering mixing uncertainty is mostly based on a metamaterial lattice structure and limited by the manufacturing precision of additive manufacturing, and the actual product performance is often different from a theoretical design result; therefore, starting from the feasibility and the manufacturing level of actual industrial manufacturing, aiming at the particle reinforced material which is widely used in industrial manufacturing, the invention establishes a component robust topological optimization method using the particle reinforced material by considering the uncertainty of the volume distribution of the particle reinforced phase in the matrix material, and popularizes the existing component robust optimization design method aiming at single material into the composite material; meanwhile, the manufacturing precision of the design result given by the single macroscopic-level component robust topological optimization to additive manufacturing in the actual manufacturing process can be met through the current level, the performance degradation of the existing multi-level robust optimization design result based on the lattice structure after actual manufacturing is avoided, and the method has high industrial feasibility.
4) In the existing component robust topology optimization method, because probability uncertainty of normal distribution is considered, a target performance statistical moment is generally calculated by adopting a Hermit integral format with integral limit of plus and minus infinity; but the boundless of the normal distribution value conflicts with the generally non-negative reasonable value of the uncertainty parameter in the actual engineering, so that the design result is unreasonable; the invention provides a numerical method for accurately estimating the mean value and the standard deviation of the target performance of a component by introducing a univariate dimension reduction method for the optimized target performance and a Laguerre integral format with the integral limit from zero to positive infinity; compared with the existing product structure performance statistical moment estimation method considering probability interval mixed uncertainty, the method is based on a bounded probability uncertainty model which is more practical, namely generalized beta distribution (GBeta distribution), has better compatibility with a mature and steady topological optimization framework, and can efficiently derive gradient information of target performance to design variables for iterative updating and optimizing.
Drawings
FIG. 1 is a flow chart of a method for robust topology optimization of a particulate reinforced material structure taking into account mixing uncertainty.
FIG. 2 is a block diagram of a high speed press of a type (with the top beam removed).
Fig. 3 is an initial design drawing of the outer link.
FIG. 4 is a schematic diagram of an outer link robust topology optimization design element.
FIG. 5 is an outer link robust topology optimization design result.
FIG. 6 is a design diagram of an outer link structure based on the results of a robust topology optimization design.
Detailed Description
The invention is further illustrated by the following figures and examples.
The related information in the figure is practical application data of the robust design of the outer connecting rod for a certain type of high-speed press, and figure 1 is a flow chart of a robust topological optimization design method of a particle reinforced material component considering mixing uncertainty.
1. The outer connecting rod for a high-speed press of a certain type manufactured by using 20% SiC particle reinforced Al365 material shown in FIG. 2 is taken as a research object, and the uncertainty of the outer connecting rod in the manufacturing and using processes is considered:
1.1) the external load applied to the external connecting rod has certain uncertainty due to the fluctuation of the rotating speed of the main shaft of the press, the density of a sliding block, the nonuniformity of a punched material and the like, but the amplitude f and the direction angle of the external load are determined because the external load is difficult to measure in the working process of the press and sufficient sample information about the external load is difficult to obtain
Figure BDA0002432132290000122
Processing as interval uncertainty;
1.2) Material Properties (Young's modulus E) of the base Material Al365 of the outer connecting rodAlV to Poisson ratioAl) Because the physical properties of raw materials are not uniform, the process fluctuation in the metallurgical process and the like have more obvious uncertainty, but sufficient sample information can be obtained by measuring a finished product, so that the method can be used for processing the bounded probability uncertainty; the SiC reinforced particles are generally prepared by a precise chemical method such as a sol-gel method, and the Young modulus and the Poisson ratio are stable, so that the nominal value of the SiC reinforced particles is used without considering the uncertainty; the volume fraction Vol (y) uncertainty theta of the SiC particle reinforced phase distributed in the Al matrix along the thickness direction (namely the y direction) can obtain sufficient sample information by measuring relevant parameters of an adding mechanism, so that the uncertainty is also treated as bounded probability uncertainty; further, the above bounded probability uncertainties are all described by random variables obeying generalized beta distribution (GBeta distribution), and the parameter information of each uncertainty variable is summarized as shown in table 1:
TABLE 1 summary table of uncertainty parameter information of external connecting rod
Figure BDA0002432132290000121
For interval variables, the uncertainty parameters are interval midpoint and radius; for bounded probability variables, the uncertainty parameters are their mean and standard deviation;
2. discretizing the design domain of the outer connecting rod, specifically;
the outer connecting rod of the high-speed press is generally small in external load component force along the axial direction of the hinge hole in the actual working process, so that the stress condition of the outer connecting rod of the high-speed press is simplified into a two-dimensional plane stress state, meanwhile, the details of parts such as chamfers, fillets and the like are removed to improve the calculation efficiency, the simplified outer connecting rod is placed in a regular rectangular design domain (the range enclosed by a dashed frame in figure 4, the size of the simplified outer connecting rod is X × Y which is 355mm × 180mm), and the rectangular design domain is divided into Nx×NyA square unit of Nx，NyAre the number of divisions along the x and y axes, respectively, and in this design, N is takenx＝710、NyEach unit is a square with a side length of 0.5mm × 0.5.5 mm, each unit is assigned a unique design variable ρ, based on the classical penalized isotropic material (SMIP) framework in topology optimizatione∈[0,1](e＝1,2,…,710×360)；
3. Discretizing the volume distribution of the SiC particle reinforced phase in the Al365 matrix of the outer connecting rod specifically comprises the following steps:
3.1) according to the discrete design domain, a coordinate system is specified, the force transmission direction of the outer connecting rod is an x axis, and the coordinate system is set as shown in figure 4, so that the coordinates of each discrete unit node can be determined; in the particle reinforced material external connecting rod, the volume fraction of the particle reinforced phase in the matrix is layered and gradually changed, namely the volume fraction of the particle reinforced phase in the matrix is only changed along the direction vertical to the force transmission direction of the connecting rod, namely the direction of the y axis in the figure; because the toughness and wear resistance of the outer connecting rod are required to be enhanced, the deformation is reduced and the instability is avoided when the high-speed press works practically, the theoretical expression Vol (y) of the volume fraction of the particle reinforced phase is selected as follows according to the practical use requirement:
Vol(y)＝20％·cos(πy/Y) Eq.33
wherein, Y is 180mm which is the length of the Y-axis direction of the rectangular design domain; the SiC particle reinforced phase has the largest theoretical volume fraction at the center of the outer connecting rod and gradually reduces to 0 towards two sides, and the volume fraction of the particle reinforced phase in the same thickness is constant; the theoretical volume fraction change mode is a continuous function of a y coordinate, and needs to be further discretized;
3.2) calculating the actual average volume fraction Vol (y) of the particle-reinforced phase in the ith (i-1, 2, …,360) layer according to the 710 × 360 unit design domain and the theoretical volume fraction change mode Vol (y) of the particle-reinforced phase which are dispersed in the step 2*(i) The following were used:
Figure BDA0002432132290000131
in the formula Eq.34, θ is the uncertainty of the volume fraction of the particle-enhanced phase; the volume fractions of the particle-enhanced phases of all the units in the i-th (i-1, 2, …,360) layer were all the above calculated Vol*(i) (i ═ 1,2, …,360), as a function of layer number i; so far the particle-enhanced phase volume fraction has been discretized;
3.3) calculating the Young's modulus of each unit in the ith (i ═ 1,2, …,360) layer using the Halpin-Tsai microstructure model
Figure BDA0002432132290000147
The physical parameters include:
3.3.1) the physical properties of the added SiC reinforcing particles are: average length of particles lG1 μm, average width wG0.4 μm and average thickness tG0.4 μm, Young's modulus EG400GPa and Poisson's ratio vG＝0.17；
3.3.2) define the following parameters:
Figure BDA0002432132290000141
3.3.3) young's modulus of each unit in ith (i ═ 1,2, …,360) layer
Figure BDA0002432132290000142
The following (in GPa) is calculated:
Figure BDA0002432132290000143
3.3.4) the poisson ratio of each unit in the ith (i ═ 1,2, …,360) layer is calculated as follows:
Figure BDA0002432132290000144
3.4) introducing a penalty factor p of 3 based on the classical isotropic material with penalty (SMIP) framework in topological optimization, and specifying a minimum allowable Young's modulus Emin1E-3GPa, the young's modulus of each unit in the i-th (i-1, 2, …,360) layer can be ultimately expressed as:
Figure BDA0002432132290000145
4. according to a classical finite element mode, physical constraints and geometric constraints are applied to the discretized structure, and the method specifically comprises the following steps:
4.1) geometrical constraint: in the original design of the outer connecting rod in FIG. 3, hinge holes phi 60 and phi 110 are reserved, and two holes (region I in FIG. 4) are provided with no material, that is, the design variable rho corresponding to all units in the holese0 [ identical to ] or; in the original design of the outer connecting rod in FIG. 3, two connecting sleeves (the left sleeve with an inner diameter phi 60 and an outer diameter phi 90, and the right sleeve with an inner diameter phi 110 and an outer diameter phi 180) are used as ring structure retaining materials, and design variables rho corresponding to all units in the two ring areas (area II in FIG. 4) are sete1 [ identical to ] or; the topology of the connecting plate between the two hinge holes is the object of the design optimization, so the robust topology optimization of the outer connecting rod is actually performed on the area excluding I and II in the design domain in fig. 4, that is, the connecting plate between the two hinge holes; after the setting is completed, as shown in fig. 4;
4.2) physical constraints: according to the classical finite element method framework, the unit corresponding to the inner wall of the phi 60 hinge hole in the figure 3 is set as a fixed constraint, the phi 110 hinge hole in the figure 3 applies a horizontal leftward radial force with an uncertainty amplitude f and an uncertainty direction angle at the center of the radial forceAfter the setting is finished asFIG. 4 is a schematic illustration;
5. taking the yield of the outer connecting rod structure as an optimization target, taking the mean value and the standard deviation of the structure yield under the worst working condition as the optimization target, and establishing a robust topological optimization design model of the particle reinforced material outer connecting rod considering the mixed uncertainty of the interval and the bounded probability as follows:
Figure BDA0002432132290000151
where ρ is (ρ)1,ρ2,…,ρ710×360)TIs (710 × 360) × 1 dimension design vector, ρmin0.001 is the minimum allowable value for each design variable given in the design; x ═ p (p)Al,νAl,θ)TIs a 3 × 1-dimensional bounded probability uncertainty vector;
Figure BDA0002432132290000152
is a 2 × 1 dimensional interval uncertainty vector;
Figure BDA0002432132290000153
is the volume of the structure corresponding to the current design vector ρ; v0710 × 360 is the design domain volume;
Figure BDA0002432132290000154
is the maximum allowable value of the structural volume given in the design;
k (x) U ═ f (i) is the outer link balance equation, where k (x) is a 2(710 × 360) × 2(710 × 360) dimensional global stiffness matrix, f (i) is a 2(710 × 360) × 1 dimensional nodal force matrix, U is a 2(710 × 360) × 1 dimensional nodal displacement matrix;
Figure BDA0002432132290000155
the yield value of the outer connecting rod made of the SiC particle reinforced Al material under the action of the interval uncertain vector I under the worst working condition is obtained;
Figure BDA0002432132290000156
respectively for worst case conditions under the influence of a bounded probabilistic uncertainty vector XYield value
Figure BDA0002432132290000157
Mean and standard deviation of;
the worst working condition of the SiC particle reinforced Al material outer connecting rod under the action of the interval uncertain vector I is determined by the following steps:
A) according to the classical finite element method, the yield of a structure taking into account both the interval and bounded probability uncertainty effects can be written as:
c(ρ,X,I)＝UTK(X)U＝F(I)TK(X)-1F(I) Eq.40
B) definition of
Figure BDA0002432132290000158
To obtain a constant vector by taking the mean of each probability variable in the bounded probabilistic uncertainty vector X, we call μXIs a mean vector of a bounded probabilistic uncertainty vector X, where
Figure BDA0002432132290000159
μθRespectively uncertainty rhoAl,νAlThe mean of θ; let bounded probability uncertainty vector X ═ μ in structure yield c (ρ, X, I)XThen at this time, the structural yield only includes the influence of the interval uncertainty I, which can be written as c (ρ, μ)XI) ═ c (ρ, I); meanwhile, the overall rigidity matrix is also a constant matrix and can be written as K (mu)X)＝K；
C) Writing a node moment array into the form of the sum of the force vectors of all externally-loaded nodes, wherein the connecting rod only contains one uncertain externally-loaded node, so that the method comprises the following steps:
Figure BDA00024321322900001510
simultaneously, the method comprises the following steps:
Figure BDA0002432132290000161
in the formula (I), the compound is shown in the specification,
Figure BDA0002432132290000162
respectively is the amplitude component of the external load F along the x-axis direction and the y-axis direction; e.g. of the typex，eyUnit node force vectors corresponding to the node acted by the external load F along the directions of x and y axes;
D) according to the linear elasticity assumption, there are:
Figure BDA0002432132290000163
in the above equation, the amplitude and the direction angle of the uncertain load are respectively derived:
Figure BDA0002432132290000164
respectively order the obtained derivative information
Figure BDA0002432132290000165
Solving to obtain the worst working condition of the particle reinforced material outer connecting rod under the action of the interval uncertain vector I
Figure BDA0002432132290000166
Yield value of worst working condition
Figure BDA0002432132290000167
Yield value of worst working condition under influence of bounded probability uncertainty vector X
Figure BDA0002432132290000168
The mean and standard deviation of (a) are determined by:
a) reduction of
Figure BDA0002432132290000169
Mu inXFor the bounded probabilistic uncertainty vector X, the yield value of the worst case at this time may be written as
Figure BDA00024321322900001610
Including probability uncertainty;
b) yield value of worst working condition by univariate dimension reduction method of multivariable function
Figure BDA00024321322900001611
Can be developed by the following formula:
Figure BDA00024321322900001612
in the formula, X＜i＞(i ═ 1,2,3) are as follows:
Figure BDA00024321322900001613
c) according to the expansion in b), for calculating the yield value of the worst working condition
Figure BDA00024321322900001614
The high-dimensional integral of the first-order and second-order origin moments can be converted into the operation of a plurality of one-dimensional integrals:
Figure BDA0002432132290000171
Figure BDA0002432132290000172
phi (X) in the formulae Eq.47, Eq.48i) Is the probability uncertainty Xi(i ═ 1,2,3) probability density function;
each one-dimensional integral in the above equation was calculated using the laguer (L aguerre) integral format:
Figure BDA0002432132290000173
wherein t is the number of points integrated by Laguerre (L aguerre), and t is 6, and x(j),λ(j)(j ═ 1,2, …,6) are the standard integration points and their corresponding weights, respectively, given by the laguerre (L aguerre) integration rule;
d) worst caseYield value
Figure BDA0002432132290000174
The mean and standard deviation of (a) can be obtained by:
Figure BDA0002432132290000175
6. an optimal criterion method (OC method) is adopted to iteratively solve the SiC particle reinforced Al material outer connecting rod robust optimization design model considering the mixed uncertainty of the interval and the bounded probability:
6.1) initializing design variables to
Figure BDA0002432132290000176
The following takes the 1 st iteration as an example to illustrate the direct solution process of the particle-reinforced connecting rod based on the optimal criterion method:
6.2) defining the weighted objective function J (p) for achieving
Figure BDA0002432132290000177
The double-target optimization:
Figure BDA0002432132290000178
6.3) target function and constraint function for each design variable rhoe(e ═ 1,2, …,710 × 360) gradients can be calculated by the following equations, respectively:
Figure BDA0002432132290000181
Figure BDA0002432132290000182
6.3.1) to obtain the objective function gradient, we need to substitute Eq.50 into Eq.52, we get:
Figure BDA0002432132290000183
6.3.2) gradient term in Eq.52
Figure BDA0002432132290000184
(e-1, 2, …,710 × 360) the specific calculation formula is as follows;
Figure BDA0002432132290000185
Figure BDA0002432132290000186
respective gradient terms in the formulae Eq.55, Eq.56
Figure BDA0002432132290000187
(e ═ 1,2, …,710 × 360) given by the classical topology optimization framework SIMP:
Figure BDA0002432132290000188
in the formula Eq.57, the compound,
Figure BDA0002432132290000189
is that
Figure BDA00024321322900001810
The reason for this is that the bounded probability uncertainty vectors in the aforementioned 4 terms are all constant values; k is a radical ofeThe element stiffness matrix of the element e can be calculated by a classical finite element theory; u. ofeThe element displacement matrix of the element e can be extracted from a node displacement matrix U obtained by solving finite element equations of each time through a classical finite element theory;
6.3.3) all the calculation results are substituted into an objective function gradient expression Eq.3-22 to obtain a final objective function gradient result, and the following steps are intercepted:
Figure BDA0002432132290000191
and has an objective function value J (rho) of 2054.920 mm;
6.4) obtaining gradient information of the objective function and the obtained constraint function according to the obtained objective function
Figure BDA0002432132290000192
Updating design variables, and intercepting the parts as follows:
ρ1＝0.294,ρ1＝0.320,…,ρ710×360-1＝0.215,ρ710×360＝0.186 Eq.59
6.5) check the difference between the objective function value in this iteration and the objective function value in the previous iteration, and repeat steps 6.1) through 6.5) since it is the first iteration that is used for the presentation, which is defined as the current objective function value 158.2399, and does not satisfy the convergence threshold of 0.01).
The final optimal solution is intercepted as follows:
ρ1＝0,…,ρ16247＝4.28E-31,…,ρ16252＝1,…,ρ710×360＝0 Eq.60
iterative optimization converges at the 379 th generation, and the volume fraction cloud images of the structure corresponding to the optimal solution and the enhanced phases in the structure are shown in fig. 5; the structural properties of the optimal solution are
Figure BDA0002432132290000193
The worst working condition corresponding to the optimal solution is
Figure BDA0002432132290000194
This value can be used for further engineering analysis; the design requirement and the working requirement of the robustness of the high-speed press outer connecting rod facing the particle reinforced material are met, so that the effectiveness of the proposed method is verified; after the contour of the robust topology optimization result is smoothed and the structures of the parts such as the chamfer and the fillet are arranged according to actual needs, the finally obtained design drawing of the outer connecting rod is shown in fig. 6.
It should be noted that the summary and the detailed description of the invention are intended to demonstrate the practical application of the technical solutions provided by the present invention, and should not be construed as limiting the scope of the present invention. Any modification and variation of the present invention within the spirit of the present invention and the scope of the claims will fall within the scope of the present invention.

Claims (6)

1. A method for robust topology optimization of a particulate reinforced material structure taking into account mixing uncertainties, characterized in that it comprises the following steps:
1) the following uncertainties in the manufacture and use of the component based on the particulate reinforcing material are considered: the material property of the base material of the component, the material property of the particle reinforced phase, the volume fraction of the particle reinforced phase distributed in the base, and the amplitude and the direction of the external load applied to the component; wherein, because it is difficult to obtain sufficient sample information about the external load, the amplitude and direction uncertainty of the external load are treated as interval uncertainty; regarding the matrix material property, the particle reinforced phase material property and the volume fraction of the particle reinforced phase distributed in the matrix with sufficient sample information as bounded probability uncertainty processing, and describing each bounded probability uncertainty parameter by adopting a random variable obeying generalized beta distribution (GBeta distribution);
2) discretizing a component design domain, specifically:
the stress condition of the component is simplified to be a two-dimensional plane stress state, the holes of the component are reserved, and meanwhile, the detailed geometric shapes are removed to improve the calculation efficiency; placing the simplified component in a regular rectangular design domain, and dividing the rectangular design domain into Nx×NyA square unit of Nx，NyThe division numbers along the directions of the x axis and the y axis are respectively; based on the classical isotropic material with penalty (SMIP) framework in topological optimization, each unit is assigned with a unique design variable rhoe∈[0,1](e＝1,2,…,NxNy)；
3) Discretizing the volume distribution of the particle reinforced phase in the component matrix, specifically:
3.1) according to a coordinate system specified by the discrete design domain, and taking the force transmission direction of the component as an x axis, each unit coordinate can be uniquely determined; in the particle reinforced material member, the volume fraction of the particle reinforced phase in the matrix is in layered gradual change, namely the volume of the particle reinforced phase in the matrixThe fraction is changed only along the force transmission direction (namely the y-axis direction) vertical to the component, the theoretical change mode is recorded as Vol (y), the change mode is a continuous function of a coordinate y, the selection is carried out according to the actual use requirement of the component, and the volume fraction of the particle reinforced phase in the same thickness is a constant; considering the uncertainty fluctuation of the volume fraction of the particle-reinforced phase in the actual production, the actual change mode is recorded as Vol*(y) with bounded probabilistic uncertainty;
3.2) calculating the ith (i-1, 2, …, N) according to the discrete component design domain and the volume fraction change mode of the particle reinforced phasey) Mean volume fraction Vol of intralamellar, particle-reinforced phase*(i) (ii) a The volume fractions of the particle-enhanced phases of all the units in the i layer are Vol*(i) Is a discrete function of layer number i;
3.3) calculation of ith (i ═ 1,2, …, N) using the hallin-Tsai microstructure modely) Young's modulus of each unit in layer
Figure FDA0002432132280000011
To poisson's ratio
Figure FDA0002432132280000012
3.4) introduce penalty factors based on the classical isotropic material with penalty (SMIP) framework in topology optimization, then the ith (i ═ 1,2, …, Ny) The young's modulus of each cell within a layer can ultimately be expressed as:
Figure FDA0002432132280000013
in the formula Eq.1, the compound,
Figure FDA0002432132280000014
Eminrespectively, 3.3) th calculated i (i ═ 1,2, …, Ny) The Young modulus of each unit in the layer and the preset minimum allowable Young modulus value are constants; p is a constant penalty factor specified in advance;
Figure FDA0002432132280000021
for the ith (i ═ 1,2, …, N) construction based on penalized isotropic material (SMIP) frameworksy) Young's modulus of each unit in the layer;
4) applying physical constraints and geometric constraints to the discretized structure, specifically:
4.1) physical constraints including the fixation or support of the structure, external loads, applied according to the classical finite element method;
4.2) geometric constraints including the specified holes in the structure and the areas where the material is forcibly retained by setting the design variable ρ for the cell covered by the holee≡ 0 and the design variable position ρ corresponding to the cell that requires forced retention of material position coveragee1 and does not change its value during the subsequent optimization;
5) the structural yield of the particle reinforced material member is an optimized design target, the structural yield under the joint influence of the interval and bounded probability mixing uncertainty is considered in a robust topological optimization framework, so the yield mean value and the standard deviation of the member under the worst working condition can be used as the representation of the optimized design target, and a robust optimized design model of the particle reinforced material member under the condition of considering the interval and bounded probability mixing uncertainty is established as shown in Eq.2:
Figure FDA0002432132280000022
in the formula Eq.2, the compound,
Figure FDA0002432132280000023
is (N)xNy) × 1 dimension design vector, ρminThe minimum allowable value of each design variable given in the design is a constant; x ═ X (X)1,X2,…,Xm)TIs a m × 1-dimensional bounded probability uncertainty vector whose elements include the material properties of the component matrix material, the material properties of the particle-enhanced phase, the uncertainty of the volume distribution of the particle-enhanced phase in the matrix material, and I ═ f1,f2,…,fn,α1,α2,…,αn)TIs 2n × 1 dimensional interval uncertainA property vector of which f1,f2,…,fnRespectively the amplitude of n uncertain external loads borne by the component, α1,α2,…,αnRespectively representing n direction angles of uncertain external load borne by the component;
Figure FDA0002432132280000024
is the volume of the structure corresponding to the current design vector ρ; v0＝NxNyIs the volume of the design domain;
Figure FDA0002432132280000025
is the volume fraction of the designed structure occupying the designed domain, which is a constant;
k (x) U ═ f (i) is the member balance equation, where k (x) is (2N)xNy)×(2NxNy) Maintaining an overall stiffness matrix, and constructing by using a classical finite element theory, wherein a specific numerical value of the overall stiffness matrix is influenced by a bounded probability uncertain vector X; f (I) is (2N)xNy) × 1 dimensional node force matrix is constructed by classical finite element theory, and its specific value is influenced by interval uncertainty vector I, U is (2N)xNy) × 1 dimension node displacement matrix;
Figure FDA0002432132280000031
the yield value of the worst working condition of the particle reinforced material component under the action of the interval uncertain vector I is calculated as follows:
A) the structural yield, which simultaneously considers the effects of interval and bounded probability uncertainty, can be written as eq.3 according to the classical finite element method:
c(ρ,X,I)＝UTK(X)U＝F(I)TK-1(X)F(I) Eq.3
B) definition of
Figure FDA0002432132280000032
To obtain a constant vector by taking the mean of each probability variable in the bounded probabilistic uncertainty vector X, we call μXIs a mean vector of a bounded probabilistic uncertainty vector X, where
Figure FDA0002432132280000033
Respectively, each uncertainty X1,X2,…,XmThe mean value of (a); let bounded probability uncertainty vector X ═ μ in structure yield c (ρ, X, I)XThen at this time, the structural yield only includes the influence of the interval uncertainty I, which can be written as c (ρ, μ)XI) ═ c (ρ, I); meanwhile, the overall rigidity matrix is also a constant matrix and can be written as K (mu)X)＝K；
C) Writing the node moment array into the form of the sum of the force vectors of all the externally-loaded nodes:
Figure FDA0002432132280000034
simultaneously, the method comprises the following steps:
Figure FDA0002432132280000035
in the formula Eq.5, fix＝ficosαi，fiy＝fisinαiRespectively an external load FiA magnitude component along the x, y directions; e.g. of the typeix，eiyRespectively corresponding to an external load FiA unit node force vector of the acting node along the x and y axis directions;
D) according to the linear elasticity assumption, the overall effect of n uncertain loads can be equivalent to the superposition of the effects of the individual effects of each load:
Figure FDA0002432132280000036
in eq.6, the amplitude and the direction angle of the uncertain load are differentiated, respectively, to give (i ═ 1,2, …, n):
Figure FDA0002432132280000037
according to Eq.7 the derivative information obtained respectively
Figure FDA0002432132280000038
Solve to obtain
Figure FDA0002432132280000039
Figure FDA00024321322800000310
I.e. the worst condition of the particle reinforced material member under the action of the interval uncertainty vector I
Figure FDA00024321322800000311
The yield value of the worst working condition can be written as
Figure FDA00024321322800000312
In the formula Eq.2, the compound,
Figure FDA00024321322800000313
respectively the yield value of the worst working condition under the influence of the bounded probability uncertainty vector X
Figure FDA00024321322800000314
The mean and standard deviation of (a) are calculated as follows:
a) reduction of
Figure FDA0002432132280000041
Mu inXFor the bounded probabilistic uncertainty vector X, the yield value of the worst case at this time may be written as
Figure FDA0002432132280000042
Including probability uncertainty;
b) yield value of worst working condition through single variable dimension reduction method of Rahman multivariable function
Figure FDA0002432132280000043
Can be developed by the following formula:
Figure FDA0002432132280000044
in equation Eq.8, m is the number of bounded probability uncertainty parameters contained in a bounded probability uncertainty vector X, X＜i＞(i ═ 1,2, …, m) defined by eq.9:
Figure FDA0002432132280000045
x in the formula Eq.9iIs the ith bounded probability uncertainty variable;
c) according to the expansion Eq.8, calculating the yield value of the worst working condition
Figure FDA0002432132280000046
The high-dimensional integral of the first-order and second-order origin moments can be converted into the operation of a plurality of one-dimensional integrals:
Figure FDA0002432132280000047
Figure FDA0002432132280000048
phi (X) in the formulae Eq.10, Eq.11i) Is the probability uncertainty Xi(i ═ 1,2, …, m) probability density function;
d) each one-dimensional integral in the equations eq.10, eq.11 is calculated using the laguerre (L aguerre) integral format:
Figure FDA0002432132280000049
in the equation Eq.12, t is the number of integration points of Laguerre (L aguerre) and is a predetermined constant, x(j),λ(j)(j ═ 1,2, …, t) are the standard integration points and their corresponding weights, respectively, given by the laguerre (L aguerre) integration rule,are all constants;
Figure FDA00024321322800000410
by reaction at X＜i＞Intermediate order bounded probability uncertainty variable XiAre respectively as
Figure FDA0002432132280000051
Is obtained by
Figure FDA0002432132280000052
Wherein
Figure FDA0002432132280000053
Are also all constants, calculated by equiprobable transformation of each integration point, i.e.
Figure FDA0002432132280000054
e) Worst condition yield value
Figure FDA0002432132280000055
The mean and standard deviation of (d) can be obtained by eq.13:
Figure FDA0002432132280000056
6) iteratively solving a robust optimization design model of the particle reinforced material component with the mixture uncertainty of the considered interval and the bounded probability of Eq.2 by adopting a standard optimization criterion method (OCmethod) in topological optimization, wherein the calculation process in each iteration process specifically comprises the following steps:
6.1) defining a weighted objective function in Eq.14 for achieving
Figure FDA0002432132280000057
The double-target optimization:
Figure FDA0002432132280000058
in equation Eq.14, J (ρ) is a defined weighting objective function; w is a weight value which is specified in advance and is a constant;
6.2) respectively calculating the target function and the constraint function for each design variable rho according to Eq.15 and Eq.16e(e＝1,2,…,NxNy) Gradient:
Figure FDA0002432132280000059
Figure FDA00024321322800000510
6.3) updating the current design variable according to the gradient information of the objective function and the constraint function and the standard optimal criterion method;
6.4) checking the difference value between the objective function value in the current iteration and the objective function value in the previous iteration, wherein for the first iteration, the difference value is defined as the objective function value of the first generation, and if the difference value is smaller than a pre-specified convergence threshold value, the convergence condition is met and an updated design variable is output; otherwise, repeating the steps 6.1) to 6.4).
2. The method for robust topology optimization of particle-reinforced material members taking into account mixing uncertainty as claimed in claim 1, wherein in step 1), the volume fraction of the particle-reinforced phase distributed in the matrix with sufficient sample information is treated as bounded probability uncertainty, as follows:
the volume fraction of the particle-reinforced phase in the matrix is changed only along the thickness direction (i.e. the y-axis direction) of the component, and the change mode has a theoretical design expression Vol (y), but the volume fraction of the particle-reinforced phase has uncertainty due to factors such as the hysteresis of a reinforcing particle adding port control system and the like in the actual manufacturing process and can be represented by the product of a deterministic theoretical expression Vol (y) and an uncertain parameter, as shown in Eq.17:
Vol*(y)＝Vol(y)·θ Eq.17
in the formula Eq.17, Vol*(y) is the manner in which the volume fraction of the particulate reinforcing phase changes during actual manufacture; theta is an uncertain parameter reflecting fluctuation of the actual volume fraction of the enhanced phase relative to a theoretical design value, is called as the volume fraction uncertainty of the particle enhanced phase, and uncertain information of the uncertain parameter can be obtained by measuring relevant parameters (feeding port response time lag, adding mechanism friction resistance and the like) of the adding mechanism.
3. A method for robust topology optimization of a particle-reinforced material member taking into account mixing uncertainty according to claim 1, wherein in step 1), random variables obeying a generalized beta distribution (GBeta distribution) are used to describe each bounded probability uncertainty parameter as follows:
1.1) for bounded probability uncertainty parameter XiS samples of the parameter are obtained through experiments, and then a sample set is constructed
Figure FDA0002432132280000061
From this sample set, the parameter X is calculated in Eq.18iValue range of (1), calculating parameter X according to Eq.19iMean and variance of (a):
Figure FDA0002432132280000062
Figure FDA0002432132280000063
1.2) describing distribution in [ a ] by adopting generalized beta distributioni,bi]Inner and mean and variance are respectively
Figure FDA0002432132280000064
Parameter X ofiFirst, the mean and variance are normalized as shown in Eq.20:
Figure FDA0002432132280000065
then, the parameter X is calculated using Eq.21iDistribution parameter α of the generalized beta distributioni,βi：
Figure FDA0002432132280000066
Recording parameter XiObey ini,bi]Internal and distribution parameter αi,βiGeneralized beta distribution of (i.e. X)i～GBeta(ai,bi|αi,βi) And the probability density function is shown as Eq.22:
Figure FDA0002432132280000071
in the formula Eq.22, fXi(. is) a parameter XiA probability density function of; (. cndot.) is a gamma function.
4. A method for robust topology optimization of particle-reinforced material components taking mixing uncertainty into account according to claim 1, characterized in that in said step 3.3), the ith (i ═ 1,2, …, N) is calculated using hallin-Tsai microstructure modely) Young's modulus of each unit in layer
Figure FDA0002432132280000072
To poisson's ratio
Figure FDA0002432132280000073
The method comprises the following specific steps:
3.3.1) the physical properties of the reinforcing particles are: average length of particles lGAverage width wGAnd an average thickness tGYoung's modulus EGEtc.;
3.3.2) define the following parameters:
Figure FDA0002432132280000074
in the formula Eq.23, EMIs the Young's modulus of the matrix material;
3.3.3) th i (i ═ 1,2, …, Ny) Young's modulus of each unit in layer
Figure FDA0002432132280000075
Calculated by eq.24:
Figure FDA0002432132280000076
in the formula Eq.24, Vol*(i) Is the ith (i ═ 1,2, …, Ny) The actual average volume fraction of the particle-enhanced phase within the layer, with probabilistic uncertainty;
3.3.4) th i (i ═ 1,2, …, Ny) The poisson ratio of each unit in the layer is calculated by Eq.25:
Figure FDA0002432132280000077
in the formula Eq.25, the compound,
Figure FDA0002432132280000078
νG、νMare respectively the ith (i ═ 1,2, …, Ny) The Poisson's ratio of each unit, reinforcing particle and matrix material in the layer.
5. A method for robust topology optimization of a particle-reinforced material member taking into account mixing uncertainty according to claim 1, characterized in that said step 6.2) is specified as follows:
6.2.1) substituting Eq.10, Eq.11 into Eq.15:
Figure FDA0002432132280000079
gradient term in equation Eq.26
Figure FDA0002432132280000081
Calculated for Eq.10, Eq.11, respectively, worst workYield value under the condition
Figure FDA0002432132280000082
First and second order origin moments vs. design variables ρe(e＝1,2,…,Nx·Ny) A gradient of (a);
6.2.2) gradient term
Figure FDA0002432132280000083
The specific calculation formula is shown as Eq.27 and Eq.28;
Figure FDA0002432132280000084
Figure FDA0002432132280000085
gradient terms in the equations Eq.27 and Eq.28
Figure FDA0002432132280000086
Given by the classical topology optimization framework SIMP:
Figure FDA0002432132280000087
in the formula Eq.29
Figure FDA0002432132280000088
Is that
Figure FDA0002432132280000089
The reason for this is that the bounded probability uncertainty vectors in the aforementioned 4 terms are all constant values; p is a constant penalty factor specified in advance; k is a radical ofeThe element stiffness matrix of the element e can be calculated by a classical finite element theory; u. ofeThe element displacement matrix of the element e can be extracted from a node displacement matrix U obtained by solving finite element equations of each time through a classical finite element theory;
6.2.3) substituting the calculated result of Eq.29 into Eq.27 and Eq.28, and further substituting the calculated result of Eq.26 and Eq.27 into an expression Eq.26 to obtain a final objective function gradient result.
6. A method for robust topology optimization of a particle reinforced material member taking into account mixing uncertainty according to claim 1, characterized in that said step 6.3) is specified as follows:
6.3.1) for iteratively determining an appropriate constraint function Lagrangian multiplier lambda for updating the design variable by using binary search (Bisection algorithmm), firstly, specifying the upper and lower bounds lambda of the value of the Lagrangian multiplier lambda1、λ21E-3, 1E9, respectively;
6.3.2) taking the current Lagrange multiplier as an average value of the upper and lower bounds of the value is shown as Eq.30:
λ＝λmiddle＝(λ1+λ2)/2 Eq.30
6.3.3) update the design variables by eq.31 (e ═ 1,2, …, NxNy)：
Figure FDA0002432132280000091
In the formula Eq.31, m is 0.2 and is the search movement step length, η is 0.5 and is the search damping coefficient, BeThe gradient information of the objective function and the constraint function is obtained, as shown in Eq.32:
Figure FDA0002432132280000092
6.3.4) calculating the volume of the structure corresponding to the updated design variable
Figure FDA0002432132280000093
And adjusting the value boundary of the Lagrange multiplier according to the violation condition of the constraint function: if it is
Figure FDA0002432132280000094
Contraction value from lower bound to λ1＝λmiddle(ii) a If it is
Figure FDA0002432132280000095
Contraction values from upper bound to λ2＝λmiddle；
6.3.5) check the convergence criterion (λ) of the binary search2-λ1)/(λ2+λ1) If the value is less than 1E-3, returning to repeat 6.3.2) to 6.3.5) to search the Lagrangian for the next time; if so, indicating that the current Lagrangian operator meets the requirement, and outputting the current design variable
Figure FDA0002432132280000096
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