CN111475976B - Robust topology optimization method for particle reinforced material member considering mixing uncertainty - Google Patents
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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 steady topology optimization design model established by the invention truly reflects the distribution characteristics of multisource uncertainty in actual engineering, and the provided 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 invention has good engineering application value.
Description
Technical Field
The invention belongs to the field of equipment structure optimization design, and relates to a method for optimizing the steady topology of a particle reinforced material component by considering mixing uncertainty.
Background
Topology optimization, 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 has 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 the 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 accurately controlled, 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 structural robust topological optimization by introducing geometric boundary uncertainty at a microscopic level of a lattice structure has certain difficulty, and part of the reason is 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
In order to solve the problem of the robust topology optimization design of the particle reinforced material member under the condition of coexistence of uncertain factors of a probability interval, the invention provides a robust topology optimization design method of the particle reinforced material member, which considers the mixed uncertainty of the interval and the bounded probability. Considering the uncertainty of the component based on the particle reinforced material in the manufacturing and using processes, regarding the insufficient external load of the sample as interval uncertainty, and regarding the sufficient component matrix property of the sample and the volume distribution of the particle reinforced phase in the matrix as bounded probability uncertainty; discretizing the volume distribution of a design domain and a particle reinforced phase in a component matrix, setting structural constraint and external load, and establishing a steady optimized design model of the particle reinforced material component considering the mixed uncertainty of an interval and a bounded probability; the model is solved iteratively using an optimal criterion method: firstly, decoupling probability interval mixed uncertainty, and determining worst working conditions by using optimized target gradient information; then, estimating the mean value and the variance of the optimization target under the worst working condition by using a univariate dimension reduction method and a Laguerre (Laguerre) integral format to construct a target function; and finally, calculating gradients of the target function and the constraint function respectively to update the design variables. Therefore, the problem of the robust topological optimization design of the particle reinforced material component under the condition of coexistence of uncertain factors of the probability interval is efficiently solved.
The invention is realized by the following technical scheme: a method for robust topology optimization of a particle reinforced material member taking into account mixing uncertainty, 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 uncertainties of the external load are treated as interval uncertainties; regarding the material property of a matrix material, the material property of a particle reinforced phase 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 the component design domain specifically comprises the following steps:
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; the simplified components are placed in a regular rectangular design domain, and the rectangular design domain is divided into N x ×N y A square unit of N x ，N y The 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 rho e ∈[0,1](e＝1,2,…,N x N y )；
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 phase y ) Average volume fraction Vol of intralaminar, particulate reinforcing phase * (i) (ii) a The volume fractions of the particle-reinforced 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 model y ) Young's modulus of each unit in layer
To poisson's ratio
3.4) introduce penalty factors based on the classical penalized isotropic material (SMIP) framework in topology optimization, then the ith (i ═ 1,2, …, N y ) The young's modulus of each unit within a layer can be ultimately expressed as:
in the formula Eq.1, the compound,
E min respectively, 3.3) th calculated i (i ═ 1,2, …, N y ) 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;
for the ith (i ═ 1,2, …, N) construction based on penalized isotropic material (SMIP) frameworks y ) 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 classical finite element method;
4.2) geometrical constraints including the specified holes in the structure and the areas where the material is forced to remain, by setting the design variable ρ corresponding to the cell covered by the hole e ≡ 0 and the design variable position ρ corresponding to the cell that requires forced retention of material position coverage e 1 and does not change its value in the subsequent optimization process;
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:
in the formula Eq.2, the compound,
is (N) x N y ) X 1-dimensional design vector, p min The minimum allowable value of each design variable given in the design is a constant; x ═ X (X) 1 ,X 2 ,…,X m ) T Is an m x 1-dimensional bounded probability uncertainty vector whose elements include material properties of the component matrix material, material properties of the particle-enhanced phase, uncertainty of the volume distribution of the particle-enhanced phase in the matrix material; i ═ f 1 ,f 2 ,…,f n ,α 1 ,α 2 ,…,α n ) T Is a 2n x 1 dimensional interval uncertainty vector, where f 1 ,f 2 ,…,f n Respectively the amplitude and alpha of n uncertain external loads borne by the component 1 ,α 2 ,…,α n Respectively representing n direction angles of uncertain external load borne by the component;
k (x) U ═ f (i) is the member balance equation, where k (x) is (2N) x N y )×(2N x N y ) 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) x N y ) The node force matrix is a multiplied by 1 dimension node force matrix and is constructed by using a classical finite element theory, and a specific numerical value of the node force matrix is influenced by an interval uncertain vector I; u is (2N) x N y ) X 1-dimensional node displacement matrix;
A) the yield of a structure considering both interval and bounded probability uncertainty effects can be written as eq.3 according to classical finite element methods:
c(ρ,X,I)＝U T K(X)U＝F(I) T K -1 (X)F(I) Eq.3
B) definition of
For a constant vector obtained by averaging each of the probability variables in the bounded probability uncertainty vector X, we call μ X Is a mean vector of the bounded probability uncertainty vector X, where
Are each uncertainty X 1 ,X 2 ,…,X m The mean value of (a); let bounded probability uncertainty vector X ═ μ in structure yield c (ρ, X, I) X Then at this time, the structural yield only includes the influence of the interval uncertainty I, which can be written as c (ρ, μ) X ,I)＝c(ρ,I) (ii) a Meanwhile, the overall rigidity matrix is also a constant matrix which 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:
simultaneously, the method comprises the following steps:
in the formula Eq.5, f ix ＝f i cosα i ，f iy ＝f i sinα i Respectively an external load F i Amplitude components along the x, y directions; e.g. of the type ix ，e iy Respectively corresponding to an external load F i A unit node force vector of the acting node along the x and y axis directions;
D) according to the linear elasticity assumption, the total action of n uncertain loads can be equivalent to the superposition of the effects of the individual actions of each load:
in eq.6, the amplitude and the direction angle of the uncertain load are differentiated, respectively, to give (i ═ 1,2, …, n):
respectively enabling the obtained derivative information according to Eq.7
To obtain a solution
(I-1, 2, …, n), i.e. the worst case of the particulate reinforcing material member under the influence of the interval uncertainty vector I
The yield value of the worst working condition can be written as
In the formula Eq.2, the compound,
respectively the yield value of the worst working condition under the influence of a bounded probability uncertainty vector X
The mean and standard deviation of (a) are calculated as follows:
a) reduction of
Mu in X For the bounded probabilistic uncertainty vector X, the yield value of the worst case at this time may be written as
Including a probability uncertainty;
b) yield value of worst working condition through single variable dimension reduction method of Rahman multivariable function
Can be developed by the following formula:
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:
x in the formula Eq.9 i Is the ith bounded probability uncertainty variable;
c) according to the expansion Eq.8, calculating the yield value of the worst working condition
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:
phi (X) in the formulae Eq.10, Eq.11 i ) Is the probability uncertainty X i (i ═ 1,2, …, m) probability density function;
d) each one-dimensional integral in equations eq.10, eq.11 is calculated using the Laguerre (Laguerre) integration format:
in the formula Eq.12, t is the number of Laguerre (Laguerre) integration points and is a constant specified in advance; x is a radical of a fluorine atom (j) ,λ (j) (j ═ 1,2, …, t) are the standard integration points and their corresponding weights, given by the Laguerre integration rule, all constant;
by reaction at X ＜i＞ Zhongjun bounded probability uncertainty variable X i Are respectively as
Is obtained by
Wherein
Are also all constants, calculated by equiprobable transformation of each integration point, i.e.
e) Worst condition yield value
The mean and standard deviation of (d) can be obtained by eq.13:
6) adopting a standard optimization criterion method (OC method) in topology optimization to iteratively solve a robust optimization design model of the particle reinforced material component of Eq.2 considering interval and bounded probability mixing uncertainty, 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
The double-target optimization:
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.16 e (e＝1,2,…,N x N y ) Gradient:
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), the volume fraction of the particle-reinforced phase distributed in the matrix with sufficient sample information is treated as a bounded probability uncertainty, which is as follows:
the volume fraction of the particle-reinforced phase in the matrix is changed only in the thickness direction (i.e. the y-axis direction) of the member in a manner of having a theoretical design expression Vol (y), but the volume fraction of the particle-reinforced phase has uncertainty in an actual manufacturing process due to factors such as hysteresis of a reinforcing particle addition port control system, and can be represented by a 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 theta can be obtained by measuring relevant parameters (feeding port response time lag, adding mechanism friction resistance and the like) of an 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) to bounded probability uncertainty parameter X i S samples of the parameter are obtained through experiments, and then a sample set is constructed
From this sample set, the parameter X is calculated in Eq.18 i Value range of (1), calculating parameter X according to Eq.19 i Mean and variance of (c):
1.2) describing distribution in [ a ] by adopting generalized beta distribution i ,b i ]Inner and mean and variance are respectively
Parameter X of i First, the mean and variance are normalized as shown in Eq.20:
then, the parameter X is calculated using Eq.21 i Of the generalized beta distribution of (a) i ,β i ：
Recording parameter X i Obey in i ,b i ]Internal and distribution parameter of alpha i ,β i Generalized beta distribution of (i.e. X) i ～GBeta(a i ,b i |α i ,β i ) And the probability density function is shown as Eq.22:
in the formula Eq.22, the compound,
is a parameter X i A probability density function of; Γ () is a gamma function.
Further, in the step 3.3), the ith (i ═ 1,2, …, N) is calculated using a hallin-Tsai microstructure model y ) Young's modulus of each unit in layer
To poisson's ratio
The method comprises the following specific steps:
3.3.1) the physical properties of the reinforcing particles are: average length of particles l G Average width w G And an average thickness t G Young's modulus E G Etc.;
3.3.2) define the following parameters:
in the formula Eq.23, E M Is the young's modulus of the matrix material;
3.3.3) th (i ═ 1,2, …, N y ) Young's modulus of each unit in layer
Calculated by eq.24:
in the formula Eq.24, Vol * (i) Is the ith (i ═ 1,2, …, N y ) The actual average volume fraction of the particle-enhanced phase within the layer, with probabilistic uncertainty;
3.3.4) th i (i ═ 1,2, …, N y ) The poisson ratio of each unit in the layer is calculated by Eq.25:
in the formula Eq.25, the compound,
ν G 、ν M are respectively the ith (i is 1,2, …, N) y ) 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:
gradient term in equation Eq.26
Worst case yield values calculated for Eq.10 and Eq.11, respectively
First and second order moments of origin for each design variable ρ e (e＝1,2,…,N x ·N y ) A gradient of (a);
6.2.2) gradient term
The specific calculation formula is shown as Eq.27 and Eq.28;
gradient terms in the equations Eq.27 and Eq.28
SIMP by classical topological optimization frameworkAnd (3) discharging:
in the formula Eq.29
Is that
The reason for this is that the bounded probability uncertainty vectors in the aforementioned 4 items are all constant values; p is a constant penalty factor specified in advance; k is a radical of e The element stiffness matrix of the element e can be calculated by a classical finite element theory; u. of e The 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 Lagrange multiplier lambda for updating design variables by using binary search (Bisection algorithm), firstly, specifying the upper and lower bounds of the values of the Lagrange multiplier lambda 1 、λ 2 1E-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＝λ middle ＝(λ 1 +λ 2 )/2 Eq.30
6.3.3) update design variables by eq.31 (e ═ 1,2, …, N) x N y )：
In equation eq.31, m ═ 0.2 is the search move step; η ═ 0.5 is the search damping coefficient; b is e Obtaining gradient information of an objective function and a constraint function, as shown in Eq.32:
6.3.4) calculating the volume of the structure corresponding to the updated design variable
And adjusting the value boundary of the Lagrange multiplier according to the violation condition of the constraint function: if it is
Contraction value from lower bound to λ 1 ＝λ middle (ii) a If it is
Contraction taking an upper bound to λ 2 ＝λ middle ；
6.3.5) check the convergence condition (λ) of the binary search 2 -λ 1 )/(λ 2 +λ 1 ) 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
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 expressed by the 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 mixed uncertainty, the invention introduces an elastic deformation hypothesis of a lead-in wire, namely the final deformation of the component under the action of uncertain external loads in a plurality of intervals can be obtained by superposing the deformations 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 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 update and optimization.
Drawings
FIG. 1 is a flow chart of a method for robust topology optimization of a particulate reinforced material member 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 the result of the outer link robust topology optimization design.
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 in 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
Processing as interval uncertainty;
1.2) Material Properties (Young's modulus E) of the base Material Al365 of the outer connecting rod Al V ratio to Poisson Al ) 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 using 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
For interval variables, the uncertainty parameters are interval midpoint and radius; for a bounded probability variable, the uncertainty parameters are its mean and standard deviation;
2. discretizing the design domain of the outer connecting rod, specifically;
the external connecting rod of the high-speed press generally has small external load component force along the axial direction of the hinge hole in the actual working process, so that the stress condition of the external connecting rod is simplified into a two-dimensional plane stress state, and meanwhile, the details of parts such as chamfers, fillets and the like are removed to improve the calculation efficiency; the simplified outer link is placed in a regular rectangular design domain (the range enclosed by the dotted line frame in fig. 4, the size of which is X × Y355 mm × 180mm), and the rectangular design domain is divided into N x ×N y A square unit of N x ，N y Are the number of divisions along the x and y axes, respectively, and in this design, N is taken x ＝710、N y Each unit is a square with a side length of 0.5mm × 0.5 mm; based on the classical isotropic material with penalty (SMIP) framework in topological optimization, each unit is assigned with a unique design variable rho e ∈[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 a particle reinforced phase in a 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 theoretical volume fraction of SiC particle reinforced phase at the center of the outer connecting rod is the largest, the theoretical volume fraction is gradually reduced to 0 towards two sides, and the volume fraction of the particle reinforced phase in the same thickness is a 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 i (i ═ 1,2, …,360) th layer according to the 710 × 360 unit design domain and the theoretical particle-reinforced phase volume fraction variation mode Vol (y) which are dispersed in the step 2 * (i) The following:
in equation Eq.34, θ is the uncertainty of the volume fraction of the particle enhancing phase; vol calculated above for all units in the ith (i: 1,2, …,360) layer * (i) (i-1, 2, …,360) is a function of layer number i; so far the volume fraction of the particulate reinforcement phase 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
The physical parameters include:
3.3.1) the physical properties of the added SiC reinforcing particles are: average length of particles l G 1 μm, average width w G 0.4 μm and average thickness t G 0.4 μm, Young's modulus E G 400GPa and Poisson's ratio v G ＝0.17；
3.3.2) define the following parameters:
3.3.3) young's modulus of each unit in ith (i ═ 1,2, …,360) layer
The following is calculated (in GPa):
3.3.4) poisson's ratio of each unit in the ith (i ═ 1,2, …,360) layer is calculated as follows:
3.4) introduce a penalty factor p of 3 based on the classical penalized isotropic material (SMIP) framework in topology optimization, and specify a minimum allowable Young's modulus E min 1E-3GPa, the young's modulus of each unit in the i-th (i-1, 2, …,360) layer can be ultimately expressed as:
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, phi 60 and phi 110 hinged holes are reserved, and two holes (region I in FIG. 4) are provided with no material, i.e. the design variables rho corresponding to all units in the holes e 0 [ identical to ] or; in the original design of the outer connecting rod in FIG. 3, two connecting sleeves (the inner diameter phi 60 and the outer diameter phi 90 of the left sleeve and the inner diameter phi 110 and the outer diameter phi 180 of the right sleeve) are arranged as ring-shaped structure retaining materials, and design variables rho corresponding to all units in the two ring areas (area II in FIG. 4) are set e 1 [ 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 a classical finite element method framework, a unit corresponding to the inner wall of a phi 60 hinge hole in figure 3 is set as a fixed constraint, and a unit corresponding to the inner wall of a phi 110 hinge hole in figure 3The center applies a horizontal leftward radial force having an uncertainty magnitude f and an uncertainty direction angle
After the setting is finished, as shown in FIG. 4;
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:
where ρ is (ρ) 1 ,ρ 2 ,…,ρ 710×360 ) T Is a (710 × 360) × 1-dimensional design vector, ρ min 0.001 is the minimum allowable value of each design variable given in the design; x ═ p (p) Al ,ν Al ,θ) T Is a 3 x 1 dimensional bounded probability uncertainty vector;
is a 2 x 1 dimensional interval uncertainty vector;
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 node displacement matrix;
is a SiC particle sizeThe yield value of the worst working condition of the external connecting rod made of the strong Al material under the action of the interval uncertain vector I;
respectively the yield value of the worst working condition under the influence of the bounded probability uncertainty vector X
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)＝U T K(X)U＝F(I) T K(X) -1 F(I) Eq.40
B) definition of
To obtain a constant vector by taking the mean of each probability variable in the bounded probabilistic uncertainty vector X, we call μ X Is a mean vector of the bounded probability uncertainty vector X, where
μ θ Respectively uncertainty rho Al ,ν Al The mean of θ; let bounded probability uncertainty vector X ═ μ in structure yield c (ρ, X, I) X Then at this time, the structural yield only includes the influence of the interval uncertainty I, which can be written as c (ρ, μ) X I) ═ c (ρ, I); meanwhile, the overall rigidity matrix is also a constant matrix which can be written as K (mu) X )＝K；
C) Writing a node moment array into a form of summation of force vectors of all external load nodes, wherein the connecting rod only contains one uncertain external load, so that the connecting rod has the following components:
simultaneously, the method comprises the following steps:
in the formula (I), the compound is shown in the specification,
respectively the amplitude components of the external load F along the directions of the x axis and the y axis; e.g. of the type x ，e y Respectively corresponding to the unit node force vectors of the node acted by the external load F along the directions of the x axis and the y axis;
D) according to the linear elasticity assumption, there are:
in the above equation, the amplitude and the direction angle of the uncertain load are respectively derived:
respectively order the obtained derivative information
Solving to obtain the worst working condition of the particle reinforced material outer connecting rod under the action of the interval uncertain vector I
Yield value of worst working condition
Yield value of worst working condition under influence of bounded probability uncertainty vector X
The mean and standard deviation of (a) are determined by:
a) reduction of
Mu in X For the bounded probability uncertainty vector X, the yield value of the worst case condition at this time can be written as
Including probability uncertainty;
b) yield value of worst working condition by using univariate dimension reduction method of multivariable function
Can be developed by the following formula:
in the formula, X ＜i＞ (i ═ 1,2,3) are as follows:
c) according to the expansion in b), used for calculating the worst working condition yield value
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:
phi (X) in the formulae Eq.47, Eq.48 i ) Is the probability uncertainty X i (i ═ 1,2,3) probability density function;
each one-dimensional integral in the above equation is calculated using the Laguerre (Laguerre) integration format:
wherein t is the number of Laguerre (Laguerre) integration points, and t is designated as 6; x is the number of (j) ,λ (j) (j ═ 1,2, …,6) are the standard integration points and their corresponding weights, respectively, given by the Laguerre (Laguerre) integration rule;
d) worst condition yield value
The mean and standard deviation of (a) can be obtained by:
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
The following takes the 1 st iteration process as an example to illustrate the flow of the particle reinforced connecting rod direct solution based on the optimal criterion method:
6.2) defining the weighted objective function J (p) for achieving
The double-target optimization:
6.3) objective function and constraint function to each design variable rho e The (e ═ 1,2, …,710 × 360) gradient can be calculated by the following equations, respectively:
6.3.1) to obtain the gradient of the objective function, we need to substitute Eq.50 into Eq.52, which yields:
6.3.2) gradient term in Eq.52
(e ═ 1,2, …,710 × 360) the specific calculation formula is as follows;
respective gradient terms in the formulae Eq.55, Eq.56
(e ═ 1,2, …,710 × 360) is given by the classical topology optimization framework SIMP:
in the formula Eq.57, the compound,
is that
The reason for this is that the bounded probability uncertainty vectors in the aforementioned 4 terms are all constant values; k is a radical of formula e The element stiffness matrix of the element e can be calculated through a classical finite element theory; u. of e The 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) substituting all the calculation results into an objective function gradient expression Eq.3-22 to obtain a final objective function gradient result, and intercepting the following steps:
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
The design variables are updated, and the truncated parts are as follows:
ρ 1 ＝0.294,ρ 1 ＝0.320,…,ρ 710×360-1 ＝0.215,ρ 710×360 ＝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 since it is the first iteration that is used for the presentation, this difference is defined as the current objective function value 158.2399, not meeting the convergence threshold of 0.01, so steps 6.1) to 6.5) are repeated.
The final optimal solution is intercepted as follows:
ρ 1 ＝0,…,ρ 16247 ＝4.28E-31,…,ρ 16252 ＝1,…,ρ 710×360 ＝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
The worst operating condition corresponding to the optimal solution is
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 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; the method comprises the following steps of (1) regarding the material properties of a base material, the material properties of a particle enhanced phase and the volume fraction of the particle enhanced phase distributed in a base body 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;
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; the simplified component is placed in a regular rectangular design field, and the rectangle is placedDesign Domain partitioning into N x ×N y A square unit of N x ，N y The division numbers along the directions of the x axis and the y axis are respectively; based on a classical isotropic material framework with penalty in topological optimization, each unit is endowed with a unique design variable rho e ∈[0,1]Wherein e is 1,2, …, N x N y ；
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 layered and gradually changed, namely the volume fraction of the particle reinforced phase in the matrix is only changed along the force transmission direction vertical to the component, 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 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 average volume fraction Vol of the particle enhanced phase in the ith layer according to the discrete component design domain and the volume fraction change mode of the particle enhanced phase * (i) Where i is 1,2, …, N y (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) calculating the Young modulus of each unit in the ith layer by using a Halpin-Tsai microstructure model
To poisson's ratio
3.4) based on the classical penalized isotropic material framework in topological optimization, by introducing penalty factors, the Young's modulus of each unit in the i-th layer can be finally expressed as:
in the formula Eq.1, the compound,
E min respectively 3.3) calculating the Young modulus of each unit in the ith layer and the preset minimum allowable Young modulus value which are constants; p is a constant penalty factor specified in advance;
the Young's modulus of each unit in the ith layer based on the penalized isotropic material framework construction;
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) geometrical constraints including the specified holes in the structure and the areas where the material is forced to remain, by setting the design variable ρ corresponding to the cell covered by the hole e ≡ 0 and the design variable position ρ corresponding to the cell that requires forced retention of material position coverage e 1 and does not change its value during the subsequent optimization;
5) the structural yield of the particle reinforced material member is an optimization design target, and the structural yield under the joint influence of the interval and bounded probability mixed uncertainty is considered in a robust topological optimization frame, so that the yield mean value and the standard deviation of the member under the worst working condition can be used as the characteristics of the optimization design target, and a robust optimization design model of the particle reinforced material member under the interval and bounded probability mixed uncertainty is established and is shown as Eq.2:
in the formula Eq.2, the compound,
is (N) x N y ) X 1-dimensional design vector, p min The minimum allowable value of each design variable given in the design is a constant; x ═ X (X) 1 ,X 2 ,…,X m ) T Is an m x 1-dimensional bounded probability uncertainty vector whose elements include material properties of the component matrix material, material properties of the particle-enhanced phase, uncertainty of the volume distribution of the particle-enhanced phase in the matrix material; i ═ f 1 ,f 2 ,…,f n ,α 1 ,α 2 ,…,α n ) T Is a 2n x 1 dimensional interval uncertainty vector, where f 1 ,f 2 ,…,f n Respectively the amplitude and alpha of n uncertain external loads borne by the component 1 ,α 2 ,…,α n Respectively representing n direction angles of uncertain external load borne by the component;
k (x) U ═ f (i) is a member balance equation, where k (x) is (2N) x N y )×(2N x N y ) Maintaining a total stiffness matrix, constructing by using a classical finite element theory, wherein a specific numerical value of the matrix is influenced by a bounded probability uncertain vector X; f (I) is (2N) x N y ) The x 1-dimensional node force matrix is constructed by using a classical finite element theory, and the specific numerical value of the x 1-dimensional node force matrix is influenced by an interval uncertain vector I; u is (2N) x N y ) X 1-dimensional node displacement matrix;
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)＝U T K(X)U＝F(I) T K -1 (X)F(I) Eq.3
B) definition of
To obtain a constant vector by taking the mean of each probability variable in the bounded probabilistic uncertainty vector X, we call μ X Is a mean vector of a bounded probabilistic uncertainty vector X, where
Are each uncertainty X 1 ,X 2 ,…,X m The mean value of (a); let bounded probability uncertainty vector X ═ μ in structure yield c (ρ, X, I) X Then at this time, the structural yield only includes the influence of the interval uncertainty I, which can be written as c (ρ, μ) X I) ═ 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 a form of the sum of force vectors of all external load nodes:
simultaneously, the method comprises the following steps:
in the formula Eq.5, f ux ＝f u cosα u ，f uy ＝f u sinα u Respectively an external load F u A magnitude component along the x, y directions; e.g. of a cylinder ux ，e uy Respectively corresponding to an external load F u Along the x, y axis of the node acted uponUnit node force vector of direction;
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:
the amplitude and the direction angle of the uncertain load are respectively derived in Eq.6, and the following steps are obtained:
respectively enabling the obtained derivative information according to Eq.7
Solve to obtain
I.e. the worst working condition of the particle reinforced material member under the action of the interval uncertain vector I
The yield value of the worst working condition can be written
In the formula Eq.2, the compound,
respectively the yield value of the worst working condition under the influence of a bounded probability uncertainty vector X
The mean and standard deviation of (a) are calculated as follows:
a) reduction of
Mu in X For the bounded probabilistic uncertainty vector X, the yield value of the worst case at this time may be written as
Including a probability uncertainty;
b) yield value of worst working condition through single variable dimension reduction method of Rahman multivariable function
Can be developed by the following formula:
in equation Eq.8, m is the number of bounded probability uncertainty parameters contained in a bounded probability uncertainty vector X, X ＜h＞ H 1,2, …, m being defined by eq.9:
x in the formula Eq.9 h Is the h bounded probability uncertainty variable;
c) according to the expansion Eq.8, calculating the worst working condition yield value
The high-dimensional integral of the first-order and second-order origin moments can be converted into a plurality of one-dimensional integral operations:
phi (X) in the formulae Eq.10, Eq.11 h ) Is the probability uncertainty X h H is a probability density function of 1,2, …, m;
d) each one-dimensional integral in equations eq.10, eq.11 is calculated using the Laguerre (Laguerre) integration format:
in the formula Eq.12, t is the number of Laguerre integral points and is a constant specified in advance; x is the number of (j) ,λ (j) J is 1,2, …, t is the standard integration point and its corresponding weight, given by the laguerre integration rule, all of which are constants;
by reaction at X ＜h＞ Intermediate order bounded probability uncertainty variable X h Are respectively as
Is obtained by
Wherein
Are also all constants, calculated by equiprobable transformation of each integration point, i.e.
e) Worst working condition yield value
The mean and standard deviation of (d) can be obtained by eq.13:
6) iteratively solving a robust optimization design model of the particle reinforced material component with the mixed uncertainty of the considered interval of Eq.2 and the bounded probability by adopting a standard optimal criterion 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
The double-target optimization:
in equation Eq.14, J (ρ) is a defined weighting objective function; w is a weight value which is designated 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.16 e Gradient:
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 the 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 components taking mixing uncertainty into account 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 changes only in the thickness direction of the member in a theoretical variation mode Vol (y), but the volume fraction of the particle-reinforced phase has uncertainty in an actual manufacturing process due to a hysteresis factor of a reinforcing particle addition port control system, and can be represented by a product of the deterministic theoretical variation mode 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 the fluctuation of the actual volume fraction of the enhancement phase relative to a theoretical design value, is called the volume fraction uncertainty of the particle enhancement phase, and uncertain information can be obtained by measuring relevant parameters of an adding mechanism: the feeding port responds to time delay and obtains friction resistance of the adding mechanism.
3. 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), random variables obeying a generalized beta distribution are used to describe respective bounded probability uncertainty parameters as follows:
1.1) to bounded probability uncertainty parameter X h S samples of the parameter are obtained through experiments, and then a sample set is constructed
From this sample set, the parameter X is calculated as Eq.18 h Value range of (1), calculating parameter X according to Eq.19 h Mean and variance of (a):
1.2) describing distribution at [ a ] by adopting generalized beta distribution i ,b i ]Inner and mean and variance are respectively
Parameter X of h The mean and variance are first normalized as shown in eq.20:
then, the parameter X is calculated using Eq.21 h Distribution parameter alpha of generalized beta distribution h ,β h ：
Recording parameter X i Obey in [ a h ,b h ]Internal and distribution parameter of alpha i ,β h Generalized beta distribution of (i.e. X) h ～GBeta(a h ,b h |α h ,β h ) And the probability density function is shown as Eq.22:
in the formula Eq.22, the compound,
is a parameter X i A probability density function of; Γ () is a gamma function.
4. A test according to claim 1The robust topology optimization method of the particle reinforced material member with the mixing uncertainty taken into consideration is characterized in that in the step 3.3), a Halpin-Tsai microstructure model is used for calculating the Young modulus of each unit in the ith layer
To poisson's ratio
The method comprises the following specific steps:
3.3.1) the physical properties of the reinforcing particles are: average length of particles l G Average width w G And an average thickness t G Young's modulus E G ；
3.3.2) define the following parameters:
in the formula Eq.23, E M Is the Young's modulus of the matrix material;
3.3.3) Young's modulus of each element in the i-th layer
Calculated by eq.24:
in the formula Eq.24, Vol * (i) Is the average volume fraction of the particulate reinforcement phase in layer i, with probabilistic uncertainty;
3.3.4) Poisson's ratio of each unit in the ith layer is calculated by Eq.25:
in the formula Eq.25, the compound,
ν G 、ν M the Poisson ratios of the units, the reinforcing particles and the matrix material in the ith layer are respectively.
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:
gradient term in equation Eq.26
Worst case yield values calculated for Eq.10 and Eq.11, respectively
First and second order origin moments vs. design variables ρ e A gradient of (a);
6.2.2) gradient term
The specific calculation formula is shown as Eq.27 and Eq.28;
gradient terms in the equations Eq.27 and Eq.28
Given by the classical topological optimization framework SIMP:
in the formula Eq.29
Is that
The reason for this is that
The bounded probability uncertainty vectors in (1) are all constant values; p is a constant penalty factor specified in advance; k is a radical of e The element stiffness matrix of the element e can be calculated by a classical finite element theory; u. of e The 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 results 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) to determine the appropriate constraint function Lagrangian multiplier λ using binary search iteration for updating the design variables, first specify the upper and lower bounds λ of the values of Lagrangian multiplier λ 1 、λ 2 1E-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＝λ middle ＝(λ 1 +λ 2 )/2 Eq.30
6.3.3) update design variable e ═ 1,2, …, N by Eq.31 x N y ：
In equation eq.31, γ ═ 0.2 is the search move step; η ═ 0.5 is the search damping coefficient; b e Obtaining gradient information of an objective function and a constraint function, as shown in Eq.32:
6.3.4) calculating the volume of the structure corresponding to the updated design variable
And adjusting the value boundary of the Lagrange multiplier according to the violation condition of the constraint function: if it is
Contraction value from lower bound to λ 1 ＝λ middle (ii) a If it is
Contraction taking an upper bound to λ 2 ＝λ middle ；
6.3.5) check the convergence criterion (λ) of the binary search 2 -λ 1 )/(λ 2 +λ 1 ) 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 yes, indicating that the current Lagrangian operator meets the requirement, and outputting the current design variable
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