CN110069864B

CN110069864B - Improved bidirectional progressive structure topology optimization method combined with variable density method

Info

Publication number: CN110069864B
Application number: CN201910336858.8A
Authority: CN
Inventors: 鲁世红; 余威; 刘文浩; 李磊; 金建; 薛澄澄; 李翔
Original assignee: Nanjing University of Aeronautics and Astronautics
Current assignee: Nanjing University of Aeronautics and Astronautics
Priority date: 2019-04-24
Filing date: 2019-04-24
Publication date: 2023-07-21
Anticipated expiration: 2039-04-24
Also published as: CN110069864A

Abstract

The invention discloses an improved bidirectional progressive structure topology optimization method combined with a variable density method. The design method comprises the steps of applying an improved bidirectional penalty function in a variable density method to a bidirectional progressive structure topology optimization method, filtering sensitivity by using a sensitivity filtering method, then further homogenizing the sensitivity by using a BESO discrete method, enabling a result to be easier to converge, and finally searching a sensitivity threshold by adopting a dichotomy for unit updating iteration until iteration converges. The method provided by the invention can effectively reduce the iteration times, save the calculation time and improve the optimization efficiency.

Description

Improved bidirectional progressive structure topology optimization method combined with variable density method

Technical Field

The invention relates to the technical field of structural equivalent static load, in particular to an improved bidirectional progressive structural topology optimization method combined with a variable density method.

Background

Xie et al in 1993 proposed a heuristic based topology optimization method, evolutionary structural optimization (Evolutionary Structure Optimization, ESO), which differs from other modeling methods based on continuous design variables in that it originates from stress design techniques, considering structurally non-functional materials within the design domain, i.e. those of low stress or low strain energy density, are inefficient and removable. Removal of material may be by changing the modulus of elasticity as a function of stress or strain energy density or by directly deleting those material spaces of low stress or strain energy density. By removing the ineffective or inefficient material step by step, the remaining structure will gradually tend to be optimized.

The evolutionary structure optimization method is an evolutionary structure optimization method capable of deleting and adding materials simultaneously, namely, the materials around a high-stress area are supplemented while deleting low-efficiency materials, and the originally designed area can be smaller, so that the calculation efficiency is improved. However, the original ESO method can only delete the defect of the unit, and the query proposes AESO (Add ition Evolutionary Structure Optimization) method, which can effectively regenerate the material unit in the solving process, and can effectively avoid the condition of irrecoverable caused by the wrong deletion of the material. Still further, combining the AESO with the ESO process constitutes a Bi-directional BESO (Bi-directional Evolutionary Structure Optimization, BESO) process that both deletes and restores material units. Liu combines ESO with genetic algorithm, and increases and decreases material units by adopting binary coding form for the material units and performing crossover, mutation and other genetic evolution operations by taking the material units as individuals.

The ESO generally adopts indexes such as stress, sensitivity and the like of the structure as evolution criteria, the theoretical formula and the thought are quite popular and easy to understand, and the ESO is easy to accept and use by engineering staff and is suitable for the problem of optimizing the structure which is difficult to express by an objective function.

Disclosure of Invention

The invention aims to solve the problems that an iteration process is unstable and multiple convergence is easy to occur in an equivalent static load method, and provides an improved bidirectional progressive structure topology optimization method combined with a variable density method.

In order to achieve the above object, the present invention discloses an improved bidirectional progressive structural topology optimization method in combination with a variable density method, the method comprising the steps of:

defining a design domain, load and boundary conditions and material parameters of a material interpolation model, and presetting an evolution rate, a filtering radius and a volume ratio;

calculating flexibility and unit sensitivity based on the new bidirectional interpolation function;

filtering the unit sensitivity by adopting a grid sensitivity filtering method, and homogenizing the filtered sensitivity by adopting a BESO discrete method;

and obtaining a sensitivity threshold based on a dichotomy, and iterating according to the sensitivity threshold to obtain a final objective function value.

The new bi-directional interpolation function is:

wherein ρ is _i For the pseudo-density value of the ith cell, q and p are coefficient factors, ρ _min Is the minimum of the pseudo-density values;

the improved BESO method is specifically as follows, the minimum compliance of the mathematical model according to the equivalent static load method obeys the volume constraint:

omega is a flexibility value, F is a stressed load, U is a displacement vector, K is a stiffness matrix and ρ is _i For the pseudo-density value of the ith cell, delta _i Is the true density value (0 or 1) of the i-th cell.

Deriving the unit pseudo-density according to the sensitivity equal to the compliance:

the method can be as follows:

for the grid sensitivity filtering method, the grid sensitivity filtering method is used for modifying sensitivity information of an objective function to prevent the occurrence of a checkerboard phenomenon, namely:

d in _i ＝r _min -H(i，j)，{i∈N|H(i，j)≤r _min }，r _min Is the minimum unit diameter d _min H (i, j) is the distance of unit i from the adjacent unit j; . r is (r) _min The smaller the unit node sensitivity is, the closer to the initial sensitivity is; r is (r) _min The greater the sensitivity of each unit node, the more nearly the same. Through proper modification of the sensitivity coefficient, the singular problem of numerical value sharpening can be solved.

In order to ensure convergence of the results after the iteration, a discrete method of BESO was decided to further homogenize the sensitivity. The specific formula is as follows:

for the sensitivity after homogenization, k is the sensitivity of the e-th unit and p (0 < p < 1) is the homogenization factor.

A dichotomy is used to find the sensitivity threshold. In the initial stage of iteration, the minimum value of the found sensitivity is marked as min, and the maximum value of the found sensitivity is marked as max. Setting the error range to be 0.00001, setting the circulation condition (max-min)/max to be more than 0.00001, taking the threshold value th= (max-min)/2, then comparing the threshold value with the sensitivity value of each unit, and setting the density of the unit with the high sensitivity value to be 1; on the other hand, the density of the unit having a small sensitivity value was set to 0.001. Finally, judging whether the sum of the density values of all the units is larger than V ^* (V ^* =f×v, volume threshold): if greater than V ^* Assigning th to min, then executing initial circulation condition (max-min)/max > 0.00001 until convergence, and ending all circulation; if less than V ^* The th is assigned to max, then the initial loop condition (max-min)/max > 0.00001 is then performed until convergence, ending all loops.

Drawings

FIG. 1 is a flow chart of an improved bi-directional progressive structural topology optimization method of the present invention in combination with a variable density method.

FIG. 2 is a graph of the change in the modified bi-directional penalty function value with cell density.

Fig. 3 is a two-dimensional cantilever beam example.

FIG. 4 is a schematic diagram of example optimization results: 4 (a) is a conventional method, and 4 (b) is a method proposed by the present invention.

Detailed Description

The invention will be described in more detail below with reference to the accompanying drawings.

The improved BESO process incorporating the variable density process is specifically described below,

the design domain, load and boundary conditions of the material interpolation model are defined, and corresponding material parameters are also defined. Given preset parameters: evolution rate ER, filter radius r _min Volume ratio f.

The minimum compliance of the mathematical model according to the equivalent static load method obeys the volume constraint:

omega is a flexibility value, F is a stressed load, U is a displacement vector, K is a stiffness matrix and ρ is _i For the pseudo density of the ith cell, delta _i Is the true density value (0 or 1) of the i-th cell.

the method can be as follows:

for the grid sensitivity filtering method, sensitivity information for modifying the objective function:

for the sensitivity after homogenization, k is the sensitivity of the e-th unit and p (0 < p < 1) is the homogenization factor. The sensitivity after filtration was again homogenized using the above formula.

According to formula V ^k+1 ＝V ^k (1.+ -. ER), iterating according to the value of the evolution rate ER until V ^k+1 ≥fV ^k Until that point.

Prior to stopping the above iteration, a dichotomy is used to find the sensitivity threshold. In the initial stage of iteration, the minimum value of the found sensitivity is marked as min, and the maximum value of the found sensitivity is marked as max. Setting the error range to be 0.00001, setting the circulation condition (max-min)/max to be more than 0.00001,taking a threshold value th= (max-min)/2, then comparing the threshold value with the sensitivity value of each unit, and setting the density of the unit with the high sensitivity value to be 1; on the other hand, the density of the unit having a small sensitivity value was set to 0.001. Finally, judging whether the sum of all the unit density values is larger than V (f is the volume threshold value): if greater than V ^* Assigning th to min, then executing initial circulation condition (max-min)/max > 0.00001 until convergence, and ending all circulation; if less than V ^* The th is assigned to max, then the initial loop condition (max-min)/max > 0.00001 is then performed until convergence, ending all loops.

Two examples are calculated below based on the original progressive structure optimization method and the method proposed by the present invention.

As shown in fig. 2, the design domain is a two-dimensional cantilever structure of 40mm×200mm, which is divided into a finite element model of 40×200 units, wherein the midpoint on the right side of the model is acted by a vertical downward static load f=1000n, and the left side of the cantilever is fixedly supported. Other optimization parameters of the calculation are shown in table 1.

TABLE 1

The results after optimization are shown in fig. 4 (a) and fig. 4 (b).

Table 2 shows the results of the conventional method compared with the results of the present method.

TABLE 2

As can be seen from comparison of the optimized results of fig. 4 (a) and fig. 4 (b), for the cantilever beam with single-side solid support, the method proposed by the present invention is similar to the optimized result of the conventional progressive structural optimization method, and the final objective function value is very close, which illustrates that the method proposed by the present invention is feasible. In the method, the iteration times are 29 times, and the iteration times of the traditional method are 36 times, and the iteration times of the method are less than the traditional iteration times, so that the optimization time is reduced, and the optimization efficiency is improved by 19.4%. Therefore, the method provided by the invention has higher efficiency than the traditional method.

Claims

1. An improved bi-directional progressive structural topology optimization method in combination with a variable density method, the method comprising the steps of:

obtaining a sensitivity threshold based on a dichotomy, and iterating according to the sensitivity threshold to obtain a final objective function value;

the new bi-directional interpolation function is:

wherein ρ is _i The pseudo density value of the ith unit, q and p are coefficient factors;

the grid sensitivity filtering method is as follows

D in _i ＝r _min -H(i，j)，{i∈N|H(i，j)≤r _min }，r _min Is the minimum unit diameter d _min H (i, j) is the distance of unit i from the adjacent unit j;

the method for calculating the compliance comprises the following steps:

min：ω(P)＝F ^T U＝U ^T KU

sbj to：P＝{ρ _i }，ρ _i ＝1 or ρ _min

F＝KU

wherein ω is a compliance value, F is a load, U is a displacement vector, K is a stiffness matrix, ρ _i For the pseudo-density value of the ith cell, delta _i Is the true density value of the ith cell, V ^* Is a volume threshold;

the discrete method is

In the method, in the process of the invention,for the sensitivity after homogenization, k is the sensitivity of the e-th unit, s is a homogenization factor, and the value range of s is more than 0 and less than 1.

2. The improved bi-directional progressive structural topology optimization method of claim 1, in combination with a variable density method,

the iterative process includes: firstly, finding out the minimum value of the sensitivity and marking the minimum value as min, finding out the maximum value of the sensitivity and marking the maximum value as max, setting the error range as 0.00001, and setting the circulation condition (max-min)/max>0.00001, taking a threshold value th= (max-min)/2, then comparing the threshold value with the sensitivity value of each unit, and setting the unit density with the sensitivity value larger than the threshold value to be 1; setting the density of units with sensitivity value smaller than the threshold value to 0.001, and finally judging whether the sum of the density values of all units is larger than V ^* ，V ^* =f×v: if greater than V ^* Assign th to min, then follow the initial loop condition (max-min)/max>0.00001, until convergence, ending the cycle; if less than V ^* The th is assigned to max, then the initial loop condition (max-min)/max > 0.00001 is performed next until convergence, ending the loop.