1 With the improvement of the domestic automotive R&D level, optimal design has been gradually applied to the vehicle development process. However, there are no definite conclusions as to how many optimization design methods can be reasonably applied in engineering.
In view of the above problems, combined with the engineering practice of vehicle development, the characteristics of several commonly used optimization design methods were studied, and different optimization design methods were compared and studied, and some experience in the application of optimization design in the automotive design process was summarized.
2 experimental design
2.1 Several commonly used test design methods
The design of experiments (DOE) is the basis for optimization work. In the optimization design of automobiles, the commonly used DOE methods include the Orthogonal Array (OA) method and the optimized Symmetric Latin Hypercube (OSLH) method.
OA arranges test samples by applying orthogonal tables. Commonly used are L 8 (2 7), L 25 (5 6), and L 12 (3 1 × 2 4). This method is a mature DOE method, which ensures the uniform distribution and orthogonality of the sample in the sample space. The so-called orthogonality, that is, each level of each factor in the test matrix occurs in the same number of times in the total number of trials, and at the same time, the number of pairs of various levels in the ordered number of pairs between the arbitrary two factors. The same number of occurrences. The general optimization design software can support OA with a maximum of 4 levels and 256 sample points.
Engineers can also customize the OA matrix according to the test rules for the characteristics of different optimization problems.
The OSLH method is a modified DOE method based on the ordinary Latin square test. It adds some judgment criteria to ensure uniform distribution of the test sample in the design space, mainly including the maximum and minimum distance criteria, the entropy criterion, and the minimization of L2. Center phase difference criteria. This method is a kind of sampling method that requires the least number of tests under the premise of the same sample level, but it sacrifices the orderly comparability of sample points and guarantees the number of sample levels. In some vehicle optimization processes, the approximate model established based on this method is less precise in the design space edge, and therefore is not conducive to the optimization result may be in the design of the edge of the space, and at the same time, the method is not suitable for the variable level is not the case .
2.2 DOE Method Selection and Definition
The main purpose of DOE is to ensure that an accurate approximate model can be established in the later period, and the size of the matrix mainly depends on the number of design variables. Under normal circumstances, it is recommended that the number of samples be at least 3 times the number of variables. For different situations where different variables exist, such as the case where there are both size variables and subsystem configuration types as variables, the OA method is usually used. In view of the fact that there are not many variables but many levels, it is necessary to accurately simulate the relationship between single variables and output indicators. OSLH is recommended as the DOE matrix. If the sample points are added at a later stage, the accuracy of the approximate model of both methods will increase to some extent. Commonly-used optimization design software such as iSIGHT and OPTIMUS can establish different DOE matrices according to requirements. OA methods can also establish DOE matrices by checking OA tables.
1 The error of the optimization result of two different DOE methods was used to compare the stiffness of a white car with a set of 42 design variables and 1681 test samples. This problem is a part of a multi-condition optimization process in which the design variables mainly include the position, angle, section size, plate thickness, and material of each structural beam. The optimization goal is to minimize the quality in the case of satisfying various performance index requirements. From the results, the difference between the predicted value obtained by the approximate model established by the DOE and the stiffness value obtained by the finite element calculation is basically within 1%, which indicates that the two DOE methods can satisfy the project if the test sample is sufficient. The accuracy of the problem.
3 approximate model
In vehicle analysis, for some optimization problems such as short calculation time and single variables, such as the optimization of the board thickness of the body-in-white, the analysis software can be used to optimize the cycle without creating an approximate model. However, in the development process, it is often a multi-objective optimization problem with long computing time. At the same time, engineers need to find the relationship between design variables and output indicators. Therefore, it is necessary to use the approximate model to complete the optimization design work, so choose one Appropriate approximate models to solve the optimization design problem are critical. The commonly used approximate models include polynomial response surface, radial basis function neural network, Kriging interpolation model and so on.
3.1 Polynomial Response Surface Model
The polynomial response surface model is a model that uses a polynomial function to fit the design space. At present, the main application is the n-th order polynomial response surface (n=1, 2, 3,...) Model. The response surface model adopted in the iSIGHT has 2nd and 3rd order. And 4th order response surface models. The model can simulate a simple functional relationship more accurately in a local range. The model can be used to perform optimization design studies for a single simple condition, such as the optimization of the stiffness of the interface of the body in white when there are enough sample points. However, the accuracy of solving non-linear problems such as collision problems is not high, so it is not suitable for multidisciplinary optimization research in the early development of automobiles. If you must apply a polynomial response surface to solve complex problems, you need to estimate in advance some of the functional expressions that the engineering problem may satisfy.
Taking the variable stiffness optimization problem of lifting gate gas spring hinge of a certain MPV and the optimization problem of a car with 100% positive 7-variables as examples, the 2nd-order and 4th-order response surface models were used for optimization calculation. The design variables of the problem mainly include the size, thickness, position, and shape of the relevant structural components. The DOE samples are controlled within 100 or less. The table compares the variation of each indicator before and after optimization of the above two problems and the error between the response surface optimization result and the finite element calculation value. From the result point of view, the polynomial response surface model can basically meet the accuracy requirements of linear problems such as stiffness or modal analysis, and the error can be controlled within 5%; however, it is adopted to solve the problem of high nonlinearity such as passive safety. The method of establishing the approximate model can not meet the accuracy requirements. In Table 2, the intrusion error of position 1 is even up to 50%. Therefore, other approximate model methods need to be sought.
Table 2 Polynomial response surface model error analysis %3.2 Radial Basis Function Neural network model RBF is a type of neural network, mainly based on the Euclidean norm (distance) between the point to be measured and the sample point. A class of functions. Radial function as a basis function, the model constructed by linear superposition is a radial basis function model, and this approximate model can pass all DOE sample points.
The so-called variable-weight coefficient curve method is used in iSIGHT software. It is an internal optimization algorithm that relies on different values ​​of C to make the approximate method torsional stiffness. Forward bending stiffness. Back bending stiffness. Overall bending stiffness quality OA 0.097 2 -0.286 5 1.420 9 0.063 5 0.005 2 OSLH -0.099 7 -0.052 7 0.443 4 -1.003 6 -0.273 8 Parameter Lifting Hinge Hinge Stiffness Optimization 50 km/h 100% Rigid Wall Collision Mass Opening Stiffness Closing Stiffness Effective Acceleration Energy Absorption Efficiency Position 1 Invasive Position 2 Intrusion Quantity variation 0.428 7 56.787 6 34.753 -8.11 17.57 -11.50 -17.06 Error 0.007 2 4.080 1 2.177 6.74 -7.40 49.53 37.61 The model is close to a different function expression. For example, if C takes 1.15, then a seven-sample DOE can be used to fit a one-dimensional step function approximation model; for C, two can fit a one-dimensional linear function by using 3 samples of DOE; C can take 3 samples to pass 7 samples of DOE. Fitting a simple harmonic function; but it is also possible to fit a step function into a sine wave function, so for this case the C value needs to be optimized. The method for optimizing the C value is: first delete a sample point, fit the approximate model with the remaining sample points, calculate the error of the model at all sample points, and then sum; all sample points are processed according to this method; And the minimum C value will eventually find an optimal C value as the coefficient of the approximate model term.
When the sample points are sufficient, the application of RBF is equivalent to the Kriging model, which can meet the engineering requirements. Table 3 compares the error between the Kriging model and the RBF model established using the iSIGHT software for predicting the stiffness of a certain body in white.
3.3 Kriging approximation model
The Kriging model is an unbiased estimation model with minimum variance estimation proposed by the South African geologist Danie Krige in 1951.
A one-dimensional expression of the Kriging model can be written as: y(x)=f(x)+z(x). The commonly used engineering development software uses the method of fixing the f(x) term as a constant, that is, the ordinary kriging method. This constant is the function of the sample space.
z(x) is obtained as a change item by calculating its covariance between different sample points. The covariance is mainly achieved by multiplying a variogram b(x) by different weights. In most cases, the variogram uses a normal distribution function, that is, one-dimensional bi(x)=exp(1) is greater than one-dimensional bi(X)=exp - dk = 1Σθk(xk -s ik)Σ2(2) In the formula, θ is the range, and the degree of the function value close to the sample point is controlled away from the sample point; si is the sample point independent variable value, i is the sample point number, and d is the number of independent variables.
The biggest advantage of this method is that you can pass the approximate model through all sample points. Because the model is based on the unbiased estimation and the principle of variance minimization, the spatial estimation value existing between sample points decreases with increasing distance from the sample point. However, the error of the model can be corrected in time through the increase of sample points, which is most suitable for the optimization design process based on finite element analysis. If the sampling is unreasonable, this method may produce erroneous estimates for models that have obvious regularity or can be expressed with a certain function. At the same time, this method is not suitable for most optimization designs based on vehicle physical tests.
4 optimization algorithm
In the development of the vehicle, two optimization algorithms, the simulated annealing algorithm and the genetic algorithm, are usually used. Both of these algorithms can solve the multi-objective global optimization problem and are not limited by the design starting point.
The iSIGHT software provides a simulated annealing algorithm (ASA) and two multi-objective genetic algorithms (NCGA and NSGAII). The essence of the multi-objective optimization algorithm is to combine each design goal with a weighted reorganization to compose a single-objective optimization problem. The difference in target weights will have a great influence on the optimization result. Therefore, the weights of the design goals need to be reasonably configured in the optimization process. .
4.1 Simulated Annealing Algorithm
ASA comes from the principle of metal annealing. In the initial stage of cooling, the internal atoms are intense and the internal energy is maximal. With the progress of cooling, the internal particles gradually become orderly. Finally, when the ambient temperature reaches the ground state, the internal energy is minimized. Similarly, during the initial iteration phase, the algorithm searches for the best design results across the entire design space. As the iteration progresses, it will gradually focus on the optimal area of ​​the design space. During the optimization process, engineers can control the search direction and speed of the optimization process based on actual problems. The adaptive ASA provided by iSIGHT can automatically adjust the values ​​of these control parameters during the optimization process.
4.2 Genetic Algorithm
In 1962, Holland proposed the idea of ​​genetic algorithm for the first time. Using simulation biogenetics and the natural selection mechanism, through the gene recombination, each individual's adaptability was improved. To a certain extent, genetic algorithms are mathematical simulations of biological evolution processes. All design variables in the optimization process can be viewed as chromosomes in biogenetics. By pre-setting, a certain number of design samples will be generated in each iteration, called "individuals". Similar to the natural world, for each iteration, the current optimization problem will evaluate all individuals and sort them in order of assessment results, so that better variable levels have more opportunities for copying and replacement. In this way, the poor "individuals" can be eliminated, and a new generation of "individuals" can be created through crossover and mutation processes. The new sample is then subjected to the next round of optimization until it finds the best value for the optimization problem.
The NCGA and NSGAII in the iSIGHT are similar in many aspects. Both algorithms divide all individuals into two “populationsâ€. The first population concentrates the high-quality design samples sought in the optimization. The second population concentration needs to pass. Crossover and variation can produce individuals for the next generation of samples. The difference lies in the generation of the next generation of individuals in the process.
NCGA's recognition of outstanding individuals only selects one design goal to evaluate, ranks individuals according to their performance in each design goal, and completes the cross-variation between individual excellent individuals, and guarantees a variety of variable levels by randomly selecting individual crossovers. Sex so as not to miss the quality variable level.
The NSGA-II uses all design objective functions to evaluate and select excellent individuals. When there are two approximate excellent bodies, the algorithm automatically separates them according to the principle of the farthest distance, avoiding crossover between each other. At the same time, the algorithm also guarantees the diversity of variable levels by randomly selecting individual crossovers.
4.3 Algorithm Application
Through the comparative study of vehicle optimization problems, it is found that in many optimization design problems, the algorithm selection for the unconstrained multi-objective optimization problem is the most difficult. The ideal unconstrained multi-objective optimization problem is the least ambiguous goal in all optimization problems. Different optimization algorithms perform differently in different optimization problems. There is no absolute optimal optimization algorithm to solve all vehicle optimization design problems. Therefore, for an optimization design problem, several different optimization algorithms are recommended for comparison.
For the multi-objective optimization design problem, under the same number of iterations, the ASA will get a more balanced optimization result than the genetic algorithm. The initial search direction or population of the optimization algorithm is determined randomly. Therefore, even for the same optimization design problem, performing the same number of iterations, the results of the two iterations of the calculation will be different. But as long as the number of iterations is sufficient, most optimization algorithms will eventually get a more consistent result. For genetic algorithms, expanding the number of individuals per generation will give better optimization results than adding algebra. In general, genetic algorithms need to use more iterations than the simulated annealing algorithm to find the optimal design results, but the specific number of iterations required varies by problem. After an optimization cycle is completed, the optimal design results must be judged and selected in conjunction with optimization results and various aspects of engineering requirements.
5 ends
From the aspects of principles and features of optimization methods, some DOEs, approximate models, and optimization algorithms commonly used in automotive engineering are analyzed and compared. Actual engineering problems are verified, and some applied optimization design methods are applied to solve engineering problems. Sexual conclusions.
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