A method for calculating the feasible solution region using a set-based design approach

Abstract

The present application provides a calculation method for finding a feasible solution region using a collective design method. The present application aims to reduce the computational load for exploring a feasible solution compared to the past. A trial vector is generated by mutation. In a case where the value of an objective function obtained by substituting the trial vector is smaller than the value of the objective function obtained by substituting a target vector, the trial vector is selected as a new feasible solution. The objective function includes a probability density function substituted into the target vector and the trial vector. The probability density function corresponds to the distribution of the feasible solution that has been found. The smaller the value of the probability density function is close to 0, the smaller the value of the objective function is.

Description

Technical Field

[0001] This specification discloses a computational method for finding the feasible solution region using a set-based design approach. Background Technology

[0002] The design of vehicle components such as suspension or powertrain mounts involves multiple design elements. For example, factors such as handling stability, vibration and noise reduction, weight reduction, and cost reduction are studied during the design process.

[0003] For example, in Patent Document 1, when the number of multiple design variables exceeds a predetermined number, design variables that contribute less to the design process are discarded. Furthermore, in Patent Document 2 and Non-Patent Document 1, feasible solutions are calculated using a set-based design method.

[0004] As methods for determining multiple design variables that meet required specifications, point-based design methods and set-based design methods are known. In the point-based design method, multiple design variables are assigned arbitrary values. Then, experiments are conducted to determine whether a prototype made according to the assigned values ​​meets the required specifications. Based on the experimental results, the prototype is improved. This process of experimentation and prototype improvement is repeated.

[0005] In ensemble design methods, allowable ranges are defined for each design variable. Then, the set of feasible solutions that simultaneously satisfy these allowable ranges is identified as the feasible region. Next, the ease of design changes or performance robustness are studied for multiple feasible solutions contained within the feasible region.

[0006] For example, in Non-Patent Literature 1, a differential evolution-based optimization method is used as a method for finding feasible regions. Specifically, in Non-Patent Literature 1, test vectors are calculated as candidates for feasible solutions. These test vectors are generated through a mutation operation in differential evolution. Then, the test vectors are compared with target vectors that already exist as feasible solutions.

[0007] For two vectors, an objective function is used for comparison. The objective function J0(x) is represented by the following equation (1). If the value of the objective function J0(x) obtained by substituting the test vector is less than (negative to the greater) the value of the objective function J0(x) obtained by substituting the target vector, the test vector is collected as a new feasible solution into the solution set. The collection of feasible solutions is called the feasible region.

[0008]

Formula 1

[0009] To decrease the value of the objective function J0(x) (increase the negative value), we only need to increase the value of the second term on the right. The function h(x) within the parentheses of the second term on the right is the Heaviside function, which is 0 when x < 0 and 1 when x > 0. Furthermore, the function g... KS (x) represents the constraint function g i (x) is the function obtained by converting it into the cost function. And the function d(x) of the second term on the right is represented by the following formula (2).

[0010]

Formula 2

[0011] Here, x′ refers to a previously found feasible solution. Taking the calculation of the test vector as an example, in equation (2), the distances between all found feasible solutions x′ and the test vector x are calculated. Then, the minimum value among the calculated distances is extracted. That is, when the test vector is further away from any existing feasible solution, the value of d(x) increases. Referring to equation (1), at this time, the value of the objective function decreases (increases in the negative direction).

[0012] Patent Document 1: Japanese Patent Application Publication No. 2022-39754 Patent Document 2: Japanese Patent Application Publication No. 2010-55466 Non-Patent Literature 1: Nishikawa et al., "A Collective Design Method Based on Differential Evolution and Its Application in Suspension Design", Proceedings of the China Automotive Technology Conference, November 2023, China Automotive Technology Conference, Vol. 54, No. 6, pp. 1121-1126. Summary of the Invention In the above equation (2), all found feasible solutions x′ are compared with the target vector and the trial vector. Therefore, as the number of feasible solutions increases, the computational load in equation (2) increases.

[0013] Therefore, this specification discloses a method for calculating the feasible solution region that can reduce the computational load when exploring feasible solutions compared to the past.

[0014] This specification discloses a computational method for finding feasible solution regions using a set-based design approach. This method includes a survival selection step. In this step, survival selection is performed on a target vector and a trial vector using an objective function, based on differential evolution. The target vector is a previously found feasible solution. The trial vector is generated through mutation. If the value of the objective function obtained by substituting the trial vector into the target vector is less than the value obtained by substituting the target vector into the target vector, the trial vector is selected as the new feasible solution. The objective function includes a probability density function. The target vector and the trial vector are substituted into the probability density function. The probability density function corresponds to the distribution of the previously found feasible solutions. The closer the value of the probability density function is to 0, the smaller the value of the objective function.

[0015] According to the above structure, the closer the probability density function is to 0, the smaller the objective function value. That is, the more the trial vector is plotted in a region where the probability of the existence of a known feasible solution is low, the higher the probability that the trial vector will be selected as a new feasible solution. Based on this operation, it is unnecessary to calculate the distance between all feasible solutions and the trial vector; a trial vector plotted far from known feasible solutions can be selected as a feasible solution.

[0016] Furthermore, in the above structure, the survival selection step can be performed multiple times. In this case, the probability density function is the sum of the existing probability density function and the new probability density function. The existing probability density function is generated based on the set of feasible solutions obtained before a predetermined number of survival selection steps. The new probability density function is generated based on the set of feasible solutions obtained after the predetermined number of survival selection steps. The new probability density function is updated each time a new feasible solution is selected through a survival selection step.

[0017] Based on the above structure, when the number of feasible solutions increases, the update of the probability density function only needs to be performed on the newly added probability density function.

[0018] Furthermore, in the above structure, when the number of feasible solutions contained in the newly added probability density function reaches a specified number, the newly added probability density function can be added to the existing probability density function.

[0019] Based on the above structure, the expansion of the newly added probability density function can be suppressed.

[0020] Invention Effects The computational method for finding feasible solution regions using the set-based design approach disclosed in this specification can reduce the computational load when exploring feasible solutions compared to the past. Attached Figure Description

[0021] Figure 1 This is a diagram illustrating the hardware structure of a computing device that performs the calculation method for the feasible solution region according to this embodiment.

[0022] Figure 2 This is a flowchart illustrating the process of determining design parameter values.

[0023] Figure 3 This is an example. Figure 2 The flowchart details the steps for exploring feasible areas.

[0024] Figure 4 This is a graph illustrating the feasible regions for variables A and B.

[0025] Figure 5 This is a diagram representing an example of a set of feasible solutions derived using existing techniques.

[0026] Figure 6 This is a diagram illustrating the set of feasible solutions obtained by the calculation method of the feasible solution region involved in this embodiment. Detailed Implementation

[0027] 1. Hardware Structure exist Figure 1 The hardware structure of the computing device is illustrated below. This computing device executes the calculation method for the feasible region involved in this embodiment. The computing device includes a computing device 10, an input unit 20A, and a display 20B.

[0028] The input unit 20A may consist of, for example, a keyboard or a mouse. The display 20B displays as described later. Figure 5 or Figure 6 The set of feasible solutions (feasible region) shown. The computing device 10 includes a CPU 10A, RAM 10B, storage device 10C, ROM 10D, and input / output controller 10E.

[0029] CPU 10A is a central processing unit, also known as a processor. RAM 10B is a volatile storage device for temporarily storing data during operation. ROM 10D is a storage device capable of reading data. Storage device 10C is a storage device capable of writing and reading data. Storage device 10C is, for example, composed of a hard disk drive (HDD) or a solid state drive (SSD).

[0030] By executing a program stored in storage device 10C or ROM 10D by CPU 10A, computing device 10 can perform tasks such as... Figure 3 The computational flow for a feasible solution is shown. For example, it can be executed... Figure 3 The functional modules of each step (S41-S48) in step S40 of the flowchart illustrated in the figure are constructed in the computing device 10.

[0031] The computing device 10 is used for the development and design of vehicle parts. For example, the computing device 10 is used in the development and design of powertrain supports or suspensions.

[0032] 2. Procedure for determining design parameter values In the feasible region calculation method of this embodiment, a set-based design method is employed. In this set-based design method, multiple design parameters become the objects of study. For example, in... Figure 2 The diagram illustrates the computational flow of the feasible solution region based on a ensemble design approach. For example, steps S10, S20, S30, and S50 are performed through agreements between the design and development team, etc. Furthermore, step S40 is performed by the computing device 10.

[0033] refer to Figure 2 Step S10 involves setting design parameters related to the vehicle parts being designed. These design parameters include various factors such as handling stability, vibration and noise reduction, weight reduction, and cost reduction. Furthermore, target performance values ​​are set for the vehicle parts being designed. For example, target performance values ​​are set for each design parameter through market research.

[0034] Next, the design parameters are screened according to the experimental design method (S20). For example, design parameters that contribute little to the target performance value are deleted.

[0035] Then, for the selected design parameters, a regression model between the design parameters and performance values ​​is created (S30). For example, a regression model between the design parameters and performance values ​​can be created using a Gaussian process or a neural network. For example, the relationship between the specified design parameters and performance values ​​is quantitatively represented by a model function.

[0036] Then, optimization processing is performed on the design parameters. Through this optimization processing, feasible regions for each design parameter are explored (S40). This exploration step is performed by the computing device 10.

[0037] First, define design variables for each design parameter. For example, for n design parameters, define design variable x. i (i=1, ..., n). Then, optimization is performed on the design variable vector x. In the optimization process, for each design variable x... i Explore feasible areas. For example... Figure 6 As shown, the feasible region is determined as a set of feasible solutions. Details of the optimization process in step S40 will be described later.

[0038] If in step S40, for each design variable x i Once the feasible region is identified, the design parameter values ​​are determined (S50). For example, the ease of design changes or performance robustness are studied by the design and development team. Finally, for each design parameter, the design values ​​are determined from the feasible region.

[0039] 3. Optimization based on set design method exist Figure 3 The detailed steps of step S40 are illustrated below. In step S40, optimization is performed according to the ensemble design method. Each step S41-S48 of step S40 is executed by the computing device 10.

[0040] Based on the regression model obtained in step S30, as shown in the following equation (3), for each design variable x i Set upper and lower limits for the required performance.

[0041]

Formula 3

[0042] The optimization process in step S40 can be understood as the process of solving the design variable vector x of the logarithmic equation (3) to minimize the objective function. As shown in equation (14) described later, a probability density function Σπ corresponding to the distribution of the feasible solutions obtained is generated in the objective function. i p i (x|θ) i )+π0p0(x|θ0). As will be discussed later, the smaller the value of the probability density function obtained by substituting the experimental vector, the higher the probability that the experimental vector will be selected as a new feasible solution.

[0043] refer to Figure 3 The optimization process is performed using differential evolution. First, initial individuals are generated based on the design variable vector x (S41). That is, an n-dimensional real-valued vector x is randomly generated. m The resulting group P, represented by the following equation (4), becomes the initial individual.

[0044]

Formula 4

[0045] Next, select the target vector x as shown in the following equation (5). i Reference vector x b x r1 x r2 Target vector x i This is a feasible solution that has already been found. Furthermore, the reference vector x... b x r1 x r2 Used to generate the mutation vector v(F) or the test vector u, which will be described later. i .

[0046]

Formula 5

[0047] Using these vectors, a mutation vector v(F) (S42) is generated by mutation as shown in the following equation (6).

[0048]

Formula 6

[0049] Where F is the scaling vector. And, the operator (hollow circle) of the latter part of the scaling vector F is the Adama product. The initial value of the scaling vector F is set according to the random number shown in the following formula (7).

[0050]

Formula 7

[0051] Furthermore, update the scaling vector F. For example, the elements F of the scaling vector F. i (i=1, ..., n) can be obtained by the following formula (8).

[0052]

Formula 8

[0053] That is, the point x′ within the feasible region that has the largest distance to its nearest neighbor becomes the element F. i The value of .

[0054] Next, through the target vector x i The binomial cross with the mutation vector v(F) generates the test vector u. i (S43). That is, through mutation and crossover, the experimental vector u is generated. i Test vector u i This is a candidate for a new feasible solution. The test vector u i The elements u ij The answer is found using the following formula (9).

[0055]

Formula 9

[0056] Here, C r Let R be the crossover rate, R ∈ {1, ..., n} be a randomly selected index, and r be the crossover rate. ij ~U(0,1) represents a uniformly random value.

[0057] For the test vector u generated by equation (9) i and target vector x i The objective function J0(x) is used for comparison (S44). This comparison is called survival selection. Figure 3 As shown in the loop of steps S42 to S48, survival selection is performed multiple times. In survival selection, a new feasible solution x is obtained. i,new The answer is obtained from the following formula (10).

[0058]

Formula 10

[0059] That is, by substituting the experimental vector u, we obtain i The value of the objective function J0(x) is less than (negative upward) and substituted into the objective vector x. i Given the value of the objective function J0(x), the test vector u i This becomes a new feasible solution.

[0060] 3. Objective function The objective function J0(x) in this embodiment is represented by the following formula (11).

[0061]

Formula 11

[0062] Furthermore, the function h(x) in the second term on the right is a Herveside function. The function g... KS (x) By using multiple constraint functions g i (x) is transformed into a cost function g KS (x) is obtained. The constraint function g i (x) is exemplified in the following equation (12). Furthermore, the KS function g KS (x) is exemplified in the following expression (13).

[0063]

Formula 12

[0064]

Formula 13

[0065] Furthermore, in equation (13), α > 0 is the adjustment based on the KS function g. KS The approximate parameter of the max operation of (x). For example, the larger the value of α, the more equivalent the solution of expression (13) is to the result of the max operation.

[0066] Furthermore, the function p(x) in equation (11) represents a finite mixture model. In the finite mixture model, the probability density functions of multiple feasible solutions are overlapped. The test vector u is then substituted into this probability density function. i and target vector x i The finite mixture model p(x) is represented by the following equation (14).

[0067]

Formula 14

[0068] In equation (14), multiple conditional probability density functions πp(x|θ) overlap. This probability density function πp(x|θ) corresponds to the distribution of the feasible solutions that have been obtained.

[0069] Figure 4 The example illustrates feasible solutions and probability density functions for variables A and B. Black dots represent the number of survival selections performed from the initial selection to the specified number of selections. Figure 3 The feasible solution at time S44. That is, black dots represent feasible solutions found at a specified number of time points. And hollow dots represent feasible solutions found after a specified number of survival choice steps. That is, hollow dots represent newly found feasible solutions after a specified number of time points.

[0070] Furthermore, the probability density function is represented by circles of different diameters. The closer to the center of the circles of different diameters, the higher the probability density (i.e., close to 1). Figure 4 The following example illustrates the probability density functions π1p1(x|θ1), π2p2(x|θ2), π3p3(x|θ3), and π0p0(x|θ0). These probability density functions π1p1(x|θ1), π2p2(x|θ2), and π3p3(x|θ3) are derived from the distribution of feasible solutions obtained at a specified number of time points. The sum of the probability density functions π1p1(x|θ1), π2p2(x|θ2), and π3p3(x|θ3) corresponds to the first term on the right-hand side of equation (14). Since it is based on the set of feasible solutions obtained before the specified number of survival selection steps, the first term on the right-hand side of equation (14) is appropriately referred to below as the existing probability density function.

[0071] The probability density distribution π0p0(x|θ0) is derived from the distribution of feasible solutions obtained after a specified number of survival selection steps. Therefore, the probability density distribution π0p0(x|θ0) will be appropriately referred to as the newly added probability density function below.

[0072] That is, the finite mixture model p(x) in equation (14) is the existing probability density function Σπ i p i (x|θ) i The sum of the finite mixture model p(x) and the newly added probability density function π0p0(x|θ0). Therefore, the finite mixture model p(x) can take values ​​from 0 to 1. Referring to the equation (11) representing the objective function, the closer the value of the finite mixture model p(x) is to 0 (the smaller), the higher the value of the second term on the right side of equation (11). The higher the value of the second term on the right side, the smaller the value of the objective function (the negative increases upwards).

[0073] refer to Figure 4 If we liken the probability density function to a mountain, then substituting the experimental vector u... i The smaller the value of p(x) in the finite mixture model obtained, the smaller the value of the test vector u. iIt is plotted at the foot of the probability density function. That is, at the test vector u i When plotted in a region where a known feasible solution appears not to exist, the value of the objective function J0(x) decreases (increases on the negative).

[0074] Here, in the conventional formulas (1) and (2), the value of the objective function decreases (increases negatively) when the experimental vector is further away from any existing feasible solution. That is, in the formulas (11) and (14) of this embodiment, the same effect as formulas (1) and (2) can be obtained.

[0075] Then, in equations (1) and (2), all the calculated feasible solutions x′ are compared with the target vector and the test vector. In contrast, in equations (11) and (14) of this embodiment, only the target vector and the test vector are substituted into the finite mixture model Σπ that overlaps the probability density functions. i p i (x|θ) i In )+π0p0(x|θ0), the same effect as that of equations (1) and (2) can be obtained. That is, in the calculation method involved in this embodiment, the computational load is reduced compared with the conventional calculation method.

[0076] After survival selection in step S44, it is determined whether the termination condition of the optimization process is met (S45). For example, as a termination condition, it is determined whether the number of generations based on differential evolution has reached a predetermined maximum number of generations. Alternatively, it is determined whether the generations in which feasible solutions do not increase (i.e., the generations in which trial vectors are rejected) are continuously generated within the specified number of generations.

[0077] If the termination condition for optimization is not met, determine the new feasible solution contained in the new probability density function π0p0(x|θ0). Figure 4 The number of hollow plotted points has reached the specified number (S46). In this step, if the number of newly added feasible solutions exceeds the specified update threshold, the new probability density function π0p0(x|θ0) is added to the existing probability density function Σπ. i p i (x|θ) i In (S47). Then, in the next step S48, the second term on the right side of the expression (14) is set to 0.

[0078] Furthermore, in step S46, if the number of newly added feasible solutions is below a predetermined update threshold, based on the newly added feasible solutions ( Figure 4 The number of hollow plotted points is used to update the newly added probability density function π0p0(x|θ0) (S48). Then, the operation process returns to step S42.

[0079] 4. Example of calculating the feasible solution region Figure 5 The example illustrates the set of feasible solutions, or feasible region, based on the computational methods involved in previous techniques using equations (1) and (2). Feasible solutions are plotted on a coordinate plane defined by variables A and B. The time required to explore the feasible region using previous techniques is 6680.40 seconds.

[0080] Figure 6 The illustrations show feasible areas based on the calculation method involved in this embodiment. Figure 6 Obtaining from the feasible region illustrated in the example Figure 5 The feasible region is equivalent to the result. Furthermore, refer to... Figure 6 The exploration time required for the feasible area involved in this embodiment is 217.24 seconds. Thus, it can be seen that in this embodiment, the time is significantly reduced while achieving the same results as conventional techniques.

[0081] Symbol Explanation 10 - Computational device, J0(x) - Objective function, p(x) - Finite hybrid model, u i -Test vector, x i -Target vector, Σπ i p i - Existing probability density function, π0p0 - New probability density function.

Claims

1. A computational method for finding a feasible solution region using a set-based design approach, characterized in that, The computational method for finding the feasible solution region using ensemble design methods includes a survival selection step. In this step, based on differential evolution, a survival selection is performed using an objective function on the target vector (considered a feasible solution) and the experimental vector generated through mutation. If the value of the objective function obtained by substituting the experimental vector is less than the value of the objective function obtained by substituting the objective vector, then the experimental vector is selected as the new feasible solution. The objective function includes a probability density function that takes the objective vector and the test vector as input and corresponds to the distribution of the calculated feasible solution. The closer the value of the probability density function is to 0, the smaller the value of the objective function.

2. The computational method for finding the feasible solution region using a set-based design approach according to claim 1, characterized in that, The survival selection step is performed multiple times. The probability density function is the sum of an existing probability density function generated based on the set of feasible solutions obtained before the predetermined number of survival selection steps and a newly generated probability density function generated based on the set of feasible solutions obtained after the predetermined number of survival selection steps. Whenever a new feasible solution is selected through the survival selection step, the newly added probability density function is updated.

3. The calculation method for finding the feasible solution region using the set-based design method according to claim 2, characterized in that, When the number of feasible solutions contained in the newly added probability density function reaches a specified number, the newly added probability density function is added to the existing probability density function.

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