Method for calculating feasible solution domains using set-based design techniques

Abstract

It is possible to reduce the computational load during the search for feasible solutions compared to the conventional art. [Solution Means] Trial vector u i is generated by mutation. Target vector x i When the value of the objective function J0(x) substituted with trial vector u i is smaller than the value of the objective function J0(x) substituted with , the trial vector u i is selected as a new feasible solution. The objective function J0(x) includes the target vector x i and the trial vector u i A probability density function Σπ i p i (x|θ i )+π0p0(x|θ0). The probability density function Σπ i p i (x|θ i )+π0p0(x|θ0) corresponds to the distribution of the feasible solutions that have already been obtained. The probability density function Σπ i p i (x|θ i )+π0p0(x|θ0), the closer the value is to 0, the smaller the value of the objective function J0(x) becomes.

Description

[Technical Field]

[0001] This specification discloses a method for calculating feasible solution regions using a set-based design method.

[0002] When designing vehicle components such as suspensions and powertrain mounts, a wide range of design elements must be considered. For example, factors such as handling stability, vibration and noise reduction, weight reduction, and cost reduction are all considered during the design phase.

[0003] For example, Patent Document 1 rejects design variables that have little design contribution when the order of multiple design variables exceeds a predetermined order. Also, Patent Document 2 and Non-Patent Document 1 calculate feasible solutions based on a set-based design method.

[0004] Two well-known methods for determining multiple design variables that satisfy the requirements are point-based design and set-based design. In point-based design, arbitrary values ​​are set for each of the multiple design variables. Then, a prototype created based on these set values ​​is experimentally tested to determine whether or not it satisfies the requirements. Based on the experimental results, the prototype is improved. In this way, the cycle of experimentation and prototype improvement is repeated.

[0005] In set-based design methods, tolerance ranges are defined for each design variable. The set of feasible solutions that simultaneously satisfy these tolerance ranges is then determined as the feasibility domain. The ease of design modification and performance robustness are then considered for the multiple feasible solutions included in the feasibility domain.

[0006] For example, Non-Patent Document 1 uses an optimization method based on differential evolution as a method for determining the feasible region. In other words, Non-Patent Document 1 calculates trial vectors as candidate feasible solutions. Trial vectors are generated by an operation called mutation in differential evolution. Then, the trial vectors are compared with target vectors that already exist as feasible solutions.

[0007] The two vectors are compared using an objective function. The objective function J0(x) is represented by the following Equation (1). When the value of the objective function J0(x) into which the trial vector is substituted is smaller (more negatively larger) than the value of the objective function J0(x) into which the target vector is substituted, the trial vector is pooled in the solution set as a new feasible solution. The set of pooled feasible solutions serves as the feasible region.

Equation

[0008] In order to make the value of the objective function J0(x) smaller (more negatively larger), the value of the second term on the right side only needs to be increased. The function h(x) in the parentheses of the second term on the right side is a Heaviside function, which takes a value of 0 when x<0 and a value of 1 when x>0. In addition, the function g KS (x) is a constraint function g i (x) converted into a cost function. Further, the function d(x) of the second term on the right side is represented by the following Equation (2).

Equation

[0009] Here, x' refers to a feasible solution found in the past. Taking the calculation for a trial vector as an example, in Equation (2), the distances between the trial vector x and all already found feasible solutions x' are determined. Then, the minimum value is extracted from the determined distances. That is, when the trial vector is farther away than any of the existing feasible solutions, the value of d(x) increases. With reference to Equation (1), at this time, the value of the objective function becomes smaller (more negatively larger).

Prior Art Documents

Patent Documents

[0010]

Patent Document 1

[0011] [Non-Patent Document 1] Nishikawa et al., "Set-Based Design Method Based on Differential Evolution and its Application to Suspension Design," Transactions of the Society of Automotive Engineers of Japan, November 2023, Society of Automotive Engineers of Japan, Transactions of the Society of Automotive Engineers of Japan, Vol. 54, No. 6, pp. 1121-1126. [Overview of the project] [Problems that the invention aims to solve]

[0012] By the way, in equation (2) above, all feasible solutions x' that have already been found 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 during the search for feasible solutions compared to conventional methods. [Means for solving the problem]

[0014] This specification discloses a method for calculating feasible solution regions using a set-based design method. This calculation method includes a survival selection step. In the survival selection step, survival selection is performed on the target vector and trial vectors using an objective function based on differential evolution. The target vector is a feasible solution that has already been obtained. The trial vector is generated by mutation. The trial vector is selected as a new feasible solution if the value of the objective function to which the trial vector is substituted is smaller than the value of the objective function to which the target vector is substituted. 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 depends on the distribution of feasible solutions that have already been obtained. 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 configuration, the closer the value of the probability density function is to 0, the smaller the value of the objective function becomes. In other words, the lower the probability of a feasible solution already being found is when a trial vector is plotted, the more likely it is that the trial vector will be selected as a new feasible solution. With this type of calculation, a trial vector plotted far from any already found feasible solution can be selected as a feasible solution without having to calculate the distance between all feasible solutions and the trial vector.

[0016] Furthermore, in the above configuration, the survival selection step may 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 the survival selection step in a predetermined number of times. The new probability density function is generated based on the set of feasible solutions obtained after the survival selection step in the predetermined number of times. Each time a new feasible solution is selected by the survival selection step, the new probability density function is updated.

[0017] With the above configuration, when the number of feasible solutions increases, the update to the probability density function only requires updating the new probability density function.

[0018] Furthermore, in the above configuration, when the number of feasible solutions included in the new probability density function reaches a predetermined number, the new probability density function may be added to the existing probability density function.

[0019] According to the above configuration, the bloat of the new probability density function is suppressed. [Effects of the Invention]

[0020] The method for calculating feasible solution regions using a set-based design method disclosed herein makes it possible to reduce the computational load during the search for feasible solutions compared to conventional methods. [Brief explanation of the drawing]

[0021] [Figure 1] This figure illustrates the hardware configuration of a calculation device that performs the feasible solution region calculation method according to this embodiment. [Figure 2] This flowchart illustrates the process for determining design specifications. [Figure 3] This diagram illustrates the details of the feasible area search step in the flowchart shown in Figure 2. [Figure 4] This diagram illustrates the executable area for variables A and B. [Figure 5] This figure shows an example of finding the set of feasible solutions based on conventional techniques. [Figure 6] This figure illustrates a set of feasible solutions obtained by the feasible solution region calculation method according to this embodiment. [Modes for carrying out the invention]

[0022] 1. Hardware Configuration Figure 1 illustrates the hardware configuration of the calculation device. This calculation device executes the method for calculating the executable area according to this embodiment. The calculation device comprises a computing device 10, an input unit 20A, and a display 20B.

[0023] The input unit 20A consists of, for example, a keyboard and a mouse. The display 20B displays a set of feasible solutions (feasible region) as shown in Figures 5 and 6, which will be described later. The computing device 10 includes a CPU 10A, RAM 10B, storage 10C, ROM 10D, and an input / output controller 10E.

[0024] The CPU 10A is the central processing unit, also called a processor. The RAM 10B is a temporary storage device, such as a volatile memory, that stores data being worked on. The ROM 10D is a storage device that allows data to be read. The storage 10C is a storage device that allows data to be written to and read from. The storage 10C consists of, for example, an HDD (Hard Disk Drive) or an SSD (Solid State Drive).

[0025] When the CPU 10A executes a program stored in the storage 10C or ROM 10D, the computing device 10 becomes capable of executing a feasible solution calculation flow, as illustrated in Figure 3. For example, functional blocks capable of executing each step (S41-S48) within step S40 of the flowchart illustrated in Figure 3 are constructed in the computing device 10.

[0026] The computing device 10 is used in the development and design of vehicle components. For example, the computing device 10 is used in the development and design of power plant mounts and suspensions.

[0027] 2. Process for determining design specifications In the method for calculating the feasible domain according to this embodiment, a set-based design method is employed. In the set-based design method, various design parameters are considered. For example, Figure 2 illustrates the process for calculating the feasible solution domain based on the set-based design method. For example, steps S10, S20, S30, and S50 are performed through discussions among the design and development team. Step S40 is performed by the computing device 10.

[0028] Referring to step S10 in Figure 2, the design specifications for the vehicle component under design are set. These design specifications include various factors such as handling stability, vibration and noise reduction, weight reduction, and cost reduction. Furthermore, target performance values ​​are determined for each vehicle component under design. For example, target performance values ​​are determined for each design specification through market research.

[0029] Next, the design parameters are screened based on experimental design (S20). For example, design parameters that have little contribution to the target performance value are removed.

[0030] Furthermore, a regression model between the design parameters and performance values ​​is created for the design parameters after screening (S30). For example, the regression model between the design parameters and performance values ​​is created using a Gaussian process or a neural network. For example, the relationship between a given design parameter and performance value is quantitatively shown by the model function.

[0031] Furthermore, an optimization process is performed on the design specifications. This optimization process searches for the feasible region for each design specification (S40). This search step is performed by the computing device 10.

[0032] First, design variables are defined for each design parameter. For example, for n types of design parameters, the design variables x i (i=1,···,n) is defined. Then, an optimization process is performed on the design variable vector x. In the optimization process, each design variable x i The feasible region is searched for. The feasible region is obtained as a set of feasible solutions, as illustrated in Figure 6. Details of the optimization process in step S40 will be described later.

[0033] In step S40, each design variable x i Once the feasible domain is determined, the design parameters are decided (S50). For example, the ease of design changes and performance robustness are considered by the design and development team. Finally, the design values ​​for each design parameter are determined from within the feasible domain.

[0034] 3. Optimization process based on set-based design methodology FIG. 3 illustrates the detailed steps of step S40. In step S40, optimization processing is performed based on a set-based design method. Each step S41-S48 of step S40 is executed by the computing device 10.

[0035] Based on the regression model obtained in step S30, as exemplified in the following Equation (3), an upper limit value and a lower limit value that satisfy the required performance are defined for each design variable x i . [Equation]

[0036] The optimization processing in step S40 can be regarded as processing for solving an optimization problem that minimizes an objective function with respect to the design variable vector x in Equation (3). As in Equation (14) described later, the objective function includes a probability density function Σπ corresponding to the distribution of already obtained feasible solutions i p i (x|θ i )+π0p0(x|θ0) is generated. As will be described later, the smaller the value of the probability density function into which the trial vector is substituted, the higher the possibility that the trial vector is selected as a new feasible solution.

[0037] With reference to FIG. 3, the optimization processing is executed based on a differential evolution method. First, initial individuals are generated based on the design variable vector x (S41). That is, an n-dimensional real-valued vector x m is randomly generated. A population P shown in the following Equation (4) formed as a result thereof serves as initial individuals. [Equation]

[0038] Next, a target vector x as exemplified in the following Equation (5) i , a base vector x b , x r1 , x r2The selected target vector x i This is a feasible solution that has already been found. Also, the base vector x b ,x r1 ,x r2 This refers to the mutation vector v(F) and trial vector u, which will be described later. i Used to generate [something].

number

[0039] Using these vectors, a mutation vector v(F) is generated by mutation as exemplified in equation (6) below (S42).

number

[0040] However, F is a scale vector. Also, the operator following the scale vector F (the white circle) is the Hadamard product. The initial value of the scale vector F is set from the random number shown in the following equation (7).

number

[0041] Furthermore, the scale vector F is updated. For example, the elements of the scale vector F F i (i=1,···,n) can be found by the following formula (8).

number

[0042] In other words, among the points x' belonging to the feasible region, the value with the maximum distance from the nearest neighbor is element F. i The value will be...

[0043] Next, the target vector x i The binary crossover between the trial vector u and the mutation vector v(F) yields the trial vector u iThis is generated (S43). In other words, through mutation and crossover, the trial vector u i The following is generated: Trial vector u i This is a candidate for a new feasible solution. Trial vector u i element u ij This can be calculated based on the following formula (9).

number

[0044] Here, C r is the crossover rate, R∈{1,···,n} is a randomly selected subscript, r ij ~U(0,1) is a uniformly random value.

[0045] The trial vector u generated by equation (9) i and target vector x i These are compared using the objective function J0(x) (S44). This comparison is called survival selection. As shown in the loop from step S42 to S48 in Figure 3, survival selection is performed multiple times. In survival selection, a new feasible solution x i,new This can be calculated based on the following formula (10).

number

[0046] In other words, the trial vector u i The value of the objective function J0(x) into which the value of the target vector x is substituted is i If the trial vector u is smaller (negatively large) than the value of the objective function J0(x) into which it is substituted, i This represents a new feasible solution.

[0047] 3. Objective function The objective function J0(x) in this embodiment is shown in the following equation (11).

number

[0048] Note that the function h(x) in the second term on the right-hand side is the Heaviside function. KS (x) is a set of multiple constraint functions g i (x) is a cost function g KS This is obtained by transforming it into (x). Constraint function g i (x) is exemplified in the following formula (12). Furthermore, the KS function g KS (x) is exemplified by the following formula (13).

number

number

[0049] In equation (13), α > 0 corresponds to the KS function g. KS This parameter adjusts the approximation of the max operation using (x). For example, the larger the value of α, the more equivalent the solution to equation (13) becomes to the result of the max operation.

[0050] Furthermore, the function p(x) in equation (11) represents a finite mixture model. In a finite mixture model, multiple probability density functions of feasible solutions are superimposed. The trial vector u is superimposed on this probability density function. i and target vector x i The following is substituted. The finite mixture model p(x) is shown in equation (14) below.

number

[0051] In equation (14), multiple conditional probability density functions πp(x|θ) are superimposed. These probability density functions πp(x|θ) correspond to the distribution of feasible solutions that have already been found.

[0052] Figure 4 illustrates feasible solutions and probability density functions for variables A and B. The black-filled plots represent feasible solutions when survival selection (S44 in Figure 3) is performed from the first to a predetermined number of times. In other words, the black-filled plots represent feasible solutions already obtained at the predetermined number of times. The white-outlined plots represent feasible solutions obtained after the survival selection step in the predetermined number of times. In other words, the white-outlined plots represent feasible solutions newly obtained after the predetermined number of times.

[0053] Furthermore, the probability density function is represented by circles of different radii. The closer to the center of the circles, the higher the probability density (i.e., closer to 1). Figure 4 illustrates the probability density functions π1p1(x|θ1), π2p2(x|θ2), π3p3(x|θ3), and π0p0(x|θ0). The probability density functions π1p1(x|θ1), π2p2(x|θ2), and π3p3(x|θ3) are determined according to the distribution of feasible solutions already found at a predetermined number of iterations. 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 found before the survival selection step at a predetermined number of iterations, the first term on the right-hand side of equation (14) will be referred to as the existing probability density function as appropriate below.

[0054] The probability density distribution π0p0(x|θ0) is determined based on the distribution of feasible solutions obtained after a predetermined number of survival selection steps. For this reason, the probability density distribution π0p0(x|θ0) will be referred to as the novel probability density function below.

[0055] In other words, the finite mixture model p(x) in equation (14) is the existing probability density function Σπ i p i (x|θ i It is the sum of ) and the new probability density function π0p0(x|θ0). Therefore, the finite mixture model p(x) can take values ​​from 0 to 1. Referring to equation (11) which shows the objective function, the closer the value of the finite mixture model p(x) is to 0 (smaller), the higher the value of the second term on the right-hand side of equation (11). The higher the value of the second term on the right-hand side, the smaller the value of the objective function becomes (larger to negative).

[0056] Referring to Figure 4, if we liken the probability density function to a mountain, the trial vector u i The value of the finite mixture model p(x) into which the value of the trial vector u is substituted is small if the base of the peak of the probability density function is small. i This indicates that the trial vector u is plotted in a region where no feasible solutions have already been found. i When plotted, the value of the objective function J0(x) becomes small (largely negative).

[0057] Here, in the prior art equations (1) and (2), the value of the objective function becomes small (negatively large) when the trial vector is farther away from any of the existing feasible solutions. In other words, in the equations (11) and (14) of this embodiment, the same effect as equations (1) and (2) can be obtained.

[0058] In equations (1) and (2), all feasible solutions x' that had already been obtained were compared with the target vector and the trial vector. In contrast, in equations (11) and (14) of this embodiment, the probability density functions are superimposed to form a finite mixture model Σπ. i p i (x|θ i By simply substituting the target vector and trial vector into )+π0p0(x|θ0), the same effect as equations (1) and (2) can be obtained. In other words, the calculation method according to this embodiment reduces the computational load compared to conventional calculation methods.

[0059] After survival selection is performed in step S44, it is determined whether the termination conditions for the optimization process are met (S45). For example, the termination conditions may be whether the number of generations obtained by differential evolution has reached a predetermined maximum number of generations. Alternatively, it may be determined whether generations in which the number of feasible solutions does not increase (i.e., generations in which the trial vector is rejected) have occurred consecutively for a predetermined number of generations.

[0060] If the termination condition for the optimization process has not been reached, it is determined whether the number of new feasible solutions (the number of white plots in Figure 4) included in the new probability density function π0p0(x|θ0) has reached a predetermined number (S46). In this step, if the number of new feasible solutions exceeds a predetermined update threshold, the new probability density function π0p0(x|θ0) is updated to match the existing probability density function Σπ i p i (x|θ i ) is added (S47). Then, in the next step S48, the second term on the right side of equation (14) is set to 0.

[0061] Furthermore, in step S46, if the number of new feasible solutions is less than or equal to a predetermined update threshold, the new probability density function π0p0(x|θ0) is updated based on the new feasible solutions (the number of outlined plots in Figure 4) (S48). The calculation process then returns to step S42.

[0062] 4. Example of calculating feasible solution regions Figure 5 illustrates the set of feasible solutions, or feasible region, calculated using the conventional method employing equations (1) and (2). The feasible solutions are plotted on a coordinate plane defined by variables A and B. The time required to search for the feasible region using the conventional method was 6680.40 seconds.

[0063] Figure 6 illustrates the executable region calculated according to this embodiment. The executable region illustrated in Figure 6 yields results equivalent to those of the executable region in Figure 5. Referring to Figure 6, the time required to search for the executable region according to this embodiment was 217.24 seconds. Thus, it can be seen that in this embodiment, the time required to obtain results equivalent to those of the prior art has been significantly reduced. [Explanation of Symbols]

[0064] 10 Computing device, J0(x) Objective function, p(x) Finite mixture model, u i Trial vector, x i Target vector, Σπ i pi Existing probability density function, π0p0 new probability density function.

Claims

1. A method for calculating feasible solution regions using a set-based design method, Based on differential evolution, it includes a survival selection step in which survival selection is performed using an objective function on a target vector, which is a feasible solution already obtained, and trial vectors generated by mutation. If the value of the objective function to which the trial vector is substituted is smaller than the value of the objective function to which the target vector is substituted, the trial vector is selected as the new feasible solution. The objective function includes a probability density function in which the target vector and the trial vector are substituted, and which corresponds to the distribution of feasible solutions already obtained. The closer the value of the probability density function is to zero, the smaller the value of the objective function becomes. A method for calculating feasible solution regions using a set-based design approach.

2. A method for calculating a feasible solution region using a set-based design method as described in claim 1, The aforementioned 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 survival selection step in a predetermined number of iterations, and a new probability density function generated based on the set of feasible solutions obtained after the survival selection step in a predetermined number of iterations. Each time a new feasible solution is selected by the survival selection step, the new probability density function is updated. A method for calculating feasible solution regions using a set-based design approach.

3. A method for calculating a feasible solution region using a set-based design method as described in claim 2, When the number of feasible solutions included in the new probability density function reaches a predetermined number, the new probability density function is added to the existing probability density function. A method for calculating feasible solution regions using a set-based design approach.

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