An optimization design method for solid models based on three-dimensional interpolation

Abstract

The present invention relates to the field of computer-aided engineering technology, and discloses an optimized design method for a solid model based on three-dimensional interpolation. By establishing a rod element model of a design domain, calculating the stress and strain of the rod using the finite element principle, and gradually removing the rod elements with lower stress through iterative calculation within the range of allowable stress and maximum flexibility constraints, an optimized rod element model is obtained. Then, according to the nominal density three-dimensional interpolation function, it is transformed into an optimized model with small cubes as basic units, which can effectively reduce the scale of the optimization calculation, reduce the computer resource occupancy rate, and improve the optimization rate.

Description

Technical Field

[0001] The present invention relates to the field of computer-aided engineering technology, and particularly relates to an optimization design method for solid models based on three-dimensional interpolation. Background Art

[0002] In the product structure design engineering, the structure of industrial equipment needs to minimize its mass while ensuring the structural strength. Therefore, the optimization design is an important means to achieve lightweight design and plays a huge role in practical engineering applications. At present, the structural optimization design mainly uses the discrete model based on volume elements for topology optimization design, but this method has a very large amount of calculation, requires a high-performance computer, and spends a lot of calculation time. Summary of the Invention

[0003] Aiming at the deficiencies of the prior art, the present invention provides an optimization design method for solid models based on three-dimensional interpolation, which can effectively reduce the scale of optimization calculation, reduce the computer resource occupancy rate, and improve the optimization rate.

[0004] To achieve the above object, the present invention provides the following technical solutions:

[0005] An optimization design method for solid models based on three-dimensional interpolation, comprising the following steps:

[0006] Step S1: Determine the shape and size of the design domain according to the usage requirements of the solid model, and use a non-overlapping arrangement of cubic unit cells closely attached to cover the entire design domain as the initial finite element model. The cubic unit cell is constructed by 28 linearly elastic rods obtained by connecting two by two the eight nodes on the cubic unit cell.

[0007] Step S2: Determine the displacement constraint region corresponding to the initial finite element model and the solid model, and set the nodes in the region to have the same displacement constraints as the solid model; determine the load region corresponding to the initial finite element model and the solid model, and evenly distribute the load value of the solid model to all the nodes in the corresponding region of the initial finite element model, with the load direction being the same as the load direction of the solid model.

[0008] Step S3: Establish the design objective, objective function, constraint conditions, and design variables of the optimization design;

[0009] Step S4: Perform iteration on the basis of the initial finite element model to obtain an iterative model, and then optimize to obtain the final finite element model;

[0010] Step S5: Statistically analyze each cubic unit cell of the final finite element model to obtain the nominal density at the eight nodes of each cubic unit cell;

[0011] Step S6: Perform three-dimensional interpolation on the cubic unit cell based on the nominal densities at the eight nodes of each cubic unit cell obtained in Step S5 to obtain the nominal density at each point within the cubic unit cell;

[0012] Step S7: Divide each cubic unit cell into n small cubes according to n×n×n, where n is an integer greater than or equal to 1; based on the nominal density at each point within the cubic unit cell obtained in Step S6, obtain the average nominal density of each small cube, compare the average nominal density of each small cube with the preset minimum threshold ρ of the average nominal density 3 and perform an operation to retain or remove the small cube, and finally obtain the optimized solid model. *

[0013] Further, in Step S3, the design goal is for the solid model to achieve the minimum mass under the condition of satisfying the constraint conditions; the objective function is the mass of the solid model; the constraint conditions are a combination of one or more of the following two constraint conditions: the local stress of the solid model is within the allowable stress range or the maximum flexibility is within the allowable flexibility range; the design variable is the presence or absence of the rod member.

[0014] Further, the specific steps of Step S4 are as follows:

[0015] Step S41: Establish a finite element model of rod elements for the solid model, where the numbers and positions of the rod elements correspond one-to-one with the numbers and positions of the linear elastic rods, and the lengths and cross-sectional areas of the rod elements are the same as those of the linear elastic rods;

[0016] Step S42: In each iteration step calculated based on the finite element principle, compare the absolute stress values of each rod element, delete the rod elements with smaller absolute stress values in the previous iteration model, and obtain the iteration model in each iteration step;

[0017] Step S43: Check whether the iteration model satisfies the constraint conditions. If it satisfies the constraint conditions, repeat Step S42 for optimized iterative calculation. If it does not satisfy the constraint conditions, stop the optimized iterative calculation and use the previous iteration model as the final optimized result of the finite element model.

[0018] Further, the method for obtaining the nominal densities at the eight nodes of each cubic unit cell in Step S5 is as follows:

[0019] Perform statistics on each cubic unit cell of the final finite element model to obtain the number of rod elements connecting each of the eight nodes of each cubic unit cell to the other seven nodes of the cubic unit cell, denoted as N i , (i = 1, 2,..., 8), convert the number of rod elements of each cubic unit cell into the nominal density of the corresponding node, that is, through ρ i = Ni , (i=1,2,…,8) is converted to the nominal density ρ of the corresponding node i , where the coordinates and nominal density of each node are recorded as P i (x i ,y i ,z i ,ρ i ),(i=1,2,…,8).

[0020] Furthermore, the specific steps of step S6 are:

[0021] Step S61: construct a basic function form of three-dimensional interpolation as ρ(x, y, z) = (ax + b)(cy + d)(ez + f), and solve the following overdetermined equations according to the nominal density at the eight nodes:

[0022]

[0023] Among them, a, b, c, d, e, and f are coefficients, and the least squares solution of the overdetermined system of equations is obtained by the least squares method, that is, six coefficients;

[0024] Step S62: determining a three-dimensional interpolation function of the nominal density of a single cubic unit cell according to the equation ρ(x, y, z)=(ax+b)(cy+d)(ez+f), thereby calculating the nominal density at each point in the single cubic unit cell;

[0025] Step S63: performing the operations of steps S61 and S62 on each cubic unit cell in sequence.

[0026] Furthermore, the specific steps of step S7 are: according to the numbering sequence of the cubic unit cells in the final finite element model obtained in step S4, each cubic unit cell is divided into n units according to n×n×n. 3 small cubes, n is an integer greater than or equal to 1; in each cubic unit cell, the average value of the nominal density at the eight vertices of the small cube is taken as the average nominal density of the small cube, and the average nominal density is less than the preset average nominal density minimum threshold ρ * The small cubes are deleted, and the rest of the small cubes are retained, and finally the optimized solid model is obtained.

[0027] It should be noted that the numbers of the cubic unit cells in the above-mentioned final finite element model (hereinafter referred to as the final numbers) follow the numbers of the initial finite element model (hereinafter referred to as the original numbers). However, due to the optimization in the intermediate process, some cubic unit cells are removed (because the corresponding rod elements have been removed). Therefore, compared with the original numbers, the final numbers have changed. There are two specific methods to obtain the final numbers. One is to directly remove the cubic unit cells with the original numbers, and the remaining cubic unit cells are not renumbered. The numbers of the cubic unit cells are unique and still have the identification function. The other is to renumber them sequentially after removal. The use of the numbers is to prevent omission of the cubic unit cells to be processed.

[0028] Preferably, the side length of the cubic unit cell is 0.1 to 0.5 times the shortest width of the initial finite element model, which can ensure the optimization calculation accuracy while improving the calculation speed. Please supplement the advantages of the selection of this numerical range here.

[0029] Preferably, the cross-section of the linear elastic rod is circular, and the area of the circular cross-section is 0.01 to 0.03 times the square of the side length of the cubic unit cell, so that the finally optimized solid model has a better filling degree and is suitable for physical production using 3D printing methods. Please supplement the advantages of the selection of this numerical range here.

[0030] Compared with the prior art, the present invention provides an optimization design method for a solid model based on three-dimensional interpolation, having the following beneficial effects: compared with the optimization using rod elements, when using solid elements for optimization, the scale of the stiffness matrix in the finite element calculation can be reduced, the occupation of computer memory can be reduced, and the requirements for computer performance can be lowered; at the same time, the smaller matrix scale can reduce the calculation time for each iteration and improve the optimization rate, which is a very suitable structural optimization method for fields such as rapid prototyping and additive manufacturing.

[0031] (2) The optimization method of the present invention establishes a rod element model of the design domain, calculates the stress and strain of the rod using the finite element principle, and gradually removes the rod elements with lower stress through iterative calculation within the range of allowable stress and maximum flexibility constraints to obtain an optimized rod element model, and then converts it into an optimized model with small cubes as the basic unit according to the nominal density three-dimensional interpolation function, which can effectively reduce the scale of the optimization calculation, reduce the computer resource occupancy rate, and improve the optimization rate. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 is the initial finite element model in the embodiment;

[0033] Figure 2 is the final finite element model in the embodiment;

[0034] Figure 3It is the nominal density contour map inside the slice with an interval of 2.5 mm after three-dimensional interpolation of a single cubic unit cell in the embodiment, successively being Figure (3a), Figure (3b), Figure (3c), Figure (3d), and Figure (3e);

[0035] Figure 4 It is the nominal density distribution map of the small cubes after dividing a single cubic unit cell in the embodiment according to 4×4×4;

[0036] Figure 5 It is the final obtained solid model after interpolation processing in the embodiment. Detailed implementation manners

[0037] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0038] Unless otherwise specifically stated, the relative arrangements, numerical expressions, and numerical values of the components and steps described in these embodiments do not limit the scope of the present invention. At the same time, it should be understood that for the convenience of description, the sizes of the various parts shown in the drawings are not drawn according to the actual proportional relationship. Technologies, methods, and devices known to those of ordinary skill in the relevant art may not be discussed in detail, but in appropriate cases, the said technologies, methods, and devices should be regarded as a part of the authorization specification. In all the examples shown and discussed here, any specific value should be interpreted as merely exemplary, rather than as a limitation. Therefore, other examples of the exemplary embodiments may also include different values. It should be noted that similar reference numerals and letters indicate similar items in the following drawings. Therefore, once an item is defined in one drawing, it does not need to be further discussed in the subsequent drawings.

[0039] In the description of the present application, it should be understood that the orientation or positional relationship indicated by the terms "center", "longitudinal", "transverse", "upper", "lower", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. is based on the orientation or positional relationship shown in the drawings, and is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and thus cannot be understood as a limitation to the protection content of the present invention.

[0040] A method for optimizing the design of a solid model based on three-dimensional interpolation according to the present invention includes the following steps:

[0041] Step 1: According to the characteristics of the cuboid entity model, the design domain is determined to be a cuboid with a length, width, and height of 100 mm, 30 mm, and 30 mm respectively. As shown in Figure 1 , a cubic unit cell is used to closely fit and arrange without overlap throughout the design domain as the initial finite element model. The cubic unit cell consists of 8 nodes and 28 linear elastic rods that connect these nodes pairwise. The side length of the cubic unit cell is 10 mm, and the cross-sections of the linear elastic rods are all circular with an area of π mm 2 .

[0042] Step 2: As shown in Figure 1 , the displacement constraint region and load region corresponding to the initial finite element model and the entity model are determined. The displacement constraint is located at the upper and lower two columns of nodes at the left end of the initial finite element model, and the load constraint is located at the lower right end of the initial finite element model. A force with a magnitude of 2 N and a vertically downward direction is applied at the node.

[0043] Step 3: Establish a mathematical model to determine the design objective, objective function, constraint conditions, and design variables of the optimization design. The design objective is for the entity model to achieve the minimum mass under the premise of meeting the constraint conditions. The objective function is the mass of the entity model. The constraint conditions are a combined constraint condition of either the local stress of the entity model being within the allowable stress range or the maximum flexibility being within the allowable flexibility range. The design variable is the presence or absence of the rod.

[0044] Step 4: Perform iterative optimization on the basis of the initial finite element model to obtain an iterative model, and then optimize to obtain the final finite element model, including the following steps;

[0045] 4.1: Establish a finite element model of rod elements for the entity model. The numbers and positions of the rod elements correspond one by one to the numbers and positions of the linear elastic rods. The lengths and cross-sectional areas of the rod elements are the same as those of the linear elastic rods;

[0046] 4.2: In each iteration step calculated based on the finite element principle, compare the absolute stress values of each rod element, and delete the rod elements with smaller absolute stress values in the previous iteration model to obtain the iterative model in each iteration step;

[0047] 4.3: Check whether the iterative model meets the constraint conditions. If it meets the constraint conditions, repeat step 4.2 for optimization iterative calculation. If it does not meet the constraint conditions, stop the optimization iterative calculation and use the previous iteration model as the optimization result of the final finite element model. As shown in Figure 2 , in the final finite element model, most of the rod elements at the front end of the protruding part of the cantilever beam are removed, and only two groups of rod elements in the horizontal and diagonal directions are retained. Also, a large number of rod elements in the middle section of the cantilever beam are removed, and only a few diagonal rod elements used to support the cantilever structure are retained.

[0048] Step 5: For each cubic unit cell in the final finite element model, count the number of rod elements connecting each of the eight nodes of the cubic unit cell to the remaining seven nodes of the cubic unit cell, denoted as N i , (i = 1, 2, …, 8), convert the number of rod elements in each cubic unit cell to the nominal density of the corresponding node, i.e., through ρ i = N i , (i = 1, 2, …, 8) to convert to the nominal density ρ of the corresponding node i , where the coordinates and nominal density of each node are recorded as P i (x i , y i , z i , ρ i ), (i = 1, 2, …, 8).

[0049] Step 6: Perform three-dimensional interpolation on the cubic unit cell according to the nominal density at the eight nodes of each cubic unit cell obtained in Step 5, so as to obtain the nominal density at each point within the cubic unit cell.

[0050] The steps for three-dimensional interpolation of a single cubic unit cell are as follows:

[0051] 6.1: Construct the basic function form of three-dimensional interpolation as ρ(x, y, z) = (ax + b)(cy + d)(ez + f), where there are six undetermined coefficients. For example, the eight interpolation nodes of the cubic unit cell in the lower left corner are P1(0, 0, 0, 3), P2(10, 0, 0, 5), P3(0, 10, 0, 3), P4(10, 10, 0, 3), P5(0, 0, 10, 1), P6(10, 0, 10, 2), P7(0, 10, 10, 0), P8(10, 10, 10, 1). Substitute the eight interpolation nodes into the function to obtain an overdetermined system of equations:

[0052]

[0053] Use the least squares method to find the least squares solution of the system of equations, and calculate that the six undetermined coefficients a, b, c, d, e, f are 0.9578, 8.1713, -0.02461, -2.799, 0.009882, -0.1452 respectively;

[0054] 6.2: Substitute the six undetermined coefficients into ρ(x, y, z) = (ax + b)(cy + d)(ez + f) to obtain the nominal density interpolation function inside a single cubic unit cell: ρ(x, y, z) = (0.9578x + 8.17130)(-0.02461y - 2.799)(0.009882z - 0.1452), and calculate the nominal density value at each point inside a single cubic unit cell, such asFigure 3 As shown; the nominal density distribution map of the interior obtained by slicing the face-centered cubic unit cell at intervals of 2.5 mm in the y-axis direction, and the nominal density distribution maps of the successive slices are shown in Figures (3a), (3b), (3c), (3d), and (3e); since the nominal density interpolation function inside each cubic unit cell can be determined according to the method of the present invention, the nominal density at any position of each small cube can be calculated based on its coordinates.

[0055] 6.3: Perform the operations in 6.1 to 6.2 on each cubic unit cell in sequence.

[0056] Step 7: Divide the single cubic unit cell of 4×4×4 into 64 small cubes, and use the average of the nominal densities at the eight vertices of the small cube as the average nominal density of the small cube. As Figure 4 shown, delete the small cubes with an average nominal density less than 2, and retain the remaining small cubes. Finally, obtain the optimized solid model, as Figure 5 shown.

[0057] It should be noted that in this application, relational terms such as first and second are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the terms "comprising", "including" or any other variant thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements not only includes those elements, but also includes other elements not expressly listed, or also includes elements inherent to such process, method, article or device. Without further limitation, an element defined by the statement "including a..." does not exclude the existence of additional identical elements in the process, method, article or device including the said element.

[0058] Although the embodiments of the present invention have been shown and described, for those of ordinary skill in the art, it can be understood that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. An optimization design method for solid models based on three-dimensional interpolation, characterized in that, It includes the following steps: Step S1: Determine the shape and size of the design domain according to the usage requirements of the solid model. Adopt a layout method in which cubic unit cells are closely fitted without overlap to cover the entire design domain as the initial finite element model. The cubic unit cell is constructed by 28 linear elastic rods obtained by connecting the eight nodes of the cubic unit cell in pairs; Step S2: Determine the displacement constraint region corresponding to the initial finite element model and the solid model, and set the nodes in the region to have the same displacement constraints as the solid model; Determine the load region corresponding to the initial finite element model and the solid model, and evenly distribute the load value of the solid model to all the nodes in the corresponding region of the initial finite element model. The load direction is the same as the load direction of the solid model; Step S3: Establish the design goal, objective function, constraint conditions, and design variables for the optimal design; Step S4: Perform iteration on the basis of the initial finite element model to obtain an iterative model, and then optimize to obtain the final finite element model; Step S5: Statistically analyze each cubic unit cell of the final finite element model to obtain the nominal density at the eight nodes of each cubic unit cell; Step S6: Perform three-dimensional interpolation on the cubic unit cells according to the nominal density at the eight nodes of each cubic unit cell obtained in Step S5 to obtain the nominal density at each point within the cubic unit cell; Step S7: Divide each cubic unit cell into n small cubes according to n×n×n, where n is an integer greater than or equal to 1; according to the nominal density at each point in the cubic unit cell obtained in the said step S6, obtain the average nominal density of each small cube, compare the average nominal density of each small cube with the preset minimum threshold ρ of the average nominal density 3 of the size, and perform operations of retaining or removing the small cube, and finally obtain the optimized solid model; * ​ The specific steps of Step S6 are as follows: Step S61: Construct the basic function form of three-dimensional interpolation as ρ(x, y, z) = (ax + b)(cy + d)(ez + f), and solve the following overdetermined system of equations according to the nominal density at the eight nodes: where a, b, c, d, e, f are coefficients, and use the least squares method to find the least squares solution of the overdetermined system of equations, that is, the six coefficients; Step S62: Determine the three-dimensional interpolation function of the nominal density of a single cubic unit cell according to the equation ρ(x, y, z) = (ax + b)(cy + d)(ez + f), so as to calculate the nominal density at each point within a single cubic unit cell; Step S63: Perform the operations of Step S61 and Step S62 on each cubic unit cell in turn.

2. The optimized design method of the solid model based on three-dimensional interpolation according to claim 1, wherein: In Step S3, the design goal is to minimize the mass of the solid model under the premise of meeting the constraint conditions; the objective function is the mass of the solid model; the constraint conditions are one or a combination of the following two constraint conditions: the local stress of the solid model is within the allowable stress range or the maximum flexibility is within the allowable flexibility range; the design variable is the presence or absence of the rod; 3. A method for optimizing the design of a solid model based on three-dimensional interpolation according to claim 1, characterized in that: The specific steps of Step S4 are as follows: Step S41: Establish a finite element model of rod elements for the solid model. The numbers and positions of the rod elements correspond one by one to the numbers and positions of the linear elastic rods. The lengths and cross-sectional areas of the rod elements are the same as those of the linear elastic rods; Step S42: In each iteration step calculated based on the finite element principle, compare the absolute stress values of each rod element, and delete the rod elements with smaller absolute stress values in the previous iteration model to obtain the iterative model in each iteration step; Step S43: Check whether the iterative model meets the constraint conditions. If it meets the constraint conditions, repeat Step S42 for optimized iterative calculation. If it does not meet the constraint conditions, stop the optimized iterative calculation and use the iterative model of the previous step as the final optimized result of the finite element model.

4. A method for optimizing the design of a solid model based on three-dimensional interpolation according to claim 1, characterized in that, The method for obtaining the nominal density at the eight nodes of each cubic unit cell in Step S5 is as follows: Statistically analyze each cubic unit cell of the final finite element model to obtain the number of rod elements connecting each of the eight nodes of each cubic unit cell to the remaining seven nodes of that cubic unit cell, denoted as N i , where \(i = 1, 2, \ldots, 8\). Convert the number of rod elements of each cubic unit cell to the nominal density of the corresponding node, that is, through \(\rho\ i = N\ i , \(i = 1, 2, \ldots, 8\) to convert to the nominal density \(\rho\ i of the corresponding node. Among them, the coordinates and nominal density of each node are recorded as \(P\ i (x\ i , y\ i , z\ i , \(\rho\ i ), \(i = 1, 2, \ldots, 8\).

5. A method for optimizing the design of a solid model based on three-dimensional interpolation according to claim 1, characterized in that: The side length of the cubic unit cell is taken as 0.1 - 0.5 times the shortest width of the initial finite element model.

6. The optimization design method of a solid model based on three-dimensional interpolation according to claim 1 is characterized in that: The cross-section of the linear elastic rod is circular, and the area of the circular cross-section is 0.01 - 0.03 times the square of the side length of the cubic unit cell.

7. A method for optimizing the design of a solid model based on three-dimensional interpolation according to claim 1, characterized in that The specific steps of step S7 are as follows: each cubic unit cell is divided into n small cubes in the order of n×n×n, where n is an integer greater than or equal to 1; within each cubic unit cell, the average of the nominal densities at the eight vertices of the small cube is taken as the average nominal density of the small cube, and the small cubes with an average nominal density less than the preset minimum threshold ρ of the average nominal density are deleted, and the remaining small cubes are retained, and finally an optimized solid model is obtained. 3 small cubes, where n is an integer greater than or equal to 1; within each cubic unit cell, the average of the nominal densities at the eight vertices of the small cube is taken as the average nominal density of the small cube, and the small cubes with an average nominal density less than the preset minimum threshold ρ * of the average nominal density are deleted, and the remaining small cubes are retained, and finally an optimized solid model is obtained.

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