Repetitive unit-based discrete truss structure layout optimization design and manufacturing method

Abstract

This invention relates to a method for optimizing the layout and manufacturing of discrete truss structures based on repetitive elements, comprising the following steps: establishing a mathematical model for truss layout optimization, direct solution based on repetitive elements, a two-step solution to first determine the element layout, 3D printing manufacturing, and integrated assembly. The beneficial effects of this invention are: it introduces additional constraints and variables specific to the characteristics of repetitive elements, proposing a direct solution method based on repetitive elements, thus giving the truss layout optimization results the characteristics of repetitive elements; this direct solution method based on repetitive elements can be used independently or to provide a mathematical model for the two-step solution to first determine the element layout; when solving large-scale problems, it proposes a two-step solution to first solve the element pattern layout and then optimize the internal structure of the element pattern, making the structural force path clearer and significantly improving computational efficiency with minimal impact on the final optimized structural volume.

Description

Technical Field

[0001] This invention belongs to the fields of structural engineering and additive manufacturing technology, and particularly relates to a method for optimizing the layout design and manufacturing of discrete truss structures based on repeatable units. Background Technology

[0002] The increasing complexity of engineering projects has led to a growing demand for the optimized design and 3D-printed integrated assembly of complex truss structures. Topology optimization of structures includes both discrete and continuous structures; the optimization of truss structures, widely used in practical engineering, falls under the category of discrete structure topology optimization.

[0003] The layout optimization of truss structures is a linear programming problem. While traditional truss layout optimization algorithms can achieve numerically optimal solutions, the complexity of the structure can lead to situations where it is difficult or even impossible to manufacture in practice. To reduce manufacturing costs, common methods include adding additional constraints such as member classification and structural complexity limits to make the final result as easy to manufacture as possible, but this makes the optimization solution more complex. Introducing repetitive elements offers a reasonable and efficient solution to simplify the layout optimization problem.

[0004] In continuum optimization, optimization methods incorporating the concept of repetitive elements are proposed, primarily including the homogenization method and the small-scale unified optimization method. The homogenization method equates micro-units to a macroscopically homogeneous medium, enabling coarse-grained finite element analysis of the entire structure at the macroscopic level. However, without introducing additional constraints, the homogenization method cannot control the connectivity between units. The small-scale unified optimization method optimizes the entire structure at a smaller scale, at the cost of high computational cost. In architectural engineering, elements are generally discrete structures with a finite size relative to the design domain. Therefore, the aforementioned repetitive element algorithms for continuum optimization cannot be directly applied to solve the layout optimization problem of truss structures.

[0005] In summary, it is essential to study a method for optimizing the layout and manufacturing of discrete truss structures based on repetitive units, so as to realize the repetitive unit layout optimization, 3D printing manufacturing and integrated assembly of complex truss structures. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for optimizing the layout and manufacturing of discrete truss structures based on repeatable units.

[0007] This structural layout optimization design and manufacturing method based on repetitive units includes the following steps:

[0008] S1. Establishment of mathematical model for truss layout optimization: First, given the structural design domain, input the dimensions, load cases and boundary constraints, and specify the element mode and the corresponding element mode complexity; use lattice discretization to process the design domain, connect any two nodes to establish the minimum connection base structure; use the mechanical equilibrium equation as the constraint condition and the minimum total volume of the members as the design objective to establish a linear optimization model for truss layout optimization.

[0009] S2. Direct solution based on repetitive elements: Set a finite number of element patterns and divide them into elements to ensure that all members belong to a certain element pattern and there are no members that cross element patterns. The element layout and the area of ​​the members at the corresponding positions of the same element pattern are exactly the same. For each member in the truss structure, use a binary variable of the activated element pattern and add repetitive element constraints to form nonlinear constraints. Transform the nonlinear programming problem into a linear programming problem so that the repetitive elements can be directly optimized and solved.

[0010] S3. First, determine the two-step solution for the unit layout: First, reduce the complexity of each unit pattern, and perform the first solution according to step S2 to obtain the activated unit pattern variables corresponding to each unit pattern. Then, based on the normal complexity of each unit pattern, Substitute the nonlinear constraint expression of each bar in step S2 into the nonlinear constraint expression of the repeating element to transform it into a linear constraint, perform a second solution, and obtain the optimization result;

[0011] S4, 3D Printing Manufacturing and Integrated Assembly: Perform 3D modeling, slice various repetitive units in the optimized model and generate printing paths, perform 3D printing manufacturing, perform integrated assembly between repetitive units, and manufacture the optimized structure.

[0012] As a preferred option, in step S1: the objective function corresponding to minimizing the total volume of the rods is:

[0013]

[0014] The expression for the constraint is,

[0015]

[0016] In the formula, Let the length vector of the rod be the length vector. Let the area vector of the rod be... For the balance matrix, Let the internal force vector of the member be . For nodal load vectors, and These are the compressive and tensile strength vectors of the member, respectively;

[0017] The three constraints in equation (2) represent the force equilibrium equation, the stress constraint of the member, and the non-negativity constraint of the member area, respectively; the design variable is the member area vector. and the internal force vector of the rod , and These are the constant matrix and constant vector generated based on the bar topology, respectively. , and These are all constants determined by the actual working conditions.

[0018] Preferably, in step S1: the balance matrix of the minimum connection basis structure when optimizing the initial state. If a solution cannot be found, increase the component length threshold and mesh density in the minimum connectivity basis structure to form the minimum connectivity basis structure of the updated optimized initial state, and then solve the problem again.

[0019] Preferably, in step S2: a binary variable for activating the element pattern is set for each member; repetitive element constraints are added to each member of the truss structure, that is, first the design domain is filled with element patterns, and then the nodes inside each element pattern are connected in pairs to form members, thus forming a repetitive element structure; to ensure that the area of ​​members of the same type of element pattern is the same at the same location, when there are repetitive element constraints in the structure... When using the unit mode, add the following constraints to each member:

[0020]

[0021]

[0022] In the formula, Let c be the cross-sectional area of ​​the member, c be the element pattern number to which the member belongs, and m be the position number of the member in the element pattern. , … This is a binary variable representing the activated element mode of the element mode to which the member belongs. It takes the value 1 or 0, where 1 indicates that the element mode is activated and 0 indicates that it is not activated. , … This represents the possible cross-sectional area corresponding to the position of the member in the element pattern; in equation (4) , … and , … Both variables are variables, and multiplying two variables together forms a nonlinear constraint.

[0023] As a preferred option, in step S2: the nonlinear constraint of equation (4) is transformed into a linear constraint using the Big M method, resulting in equation (5).

[0024]

[0025] In the formula, M is a constant; each row in formula (5) represents a constraint of a unit pattern on the member at that location; when When the first unit mode is activated, the corresponding unit mode is used. ,the remaining At this point, the first row of constraint (5) becomes Other inequalities relax and actually have no effect; when When the second unit mode of the given unit mode is activated, that is... And the rest At this point, the second row of constraint (5) becomes Other inequality relaxations have no effect; similarly, when a certain element mode is activated, the constraints of the corresponding row of that element take effect, and the inequality constraints of other rows are relaxed.

[0026] Preferably, step S3 specifically involves: first, reducing the complexity of the cell pattern, that is, reducing the number of nodes based on the normal cell pattern structure to obtain a simplified cell pattern structure; then, performing the first solution using the method in step S2 to obtain the cell pattern variables activated by each cell pattern. This is equivalent to obtaining the layout of the element pattern for each member in the design domain; then, the element pattern complexity is set to normal and the structure is regenerated, using the results obtained from the first solution. Substituting into equation (4) transforms the optimization problem into a linear programming problem, allowing for a second solution.

[0027] As a preferred option, step S4 specifically involves: extracting the structural information of the repeating units based on the optimization results. The structural information includes the repeating unit pattern, the location of the repeating units, the connection of the repeating units, and the cross-sectional dimensions of the repeating unit members. After the members of the repeating units are assembled and nodes are generated, a 3D solid model is established. Then, the various repeating units in the solid model are sliced ​​and printing paths are generated for 3D printing. Finally, the repeating units are connected to each other for integrated assembly to manufacture the optimized structure.

[0028] The beneficial effects of this invention are:

[0029] 1) The discrete truss structure layout optimization design and manufacturing method based on repetitive elements provided by this invention introduces additional constraints and variables for the characteristics of repetitive elements, and proposes a direct solution method based on repetitive elements, so that the truss layout optimization results have the characteristics of repetitive elements and are relatively easy to manufacture; the direct solution method based on repetitive elements can be used directly on its own, or it can provide a mathematical model for the two-step solution of first determining the element layout.

[0030] 2) The discrete truss structure layout optimization design and manufacturing method based on repetitive elements provided by this invention, on the basis of direct solution based on repetitive elements, proposes a two-step solution to address the problem of excessive computation time when solving large-scale problems. The solution first solves the element pattern layout and then optimizes the internal structure of the element pattern. This makes the force path of the structure clearer and significantly improves the computational efficiency while having little impact on the volume of the final optimized structure.

[0031] 3) The discrete truss structure layout optimization design and manufacturing method based on repetitive elements provided by this invention achieves the repetitiveness of elements and structural regularity of the layout optimization results by setting a finite number of element patterns and adding repetitive element constraints based on the mathematical model of truss layout optimization; it achieves fast and efficient solution based on repetitive element layout optimization by first simplifying the element complexity and then solving the normal element complexity in two steps; and it realizes the 3D printing optimization design and integrated assembly manufacturing of complex discrete truss optimization structures through 3D modeling, repetitive element model slicing, printing path generation and integrated assembly production. Attached Figure Description

[0032] Figure 1 This is a detailed flowchart of the discrete truss layout optimization design and manufacturing method based on repeatable units according to the present invention;

[0033] Figure 2a This is a diagram illustrating a unit complexity of 2×2;

[0034] Figure 2b This is a diagram illustrating the 3×3 unit complexity.

[0035] Figure 2c This is a diagram illustrating a 4×4 unit complexity.

[0036] Figure 3 This is a schematic diagram of a cantilever beam structure model;

[0037] Figure 4 This is a schematic diagram of the ordinary layout optimization result in step S1 of the cantilever beam structure;

[0038] Figure 5a This is a schematic diagram of the layout optimization results obtained by directly solving step S2 of the cantilever beam structure.

[0039] Figure 5b For cantilever beam structures, step S2 directly solves for four types of repetitive elements with a complexity of 4×4 and a number of element patterns of 4.

[0040] Figure 6a This is the first solution result of the two-step solution in step S3 when the simplified and normal element mode complexities of the cantilever beam structure are 2×2 and 4×4, respectively.

[0041] Figure 6b This is the result of the second solution for the cantilever beam structure;

[0042] Figure 7a This is the first solution result of the two-step solution in step S3 when the simplified and normal element mode complexities of the cantilever beam structure are 3×3 and 4×4, respectively.

[0043] Figure 7b This is the result of the second solution for the cantilever beam structure;

[0044] Figure 8a This is the second solution result of the two-step solution in step S3 when the simplified and normal element mode complexities of the cantilever beam structure are 3×3 and 6×6, respectively.

[0045] Figure 8b This is the second solution result of the two-step solution in step S3 when the simplified and normal element mode complexities of the cantilever beam structure are 3×3 and 8×8, respectively.

[0046] Figure 9 This is a schematic diagram of a frame-braced structure model under horizontal wind load;

[0047] Figure 10a This is the second step of the solution in step S3 when the number of repetitive element patterns in the frame-support structure is 1.

[0048] Figure 10b This is the result of the second step of the two-step solution in step S3 when the number of repetitive element patterns in the frame-support structure is 4. Detailed Implementation

[0049] The present invention will be further described below with reference to embodiments. The description of the embodiments below is only for the purpose of helping to understand the present invention. It should be noted that those skilled in the art can make several modifications to the present invention without departing from the principle of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

[0050] Example 1

[0051] As one example, such as Figure 1 As shown, a method for optimizing the layout and manufacturing of discrete truss structures based on repetitive units is presented. 3D printing technology, also known as additive manufacturing, generates structures by accumulating materials layer by layer, greatly expanding the flexibility of structure manufacturing. Optimization of the structural layout based on repetitive units yields better results, including the arrangement and combination of repetitive units. Furthermore, for a few complex configurations of repetitive units, 3D printing and integrated assembly are used to achieve the integrated manufacturing of the optimized complex truss structure. The specific steps include:

[0052] S1. Establishment of the mathematical model for truss layout optimization: First, given the structural design domain, input constraints and parameters, establish the minimum connection basis structure, using the mechanical equilibrium equation as constraints and minimizing the total volume of the members as the design objective, and establish a linear optimization model; specifically including the following steps:

[0053] S1.1 Input design conditions and parameters: Input the design domain size, load case and boundary constraints, and specify the element mode and corresponding element complexity;

[0054] S1.2 Establishing the minimum connectivity basis structure: Discretize the design domain using a uniform lattice and connect any two nodes to form the minimum connectivity basis structure;

[0055] S1.3 Establishing a mathematical model for layout optimization: The underlying principle of the optimization algorithm is a mathematical optimization problem. With the mechanical equilibrium equation as the constraint and the minimum total volume of the members as the design objective, a linear optimization model for truss layout optimization is established.

[0056] The objective function is

[0057]

[0058] The expression for the constraint is

[0059]

[0060] In the formula, Let the length vector of the rod be the length vector. Let the area vector of the rod be... For the balance matrix, Let the internal force vector of the member be . For nodal load vectors, and These are the compressive and tensile strength vectors of the member, respectively;

[0061] The objective function (1) represents the optimization objective of minimizing the volume, and the constraint equation (2) represents the force balance equation, the stress constraint of the member, and the non-negativity constraint of the member area. This problem is a linear programming problem, and the design variable is the member area vector. and the internal force vector of the rod , and These are the constant matrix and constant vector generated based on the bar topology, respectively. , and These are all constants determined by the actual working conditions.

[0062] After a given design domain, the truss layout optimization determines a structure based on equations (1) and (2) and calculates the required constants. To ensure that the optimization results can reach the theoretical optimum, the simplest way to determine the structure is to connect all nodes in the design domain in pairs so that the structure includes all possible members and establish the minimum connection base structure. Although the structure obtained by the optimization of ordinary truss layout saves materials and has reasonable stress, its structure is often too complex and difficult to manufacture.

[0063] When the equilibrium matrix B of the minimum connectivity basis structure cannot be solved when optimizing the initial state, the component length threshold and mesh density in the minimum connectivity basis structure are increased to form an updated minimum connectivity basis structure when optimizing the initial state, and the solution is performed again.

[0064] S2. Direct Solution Based on Repetitive Elements: A finite number of element patterns are set, and elements are divided according to these patterns to ensure that all members belong to a specific element pattern and there are no members that cross element patterns. Repetitive element constraints are added to each member in the truss structure, and direct optimization is performed. Specifically...

[0065] S2.1 Setting a Finite Number of Element Patterns: A finite number of element patterns are set so that all members belong to a specific element pattern and there are no members that cross element patterns. The layout and corresponding area of ​​members within the same element pattern are identical. Truss layout optimization based on repetitive elements has two characteristics: first, all members belong to a specific element pattern, and there are no members that cross element patterns; second, all elements follow a finite number of element patterns, and the layout and corresponding area of ​​members within the same element pattern are identical. To achieve a repetitive element structure, the design domain is first filled with elements, and then the nodes within each element are connected in pairs to form members. The number of members in the repetitive element structure is significantly reduced compared to the truss optimization in step S1.

[0066] S2.2 Add repetitive element constraints: Each member adopts a binary variable of the active element mode, and repetitive element constraints are added to each member of the truss structure;

[0067] To ensure that members of the same element type have the same area at the same location, taking the case where there are n element types in the structure as an example, the following constraints are added to each member:

[0068]

[0069]

[0070] In the formula, Let c be the cross-sectional area of ​​the member, c be the element pattern number to which the member belongs, and m be the position number of the member in the element pattern. , … This is a binary variable representing the activated element mode of the element mode to which the member belongs. It takes the value 1 or 0, where 1 indicates that the element mode is activated and 0 indicates that it is not activated. , … This represents the possible cross-sectional area corresponding to the position of the member in the element pattern; in equation (4) , … and , … Both variables are variables, and the multiplication of two variables forms a nonlinear constraint, which leads to difficulties in solving the problem. Therefore, it is necessary to transform it into a linear problem so that it can be solved directly.

[0071] S2.3 Direct solution based on repetitive elements: The big M method is used to transform nonlinear constraints into linear constraints, that is, to replace equation (4) with equation (5):

[0072]

[0073] In the formula, M is a given large constant; each row in equation (5) represents a constraint of an element pattern on the member at that location; when When the first unit mode is activated, ,the remaining At this point, the first row of constraint (5) becomes Other inequalities relax and actually have no effect; when When the second type of unit is activated in the corresponding unit mode, that is... And the rest At this point, the second row of constraint (5) becomes Other inequality relaxations have no effect; by analogy, when a certain element mode is activated, only the constraints of the corresponding row of the element are effective, and the inequality constraints of other rows are relaxed; Equation (5) can achieve the same effect as Equation (4), and Equation (5) is still a linear constraint, which does not change the characteristics of linear programming in the optimization problem and is easy to solve.

[0074] The truss layout optimization based on repetitive units is a linear programming problem with equation (1) as the objective function and equations (2), (3) and (5) as constraints.

[0075] Truss layout optimization based on repetitive elements is a mixed-integer programming problem, including general continuous variables and... The problem introduces existing binary integer variables; it uses the commercial solver gurobi, which is well-suited for this type of problem, for direct solving; the solver has a Python-oriented interface for easy program calling.

[0076] S3. First, determine the two-step solution for the cell layout: First, set a lower-level cell pattern complexity, and use step S2 to perform the first solution to obtain the activated cell pattern variables. Substituting into equation (4) and discarding the constraint in equation (5), we transform it into a linear constraint and perform a second solution; specifically...

[0077] S3.1 Simplifying the First Solution of Element Complexity: Due to the characteristics of mixed-integer programming, the method in step S2 may experience excessively long computation times as the design domain increases, the number of element patterns increases, or the element pattern complexity increases. In mixed-integer programming, both the number of integer variables and the number of continuous variables affect the solution time. The number of integer variables depends on the number of element patterns and the number of elements in the structure. These two parameters are determined by the specific problem and design requirements and should not be arbitrarily modified. The number of continuous variables depends on the number of members in the structure, which is determined by the number of repetitive elements and the element pattern complexity. The only variable that can be improved during the calculation process is the element pattern complexity. Therefore, the element pattern complexity is first set to a low level, and step S2 is used for the first solution, which is a mixed-integer linear programming problem, to obtain the element pattern variables activated by each element pattern. ;

[0078] like Figures 2a-2c The figures show the typical complexities of 2×2 units, 3×3 units, and 4×4 units, respectively, and the unit pattern complexity... The complexity of the element pattern represents the number of nodes in the horizontal and vertical directions. The higher the complexity of the element pattern, the higher the degree of freedom of the structure, and theoretically, a better objective function value will be obtained. However, this will result in a larger number of members in the structure and a longer solution time.

[0079] First, the complexity of the unit pattern is set to a low level, and the first solution is performed using the method in step S2 to obtain the unit pattern variables activated by each unit pattern. This is equivalent to obtaining the layout of each unit pattern in the design domain;

[0080] S3.2 Second solution for normal unit complexity: Then set the unit mode complexity to normal unit mode and regenerate the structure, using the solution obtained in the first solution. Substituting into equation (4) and discarding the constraints of equation (5), since at this time Since the constant vector is already determined, equation (4) will not change the characteristics of linear programming in the optimization problem. At this time, there are no binary integer variables in the problem, that is, the second solution is not a mixed integer programming problem. The calculation time required is much less than the first solution, and the time occupied in the entire solution process can be almost ignored.

[0081] The time taken in mixed integer programming is a decisive factor in the overall optimization time. In the first solution of the improved algorithm in step S3, due to the lower complexity of the element patterns and the smaller number of continuous variables, the computation time can be significantly reduced compared to the first solution in step S2. However, the result of the improved algorithm in step S3 cannot be guaranteed to be completely consistent with the algorithm described in step S2, because the first solution only determines the layout of multiple element patterns and does not give the internal structure of the specific element patterns. Generally, members with similar stress levels under load will adopt the same type of element pattern, but the number of element patterns and the complexity of the element patterns in the first solution may affect the result of the element pattern layout.

[0082] Increasing the number of cell patterns is more beneficial for obtaining a better structure than increasing the complexity of normal cell patterns. However, when conditions permit, the complexity of simplified cell patterns in two-step solutions should not be too low, otherwise the layout of cell patterns may deviate from the optimal.

[0083] S4, 3D Printing Manufacturing and Integrated Assembly: 3D modeling is performed, various repetitive units in the solid model are sliced ​​and printing paths are generated, 3D printing is carried out, and integrated assembly is performed between the repetitive units to manufacture the optimized structure; specifically...

[0084] S4.1 3D Printing of Repetitive Units: 3D modeling is performed using Rhino software. Cura software is used to slice various repetitive units in the resulting solid model and generate printing paths. Structural information of the repetitive units is extracted based on the optimization results. This information includes the repetitive unit pattern, location, connection, and cross-sectional dimensions of the repetitive unit members. After assembling the repetitive units and generating nodes, a 3D solid model is established. Then, various repetitive units in the solid model are sliced, printing paths are generated, and 3D printing is performed.

[0085] S4.2 Integrated assembly of repetitive units: Repetitive units are connected by welding, bolting, or other methods to achieve integrated assembly and manufacture an optimized structure.

[0086] Example 2

[0087] As another embodiment, this embodiment demonstrates the optimized design and manufacturing of the cantilever beam problem through the general layout optimization in step S1 and the direct solution based on repetitive elements in step S2 of embodiment one.

[0088] like Figure 3As shown, the design domain of the cantilever beam is 6 units wide and 3 units high. Each element has a height and width of 1 unit, resulting in a design domain of 6 × 3 = 18 elements. A vertically downward unit load is applied at the upper right corner of the design domain, and horizontal and vertical degrees of freedom are constrained at the upper and lower nodes on the left side. All quantities are dimensionless, the structural self-weight is ignored, and the allowable tensile and compressive stresses of the members are taken as [value missing]. Design variable bar area vector and the internal force vector of the rod All initial values ​​were set to 0; the calculations were performed on a workstation with an Intel i7-12700K CPU (3.61GHz) and 32GB of RAM.

[0089] In this embodiment, although the structure obtained by the ordinary layout optimization in step S1 saves materials and has reasonable stress distribution, its structure is often too complex and difficult to manufacture. The schematic diagram of the optimization result is shown below. Figure 4 As shown; light-colored bars represent tension, dark-colored bars represent compression, and the bar thickness represents its cross-sectional area. Bars with an area smaller than a threshold will not be displayed. This threshold is 1 / 1000 of the area of ​​the largest bar in the structure.

[0090] Based on this, step S2 is implemented in this embodiment. The unit complexity is set to 4×4, and the number of repetitive unit patterns is set to 4, based on the direct solution of the repetitive units. A schematic diagram of the direct optimization result is shown below. Figure 5a As shown in the diagram, there are four types of repetitive unit structures. Figure 5b As shown; the directly solved structural volume is 32, and the solution time is 20498s; the optimized structural force transmission path is similar to that of ordinary layout optimization, but since there are no cross-element mode members, the member lengths are not significantly different, and the number of members connected to a node is limited, and there are no overly complex nodes. Compared with the ordinary layout optimization result obtained in step S1, step S2 greatly reduces the manufacturing difficulty of the structure.

[0091] This embodiment demonstrates the application of some steps in the present invention in truss structure layout optimization, showing that the equations established in step S2 based on repetitive elements can be directly used for the optimization solution of repetitive elements, achieving good results.

[0092] Example 3

[0093] As another embodiment, based on the discrete truss structure layout optimization design and manufacturing method based on repeatable elements proposed in Embodiment 1, for the same cantilever beam problem in Embodiment 2, the two-step solution in step S3 is further performed by using the direct solution method based on repeatable elements in step S2.

[0094] The complexity of the normal unit pattern is set to 4×4, the number of repetitive unit patterns is set to 4, and the simplified unit complexity is set to 2×2 and 3×3 respectively when solving the two steps of step S3.

[0095] In this embodiment, the optimized results of the two-step solution in step S3 are illustrated when the simplified and normal unit mode complexities are 2×2 and 4×4, respectively. Figure 6a and Figure 6b As shown in the diagram, the optimized results of the two-step solution in step S3 are illustrated when the complexities of the simplified and normal cell modes are 3×3 and 4×4, respectively. Figure 7a and Figure 7b As shown;

[0096] The specific optimization results are as follows:

[0097]

[0098] When the simplified unit pattern has a complexity of 2×2: the structure volume directly solved in step S2 is 32, and the solution time is 20498s. The structure volumes solved in the two steps of step S3 are 34.400 and 33.688, respectively. Compared with the direct solution, the volume increases by 5.28%, but the solution time is greatly reduced to only 25s, which is about 1 / 800 of the direct solution, and the computational efficiency is significantly improved.

[0099] When the simplified unit pattern has a complexity of 3×3: the structure volumes solved in the two steps of step S3 are 33 and 32 respectively. The final volume is consistent with the volume solved directly, but the calculation time is 4241s, which is about 1 / 5 of the direct solution, and the calculation efficiency is significantly improved.

[0100] In this embodiment, the structural layout and shape differ significantly when the simplified unit pattern complexity is 2×2 and 3×3. Increasing the simplified unit pattern complexity makes the unit layout result solved in the first step closer to the ideal situation, so that the final structure is closer to the result of direct solution. Therefore, when conditions permit, ensuring the complexity of the simplified unit pattern is beneficial to obtaining a solution that is closer to the optimal layout.

[0101] When the simplified unit pattern complexity is 3×3: the second solution results of the two steps in step S3 when the normal unit pattern complexity is 6×6 and 8×8 are respectively as follows: Figure 8a , Figure 8b As shown, the general layout of the structure is similar to that of the normal cell pattern with a complexity of 4×4, but the internal details of the cell pattern are more complex. The final volumes of the corresponding structures are 31.486 and 31.530, respectively, which are only about 1.5% different from the cell pattern with a complexity of 4×4, but the computation time increases rapidly.

[0102] This embodiment demonstrates the application of some steps in the present invention in truss structure layout optimization. It shows that by using the solution equation based on repetitive elements established in step S2 to perform the two-step solution in step S3, a good result is obtained. The two-step solution, which first solves the element pattern layout and then optimizes the internal structure of the element pattern, makes the force path of the structure clearer and significantly improves the computational efficiency with little impact on the final optimized structure volume. It can be used to solve problems with excessively long computation time when solving large-scale problems.

[0103] Example 4

[0104] The optimization approach for repetitive element structures in truss-like discrete structures is as follows: First, given a structural design domain, a finite number of element patterns are set and elements are divided. The final overall structure consists of multiple complex repetitive element configurations. This allows for the mass production of a few element patterns using 3D printing and simple integrated assembly. However, current research on the optimization of truss repetitive element structures is limited and does not address layout optimization. Most current repetitive element structures use the same element pattern within the same sub-design domain, i.e., the number of repetitive element patterns is set to 1. This is partly determined by the characteristics of homogenization methods, but there is no such restriction in truss structures. The element pattern selected for each member is not necessarily related to its position, making the final structural layout more flexible. Therefore, the improved application of reasonable and effective repetitive element algorithms is a crucial factor in truss structure layout optimization.

[0105] As another embodiment, this embodiment uses the discrete truss structure layout optimization design and manufacturing method based on repetitive elements proposed in Embodiment 1 to solve the frame-braced structure problem under horizontal wind load in two steps, and compares the differences in results for different element mode numbers.

[0106] like Figure 9 As shown, the design domain of the frame-braced structure under horizontal wind load is 4 wide and 12 high, the unit height is 2 and the width is 1, and the design domain has a total of 6×3=18 units. There are horizontal unit loads acting on both sides at heights of 4, 8 and 12. The horizontal and vertical degrees of freedom are constrained at the nodes at both ends of the bottom edge. The complexity of the simplified unit pattern is set to 2×2, the complexity of the normal unit pattern is 6×6, and the number of repetitive unit patterns is set to 1 and 4 respectively.

[0107] In this embodiment, the optimization results of the two-step solution in step S3 are illustrated when the number of repetitive unit patterns is 1 and 4, respectively. Figure 10a and Figure 10b As shown; the specific optimization results are as follows:

[0108]

[0109] Step S3 is solved twice: the number of repetitive element patterns is 1, that is, all element patterns have the same element structure, and the structure volume of the second solution is 1082.47; the number of repetitive element patterns is 4, that is, there are four different element structures, and the structure volumes of the first and second solutions are 300.43 and 295.33, respectively, with a solution time of 1852s.

[0110] It can be seen that increasing the number of repeating element patterns reduces the structural volume to only about 30% of that when the number of repeating element patterns is 1. The optimization effect is significant, indicating that under heavy loads, appropriately increasing the number of repeating element patterns can better leverage the advantage of saving materials.

[0111] As demonstrated in Examples 2, 3, and 4, the discrete truss structure layout optimization design and manufacturing method based on repetitive elements provided by this invention introduces additional constraints and variables tailored to the characteristics of repetitive elements. The proposed direct solution based on repetitive elements makes the truss layout optimization results relatively easy to manufacture due to the repetitive element characteristics. To address the issue of excessively long computation times when solving large-scale problems, a two-step solution is implemented: first solving the element pattern layout, then optimizing the internal structure of the element pattern. This makes the structural force path clearer and significantly improves computational efficiency with minimal impact on the final optimized structure volume. Thus, it enables 3D printing optimization design and integrated assembly manufacturing of complex discrete truss optimization structures. Furthermore, practical verification has shown the effectiveness of this invention.

Claims

1. A structural layout optimization design and manufacturing method based on repetitive units, characterized in that, Includes the following steps: S1. Establishment of mathematical model for truss layout optimization: First, given the structural design domain, input the dimensions, load cases and boundary constraints, and specify the element mode and the corresponding element mode complexity; The design domain is discretized using a lattice, and a minimum connection basis structure is established by connecting any two nodes. A linear optimization model for truss layout optimization is established with mechanical equilibrium equations as constraints and the minimum total volume of the members as the design objective. S2. Direct solution based on repetitive elements: Set a finite number of element patterns and divide them into elements to ensure that all members belong to a certain element pattern and there are no members that cross element patterns. The element layout and the area of ​​the members at the corresponding positions of the same element pattern are exactly the same. For each member in the truss structure, use a binary variable of the activated element pattern and add repetitive element constraints to form nonlinear constraints. Transform the nonlinear programming problem into a linear programming problem so that the repetitive elements can be directly optimized and solved. S3. First, determine the two-step solution for the unit layout: First, reduce the complexity of each unit pattern, and perform the first solution according to step S2 to obtain the activated unit pattern variable t for each unit pattern. c Then, based on the normal complexity of each unit pattern, t c Substitute the nonlinear constraint expression of each bar in step S2 into the nonlinear constraint expression of the repeating element to transform it into a linear constraint, perform a second solution, and obtain the optimization result; S4, 3D Printing Manufacturing and Integrated Assembly: Perform 3D modeling, slice various repetitive units in the optimized model and generate printing paths, perform 3D printing manufacturing, and manufacture the optimized structure by integrating the repetitive units.

2. The structural layout optimization design and manufacturing method based on repetitive units according to claim 1, characterized in that, In step S1: the objective function corresponding to minimizing the total volume of the rods is, The expression for the constraint is, In the formula, l is the member length vector, a is the member area vector, B is the equilibrium matrix, q is the member internal force vector, f is the nodal load vector, and σ is the member load vector. c and σ t These are the compressive and tensile strength vectors of the member, respectively; The three constraints in equation (2) represent the force equilibrium equation, the stress constraint of the member, and the non-negativity constraint of the member area, respectively; the design variables are the member area vector a and the member internal force vector q, B and l are the constant matrix and constant vector generated according to the member topology, respectively, f and σ c and σ t These are all constants determined by the actual working conditions.

3. The structural layout optimization design and manufacturing method based on repetitive units according to claim 2, characterized in that, In step S1: When the equilibrium matrix B of the minimum connection basis structure at the initial state cannot be solved, the component length threshold and mesh density in the minimum connection basis structure are increased to form the updated minimum connection basis structure at the initial state, and the solution is performed again.

4. The structural layout optimization design and manufacturing method based on repetitive units according to claim 1, characterized in that, In step S2: A binary variable is set to activate the element pattern for each member. Repeatable element constraints are added to each member of the truss structure. This means first filling the design domain with element patterns, and then connecting the nodes within each element pattern pairwise to form members, thus creating repeatable elements. To ensure that members of the same type have the same area at the same location, when there are n element patterns in the structure, the following constraints are added to each member: t c1 +t c2 +…+t cn =1 (3) a i =a m1 ·t c1 +a m2 ·t c2 +…+a mn ·t cn (4) In the formula, a i t is the cross-sectional area of ​​the member, c is the element pattern number to which the member belongs, and m is the position number of the member in the element pattern; c1 t c2 、…、t cn This is a binary variable representing the activated element pattern of the element pattern to which the member belongs. Its value is either 1 or 0, where 1 indicates an activated element pattern and 0 indicates an inactive one. m1 a m2 ... a mn Let a be the possible cross-sectional area corresponding to the position of the member in the element pattern; in equation (4) a m1 a m2 ... a mn and t c1 t c2 、…、t cn Both variables are variables, and multiplying two variables together forms a nonlinear constraint.

5. The structural layout optimization design and manufacturing method based on repetitive units according to claim 4, characterized in that, In step S2: The Big M method is used to transform the nonlinear constraint of equation (4) into a linear constraint, resulting in equation (5). In the formula, M is a constant; each row in formula (5) represents a constraint of a unit pattern on the member at that location; when a i When the first unit mode is activated, t c1 =1, the rest t cx =0, at this time the first row of constraint (5) becomes a i =a m1 Other inequalities relax and actually have no effect; when a i When the second unit mode is activated, i.e., t c2 =1, while the rest t cx =0, at this time the second row of constraint (5) becomes a i =a m2 Other inequality relaxations have no effect; similarly, when a certain element mode is activated, the constraints of the corresponding row of that element take effect, and the inequality constraints of other rows are relaxed.

6. The structural layout optimization design and manufacturing method based on repetitive units according to claim 4, characterized in that, Step S3 specifically involves: first, reducing the complexity of the cell pattern, that is, reducing the number of nodes based on the normal cell pattern structure to obtain a simplified cell pattern structure; then, using the method in step S2, performing the first solution to obtain the cell pattern variable t activated by each cell pattern. c This is equivalent to obtaining the layout of the element pattern for each member in the design domain; then, the element pattern complexity is set to normal and the structure is regenerated, and the t obtained from the first solution is used. c Substituting into equation (4) transforms the optimization problem into a linear programming problem, allowing for a second solution.

7. The structural layout optimization design and manufacturing method based on repetitive units according to claim 1, characterized in that, Step S4 specifically involves: extracting the structural information of repetitive units based on the optimization results. The structural information includes the repetitive unit pattern, the location of the repetitive units, the connection of the repetitive units, and the cross-sectional dimensions of the repetitive unit members. After the repetitive units are assembled and nodes are generated, a 3D solid model is established. Then, the various repetitive units in the solid model are sliced ​​and printing paths are generated for 3D printing. Finally, the repetitive units are connected to each other for integrated assembly, and the optimized structure is manufactured.

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