A Multi-Material Topology Optimization Method Based on Negative Mapping Interpolation

Abstract

This invention relates to the fields of structural optimization design and computational mechanics, specifically to a multi-material topology optimization method based on negative mapping interpolation, comprising the following steps: S1. Design domain initialization; S2. Objective function establishment; S3. Negative mapping interpolation construction; S4. Unified expression of material distribution; S5. Optimization solution; S6. Interface evolution and update; S7. Convergence determination and result output. This invention establishes a repulsion mechanism between materials through negative mapping interpolation, achieving strict partitioning of multiple materials; avoiding intermediate density problems and obtaining a clear topological structure; realizing coordinated control of material distribution and interface evolution, improving the physical rationality of the structure; eliminating the need for complex sensitivity analysis, and ensuring a stable and efficient optimization process; and can be extended to microstructure design and multi-scale optimization problems.

Description

Technical Field

[0001] This invention relates to the fields of structural optimization design and computational mechanics, specifically to a multi-material topology optimization method based on negative mapping interpolation. Based on the consistency of material distribution and interface based on negative mapping interpolation (NMI), it can be applied to the fields of multi-material structural design and multi-scale optimization. Background Technology

[0002] Multimaterial topology optimization aims to achieve the optimal spatial distribution of various materials (including solid materials and voids) within a given design domain to improve the mechanical properties of structures. However, most existing methods use density-based interpolation models (such as the SIMP method) to describe the material distribution and characterize the structural interfaces through level set functions or projection methods.

[0003] Within the aforementioned technical framework, the following problems typically exist:

[0004] 1. Material overlap problem: Different material density variables may simultaneously take non-zero values ​​in the same element, resulting in material distribution that does not conform to physical reality;

[0005] 2. Gray-scale unit problem: The optimization results contain a large number of intermediate density areas, resulting in a blurred interface;

[0006] 3. Uncontrollable interface problem: Material distribution and interface evolution are independent of each other, making precise control difficult;

[0007] 4. Insufficient manufacturability: The generated structure has unclear boundaries, making it difficult to use directly in engineering manufacturing.

[0008] Therefore, how to construct a framework that can unify material distribution and interface expression, and effectively eliminate material overlap and gray-scale unit topology optimization methods, has become a key technical problem that urgently needs to be solved in this field. Summary of the Invention

[0009] The purpose of this invention is to provide a multi-material topology optimization method based on negative mapping interpolation to solve the problems of material overlap, gray-scale units and poor structural manufacturability caused by the inconsistency between material distribution and interface description in the prior art.

[0010] To achieve the above-mentioned objectives, the technical solution adopted by the present invention is as follows:

[0011] A multi-material topology optimization method based on negative mapping interpolation includes the following steps:

[0012] S1. Design domain initialization;

[0013] Establish a finite element discrete model within a given design domain and define level set functions corresponding to multiple materials;

[0014] S2. Establishing the objective function;

[0015] Construct an optimization model with the goal of minimizing structural flexibility or achieving optimal equivalent performance, and impose volume fraction constraints;

[0016] S3. Negative mapping interpolation construction;

[0017] Negative mapping is applied to the level set functions corresponding to each material to construct material indicator functions, ensuring that each spatial location is occupied by only one material; a two-phase subproblem is then considered. .set up For the node level set function The set of overlapping cells detected. For any Let the dominant phase and the suppressed phase be respectively... and .

[0018] Then for The node level set function is updated by cell-level negative mapping as follows:

[0019]

[0020] in, and Let i and j represent the level set functions corresponding to different materials, respectively, and let i and j represent the node indices of the level set functions. and These represent the density fractions of materials m and n within the same unit e, respectively.

[0021] right Aggregating all elements within a set can be done using a global operator:

[0022]

[0023] in, For the negative mapping global operator for materials m and n, Let m be the level set function of materials m and n before negative mapping interpolation and before updating. Let m be the density fraction of material m and material n within the element. For the level set functions of materials m and n after negative mapping interpolation;

[0024] And for For units outside the set, their level set function remains unchanged.

[0025] Special case: When taking a specific value At that time, the hard sign flip form is obtained. This form is consistent with Equation (1) used to completely eliminate interface overlap.

[0026] S4. Material distribution is expressed uniformly;

[0027] Based on the negative mapping interpolation results, a mapping relationship between material distribution and interface location is established to achieve a unified description of material regions and interfaces;

[0028] S5. Optimize the solution;

[0029] By combining the proportional topology optimization method with the alternating active phase algorithm, the multi-material problem is decomposed into several bi-material sub-problems for iterative updating;

[0030] S6. Interface Evolution and Updates;

[0031] By updating the level set function, the material interface can be dynamically evolved during the optimization process;

[0032] S7. Convergence determination and result output;

[0033] When the objective function or design variables meet the convergence condition, the final topology is output.

[0034] As a preferred option:

[0035] The negative mapping interpolation achieves the mutual exclusion relationship between materials by performing a sign mapping on the level set function;

[0036] The optimization process incorporates proportional filtering or energy filtering to improve stability.

[0037] The method can be extended to multi-scale topology optimization, and macro-micro coupling is achieved through homogenization theory.

[0038] Compared with the prior art, the advantages of the present invention are as follows:

[0039] 1. Eliminate material overlap: Establish a repulsion mechanism between materials through negative mapping interpolation to achieve strict partitioning of multiple materials;

[0040] 2. Eliminate grayscale units: Avoid intermediate density issues and obtain a clear topological structure at the interface;

[0041] 3. Unification of material and interface representation: Achieving coordinated control of material distribution and interface evolution to improve the physical rationality of the structure;

[0042] 4. Improved computational efficiency: No complex sensitivity analysis is required; the optimization process is stable and efficient.

[0043] 5. It has good scalability: it can be extended to microstructure design and multi-scale optimization problems. Attached Figure Description

[0044] Figure 1 This is a schematic diagram of the method flow of the present invention;

[0045] Figure 2 The unit of material overlap in the iteration of the embodiment;

[0046] Figure 3 Examples Material distribution diagram;

[0047] Figure 4 Examples Material distribution diagram;

[0048] Figure 5 Examples A schematic diagram of the level set function;

[0049] Figure 6 Examples A schematic diagram of the level set function;

[0050] Figure 7 The unit of material overlap in the iteration of the embodiment;

[0051] Figure 8 Examples Material distribution diagram;

[0052] Figure 9 Examples Material distribution diagram;

[0053] Figure 10 Examples A schematic diagram of the level set function;

[0054] Figure 11 Examples A schematic diagram of the level set function. Detailed Implementation

[0055] To make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and examples.

[0056] According to such Figure 1 The process shown involves establishing a finite element model within a two-dimensional design domain, dividing the domain into several elements, and defining multiple material level set functions. The material functions are processed using a negative mapping interpolation method to ensure that only one material exists within each element, thus forming a clear material interface. A negative mapping interpolation strategy is introduced for the interface level set functions, and its expression is shown in (1).

[0057]

[0058] in, and Let i and j represent the level set functions corresponding to different materials, and i and j represent the node indices (coordinates) of the level set functions. and These represent the density fractions of materials m and n within the same unit e, respectively.

[0059] The specific implementation steps of the negative mapping interpolation strategy are as follows:

[0060] First, when the calculation results tend to stabilize, traverse all elements within the design domain and filter out elements with overlapping materials, such as... Figures 2 to 6 As shown. Figure 3 and Figure 4 They were given respectively and Corresponding material distribution diagram; Figure 5 and Figure 6 They were shown respectively and The level set function representation. Where the red and blue regions respectively represent... and The region of level set functions, where the green region represents the region that satisfies The contour lines of the horizontal set are visible. Material overlap can be clearly observed within this region.

[0061] Subsequently, all nodes belonging to elements where material overlap occurs are selected. For each node in such an element, the following operation is performed: the level set function of one of the material phases is maintained. The value remains unchanged, such as Figure 8 and Figure 10 As shown; for another material phase, the level set function value at this node is determined using equation (1). Interpolation is its negative, such as Figure 9 and Figure 11 As shown.

[0062] After completing the above operations, and The material distribution diagrams are as follows: Figure 7 , Figure 8 and Figure 9 As shown, it is clear that the material overlap in this area has been eliminated.

[0063] The final result is a topological structure with no material overlap, no grayscale areas, and a clear interface.

[0064] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the implementation methods of the present invention, and should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of the present invention.

Claims

1. A multi-material topology optimization method based on negative mapping interpolation, characterized in that, Includes the following steps: S1. Design domain initialization; Establish a finite element discrete model within a given design domain and define level set functions corresponding to multiple materials; S2. Establishing the objective function; Construct an optimization model with the goal of minimizing structural flexibility or achieving optimal equivalent performance, and impose volume fraction constraints; S3. Negative mapping interpolation construction; Negative mapping is performed on the level set function corresponding to each material to construct the material indicator function, so that each spatial location is occupied by only one material. S4. Material distribution is expressed uniformly; Based on the negative mapping interpolation results, a mapping relationship between material distribution and interface location is established to achieve a unified description of material regions and interfaces; S5. Optimize the solution; By combining the proportional topology optimization method with the alternating active phase algorithm, the multi-material problem is decomposed into several bi-material sub-problems for iterative updating; S6. Interface Evolution and Updates; By updating the level set function, the material interface can be dynamically evolved during the optimization process; S7. Convergence determination and result output; When the objective function or design variables meet the convergence condition, the final topology is output.

2. The multi-material topology optimization method based on negative mapping interpolation according to claim 1, wherein, The specific method for S3 is as follows: The level set function corresponding to each material is processed by negative mapping, a material indicator function is constructed, and each spatial position is occupied by only one material; considering a two-phase sub-problem ; let be the set of cells detected by the node level set function to overlap; for any , the dominant phase and the suppressed phase are respectively and ; Then for The node level set function is updated by performing a cell-level negative mapping as follows: ； in, and Let i and j represent the level set functions corresponding to different materials, respectively, and let i and j represent the node indices of the level set functions. and These represent the density fractions of materials m and n within the same unit e, respectively. right Aggregating all elements within a set can be done using a global operator: ； in, For the negative mapping global operator for materials m and n, Let m be the level set function of materials m and n before negative mapping interpolation and before updating. Let m be the density fraction of material m and material n within the element. For the level set functions of materials m and n after negative mapping interpolation; And for For units outside the set, their level set function remains unchanged.

3. The multi-material topology optimization method based on negative mapping interpolation according to claim 2, characterized in that, S3 also includes special cases: when a specific value is taken. At that time, the hard sign flip form is obtained. This form is consistent with Equation (1) used to completely eliminate interface overlap.

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