Topology optimization design model reduction method based on domain decomposition and neural networks

Abstract

A method for reducing the order of a topology optimization design model based on domain decomposition and neural networks is disclosed. In topology optimization of a structure, the mechanical field of that structure is needed to evaluate the objective function and / or constraints. In the model reduction method for efficiently computing the mechanical field, the fine-scale structure of the structure is first coarsened to provide a coarse-scale structure. The finite element method is applied to the coarse-scale structure to obtain the coarse-scale mechanical field. The fine-scale mechanical field is then derived from the coarse-scale mechanical field, rather than directly computing it from the fine-scale structure using the finite element method. This allows the use of a fine-scale mechanical field with higher accuracy than the coarse-scale mechanical field, while simultaneously saving computational costs. When generating the fine-scale mechanical field, an artificial neural network called MapNet is used to map the coarse-scale mechanical field to the fine-scale mechanical field. MapNet can be implemented using convolutional and deconvolutional layers.

Description

[0001] Cross-references to related applications

[0002] This application claims priority to U.S. Provisional Patent Application No. 63 / 286,566, filed December 7, 2021, and U.S. Non-Provisional Patent Application No. 18 / 057,249, filed November 21, 2022, the disclosures of which are incorporated herein by reference in their entirety. Technical Field

[0003] This invention generally relates to topology optimization in large-scale designs. In particular, it relates to an artificial neural network-based model reduction technique for calculating the mechanical fields of structures, and its application in topology optimization in large-scale designs, which significantly reduces the expensive computational cost of numerical evaluation of objective functions and constraints. Background Technology

[0004] From the design of industrial products such as aircraft components to the design of advanced materials like metamaterials, structural design has always played a crucial role in engineering. Particularly in mechanical engineering, a common design method is topology optimization. This method systematically optimizes material distribution within a defined design range to achieve design objectives while subject to certain design constraints. However, because topology optimization requires repeated evaluation of the objective function and constraints—typically achieved through numerical simulations such as finite element method (FEM) analysis—the computational cost of topology optimization can be extremely high for large-scale designs. For instance, designing an aircraft wing with gigahertz resolution might require 8,000 central processing unit units running simultaneously for several days.

[0005] Accelerating large-scale simulations is an effective method for speeding up topology optimization processes and reducing computational costs for large-scale design. Starting with the conventional order reduction methods used in the early years, various approaches have been developed over the years. With the rapid development of deep learning methods in recent years, deep learning models (such as artificial neural networks) have been adopted in various physics fields. One advantage of artificial neural networks is that once constructed, their predictions can be computed rapidly on a millisecond timescale. This advantageous property of artificial neural networks has been used in large-scale analysis, using them as surrogate models to replace time-consuming numerical simulations, such as FEM. Surrogate models based on artificial neural networks have been widely adopted in structural mechanics to predict the mechanical response of structures. For example, White et al. (2019), Tan et al. (2019), Lee et al. (2020), and Nie et al. (2020) used this approach. These studies utilized fully connected neural networks or convolutional neural networks to predict mechanical fields, such as the stress / strain fields of structures subjected to various loads.

[0006] Another approach to accelerating the design process is to apply deep learning models to directly predict optimized or near-optimal structures, partially or completely skipping the optimization process. This can be seen in the work of Sosnovik et al. (2019), Yu et al. (2019), Kollmann et al. (2020), and Ates et al. (2021). In these deep learning models, the input is either an intermediate structure generated during topology optimization or the initial design, while the output is the optimized structure; essentially, the structural design is treated as an image. Therefore, these models can be used as black boxes, requiring no prior knowledge associated with the design problem and reducing computational time by skipping the design process. However, to train the model, a large number of topology optimization designs must be generated in advance, which in turn requires significant FEM computation.

[0007] Despite significant interest and effort in developing efficient machine learning-based topology optimization methods, existing methods / models suffer from two main drawbacks. The first is the need for massive amounts of training data. Since most of the required data is generated through time-consuming simulations (such as FEM), the time required to generate training data increases with the problem size. This is impractical for large-scale topology design. Several methods have been proposed to reduce the amount of training data. For example, in Qian and Ye (2021), only designs from the early stages of optimization were used as training data. Data from later stages of optimization were omitted because most of these designs were similar to those in previous iterations and therefore did not contribute much to the learning process of the machine learning model. However, even with reduced training data, costly FEM computations are still required. This can be problematic for large-scale topology optimization designs (e.g., designs with gigabit-level resolution) due to the high computational demands and potentially limited availability. In another study by Chi et al. (2021), an online update scheme was proposed to avoid long offline training sessions, thus mitigating the need for large amounts of training data. However, because it utilizes an online training scheme, the model needs to be trained in real time for each new problem. Therefore, if it is necessary to repeatedly solve problems with different conditions, such as compliance minimization design problems with different applied load locations and volume fraction requirements, the efficiency of this method will decrease.

[0008] Another drawback of most of the studies mentioned above is the poor transferability of the trained models; their predictive accuracy drops significantly when applied to "unseen" settings. Therefore, most existing models have a rather narrow scope of application, making it difficult to generalize to problems with different settings, especially those with different domain shapes and sizes. A major reason for this narrow scope is that in these models, the machine learning model maps the entire domain, represented as an image, along with possible boundary / loading conditions, to its corresponding output, such as a stress field or design scheme. Adapting to different domains and conditions requires a large amount of representative training data, which is challenging, if not entirely impossible, for large-scale design.

[0009] One approach to reduce the amount of training data and / or improve transferability is to incorporate physics into the machine learning process or model. An example of reducing the amount of training data is the Physics Information Neural Network (PINN) method, which has recently garnered significant attention. In PINN, physics is embedded in the loss function. Therefore, very little or no training data is required. Its drawback is a substantial increase in training time due to the increased complexity of the training, which is essentially a multi-objective optimization problem. For improving transferability, see the work of Wang et al. (2021), where the neural network maps the mechanical fields (such as displacement and / or stress / strain fields) of the initial design to the corresponding design scheme. The input to the artificial neural network model contains some physical properties of the problem. Therefore, the model has strong generalization ability. Even if the model is trained under a specific boundary condition, it can predict design schemes for the same problem with different boundary conditions. However, the quality of the generated design schemes still needs improvement, and the model seems to be applicable only to a fixed design domain.

[0010] There is a need in this field for a technique based on artificial neural networks that can significantly reduce the computational costs encountered in the expensive numerical evaluation of objective functions and constraints, in order to accelerate the topology optimization process for large-scale designs. Summary of the Invention

[0011] A first aspect of the invention provides a first method executed by a computer for calculating the mechanical field of a structure.

[0012] In the first method, the structure is discretized by dividing it into multiple fine-scale elements using a fine-scale grid. The fine-scale structure is then coarsened to produce a coarse-scale structure, such that the coarse-scale structure consists of multiple coarse-scale elements, where the number of corresponding coarse-scale elements is less than the number of corresponding fine-scale elements in the multiple fine-scale elements. An FEM is applied to this coarse-scale structure to compute the coarse-scale mechanical field of the structure. The fine-scale mechanical field of the structure is then obtained from this coarse-scale mechanical field. Obtaining the fine-scale mechanical field from the coarse-scale mechanical field involves: fragmenting the coarse-scale mechanical field into multiple fragments, whereby each fragment has a fragment boundary on the coarse-scale mechanical field, such that each fragment is a local portion of the coarse-scale mechanical field located within the fragment boundary; mapping the local portion of the coarse-scale mechanical field to the corresponding local portion of the fine-scale mechanical field using an artificial neural network, thereby computed the corresponding local portion of the fine-scale mechanical field for the multiple fragments; and combining the corresponding local portions of the fine-scale mechanical field to generate the fine-scale mechanical field. The generated fine-scale mechanical field is then set as the mechanical field of the structure. Therefore, this method allows the use of fine-scale mechanical fields with higher accuracy compared to coarse-scale mechanical fields as mechanical fields, without the need to use FEM to directly calculate the entire fine-scale mechanical field from the fine-scale structure to save computational costs.

[0013] Artificial neural networks are exemplarily implemented as multi-participant path neural networks (MapNet).

[0014] Preferably, MapNet comprises multiple convolutional layers, multiple deconvolutional layers, and residual blocks. Each of the convolutional and deconvolutional layers has a 3×3 filter size, a 2×2 stride except for the last layer of MapNet, and the activation function ReLU. The last layer of MapNet has a 1×1 stride. A key feature of MapNet is that its input consists of a local portion of a coarse-scale mechanical field and a corresponding local portion of a fine-scale density field within the fragment boundaries, where the fine-scale density field is defined by a fine-scale structure. Another feature of MapNet is the addition of a density field downsampled from the fine-scale density field at each deconvolutional layer, representing the density field at the same scale. Both of these features are unique and crucial for achieving high accuracy when computing the corresponding local portions of the fine-scale mechanical field.

[0015] When mapping a local portion of a coarse-scale mechanical field to a corresponding local portion of a fine-scale mechanical field, preferably, the artificial neural network predicts the corresponding local portion of the fine-scale mechanical field based on the fine-scale density field and the local portion of the coarse-scale mechanical field. The fine-scale structure defines the fine-scale density field.

[0016] In the process of coarsening the fine-scale structure to generate the coarse-scale structure, the coarse-scale structure is obtained by scaling down the fine-scale density field to obtain the coarse-scale density field. The fine-scale structure defines the fine-scale density field, and the coarse-scale density field defines the coarse-scale structure.

[0017] In the process of fragmenting a coarse-scale mechanical field into multiple fragments, there may or may not be fragment overlap between corresponding fragments in the multiple fragments.

[0018] A second aspect of the invention provides a second computer-executed method for performing topology optimization of a structure according to design requirements. The design requirements specify minimizing or maximizing an objective function subject to one or more constraints.

[0019] The second method includes the following steps: (a) selecting a candidate structure for testing whether the candidate structure meets the design requirements; (b) calculating the mechanical field of the candidate structure according to any embodiment of the first method of this disclosure; (c) using the calculated mechanical field to evaluate the objective function, one or more constraints, or both the objective function and one or more constraints; (d) determining whether the candidate structure meets the design requirements; and (e) if the candidate structure does not meet the design requirements, updating the candidate structure and repeating steps (b) to (e), otherwise setting the candidate structure that meets the design requirements as the structure obtained by topology optimization.

[0020] In some embodiments, the objective function and one or more constraints are related to a design problem that minimizes structural compliance. The mechanical field is the strain energy field.

[0021] In some embodiments, the objective function and one or more constraints relate to a design problem that minimizes thermal compliance. The mechanical field is the temperature field.

[0022] Other aspects of this disclosure are disclosed through the embodiments described below. Attached Figure Description

[0023] Figure 1A Examples of design domains considered in the design problem of minimizing structural flexibility are shown, including (a) a cantilever beam with a load applied at the upper right corner, (b) a cantilever beam with multiple loads, (c) an L-shaped beam, and (d) a bridge.

[0024] Figure 1B The design domains considered in the design problem of minimizing thermal compliance are shown under two different boundary condition settings: (a) with a small fixed temperature region; and (b) with a large fixed temperature region.

[0025] Figure 2 A schematic diagram of a conventional FEM-based topology optimization design framework is shown.

[0026] Figure 3 A schematic diagram of a MapNet-based topology optimization design framework according to an exemplary embodiment of the present invention is shown.

[0027] Figure 4 This illustrates the coarsening process of converting fine-scale structures into coarse-scale structures when performing structural topology optimization.

[0028] Figure 5 This illustrates the fragmentation process of pruning a domain into smaller subdomains (fragments) when performing structural topology optimization.

[0029] Figure 6 A comparison between non-overlapping and overlapping fragmentation processes is shown.

[0030] Figure 7 Similar fragments from cut-down L-shaped beams and cantilever beam structures are shown.

[0031] Figure 8 A schematic diagram of an exemplary architecture of MapNet is shown.

[0032] Figure 9 The fragmentation, MapNet prediction, and reconstruction process is shown, starting with the coarse-scale strain energy field and the fine-scale density field, and finally obtaining the fine-scale strain energy field.

[0033] Figure 10 Samples of the distributions of coarse-scale and fine-scale strain energy fields obtained from fragments of a cantilever beam design problem are shown.

[0034] Figure 11 The predictions of the strain energy field (fragments) by MapNet trained with different amounts of training data (cantilever beam problem) are shown.

[0035] Figure 12 The comparison between MapNet's predictions of strain energy fields (fragments) and the actual strain energy fields trained in different ways is shown, including training without using fragment data and without using density field, training with only fragment data, and training with both fragment data and density field.

[0036] Figure 13 The comparison of MapNet's predictions for the strain energy field is shown in two cases: the first case is where MapNet neither performs fragmentation nor uses fine-scale density for training, and the second case is where MapNet performs fragmentation and uses fine-scale density for training.

[0037] Figure 14The comparison shows (a) between predictions of the strain energy field by MapNet trained with fine-scale density at different fragment clipping scales, and (b) between magnified views of the predicted strain energy field.

[0038] Figure 15 An example of a non-uniqueness problem is shown.

[0039] Figure 16 The comparison between MapNet's predictions of the strain energy field after non-overlapping fragmented training and after overlapping fragmented training is shown.

[0040] Figure 17 A comparison is shown between the cantilever beam topology optimization results obtained by conventional FEM-based methods and the MapNet method of this disclosure.

[0041] Figure 18 A comparison is shown between the topology optimization results for a cantilever beam subjected to multiple loads obtained by the FEM-based method and the MapNet method of this disclosure.

[0042] Figure 19 A comparison is shown between the L-beam topology optimization results obtained by the FEM-based method and the MapNet method of this disclosure.

[0043] Figure 20 A comparison is shown between the topology optimization results for the bridge design problem obtained by a method based solely on FEM and the MapNet method of this disclosure.

[0044] Figure 21 A comparison is shown between the cantilever beam topology optimization results obtained by a method based solely on FEM and the MapNet method of this disclosure (both using the SIMP method).

[0045] Figure 22 A comparison is shown between the topology optimization results obtained by a method based solely on FEM and the MapNet method of this disclosure (both using the SIMP method) for (a) the L-beam problem and (b) the bridge design problem.

[0046] Figure 23 A comparison is shown between the topology optimization results obtained by the FEM-based method and the MapNet method of this disclosure for a thermal problem with a small fixed temperature range.

[0047] Figure 24 A comparison is shown between the topology optimization results obtained by the FEM-based method and the MapNet method of this disclosure for thermal problems with large fixed temperature regions.

[0048] Figure 25 A flowchart illustrating exemplary steps of a first method executed by a computer according to this disclosure is shown for calculating the mechanical field of a structure, the method calculating the fine-scale mechanical field of the structure from the coarse-scale mechanical field of the structure to save computational costs.

[0049] Figure 26 A flowchart illustrating exemplary steps for calculating a fine-scale mechanical field from a coarse-scale mechanical field is shown, using an artificial neural network in the calculation.

[0050] Figure 27 A flowchart illustrating exemplary steps of a second method executed by a computer is shown for performing topology optimization of a structure according to design requirements, which specify minimizing or maximizing an objective function subject to one or more constraints.

[0051] Those skilled in the art will understand that the elements in the accompanying drawings are illustrated for simplicity and clarity and are not necessarily depicted to scale. Detailed Implementation

[0052] In this disclosure, an artificial neural network-based model reduction method is proposed, disclosed, and developed to significantly reduce the computational cost of expensive numerical evaluations of objective functions and constraints, thereby accelerating the topology optimization process for large-scale designs. In particular, the method is scalable and applicable to different problem settings, including varying domain sizes and shapes, without requiring retraining of the deep learning model.

[0053] The following provides an overview of the disclosed method. In this method, the objective function and / or constraints are evaluated on a coarser mesh, rather than the original mesh, using conventional methods such as FEM. Then, an artificial neural network model called Multi-Participation Path Neural Network (MapNet) is used to map the coarse-scale mechanical field to a fine-scale mechanical field. A key advantage of this approach is the readily available coarse-scale field, which contains physical information such as boundary conditions and loading conditions. If the boundary / loading conditions change, the coarse-scale field changes accordingly. Training MapNet is easier and requires less fine-scale data compared to artificial neural network models that map structures to their fine-scale mechanical fields, as the neural network only needs to learn the relationship between the coarse-scale mechanical field and its corresponding fine-scale mechanical field. To improve transferability and scalability, a domain decomposition approach is utilized in the disclosed method. The problem domain is decomposed into a set of small subdomains or fragments, and the neural network is configured to perform mapping on each small subdomain rather than the entire problem domain. The predicted fields for each subdomain are then combined to form the field of the original domain. A key advantage of this approach is that many different domains with varying sizes and shapes can potentially be decomposed into similar sets of subdomains / fragments, making it easier to transfer MapNet trained on data from a specific design problem to different problems. Furthermore, the increased amount of training data, due to the ability to break down a single dataset into many subdomains, enhances the accuracy of the neural network.

[0054] The following sections first describe an exemplary design problem used to demonstrate the performance of the disclosed method. Next, a detailed implementation of the disclosed method is discussed. Then, the accuracy and efficiency of the disclosed method are demonstrated on various design problems with different design domains and boundary conditions, using results obtained from conventional topology optimization methods as a benchmark. Finally, various embodiments of the invention are developed and described in detail.

[0055] A. Problem Statement

[0056] The disclosed model reduction method based on artificial neural networks is illustrated by implementing topology optimization on two benchmark design problems: a design problem that minimizes structural compliance and a design problem that minimizes thermal compliance.

[0057] The design problem of minimizing structural flexibility can be mathematically represented in discrete form as follows:

[0058]

[0059]

[0060] KU = F

[0061] Where: x is the design variable, representing the "density" of the elements; C(x) is the objective function; U is the displacement vector; K is the global stiffness matrix; F is the loading vector; u i is the displacement vector of the i-th element; k0 is the local stiffness matrix associated with the element occupied by solid material; V(x) represents the volume of the structure, and V0 is the volume of the entire design domain; f is the desired volume fraction. As an illustrative example to demonstrate the disclosed method, the volume fraction constraint is set to 0.4 for all cases. Furthermore, to demonstrate the disclosed artificial neural network-based model reduction method, two common topology optimization methods are used: Two-Way Progressive Structure Optimization (BESO) and Penalized Solid Isotropic Material (SIMP) to perform topology optimization design. The element density x is 0 or 1 in BESO and a value between 0 and 1 in SIMP, where 0 represents voids and 1 represents the solid phase of the structure. The design objective is to optimize the topology, i.e., density, of the structure within the design domain such that the compliance (strain energy) of the structure is minimized under a given loading and volume fraction. Multiple design cases with different domains, boundary conditions, and loading conditions are selected to demonstrate the transferability of this method. The first design problem is... Figure 1A Subgraph (a) shows the classic two-dimensional cantilever beam design problem. The design domain is a square with a fixed left boundary, and a vertically distributed load is applied to the upper right boundary of the domain. The second design problem is essentially the same as the cantilever beam design, except that instead of a single load, three dispersed forces are applied at the center of the top boundary, the center of the right boundary, and the center of the bottom boundary, respectively, as shown in Figure (a). Figure 1A As shown in sub-diagram (b). In addition, two other design problems were considered, namely, respectively in... Figure 1A The L-shaped beam and bridge designs are shown in subgraphs (c) and (d). These two cases have different domain shapes and dimensions. Furthermore, the boundary conditions and loading conditions are completely different from those in the cantilever beam design problem. For the beam designs in (a), (b), and (c), the distributed load is applied over a square region of L / 16 by L / 16, while for the bridge design problem, the load is distributed over the entire top boundary.

[0062] To demonstrate the generality of the disclosed artificial neural network-based model reduction method, it is also used to solve a design problem of minimizing thermal compliance. In this problem, the design objective is to minimize thermal compliance under given heat loading and boundary conditions. The mathematical model of the design problem is given by the following equation:

[0063]

[0064]

[0065] K c T = F

[0066] 0≤x≤1

[0067] Where: x is the design variable, representing the element's "density"; C(x) is the objective function; T is the temperature field; K c is the conductivity matrix; F is the heat load including contributions from boundary heat flux and internal heat generation / loss. The SIMP method is used for this problem, and the design domain is square. This design problem is based on the problems presented in the studies of M. BENDSOE and O. SIGMUND (2003) and X. QU, N. PAGALDIPDITTI, R. FLEURY and J. SAIKI (2016), where the design domain is isolated at all edges except for the heat sink located at the center of the top boundary. The entire plate is subjected to distributed heating. Two different boundary conditions are considered, the first with a smaller fixed temperature region and the second with a larger fixed temperature region. Figure 1B The length of the boundary with a fixed temperature is shown in the figure.

[0068] B. Research Methods

[0069] As described above, in the disclosed model reduction method based on artificial neural networks, the evaluation of the objective function and / or constraints in each topology optimization iteration is performed using MapNet, rather than numerical simulation used in conventional topology optimization methods. The main evaluation process can be divided into five steps. First, the fine-scale structure is scaled down to obtain its coarse-scale structure (coarsening), and FEM calculations are performed on the coarse-scale structure to obtain the fields required to evaluate the objective function and / or constraints. Next, the entire domain / field is decomposed into a set of smaller fragments (fragmentation). Then, MapNet is used to map the coarse-scale field of each fragment to the corresponding fine-scale field. Finally, the fragments of the fine-scale field are combined to form the fine-scale field of the original domain (reassembly). This process is performed in... Figure 3 As shown in the flowchart, and with Figure 2 The following section compares the conventional FEM-based topology optimization methods with those shown. Each process is described in detail in the following paragraphs.

[0070] coarsening by having N F The fine-scale density field of the structure of each element is scaled down to have N C To perform this using a coarse-scale density field of N elements, where N C Should be compared to N F Much smaller to achieve high efficiency. The density values ​​of coarse-scale elements are obtained by adjusting N. F / N C It is obtained by averaging the density values ​​of elements at a fine scale. Figure 4An example of the conversion from a fine-scale structure to a coarse-scale structure is shown. In the second step, FEM calculations are performed on the coarse-scale structure to obtain its physical fields, such as stress / strain fields. Due to the large scale difference between the coarse and fine scales, simulation time can be significantly reduced.

[0071] Next, the entire field will be fragmented, such as... Figure 5 As shown in the diagram, using a sample of the density field obtained from the cantilever beam design problem as an example, the fine-scale structure is trimmed into 64 smaller non-overlapping fragments. However, as an alternative to non-overlapping trimming, overlapping trimming can also be introduced further, such as... Figure 6 As shown in the diagram. In fact, overlapping clipping is more advantageous because when the fragments are combined to form the entire field, the likelihood of discontinuities at the edges of each fragment is reduced, thus making the entire field smoother.

[0072] As mentioned earlier, using MapNet to map fragment fields rather than the entire domain helps improve portability. This is because fragments from different designs are more likely to be similar to each other, even if the overall designs are completely different. Consider... Figure 7 A comparison between the cantilever beam and the L-shaped beam shown in the diagram clearly demonstrates the distinct differences between the two structures when viewed as a whole. However, as the domain is fragmented, more and more similar fragments emerge, such as... Figure 7 Fragments are labeled with the same connecting line. Therefore, if MapNet is trained with clipped cantilever beam data, the trained MapNet may provide accurate predictions of L-shaped beams without any retraining. This advantage is even more pronounced in design cases with varying domain sizes and shapes. Furthermore, the fragmentation process can increase the amount of training data for MapNet. For example, as... Figure 5 As shown, a structural sample is fragmented into 64 samples, providing 64 times the number of samples that can be used to train MapNet. It should be noted that the fragmentation method is only possible in the specific way MapNet is constructed. Since MapNet predicts fine-scale fields from coarse-scale fields, it does not require additional input regarding external constraints of the problem (such as boundary conditions or loading conditions). However, if the constructed artificial neural network maps the density field of the structure to the corresponding mechanical field, fragmentation cannot be performed because additional input boundary / loading conditions are required.

[0073] After fragmentation, MapNet is used to map the coarse-scale field of each fragment to its fine-scale field. The architecture of MapNet is as follows: Figure 8As shown, each of the convolutional and deconvolutional layers has a filter size of 3×3, a stride of 2×2 (except for the last layer which has a stride of 1×1), and the activation function is ReLU. A key feature of MapNet is that its input consists of a local portion of the coarse-scale mechanical field within the fragment boundary and a corresponding local portion of the fine-scale density field within the same fragment boundary, where the fine-scale density field is defined by a fine-scale structure. Another feature is the addition of a density field downsampled from the fine-scale density field at each deconvolutional layer. Both of these features are unique and crucial for achieving high accuracy when computing the corresponding local portions of the fine-scale mechanical field. Details of the MapNet architecture can be found in the MapNet code in the GitHub repository. The coarse-scale grid size in the MapNet architecture is chosen to be 1 / 16 of the fine-scale grid size in both width and height. For ease of training MapNet, the data used as input and output are represented using two-dimensional arrays, where each element represents the field value at the corresponding location in the two-dimensional design problem. However, due to MapNet's convolutional architecture, a dataset with N samples is shaped into (N×W×H×1), where W and H represent the width and height of each array, respectively. The inventors experimentally discovered that applying the concepts of U-Net to MapNet can improve performance. Furthermore, incorporating fine-scale structures into different deconvolutional layers also helps improve prediction accuracy.

[0074] To train MapNet, results obtained from early iterations of a FEM-based topology optimization design can be used. Specifically, for Figure 1A The cantilever beam design problem shown in subgraph (a) is subjected to conventional FEM-based topology optimization. Fine-scale strain energy fields and density fields are extracted during each iteration to form a training dataset. Results from earlier iterations of topology optimization are used as training data because the structure's topology or density field undergoes major changes only during early iterations, while later iterations involve only local fine-tuning. The trained MapNet is then applied to other design problems. Fragments of the fine-scale strain energy fields predicted by MapNet are recombined back to the original scale through a recombination process. For non-overlapping recombinations, the process is straightforward, simply combining the predictions edge-to-edge. For overlapping cases, each fragment is also combined with adjacent fragments, but the average is taken at each overlapping region (where multiple fragments overlap).

[0075] C. Results and Discussion

[0076] The following section first analyzes the performance of MapNet, and then discusses the performance of the disclosed method on several common two-dimensional design problems of topology optimization.

[0077] C.1 MapNet Performance Evaluation

[0078] As shown in subgraph (a) of Figure 1, the accuracy of MapNet is studied using a cantilever beam design problem with a volume fraction constraint of 0.4 and a distributed area load applied to the upper right corner. The design resolution N is chosen as... F =512×512 is used as the fine-scale grid, while N is chosen as the coarse-scale grid. C =32×32. A standard FEM-based topology optimization was run to produce a design that converged within 200 iterations. Detailed values ​​for each parameter used in the FEM and topology optimization algorithms can be found in the code available in the Github repository. The data obtained from this topology optimization process was used to train MapNet following the procedure described in the previous section. The impact of the amount of training data on the accuracy of the neural network is analyzed first. Furthermore, the performance of MapNet is compared with MapNet without fragmentation and without embedding fine-scale density in the neural network to demonstrate the effectiveness of the disclosed method. Next, the fragmentation method is analyzed by studying the impact of different fragment sizes on performance. Finally, the overlap technique is discussed and applied to the fragmentation process.

[0079] C.1.1 Number of Training Data

[0080] The topology optimization process for the cantilever beam design is run N times to obtain three sets of fine-scale data with total numbers N = 40, 60, and 100. For example, if a neural network is to be trained with 40 data points, the topology optimization can only run a maximum of 40 iterations. The coarse-scale strain energy field required for training MapNet is obtained by performing FEM on the coarse-scale structure, which is obtained by reducing the scale of the fine-scale structure as described above. Then, the strain energy field is clipped into small non-overlapping fragments using a clipping scale of 16, with each sample of the coarse-scale strain energy field clipped from its original 32×32 size to 256 samples of 2×2 fragments. The fine-scale density field and strain energy field are also clipped from their original 512×512 size to 256 samples of 32×32 non-overlapping fragments. Therefore, the actual number of training data for MapNet is N. 训练 =256×40, 256×60, and 256×100. The cutting process is as follows: Figure 9 As shown in the image. To illustrate this more clearly, Figure 9The cropping scale used was 8. Using these cropped data, MapNet was trained using the ADAM optimizer and a learning rate of 1e-4. Due to computational limitations, the number of iterations was set to 1000 steps. To simplify the training process of the neural network, the input and output values ​​were normalized to the range of 0 to 1. The normalization factors for the coarse-scale strain energy field and the fine-scale strain energy field were chosen to be 1e-4 and 1e-6, respectively, both derived from... Figure 10 The distribution shown in the figure is obtained.

[0081] After MapNet was trained, 100 datasets not included in the training dataset were used to evaluate MapNet's accuracy. In this case, data from iterations 100 to 200 of a standard FEM-based topology optimization process were selected as test data. Each dataset was pruned into 256 fragments. The fine strain energy field predicted by MapNet for each fragment was compared with the strain energy field obtained directly from FEM-based topology optimization using FEM (real data). The comparison results are shown in [the original text]. Figure 11 As shown in the figure. For clarity, the strain energy field is plotted on a logarithmic scale with an offset of 1e-8, i.e., U 图 =log(U 原始 +10 -8 To provide a qualitative measurement of error, the mean square error is calculated and listed in Table 1. The mean square error is defined as...

[0082]

[0083] Where N is the total number of test fragments, and These refer to the strain energy of the i-th element, predicted by MapNet and obtained by FEM, respectively. From Figure 11 As shown in Table 1, the mean squared error decreases with the increase in the amount of training data, thus MapNet's accuracy improves as expected. Considering both accuracy and efficiency, all analyses published herein used MapNet trained with data obtained from 60 topology optimization iterations. To demonstrate the impact of fragmentation and the inclusion of a fine-scale density field in artificial neural networks, the predicted fine-scale strain energy field of fragmentation was compared with the strain energy field obtained by MapNet trained without fragmentation (i.e., MapNet directly maps a 32×32 coarse-scale field to a 512×512 fine-scale field), and this was compared with the strain energy field obtained by MapNet without incorporating a fine-scale density field. Figure 12The comparison results are shown in the figure. As can be seen from the figure, fragmentation and incorporating fine-scale density significantly improve prediction accuracy. In particular, adding a fine-scale density field produces clearer and more defined edges. This is because the fine-scale density field contains information about the general shape of the structure and can provide a filtering effect on the prediction.

[0084] Number of training data 100 60 40 Mean square error 0.00029 0.00058 0.00072

[0085] Table 1: Comparison of mean squared errors of MapNet trained with different amounts of training data

[0086] To study the performance of MapNet, it is also important to examine the prediction accuracy of the entire field (i.e., the original 512×512 field obtained after combining all the 32×32 fragment fields). This process is called recombination, and it is also... Figure 9 As shown in the image. Figure 13 The second row shows the reconstructed strain energy field predicted by MapNet, which was trained using fragmented data and fine-scale density. In addition to visual comparisons, quantitative comparisons were also made by calculating the mean squared error of the predictions, and the results are listed in Table 2. Figure 13 Table 2 also shows the predictions given by MapNet trained without fragmentation and without using a density field for comparison. It can be observed that the fine-scale strain energy field predicted using the published method is also better in the recombined form.

[0087]

[0088] Table 2: Comparison of mean squared errors between networks trained without fragmentation and fine-scale density and networks trained with both.

[0089] C.1.2 Fragmentation - Clipping Scale

[0090] The results shown in the previous section demonstrate that fragmentation improves the prediction accuracy of neural networks. In these results, fragmentation using a pruning scale of 16 increased the training data by a factor of 256. Using a larger pruning scale (e.g., 32) would further increase the amount of training data and should further improve MapNet's accuracy. However, this research shows that this is not necessarily the case. This study was conducted by training MapNet with three different pruning scales: 8, 16, and 32. Figure 14 The results, presented in the recombined form, were compared, and the prediction errors for all cases are listed in Table 3.

[0091] Cutting dimensions 8 16 32 Mean square error 0.00088 0.00058 0.00152

[0092] Table 3: Comparison of mean squared errors of MapNet trained with different fragment pruning scales

[0093] By comparing the results obtained from pruning scales of 8 and 16, it can be observed that the mean squared error of MapNet decreases as the pruning scale increases. This is due to the aforementioned explanation; as the pruning scale increases, more fragments are generated, thus increasing the amount of training data used to train MapNet. However, when the scale increases to 32, corresponding to a fragment size of 1×1 for coarse-scale data and 16×16 for fine-scale data, performance degrades and the error rate increases. One reason for the performance degradation is the unsmooth boundaries between adjacent fragments, such as... Figure 14 As shown in subfigure (b), since the fine strain energy of the entire region is obtained by simply rearranging all non-overlapping fragments, the values ​​at the fragment edges are expected to be discontinuous between adjacent fragments. Another reason may be that the probability of non-uniqueness increases as samples are pruned into smaller fragments. Non-uniqueness refers to the situation where the same input to a neural network corresponds to different output values, essentially having a one-to-many mapping, which makes training the neural network more difficult. For example, as... Figure 10 As shown, the coarse-scale strain energy ranges from 0 to 1.5e-4. Therefore, any two values ​​with a difference less than 1e-12 are considered identical. By using this criterion and considering fragment data with a clipping scale of 32 and a fragment size of 1×1, it was found that two fragments with the same coarse-scale strain energy value of 2.57e-5 and the exact same density field have two different fine-scale strain energy fields, as shown. Figure 15 As shown in the diagram. Therefore, from a cost-benefit perspective, the pruning scale should not be too large to avoid non-uniqueness issues during MapNet training. In this case, the optimal pruning scale is 16.

[0094] C.1.3 Fragmentation - Overlapping Fragments

[0095] Although the results obtained by performing fragmentation are significantly better than those obtained without it, closer inspection reveals that, for reasons explained in the previous section, the fine-scale strain energy field predicted in the reconstituted form is not very smooth at the boundaries between the two fragments. To overcome this problem, Chapter B ( Figure 6 The overlapping method described in [reference]. Figure 15During fragmentation, overlap is performed by pruning the domain at smaller intervals. Given an original domain of 32×32 with a pruning scale of 16, the domain is pruned at 2-element intervals. For overlaps, the domain is pruned at 1-element intervals. Therefore, a 32-element side length will be pruned into 31 fragments instead of the previous 16. Similarly, after fragmenting the fine-scale strain energy predicted via MapNet, the fragments are also combined at 16-element intervals, averaging the results in any overlapping portions. Figure 16 The results obtained using overlapping fragmentation are shown in Figure 1, and the mean squared error is listed in Table 4. A comparison with previous results obtained using non-overlapping fragmentation shows that the predicted fine-scale strain energy field is smoother when using overlapping fragmentation, and the error of MapNet is further reduced.

[0096] <![CDATA[The number of training data (N 训练 )]]> Overlapping fragmentation Non-overlapping fragmentation Mean square error 0.0004 0.0006

[0097] Table 4: Comparison of mean squared errors between MapNet trained with overlapping fragments and MapNet trained with non-overlapping fragments.

[0098] C.2 Application of MapNet in Topology Optimization Design

[0099] We developed and trained MapNet using the methods described in the previous section, and then used it to perform a topology optimization process, while also studying the methods ( Figure 3 Following the process described in the flowchart in Figure 1, the topology optimization process for the cantilever beam design with a single load, as shown in subgraph (a) of Figure 1, is executed. The BESO algorithm is used in this case. For clarity, it should be noted that the MapNet used in all structural designs presented in this disclosure was trained using only 60 topology optimization data points obtained from the first 60 iterations of the cantilever beam design shown in subgraph (a) of Figure 1, with a pruning scale of 16 and overlapping fragmentation. Figure 17 The optimized structure obtained through the improved topology optimization process (MapNet topology optimization process) will be compared with the optimized structure obtained through the conventional FEM-based topology optimization process. Figure 17The report also provides the final objective, namely the flexibility associated with the two structures. Furthermore, the optimized structures obtained from MapNet without using fragmentation and fine-scale density fields demonstrate the superiority of the current MapNet architecture. The first thing that becomes apparent through comparison is that MapNet with fragmentation performs significantly better results than without it. Simultaneously, the optimized structures obtained from MapNet topology optimization are very similar to those obtained using conventional methods. In the MapNet topology optimization process, only the coarse-scale field needs to be computed using FEM, thus significantly reducing the total computation time required to complete the entire topology optimization process. Table 5 details the time for each relevant stage in each case, as well as the total time required for each iteration in the topology optimization process. The computation times shown are based on calculations performed on a 16-core Intel Xeon CPU E5-2687Wv2 (3.40GHz). As can be seen from Table 5, using MapNet topology optimization reduces computation time by a factor of 300 in a single topology optimization iteration. Considering that the topology optimization design in this cantilever beam case requires 200 iterations to converge, while the training data is generated using only the first 60 iterations, the total time saved in completing one topology optimization process is significant.

[0100]

[0101] Table 5: Time comparison of completing a single topology optimization iteration using FEM alone and the MapNet proposed in this disclosure.

[0102] C.3 MapNet Portability

[0103] This section demonstrates the transferability of MapNet through several benchmark design problems, which differ from the cantilever beam design case used to train MapNet. These are... Figure 1A The diagram illustrates the design problems of cantilever beams with multiple forces, L-beams, and bridges. Specifically, a MapNet trained using the previous simple cantilever design was directly used to perform the topology optimization process, thus solving the three new design problems without any retraining.

[0104] Figure 1A The first two design problems shown in subgraphs (a) and (b) have the same square design domain, with a fine-scale grid of 512×512 and a coarse-scale grid of 32×32. The BESO algorithm was used as the topology optimization method for all three design cases. Figure 18 and Figure 19The optimized structures and their final compliances obtained through conventional FEM-based topology optimization and MapNet-based topology optimization are shown respectively. As the results demonstrate, the optimized structures obtained using the MapNet topology optimization method are similar to those obtained using the FEM topology optimization method, and have similar final compliances.

[0105] Figure 1A The boundary conditions, domain shape, and dimensions of the bridge design problem shown in subgraph (d) differ significantly from all other design cases. The design domain is distributed into elements of 768 (width) × 384 (height). Figure 20 The diagram shows the design results obtained through FEM-based topology optimization. In this case, a coarse-scale mesh was chosen as N. C = 48×24. During fragmentation, each sample of the coarse-scale strain energy field is clipped into 47×23 2×2 fragments, including one element of overlap clipping. The fine-scale density field is clipped into 47×23 distinct fragments of size 32×32. These inputs are then fed into MapNet to predict the fine-scale strain energy field for all 47×23 fragments of size 32×32. These fine-scale strain energy fields are then combined to form the entire field of the original domain, i.e., the field of size 768×384, and the topology optimization process is performed as described previously. Although the design domain of the bridge is completely different from that of the cantilever beam, the previously trained MapNet can still be directly applied to this problem by using the fragmentation process, because it provides predictions of fragments / building blocks of the structure, rather than predictions of the entire structure. Therefore, its generalization ability is greatly improved. Figure 20 The results also show the optimization results obtained through a MapNet-based topology optimization process. The results demonstrate that MapNet, trained on cantilever beam data, once again exhibits excellent performance, providing optimized structures and flexibility that closely resemble the results obtained with real data.

[0106] It should be noted that the strain energy field needs to be properly normalized before being input into MapNet; otherwise, the predicted fine-scale strain energy field will have significant errors. Since the strain energy range may differ for different problems, different normalization factors should be determined and used for different problems. This can be done by observing the coarse and fine strain energy fields in the first few iterations (e.g., five iterations of a topology optimization process based on FEM). In the first two design cases, the normalization factor is the same as that for a simple cantilever beam design with a single load. In the bridge design case, it was found that the normalization factor should be set to 1e-7 and 1e-9 for coarse-scale and fine-scale data, respectively.

[0107] To examine efficiency, the computation time required to perform MapNet-based topology optimization in the three new design problems was calculated. For the cantilever beam and L-beam with multiple external loads, the time savings per design iteration are similar to those listed in Table 4, as both have the same mesh size (512×512) as the previous cantilever beam problem. For the bridge design problem, the time savings per iteration are slightly different, approximately 250 times, as listed in Table 6.

[0108]

[0109] Table 6: Comparison of single-iteration computation time for topology optimization using the simple FEM method and the MapNet method of this disclosure for bridge design problems.

[0110] Application of C.4SIMP in MapNet

[0111] In this section, the disclosed method is used to perform a SIMP-based topology optimization process to demonstrate that the disclosed method is not limited by the type of topology optimization method. In the SIMP method, the density field contains intermediate values, thus constructing a new MapNet. Following the process described in previous sections and again using a cantilever beam design with a single load to generate training data, it was found that MapNet requires a minimum of 60 training data points from the topology optimization process to achieve satisfactory prediction accuracy. The performance of MapNet is then evaluated by directly executing the SIMP method with MapNet and comparing the resulting design with that obtained using the FEM-based SIMP method in the same case. Figure 21 As shown, satisfactory results were obtained.

[0112] Next, to demonstrate transferability, the MapNet-based SIMP method is used directly to design L-shaped beams and bridges. Figure 22 In this paper, the optimization results for each design problem are compared with the design solutions obtained through the FEM-based SIMP method. These results further demonstrate the good transferability of the disclosed method.

[0113] C.5 Application to thermal problems

[0114] The disclosed method is also applied to the design of structures with minimal thermal flexibility to further demonstrate its performance. Two design problems are described in Section A above. The first design problem considered is having a small, fixed temperature region at the center of the top boundary (…). Figure 1BThe subgraph (a) is shown. The domain size for this design problem is also chosen to be 512 × 512, and the volume fraction constraint is chosen to be 0.4. This problem is first solved using the FEM-based SIMP method, with a filter radius of 16. The computation converges in approximately 100 iterations. Figure 23 The subplot on the left shows the optimization results. A new MapNet with the same architecture as the previous MapNet was built and trained using data obtained from the initial iterations of FEM-based topology optimization. Specifically, training MapNet with 40 topology optimization datasets was found to be sufficient to achieve satisfactory performance. The coarse-scale grid size for this problem is also 32×32. By observing the distribution and extent of thermal compliance obtained from the first five iterations of FEM-based topology optimization, normalization factors for this thermal problem were chosen to be 200 and 5 for coarse-scale and fine-scale thermal compliance, respectively. Figure 3 The process shown in the figure uses a trained MapNet to perform a topology optimization process and uses MapNet-based topology optimization to minimize thermal flexibility. Figure 23 The optimization results are shown in the figure, and it can be observed that the optimization results are close to those of the topology optimization based on FEM.

[0115] To demonstrate the transferability of the trained MapNet, the second hot problem is solved using MapNet trained on data from the first hot problem. Figure 1B The subgraph (b) shows a large fixed-temperature region at the top of the boundary. The design results are presented together with those obtained from FEM-based topology optimization. Figure 24 Although the actual optimization results for the second hot problem appear significantly different from those for the first hot problem, the fact that the MapNet-based method produced similar optimization results further demonstrates MapNet's good transferability.

[0116] D. Extension of the disclosed method

[0117] While the published method demonstrates high efficiency in solving design problems in the two-dimensional domain, it can potentially be extended to solve three-dimensional problems. Since the computational time required in three-dimensional simulations is significantly greater than that in two-dimensional simulations, the time savings could be substantial if a large difference is chosen between coarse and fine scales.

[0118] E. Details of the embodiments of the present invention

[0119] The following details embodiments of the present invention, based on the details, examples, and applications disclosed above regarding the model reduction method based on artificial neural networks. It should be noted that, according to the above description, the model reduction method based on artificial neural networks can be implemented in a computer system through appropriate programming.

[0120] A first aspect of the invention provides a first method executed by a computer for calculating the mechanical field of a structure. The mechanical field is related to the design problem considered in structural topology optimization. If the design problem is minimizing structural compliance, the mechanical field can be a stress-strain field or a strain energy field. If the design problem is minimizing thermal compliance, the mechanical field can be chosen as a temperature field.

[0121] Figure 25 A flowchart illustrating exemplary steps of a first method executed by a computer is shown. The first method includes steps 2510, 2520, 2530, 2540, and 2550.

[0122] In step 2510, a fine-scale structure is prepared or established for modeling the structure. A fine-scale structure is a geometric model obtained by dividing the structure into multiple fine-scale elements, such that the structure is composed of multiple fine-scale elements.

[0123] The fine-scale structure can be represented by listing the position coordinates of elements at different fine scales. It should be noted that the density field (called the fine-scale density field) can be calculated from the fine-scale structure, and vice versa. In this sense, the fine-scale structure and the fine-scale density field are equivalent. The fine-scale structure can be defined by the fine-scale density field, and vice versa. In some embodiments, the fine-scale structure is the fine-scale density field.

[0124] After obtaining the fine-scale structure in step 2510, the fine-scale structure is coarsened in step 2520 to generate a coarse-scale structure. Specifically, the generated coarse-scale structure consists of multiple coarse-scale elements, wherein the number of corresponding coarse-scale elements is chosen to be less than the number of corresponding fine-scale elements in the multiple fine-scale elements. The coarsening process is described in detail in Section B above, and an example of the coarsening process is given in Section C.

[0125] Similar to fine-scale structures, coarse-scale structures are geometric models obtained by dividing the structure into multiple coarse-scale elements, and can be represented by listing the position coordinates of different coarse-scale elements. The density field (called the coarse-scale density field) can be calculated from the coarse-scale structure, and vice versa. In this sense, coarse-scale structures and coarse-scale density fields are equivalent. A coarse-scale structure can be defined by a coarse-scale density field, and vice versa. In some embodiments, the coarse-scale structure is a coarse-scale density field.

[0126] In step 2520, preferably, a coarse-scale density field is generated by reducing the scale of the fine-scale density field, thereby coarsening the fine-scale structure to generate a coarse-scale structure.

[0127] After obtaining the coarse-scale structure in step 2520, in step 2530, FEM is applied to the coarse-scale structure to calculate the coarse-scale mechanical field of the structure.

[0128] In step 2540, advantageously, the fine-scale mechanical field of the structure is calculated from the coarse-scale mechanical field, rather than directly from the fine-scale structure using FEM. This reduces the computational cost of obtaining the fine-scale mechanical field. It should be noted that both the coarse-scale and fine-scale structures are geometric models, and the coarse-scale structure has a lower model order than the fine-scale structure. Therefore, obtaining the fine-scale mechanical field from the coarse-scale structure instead of the fine-scale structure reduces computational requirements by utilizing model order reduction.

[0129] Figure 26 A flowchart illustrating an exemplary step 2540 is shown. Step 2540 includes steps 2610, 2620, and 2630.

[0130] In step 2610, the coarse-scale mechanical field is fragmented into multiple fragments. Each fragment has a fragment boundary on the coarse-scale mechanical field, such that the individual fragment is a local portion of the coarse-scale mechanical field located within that fragment boundary. For example, Figure 5 The non-overlapping fragment 520 has a fragment boundary 525 and is a local portion of the field (domain) 590 located within that boundary 525. In another example, Figure 6 The overlapping fragments 620 in the fragmentation process have fragment boundaries 625. The fragmentation process is described in detail in Section B, and an example of the fragmentation process is given in Section C. It should be noted that, as described in Section B, fragment overlap between corresponding fragments in multiple fragments may or may not exist.

[0131] In step 2620, an artificial neural network is used to map or estimate the corresponding local portion of the coarse-scale mechanical field to, or to estimate the corresponding local portion of the fine-scale mechanical field. The corresponding local portion of the fine-scale mechanical field is the portion of the fine-scale mechanical field located within the fragment boundary of a single fragment. Further, this mapping procedure is used to process all fragments among a plurality of fragments. As a result, the corresponding local portions of the fine-scale mechanical fields for the plurality of fragments are computed. Advantageously, the artificial neural network is implemented, exemplarily, with MapNet. Details of MapNet can be found in sections B and C. In some embodiments, MapNet includes multiple convolutional layers and multiple deconvolutional layers. Each of the convolutional and deconvolutional layers has a 3×3 filter size, a 2×2 stride (except for the last layer of MapNet), and an activation function ReLU. The last layer of MapNet has a 1×1 stride.

[0132] In step 2630, the corresponding local portions of the fine-scale mechanical field are combined (recombined) to generate the fine-scale mechanical field. Specifically, the corresponding local portions are combined using the recombination process described in Sections B and C. In short, for non-overlapping fragmentation, the recombination process is completed by combining the corresponding local portions of the fine-scale mechanical field edge-to-edge. For overlapping fragmentation, the recombination process also combines the fragments, but at each point in the overlapping region (where multiple fragments overlap), the average value of the points on the aforementioned multiple fragments is taken.

[0133] refer to Figure 25 After generating the fine-scale mechanical field in step 2540, the generated fine-scale mechanical field is set as the mechanical field of the structure in step 2550. In this way, the disclosed method allows the use of a fine-scale mechanical field with higher accuracy compared to a coarse-scale mechanical field as the mechanical field, without having to use FEM to directly calculate the entire fine-scale mechanical field from the fine-scale structure, thus saving computational costs.

[0134] The mechanical field of the structure obtained in step 2550 can be advantageously used for topology optimization of the structure.

[0135] A second aspect of the invention provides a second computer-executed method for performing topology optimization of a structure according to design requirements. The design requirements specify minimizing or maximizing an objective function under one or more constraints. If the objective function and one or more constraints relate to a design problem of minimizing structural compliance, the mechanical field can be set as a strain energy field or a stress-strain field. Similarly, if the objective function and one or more constraints relate to a design problem of minimizing thermal compliance, the mechanical field can be set as a temperature field.

[0136] Figure 27A flowchart illustrating exemplary steps of a second method executed by a computer is shown. The second method includes steps 2710, 2720, 2730, 2740, 2745, 2750, and 2760.

[0137] Step 2710 is the initialization step. In step 2710, a candidate structure is selected to test whether the candidate structure meets the design requirements.

[0138] In step 2720, the mechanical field of the candidate structure is calculated according to any embodiment of the first method performed by a computer as disclosed above.

[0139] In step 2730, the calculated mechanical field of the candidate structure is used to evaluate the objective function, one or more constraints, or both the objective function and one or more constraints.

[0140] In step 2740, it is determined whether the candidate structure meets the design requirements based on the evaluated objective function and one or more constraints.

[0141] If it is determined that the candidate structure does not meet the design requirements (decision step 2745), the candidate structure is updated in step 2750, and steps 2720, 2730, 2740 and 2745 are repeated to process the updated candidate structure.

[0142] If it is determined that the candidate structure meets the design requirements (decision step 2745), then in step 2760, the candidate structure that meets the design requirements is set as the structure obtained through topology optimization.

[0143] As mentioned above, both the first and second methods can be implemented using a computer system. This computer system can be implemented using one or more computers. A single computer can be a general-purpose computer, a workstation, a computing server, a distributed server in a computing cloud, a laptop computer, a mobile computing device, etc.

[0144] This disclosure may be implemented in other specific forms without departing from the spirit or essential characteristics thereof. Therefore, these embodiments are to be regarded in all respects as illustrative rather than restrictive. The scope of the invention is indicated by the appended claims rather than by the foregoing description, and therefore all variations within the meaning and equivalent scope of the claims are intended to be included therein.

[0145] References

[0146] The following is a list of references cited in this specification. The entire contents of each of these references are incorporated herein by reference.

[0147] T. GOEL and WJROUX (2010), “Topology optimization for design engineering products”, U.S. Patent Application Publication No. 2010 / 0262406.

[0148] Y. LUO, Z. KANG and P. LIU (2021), “A structural topology optimization method based on material field reduction series expansion”, U.S. Patent Application Publication No. 2021 / 073428.

[0149] M.RAISSI, P.PERDIKARIS, and GEKARNIADAKIS (2021), “Physical Information Learning Machine”, U.S. Patent No. 10,963,540.

[0150] MYWANG (2021), “A B-spline high-resolution cell level set method for 3D cell structure modeling and topology optimization”, U.S. Patent No. 10,984,593.

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[0152] MPBENDSOE and O. SIGMUND (2003). Topology Optimization: Theory, Methods, and Applications. Springer Science & Business Media.

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[0154] HTKOLLMANN, DWABUEIDDA, S. KORIC, E. GULERYUZ, and NASOBH (2020). Deep learning for topology optimization of two-dimensional metamaterials. Materials Design. 196, 109098.

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Claims

1. A method executed by a computer for calculating the mechanical field of a structure, the method comprising: The structure is modeled using a fine-scale structure, wherein the fine-scale structure is obtained by dividing the structure into multiple fine-scale elements; The fine-scale structure is coarsened to produce a coarse-scale structure, such that the coarse-scale structure is composed of a plurality of coarse-scale elements, wherein the number of corresponding coarse-scale elements in the plurality of coarse-scale elements is less than the number of corresponding fine-scale elements in the plurality of fine-scale elements. The finite element method is applied to the coarse-scale structure to calculate the coarse-scale mechanical field of the structure. Obtaining the fine-scale mechanical field of the structure from the coarse-scale mechanical field, wherein obtaining the fine-scale mechanical field from the coarse-scale mechanical field includes: The coarse-scale mechanical field is fragmented into multiple fragments, such that each fragment has a fragment boundary on the coarse-scale mechanical field, and the individual fragment is a local part of the coarse-scale mechanical field located within the fragment boundary. An artificial neural network is used to map a local portion of the coarse-scale mechanical field to a corresponding local portion of the fine-scale mechanical field, thereby calculating the corresponding local portion of the fine-scale mechanical field for the multiple fragments; as well as The corresponding local portions of the fine-scale mechanical field are combined to generate the fine-scale mechanical field; as well as The generated fine-scale mechanical field is set as the mechanical field of the structure, thereby allowing the use of a fine-scale mechanical field with higher accuracy than the coarse-scale mechanical field as the mechanical field, without having to use the finite element method to directly calculate the entire fine-scale mechanical field from the fine-scale structure to save computational costs.

2. The method according to claim 1, wherein the artificial neural network is implemented as a multi-participant path neural network, i.e., MapNet.

3. The method of claim 2, wherein the MapNet comprises a plurality of convolutional layers, a plurality of deconvolutional layers, and residual blocks, wherein each of the convolutional and deconvolutional layers has a 3×3 filter size and an activation function ReLU, each of the convolutional and deconvolutional layers except the last layer of the MapNet has a 2×2 stride, the last layer of the MapNet has a 1×1 stride, wherein the input of the MapNet is a local portion of the coarse-scale mechanical field and a corresponding local portion of the fine-scale density field within the fragment boundary, the fine-scale density field being defined by the fine-scale structure, and wherein a density field of the same scale downsampled from the fine-scale density field is added at each deconvolutional layer.

4. The method of claim 1, wherein in the process of mapping a local portion of the coarse-scale mechanical field to a corresponding local portion of the fine-scale mechanical field, the artificial neural network predicts the corresponding local portion of the fine-scale mechanical field based on the fine-scale density field and the local portion of the coarse-scale mechanical field, wherein the fine-scale structure defines the fine-scale density field.

5. The method of claim 1, wherein in the process of coarsening the fine-scale structure to generate the coarse-scale structure, the coarse-scale structure is obtained by scaling down the fine-scale density field to obtain the coarse-scale density field, wherein the fine-scale structure defines the fine-scale density field, and wherein the coarse-scale density field defines the coarse-scale structure.

6. The method of claim 1, wherein during the fragmentation of the coarse-scale mechanical field into a plurality of fragments, there is fragment overlap among the corresponding fragments of the plurality of fragments.

7. The method of claim 1, wherein during the fragmentation of the coarse-scale mechanical field into a plurality of fragments, there is no fragment overlap between corresponding fragments of the plurality of fragments.

8. A computer-executed method for performing topology optimization of a structure according to design requirements, wherein the design requirements specify minimizing or maximizing an objective function subject to one or more constraints, the method comprising the following steps: (a) Select a candidate structure to test whether the candidate structure meets the design requirements; (b) Calculate the mechanical field of the candidate structure using the method of any one of claims 1-7; (c) Use the calculated mechanical field to evaluate the objective function and / or one or more constraints; (d) Determine whether the candidate structure meets the design requirements; as well as (e) If the candidate structure does not meet the design requirements, the candidate structure is updated and steps (b) to (e) are repeated; otherwise, the candidate structure that meets the design requirements is set as the structure obtained by topology optimization.

9. The method of claim 8, wherein the objective function and the one or more constraints are related to a design problem of minimizing structural flexibility, and wherein the mechanical field is a strain energy field.

10. The method of claim 8, wherein the objective function and the one or more constraints are related to a design problem of minimizing thermal compliance, and wherein the mechanical field is a temperature field.

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