A ship structure strain field inversion method and system based on a graph neural network

Abstract

The present application belongs to the field of ship strain field inversion, and specifically discloses a ship structure strain field inversion method and system based on a graph neural network, which comprises: taking the full-ship structure finite element calculation results as original data, based on the graph neural network, introducing a multi-information attention mechanism, designing a composite objective function, and guiding the model to learn the strain propagation law; based on the multi-working-condition simulation data, iteratively optimizing the parameters, establishing a ship global strain field inversion model, and realizing high-precision reconstruction under sparse observation. The present application establishes the connection between the ship local strain point and the strain field through an intelligent algorithm, can realize the reconstruction of the entire ship structure strain field from a few strain monitoring points, and can further analyze the ship structure strain field and stress field state according to the knowledge of structural mechanics, thereby providing a new method for the current ship structure health monitoring strain field inversion.

Description

Technical Field

[0001] This invention belongs to the field of ship strain field inversion, and more specifically, relates to a method and system for ship structural strain field inversion based on graph neural networks. Background Technology

[0002] As a primary means of cargo transportation, ships operate under the constant stress of wind, waves, and currents. Structural damage to ships can lead to significant economic losses and even endanger crew safety. Ship structural health monitoring can provide real-time information on the stress state of the hull structure, offering early warnings of potential structural failures and preventing navigational accidents caused by sudden damage, thus protecting the lives of crew and the safety of ship property. Furthermore, ship structural health monitoring reduces unnecessary repairs, accurately assessing maintenance needs through data, lowering operating costs, and extending the ship's lifespan. In addition, ship structural health monitoring meets the intelligent requirements of maritime regulations for safe ship operation, providing technical support for the continuous safe operation of merchant ships and special-purpose vessels navigating in harsh marine environments.

[0003] However, one of the current bottlenecks in ship structural health monitoring is how to efficiently and accurately obtain the stress state of the entire ship structure for structural health analysis. Currently, surrogate model methods are commonly used, with training data typically derived from finite element method (FEM) calculations of the entire ship. The input is simplified ideal loads from wind, waves, and current, and the output is a stress field. However, the calculated loads of this method differ from the actual environmental loads, and since the finite element structure is an ideal structure, the differences between it and the real structure cannot be reflected in the FEM calculation results.

[0004] The inverse finite element method (IEM) is a relatively new approach in academia. Its mechanism combines shell mechanics with the finite element method. It derives the relationship between discrete strain and displacement fields of individual elements by deriving the strain and deformation geometry equations of each element, thus obtaining the element stiffness matrix. Then, it obtains the overall stiffness of the entire structure through element assembly and other methods. Finally, it uses a few collected strain point values ​​to numerically infer the structural strain field. The advantage of this method is that it avoids the simplification problem of external wind, wave, and current loads encountered in the finite element method. However, due to its similarity to the finite element method, it suffers from low efficiency in solving the strain field of the entire ship structure, making it unsuitable for the rapid feedback and suggestions required by intelligent ships. Furthermore, this method is purely based on mechanical theory, and the derivation of the structural stiffness matrix cannot avoid ideal simplification, making it difficult to reflect the differences between the real ship structure and the ideal structure. Summary of the Invention

[0005] In view of the above-mentioned defects or improvement needs of the existing technology, the present invention provides a method and system for inverting the strain field of ship structure based on graph neural network, the purpose of which is to accurately and efficiently obtain the stress state of the entire ship structure.

[0006] To achieve the above objectives, according to one aspect of the present invention, a method for inverting the strain field of a ship structure based on a graph neural network is proposed, comprising the following steps:

[0007] (1) Model building stage:

[0008] Based on the finite element model of the ship structure, finite element simulations were performed under different sea conditions to obtain nodal coordinates and nodal strains.

[0009] Based on the mechanical sensitivity of key structural areas of the ship, some nodes were selected as observation nodes, and the remaining nodes were non-observation nodes.

[0010] The node strain X after masking is obtained by covering the node strain of non-observed nodes with a mask vector mask; a graph structure is constructed based on the node coordinate pos; and the nearest graph distance dist from each node to the nearest observed node is calculated based on the graph structure.

[0011] The node coordinates pos, node strain X, mask vector mask, and nearest on-graph distance dist are concatenated to form the node feature F;

[0012] The graph neural network is trained based on node features F and graph structure. The node features are iteratively updated to realize the propagation of strain information from observed nodes to non-observed nodes. The graph neural network parameters are updated according to the deviation between the predicted values ​​and the actual values ​​of the non-observed node features. The trained graph neural network is the ship strain feature inversion model.

[0013] (2) Model application stage:

[0014] Using the nodes where the ship's sensors are located as observation nodes, the strain of the observation nodes is obtained; based on the strain of the observation nodes and the coordinates of all nodes, node features are formed; the node features are input into the ship strain feature inversion model, and all node features are output to obtain the strain field of the entire ship structure.

[0015] As a further preferred method, training the graph neural network based on the graph structure is performed based on node features F, including:

[0016] All node features are embedded into the graph structure nodes and mapped to the hidden space to obtain node mapping features; nodes are divided into target nodes and neighbor nodes according to the nearest neighbor relationship represented by the graph structure; the mapping features of neighbor nodes are weighted and aggregated based on the attention mechanism, and the aggregated neighbor node mapping features are fused with the target node mapping features to obtain new target node mapping features; after multi-level aggregation of node mapping features, they are mapped back to the original feature space to obtain node feature prediction values.

[0017] As a further preferred method, node features are mapped to the latent space to obtain node mapping features, specifically as follows:

[0018]

[0019] in, For node mapping features, It is a non-linear activation function. The node feature transformation matrix, This is the bias corresponding to the transformation.

[0020] As a further preferred approach, the mapping features of neighbor nodes are weighted and aggregated based on a multi-information edge-weighted attention mechanism. Specifically, for the target node... i The aggregated neighbor node mapping features as follows:

[0021]

[0022] in, For the target node i The number of neighboring nodes; , Target nodes i No. k The node mapping features of each neighboring node and the corresponding attention score.

[0023] As a further preferred option, the attention score of a node is calculated as follows:

[0024]

[0025]

[0026]

[0027]

[0028] in, The node attention score is after softmax normalization. , , These are the node's coordinate attention, distance attention, and fusion attention scores, respectively. Represents a non-linear activation function. This indicates a gating fusion mechanism; , These are the mapping matrices for the coordinate and distance transformations of the graph neural network, respectively. , These are the coordinate and distance mapping biases of the graph neural network, respectively. , These are the coordinates of the target node and its neighboring nodes, respectively. , These are the nearest graph distances for the target node and its neighboring nodes, respectively.

[0029] As a further preferred approach, the aggregated neighbor node mapping features are fused with the target node mapping features to obtain new target node mapping features, specifically:

[0030]

[0031] in, , The original target node mapping features and the updated target node mapping features are respectively used. Update the transformation matrix for the target node features. This is the bias corresponding to the transformation.

[0032] As a further preferred option, the graph structure is constructed based on node coordinates pos, including:

[0033] Based on the node coordinates pos, the local nearest neighbor set of the node is constructed using the K-nearest neighbor method; then, through BFS search, the higher-order nearest neighbor set of the node is searched based on the local neighborhood set, thus obtaining the graph structure.

[0034] As a further preferred method, after obtaining the nodal coordinates and nodal strain through finite element simulation, the nodal coordinates are normalized using the min-max normalization method, and the nodal strain is normalized using the z-score normalization method.

[0035] As a further preferred option, the loss function used when training the graph neural network is... for:

[0036]

[0037]

[0038] in, , These are the mean squared error loss (MSE) and propagation regularization, respectively. Weighted parameters; , These represent the number of target nodes and the number of neighboring nodes corresponding to a given target node, respectively. , Let i and j represent the predicted node feature values ​​of the target node i and its neighboring node j, respectively. This represents the L2 norm.

[0039] According to another aspect of the present invention, a ship structure strain field inversion system based on graph neural networks is provided, including a processor, the processor being used to execute the above-described ship structure strain field inversion method based on graph neural networks.

[0040] In summary, compared with the prior art, the above-described technical solutions conceived by this invention mainly possess the following technical advantages:

[0041] 1. This invention uses the finite element calculation results of the entire ship structure as the raw data, and with the help of graph neural networks, captures the strain propagation characteristics of the ship structure in space, establishes the mapping relationship between local strain points and strain fields of the ship, and can invert the strain field of the entire ship structure from the strain of a few observation nodes, realizing high-precision and efficient inversion of the ship strain field under different working conditions, which can support further analysis of the health status of the ship structure.

[0042] 2. Based on traditional graph neural networks, this invention proposes a graph neural network that integrates multi-information attention by designing target-related information features. This method can effectively avoid the problem of oversmoothing of node features caused by deep models, and achieve a dual improvement in the accuracy of inversion numerical values ​​and the ability to represent changes in strain space. Attached Figure Description

[0043] Figure 1 This is a schematic diagram of the process of the ship structure strain field inversion method based on graph neural network according to an embodiment of the present invention.

[0044] Figure 2 This is a schematic diagram of a finite element model of a ship structure according to an embodiment of the present invention.

[0045] Figure 3 This is a schematic diagram illustrating the computational principle of the semi-supervised heterogeneous graph neural network node feature inversion method according to an embodiment of the present invention.

[0046] Figure 4 The image shows the predicted strain field results of the test sample in an embodiment of the present invention.

[0047] Figure 5 The graph shows the error results of the calculations used to verify the embodiments of the present invention. Detailed Implementation

[0048] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0049] This invention provides a method for inverting the strain field of a ship structure based on a graph neural network, such as... Figure 1 and Figure 3 As shown, it includes the following steps:

[0050] S1. Construct a finite element model of the ship structure and perform finite element simulations under different sea conditions to obtain nodal coordinates and nodal strain data.

[0051] S11. Establish a finite element model of the ship structure: Taking the experimental hull as the object, establish a finite element simulation 3D structural model of the ship. The mesh form of its plates and strong ribs is shell element. The mesh size is such that the longitudinal dimension is no greater than 1 rib spacing and the transverse dimension is no greater than the longitudinal rib spacing.

[0052] S12. Finite Element Simulation: The wave spectrum of a typical ship operating area is selected as the environmental condition. The wave parameters include wave height, frequency, and wave direction, where wave height covers from level 0 to the highest wave level that allows ship navigation. n h Species; frequency is related to wavelength, and the calculated wavelength covers 1 / 3. L (Captain) ~1 Ln l Types; wave direction covers 0°~360° and above α Fixed interval n θ Species. Hydrodynamic simulation calculations were used. n h × n l × n θ Hydrodynamic loads on the structural surface mesh under various working conditions were applied to the finite element simulation of the ship's hull structure. The plane strain in three directions was extracted from the mesh nodes of the entire ship's finite element model. Then, the deformation results of the node in six directions were calculated: linear displacement along the X, Y, and Z axes. and angular displacement about the three axes The deformation results in these six directions are used as the nodal strain.

[0053] S13. Data Normalization: Considering that the strain characteristics of the original simulation data exhibit a long-tailed Gaussian distribution, the z-score normalization method is used to normalize the strain characteristics of the nodes; the nodes of the simulation data point cloud exhibit a near-uniform distribution in three-dimensional space, and the min-max normalization method is used to normalize the node coordinates.

[0054] By adopting a component normalization strategy, the convergence speed of the graph neural network parameters was improved, and the model's ability to fit nonlinear strain-coordinate mapping relationships was enhanced, providing a foundation for the spatial feature learning of the subsequent attention mechanism.

[0055] S2. Based on the nodal strain and nodal coordinates obtained from finite element simulation, a training set and a test set for the graph neural network model are formed.

[0056] S21. Based on the mechanical sensitivity of key structural regions of the ship, some nodes are selected as observation nodes, and the remaining nodes are non-observation nodes. For observation nodes, their nodal strain is preserved; for non-observation nodes, their nodal strain is covered by a mask vector (bool vector). Through the above processing, the nodal strain characteristics of the original finite element simulation are preserved. This is transformed into sparse observation strain data, i.e., nodal strain X.

[0057] Specifically, based on the stress distribution characteristics of ship structures and engineering monitoring requirements, a structured masking strategy is adopted to process the original finite element simulation data: Based on prior knowledge related to key ship structures, 50 observed nodes are selected from a large number of nodes (approximately 47,000 in this embodiment), retaining their true strain characteristic values; subsequently, zero-padding masking is used to mask the remaining nodes (non-observed nodes), simulating a sparse observation scenario with limited sensor deployment in actual engineering. This processing method, while ensuring the physical rationality of the data, provides training samples that meet engineering constraints for the graph neural network by controlling the number of observed nodes (50 nodes) and the construction of the feature distribution in the non-observed area. This enables the model to effectively learn the spatial propagation law of strain characteristics from the observed area to the non-observed area and significantly improves its generalization ability under limited sensor deployment conditions.

[0058] S22. Construct a graph structure based on node coordinates pos; and based on the graph structure, calculate the nearest graph distance dist from each node to the nearest observed node.

[0059] Furthermore, considering that the hull structure is a typical manifold structure, using the K-nearest neighbor method based on Euclidean distance to calculate the nearest neighbor relationship between nodes will lead to a large error. This invention proposes a local neighborhood-based nearest neighbor calculation method, that is, based on the node coordinates pos, a local nearest neighbor set of each node is constructed using the local K-nearest neighbor method; using BFS search, a higher-order nearest neighbor set of the node is searched on the local nearest neighbor set to obtain the higher-order graph structure of the ship structure.

[0060] Specifically, a graph-structured network is constructed by exploring the local neighborhoods of nodes. Based on this network, the higher-order neighbors of the target node are expanded using the nearest neighbor relationships between nodes. This process can be represented as:

[0061]

[0062]

[0063] in, Represented as nodes A set with K nearest neighbors. For the Euclidean distance operator, rank( () are arranged in order. Represented as nodes The m-order extended neighbor set, In local areas Remove nodes based on The set of nodes that can be reached after m-1 hops.

[0064] Based on the graph structure and the mask vectors of observed / unobserved nodes, the graph distance dist from the nearest observed node is calculated.

[0065] S23. Concatenate the node coordinates pos, node strain X, mask vector mask, and nearest on-graph distance dist to form the node feature F:

[0066]

[0067] in, pos represents the node strain characteristics (after masking), mask represents the node's 3D coordinates, mask represents the node's observed / unobserved mask vector (0 corresponds to unobserved nodes, 1 corresponds to observed nodes), and dist represents the nearest on-graph distance from the node to the nearest observed node.

[0068] S3. Based on the above graph structure, a graph neural network with multi-information attention is introduced to invert the features of non-observed nodes. Specifically, the graph neural network is trained based on the node features F and the constructed graph structure. The message passing mechanism of the graph neural network is used to realize the propagation of strain information from the observed nodes to the non-observed areas. After multiple iterations, the global node update features of the ship structure are obtained. The graph neural network loss is calculated by combining the true value of finite element simulation with the update features. Then, the relevant parameters of the graph neural network are updated using backpropagation. The trained graph neural network is the ship strain feature inversion model.

[0069] Furthermore, after the node features are input into the graph neural network:

[0070] All node features are embedded into the nodes of the graph structure and mapped to the latent space to obtain a high-dimensional representation of the node features, i.e., node-mapped features. :

[0071]

[0072] The nodes are divided into target node i and neighbor nodes j according to the graph structure. The mapping features of the neighbor nodes are weighted and aggregated based on a multi-information edge-weight attention mechanism.

[0073]

[0074] in, For the target node i The number of neighboring nodes; , Target nodes i No. k The node mapping features of each neighboring node and their corresponding attention scores. Attention scores. Based on the multi-attention computation and fusion mechanism, the following is determined:

[0075]

[0076]

[0077]

[0078]

[0079] in, , , These represent the coordinate attention, distance attention, and fusion attention scores of the node, respectively. , , This is represented by the non-linear activation function, weights, and biases of a neural network. This indicates a gating fusion mechanism. This represents the attention score after softmax normalization.

[0080] The aggregated neighbor node mapping features are fused with the target node mapping features to obtain the updated target node mapping features:

[0081]

[0082] in, , The original target node mapping features and the updated target node mapping features are respectively.

[0083] Through the multi-layer stacking of the above process, after multi-layer feature aggregation, the aggregated nodes are mapped to features. Mapping to the original feature space:

[0084]

[0085] in For the predicted values ​​of nodal strain characteristics, Predict the feature mapping matrix for the nodes.

[0086] Based on this, the model hyperparameters are adjusted to ensure the accuracy of the trained model. This invention, building upon graph neural networks, utilizes a message-passing mechanism to calculate and fuse coordinate- and distance-based attention, achieving non-uniform aggregation of node features and solving the heterogeneity problem of traditional graph neural networks.

[0087] By stacking multiple layers of this process, the model can capture higher-order topological relationships, thereby enabling the propagation of strain information from observed nodes to non-observed nodes and ensuring that the propagation process conforms to actual physical laws.

[0088] Specifically, the output of each layer of the graph neural network model can be represented as:

[0089]

[0090]

[0091] Where L represents the number of model layers, The model's attention score based on multiple information at layer l. Calculate the function for attention information; This represents the output of the model at layer l+1. This is the node feature update function. Through experiments, the number of model stacking layers L is set to achieve high-precision inversion of strain results and high-precision characterization of nodal strain details.

[0092] Furthermore, during model training, a multi-information attention mechanism is introduced to effectively alleviate the problem of overly smooth node features caused by the increase in depth of graph neural networks. Based on this, a composite loss function including mean squared error (MSE) and propagation regularization is constructed. By constraining the spatial distribution characteristics during feature aggregation, the model's inversion accuracy and physical plausibility for strain features of unobserved nodes are further improved.

[0093] Mean Squared Error (MSE) is used to constrain the model's prediction accuracy. Furthermore, in deep graph neural networks, the features of adjacent nodes become increasingly similar with each aggregation, preventing the model from learning the spatial variation patterns of node features. Therefore, propagation regularization is introduced. Constrain the model output:

[0094]

[0095] The final loss function used during training is obtained. for:

[0096]

[0097] in, , These are the mean squared error loss (MSE) and propagation regularization, respectively. Weighted parameters; , These represent the number of target nodes and the number of neighboring nodes corresponding to a given target node, respectively. , Let i and j represent the predicted node feature values ​​of the target node i and its neighboring node j, respectively. The L2 norm represents the difference between the predicted feature values ​​of the target node and its neighboring nodes.

[0098] Furthermore, the ship strain characteristic inversion model is tested: based on finite element simulation test data, the same preprocessing method as the training data is used to extract nodal features; the extracted nodal features are input into the ship strain characteristic inversion model, and the strain values ​​of all nodes are calculated and output through the model, thereby realizing the accurate inversion of strain of non-observed nodes, and finally constructing the strain field of the entire ship structure.

[0099] Specifically, using percentage root mean square error The average results of the model tests are quantified and combined with the absolute error of each node. The model's ability to learn about local strain details in ships was evaluated. During the testing phase, a preprocessed finite element simulation test set was used. The average root mean square error (RMSE) across 10 test samples was 4.732. This indicates that the model has high prediction accuracy and reliability, with its output showing a small deviation from the true values, effectively reflecting the strain characteristic distribution under actual working conditions.

[0100] The ship structure strain field inversion method of this invention is in Figure 2 The strain inversion thermogram and the absolute error thermogram of nodal features under the finite element simulation data sample are as follows: Figure 4 , Figure 5 As shown.

[0101] S4. Application of the Ship Strain Feature Inversion Model: Using the nodes where ship sensors are located as observation nodes, the strain of the observation nodes is obtained; based on the strain of the observation nodes and the coordinates of all nodes, node features are formed; the node features are input into the ship strain feature inversion model, and all node features are output, thereby inverting the strain of a large number of non-observation nodes to obtain the strain field of the entire ship structure. This invention can effectively solve the problem of insufficient coverage of observation nodes in traditional monitoring, and significantly improve the integrity and prediction accuracy of the ship structure strain field.

[0102] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for inverting the strain field of a ship structure based on a graph neural network, characterized in that, include: (1) Model building stage: Based on the finite element model of the ship structure, finite element simulations were performed under different sea conditions to obtain nodal coordinates and nodal strains. Based on the mechanical sensitivity of key structural areas of the ship, some nodes were selected as observation nodes, and the remaining nodes were non-observation nodes. The nodal strain X after masking is obtained by covering the nodal strain of non-observed nodes with a mask vector mask. Construct a graph structure based on node coordinates pos, and calculate the nearest graph distance dist from each node to the nearest observed node based on the graph structure. The node coordinates pos, node strain X, mask vector mask, and nearest on-graph distance dist are concatenated to form the node feature F; The graph neural network is trained based on node features F and graph structure, and the node features are iteratively updated to realize the propagation of strain information from observed nodes to unobserved nodes. The graph neural network parameters are updated according to the deviation between the predicted values ​​and the actual values ​​of the unobserved node features. The trained graph neural network is the ship strain characteristic inversion model; Training a graph neural network includes: All node features are embedded into the graph structure nodes and mapped to the hidden space to obtain node mapping features; nodes are divided into target nodes and neighbor nodes according to the nearest neighbor relationship represented by the graph structure; the mapping features of neighbor nodes are weighted and aggregated based on the attention mechanism, and the aggregated neighbor node mapping features are fused with the target node mapping features to obtain new target node mapping features; after multi-level aggregation of node mapping features, they are mapped back to the original feature space to obtain node feature prediction values; The mapping features of neighbor nodes are weighted and aggregated based on a multi-information edge-weighted attention mechanism. Specifically, for the target node... i Its aggregated neighbor node mapping features as follows: in, For the target node i The number of neighboring nodes; , Target nodes i No. k The node mapping features of each neighboring node and their corresponding attention scores; the attention score of a node is calculated as follows: in, The node attention score is after softmax normalization. , , These are the node's coordinate attention, distance attention, and fusion attention scores, respectively. Represents a non-linear activation function. This indicates a gating fusion mechanism; , These are the mapping matrices for the coordinate and distance transformations of the graph neural network, respectively. , These are the coordinate and distance mapping biases of the graph neural network, respectively. , These are the coordinates of the target node and its neighboring nodes, respectively. , These are the nearest graph distances between the target node and its neighboring nodes, respectively. (2) Model application stage: Using the nodes where the ship's sensors are located as observation nodes, the strain of the observation nodes is obtained; based on the strain of the observation nodes and the coordinates of all nodes, node features are formed; the node features are input into the ship strain feature inversion model, and all node features are output to obtain the strain field of the entire ship structure.

2. The method for inverting the strain field of ship structures based on graph neural networks as described in claim 1, characterized in that, The node mapping features are obtained by mapping node features to the latent space, specifically: in, For node mapping features, It is a non-linear activation function. The node feature transformation matrix, This is the bias corresponding to the transformation.

3. The method for inverting the strain field of ship structures based on graph neural networks as described in claim 1, characterized in that, The aggregated neighbor node mapping features are fused with the target node mapping features to obtain new target node mapping features, specifically: in, , The original target node mapping features and the updated target node mapping features are respectively used. Update the transformation matrix for the target node features. This is the bias corresponding to the transformation.

4. The method for inverting the strain field of ship structures based on graph neural networks as described in claim 1, characterized in that, Constructing a graph structure based on node coordinates pos includes: Based on the node coordinates pos, the local nearest neighbor set of the node is constructed using the K-nearest neighbor method; then, through BFS search, the higher-order nearest neighbor set of the node is searched based on the local neighborhood set, thus obtaining the graph structure.

5. The method for inverting the strain field of ship structures based on graph neural networks as described in claim 1, characterized in that, After obtaining the nodal coordinates and nodal strain through finite element simulation, the nodal coordinates are normalized using the min-max normalization method, and the nodal strain is normalized using the z-score normalization method.

6. The method for inverting the strain field of a ship structure based on a graph neural network as described in any one of claims 1-5, characterized in that, The loss function used when training a graph neural network for: in, , These are the mean squared error loss (MSE) and propagation regularization, respectively. Weighted parameters; , These represent the number of target nodes and the number of neighboring nodes corresponding to a given target node, respectively. , Let i and j represent the predicted node feature values ​​of the target node i and its neighboring node j, respectively. This represents the L2 norm.

7. A ship structure strain field inversion system based on graph neural networks, characterized in that, Includes a processor for executing the ship structure strain field inversion method based on graph neural networks as described in any one of claims 1-6.

Images