A lattice structure mechanical property prediction method for part manufacturing

Abstract

The application discloses a lattice structure mechanical property prediction method for part manufacturing. In view of the bottleneck of the traditional method in precision, efficiency and speed, the method realizes optimization through four modules: a structure generation module generates VTK format data; a feature extraction module analyzes five types of features such as point coordinate statistics and element topology; a deep cross network prediction module is processed through a cross network and a deep network double path, and outputs porosity and compression modulus; and an optimization module optimizes printing parameters according to the prediction result. The method can accurately capture the interactive effect of micro-configuration and process parameters, accelerate the whole process of 'design-prediction-optimization', and is suitable for performance prediction of core components of high-end equipment.

Description

Technical Field

[0001] This invention relates to the fields of additive manufacturing, deep learning, and computational mechanics, specifically to a method for predicting the mechanical properties of lattice structures for parts manufacturing. Background Technology

[0002] In the fields of robotics and intelligent manufacturing, lattice structures are three-dimensional lightweight structures composed of periodically arranged micro-units, forming a spatial network through rods and nodes. As a key technology for achieving a balance between lightweight design and high performance, they are widely used in core components such as collaborative robot end effectors, precision guide rails and sliders, and automated equipment frames. These applications place stringent demands on performance prediction. For example, robotic arm end effectors must simultaneously meet comprehensive requirements for lightweight design, high rigidity, and high-precision positioning; high-speed guide rails and sliders must possess rapid dynamic response capabilities and maintain structural stability under long-term cyclic loads. Traditional performance prediction methods are currently at a bottleneck in terms of accuracy, efficiency, and speed, making it difficult to meet the technical demands of intelligent manufacturing for flexibility and rapid iteration.

[0003] Insufficient prediction accuracy directly impacts the performance of high-end equipment. Finite element analysis, when dealing with complex lattice topologies, suffers from significant deviations in stress distribution predictions due to the trade-off between mesh generation accuracy and computational cost, particularly noticeable in element connection regions. Empirical formulas oversimplify structure-performance relationships, employing linear fitting with a single density parameter, failing to reflect the differences in mechanical behavior across various topological configurations. Ordinary machine learning models, limited by feature dimensions and using only a few geometric parameters, cannot capture the microscopic configurational features of lattice structures, such as element arrangement patterns and bounding box proportions, resulting in poor stability of performance prediction results.

[0004] Inefficient prediction processes prolong product development cycles. Traditional parameter optimization employs orthogonal experimental design, adjusting only a small number of process parameters at a time, failing to reveal the coupling effects between multiple parameters. When multiple process parameters change collaboratively, nonlinear performance responses may occur, which are difficult to capture using simple controlled variable methods. Multi-objective optimization requires building separate predictive models for different performance indicators, making it impossible to coordinate the trade-off between lightweighting and dynamic performance, trapping the design process in a cycle of repeated trial and error. This inefficient optimization process results in lengthy process debugging cycles for core robot components, making it difficult to adapt to customized production needs.

[0005] Slow prediction speed hinders the real-time response of smart manufacturing. Traditional numerical simulation methods require significant computational resources for a single performance evaluation of complex lattice models, making it impossible to support rapid comparison of multiple scenarios. Physical experimental methods face even longer process cycles, with the complete process from structure fabrication to mechanical testing involving multiple stages, and cannot process multiple sets of parameter schemes in parallel. This slowness leads to difficulties in real-time feedback development, restricting the flexible production capabilities of smart manufacturing.

[0006] This invention addresses the performance prediction needs in the fields of robotics and intelligent manufacturing. Through multi-dimensional feature extraction, deep cross-network modeling, and rapid optimization loop, it solves the core problems of insufficient accuracy, low efficiency, and slow speed of traditional methods. Its technical advantages lie in achieving precise characterization of microstructures, capturing the interaction effects of process parameters, and ultimately accelerating the entire "design-prediction-optimization" process. This innovative solution is particularly suitable for the rapid design of high-end equipment such as collaborative robots and precision automated equipment, providing key technical support for flexible production in intelligent manufacturing. Summary of the Invention

[0007] This invention designs a method for predicting the mechanical properties of lattice structures based on deep cross-networks, targeting truss-type lattice structures to achieve high-precision prediction of mechanical properties in the fields of robotics and intelligent manufacturing. First, the proposed method for predicting the mechanical properties of lattice structures for parts manufacturing integrates four modules: a structure generation module, a feature extraction module, a deep cross-network prediction module, and an optimization module. This design upgrades traditional experience-driven lattice structure design to a data-driven, precise development model. Second, based on the working characteristics of core robot components, key process parameters are input to generate VTK format lattice structure data. Geometric and topological information is analyzed from the VTK data, extracting five key features to comprehensively quantify the physical properties of the lattice structure, providing a data foundation for parts quality control. Then, the cross-network explicitly models the nonlinear interaction effects of process parameters, generating higher-order cross terms while retaining interpretability. Next, the deep network learns local features such as unit arrangement patterns and connection methods through fully connected layers to achieve dimensionality reduction and noise filtering. Finally, the dual-path outputs are concatenated and input into a linear layer, simultaneously outputting porosity and compressibility modulus, directly linking structural performance with manufacturing parameters. This design combines feature engineering and deep learning to provide a high-precision, high-efficiency, innovative, and easily implementable lattice structure design solution for the robotics and intelligent manufacturing fields. It drives the upgrade of truss-type parts from traditional experience-based design to data-driven precision manufacturing, meeting the core needs of this field for lightweight, high strength, and customization. To achieve the above objectives, the following steps are taken:

[0008] Step 1: The structure generation module generates an Octet truss lattice suitable for STL format based on the requirements of line spacing, layer height, printing speed, and printing temperature of the robot's core components, and then converts the STL mesh into a VTK format string.

[0009] Step 2: Feature extraction module. First, the robot part in VTK format converted by the structure generation module is parsed for point coordinates and unit topology. Then, the point coordinate statistical features, unit topology features, bounding box geometric features, structure density features and complexity features are obtained.

[0010] Step 2.1: Perform point coordinate analysis and cell topology analysis using VTK data, then obtain point coordinate statistical features, bounding box geometric features, cell topology features, structure density features, and complexity features. The expressions involved are as follows:

[0011] f points =[μ x ,μ y ,μ z ,σ x ,σ y ,σ z ,x min ,y min ,z min ,x max ,y max ,z max ] T ,

[0012] Where f points It is a statistical feature of point coordinates, μ x It is the average value of the x-coordinate, μ y It is the average value of the y-coordinate, μ z It is the average value of the z-coordinate, σ x σ is the standard deviation of the x-coordinate. y σ is the standard deviation of the y-coordinate. z It is the standard deviation of the z-coordinate, x min It is the minimum value of the x-coordinate of all points, y min It is the minimum value of the y-coordinate of all points, z min It is the minimum value of the z-coordinate of all points, x max It is the maximum value of the x-coordinates of all points, y max It is the maximum value of the y-coordinates of all points, and the z-coordinates are... max It is the maximum value of the z-coordinate of all points;

[0013] f bbox =[Δx,Δy,Δz,ΔxΔyΔz] T ,

[0014] Where f bboxΔx represents the geometric features of the bounding box, Δy represents the length of the bounding box along the x-axis, Δz represents the length of the bounding box along the y-axis, and Δz represents the volume of the bounding box.

[0015] f cell =[μ edge ,σ edge ,μ area ,∑area] T ,

[0016] Where f cell It is a unit topological feature, μ edge It is the average of the lengths of all edges of all elements, σ edge It is the standard deviation of the lengths of all edges of all elements, μ area ∑area is the average area of ​​all surfaces of all elements, and ∑area is the sum of the areas of all surfaces of all elements;

[0017]

[0018] Where f density It is a structural density characteristic, N points It is the total number of dots in the dot matrix structure, that is, the number of dots in all VTK data, A total V is the sum of the surface areas of all elements in the lattice structure, and V is the volume of the smallest axis-aligned bounding box that encloses the lattice structure.

[0019] f complexity =[N points N faces ] T ,

[0020] Where f complexity It is a complexity feature, N points N is the total number of dots in the lattice structure. faces It is the total number of faces in the lattice structure;

[0021] x0 = [f point ,f bbox ,f cell ,f density ,f complexity ] T ,

[0022] Where x0 is the initial input feature vector of the cross-network.

[0023] Step 3: The deep cross-network prediction module first receives the three-dimensional structural data of the robot part and extracts the feature vector from it. Then, the feature vector is input into the cross-network and the deep network. Then, the outputs of the cross-network and the deep network are concatenated into a combined feature vector. Finally, the combined feature vector is input into the linear output layer to obtain two mechanical properties, porosity and compressive modulus, that match the manufacturing requirements of the robot part.

[0024] Step 3.1: Input the feature vectors into the cross-network. The weights and biases of the cross layers are initialized using a truncated normal distribution. The cross-network explicitly captures the interaction effects of line spacing, layer height, velocity, and temperature. At the same time, the 3-layer cross-network generates 4th-order feature cross terms to process density and bounding boxes. The relevant expressions are as follows:

[0025] x l+1 =x0⊙(W l T x l +b l )+x l ,

[0026] Where x l+1 x is the output feature vector of the l-th layer of the cross-network, and x0 is the initial input feature vector of the cross-network.

[0027] W represents multiplication. l T It is the transpose of the weight matrix of the l-th layer of the cross-network, x l b is the input feature vector of the l-th layer of the cross-network. l It is the bias vector of the l-th layer of the cross-network;

[0028] w l ~TruncatedNormal(μ=0,σ=0.01),

[0029] b l ~TruncatedNormal(μ=0,σ=0.01),

[0030] Where w l b is the weight matrix of the l-th layer of the cross-network. l is the bias vector of the l-th layer of the cross-network, TruncatedNormal is the truncated normal distribution, μ is the mean of the truncated normal distribution, and σ is the standard deviation of the truncated normal distribution;

[0031] Step 3.2: Input the feature vector into the deep network. The deep network contains three fully connected layers, each followed by a ReLU activation function, with decreasing dimensionality between layers. Learn the unit topology. The relevant expressions are as follows:

[0032]

[0033] d→128→64→32,

[0034] Where h1 is the hidden layer output of the first layer of the deep network, h1∈R 128 h2 is the hidden output of the second layer of the deep network, h2∈R 64 h3 is the hidden output of the third layer of the deep network, h3∈R 64 , It is the weight matrix of the first layer of the deep network. This is the weight matrix of the second layer of the deep network. This is the weight matrix of the third layer of the deep network. It is the bias vector of the first layer of the deep network. It is the bias vector of the second layer of the deep network. It is the bias vector of the third layer of the deep network. x0 is the input feature vector of the deep network, and d is the input feature dimension;

[0035] Step 3.3: Concatenate the outputs of the cross-network and the deep network into a combined feature vector:

[0036] z = [x l ,h k ] T ,

[0037] Where x l It is the final output of the cross-network, h k is the final output of the deep network, and z is the fused feature vector;

[0038] Step 3.4: Input the combined feature vector into the linear output layer to obtain the predicted values ​​of the two mechanical properties, porosity and compressive modulus.

[0039]

[0040] in This is the original predicted value of porosity. This is the original predicted value of the compressive modulus, W. out Here, z is the weight matrix of the output layer, and b is the fused feature vector. out This is the bias vector of the output layer, porosity is the final predicted value of porosity, and modulus is the final predicted value of compressive modulus. It is the Sigmoid activation function. It is the Softplus activation function.

[0041] Step 4: Optimize the module to optimize printing parameters based on predicted performance and robot part manufacturing constraints. Attached Figure Description

[0042] Figure 1 This is an overall flowchart of an embodiment of the present invention; Detailed Implementation

[0043] To more clearly illustrate the purpose, technical solution, and advantages of this invention, the following detailed description of the invention will be provided with the aid of the accompanying drawings and specific embodiments.

[0044] Figure 1 This is a flowchart of an embodiment. This embodiment provides a method for predicting the mechanical properties of lattice structures for parts manufacturing. The specific process includes: generating truss-type lattice structure data in VTK format through a structure generation module; extracting five key features from the VTK data through a feature extraction module to generate multi-dimensional feature vectors; performing dual-path processing through a deep cross-network prediction module; explicitly modeling the interaction effects of process parameters through the cross-network path; and finally, predicting the mechanical properties porosity and compressive modulus through feature fusion output.

[0045] A method for predicting the mechanical properties of lattice structures for parts manufacturing includes the following steps:

[0046] Step 1: The structure generation module generates an Octet truss lattice suitable for STL format based on the requirements of line spacing, layer height, printing speed, and printing temperature of the robot's core components, and then converts the STL mesh into a VTK format string.

[0047] Step 2: Feature extraction module. First, the robot part in VTK format converted by the structure generation module is parsed for point coordinates and unit topology. Then, the point coordinate statistical features, unit topology features, bounding box geometric features, structure density features and complexity features are obtained.

[0048] Step 2.1: Perform point coordinate analysis and cell topology analysis using VTK data, then obtain point coordinate statistical features, bounding box geometric features, cell topology features, structure density features, and complexity features. The expressions involved are as follows:

[0049] f points =[μ x ,μ y ,μ z ,σ x ,σ y ,σ z ,x min ,y min ,z min ,x max,y max ,z max ] T ,

[0050] Where f points It is a statistical feature of point coordinates, μ x It is the average value of the x-coordinate, μ y It is the average value of the y-coordinate, μ z It is the average value of the z-coordinate, σ x σ is the standard deviation of the x-coordinate. y σ is the standard deviation of the y-coordinate. z It is the standard deviation of the z-coordinate, x min It is the minimum value of the x-coordinate of all points, y min It is the minimum value of the y-coordinate of all points, z min It is the minimum value of the z-coordinate of all points, x max It is the maximum value of the x-coordinates of all points, y max It is the maximum value of the y-coordinates of all points, and the z-coordinates are... max It is the maximum value of the z-coordinate of all points;

[0051] f bbox =[Δx,Δy,Δz,ΔxΔyΔz] T ,

[0052] Where f bbox Δx represents the geometric features of the bounding box, Δy represents the length of the bounding box along the x-axis, Δz represents the length of the bounding box along the y-axis, and Δz represents the volume of the bounding box.

[0053] f cell =[μ edge ,σ edge ,μ area ,∑area] T ,

[0054] Where f cell It is a unit topological feature, μ edge It is the average of the lengths of all edges of all elements, σ edge It is the standard deviation of the lengths of all edges of all elements, μ area ∑area is the average area of ​​all surfaces of all elements, and ∑area is the sum of the areas of all surfaces of all elements;

[0055]

[0056] Where f density It is a structural density characteristic, N points It is the total number of dots in the dot matrix structure, that is, the number of dots in all VTK data, A totalV is the sum of the surface areas of all elements in the lattice structure, and V is the volume of the smallest axis-aligned bounding box that encloses the lattice structure.

[0057] f complexity =[N points N faces ] T ,

[0058] Where f complexity It is a complexity feature, N points N is the total number of dots in the lattice structure. faces It is the total number of faces in the lattice structure;

[0059] x0 = [f point ,f bbox ,f cell ,f density ,f complexity ] T ,

[0060] Where x0 is the initial input feature vector of the cross-network.

[0061] Step 3: The deep cross-network prediction module first receives the three-dimensional structural data of the robot part and extracts the feature vector from it. Then, the feature vector is input into the cross-network and the deep network. Then, the outputs of the cross-network and the deep network are concatenated into a combined feature vector. Finally, the combined feature vector is input into the linear output layer to obtain two mechanical properties, porosity and compressive modulus, that match the manufacturing requirements of the robot part.

[0062] Step 3.1: Input the feature vectors into the cross-network. The weights and biases of the cross layers are initialized using a truncated normal distribution. The cross-network explicitly captures the interaction effects of line spacing, layer height, velocity, and temperature. At the same time, the 3-layer cross-network generates 4th-order feature cross terms to process density and bounding boxes. The relevant expressions are as follows:

[0063] x l+1 =x0⊙(W l T x l +b l )+x l ,

[0064] Where x l+1 is the output feature vector of the l-th layer of the cross-network, x0 is the initial input feature vector of the cross-network, ⊙ denotes multiplication, and W l T It is the transpose of the weight matrix of the l-th layer of the cross-network, x l b is the input feature vector of the l-th layer of the cross-network. l It is the bias vector of the l-th layer of the cross-network;

[0065] w l ~TruncatedNormal(μ=0,σ=0.01),

[0066] b l ~TruncatedNormal(μ=0,σ=0.01),

[0067] Where w l b is the weight matrix of the l-th layer of the cross-network. l is the bias vector of the l-th layer of the cross-network, TruncatedNormal is the truncated normal distribution, μ is the mean of the truncated normal distribution, and σ is the standard deviation of the truncated normal distribution;

[0068] Step 3.2: Input the feature vector into the deep network. The deep network contains three fully connected layers, each followed by a ReLU activation function, with decreasing dimensionality between layers. Learn the unit topology. The relevant expressions are as follows:

[0069]

[0070] d→128→64→32,

[0071] Where h1 is the hidden layer output of the first layer of the deep network, h1∈R 128 h2 is the hidden output of the second layer of the deep network, h2∈R 64 h3 is the hidden output of the third layer of the deep network, h3∈R 64 , It is the weight matrix of the first layer of the deep network. This is the weight matrix of the second layer of the deep network. This is the weight matrix of the third layer of the deep network. It is the bias vector of the first layer of the deep network. It is the bias vector of the second layer of the deep network. It is the bias vector of the third layer of the deep network. x0 is the input feature vector of the deep network, and d is the input feature dimension;

[0072] Step 3.3: Concatenate the outputs of the cross-network and the deep network into a combined feature vector:

[0073] z = [x l ,h k ] T ,

[0074] Where x l It is the final output of the cross-network, hk is the final output of the deep network, and z is the fused feature vector;

[0075] Step 3.4: Input the combined feature vector into the linear output layer to obtain the predicted values ​​of the two mechanical properties, porosity and compressive modulus.

[0076]

[0077] in This is the original predicted value of porosity. This is the original predicted value of the compressive modulus, W. out Here, z is the weight matrix of the output layer, and b is the fused feature vector. out This is the bias vector of the output layer, porosity is the final predicted value of porosity, and modulus is the final predicted value of compressive modulus. It is the Sigmoid activation function. It is the Softplus activation function.

[0078] Step 4: Optimize the module to optimize printing parameters based on predicted performance and robot part manufacturing constraints.

Claims

1. A method for predicting the mechanical properties of lattice structures for parts manufacturing, characterized in that, Includes the following steps: Step 1: The structure generation module generates an Octet truss lattice suitable for STL format based on the requirements of line spacing, layer height, printing speed, and printing temperature of the robot's core components, and then converts the STL mesh into a VTK format string. Step 2: Feature extraction module. First, the robot part VTK format string converted by the structure generation module is parsed for point coordinates and unit topology. Then, the statistical features f of the point coordinates are obtained. points , unit topological features f cell Bounding box geometric features f bbox Structural density characteristics f density and complexity feature f complexity ; Step 3: The deep cross-network prediction module first receives the 3D structural data of the robot part and extracts the feature vector x0 = [f point ,f bbox ,f cell ,f density ,f complexity ] T Next, the feature vector is input into the cross network and the deep network. Then, the outputs of the cross network and the deep network are concatenated into a combined feature vector. Finally, the combined feature vector is input into the linear output layer to obtain two mechanical properties, porosity and compressive modulus, that match the requirements for manufacturing robot parts. Step 4: Optimize the module to optimize printing parameters based on predicted performance and robot part manufacturing constraints.

2. The method for predicting the mechanical properties of lattice structures for parts manufacturing according to claim 1, characterized in that, The feature extraction module in step 2 first performs point coordinate parsing and cell topology parsing on the VTK format string of the robot part converted by the structure generation module, and then obtains point coordinate statistical features, cell topology features, bounding box geometric features, structure density features, and complexity features. The specific implementation steps are as follows: Step 2.1: Perform point coordinate analysis and cell topology analysis using VTK data, then obtain point coordinate statistical features, bounding box geometric features, cell topology features, structure density features, and complexity features. The expressions involved are as follows: f points =[μ x ,m y ,m z ,s x ,s y ,s z ,x min ,y min ,z min ,x max ,y max ,z max ] T , Where f points It is a statistical feature of point coordinates, μ x It is the average value of the x-coordinate, μ y It is the average value of the y-coordinate, μ z It is the average value of the z-coordinate, σ x σ is the standard deviation of the x-coordinate. y σ is the standard deviation of the y-coordinate. z It is the standard deviation of the z-coordinate, x min It is the minimum value of the x-coordinate of all points, y min It is the minimum value of the y-coordinate of all points, z min It is the minimum value of the z-coordinate of all points, x max It is the maximum value of the x-coordinates of all points, y max It is the maximum value of the y-coordinates of all points, z max It is the maximum value of the z-coordinate of all points; f bbox =[Δx,Δy,Δz,ΔxΔyΔz] T , Where f bbox Δx represents the geometric features of the bounding box, Δy represents the length of the bounding box along the x-axis, Δz represents the length of the bounding box along the y-axis, and Δz represents the volume of the bounding box. f cell =[μ edge ,s edge ,m area ,∑area] T , Where f cell It is a unit topological feature, μ edge It is the average of the lengths of all edges of all elements, σ edge It is the standard deviation of the lengths of all edges of all elements, μ area ∑area is the average area of ​​all surfaces of all elements, and ∑area is the sum of the areas of all surfaces of all elements; Where f density It is a structural density characteristic, N points It is the total number of dots in the dot matrix structure, that is, the number of dots in all VTK data, A total V is the sum of the surface areas of all elements in the lattice structure, and V is the volume of the smallest axis-aligned bounding box that encloses the lattice structure. f complexity =[N points ,N faces ] T , Where f complexity It is a complexity feature, N points N is the total number of dots in the lattice structure. faces It is the total number of faces in the lattice structure; x0=[f point ,f bbox ,f cell ,f density ,f complexity ] T , Where x0 is the initial input feature vector of the cross-network.

3. The method for predicting the mechanical properties of lattice structures for parts manufacturing according to claim 1, characterized in that, The deep cross-network prediction module described in step 3 first receives the three-dimensional structural data of the robot part and extracts feature vectors from it. Then, it inputs the feature vectors into the cross-network and the deep network. Next, it concatenates the outputs of the cross-network and the deep network into a combined feature vector. Finally, it inputs the combined feature vector into the linear output layer to obtain the two mechanical properties, porosity and compressive modulus, that match the manufacturing requirements of the robot part. The specific implementation is as follows: Step 3.1: Input the feature vectors into the cross-network. The weights and biases of the cross layers are initialized using a truncated normal distribution. The cross-network explicitly captures the interaction effects of line spacing, layer height, velocity, and temperature. At the same time, the 3-layer cross-network generates 4th-order feature cross terms to process density and bounding boxes. The relevant expressions are as follows: x l+1 =x0⊙(W l T x l +b l )+x l , Where x l+1 is the output feature vector of the l-th layer of the cross-network, x0 is the initial input feature vector of the cross-network, ⊙ denotes multiplication, and W l T It is the transpose of the weight matrix of the l-th layer of the cross-network, x l b is the input feature vector of the l-th layer of the cross-network. l It is the bias vector of the l-th layer of the cross-network; w l ~TruncatedNormal(μ=0,σ=0.01), b l ~TruncatedNormal(μ=0,σ=0.01), Where w l b is the weight matrix of the l-th layer of the cross-network. l is the bias vector of the l-th layer of the cross-network, TruncatedNormal is the truncated normal distribution, μ is the mean of the truncated normal distribution, and σ is the standard deviation of the truncated normal distribution; Step 3.2: Input the feature vector into the deep network. The deep network contains three fully connected layers, each followed by a ReLU activation function, with decreasing dimensionality between layers. Learn the unit topology. The relevant expressions are as follows: d→128→64→32， Where h1 is the hidden layer output of the first layer of the deep network, h1∈R 128 h2 is the hidden output of the second layer of the deep network, h2∈R 64 h3 is the hidden output of the third layer of the deep network, h3∈R 64 , It is the weight matrix of the first layer of the deep network. It is the weight matrix of the second layer of the deep network. This is the weight matrix of the third layer of the deep network. It is the bias vector of the first layer of the deep network. It is the bias vector of the second layer of the deep network. It is the bias vector of the third layer of the deep network. x0 is the input feature vector of the deep network, and d is the input feature dimension; Step 3.3: Concatenate the outputs of the cross-network and the deep network into a combined feature vector: z=[x l ,h k ] T , Where x l It is the final output of the cross-network, h k is the final output of the deep network, and z is the fused feature vector; Step 3.4: Input the combined feature vector into the linear output layer to obtain the predicted values ​​of the two mechanical properties, porosity and compressive modulus. in This is the original predicted value of porosity. It is the original predicted value of the compressive modulus, W. out Here, z is the weight matrix of the output layer, and b is the fused feature vector. out This is the bias vector of the output layer, porosity is the final predicted value of porosity, and modulus is the final predicted value of compressive modulus. It is the Sigmoid activation function. It is the Softplus activation function.

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