Machine learning method for physical-electric coupling characteristics of dielectric elastomer under prestretching
By using machine learning methods, combining physical information neural networks and Kolmogorov-Arnold networks, constitutive models of dielectric elastomers are automatically discovered, solving the problem that existing models cannot describe the nonlinear response of dielectric elastomers and achieving high-precision material property description.
Patent Information
- Application Number
- CN202510924176.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-07-04
- Publication Date
- 2025-10-17
AI Technical Summary
Existing constitutive models cannot fully capture the complex nonlinear response of dielectric elastomers under electromechanical coupling, resulting in inaccurate descriptions of material properties.
A machine learning approach is used to automatically discover the constitutive model of dielectric elastomer materials through the joint training of physical information neural networks and Kolmogorov-Arnold networks, and an explicit expression of the strain energy function is generated using experimental datasets.
It achieves high-precision description of dielectric elastomer materials under electromechanical coupling, simplifies the process of establishing constitutive models, and reduces the problem of inapplicable models due to insufficient experience or inadequate testing.
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Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of constitutive modeling of dielectric elastomer materials in solid mechanics, and particularly relates to a machine learning method for force-electric coupling characteristics of dielectric elastomers under pre-stretching. BACKGROUND
[0002] As an electroactive polymer, dielectric elastomer material has the advantages of low density and low modulus, large driving strain, fast driving speed, and high specific energy density. Due to these characteristics, dielectric elastomer material can be processed into actuators, sensors and energy harvesters. It is expected to play a major role in robotics, aerospace technology, biomedicine and energy harvesting. The core part of these actuators, sensors and energy harvesters is a layer of dielectric elastomer with flexible electrodes on the upper and lower surfaces. In order to enhance the mechanical properties of the material and optimize its electrical characteristics, the material will be pre-stretched during the manufacturing process. There are some theoretical studies on the analysis of the dielectric elastomer electrically induced deformation experiment after pre-stretching, such as Neo-Hookean model, Arruda-Boyce model, Gent model, etc. However, there is no model that can perfectly fit the experiment. It is still of great significance to re-examine the constitutive relationship of dielectric elastomer material.
[0003] Since dielectric elastomer material is similar to rubber when the dielectric property is not considered, many constitutive models for rubber materials, such as hyperelastic models and viscoelastic models, are generally applied to the theoretical analysis of dielectric elastomers. Therefore, the research on the constitutive model of dielectric elastomer material is based on the rich experience of hyperelastic or viscoelastic constitutive and experimental test experience of researchers. It is worth emphasizing that the constitutive model of dielectric elastomer not only needs to consider the mechanical response of the material, but also should cover its electrical characteristics. Therefore, engineers must have certain experience in electro-mechanical coupling model and experimental test ability to select the appropriate constitutive model. In practical work, engineering and technical personnel often need to compare and calibrate multiple constitutive models to accurately describe the complex nonlinear response of the material under the action of different loads and electric fields. In each model calibration, the feedback of experimental data is relied on, and the appropriate parameter calibration method is selected. However, for materials with highly complex electro-mechanical coupling behavior, the existing constitutive model cannot completely capture the characteristics of experimental data. Therefore, how to select and construct a reasonable constitutive model to accurately describe the mechanical response of dielectric elastomer under the action of electro-mechanical coupling is still an important and challenging problem in current research. SUMMARY
[0004] In order to overcome the defects existing in the prior art, the purpose of the present application is to provide a machine learning method for the dielectric elastomer force-electric coupling characteristics under pre-stretching, which automatically discovers the constitutive model of the dielectric elastomer material from the experimental data set by using the machine learning method, so as to reflect the performance of the dielectric elastomer material.
[0005] In order to achieve the above-mentioned purpose, the technical scheme adopted by the present application is:
[0006] The machine learning method for the dielectric elastomer force-electric coupling characteristics under pre-stretching comprises the following steps:
[0007] Step one: obtaining the voltage and stretch ratio experimental data set of the dielectric elastomer film after pre-stretching and then applying voltage, which is used as the input data of the physical information neural network in step two and the Kolmogorov-Arnold network in step three;
[0008] Step two: physical information neural network building; used for predicting the mapping relationship of invariants I1, I2 and strain energy function W(I1, I2), and used for the optimization training of the Kolmogorov-Arnold network in step three;
[0009] Step three: physical information neural network building; used for predicting the mapping relationship of invariants I1, I2 and strain energy function W(I1, I2), and generating the dielectric elastomer material constitutive model reflecting the nonlinear behavior of the dielectric elastomer film;
[0010] Step four: based on the voltage and stretch ratio experimental data set of the dielectric elastomer film after pre-stretching and then applying voltage obtained in step one, the Kolmogorov-Arnold network is used for training to determine the optimal network parameters to describe the nonlinear force-electric coupling characteristics of the dielectric elastomer film under the action of the applied electric field.
[0011] The step one is specifically:
[0012] Firstly, a dielectric elastomer film clamped between two flexible electrodes is subjected to relevant mechanical test experiments, a uniaxial pre-stretching test is carried out, and the elongation ratio of the pre-stretching state generated by the external force P pre-stretching is obtained After continuing to apply the electric field E, the elongation ratio after the action of P continues And the voltage The experimental data set of the dielectric elastomer is constituted;
[0013] Based on the dielectric elastomer theory, the corresponding deformation invariant Under the combination of different stretch ratios
[0014]
[0015] where I1, I2, I3 are three invariants of Cauchy-Green deformation tensor, Due to the incompressibility of the material, I3 = 1.
[0016] The step two is specifically:
[0017] Building a physical information network W net , to predict the strain energy function W(I1, I2);
[0018] W pred (I1, I2) = W net (I1, I2; w, b) (2)
[0019] where W pred is the predicted strain energy function based on two invariants I1, I2, W net represents a fully connected neural network, w represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network;
[0020] Combined with the automatic differentiation calculation ability of the neural network, the following parameters of the constitutive model are calculated
[0021]
[0022] When three-dimensional tension, the deformation tensor is
[0023]
[0024] The Cauchy stress is
[0025]
[0026] Assuming that the strain energy function W is only a function of the first invariant I1, p is the hydrostatic pressure determined by the boundary conditions, and I is the unit matrix; The Cauchy stress is written in component form as
[0027]
[0028] To eliminate the hydrostatic pressure p, the first and second terms of equation (6) are subtracted from the third term, respectively
[0029]
[0030] Under the action of external force pre-tension, the real stress in the two directions in the plane is
[0031]
[0032] The relationship between the real electric field and the real electric displacement is
[0033] E = D / ε, (9)
[0034] Under the combined action of uniaxial tension and electric field, the in-plane equilibrium equation of the ideal dielectric elastomer thin film is
[0035]
[0036] σ P is the in-plane real stress generated by the pre-stretching action of external force P; ε is the dielectric constant of the dielectric elastomer thin film, which is a constant independent of deformation; E is the real electric field under the action of voltage, E = V / H, H = h / (λ1λ2); W is the free energy function. The equilibrium equation reflects the situation that the dielectric elastomer thin film reaches equilibrium state under the combined action of the in-plane real stress σ P generated by the pre-stretching action of external force P and the in-plane equivalent Maxwell stress εE 2 generated by the applied voltage V.
[0037] The above prior knowledge is incorporated into the loss function of the physical information neural network, and the loss function Loss of the physical information neural network is calculated, which is defined as the mean square error Loss of the real value of the voltage at the stretching ratio of the measurement point and the predicted value V pred .
[0038]
[0039] Wherein is the real value of the voltage at the measurement point, V pred is the predicted value of the voltage at the measurement point, N P is the sampling point number at the nominal stress data point .
[0040] The step three is specifically:
[0041] A Kolmogorov-Arnold network is built to predict the explicit expression of the strain energy function W(I1, I2);
[0042] The strain energy density function W(I1, I2) of the dielectric elastomer material is decomposed into the Kolmogorov-Arnold expression form:
[0043]
[0044] Wherein the function φ q,p has trainable parameters, φ q,p : And Φ q :
[0045] Φ = {φ q,p}, p = 1, 2, …, n inq = 1,2...,n out , (13)
[0046] where n in is the number of nodes in the i-th layer of the computational graph, (l,i) denotes the i-th neuron in the l-th layer, x l,i denotes the activation value of the (l,i)-neuron, and between layer l and layer l+1, there are n l n l+1 activation functions: the activation function connecting (l,i)- and (l+1,j)- is denoted as
[0047]
[0048] in matrix form
[0049]
[0050] where Φ l is the matrix of functions corresponding to the l-th layer of the Kolmogorov-Arnold network; a general Kolmogorov-Arnold network is a combination of L layers: given an input vector X, the output of the Kolmogorov-Arnold network is
[0051]
[0052] Setting of the network activation function: includes a base function b(x) such that the activation function b(x) is the sum of the base function and a spline function;
[0053] φ(x) = w(b(x) + spline(x)) (17)
[0054] where
[0055] b(x) = silu(x) = x / (1 + e -x ) (18)
[0056] spline(x) is parameterized as a linear combination of B-splines such that
[0057]
[0058] where c i is trainable, and in principle, w is redundant since it can be absorbed into b(x) and spline(x). However, this w factor is still included to better control the overall magnitude of the activation function.
[0059] Initialization of the scaling: each activation function is initialized to have a spline(x) ~ 0, and w is initialized according to Xavier initialization.
[0060] Spline grid update: To solve the problem that the spline is defined on a bounded region, but the activation values can evolve from the fixed region during training, each grid is dynamically updated according to the input activation function.
[0061] Pruned Kolmogorov-Arnold networks are more interpretable than non-pruned ones. To make Kolmogorov-Arnold networks maximally interpretable, some simplification techniques are used;
[0062] Sparsification: Kolmogorov-Arnold networks use L1 regularization of linear weights to support sparsity
[0063] The L1 norm of an activation function is defined as its average amplitude over N p inputs
[0064]
[0065] Then, for a Kolmogorov-Arnold network layer Φ with n in inputs and n out outputs, the L1 norm of Φ is defined as the sum of the L1 norms of all activation functions, i.e.,
[0066]
[0067] In addition, the entropy of Φ is defined as
[0068]
[0069] The total training objective is the prediction loss l pred of all Kolmogorov-Arnold network layers plus L1 and entropy regularization:
[0070]
[0071] where μ1, μ2 are relative amplitudes, usually set as μ1 = μ2 = 1, and λ controls the overall regularization amplitude.
[0072] Visualization: The transparency of the activation function φ l,i,j is set to be proportional to tanh(βA l,i,j ), where β = 3. Therefore, functions with small amplitudes will gradually disappear so that we can focus on important functions.
[0073] Pruning: After training with sparsity penalty, to prune the network to a smaller subnetwork. Sparsify the Kolmogorov-Arnold network on the node layer (not the edge layer). For each node (say the i-th neuron in the l-th layer), define its incoming and outgoing scores as
[0074]
[0075] If both the incoming and outgoing scores are greater than a threshold hyperparameter 0 = 10 -2 , the node is considered important. All unimportant neurons are pruned.
[0076] Symbolization: When suspecting that some activation functions are actually symbolic (e.g., cos or log), we provide an interface to set them to the specified symbolic form, fix_symbolic(l, i, j, f) can set the (l, i, j) activation to f. However, one cannot simply set the activation function to the exact symbolic formula, because its input and output can have shifts and scalings. Therefore, obtain the pre-activation x and post-activation y from the samples, and fit the affine parameters (a, b, c, d) such that y ~ cf(ax + b) + d. The fitting is done by an iterative grid search of a, b and linear regression.
[0077] The fourth step is specifically:
[0078] When the loss function Loss of the physical information neural network and the loss function of the Kolmogorov-Arnold network meet the conditions at the same time, stop the network training;
[0079] Considering the differences in training efficiency and convergence effect of the physical information network and the Kolmogorov-Arnold network, an iterative alternating training scheme can be adopted, first reduce the loss function Loss of the physical information network, then train the loss function Loss of the Kolmogorov-Arnold network, and realize joint training of the network through iterative algorithm. In the back propagation phase, for the physical information network, the system calculates the gradient of the loss to the network parameters through automatic differentiation, and the parameter update adopts adaptive optimization algorithm (such as Adam optimizer), and applies Xavier initialization to the linear weight, so that the loss function Loss is less than a certain threshold (such as 10 -4 ), stop the network training. For the Kolmogorov-Arnold network, the system calculates the gradient of the loss to the network parameters through automatic differentiation, including the linear combination weight and the output layer weight w l,iand univariate basis function internal parameters (such as spline control point coordinates, polynomial coefficients), wherein the basis function gradient needs to consider its local support characteristics, such as the node interval gradient of the spline function is only locally non-zero. The parameter update adopts an adaptive optimization algorithm (such as the Adam optimizer), applies Xavier initialization to the linear weight and superimposes an L2 regularization term to prevent overfitting, and applies a smoothness constraint to the basis function parameters to ensure that the strain energy function form is reasonable. In addition, if the validation set loss does not decrease for consecutive rounds, an early stopping mechanism is triggered to terminate training, and finally the network parameters are simultaneously satisfied by iterative optimization to meet the loss minimization and the theoretical completeness of the function representation, realizing high-precision approximation under data-driven.
[0080] The beneficial effects of the present application are:
[0081] In summary, the method proposed in the present application realizes the joint training of the experimental data set obtained by mechanical test experiment, the physical information neural network and the Kolmogorov-Arnold network, directly obtains the explicit expression of the strain energy function in the constitutive model of the dielectric elastomer polymer material, and simultaneously determines the model parameters in the constitutive model.
[0082] The present application realizes an automatic modeling scheme for the force-electric coupling characteristics of dielectric elastomer polymers based on experimental data sets. The constitutive model is completely trained by a neural network, and the strain energy function expression of the model result may be different from all the models of the traditional constitutive model of dielectric elastomer polymers. The more possibilities of the constitutive model enable it to automatically discover new constitutive models of dielectric elastomer polymers and have the ability to predict complex nonlinear behaviors that existing constitutive models cannot handle.
[0083] The physical information machine learning method for the force-electric coupling characteristics of dielectric elastomers under pre-stretch provided by the present application does not require a large amount of experimental data, thereby significantly simplifying the establishment process of the material constitutive model and reducing the problem of unsuitable models caused by insufficient experience or insufficient testing. BRIEF DESCRIPTION OF DRAWINGS
[0084] Figure 1 Physical information machine learning method for force-electric coupling characteristics of dielectric elastomers under pre-stretch.
[0085] Figure 2 Based on the dielectric elastomer test case of the present method, the constitutive and voltage prediction diagram of a dielectric elastomer under uniaxial tension of a certain material. DETAILED DESCRIPTION
[0086] The present application will be further described in detail below with reference to the accompanying drawings.
[0087] The application discloses a physical information machine learning method for electric coupling characteristics of a pre-stretched dielectric elastomer, and realizes an artificial intelligence neural network automatic modeling scheme of a constitutive model of a dielectric elastomer material based on an experimental data set.
[0088] The application will be described in detail in the following steps, and the specific operation process is shown in the following figure: Figure 1
[0089] Step one: experimental data set acquisition, used for generating input data of step two and step three; first, a dielectric elastomer film clamped between two flexible electrodes is subjected to relevant mechanical test experiments, uniaxial tension test is carried out, and then voltage is applied, so that an experimental data set of the dielectric elastomer material composed of the tensile ratio and the voltage is obtained;
[0090] Taking the uniaxial tension test as an example, a flat test piece for testing is prepared, uniaxial tension test is carried out, and an experimental data set of the dielectric elastomer material composed of the tensile ratio and the voltage is obtained, as shown in the following figure: Figure 1 Specifically, a dielectric elastomer film is clamped between two flexible electrodes to carry out relevant uniaxial tension test, in the reference state, the film is not subjected to mechanical force and voltage, the thickness of the film is H, and the length is L1 and L2. When the dielectric high-elastic polymer is subjected to external force P in the pre-stretching state, the thickness becomes h1, and the in-plane direction is respectively expanded by λ p2 times. Finally, voltage is applied, and the final equilibrium state of the dielectric high-elastic polymer under the coupling of the electric field and the force is obtained, on the basis of the in-plane pre-stretching, the horizontal direction maintains the external force P, the thickness direction adds the voltage V, the thickness becomes h, and the in-plane two directions are respectively expanded by λ1 times and λ2 times. Finally, the experimental data of the voltage under different tensile ratio combination parameters are obtained, so that the experimental data set of the dielectric elastomer film composed of the tensile ratio and the voltage is obtained.
[0091] Based on the dielectric elastomer theory, the corresponding deformation invariants under different tensile ratio combination parameters are calculated. Taking the uniaxial tension test as an example, the calculation formula of the corresponding deformation invariants under different tensile ratio combination parameters is as follows:
[0092]
[0093] Wherein, I1, I2 and I3 are three invariants of the Cauchy-Green deformation tensor, Due to the incompressibility of the material, I3 = 1.
[0094] Step two: physical information network building; for predicting the mapping relationship of I1, I2 and strain energy function W(I1, I2); building a physical information network W net , to predict the strain energy function W(I1, I2).
[0095] First, build a fully connected neural network W net (w, b) to predict the strain energy function W(I1, I2), where the neural network input is the deformation invariant I1, I2 of the hyperelastic material, and the output is the strain energy function W(I1, I2) pred , as shown in the attached Figure 1 .
[0096] W pred (I1, I2) = W net (I1, I2; w, b) (2)
[0097] Where W pred is the predicted strain energy function based on two invariants I1, I2, W net represents a fully connected neural network, w represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network.
[0098] Combined with the automatic differentiation calculation ability of the neural network, the following parameters of the constitutive model are calculated
[0099]
[0100] When three-dimensional tension, the deformation tensor is
[0101]
[0102] The Cauchy stress is
[0103]
[0104] Assuming that the strain energy function W is only a function of the first invariant I1, p is the hydrostatic pressure determined by the boundary conditions, and I is the unit matrix. The Cauchy stress is written in component form as
[0105]
[0106] To eliminate the hydrostatic pressure p, the first and second terms of equation (6) are subtracted from the third term
[0107]
[0108] Under the action of external force pre-stretch, the real stress in the two directions in the plane is
[0109]
[0110] The relationship between true electric field and true electric displacement is:
[0111] E=D / ε, (9)
[0112] Under the combined action of uniaxial tension and electric field, the in-plane equilibrium equation of an ideal dielectric elastic film is:
[0113]
[0114] σ P is the true stress in the plane caused by the pre-tensioning action of the external force P; ε is the dielectric constant of the dielectric elastomer film, which is a constant that does not depend on deformation; E is the true electric field under the action of voltage, E = V / H, H = h / (λ1λ2); W is the free energy function. This equilibrium equation reflects the true stress σ in the plane caused by the pre-tensioning action of the external force P. P and the in-plane equivalent Maxwell stress εE generated by the applied voltage V 2 The situation where the dielectric elastomer film reaches equilibrium under the combined action.
[0115] The loss function of the physical information neural network is defined as the stretching ratio of the measurement point The true value of the voltage at and the predicted value V pred Mean square error Loss
[0116]
[0117] in is the true value of the voltage at the measurement point, V pred is the predicted value of the voltage at the measurement point, N P is the nominal stress data point The number of sampling points.
[0118] Step 3: Kolmogorov-Arnold network; used to generate a constitutive model that can reflect the nonlinear behavior of dielectric elastomer films;
[0119] Build a Kolmogorov-Arnold network to predict the explicit expression of the strain energy function W(I1,I2); decompose the strain energy density function W(I1,I2) of the dielectric elastomer material into a Kolmogorov-Arnold representation:
[0120]
[0121] where the function φ q,p With trainable parameters, φ q,p : and Φ q :
[0122] Φ = {φ q,p}, p = 1,2,...,n in , q = 1,2,...,n out , (13)
[0123] where n in is the number of nodes in the i-th layer of the computational graph. (l,i) denotes the i-th neuron in the l-th layer, with x l,i denoting the activation value of the (l,i)-neuron. Between layer l and layer l+1, there are n l n l+1 activation functions: the activation function connecting (l,i)- and (l+1,j)- is denoted by
[0124]
[0125] in matrix form
[0126]
[0127] where Φ l is the matrix of functions corresponding to the l-th layer of the Kolmogorov-Arnold network. A general Kolmogorov-Arnold network is a combination of L layers: given an input vector X, the output of the Kolmogorov-Arnold network is
[0128]
[0129] Setting of the network activation function: includes a base function b(x) such that the activation function b(x) is the sum of the base function and a spline function.
[0130] φ(x) = w(b(x) + spline(x)) (17)
[0131] where
[0132] b(x) = silu(x) = x / (1 + e -x ) (18)
[0133] spline(x) is parameterized as a linear combination of B-splines, such that
[0134]
[0135] where c i is trainable. In principle, w is redundant, as it can be absorbed into b(x) and spline(x). However, this w factor is still included to better control the overall amplitude of the activation function.
[0136] Initialization of the scaling: Each activation function is initialized to have a spline(x) ~ 0. w is initialized according to Xavier initialization.
[0137] Updating of the spline line grid: To solve the problem that the spline is defined on a bounded region, but the activation values can evolve out of the fixed region during training, each grid is dynamically updated according to the input activation function.
[0138] Pruned Kolmogorov-Arnold networks are more interpretable than non-pruned ones. To make the Kolmogorov-Arnold network maximally interpretable, some simplification techniques are used,
[0139] Sparsification: The Kolmogorov-Arnold network uses L1 regularization of the linear weights to support sparsity
[0140] The L1 norm of an activation function is defined as its average amplitude over N p inputs
[0141]
[0142] Then, for a Kolmogorov-Arnold network layer Φ with n in inputs and n out outputs, the L1 norm of Φ is defined as the sum of the L1 norms of all activation functions, i.e.,
[0143]
[0144] In addition, the entropy of Φ is defined as
[0145]
[0146] The total training objective is the prediction loss l pred of all Kolmogorov-Arnold network layers plus the L1 and entropy regularization:
[0147]
[0148] where μ1, μ2 are relative amplitudes, usually set as μ1 = μ2 = 1, and λ controls the overall regularization amplitude.
[0149] Visualization: The transparency of an activation function φ l,i,j is set to be proportional to tanh(βA l,i,j ), where β = 3. Thus, functions with small amplitudes will gradually disappear so that we can focus on important functions.
[0150] Pruning: After training with sparsity penalty, to prune the network to a smaller subnetwork. Sparsify the Kolmogorov-Arnold network on the node layer (not the edge layer). For each node (say the i-th neuron in the l-th layer), define its incoming and outgoing scores as
[0151]
[0152] If both the incoming and outgoing scores are greater than a threshold hyperparameter θ = 10 -2 , the node is considered important. All unimportant neurons are pruned.
[0153] Symbolization: When suspecting that some activation functions are actually symbolic in nature (e.g. cos or log), we provide an interface to set them to the specified symbolic form, fix_symbolic(l, i, j, f) can set the (l, i, j) activation to f. However, one cannot simply set the activation function to the exact symbolic formula, because its input and output can have shifts and scalings. Therefore, obtain the pre-activation x and post-activation y from the samples, and fit the affine parameters (a, b, c, d) such that y ≈ cf(ax + b) + d. The fitting is done by an iterative grid search over a, b and linear regression.
[0154] Step four: neural network training, find the optimal network parameters;
[0155] Based on the experimental data set of the dielectric elastomer material obtained in step one, the network is trained, and the physical information neural network in step two and the Kolmogorov-Arnold network in step three are jointly trained;
[0156] When the loss function Loss of the physical information neural network and the loss function of the Kolmogorov-Arnold network reach the condition at the same time, stop the network training;
[0157] Considering the differences in training efficiency and convergence effect of the physical information network and the Kolmogorov-Arnold network, an iterative alternating training scheme can be adopted, first reduce the loss function Loss of the physical information network, and then train the loss function Loss of the Kolmogorov-Arnold network, and realize the joint training of the network through the iterative algorithm. In the back propagation phase, for the physical information network, the system calculates the gradient of the loss to the network parameters by automatic differentiation, and the parameter update adopts the adaptive optimization algorithm (such as Adam optimizer), and the linear weight applies Xavier initialization, so that the loss function Loss is less than a certain threshold (such as 10 -4), stop network training. For the Kolmogorov-Arnold network, the system calculates the gradient of the loss to the network parameters through automatic differentiation, including the linear combination weights and the output layer weights w l,i As well as the internal parameters of the univariate basis functions (such as the coordinates of the spline control points and the polynomial coefficients), the gradient of the basis functions must take into account their local support characteristics. For example, the node interval gradient of the spline function is only locally non-zero. The parameters are updated using an adaptive optimization algorithm (such as the Adam optimizer). Xavier initialization is applied to the linear weights and an L2 regularization term is superimposed to prevent overfitting. Smoothness constraints are imposed on the basis function parameters to ensure the reasonable shape of the strain energy function. In addition, if the validation set loss does not decrease for multiple consecutive rounds, the early stopping mechanism is triggered to terminate the training. Ultimately, through iterative optimization, the network parameters are optimized to simultaneously meet the requirements of loss minimization and theoretical completeness of the function representation, achieving high-precision approximation under data-driven conditions.
[0158] Through the above steps, the final output of the machine learning network is the strain energy function W of the dielectric elastomer obtained by the constitutive modeling method. * The expression of (I1, I2) generates a dielectric elastomer constitutive model that can reflect the nonlinear behavior of the dielectric elastomer. And the strain energy function W of the constitutive model is * The model parameters in (I1, I2) are simultaneously determined by a deep regression network. The resulting constitutive model of dielectric elastomers can be verified by comparison with experimental datasets.
[0159] In summary, the method proposed in the present invention realizes the joint training of the experimental data set obtained through mechanical testing, the physical information neural network and the Kolmogorov-Arnold network, directly obtains the explicit expression of the strain energy function in the constitutive model of the dielectric elastomer, and simultaneously determines the model parameters in the constitutive model.
[0160] Step 1 is used to generate input data for the physical information network in step 2 and the deep regression network in step 3; the physical information network in step 2 obtains the mapping relationship between I1, I2 and the strain energy function W(I1, I2), which is used for optimization training of the deep regression network in step 3; the Kolmogorov-Arnold network in step 3 ultimately generates a constitutive model that can reflect the nonlinear behavior of the dielectric elastomer film;
[0161] Example:
[0162] The invention provides an application of the dielectric elastomer constitutive model scheme in constitutive and deformation analysis of a dielectric elastomer of a certain material.
[0163] A uniaxial tensile test is performed on a dielectric elastomer film to obtain the tensile ratio and voltage The experimental data set of the dielectric elastomer is composed of points as shown in the attached Figure 2
[0164] Through the dielectric elastomer constitutive model scheme of the present application, combined with the training of deep regression and physical information network, the explicit expression of the strain energy function in the constitutive model of the dielectric elastomer is directly obtained as
[0165]
[0166] And the model parameters in the constitutive model are determined as C 10 = 6869.37; C 20 = 1.27;
[0167] Based on the network model training, the constitutive model of the dielectric elastomer can be further calculated to obtain Cauchy stress and in-plane true stress, and finally obtain the voltage, and the results are shown by the lines as shown in the attached Figure 2 Figure 2 Compared with the experimental data set, as shown in the attached Figure 2 , it can be seen that the constitutive model well predicts the "N" shaped experimental data rule obtained by the mechanical test, which verifies the obtained dielectric elastomer constitutive model.
[0168] The method proposed in the present application, through the experimental data set obtained by the mechanical test, combined with the joint training of the physical information neural network and the Kolmogorov-Arnold network, directly obtains the explicit expression of the strain energy function in the constitutive model of the dielectric elastomer, and simultaneously determines the model parameters in the constitutive model. In particular, the constitutive model is completely trained by the neural network, and the constitutive model expression has more possibilities, which enables it to predict some complex nonlinear behaviors that existing constitutive models cannot handle.
Claims
1. A machine learning method for the electromechanical coupling characteristics of dielectric elastomers under pre-tension, characterized by: The following steps are included: Step 1: Obtaining an experimental data set of voltage and stretch ratio obtained by applying voltage to a dielectric elastomer film after pre-stretching, which is used to generate input data for the physical information neural network in step 2 and the Kolmogorov-Arnold network in step 3; Step 2: Construction of a physical information neural network; used to predict the mapping relationship between the invariants I1, I2 and the strain energy function W(I1, I2), which is used for the optimization training of the Kolmogorov-Arnold network in step 3; Step 3: Construction of a physical information neural network; used to predict the mapping relationship between the invariants I1, I2 and the strain energy function W(I1, I2), and generate a dielectric elastomer material constitutive model that can reflect the nonlinear behavior of the dielectric elastomer film; Step 4: Based on the experimental data set of voltage and stretch ratio obtained by applying voltage to the dielectric elastomer film after pre-stretching obtained in Step 1, a Kolmogorov-Arnold network is trained to determine the optimal network parameters to describe the nonlinear electromechanical coupling characteristics of the dielectric elastomer film under the action of an external electric field.
2. The machine learning method for the electromechanical coupling characteristics of a dielectric elastomer under pre-tension according to claim 1, characterized in that: The step 1 is specifically as follows: First, a mechanical test was carried out on a dielectric elastomer film sandwiched between two flexible electrodes. A uniaxial pre-stretching test was performed to obtain the elongation ratio of the pre-stretched state caused by the external force P pre-stretching. After the electric field E is applied again, the elongation ratio after P continues to act is and voltage Experimental dataset of constructed dielectric elastomers; Calculation of different stretching ratios based on dielectric elastomer theory The corresponding deformation invariants under the combined parameters; Among them, I1, I2, I3 are the three invariants of the Cauchy Green deformation tensor, Due to the incompressibility of the material, I3=1.
3. The machine learning method for the electromechanical coupling characteristics of a dielectric elastomer under pre-tension according to claim 2, characterized in that: The step 2 is specifically as follows: Build a physical information network W net , to predict the strain energy function W(I1,I2); W pred (I1,I2)=W net (I1,I2;w,b) (2) Among them, W pred is the predicted strain energy function based on the two invariants I1 and I2, W net represents a fully connected neural network, w represents the weight of the fully connected neural network, and b represents the bias of the fully connected neural network; Combining the automatic differentiation computing capability of the neural network, the following parameters of the constitutive model are calculated: When stretched in three directions, the deformation tensor is: The Cauchy stress is: Assume that the strain energy function W is only a function of the first invariant I1, p is the hydrostatic pressure determined by the boundary conditions, and I is the identity matrix; the Cauchy stress is written in component form as: To eliminate the hydrostatic pressure p, the third term is subtracted from the first and second terms of Equation (6) to obtain: Under the action of external pre-tension, the true stress in two directions in the plane is The relationship between true electric field and true electric displacement is: E=D / ε, (9) Under the combined action of uniaxial tension and electric field, the in-plane equilibrium equation of an ideal dielectric elastomer film is: σ P is the true in-plane stress generated by the pre-stretching action of the external force P; ε is the dielectric constant of the dielectric elastomer film; E is the true electric field under the action of voltage, E = V / H, H = h / (λ1λ2); W is the free energy function; Incorporate the above prior knowledge into the loss function of the physical information neural network and calculate the loss function Loss of the physical information neural network, which is defined as the stretching ratio of the measurement point The true value of the voltage at and the predicted value V pred The mean square error Loss: in is the true value of the voltage at the measurement point, V pred is the predicted value of the voltage at the measurement point, N P is the nominal stress data point The number of sampling points.
4. The machine learning method for the electromechanical coupling characteristics of a dielectric elastomer under pre-tension according to claim 3, characterized in that: The step three is specifically as follows: Build a Kolmogorov-Arnold network to predict the explicit expression of the strain energy function W(I1,I2); Decompose the strain energy density function W(I1,I2) of the dielectric elastomer material into the Kolmogorov-Arnold representation: where the function φ q,p With trainable parameters, and Φ={φ q,p }, p=1,2,…,n in , q=1,2…,n out , (13) where n in is the number of nodes in the i-th layer of the computational graph, (l,i) represents the i-th neuron in the l-th layer, and x l,i Represents the activation value of (l,i)-neuron. Between layer l and layer l+1, there are n l n l+1 Activation function: The activation function connecting (l,i)- and (l+1,j)- is expressed as: Written in matrix form: where Φ l is the function matrix corresponding to the lth layer of the Kolmogorov-Arnold network; the Kolmogorov-Arnold network is a combination of L layers: given an input vector X, the output of the Kolmogorov-Arnold network is Setting of network activation function: including a basis function b(x), so that the activation function b(x) is the sum of the basis function and the spline function; φ(x)=w(b(x)+spline(x)) (17) in b(x)=shadow(x)=x / (1+e -x ) (18) spline(x) is parameterized as a linear combination of B-splines such that where c i It is trainable.
5. The machine learning method for determining the electromechanical coupling characteristics of a dielectric elastomer under pre-tension according to claim 4, characterized in that Find an explicit expression for the strain energy function and initialize the scaling: Each activation function is initialized to have a spline spline(x)≈0, and w is initialized according to Xavier initialization; Update of the spline mesh: Sparsification: Kolmogorov-Arnold networks use L1 regularization of linear weights to support sparsity; The L1 norm of the activation function is defined as its p Average amplitude over inputs: Then, for n in input and n out The L1 norm of the Kolmogorov-Arnold network layer Φ with 1 output is defined as the sum of the L1 norms of all activation functions, that is, In addition, the entropy of Φ is defined as: The total training target is the prediction loss l for all Kolmogorov-Arnold network layers pred Add L1 and entropy regularization: Where μ1 and μ2 are relative amplitudes, usually set to μ1 = μ2 = 1, and λ controls the overall regularization amplitude; Visualization: Setting the activation function φ l,i,j The transparency and tanh(βA l,i,j ) are proportional; Pruning: After training with a sparsification penalty, the Kolmogorov-Arnold network is sparsified at the node level. For each node, its incoming and outgoing scores are defined as: If both incoming and outgoing scores are greater than a threshold hyperparameter θ = 10 -2 , then the node is considered important and all unimportant neurons are pruned; Symbolization: fix_symbolic(l,i,j,f) sets the activation of (l,i,j) to f when it is suspected that some activation function is actually symbolic; obtain the pre-activation x and post-activation y from the sample and fit the affine parameters (a,b,c,d) such that y ≈ cf(ax+b)+d. The fitting is done by iterative grid search and linear regression on a and b.
6. The machine learning method for the electromechanical coupling characteristics of a dielectric elastomer under pre-tension according to claim 5, characterized in that: The step 4 is specifically as follows: When the loss function Loss of the physical information neural network and the loss function of the Kolmogorov-Arnold network meet the conditions at the same time, stop network training; Adopting an iterative alternating training scheme, the loss function of the physical information network is first reduced, and then the loss function of the Kolmogorov-Arnold network is trained, and the network joint training is achieved through an iterative algorithm; In the backpropagation phase, for the physical information network, the system calculates the gradient of the loss to the network parameters through automatic differentiation. The parameters are updated using an adaptive optimization algorithm. Xavier initialization is applied to the linear weights to make the loss function less than a certain threshold, and the network training is stopped. For the Kolmogorov-Arnold network, the system calculates the gradient of the loss with respect to the network parameters by automatic differentiation, including the linear combination weights and the output layer weights w l,i and the internal parameters of the univariate basis functions, where the gradients of the basis functions need to take into account their local support characteristics; An adaptive optimization algorithm is used for parameter update. Xavier initialization is applied to linear weights and an L2 regularization term is superimposed to prevent overfitting. Smoothness constraints are imposed on the basis function parameters to ensure the reasonable shape of the strain energy function. In addition, if the validation set loss does not decrease for multiple consecutive rounds, an early stopping mechanism is triggered to terminate training. Ultimately, through iterative optimization, the network parameters are optimized to simultaneously meet the requirements of loss minimization and theoretical completeness of function representation, achieving high-precision approximation driven by data.