A machine learning based plasticity model using proper orthogonal decomposition
Dengpeng Huang, Jan Niklas Fuhg, Christian Weißenfels, Peter Wriggers
AA machine learning based plasticity model using proper orthogo-nal decomposition
Dengpeng Huang · Jan Niklas Fuhg · Christian Weißenfels · Peter Wriggers
Abstract
Data-driven material models have many advantages over classical numerical approaches, suchas the direct utilization of experimental data and the possibility to improve performance of predictionswhen additional data is available. One approach to develop a data-driven material model is to usemachine learning tools. These can be trained offline to fit an observed material behaviour and then beapplied in online applications. However, learning and predicting history dependent material models, suchas plasticity, is still challenging. In this work, a machine learning based material modelling framework isproposed for both elasticity and plasticity. The machine learning based hyperelasticity model is developedwith the Feed forward Neural Network (FNN) directly whereas the machine learning based plasticitymodel is developed by using of a novel method called Proper Orthogonal Decomposition Feed forwardNeural Network (PODFNN). In order to account for the loading history, the accumulated absolute strainis proposed to be the history variable of the plasticity model. Additionally, the strain-stress sequencedata for plasticity is collected from different loading-unloading paths based on the concept of sequencefor plasticity. By means of the POD, the multi-dimensional stress sequence is decoupled leading toindependent one dimensional coefficient sequences. In this case, the neural network with multiple outputis replaced by multiple independent neural networks each possessing a one-dimensional output, whichleads to less training time and better training performance. To apply the machine learning based materialmodel in finite element analysis, the tangent matrix is derived by the automatic symbolic differentiationtool AceGen. The effectiveness and generalization of the presented models are investigated by a seriesof numerical examples using both 2D and 3D finite element analysis.
Keywords
Machine Learning · Artificial Neural Network · Plasticity · Proper Orthogonal Decomposi-tion · Finite Element Method
With the development of data mining technology, machine learning algorithms, high performance com-puting and robust numerical methods, data-driven computational modelling play an important role in
D. Huang ( (cid:66) ), J. N. Fuhg, C. Weißenfels, P. WriggersInstitute of Continuum Mechanics, Leibniz University of Hannover, Appelstr. 11, 30167 Hannover, GermanyE-mail: [email protected] a r X i v : . [ c s . C E ] J a n not only accurate but also fast predictions of complex industrial processes. In particular, accurate ma-terial models are key parts in structure analysis. In the past years, tremendous effort has been madein developing material models, see e.g. the review of models by Cao [2017] for metal forming processes.However, the proposed models show limitations in generalization or accuracy in some cases when themodel is applied to engineering problems.As a data-driven approach, the Machine Learning (ML) based material modelling provides an al-ternative tool to narrow the gap between experimental data and material models. By use of the MLtechnology, such as artificial neural networks, see e.g. Hassoun et al. [1995], or Gaussian Processes, seee.g. Rasmussen [2003], constitutive equations can be approximated by using experimental data withoutpostulation of a specific constitutive model. An advantage of machine learning based material models isthat they can iteratively be improved if more experimental data are available, which yields more flexibleand sustainable material descriptions. For a review of machine learning in computational mechanics, seeOishi and Yagawa [2017] and references therein.In order to replace the classical constitutive model in computational mechanics by data-driven mod-elling, multiple approaches have been proposed in the literature. The model-free data-driven computingparadigm proposed by Kirchdoerfer and Ortiz [2016], Kirchdoerfer and Ortiz [2018], Eggersmann et al.[2019] and Stainier et al. [2019], conducts the computing directly from experimental material data underthe constraints of conservation laws, which bypasses the empirical material modelling step. This ap-proach works without constitutive model and seeks to find the closest possible state from a prespecifiedmaterial data set. A manifold learning approach is proposed by Iba˜nez et al. [2017], Ibanez et al. [2018]and Ib´a˜nez et al. [2019], where the so-called constitutive manifold is constructed from collected data.A self-consistent clustering approach has been developed to predict the behaviour of heterogeneous ma-terials under inelastic deformation, see Liu et al. [2016] and Shakoor et al. [2019]. Tang et al. [2019]proposed a mapping approach, where one-dimensional data are mapped into three-dimensions for non-linear elastic material modelling without the construction of an analytic mathematical function for thematerial equation. Since the performance of the data-driven computing is highly determined by qualityand completeness of the available data, data completion and data uncertainties have been investigated,see Ayensa-Jim´enez et al. [2018] and Ayensa-Jimnez et al. [2019].In addition to the data-driven approaches mentioned above, the artificial neural network as a machinelearning approach has been applied to approximate the constitutive model based on data as well, seeGhaboussi and Sidarta [1998], Hashash et al. [2004], and Lefik and Schrefler [2003]. In order to fit aconstitutive material equation, the neural network is trained offline using experimental data collectedfrom different loading paths. Afterwards, the network based model is applied online for testing andapplications. A nested adaptive neural network has been applied in Ghaboussi et al. [1998], Ghaboussiand Sidarta [1998] for modelling the constitutive behaviour of geomaterials. In Hashash et al. [2004],a feed forward neural network based constitutive model is implemented in finite element analysis tocapture the nonlinear material behaviour, where the consistent material tangent matrix is derived andevaluated. Artificial neural networks are also applied as incremental non-linear constitutive models inLefik and Schrefler [2003] for finite element applications. Furthermore, this approach has been appliedto predict the stress-strain curves and texture evolution of polycrystalline metals by Ali et al. [2019].Instead of the offline training, the neural network based constitutive model can be trained online byauto-progressive algorithms as well, see Pabisek [2008] and Ghaboussi et al. [1998]. Lastly, artificialneural networks have been applied to the heterogeneous material modelling, such as Le et al. [2015], Luet al. [2019], Li et al. [2019], Liu et al. [2019] and Yang et al. [2019].The data-driven model free approach conducts calculations directly from the data, which bypassesthe model on one hand but highly relies on the quality and completeness of the data on the otherhand. The machine learning approaches mentioned above apply the previous strain and stress as historyvariables, which introduces extra errors for the elastic stage of inelastic deformation, and thus affects thecapabilities to capture the load history in real applications. Additionally, the derivation of the tangentmatrix for the neural network based model is complex when changing the network architecture. Thus,there are many issues present in machine learning based material modelling approaches, such as the datacollection strategy, the selection of history variables and the applications in finite element analysis.The objective of this work is to develop a machine learning based hyperelastic and plasticity modelsfor finite element applications as well as a corresponding data collection strategy. To simplify the datacollection process from experiments, only strain components act as input data and only stress com-ponents represent output data. Instead of using the previous strain and stress as history variables inplasticity, in this work, the accumulated absolute total strain is applied as history variable to distinguishdifferent loading paths. This variable can be computed from preexisting input data without additionaleffort as e.g. different experiments. Due to its history dependence, the training data for plasticity willbe sequential data sets obtained under different loading-unloading paths. Since the isotropic plasticitycan be formulated in the principle space, the training sequence data is collected only from tension andcompression tests, which simplifies the data collection. A novel method called Proper Orthogonal De-composition Feedforward Neural Network (PODFNN) is proposed in combination with the introducedhistory variable for predicting the stress sequences in case of plasticity. By means of the Proper Or-thogonal Decomposition (POD), the stress sequence is transformed into multiple independent coefficientsequences, where the stress at any time step can be recovered by a linear combination of the coefficientsand the basis.The presented approach decomposes the strain-stress relationship into multiple independent neuralnetworks with only one output, which significantly decreases the complexity of the model. In orderto apply the machine learning based model in finite element analysis, the tangent matrix has to becomputed. It is derived by the symbolic differentiation tool AceGen, see Korelc and Wriggers [2016].The effectiveness and generalization of the machine learning based plasticity model is validated in 2Dand 3D using several applications.This paper is structured as follows: In Section 2, the machine learning based material modellingframework is presented. Then the Feed forward Neural network (FNN) is applied to learn the hyper-elastic material law in Section 3. In Section 4, the data collection strategy for plasticity is proposed.Based on the training data, the machine learning based plasticity model is developed and validated infinite element analysis in Section 5, which is followed by the conclusions in Section 6. To develop a data-driven material model by means of machine learning technology, three steps arenecessary: data collection, machine learning and validation, see Fig. 1. As a fundamental ingredient fordata-driven models, the data, representing the material behaviour, have to be collected firstly. Accordingto the specific problem, the training data can be collected from experiments and simulations.In this work, strain-stress data are employed as the input and output of the data-driven model.For the plasticity model, the strain-stress data will be collected for specific loading paths and stored assequences. Depending on the problem, the training data usually have to be preprocessed utilizing datascaling, data decomposition and data arrangement.The second step is to fit the constitutive equation related to the data by means of the ML technology.The Artificial Neural Network (ANN) as a machine learning technology will be employed in this work.The hyperparameters of the neural network based model have to be selected according to the data andthe accompanying accuracy requirements. Once the model is trained, the describing parameters will beused and stored for the material description of the developed model.The final step is to validate the accurate reproduction of the ML based material model. To do so, theML based material model is compared with a standard material model within several finite element ap-plications. By deriving the tangent matrix and residual vector, the ML based model can be incorporatedinto a FEM code. The performance of the developed model will be evaluated by benchmark tests. If theaccuracy of the material model can not meet the necessary requirements, the model hyperparameterswill be optimized or supplemental data will be collected. Therefore, the machine learning based frame-work is an open system and the accuracy of the developed model can be improved iteratively during itsapplication.
Data Collection
Loading conditionsData from experimentsData from simulationsData preprocessing εσ Machine Learning
Artificial neural networksNetwork architectureTraining algorithm ... σ σ σ ε ε ε Validation
Derive the material tangentIncorporartion in FEM codeBenchmark testsModel valitation σ = f ( ε , W ) F Data supplementAccuracy improvement
Material model:Figure 1: The data-driven material modelling framework.
Before the plasticity model is developed in detail, a ML based hyper-elasticity model is presuited byutilizing feed forward neural networks (FNNs) in this section.
Feed forward neural network is a fundamental machine learning technology. A deep FNN is composedof several connected layers of artificial neurons and biases, where the data is fed into an input layer andthen flows through some hidden layers. The output is finally predicted at an output layer, as shown inFig. 2. The neurons from different layers are fully connected through the weights w . In the predictionphase, the data flows in one way from the input layer to the output layer. In the training phase, theglobal error defined by the mean-squared differences between the target value and the FNN output willbe back-propagated through the hidden layers. This step is performed in order to update the weights,where the objective is to minimize the global error.Input layer Output layerHidden layers Targetsb b b i i i m bias o o o n t t t n E ( w ) = n n (cid:80) j =1 [ o j ( w ) − t j ] Figure 2: The feed forward neural network.At each neuron, an activation function is attached, see Fig. 3. The output of each neuron is computedby multiplying the outputs from the previous layer with the corresponding weights. For the neuron j inthe layer k , the data of the previous layer k − j in layer k is computed as o kj = f s (cid:32) N (cid:88) i =1 w ij o k − i + b k − i (cid:33) , (1)where N is the number of neurons in the previous layer k − w ij is the weight connecting neurons i and j , o k − i is the output of the neuron i in layer k − b k − i is its bias. A common choice forthe activation function is the sigmoid function f s ( x ) = 21 + e − x − . (2)The specific architecture of the FNN, such as the number of layers and the number of neurons ineach layer, has to be determined according to the complexity of the data set. (cid:80) f ( x ) w j w j w j o kj o k − o k − o k − ... ... Figure 3: The artificial neuron.
In the training phase, the weighs of neural network will be initialized firstly, see Nguyen and Widrow[1990], which is followed by the weights updating using a training algorithm such that the global error isminimized. The global error, also named as loss function or network performance, is defined accordingto the difference between the network prediction and the target data as shown in Fig. 2. The meansquared error is used to measure the loss E ( w ) = 1 N N (cid:88) i =1 [ o i ( w ) − t i ] = 1 N N (cid:88) i =1 e i , (3)where N is the number of outputs, o i is the i -th output, w is the vector that contains the weights ofneural network, and t i is the i -th target value. Training a feed forward neural network is an optimizationproblem, where the global error is treated as the objective function. To minimize the global error, theLevenberg-Marquardt algorithm is applied to update the weighs, see Hagan [1994], w n +1 = w n − ( J T J + µ I ) − J T · e , (4)in which w n +1 is the weight vector in iteration n + 1, µ is a parameter to adaptively control the speed ofconvergence, and J is the Jacobian matrix that contains the derivatives of network errors with respectto the network weights J ij = ∂e i ∂w nj . (5)In the training process, many iterations are required to update the weights until the stopping criteria isfulfilled, where one iteration is also known as one epoch. To approximate hyperelastic behaviour by the FNN for finite element applications, the first task is todetermine the input and output variables for the neural network. Since the loading and the unloadingcurve coincide for the elastic deformation, as shown in Fig. 4, the relationship between the strain spaceand stress space can be seen as a one-to-one mapping. Hence, the strain-stress mapping can be approx-imated by the FNN without considering the loading history. ε εO ε ε ε t σ σ σ σ t σ Figure 4: The loading and unloading curve for hyperelasticity.Instead of using experimental data, the training data are collected here from an analytical model,which allows us to test the performance of the ML based model by comparing its simulation results withthat by an analytical model. As an example of hyperelasticity, the non-linear neo-Hookean model isapplied as the target model to learn σ = 12 λJ ( J − I + µJ ( b − I ) , (6)where the Cauchy stress σ and the left Cauchy Green tensor b are symmetric tensors. For the 2Dproblem, the inputs of the model can be chosen as the strain components ( J, b , b , b ), whereas theoutputs are chosen as the stress components ( σ , σ , σ ). According to the number of input and theoutput, the architecture of the FNN can be determined as 4 − n − n neurons is applied for this hyperelastic law. The input data is generated by taking equally spacedpoints within the given range of strain space. The stresses as output data can be computed from theneo-Hookean model in equation (6) accordingly.Before training the FNN, the generated data is scaled to the range ( − ,
1) such that training isaccelerated. Then the neural network is trained until the stopping criteria is reached. After training,the weights w and bias b s will be saved as the model parameters. The ML based hyper-elasticity modelis thus expressed as σ NN = F N N ( b , J, w , b s ) , (7)where σ NN is the predicted Cauchy stress by the FNN. The ML based model can be used in the same way as the classical constitutive model in the finite elementanalysis. The residual for the static problem is given by R ( u ) = f − (cid:90) Ω B T σ NN d Ω , (8) f = (cid:90) Ω N ρ ˆ b dv − (cid:90) ∂ Ω N ˆ t da, (9)in which B is the gradient of shape functions N , ρ is the density, σ NN is the stress computed from themachine learning based model, ˆ b and ˆ t are the body force and the surface traction respectively. Dueto the non-linearity, the Newton Rapson iterative solution scheme is applied. The tangent matrix iscomputed by taking the derivative of residual in terms of displacement K T = ∂ R ( u ) ∂ u . (10)The derivation of the tangent matrix for the neural network based model requires the computationof derivatives by the chain rule, which will be complex if the number of neuron is very large. In thiswork, the automatic differentiation tool AceGen, see Korelc and Wriggers [2016], based on the symboliccomputing in Mathematica is applied, by which the tangent matrix and residual vector can be derivedautomatically. The material parameters for the neo-Hookean model used in the training data collection are set as E = 700 N/mm , ν = 0 . . h m s .The first example is the uniaxial compression test of a plate in 2D. As shown in Fig. 5, the pressureis imposed on the top surface of the plate, the bottom of the plate is fixed in vertical direction. Thedistributed load is given as q = − M P a .The final deformation of the plate computed with the ML based model is compared with the outcomeof using the neo-Hooken model in equation (6). It can be see from Fig. 6 that the displacements invertical direction are very close. The computation time with analytical hyperelastic model is 8 . s ,whereas the computation time with the ML based model is 9 . s on the same computer.In order to further validate the generalization, a second test case, the Cook’s membrane problem,is conducted. The tapered beam is clamped at the left end and loaded at the right end by a constantdistributed vertical load q = 5 M pa , as depicted in Fig. 7. The geometric domain of the structure isdiscretized by 40 quadratic 9-node quadrilateral elements leading to 189 nodes.With the same model as trained in the first test, the final deformation of the membrane is computedand compared. As shown in Fig. 8, the vertical displacement in both cases are very close to eachother, which highlights the proficient generalization capabilities of the ML based elasticity model. Thecomputation time with analytical hyperelastic model is 15 . s , whereas the computation time with theML based model is 19 . s on the same computer. W = 4 mm L = 10 mmq Figure 5: Compression of the plate. (a) Neo-Hookean model (b) With ML based model
Figure 6: Deformed state of the plate.To this end, it proves that the FNN works well for approximating the nonlinear elastic behaviouras shown in the above results. Elastic deformation is a history independent process and the stressdepends only on current kinematic variables. However, for plastic material behaviour, the response ofthe deformed material depends not only on the current deformation but also on its loading history. Sincethe FNN does not have any inherent ability to record loading history, the current approach needs to beimproved. Furthermore, the collection process of the training data for plasticity needs to be differentfrom elasticity.0 xy mm mm mmq Figure 7: Cook’s membrane problem. (a) Neo-Hookean model (b) With ML based model
Figure 8: Deformed state of the Cook’s membrane.
The aim of this part is to develop a data-driven material model which can be used to computationallyreproduce the plastic material behavior by means of machine learning tools. Collecting data fromexperiments is a key part for data-driven material model. In experiments, only the total strain andstress data of a specimen can be collected, which means the classical concept of elastic-plastic splittingto total strain can not be applied in the data-driven model. This leads to the questions: how to buildthe data-driven model using the total strain and stress data available from experiment? and what kind1of experiments have to be conducted to collect data?
Since the plastic flow depends not only on the current stress state but also on the loading history, theplastic deformation is a history dependent process. For 1D plastic deformations, the loading and un-loading curves do not coincide as shown in Fig. 9, where the strain and stress data are time series ofdata sets and can be seen as sequences. ε εO ε ε ε σ σ σ σ σσ σ ε ( ε )Figure 9: The loading and unloading curve for plasticity.Along a 1D loading-unloading path, the strain and stress data sets of one material point have a strictsequential order and can be written as two sets of corresponding sequences { ε , ε , ε , ..., ε t , ... } ⇔ { σ , σ , σ , ..., σ t , ... } , (11)where ε t and σ t are the total strain and stress at time t collected from the experiment. Thus, thebasic data unit for a plasticity model is not a strain-stress pair but a strain-stress sequence pair. Eachstrain-stress sequence pair refers to one loading-unloading path and thus sequence data collected fromdifferent loading-unloading paths are required to train a data-driven plasticity model.In machine learning, the classical equation of plasticity will be replaced with a ML based plasticitymodel driven by experimental data as σ t = f ML ( ε t , h t ) , (12)2where h t is a history variable for distinguishing loading history and ε t is the total strain. In this data-driven model, both the input and the output are sequence data. To build a ML based plasticity model,the history data as well as the strain-stress data have to be obtained from experiments.The choice of history variable is crucial for a successful prediction of sequences. In the literature, thestress and strain in the last step are applied as the history information together with the current strainin the input, see Hashash et al. [2004]. However, the previous strain and previous stress are not enoughto distinguish the loading history in real applications, since any error in the predicted previous stressby the ML tool will introduce an extra error into the system in an accumulate way. In this work, theaccumulated absolute strain is applied in the input as history variable. The accumulated absolute straincomponent ε tacc,j at time step t can be computed for the j -th strain component as h tj := ε tacc,j = (cid:40)(cid:80) ti =3 | ε i − j − ε i − j | , t (cid:62) , , t = 1 , , (13)where ε i − j and ε i − j are total strain components at time step i − i − | ε i − j − ε i − j | is the absolute increment ofstrain component j from time step i − i −
1, which is necessary to distinguish the loading historywhen tension and compression loadings are mixed, such as in loading-unloading path.For the 1D case, depicted in Fig. 9, a monotonic loading (e.g. from σ to σ ) and a mixed loading-unloading (e.g. from σ to σ ) may lead to the same total strain (e.g. ε = ε ). To distinguish monotonicloading from mixed loading-unloading, the absolute value of strain increment | ε i − − ε i − | is applied inequation (13), which leads to different accumulated absolute strain for different paths, e.g. ε acc = (cid:88) i =3 | ε i − − ε i − | (14)= (cid:88) i =3 | ε i − − ε i − | + (cid:88) i =5 | ε i − − ε i − | (15)= ε acc + (cid:88) i =5 | ε i − − ε i − | > ε acc . (16)Note that | ε i − − ε i − | is applied instead of | ε i − ε i − | in equation (13), since (cid:80) ti =3 | ε i − ε i − | is equalto ε t for monotonic loading, and it is not a history variable but the current input. The advantage ofapplying the accumulated absolute strain as the history variable is that it can be obtained from theexisting experimental data without the effort to collect them additionally. To collect the strain-stress sequence data from experiments, the loading paths required in experimentshave to be investigated. Since isotropic plasticity can be formulated in the principle strain-stress space,the ML based plasticity model can be formulated in the principle space as well, where the input andoutput of the model are principle stain and principle stress respectively. Therefore only the principlestrain and principle stress are required to be collected from the experiments.To collect the principle strain and principle stresses, the multi-axial loading tests can be conductedon special designed specimens, such as the biaxial experiments conducted by Mohr et al. [2010] and the3loading paths suggested by Goel et al. [2011]. The von Mises yield surface covering different stress statesis shown in Fig. 10 for the 2D case. In order to learn the plasticity behavior by e.g. artificial neuralnetworks, yielding as well as hardening effects have to be captured implicitly. To fully describe the stressstates existed in the deformed structures, the data have to be collected from several tests under differentloading-unloading paths. However, only biaxial tests for 2D and triaxial tests for 3D are required tocollect data in principle space. Equal biaxial tensionPure shearEqual biaxialcompression UniaxialTensionUniaxial compressionUniaxial compressionPure shearUniaxial tension σ σ O Figure 10: Stress states in 2D.In experiments, the principle strain and stress data can be collected within a homogeneous region atone point within a specimen, which can be descried by a quadrilateral region for 2D case depicted inFig. 11. The biaxial loading-unloading paths at this region can be characterized by the displacementsof the edges connected to point A . y xu x u y A A (cid:48)
Figure 11: Quadrilateral region within a specimen to collect data with biaxial loading.For each biaxial loading-unloading path, the point A moves from its original position to A (cid:48) for load-4ing and then goes back to original position for unloading, during which the displacements ( u x , u y ) willincrease linearly from zero to a specific value obeying (cid:113) u x + u y = r i , ( i = 1 , , ... ) and then decreaseto zero. max ( r i ) has to be large enough to capture as much loading range of plastic deformation aspossible. As shown in Fig. 12, the loading-unloading paths are just determined by setting a radius r i and different values of the angles φ . Since the unloading can start from different positions, multiplecircles with radius r i have to be applied in data collection. u y u x φO P i P j P k Loading-unloading path OP i r r r Figure 12: Loading-unloading paths for data collection in 2D.For the 3D case, data can be collected from a cubic region, as shown in Fig. 13, where the triaxialloading-unloading paths at this point can be characterized by the displacement ( u x , u y , u z ) at the point A .The loading-unloading path for the 3D case can be generated in the spherical coordinate system asshown in Fig. 14, where the displacement of point A obeys (cid:113) u x + u y + u z = r i , ( i = 1 , , , ... ). Theloading path OP i is distinguished by the angles φ and θ with radius r i . By looping the loading patharound the sphere, the whole range of stress states can be included.For example, along the loading path OP i in Fig. 14, the strain-stress sequence data will be collectedfirstly as ˆ ε = ε ε ... ε t ...ε ε ... ε t ...ε ε ... ε t ... × np , σ opi = σ σ ... σ t ...σ σ ... σ t ...σ σ ... σ t ... × np , (17)where np is the number of data point on the loading path OP i , ε ti ( i = 1 , ,
3) are the principle strains attime t measured in the cubic region within the specimen, σ ti ( i = 1 , ,
3) are the principle stresses at time t in that region, ˆ ε is the strain sequence and σ opi is the stress sequence of path OP i . Then the history5 zx u x u z A A (cid:48) yu y Figure 13: Cubic region within a specimen to collect data with triaxial loading. u x O u y u z θ P i Loading-unloading path OP i φ Figure 14: Loading-unloading paths for data collection in 3D.data, accumulated absolute strain ε tacc , are computed from the strain sequence ˆ ε using equation (13).The final strain sequence data of path OP i are obtained by combining the total strain sequence and the6history variable sequence as ε opi = ε ε ... ε t ...ε ε ... ε t ...ε ε ... ε t ...ε acc, ε acc, ... ε tacc, ...ε acc, ε acc, ... ε tacc, ...ε acc, ε acc, ... ε tacc, ... × np , (18)where ε tacc, , ε tacc, and ε tacc, are computed from the strain components ε t , ε t and ε t respectivelyaccording to equation (13). Finally, the input and output data are obtained by combining the sequencesfrom all of loading paths OP i , ( i = 1 , , ..., nl ) as I ε = (cid:2) ε op ε op ... ε opi ... (cid:3) × M , O σ = (cid:2) σ op σ op ... σ opi ... (cid:3) × M , (19)where nl is the number of loading path and M = np × nl . After data collection, the feed forward neural network is used to learn the relationship within the data.The neural network will approximate a mapping between inputs and outputs. However, the accuracy ofthe approximation depends on the complexity of the relationship. As a widely used technique in modelorder reduction, the Proper Orthogonal Decomposition (POD) provides an approach to decouple thetraining data, which simplifies the training and increases accuracy.
The Proper Orthogonal Decomposition (POD) in combination with machine-learning tools, such asGaussian Processes (Xiao et al. [2010]) and Long-Short-Term memory network (Mohan and Gaitonde[2018]), as a reduced order model has been used to surrogate model generation of fluid dynamic systems.Here we introduce a novel combination framework, where POD and FNNs are combined for preprocessingand prediction of sequence data. We call this approach Proper Orthogonal Decomposition Feed forwardNeural Network (PODFNN). Using POD, the time series vector variables can be represented with areduced number of modes neglecting higher order modes if the error is acceptable. Thus, by use of thePOD, the problem will be decoupled into a combination of several different modes.As time series data, the stress sequence in training data can be rewritten as a snapshot matrix O σ = (cid:2) σ op σ op ... σ opi ... (cid:3) = σ σ ... σ k ...σ σ ... σ k ...σ σ ... σ k ... × M , (20)where each column of the matrix is a snapshot and can be written as a vector o kσ = (cid:2) σ k σ k σ k (cid:3) T .Using POD, any snapshot o kσ can be represented by a linear combination of the basis o kσ = ¯ σ + m (cid:88) i =1 α ki ϕ i , (21)7where ϕ i = (cid:2) φ k φ k φ k (cid:3) T is the i -th basis vector, α ki is the coefficient, m is the number of POD modeand ¯ σ = (cid:2) ¯ σ ¯ σ ¯ σ (cid:3) T is the mean value vector of the snapshot matrix. Since ¯ σ , ϕ i are constants, andthe bases ϕ i are independent with each other, the stress sequence can thus be decoupled into independentcoefficient sequences.The components of mean value vector ¯ σ of the snapshot matrix are computed as¯ σ = 1 M M (cid:88) i =1 σ i , ¯ σ = 1 M M (cid:88) i =1 σ i , ¯ σ = 1 M M (cid:88) i =1 σ i . (22)To find the basis vectors and its coefficients, the deviation matrix is firstly computed as O devσ = σ − ¯ σ σ − ¯ σ ... σ k − ¯ σ ...σ − ¯ σ σ − ¯ σ ... σ k − ¯ σ ...σ − ¯ σ σ − ¯ σ ... σ k − ¯ σ ... × M . (23)Then, by applying Singular Value Decomposition (SVD) to the deviation matrix O devσ = U SV T , (24)where U and V are the unitary matrices, S is the diagonal matrix with non-negative real numbers onthe diagonal, the basis vectors ϕ i can be determined from the non-zero columns of matrix U Φ = (cid:2) ϕ ϕ ... ϕ m (cid:3) = (cid:2) u u ... u m (cid:3) × m , (25)where u m is the m -th non-zero column of matrix U and m is equal to the rank of O devσ .The coefficients α k = (cid:2) α k α k ... α km (cid:3) T can be obtained by projecting the snapshot o kσ on thebasis matrix Φ α k = Φ T o kσ . (26)Since the stress components are independent, the rank of matrix O devσ is 3 in this work ( m = 3). Thestress sequence in equation (19) will be represented by 3 independent coefficient sequences O σ = σ σ ... σ k ...σ σ ... σ k ...σ σ ... σ k ... × M P OD → (cid:2) α α ... α k ... (cid:3) × M (cid:2) α α ... α k ... (cid:3) × M (cid:2) α α ... α k ... (cid:3) × M , . (27)Therefore, the training data composed by the strain-stress sequences in equation (19) for plasticity modelis transformed into training data composed by strain-coefficient sequences and can be written as (cid:2) ε op ε op ... ε opi ... (cid:3) × M ⇔ (cid:2) α α ... α k ... (cid:3) × M (cid:2) α α ... α k ... (cid:3) × M (cid:2) α α ... α k ... (cid:3) × M , (28)where the three coefficient sequences are independent from each other.8 Once the training data is prepared, FNNs are applied to learn the mapping between the strain sequenceand the coefficient sequences in equation (28). Since the coefficients in the POD representation areindependent, each coefficient can be predicted by one FNN, as shown in Fig. 15, where the originalstrain-stress mapping approximated by one complex neural network is decoupled into three independentstain-coefficient mappings approximated by simpler neural networks. σ t σ t σ t FNN ε tacce, α ti ε tacce, ε tacce, ε t ε t ε t FNNi ( i = 1 , , . ) ε tacce, ε tacce, ε tacce, ε t ε t ε t Figure 15: Decoupling the strain-stress mapping (left) into independent strain-coefficient mappings(right) by POD.After training, the coefficient α tNN,i at time t will be predicted by the feed forward neural network F N N i as α tNN,i = F N N i ( ε t , ε tacc , w , b s ) , ( i = 1 , , , (29)where i indicates the number of the coefficient, ε t = (cid:2) ε t ε t ε t (cid:3) T is the current strain, ε tacc = (cid:2) ε tacc, ε tacc, ε tacc, (cid:3) T is the accumulated absolute strain, w is the weight matrix and b s is the bias ofneural network. Once the coefficient α tNN is predicted by FNNs as described in equation (29), the principle stress canbe recovered from the POD representation as˜ σ tP ODF NN = ¯ σ + (cid:88) i =1 α tNN,i ϕ i . (30)Finally, the Cauchy stress is obtained by transforming the principle stress into the general space σ tP ODF NN = Q ˜ σ tP ODF NN Q T . (31)The formulation of the ML based plasticity model defines a constitutive function in the following format σ tP ODF NN = P ODF N N ( ε t , ε tacc , w , b s ) , (32)where ε t is the current total strain, ε tacc is the history variable. To apply the ML based plasticity modelin the finite element analysis, the residual vector and tangent matrix have to be derived. The automaticsymbolic differentiation tool AceGen is applied to derive the tangent matrix again.9 In the following, the performance of the developed ML based plasticity model is evaluated using finiteelement applications.
Apart from collecting the training data from experiments, simulation data using the von Mises plasticitymodel can be collected as well to train the ML tool, in this way the performance of the ML basedplasticity model can be verified by comparing to the analytical model. In this work, the strain-stresssequences are collected at a Gauss point of a finite element under specific loading paths described above.The strain is computed as the symmetric part of the displacement gradient for small deformations ε = 12 ( H + H T ) , (33)where H is the displacement gradient with H = Grad u . Using spectral decomposition, the strain canbe formulated as ε = Q · Λ · Q T , (34)where Λ is the principle strain and Q is the rotation matrix obtained from the eigendirections. Theprinciple strain Λ along different loading paths will serve as input data to train the ML based plasticitymodel.For small strain plasticity, the additively decomposition of strain into elastic part and plastic part isassumed Λ = Λ e + Λ p , (35)where Λ e and Λ p are the elastic strain and plastic strain respectively. The plastically admissible stressis given by Σ = ρ ∂ψ∂ Λ e , (36)where ψ is the free energy function. The principle stress Σ will serve as the output data, correspondingto the principle strain as input. By assuming the von Mises yield criteria, the yield function is writtenas f = (cid:114) (cid:107) Σ dev (cid:107) − σ y ( α ) , (37)where Σ dev is the deviatoric stress with Σ dev = Σ − tr Σ · and α is the isotropic hardening variable.By use of the associated plastic flow rule, the evolution equations for the principle plastic strain and thehardening variable are formulated as ˙ Λ p = ˙ γ ∂f∂ Σ dev , ˙ α = ˙ γ ∂f∂A (38)where γ is the plastic multiplier and A is the thermodynamic force conjugate with α . The plastic flowhas to full fill the Kuhn-Tucker conditions f (cid:54) , ˙ γ (cid:62) , ˙ γf = 0 . (39)0 To test the performance of the ML based plasticity model, the 1D uniaxial tension and compression testis conducted firstly. The von Mises plasticity with the linear isotropic hardening is applied as the targetmodel. The material parameters of the plasticity model are set as: Young’s modulus E = 700 N/mm ,yield stress σ y = 100 M P a and isotropic hardening parameter H iso = 10. To prepare the training data,11 sets of strain-stress sequence data are collected from the target model with strain increments beingincreased linearly from 0 .
02 to 0 .
03. The stress sequences are then transformed into the coefficientsequence by the POD.The feed forward neural network with the architecture of (2-20-20-1) is applied to predict the coef-ficient, where the input layer containing 2 neurons is connected with two hidden layers containing 20neurons each. The output layer contains one neuron. The input of the neural network is the total straintogether with the accumulated absolute strain, and the output of the neural network is the coefficienttransformed from the stress sequence. The Levenberg-Marquardt algorithm is applied as the optimizerin training. The weights are initialized by the Nguyen-Widrow method. After 4000 epochs, the meansquared error decreased to 0 . S t r e ss Strain PlasticityNN modelFigure 16: Uniaxial tension and compression.1
To evaluate the ML based plasticity model, benchmark tests in 2D are presented. The von Misesplasticity with an exponential isotropic hardening law σ y = y + y (0 . γ ) . is set as the targetmodel. The material parameters are set as: Young’s modulus E = 1 N/mm , Poison’s ratio ν = 0 . y = 0 . M P a . To collect the training data, 122 loading-unloading pathsevenly distributed within the circles ( r = 0 . , r = 0 . φ . Then the collected stress sequence data is transformedinto the coefficient sequences using POD.Since there are two coefficients referring to the two principle stress components in the 2D case, twoFNNs will be required to predict the coefficients. In this part, the same network architecture (4-20-20-1) is applied for the two FNNs, where the input layer containing 4 neurons is connected with twohidden layers containing 20 neurons each. The output layer contains always one neuron. The total straintogether with the accumulated absolute strain are applied as the input of the neural network. The outputof the neural network is the coefficient transformed from the stress sequences. The Levenberg-Marquardtalgorithm is applied as the optimizer as well. The training progress is terminated when the gradient oferror is less than 10 − , where the mean squared error is decreased to 7 . × − for the first FNN and6 . × − for the second FNN.After the training process, the weighs and biases of the neural network are output as the constantmodel parameters, by which the Cauchy stress is recovered according to the POD formulation. Thetangent matrix and the residual vector are derived using the symbolic differentiation tool AceGen again.Firstly, the 2D ML based plasticity model is tested by the Cook’s membrane problem. The beam isclamped at the left end and loaded at the right end by a constant distributed vertical load q = 0 . M pa ,as depicted in Fig. 7. In the unloading process, the direction of vertical load is changed to be negative.The geometric domain of the structure is discretized by 40 quadratic 9-node quadrilateral elementsleading to 189 nodes.Before unloading, the final deformation state of the beam using the proposed ML based plasticitymodel is compared with the target plasticity model, which is depicted in Fig. 17. It can be observedthat the vertical displacement of the structure is almost identical.The load displacement curve of the upper node (48 ,
60) at the right end of the cantilever beam isplotted in Fig. 18. The figure shows that the ML based plasticity model captures the loading andunloading behaviour very well.The second example to test the ML based plasticity model is a punch test as shown in Fig. 19, wherethe vertical displacement boundary condition ( u = 0 . mm ) is imposed on the top of the block andthe bottom of the block is only fix in the vertical direction. In the unloading process, the direction ofvertical displacement boundary is changed to be positive. The block is discretized with 100 quadratic9-node quadrilateral elements leading to 441 nodes.Before unloading, the final deformation state of the block with ML based plasticity model is comparedwith that using the target plasticity model, which are depicted in Fig. 20. It can be observed that thehorizontal displacement of the structure is very close for the two models.The load displacement curve of the upper node (0 ,
1) at the left end of the block is plotted in Fig.21, where it can be seen that the ML based model follows the plasticity model well both in loading and2 (a) With plasticity model (b) With NN model
Figure 17: Final deformation state of the 2D Cook’s membrane.00.010.020.030.04 0 2 4 6 8 10 12 F o r ce F y Displacement
U y
PlasticityNN modelFigure 18: Load deflection curve of the 2D Cook’s membrane.unloading.
In this section, the ML based plasticity model is extended to 3D applications. To generate the trainingdata, one hexahedron finite element is applied to different loading situations as described in Fig. 13. Thevon Mises plasticity with an exponential isotropic hardening law σ y = y + y (0 . γ ) . is set as31 mm u mm xy Figure 19: 2D punch problem. (a) With plasticity model (b) With NN model
Figure 20: Final deformation state of the 2D block.the target model. The material parameters are set as: Young’s modulus E = 10 N/mm , Poison’s ratio ν = 0 .
33, and the initial yield stress y = 0 . M P a . During the training data preparation, strain-stresssequences along 8100 loading paths are generated based on the sphere ( r = 0 .
02) in Fig. 14, where 90values are assigned to the angles φ and θ respectively. Since the huge amount of data have to be collectedfor unloading in 3D, only the loading data is collected here and the unloading is not considered in thispart.In the three-dimensional case, three FNNs are required to predict the coefficients, which are cor-responding to the principle Cauchy stress components. The same network architecture (6-16-16-1) isemployed for all FNNs, where three total strain components together with three accumulated absolutestrains are applied as the input of the networks. The output of the neural network is the coefficient.The weights are initialized by the Nguyen-Widrow method. During training, the Levenberg-Marquardt400.00020.00040.00060.00080.0010.0012 0 0.02 0.04 0.06 0.08 F o r ce F y Displacement
U y
PlasticityNN modelFigure 21: Load deflection curve of the 2D block.algorithm is applied as the optimizer, where the training progress is terminated when the gradient ofglobal error is less than 10 − . The mean squared errors are decreased to 5 . × − , 1 . × − and 6 . × − for the first, second and third FNN respectively. The training process costs time of1 h m s , 1 h m s and 52 m s for the first, second and third FNN respectively.To evaluate the performance of the POD representation, the performances of training strain-stressmodel with one FNN(6-16-16-3) and training three strain-coefficient models with three FNNs(6-16-16-1)are compared. Without POD, only one FNN is required to approximate the mapping, where the outputincludes 3 stress components. With POD, three independent FNNs will be applied, where the outputof FNN includes only one POD coefficient. The training performances within 2000 epochs are shown inFig. 22. It can be seen that the average of the mean squared errors of the three FNNs is smaller thanthat without POD. Additionally, training a FNN with architecture of (6-16-16-3) costs computation timeof 4 h m s whereas the average training time of the three FNNs (6-16-16-1) is 1 h m
27. It can beobserved that the POD approach leads to less training time and better training performance.The first example to test the 3D machine learning based plasticity model is the necking of a bar asshown in Fig. 23, where the left end of the bar is fixed and the displacement boundary u = 0 . mm is imposed at the right end along its axial direction. An artificial imperfection is set in the center ofthe bar to trigger the necking, where the radius at the center is chosen to be R c = 0 . R . The bar isdiscretized with 200 quadratic 27-node elements leading to 2193 nodes.Fig. 24 shows the final deformation of the bar after tension, where only one quarter of the bar iscomputed due to the symmetry. It can be observed that the amounts of the necking computed by thetwo models are close to each other.The load displacement curve of the bar under the uniaxial tension is plotted in Fig. 25, where the5 Epoch -8 -7 -6 -5 -4 M ean S qua r ed E rr o r ( M SE ) Figure 22: MSE of training FNN(6-16-16-3) and the average MSE of training 3 FNNs(6-16-16-1). L = 10 mmR = 0 . y zx u z = u u z = 0 Figure 23: Geometry and boundary conditions of the bar.neural network based model follows the plasticity model quite well.The second example is the punch test, where the vertical displacement boundary condition ( u =0 . mm ) is imposed on the top of the block and the bottom of the block is only fixed in the verticaldirection, as shown in Fig. 26. The block is discretized with 100 quadratic 27-node elements leading to441 nodes.Fig. 27 shows the final deformation of the block after compression. It can be observed that thedisplacements in the horizontal direction computed by the two models are close to each other. The loaddisplacement curve of the block under compression is plotted in Fig. 28. It can be seen that the neuralnetwork based model follows the plasticity model quite well.The last example for the 3D neural network based model is the Cook’s membrane problem. The3D beam is clamped at the left end and loaded at the right end by a constant distributed vertical load q = 0 . M P a , as depicted in Fig. 29. By use of the quadratic finite element with 27 nodes, the beamis discretized using 1080 elements leading to 10309 nodes.6 (a) With plasticity model(b) With NN model
Figure 24: Final deformation state of the cylindrical bar.0.0e01.0e-42.0e-43.0e-44.0e-4 0 0.01 0.02 0.03 0.04 0.05 F o r ce F z Displacement U z PlasticityNN modelFigure 25: Load deflection curve of the cylindrical bar.Fig. 30 shows the final deformation of the Cook’s membrane. It can be observed that the displace-ments in the vertical direction computed by the two models are almost identical. The load displacementcurve of the upper node (48,0,60) is plotted in Fig. 31. It can be seen that the machine learning based71 mm u mm mmxz Figure 26: 3D punch problem. (a) With plasticity model (b) With NN model
Figure 27: Final deformation state of the 3D punch problem.model follows the plasticity model quite well.
In this work, a machine learning based material modelling approach for hyper-elasticity and plasticityis proposed. Common tools such as FNNs show proficient performances for capturing the mappingbetween strain and stress in the case of elasticity. However, history variables are required to distinguishthe loading history in case of plasticity. In this context, FNNs show subpar performances. Thus, theaccumulated absolute strain is proposed to be the history variable, which captures the loading historywell without requirement for additional data. Here we present a novel method called Proper Orthogonal80.0e02.0e-44.0e-46.0e-48.0e-41.0e-31.2e-3 0 0.02 0.04 0.06 0.08 0.1 0.12 F o r ce F z Displacement U z PlasticityNN modelFigure 28: Load deflection curve of the 3D punch problem. xz mm mm mmq mmyu z = 0 u y = 0 Figure 29: 3D Cook’s membrane problem.Decomposition Feed forward Neural Network (PODFNN), which in combination with the introducedhistory variable is able to overcome this problem. By use of the POD, less training time and bettertraining performance are obtained in the network training. Additionally, it has been shown that thetraining data collected only from the multi-axial loading tests are enough to capture the von Misesyield surface and the hardening law. The automatic symbolic differentiation tool AceGen provides avery convenient way to derive the tangent matrix for the machine learning based material model. The9 (a) With plasticity model (b) With NN model Figure 30: Final deformation state of the 3D Cook’s membrane.00.040.080.120.16 0 2 4 6 8 10 12 F o r ce F z Displacement U z PlasticityNN modelFigure 31: Load deflection curve of the 3D Cook’s membrane.generalization and accuracy of the presented model as well as the data generation strategy have beenverified by finite element applications both in 2D and 3D.
Acknowledgements
The first author would like to thank the China Scholarship Council (CSC) and the Graduate Academyof Leibniz Universit¨at Hannover for the financial support. The second author acknowledges the financial0support from the Deutsche Forschungsgemeinschaft under Germanys Excellence Strategy within theCluster of Excellence PhoenixD (EXC 2122, Project ID 390833453). The last author acknowledgesthe support of Deutsche Forschungsgemeinschaft for the project C2 within the collaborative researchcenter/Transregio TR73.1
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