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A machine learning based plasticity model using proper orthogonal decomposition
Dengpeng Huang, Jan Niklas Fuhg, Christian Weißenfels, Peter Wriggers
TL;DR
History-dependent plasticity remains difficult for machine-learning material models because load history must be represented while supporting finite-element use. The paper proposes PODFNN with accumulated absolute strain and sequential loading data, and reports validated accuracy and generalization in 2D and 3D finite-element applications.
Problem
Machine-learning models must represent loading history accurately when predicting history-dependent plasticity from available material data.
Method
The framework uses accumulated absolute strain as the plasticity history variable and PODFNN to transform stress sequences into independent coefficient sequences learned by separate neural networks.
Results
The model's generalization and accuracy, together with its data-generation strategy, were verified through finite-element applications in both 2D and 3D.
Takeaways & Limitations
POD reduces training time and improves training performance, while the accumulated absolute strain captures loading history without requiring additional data.
Takeaways & Limitations
The approach assumes additive elastic-plastic strain decomposition for small-strain plasticity and depends on the quality and completeness of the data.
Abstract
from arXiv · showhide
Data-driven material models have many advantages over classical numerical approaches, such as the direct utilization of experimental data and the possibility to improve performance of predictions when additional data is available. One approach to develop a data-driven material model is to use machine learning tools. These can be trained offline to fit an observed material behaviour and then be applied in online applications. However, learning and predicting history dependent material models, such as plasticity, is still challenging. In this work, a machine learning based material modelling framework is proposed for both elasticity and plasticity. The machine learning based hyperelasticity model is developed with the Feed forward Neural Network (FNN) directly whereas the machine learning based plasticity model is developed by using of a novel method called Proper Orthogonal Decomposition Feed forward Neural Network (PODFNN). In order to account for the loading history, the accumulated absolute strain is proposed to be the history variable of the plasticity model. Additionally, the strain-stress sequence data for plasticity is collected from different loading-unloading paths based on the concept of sequence for plasticity. By means of the POD, the multi-dimensional stress sequence is decoupled leading to independent one dimensional coefficient sequences. In this case, the neural network with multiple output is replaced by multiple independent neural networks each possessing a one-dimensional output, which leads to less training time and better training performance. To apply the machine learning based material model in finite element analysis, the tangent matrix is derived by the automatic symbolic differentiation tool AceGen. The effectiveness and generalization of the presented models are investigated by a series of numerical examples using both 2D and 3D finite element analysis.
1 Introduction
The section motivates machine-learning material models as alternatives for improving data use and prediction, while identifying unresolved challenges in generalization, history dependence, data collection, and finite-element integration. The paper addresses these challenges with a history variable, sequential plasticity data, and PODFNN decomposition.
- Existing material models can have limited generalization or accuracy when applied to engineering problems.
- Machine-learning material models approximate constitutive equations from experimental data without prescribing a specific constitutive model.
- Previous neural-network approaches use prior strain and stress as history variables, creating elastic-stage errors and complicating load-history capture.
- The proposed framework uses accumulated absolute total strain as a plasticity history variable and collects sequential strain-stress data from different loading-unloading paths.
- PODFNN decomposes the strain-stress relationship into independent one-output neural networks, while AceGen derives the tangent matrix for finite-element applications.
2 Data-driven material modelling framework
The framework collects and preprocesses material data, fits constitutive mappings with artificial neural networks, and validates the resulting model in finite-element applications. Its open-system workflow supports iterative improvement through hyperparameter optimization or supplemental data.
- Data-driven modelling proceeds through data collection, machine learning, and validation using experimental or simulated material behaviour.
- Plasticity strain-stress data are collected for specific loading paths and stored as sequences before preprocessing.
- Artificial neural networks fit the constitutive equation, with hyperparameters selected according to the data and required accuracy.
- Validation compares the machine-learning material model with a standard model in finite-element applications using a derived tangent matrix and residual vector.
- The framework is an open system whose accuracy can be improved iteratively by optimizing hyperparameters or collecting supplemental data.
3 Machine Learning (ML) based hyperelasticity
The section develops a feed-forward-neural-network model for hyperelasticity and integrates it into finite element analysis. The model learns a history-independent strain–stress mapping and produces deformations close to analytical solutions in two numerical tests.
- 3.1 Feed forward neural network: FNNs propagate inputs through connected neuron layers, while training back-propagates mean-squared error to update weights and minimize global error.The architecture depends on dataset complexity, including the number of layers and neurons.
- 3.3 Data collection for the ML based hyperelasticity: Hyperelasticity is approximated with an FNN because elastic strain–stress behavior is a one-to-one mapping independent of loading history.The network uses strain components as inputs and stress components as outputs.
- 3.4 The residual and tangent: The model is embedded in finite element analysis through a residual based on predicted stress, with the tangent matrix derived automatically using AceGen.Automatic differentiation avoids complex manual chain-rule calculations for large networks.
- 3.5 Testing the ML based hyperelasticity model in FEM: After 14082 training iterations, the 4-10-3 FNN reached mean squared error 0.0326, requiring 6h40m55s of training.The training data came from a neo-Hookean model with E = 700N/mm2 and ν = 0.499.
- 3.5 Testing the ML based hyperelasticity model in FEM: In plate compression, ML and neo-Hookean vertical displacements were very close, with computation times of 9.75s and 8.14s, respectively.The comparison used the same computer and a distributed load q0 = −20MPa.
- 3.5 Testing the ML based hyperelasticity model in FEM: In Cook’s membrane, the same trained model produced vertical displacements very close to the analytical model, supporting generalization to a second test case.Computation times were 19.88s for the ML model and 15.72s for the analytical model.
4 Data collection strategy for plasticity
The plasticity model uses strain-stress sequences from multiple loading-unloading paths because plastic deformation depends on loading history. Accumulated absolute strain distinguishes paths using experimentally available total-strain data, while biaxial and triaxial paths sample 2D and 3D stress states.
- Sequence concept for plasticity: Plasticity data are organized as corresponding strain-stress sequence pairs, with each pair representing one loading-unloading path.Training requires sequence data collected from different loading-unloading paths.
- History variable: Accumulated absolute strain is used as the history variable to distinguish loading paths from total-strain data.It captures differences between monotonic and mixed loading-unloading paths and can be computed from existing experimental inputs.
- History variable: The history variable uses absolute strain increments because mixed tension and compression loadings require path-sensitive accumulation.The absolute increment distinguishes loading histories when tension and compression are combined.
- 2D data collection: For 2D plasticity, strain and stress data are collected from a homogeneous quadrilateral region under multiple biaxial loading-unloading paths.Paths vary by radius and angle, and multiple circles are needed because unloading can begin from different positions.
- 3D data collection: For 3D plasticity, a cubic region is sampled along triaxial paths generated in spherical coordinates to cover the stress-state range.Path directions are defined by angles and radii, and strain-stress sequences are collected along each path.
5 Machine learning based plasticity
Feed forward neural networks are used to learn material mappings, while Proper Orthogonal Decomposition decouples complex training data before prediction. This preprocessing simplifies the relationship presented to the networks and increases training accuracy.
- PODFNN framework: Proper Orthogonal Decomposition decouples training data before FNN prediction, simplifying the learned relationship and increasing approximation accuracy.The approach is motivated by the dependence of FNN accuracy on relationship complexity.
5.1 Decouple the stress data by POD
The stress sequence is represented as a snapshot matrix and transformed with POD based on its singular-value decomposition. This yields a reduced set of independent coefficient sequences for learning.
- POD representation: POD represents time-series stress data with a reduced number of modes when the resulting approximation error is acceptable.Higher-order modes can be neglected under the chosen error tolerance.
- Snapshot matrix: The stress sequence is organized as a snapshot matrix whose columns are individual stress snapshots.The deviation matrix is then formed to compute POD basis vectors and coefficients.
- Coefficient sequences: The stress sequence is decoupled into independent coefficient sequences because the POD bases are independent.In this work, three independent coefficient sequences represent the stress sequence.
- POD basis construction: Singular Value Decomposition supplies unitary matrices and a diagonal matrix used to determine the POD basis vectors.The basis vectors are obtained from non-zero columns of the SVD matrix U.
5.2 Prediction of coefficients using FNN
After POD preprocessing, separate FNNs learn the mappings from strain sequences to individual coefficient sequences. The predicted coefficients are subsequently used to reconstruct the principle stress.
- Coefficient prediction: Each independent POD coefficient is predicted by a separate one-output FNN instead of one complex multi-output network.This decouples the original strain-stress mapping into simpler strain-coefficient mappings.
- Network inputs and outputs: The FNN inputs include accumulated absolute strain, and each network outputs its corresponding POD coefficient at the current time.The accumulated absolute strain is explicitly included in the network input.
5.3 PODFNN based plasticity model
The PODFNN plasticity model represents the constitutive relation through POD-based coefficient predictions and recovers stress for finite element use. Its tangent matrix is derived automatically with AceGen.
- The predicted stress is reconstructed from the POD representation and then transformed from principal to general space to obtain Cauchy stress.
- PODFNN formulates the plasticity model through a constitutive function mapping strain and history information to stress.
- AceGen automatically derives the residual vector and tangent matrix required to apply the machine learning plasticity model in finite element analysis.
5.4 Testing the ML based plasticity model in FEM
The machine learning plasticity model is trained from strain–stress sequences generated along prescribed loading paths and evaluated in uniaxial, 2D, and 3D finite element tests. Across these tests, its responses closely follow the target plasticity model, while POD reduces training cost and error.
- Data collection and model setup: Training sequences are generated from loading paths using principal strain as input and principal stress as output, with small-strain plasticity assumptions and von Mises constitutive relations.
- Uniaxial tension and compression: The uniaxial model uses total strain and accumulated absolute strain as inputs to an FNN that predicts a POD-transformed stress coefficient.
- Uniaxial tension and compression: The predicted stress follows the exact target solution for a testing strain increment of 0.15, supporting the history variable’s ability to capture cyclic loading.
- 2D finite element applications: The 2D Cook’s membrane model produces nearly identical final vertical displacement and captures the target model’s loading and unloading curve well.
- 2D finite element applications: The 2D block model gives close horizontal displacements and follows the target load-displacement response during loading and unloading.
- POD training performance: POD training lowers average time from 4h22m43s for one three-output FNN to 1h26m27s for three one-output FNNs, while also reducing mean squared error.
- 3D finite element applications: In 3D applications, the machine learning model closely matches target displacements and load-displacement responses for punch, Cook’s membrane, and bar-necking problems.
6 Conclusions
The paper proposes a machine learning framework for hyperelasticity and plasticity that combines accumulated absolute strain with PODFNN. The approach reduces training time, improves training performance, and is validated in 2D and 3D finite element applications.
- The proposed framework uses FNNs for hyperelasticity and introduces accumulated absolute strain as a history variable for plasticity.
- PODFNN combines the history variable with POD to address plasticity’s loading-history dependence using simpler coefficient-prediction networks.
- POD produces less training time and better training performance for the neural-network material model.
- Training data collected from multiaxial loading tests is reported to capture the von Mises yield surface and hardening law.
- AceGen provides automatic symbolic differentiation for deriving the tangent matrix needed in finite element applications.
- The model’s generalization and accuracy are verified through finite element applications in both 2D and 3D.