PaperlayerView exact revision on arXiv

Source-linked AI summary

Physics informed neural networks for continuum micromechanics

Alexander Henkes, Henning Wessels, Rolf Mahnken

arXiv:2110.07374v2cs.LG

TL;DR

The paper studies PINNs for elastostatic continuum micromechanics, where sharp material interfaces challenge global neural-network approximations. It introduces adaptive training and domain decomposition, showing that cPINNs can resolve complex nonlinear fields in heterogeneous microstructures.

  • Problem

    Sharp material phase transitions make global PINN approximations difficult to optimize, while inhomogeneous elastostatics also requires material-parameter derivatives and displacement second derivatives.

  • Method

    The paper develops a data-free PINN approach for linear elastic continuum micromechanics using strong-form equations, multi-output fields, loss scaling, hard boundary conditions, adaptive sampling, and domain decomposition.

  • Results

    The cPINN approach accurately resolves complex nonlinear stress, displacement, and energy fields in heterogeneous microstructures, including real-world μCT-scan geometries.

  • Takeaways & Limitations

    PINN and cPINN methods can capture complex elastostatic physics and highly nonlinear tensorial fields in continuum micromechanics.

  • Takeaways & Limitations

    The approaches’ full potential remains uninvestigated, leaving uncertainty quantification and inverse problems for future work.

Abstract

from arXiv · show

Recently, physics informed neural networks have successfully been applied to a broad variety of problems in applied mathematics and engineering. The principle idea is to use a neural network as a global ansatz function to partial differential equations. Due to the global approximation, physics informed neural networks have difficulties in displaying localized effects and strong non-linear solutions by optimization. In this work we consider material non-linearities invoked by material inhomogeneities with sharp phase interfaces. This constitutes a challenging problem for a method relying on a global ansatz. To overcome convergence issues, adaptive training strategies and domain decomposition are studied. It is shown, that the domain decomposition approach is able to accurately resolve nonlinear stress, displacement and energy fields in heterogeneous microstructures obtained from real-world $μ$CT-scans.

1. Introduction

The paper develops PINNs for inhomogeneous continuum micromechanics, where heterogeneous microstructures and sharp phase transitions challenge global neural-network approximations. It studies adaptive sampling and domain decomposition, including validation on a real-world wood-plastic-composite µCT scan.

  • Motivation: Heterogeneous microstructures increase computational demands because macroscopic behavior depends on the underlying material distribution.This requires numerical resolution of displacement, stress, and strain tensor fields when analytical solutions are unavailable.
  • Motivation: FEM remains widely used, but multiscale problems, uncertainty quantification, and inverse problems remain challenging.
  • Related methods: PINNs reduce data requirements by solving governing PDEs using initial and boundary conditions rather than large training datasets.
  • Research gap: The paper targets displacement and stress fields from inhomogeneous material distributions without additional data, addressing sharp parameter jumps and highly nonlinear solutions.Exact microscale stress fields are important for full-field homogenization.
  • Contributions: The proposed framework combines PINN elastostatics with adaptive collocation sampling and domain decomposition to improve convergence.The study also discusses topology, loss calculation, boundary conditions, and scaling methods.
  • Validation: A material network smooths voxelized µCT data, enabling the method to solve an inhomogeneous elastostatic boundary-value problem for a wood-plastic composite.

2. Governing equations of linear elastic micromechanics

The paper formulates linear elastic micromechanics as an elastostatic boundary-value problem on a symmetric, zero-centered square unit cell with spatially varying material parameters. Displacement, strain, stress, constitutive behavior, and boundary conditions are linked through the governing equations.

  • Problem setting: The governing PDEs describe elastostatics under linear elastic material behavior on a symmetric, zero-centered square unit cell.
  • Boundary conditions: Dirichlet sections prescribe displacements, whereas Neumann sections prescribe tractions related to stress and the boundary normal.
  • Constitutive structure: The displacement field relates to strain through the kinematic relation, while stress and strain are connected by a linear-elastic constitutive law.
  • Material law: Lamé’s first and second constants, λ(x) and µ(x), are scalar fields representing spatially varying material properties.
  • Governing fields: The displacement field u(x) is the primary variable in the balance of linear momentum.
  • Inhomogeneous formulation: For inhomogeneous materials, substituting the material law and kinematic relation into momentum balance produces the inhomogeneous Navier-Cauchy equation, including chained derivatives.

3. Artificial neural networks and physics informed neural networks

The paper represents PDE solutions with feed-forward neural networks and trains them through collocation-based minimization of PDE and boundary residuals. Hard boundary conditions instead encode admissibility directly into the network and leave only the PDE residual to optimize.

  • PINNs: PINNs use global neural-network ansatz functions to approximate solutions of boundary-value problems, unlike traditional methods using local ansatz functions.This produces a global solution through neural-network training.
  • Artificial neural networks: The paper uses densely connected feed-forward ANNs that compose nonlinear layers mapping input vectors x to output vectors y.
  • Artificial neural networks: An ANN topology is determined by its layers, neural units, and activation functions.
  • Optimization: PINN training discretizes the domain and boundary into collocation points and optimizes network parameters against an objective function.
  • Optimization: The PINN loss combines a PDE residual with a boundary-condition discrepancy, whose spatial derivatives are obtained by automatic differentiation.
  • Boundary conditions: With hard boundary conditions, optimization simplifies to minimizing only the PDE residual rather than a separate boundary loss.
  • Boundary conditions: Hard boundary conditions use N(x; θ) = G(x) + D(x) ˜N(x; θ), where D(x) vanishes on the boundary and automatically suppresses the unrestricted network output.

4. Physics informed neural networks for linear elasticity

The PINN directly solves the strong-form linear elastostatic problem by predicting displacement and stress fields, with residual-based losses and internally scaled quantities. For heterogeneous materials, smoothing material jumps enables qualitative global solutions, while interface errors remain and motivate further techniques.

  • The PINN directly solves the linear elastostatic balance law in strong form.
  • The network can output both displacement and stress components, avoiding second-order displacement derivatives in the balance law.In two dimensions, the outputs comprise ux, uy, σxx, σyy, and σxy.
  • With hard boundary conditions, the loss contains residual terms for local balance, constitutive and kinematic relations, and global work balance.The work-balance term enforces global conservation, while the other residuals represent local laws.
  • Internal rescaling places displacement, stress, and material parameters on comparable numerical scales during loss calculation.Without loss scaling, the method cannot converge for large material contrasts; scaling changes optimization weighting rather than the linear PDE itself.
  • 4.3.1. Homogeneous plate - PINN: 2.84221187 × 10^-6 is the homogeneous plate’s L2-norm of work-balance loss, indicating energy conservation despite marginally non-homogeneous contour fields.
  • 4.3.2. Single inclusion - PINN: For a single inclusion, the PINN captures material inhomogeneity, but the highest residual occurs at phase interfaces and the sudden material jump remains problematic.The reported maximum residual is max(R) = 1.782237 × 10^-2, while the mean residual is E(R) = 4.1985224 × 10^-4.

5. Adaptivity and domain decomposition

The paper studies adaptive collocation and conservative PINNs with domain decomposition to reduce errors and optimization difficulty near phase transitions. Domain decomposition consistently outperforms standard PINNs, whereas the proposed adaptive scheme becomes non-convergent at higher collocation-point counts.

  • 5.1. Adaptive collocation points: Adaptive collocation targets points with large errors, following an h-refinement strategy motivated by local mesh refinement in FEM.The method combines regular and adaptively selected points, with adaptive samples chosen from random points according to their error measures.
  • 5.1. Adaptive collocation points: Gradient clipping stabilizes adaptive training by limiting large optimization gradients after each adaptive loop begins.The gradients are scaled when they exceed a threshold, preventing exploding gradients and large changes to already learned weights.
  • 5.2. Conservative PINNs: cPINNs localize the solution by assigning a dedicated ANN to each subdomain and assembling the global solution from local subnet solutions.The domain is decomposed into N subdomains, while interface conditions enforce solution equality and flux continuity across shared boundaries.
  • 5.2. Conservative PINNs: Interface residuals for solution and flux continuity are added to the total cPINN loss, with an interface weight factor ψ enforcing compatibility.Hard boundary conditions must also be adapted for ANNs sharing portions of the external boundary.
  • 5.3. Convergence comparison: Adaptive variants become non-convergent from nd = 642 collocation points upward, with adaptive cPINNs having the highest errors for nd > 82.The authors attribute this behavior to excessive resolution and increased optimization instability from the adaptive algorithm’s dynamic nature.

6. µCT-scan of a heterogeneous microstructure

The paper applies cPINNs to a two-phase wood-plastic-composite microstructure reconstructed from a filtered and binarized µCT scan. A learned material network provides a smooth material distribution, and the 4x4 cPINN resolves the resulting heterogeneous fields with lower residuals than standard PINNs.

  • 6. µCT-scan of a heterogeneous microstructure: The µCT scan represents a two-phase wood-plastic composite comprising a polymer matrix and wood short-fiber inclusions.Gaussian filtering and binarization convert the noisy voxelized grayscale image into a two-phase material representation.
  • 6.1. Material network: A material network learns the spatial material phase from coordinate–phase pairs and provides a smooth, infinitely differentiable extension of voxelized image data.Its output is constrained between λmin and λmax for λ(x), with an analogous formulation for µ(x), so the predicted parameters remain physically admissible.
  • 6.1. Material network: The material network produces a smooth prediction of the µCT-derived material distribution.The predicted extension is shown alongside the filtered and binarized scan.
  • 6.2.1. Domain split - cPINN: The 4x4 cPINN is selected for the µCT calculation because it balances localization against the added optimization complexity of additional subnetworks and interface terms.Increasing the split from 4x4 to 5x5 raises the subnet count from 16 to 25 and causes an error jump.
  • 6.2.2. µCT-scan - cPINN: The µCT calculation uses a 4x4 cPINN with θ ≈ 2.7 × 10^4 parameters and nd = 256^2 regular-grid collocation points.The training uses up to 2 × 10^4 iterations; reported residual statistics include max(R) = 9.249407 and mean E(R) = 1.20354064 × 10^-1.

7. Conclusion and outlook

The proposed PINN and cPINN approaches address nonlinear elastostatics in heterogeneous microstructures by combining multi-output fields, loss scaling, hard boundary conditions, work balance, adaptive sampling, and domain decomposition. Numerical examples show accurate field resolution, while the method remains computationally less competitive than FEM or FFT for large forward problems and leaves extensions such as uncertainty quantification and inverse problems open.

  • Method: The multi-output PINN predicts stress and displacement fields directly, avoiding second-order derivatives in the governing equations.Loss scaling addresses multi-scale residuals, while hard boundary conditions simplify optimization and permit smaller networks.
  • Method: Work balance adds a global constraint to point-wise residuals, and adaptive sampling targets difficult regions near sharp material transitions.These strategies are designed to improve optimization and accuracy for heterogeneous material parameters.
  • Numerical results: The cPINN with regular sampling achieved the highest accuracy among the compared standard, adaptive, and decomposed PINN variants.Adaptive sampling performed better with few collocation points but became less effective on finer grids.
  • Numerical results: The cPINN resolved complex stress, displacement, and energy fields in heterogeneous microstructures, including a real-world µCT-scan of WPC.The real-world example also showed an optimal number of domain splits, indicating a trade-off in decomposition.
  • Limitations and outlook: For large forward problems, the PINN approach is not competitive with FEM or FFT-based methods in computational time.The optimization required by the neural-network formulation can last substantially longer as problem size increases.
  • Limitations and outlook: Future work includes incorporating more complex material laws and uncertainty, and extending cPINNs with parallel or XPINN approaches for larger multi-scale problems.The paper also identifies uncertainty quantification and inverse problems as open areas for further investigation.
Loading 2110.07374v2…