Source-linked AI summary
Meshless physics-informed deep learning method for three-dimensional solid mechanics
Diab W. Abueidda, Qiyue Lu, Seid Koric
TL;DR
The paper addresses the need to solve solid-mechanics partial differential equations without relying on costly discretization-based data generation. It develops a meshfree deep collocation method that trains neural networks against governing equations and boundary conditions, and reports qualitative and quantitative responses across several material models.
Problem
Deep-learning surrogate models for computational mechanics commonly require large, problem-specific datasets generated using discretization methods such as finite element analysis.
Method
The deep collocation method trains a feedforward neural network to satisfy strong-form governing equations and essential and natural boundary conditions using optimized loss functions.
Results
The method captures material responses qualitatively and quantitatively for linear elasticity, neo-Hookean hyperelasticity with large deformation, and von Mises plasticity with isotropic and kinematic hardening.
Takeaways & Limitations
After training, DCM can provide approximate solutions almost instantly at domain points from their spatial coordinates without generating FEM training data or defining mesh connectivity.
Takeaways & Limitations
DCM optimization is generally nonconvex, and the study assumes negligible body and inertial forces while using feedforward neural networks.
Abstract
from arXiv · showhide
Deep learning and the collocation method are merged and used to solve partial differential equations describing structures' deformation. We have considered different types of materials: linear elasticity, hyperelasticity (neo-Hookean) with large deformation, and von Mises plasticity with isotropic and kinematic hardening. The performance of this deep collocation method (DCM) depends on the architecture of the neural network and the corresponding hyperparameters. The presented DCM is meshfree and avoids any spatial discretization, which is usually needed for the finite element method (FEM). We show that the DCM can capture the response qualitatively and quantitatively, without the need for any data generation using other numerical methods such as the FEM. Data generation usually is the main bottleneck in most data-driven models. The deep learning model is trained to learn the model's parameters yielding accurate approximate solutions. Once the model is properly trained, solutions can be obtained almost instantly at any point in the domain, given its spatial coordinates. Therefore, the deep collocation method is potentially a promising standalone technique to solve partial differential equations involved in the deformation of materials and structural systems as well as other physical phenomena.
1 Introduction
Computational solid mechanics relies on numerical methods that can be computationally expensive, while data-driven deep-learning surrogates commonly require discretized training data. This study proposes the meshfree deep collocation method (DCM), which uses deep neural networks to satisfy governing equations and boundary conditions directly.
- Conventional computational mechanics methods for solving governing partial differential equations include isogeometric analysis, meshfree methods, and finite element analysis.
- Deep-learning surrogate models typically require large, problem-specific datasets generated through discretization methods such as finite element analysis.Complex problems require larger datasets to achieve accurate response prediction.
- Prior deep-learning approaches for partial differential equations include the deep Ritz method and deep energy methods for problems with suitable variational or energy formulations.
- The proposed DCM trains a deep neural network to find displacement fields satisfying the strong-form partial differential equation and essential and natural boundary conditions.The loss function is minimized with optimizers such as Adam or L-BFGS.
- DCM is meshfree, avoids tangent-modulus assembly, and does not necessarily require defining a potential-energy functional.
2 Method
The method uses a dense neural network trained by deep collocation to approximate displacement fields satisfying governing PDEs and boundary conditions. Its meshfree formulation minimizes a strong-form loss using sampled collocation points and gradient-based optimization.
- 2.1 Introduction to dense neural networks: The dense feedforward network maps spatial coordinates to predictions through linked neuron layers, with depth and layer widths chosen according to problem complexity.Weights and biases are updated during training, while differentiable activation functions help represent complex input-output relationships.
- 2.1 Introduction to dense neural networks: The universal approximation theorem supports using feedforward networks to approximate continuous functions on compact domains with arbitrary accuracy under stated activation-function conditions.The theorem does not determine training success, required neuron counts, or feasibility of estimating network parameters.
- 2.2 Deep collocation method: The DCM trains a neural network whose approximate solution satisfies the PDE together with essential and natural boundary conditions.The loss combines mean-square residual and boundary-condition terms, with penalty coefficients weighting their contributions.
- 2.2 Deep collocation method: For the studied quasi-static examples, spatial coordinates are mapped to displacement fields, from which strains and stresses are computed to evaluate the strong-form residual.The inertia term is assumed zero, so initial-condition loss is omitted.
- 2.2 Deep collocation method: Random points sampled from the material domain and boundary surfaces are used as meshfree collocation points for minimizing the loss.The approach avoids tangent-modulus assembly and solving a linear system, and does not require a potential-energy definition.
- 2.2 Deep collocation method: Training uses backpropagation with Adam followed by L-BFGS optimization, while architecture and hyperparameters are selected through trial and fine-tuning.The optimization problem is generally nonconvex, so local minima and saddle points remain possible.
3 Elasticity
The elasticity example applies the strong-form DCM to a small-deformation 3D bending beam and compares its displacement and stress predictions with FEM. The method converges during training and reports an L2-error of 0.11 on independently sampled points.
- Formulation: The example considers a homogeneous, isotropic elastic body under small deformation, with equilibrium expressed through stress divergence and boundary conditions.The infinitesimal strain tensor is related to stress through the Lamé-constant constitutive law.
- Problem setup: The 3D beam has L = 4 m and D = H = 1 m, with uy = C = 0.25 m prescribed on one end and zero displacement on the other.Boundary degrees of freedom are enforced explicitly because the strong-form formulation includes traction and displacement boundary terms.
- Training: The training samples 7500 interior points and 4000 points on each boundary, while Figure 5 tracks loss-function convergence.The convergence plot accompanies optimization of the DCM loss for the beam problem.
- Results: Figure 6 compares DCM and FEM vertical-displacement contours, displacement along the AB path, and von Mises stress contours.These comparisons evaluate both field distributions and a pathwise displacement profile.
- Results: 0.11 is the reported L2-error between DCM predictions at independently sampled points and corresponding FEM results.The evaluation points differ from those used to train the neural network.
- Assumption: Because the constitutive model is path-independent, the solution can be obtained using one pseudo-time step.The authors note that resampling points during optimization is possible for this case but not for elastoplastic problems.
4 Hyperelasticity
The hyperelasticity example extends the strong-form meshfree DCM to finite-deformation neo-Hookean behavior. It computes deformation measures and stresses through automatic differentiation and compares large-displacement predictions with FEM.
- Formulation: The formulation uses the strong form ∇X · P = 0 in the material domain together with prescribed displacement boundary conditions.P is the first Piola–Kirchhoff stress and N is the outward normal in the initial configuration.
- Constitutive model: For neo-Hookean behavior, the constitutive model uses deformation gradient F, right Cauchy–Green tensor C, J = det(F), and first invariant I1 = trace(C).The Helmholtz free energy is expressed in terms of J and I1, and P follows from the constitutive relation.
- Algorithm: Algorithm 2 maps sampled material points through the DNN, computes deformation quantities and P, evaluates strong-form and boundary losses, and updates network parameters.The procedure uses Adam followed by L-BFGS optimization.
- Scope: The same solution procedure can be adapted, with slight changes, to other hyperelastic constitutive models such as Mooney–Rivlin and Arruda–Boyce.The paper presents neo-Hookean elasticity as the worked hyperelastic example.
- Results: Figure 7 compares DCM and FEM vertical-displacement contours, displacement along the AB path, and von Mises stress contours.The figure organizes the comparison across full-field contours, a pathwise displacement profile, and stress.
- Problem setup: The cantilever beam imposes uy = C = 1.0 m to induce large deformation, using 7500 interior points and 4000 points on each boundary.The DCM displacement contours are compared with FEM results for this finite-deformation case.
5 Plasticity
The plasticity example applies the DCM to a path-dependent J2 elastoplastic beam with linear isotropic and kinematic hardening. It uses pseudo-time stepping and compares displacement, stress, and plastic-strain fields with FEM.
- Formulation: The constitutive model is J2 plasticity with linear isotropic and kinematic hardening, applied to a cantilever beam without inertial or body forces.The formulation is based on the strong form for elastoplastic equilibrium.
- Constitutive model: Under small deformation, total strain is additively decomposed into elastic and plastic components.This decomposition provides the kinematic basis for the elastoplastic constitutive update.
- Constitutive model: The von Mises yield condition uses deviatoric stress, back stress, equivalent plastic strain, and isotropic and kinematic hardening constants.KKT and consistency conditions define plastic evolution, including updates to the internal variables.
- Algorithm: The radial return mapping algorithm updates the plastic state from the previous integration-point state and the current strain.The procedure first computes a trial state before updating the internal variables and stresses.
- Algorithm: Elastoplastic optimization is repeated over M pseudo-time steps, carrying optimized weights and biases from step t into step t + 1.This continuation is described as a form of transfer learning and reflects path dependence.
- Problem setup: The beam case uses C = 0.2 m with 7500 interior points and 4000 points on each boundary, and Figure 8 compares DCM and FEM fields at the final load increment.The plotted outputs include vertical displacement, von Mises stress, and plastic-strain components εp_11 and εp_22.
- Algorithm: Algorithm 3 obtains displacement from the DNN, computes strain and stress derivatives by automatic differentiation, evaluates losses, and updates the network during each step.The loop includes interior equilibrium and boundary-condition evaluations.
6 Discussion, conclusions, and future directions
The deep collocation method combines collocation with deep learning to solve deformation PDEs without data generation or mesh generation, producing rapid coordinate-based solutions after training. Its broader applicability remains bounded by modeling assumptions and unresolved choices involving optimization, sampling, architectures, geometries, and boundary-condition enforcement.
- Method and scope: The method solves linear elasticity, neo-Hookean hyperelasticity, and von Mises plasticity using physics laws rather than generated training data.It merges collocation and deep learning, with physics laws used to obtain solutions.
- Method and scope: After training, the DNN can provide high-quality solutions almost instantly at arbitrary domain points from their spatial coordinates.The meshfree formulation also avoids node connectivity and mesh generation, including associated element-distortion and volumetric-locking concerns.
- Open methodological issues: Model accuracy depends on neural-network architecture, hyperparameters, and sampling choices, while nonconvex optimization can trap training in local minima.The study used Monte Carlo sampling and identifies alternative sampling strategies and architecture-optimization methods as open directions.
- Limitations and future directions: Further evaluation is needed for alternative sampling schemes, irregular geometries, and formulations using Lagrange multipliers instead of penalty-enforced boundary conditions.These directions are presented as ways to study accuracy and constraint enforcement in broader settings.
- Limitations and future directions: The study assumes negligible body and inertial forces and uses feedforward networks, limiting direct coverage of force-inclusive and history-dependent problems.Future work proposes sequence-learning architectures for path-dependent plasticity and viscoplasticity.