Details
| Original language | English |
|---|---|
| Article number | 114096 |
| Journal | Computer Methods in Applied Mechanics and Engineering |
| Volume | 386 |
| Early online date | 23 Aug 2021 |
| Publication status | Published - 1 Dec 2021 |
Abstract
In this work, we present a Parametric Deep Energy Method (P-DEM) for elasticity problems accounting for strain gradient effects. The approach is based on physics-informed neural networks (PINNs) for the solution of the underlying potential energy. Therefore, a cost function related to the potential energy is subsequently minimized. P-DEM does not need any classical discretization and requires only a definition of the potential energy, which simplifies the implementation. Instead of training the model in the physical space, we define a parametric/reference space similar to isoparametric finite elements, which is in our example a unit square. The inputs are naturally normalized preventing the vanishing gradient problem and leading to much faster convergence compared to the original DEM. Forward–backward mapping is established by means of NURBS basis functions. Another advantage of this approach is that Gauss quadrature can be employed to approximate the total potential energy, which is the loss function calculated in the parametric domain. Backpropagation available in PyTorch with automatic differentiation is performed to calculate the gradients of the loss function with respect to the weights and biases. Once the network is trained, a numerical solution can be obtained in the reference domain and then is mapped back to the physical domain. The performance of the method is demonstrated through various numerical benchmark problems in elasticity and compared to analytical solutions. We also consider strain gradient elasticity, which poses challenges to conventional finite elements due to the requirement for C1 continuity.
Keywords
- Deep energy method, Elasticity, Neural networks (NN), Partial differential equations (PDEs), Physics-informed neural networks (PINNs), Strain gradient elasticity
ASJC Scopus subject areas
- Engineering(all)
- Computational Mechanics
- Engineering(all)
- Mechanics of Materials
- Engineering(all)
- Mechanical Engineering
- Physics and Astronomy(all)
- General Physics and Astronomy
- Computer Science(all)
- Computer Science Applications
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In: Computer Methods in Applied Mechanics and Engineering, Vol. 386, 114096, 01.12.2021.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Parametric deep energy approach for elasticity accounting for strain gradient effects
AU - Nguyen-Thanh, Vien Minh
AU - Anitescu, Cosmin
AU - Alajlan, Naif
AU - Rabczuk, Timon
AU - Zhuang, Xiaoying
N1 - Funding Information: The authors extend their appreciation to the Distinguished Scientist Fellowship Program (DSFP) at King Saud University, Saudi Arabia for funding this work.
PY - 2021/12/1
Y1 - 2021/12/1
N2 - In this work, we present a Parametric Deep Energy Method (P-DEM) for elasticity problems accounting for strain gradient effects. The approach is based on physics-informed neural networks (PINNs) for the solution of the underlying potential energy. Therefore, a cost function related to the potential energy is subsequently minimized. P-DEM does not need any classical discretization and requires only a definition of the potential energy, which simplifies the implementation. Instead of training the model in the physical space, we define a parametric/reference space similar to isoparametric finite elements, which is in our example a unit square. The inputs are naturally normalized preventing the vanishing gradient problem and leading to much faster convergence compared to the original DEM. Forward–backward mapping is established by means of NURBS basis functions. Another advantage of this approach is that Gauss quadrature can be employed to approximate the total potential energy, which is the loss function calculated in the parametric domain. Backpropagation available in PyTorch with automatic differentiation is performed to calculate the gradients of the loss function with respect to the weights and biases. Once the network is trained, a numerical solution can be obtained in the reference domain and then is mapped back to the physical domain. The performance of the method is demonstrated through various numerical benchmark problems in elasticity and compared to analytical solutions. We also consider strain gradient elasticity, which poses challenges to conventional finite elements due to the requirement for C1 continuity.
AB - In this work, we present a Parametric Deep Energy Method (P-DEM) for elasticity problems accounting for strain gradient effects. The approach is based on physics-informed neural networks (PINNs) for the solution of the underlying potential energy. Therefore, a cost function related to the potential energy is subsequently minimized. P-DEM does not need any classical discretization and requires only a definition of the potential energy, which simplifies the implementation. Instead of training the model in the physical space, we define a parametric/reference space similar to isoparametric finite elements, which is in our example a unit square. The inputs are naturally normalized preventing the vanishing gradient problem and leading to much faster convergence compared to the original DEM. Forward–backward mapping is established by means of NURBS basis functions. Another advantage of this approach is that Gauss quadrature can be employed to approximate the total potential energy, which is the loss function calculated in the parametric domain. Backpropagation available in PyTorch with automatic differentiation is performed to calculate the gradients of the loss function with respect to the weights and biases. Once the network is trained, a numerical solution can be obtained in the reference domain and then is mapped back to the physical domain. The performance of the method is demonstrated through various numerical benchmark problems in elasticity and compared to analytical solutions. We also consider strain gradient elasticity, which poses challenges to conventional finite elements due to the requirement for C1 continuity.
KW - Deep energy method
KW - Elasticity
KW - Neural networks (NN)
KW - Partial differential equations (PDEs)
KW - Physics-informed neural networks (PINNs)
KW - Strain gradient elasticity
UR - http://www.scopus.com/inward/record.url?scp=85113281461&partnerID=8YFLogxK
U2 - 10.1016/j.cma.2021.114096
DO - 10.1016/j.cma.2021.114096
M3 - Article
AN - SCOPUS:85113281461
VL - 386
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
SN - 0045-7825
M1 - 114096
ER -