Meshless physics‐informed deep learning method for three‐dimensional solid mechanics

Abstract

Deep learning (DL) and the collocation method are merged and used to solve partial differential equations (PDEs) 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 DL 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 DCM is potentially a promising standalone technique to solve PDEs involved in the deformation of materials and structural systems as well as other physical phenomena.

Enhanced physics-informed neural networks for hyperelasticity

“…The L 2 -error is 0.0091. Also, we compare the solution obtained using the proposed PINN model with PINN models based on the seminal deep energy method (DEM) [23] and deep collocation method (DCM) [15]. Figure 10 shows the solutions obtained using the DEM and and DCM.…”

“…Another direction in which deep learning is used to capture physical phenomena is physics-informed neural networks (PINNs), which are of eminent interest to industry and academia [14,15,16,17]. PINNs have been used to approximate solutions for boundary value problems without necessitating the use of labeled data due to their abilities as universal approximators [18].…”

Optimizing Hyperparameters and Architecture of Deep Energy Method

“…Machine learning methods are rapidly developing as an alternative to traditional approaches to overcome the issues mentioned above. Machine learning methods can be classified as data-driven models [6][7][8][9][10][11][12][13][14][15][16] and physics-informed neural networks (PINNs) [17][18][19][20][21][22][23][24] . In data-driven models, data from experimental and computational results are used to train the models.…”

Neural Networks to Solve Partial Differential Equations: A Comparison With Finite Elements

“…The advent of low-cost high-performance hardware as well as the emergence of standardised machinelearning packages including user-friendly NN-libraries (e.g. scikit-learn [4], tensorflow by Google [5], pytorch by Facebook [6]) powered a recent revival of this concept with the appearance of several papers [7]- [16] and also a few opensource projects [17]- [20]. It is worth to mention at this stage that many of these studies have a common denominator in their motivation, namely that the NN-approach to solve PDEs can offer several advantages over established techniques like FEM in specific contexts: 1) While the computational power required to train a NN can be significantly larger than that needed to solve a FEM problem, the need for a re-mesh upon change of the underlying geometry can give the upper hand to the NN approach in those cases where the geometry is not fixed.…”