Physics-informed radial basis network (PIRBN): A local approximation neural network for solving nonlinear PDEs

Our recent intensive study has found that physics-informed neural networks (PINN) tend to be local approximators after training. This observation leads to this novel physics-informed radial basis network (PIRBN), which can maintain the local property throughout the entire training process. Compared to deep neural networks, a PIRBN comprises of only one hidden layer and a radial basis "activation" function. Under appropriate conditions, we demonstrated that the training of PIRBNs using gradient descendent methods can converge to Gaussian processes. Besides, we studied the training dynamics of PIRBN via the neural tangent kernel (NTK) theory. In addition, comprehensive investigations regarding the initialisation strategies of PIRBN were conducted. Based on numerical examples, PIRBN has been demonstrated to be more effective and efficient than PINN in solving PDEs with high-frequency features and ill-posed computational domains. Moreover, the existing PINN numerical techniques, such as adaptive learning, decomposition and different types of loss functions, are applicable to PIRBN. The programs that can regenerate all numerical results can be found at https://github.com/JinshuaiBai/PIRBN.Comment: 48 pages, 26 figure

Physics-informed neural network for friction-involved nonsmooth dynamics problems

Friction-induced vibration (FIV) is very common in engineering areas. Analysing the dynamic behaviour of systems containing a multiple-contact point frictional interface is an important topic. However, accurately simulating nonsmooth/discontinuous dynamic behaviour due to friction is challenging. This paper presents a new physics-informed neural network approach for solving nonsmooth friction-induced vibration or friction-involved vibration problems. Compared with schemes of the conventional time-stepping methodology, in this new computational framework, the theoretical formulations of nonsmooth multibody dynamics are transformed and embedded in the training process of the neural network. Major findings include that the new framework not only can perform accurate simulation of nonsmooth dynamic behaviour, but also eliminate the need for extremely small time steps typically associated with the conventional time-stepping methodology for multibody systems, thus saving much computation work while maintaining high accuracy. Specifically, four kinds of high-accuracy PINN-based methods are proposed: (1) single PINN; (2) dual PINN; (3) advanced single PINN; (4) advanced dual PINN. Two typical dynamics problems with nonsmooth contact are tested: one is a 1-dimensional contact problem with stick-slip, and the other is a 2-dimensional contact problem considering separation-reattachment and stick-slip oscillation. Both single and dual PINN methods show their advantages in dealing with the 1-dimensional stick-slip problem, which outperforms conventional methods across friction models that are difficult to simulate by the conventional time-stepping method. For the 2-dimensional problem, the capability of the advanced single and advanced dual PINN on accuracy improvement is shown, and they provide good results even in the cases when conventional methods fail.Comment: 38 Pages, 24 figure

An introduction to programming Physics-Informed Neural Network-based computational solid mechanics

Physics-informed neural network (PINN) has recently gained increasing interest in computational mechanics. In this work, we present a detailed introduction to programming PINN-based computational solid mechanics. Besides, two prevailingly used physics-informed loss functions for PINN-based computational solid mechanics are summarised. Moreover, numerical examples ranging from 1D to 3D solid problems are presented to show the performance of PINN-based computational solid mechanics. The programs are built via Python coding language and TensorFlow library with step-by-step explanations. It is worth highlighting that PINN-based computational mechanics is easy to implement and can be extended for more challenging applications. This work aims to help the researchers who are interested in the PINN-based solid mechanics solver to have a clear insight into this emerging area. The programs for all the numerical examples presented in this work are available on https://github.com/JinshuaiBai/PINN_Comp_Mech.Comment: 32 pages, 20 figures are include in this manuscrip

A data-driven smoothed particle hydrodynamics method for fluids

This thesis proposed a novel Data-Driven Smoothed Particle Hydrodynamics (DDSPH) method that, instead of applying the empirical rheological models, utilizes discrete experimental datasets to close the Navier-Stokes equations for hydrodynamic modelling. Besides, the chained hashing algorithm is applied to improve the efficiency of the data retrieval and the robustness of the method with respect to the noisy data is achieved via adding a variable that qualifies the relevance of data points to the clusters. The proposed DDSPH method introduces a new avenue for hydrodynamic modelling and has great potential for modelling complex fluids with highly nonlinear rheological relationships

Physics-guided deep learning framework for computational mechanics

This thesis developed a novel, effective and robust numerical framework based on the physics-guided deep learning technique for a wide range of mechanics modelling. In the thesis, thorough investigations regarding the proposed framework have been conducted from both theoretical and numerical aspects. It has been demonstrated that the proposed framework has great advantages over the traditional numerical methods when facing challenges, such as nonlinearity and free-surface tracking problems. The proposed framework demonstrated possibilities of integrating state-of-the-art deep learning techniques into computational mechanics and opened a new avenue for mechanics modelling

Effective enhancement of a carbon nanothread on the mechanical properties of the polyethylene nanocomposite

The mechanical performance of nanomaterial-reinforced polymer nanocomposites is a prerequisite for their engineering implementations, which is largely determined by the interfacial load transfer efficiency. This work investigates the role of the nanofillers via molecular dynamics simulations under different deformation scenarios, mimicking a maximum and minimum load transfer scenario from the polymer matrix. On the basis of the polyethylene (PE) nanocomposite reinforced by a new nanofiller-carbon nanothread (NTH), we find that the loading conditions dominantly determine its enhancement effect on the mechanical properties of the PE nanocomposite. Under tensile deformation, the ultimate tensile strength of the PE nanocomposite receives around 61 to 211% increment when the filler deforms simultaneously with the PE matrix. However, such enhancement is largely suppressed when the NTH is deforming nonsimultaneously. Similar results are observed from the compressive deformation. Specifically, both morphology and functionalization are found to alter the enhancement effect from the NTH fillers, while also relying on the loading directions. Overall, this work provides an in-depth understanding of the role of the nanofiller. The observations signify the importance of establishing effective load transfer at the interface, which could benefit the design and fabrication of high-performance polymer nanocomposites.</p

A Physics-Informed Neural Network technique based on a modified loss function for computational 2D and 3D solid mechanics

Despite its rapid development, Physics-Informed Neural Network (PINN)-based computational solid mechanics is still in its infancy. In PINN, the loss function plays a critical role that significantly influences the performance of the predictions. In this paper, by using the Least Squares Weighted Residual (LSWR) method, we proposed a modified loss function, namely the LSWR loss function, which is tailored to a dimensionless form with only one manually determined parameter. Based on the LSWR loss function, an advanced PINN technique is developed for computational 2D and 3D solid mechanics. The performance of the proposed PINN technique with the LSWR loss function is tested through 2D and 3D (geometrically nonlinear) problems. Thoroughly studies and comparisons are conducted between the two existing loss functions, the energy-based loss function and the collocation loss function, and the proposed LSWR loss function. Through numerical experiments, we show that the PINN based on the LSWR loss function is effective, robust, and accurate for predicting both the displacement and stress fields. The source codes for the numerical examples in this work are available at https://github.com/JinshuaiBai/LSWR_loss_function_PINN/

Sliding behaviour of carbon nanothread within a bundle embedded in polymer matrix

Interfacial shear strength determines the load transfer efficiency between nanofillers and polymer matrix, and thus the overall mechanical performance of polymer nanocomposites. Through atomistic simulations, this work systematically assessed the sliding behaviours of one-dimensional carbon nanothread (NTH) from a bundle configuration embedded in the poly (methyl methacrylate) (PMMA) matrix. It is found that the functionalized NTH exhibits remarkably enhanced interfacial shear strengths due to the strong mechanical interlocking effect, about an order of magnitude higher than the pristine sample. However, excess volume is generated within a functionalized bundle structure, leading to a weaker non-bonded interaction for a functionalized NTH within the bundle than that only embedded in the polymer matrix. With covalent cross-linking, the surrounding NTHs can be simultaneously dragged out from the PMMA matrix while pulling out the central NTH. Such phenomenon attributes to the filler-filler interactions and filler-matrix interactions. Further studies reveal that no extra free volume is generated within the bundle structure at high temperatures, which retains the effective load transfer efficiency. In comparison, free volume tends to be generated at the interface between NTHs and PMMA matrix at a higher temperature. These findings could be beneficial for fibre design and high-performance polymer nanocomposites with 1D nanomaterials.</p

Mechanical Properties of Single-Layer Diamond Reinforced Poly(vinyl alcohol) Nanocomposites through Atomistic Simulation

Low-dimensional carbon nanostructures are ideal nanofillers to reinforce the mechanical performance of polymer nanocomposites due to their excellent mechanical properties. Through molecular dynamics simulations, the mechanical performance of poly(vinyl alchohol) (PVA) nanocomposites reinforced with a single-layer diamond – diamane is investigated. It is found the PVA/diamane exhibits similar interfacial strengths and pull-out characteristics with the PVA/bilayer-graphene counterpart. Specifically, when the nanofiller is fully embedded in the nanocomposite, it is unable to deform simultaneously with the PVA matrix due to the weak interfacial load transfer efficiency, thus the enhancement effect is not significant. In comparison, diamane can effectively promote the tensile properties of the nanocomposite when it has a laminated structure as it deforms simultaneously with the matrix. With this configuration, the interlayer sp3 bonds endows diamane with a much higher resistance under compression and shear tests, thus the nanocomposite can reach very high compressive and shear stress. Overall, enhancement on the mechanical interlocking at the interface as triggered by surface functionalization is only effective for the fully embedded nanofiller. This work provides a fundamental understanding of the mechanical properties of PVA nanocomposites reinforced by diamane, which can shed lights on the design and preparation of next generation high-performance nanocomposites.</p

Tensile Performance of Polymer Nanocomposites with Randomly Dispersed Carbon Nanothreads

Low-dimensional nanostructures have been widely used as reinforcements for polymer nanocomposites. However, a majority of studies have considered the samples containing a single nanofiller or perfectly aligned nanofillers, which is usually not the case for the as-synthesized samples. Through molecular dynamics simulation, this work systematically assessed the tensile performance of poly(methyl methacrylate) (PMMA) nanocomposites with randomly dispersed carbon nanothreads (NTHs)─a new one-dimensional ultrathin nanofiller. It is found that NTH causes deterioration on the tensile properties due to the introduction of continuous lower-density interphases surrounding the filler, while such negative impacts can be turned into positive enhancements by functionalization. In particular, the cross-linking at the NTH/PMMA interface is able to effectively promote the enhancement effect from the nanofiller. Our results show that the samples with random but vertically aligned NTHs outperform their counterpart with randomly dispersed NTHs. Overall, it is shown that the enhancement effect is significantly influenced by the NTH dispersion, length, weight fraction, functionalization, and cross-linking. This study provides a comprehensive understanding of the influence on the tensile performance of polymer nanocomposites from the dispersion of nanofillers, which should benefit on the fabrication and application of high-performance polymer nanocomposites.</p