Fleet Prognosis with Physics-informed Recurrent Neural Networks

Services and warranties of large fleets of engineering assets is a very profitable business. The success of companies in that area is often related to predictive maintenance driven by advanced analytics. Therefore, accurate modeling, as a way to understand how the complex interactions between operating conditions and component capability define useful life, is key for services profitability. Unfortunately, building prognosis models for large fleets is a daunting task as factors such as duty cycle variation, harsh environments, inadequate maintenance, and problems with mass production can lead to large discrepancies between designed and observed useful lives. This paper introduces a novel physics-informed neural network approach to prognosis by extending recurrent neural networks to cumulative damage models. We propose a new recurrent neural network cell designed to merge physics-informed and data-driven layers. With that, engineers and scientists have the chance to use physics-informed layers to model parts that are well understood (e.g., fatigue crack growth) and use data-driven layers to model parts that are poorly characterized (e.g., internal loads). A simple numerical experiment is used to present the main features of the proposed physics-informed recurrent neural network for damage accumulation. The test problem consist of predicting fatigue crack length for a synthetic fleet of airplanes subject to different mission mixes. The model is trained using full observation inputs (far-field loads) and very limited observation of outputs (crack length at inspection for only a portion of the fleet). The results demonstrate that our proposed hybrid physics-informed recurrent neural network is able to accurately model fatigue crack growth even when the observed distribution of crack length does not match with the (unobservable) fleet distribution.Comment: Data and codes (including our implementation for both the multi-layer perceptron, the stress intensity and Paris law layers, the cumulative damage cell, as well as python driver scripts) used in this manuscript are publicly available on GitHub at https://github.com/PML-UCF/pinn. The data and code are released under the MIT Licens

Adaptive hybrid discontinuous methods for fluid and wave problems

This PhD thesis proposes a p-adaptive technique for the Hybridizable Discontinuous Galerkin method (HDG). The HDG method is a novel discontinuous Galerkin method (DG) with interesting characteristics. While retaining all the advantages of the common DG methods, such as the inherent stabilization and the local conservation properties, HDG allows to reduce the coupled degrees of freedom of the problem to those of an approximation of the solution de¿ned only on the faces of the mesh. Moreover, the convergence properties of the HDG solution allow to perform an element-by-element postprocess resulting in a superconvergent solution. Due to the discontinuous character of the approximation in HDG, p-variable computations are easily implemented. In this work the superconvergent postprocess is used to de¿ne a reliable and computationally cheap error estimator, that is used to drive an automatic adaptive process. The polynomial degree in each element is automatically adjusted aiming at obtaining a uniform error distribution below a user de¿ned tolerance. Since no topological modi¿cation of the discretization is involved, fast adaptations of the mesh are obtained. First, the p-adaptive HDG is applied to the solution of wave problems. In particular, the Mild Slope equation is used to model the problem of sea wave propagation is coastal areas and harbors. The HDG method is compared with the continuous Galerkin (CG) ¿nite element method, which is nowadays the common method used in the engineering practice for this kind of applications. Numerical experiments reveal that the e¿ciency of HDG is close to CG for uniform degree computations, clearly outperforming other DG methods such as the Compact Discontinuous Galerkin method. When p-adaptivity is considered, an important saving in computational cost is shown. Then, the methodology is applied to the solution of the incompressible Navier-Stokes equations for the simulation of laminar ¿ows. Both steady state and transient applications are considered. Various numerical experiments are presented, in 2D and 3D, including academic examples and more challenging applications of engineering interest. Despite the simplicity and low cost of the error estimator, high e¿ciency is exhibited for analytical examples. Moreover, even though the adaptive technique is based on an error estimate for just the velocity ¿eld, high accuracy is attained for all variables, with sharp resolution of the key features of the ¿ow and accurate evaluation of the ¿uid-dynamic forces. In particular, high degrees are automatically located along boundary layers, reducing the need for highly distorted elements in the computational mesh. Numerical tests show an important reduction in computational cost, compared to uniform degree computations, for both steady and unsteady computations

Shock capturing computations with stabilized Powell-Sabin elements

International audienceIn the last recent years, thanks to the increasing power of the computational machines , the interest in more and more accurate numerical schemes is growing. Methods based on high-order approximations are nowadays the common trend in the computational research community, in particular for CFD applications. This work is focused on Powell-Sabin (PS) finite elements, a finite element method (FEM) based on PS splines. PS splines are piecewise quadratic polynomials with a global C1 continuity , defined on conforming triangulations. Despite its attractive characteristics, so far this scheme hasn't had the attention it deserves. PS splines are adapted to unstructured meshes and, contrary to classical tensor product B-splines, they are particularly suited for local refinement , a crucial aspect in the analysis of highly convective and anisotropic equations. The additional global smoothness of the C1 space has a beneficial stabilization effect in the treatment of advection-dominated equations and leads to a better capturing of thin layers. Finally, unlike most of other typology of high-order finite elements, the numerical unknowns in PS elements are located in the vertices of the triangulation, leading to an easy treatment of the parallel aspects. Some geometrical issues related to PS elements are discussed here, in particular, the generation of the control triangles and the imposition of the boundary conditions. The PS FEM method is used to solve the compressible Euler equation in supersonic regime. A classical shock-capturing technique is used to reduce the oscillation around the discontinuity, while a variational multiscale formulation is used to introduce numerical diffusion in the streamwise direction. Some typical numerical examples are used to evaluate the performance of the PS discretization

Avances en la simulación numérica de la propagación de oleaje en zonas costeras

Se presentan los últimos avances en la simulación numérica de la propagación de oleaje en zonas costeras. Las nuevas herramientas numéricas permiten obtener de manera precisa y eficiente las características de la ola en las zonas de interés. A partir de un modelo reducido es posible simular en tiempo real la propagación para cualquier oleaje predominante

Hybridizable Discontinuous Galerkin with degree adaptivity for the incompressible Navier-Stokes equations

A degree adaptive Hybridizable Discontinuous Galerkin (HDG) method for the solution of the incompressible Navier-Stokes equations is presented. The key ingredient is an accurate and computationally inexpensive a posteriori error estimator based on the super-convergence properties of HDG. The error estimator drives the local modification of the approximation degree in the elements and faces of the mesh, aimed at obtaining a uniform error distribution below a user-given tolerance in a given output of interest. Three 2D numerical examples are presented. High efficiency of the proposed error estimator is found, and an important reduction of the computational effort is shown with respect to non-adaptive computations, both for steady state and transient simulations

Hybridizable discontinuous Galerkin p-adaptivity for wave propagation problems

A p-adaptive hybridizable discontinuous Galerkin method for the solution of wave problems is presented in a challenging engineering problem. Moreover, its performance is compared with a high-order continuous Galerkin. The hybridization technique allows to reduce the coupled degrees of freedom to only those on the mesh element boundaries, whereas the particular choice of the numerical fluxes opens the path to a superconvergent postprocessed solution. This superconvergent postprocessed solution is used to construct a simple and inexpensive error estimator. The error estimator is employed to obtain solutions with the prescribed accuracy in the area (or areas) of interest and also drives a proposed iterative mesh adaptation procedure. The proposed method is applied to a nonhomogeneous scattering problem in an unbounded domain. This is a challenging problem because, on the one hand, for high frequencies, numerical difficulties are an important issue because of the loss of the ellipticity and the oscillatory behavior of the solution. And on the other hand, it is applied to real harbor agitation problems. That is, the mild slope equation in frequency domain (Helmholtz equation with nonconstant coefficients) is solved on real geometries with the corresponding perfectly matched layer to damp the diffracted waves. The performance of the method is studied on two practical examples. The adaptive hybridizable discontinuous Galerkin method exhibits better efficiency compared with a high-order continuous Galerkin method using static condensation of the interior nodes

Hybrid Physics-informed Neural Networks for Dynamical Systems

Ordinary differential equations can describe many dynamic systems. When physics is well understood, the time-dependent responses are easily obtained numerically. The particular numerical method used for integration depends on the application. Unfortunately, when physics is not fully understood, the discrepancies between predictions and observed responses can be large and unacceptable. In this thesis, we show how to directly implement integration of ordinary differential equations through recurrent neural networks using Python. We leveraged modern machine learning frameworks, such as TensorFlow and Keras. Besides offering basic models capabilities (such as multilayer perceptrons and recurrent neural networks) and optimization methods, these frameworks offer powerful automatic differentiation. With that, our approach\u27s main advantage is that one can implement hybrid models combining physics-informed and data-driven kernels, where data-driven kernels are used to reduce the gap between predictions and observations. In order to illustrate our approach, we used two case studies. The first one consisted of performing fatigue crack growth integration through Euler\u27s forward method using a hybrid model combining a data-driven stress intensity range model with a physics-based crack length increment model. The second case study consisted of performing model parameter identification of a dynamic two-degree-of-freedom system through Runge-Kutta integration. Additionally, we performed a numerical experiment for fleet prognosis with hybrid models. The problem consists of predicting fatigue crack length for a fleet of aircraft. The hybrid models are trained using full input observations (far-field loads) and very limited output observations (crack length data for only a portion of the fleet). The results demonstrate that our proposed physics-informed recurrent neural network can model fatigue crack growth even when the observed distribution of crack length does not match the fleet distribution

Lithium-ion Battery Prognosis with Variational Hybrid Physics-informed Neural Networks

Lithium-ion batteries are an increasingly popular source of power for many electric applications. Applications range from electric cars, driven by thousands of people every day, to existing and future air vehicles, such as unmanned aircraft vehicles (UAVs) and urban air mobility (UAM) drones. Therefore, robust modeling approaches are essential to ensure high reliability levels by monitoring battery state-of-charge (SOC) and forecasting the remaining useful life (RUL). Building principled-based models is challenging due to the complex electrochemistry that governs battery operation, which would entail computationally expensive models not suited for prognosis and health management applications. Alternatively, reduced-order models can be used and have the advantage of capturing the overall behavior of battery discharge, although they suffer from simplifications and residual discrepancy. We propose a hybrid solution for Li-ion battery discharge and aging prediction that directly implements models based on first-principle within modern recurrent neural networks. While reduced-order models describe part of the voltage discharge under constant or variable loading conditions, data-driven kernels reduce the gap between predictions and observations. We developed and validated our approach using the NASA Prognostics Data Repository Battery dataset, which contains experimental discharge data on Li-ion batteries obtained in a controlled environment. Our hybrid model tracks aging parameters connected to the residual capacity of the battery. In addition, we use a Bayesian approach to merge fleet-wide data in the form of priors with battery-specific discharge cycles, where the battery capacity is fully available (complete data) or only partially available (censored data). The model\u27s predictive capability is monitored throughout battery usage. This way, our proposed approach indicates when significant updates to the hybrid model are needed. Our Bayesian implementation of the hybrid variational physics-informed neural network can reliably predict the battery\u27s future residual capacity, even in cases where previous battery usage history is unknown

Shock capturing computations with stabilized Powell-Sabin elements

International audienceIn the last recent years, thanks to the increasing power of the computational machines , the interest in more and more accurate numerical schemes is growing. Methods based on high-order approximations are nowadays the common trend in the computational research community, in particular for CFD applications. This work is focused on Powell-Sabin (PS) finite elements, a finite element method (FEM) based on PS splines. PS splines are piecewise quadratic polynomials with a global C1 continuity , defined on conforming triangulations. Despite its attractive characteristics, so far this scheme hasn't had the attention it deserves. PS splines are adapted to unstructured meshes and, contrary to classical tensor product B-splines, they are particularly suited for local refinement , a crucial aspect in the analysis of highly convective and anisotropic equations. The additional global smoothness of the C1 space has a beneficial stabilization effect in the treatment of advection-dominated equations and leads to a better capturing of thin layers. Finally, unlike most of other typology of high-order finite elements, the numerical unknowns in PS elements are located in the vertices of the triangulation, leading to an easy treatment of the parallel aspects. Some geometrical issues related to PS elements are discussed here, in particular, the generation of the control triangles and the imposition of the boundary conditions. The PS FEM method is used to solve the compressible Euler equation in supersonic regime. A classical shock-capturing technique is used to reduce the oscillation around the discontinuity, while a variational multiscale formulation is used to introduce numerical diffusion in the streamwise direction. Some typical numerical examples are used to evaluate the performance of the PS discretization

Hybridizable discontinuous Galerkin p-adaptivity for wave propagation problems

A p-adaptive hybridizable discontinuous Galerkin method for the solution of wave problems is presented in a challenging engineering problem. Moreover, its performance is compared with a high-order continuous Galerkin. The hybridization technique allows to reduce the coupled degrees of freedom to only those on the mesh element boundaries, whereas the particular choice of the numerical fluxes opens the path to a superconvergent postprocessed solution. This superconvergent postprocessed solution is used to construct a simple and inexpensive error estimator. The error estimator is employed to obtain solutions with the prescribed accuracy in the area (or areas) of interest and also drives a proposed iterative mesh adaptation procedure. The proposed method is applied to a nonhomogeneous scattering problem in an unbounded domain. This is a challenging problem because, on the one hand, for high frequencies, numerical difficulties are an important issue because of the loss of the ellipticity and the oscillatory behavior of the solution. And on the other hand, it is applied to real harbor agitation problems. That is, the mild slope equation in frequency domain (Helmholtz equation with nonconstant coefficients) is solved on real geometries with the corresponding perfectly matched layer to damp the diffracted waves. The performance of the method is studied on two practical examples. The adaptive hybridizable discontinuous Galerkin method exhibits better efficiency compared with a high-order continuous Galerkin method using static condensation of the interior nodes.Peer ReviewedPostprint (author’s final draft