Digital twin can be defined as a virtual representation of a physical asset enabled through data and simulators for real-time prediction, optimization, monitoring, controlling, and improved decision making. Recent advances in computational pipelines, multiphysics solvers, artificial intelligence, big data cybernetics, data processing and management tools bring the promise of digital twins and their impact on society closer to reality. Digital twinning is now an important and emerging trend in many applications. Also referred to as a computational megamodel, device shadow, mirrored system, avatar or a synchronized virtual prototype, there can be no doubt that a digital twin plays a transformative role not only in how we design and operate cyber-physical intelligent systems, but also in how we advance the modularity of multi-disciplinary systems to tackle fundamental barriers not addressed by the current, evolutionary modeling practices. In this work, we review the recent status of methodologies and techniques related to the construction of digital twins mostly from a modeling perspective. Our aim is to provide a detailed coverage of the current challenges and enabling technologies along with recommendations and reflections for various stakeholders.

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Digital Twin: Values, Challenges and Enablers From a Modeling Perspective

This work proposes an approach that combines a library of component-based reduced-order models with Bayesian state estimation in order to create data-driven physics-based digital twins. Reduced-ordermodeling produces physics-based computationalmodels that are reliable enough for predictive digital twins, while still being fast to evaluate. In contrast with traditional monolithic techniques for model reduction, the component-based approach scales efficiently to large complex systems, and provides a flexible and expressive framework for rapid model adaptation—both critical features in the digital twin context. Data-driven model adaptation and uncertainty quantification is formulated as a Bayesian state estimation problem, in which sensor data is used to infer which models in the model library are the best candidates for the digital twin. This approach is demonstrated through the development of a digital twin for a 12ft wingspan unmanned aerial vehicle. Offline, we construct a library of pristine and damaged aircraft components. Online, we use structural sensor data to rapidly adapt a physics-based digital twin of the aircraft structure. The data-driven digital twin enables the aircraft to dynamically replan a safe mission in response to structural damage or degradation.

Data‐driven physics‐based digital twins via a library of component‐based reduced‐order models

This work develops a methodology for creating and updating data-driven physics-based digital twins, and demonstrates the approach through the development of a structural digital twin for a 12ft wingspan unmanned aerial vehicle. The digital twin is built from a library of componentbased reduced-ordermodels that are derived fromhigh-fidelity finite element simulations of the vehicle in a range of pristine and damaged states. In contrast with traditional monolithic techniques for model reduction, the component-based approach scales efficiently to large complex systems, and provides a flexible and expressive framework for rapid model adaptation—both critical features in the digital twin context. The digital twin is deployed and updated using interpretable machine learning. Specifically, we use optimal trees—a recently developed scalable machine learning method—to train an interpretable data-driven classifier. In operation, the classifier takes as input vehicle sensor data, and then infers which physics-based reduced models in the model library are the best candidates to compose an updated digital twin. In our example use case, the data-driven digital twin enables the aircraft to dynamically replan a safe mission in response to structural damage or degradation.

Toward predictive digital twins via component-based reduced-order models and interpretable machine learning

A digital twin can be defined as an adaptive model of a complex physical system. Recent advances in computational pipelines, multiphysics solvers, artificial intelligence, big data cybernetics, data processing and management tools bring the promise of digital twins and their impact on society closer to reality. Digital twinning is now an important and emerging trend in many applications. Also referred to as a computational megamodel, device shadow, mirrored system, avatar or a synchronized virtual prototype, there can be no doubt that a digital twin plays a transformative role not only in how we design and operate cyber-physical intelligent systems, but also in how we advance the modularity of multi-disciplinary systems to tackle fundamental barriers not addressed by the current, evolutionary modeling practices. In this work, we review the recent status of methodologies and techniques related to the construction of digital twins. Our aim is to provide a detailed coverage of the current challenges and enabling technologies along with recommendations and reflections for various stakeholders.

Digital Twin: Values, Challenges and Enablers

Domain-decomposition least-squares Petrov–Galerkin (DD-LSPG) nonlinear model reduction

A component-based hybrid reduced basis/finite element method for solid mechanics with local nonlinearities

A component-based approach is introduced for fast and flexible solution of parameter-dependent symmetric eigenproblems. Considering a generalized eigenproblem with symmetric stiffness and mass operators, we start by introducing a “ σ-shifted” eigenproblem where the left hand side operator corresponds to an equilibrium between the stiffness operator and a weighted mass operator, with weight-parameter σ>0. Assuming that σ=λ
 n
 >0, the nth real positive eigenvalue of the original eigenproblem, then the shifted eigenproblem reduces to the solution of a homogeneous linear problem. In this context, we can apply the static condensation reduced basis element (SCRBE) method, a domain synthesis approach with reduced basis (RB) approximation at the intradomain level to populate a Schur complement at the interdomain level. In the Offline stage, for a library of archetype subdomains we train RB spaces for a family of linear problems; these linear problems correspond to various equilibriums between the stiffness operator and the weighted mass operator. In the Online stage we assemble instantiated subdomains and perform static condensation to obtain the “ σ-shifted” eigenproblem for the full system. We then perform a direct search to find the values of σ that yield singular systems, corresponding to the eigenvalues of the original eigenproblem. We provide eigenvalue a posteriori error estimators and we present various numerical results to demonstrate the accuracy, flexibility and computational efficiency of our approach. We are able to obtain large speed and memory improvements compared to a classical Finite Element Method (FEM), making our method very suitable for large models commonly considered in an engineering context.

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Component-based reduced basis for parametrized symmetric eigenproblems

United States. Air Force Office of Scientific Research (OSD/AFOSR/MURI Grant FA9550-09-1-0613)

We present an SCRBE PDE App framework for accurate and interactive calculation and visualization of the parametric dependence of the pressure field and associated Quantities of Interest (QoI)—such as impedance and transmission loss—for an extensive family of acoustic duct models. The Static Condensation Reduced Basis Element (SCRBE) partial differential equation (PDE) numerical approach incorporates several principal ingredients: component-to-system model construction, underlying “truth” finite element PDE discretization, (Petrov)-Galerkin projection, static condensation at the component level, parametrized model-order reduction for both the inter-component (port) and intra-component (bubble) degrees of freedom, and offline-online computational decompositions; we emphasize in this paper reduced port spaces and QoI evaluation techniques, especially frequency sweeps, particularly germane to the acoustics context. A PDE App constitutes a Web User Interface (WUI) implementation of the online, or deployed, stage of the SCRBE approximation for a particular parametrized model: User model parameter inputs to the WUI are interpreted by a PDE App Server which then invokes a parallel cloud-based SCRBE Online Computation Server for calculation of the pressure and associated QoI; the Online Computation Server then downloads the spatial field and scalar outputs (as a function of frequency) to the PDE App Server for interrogation and visualization in the WUI by the User. We present several examples of acoustic-duct PDE Apps: the exponential horn, the expansion chamber, and the toroidal bend; in each case we verify accuracy, demonstrate capabilities, and assess computational performance.

PDE Apps for Acoustic Ducts: A Parametrized Component-to-System Model-Order-Reduction Approach

In one aspect, a method for modeling and analyzing a physical system comprising a plurality of components includes constructing, by a computing device, a model of the plurality of components. The computing device determines that the at least one component in the plurality of components represents a region for which at least a first portion of an associated partial differential equation is linear. The computing device accesses one of a plurality of datasets, the accessed dataset comprising a representation of the first portion of the partial differential equation. The computing device determines that a subset of the plurality of components encapsulates a region for which a second portion of the associated partial differential equation is non-linear. The computing device generates a combined output based on the partial differential equation combining the first portion and the second portion.

Loi Nguyen

Papers

A component-based hybrid reduced basis/finite element method for solid mechanics with local nonlinearities

Component-based reduced basis for parametrized symmetric eigenproblems

Component-based reduced basis for parametrized symmetric eigenproblems

PDE Apps for Acoustic Ducts: A Parametrized Component-to-System Model-Order-Reduction Approach

Methods and Systems for Constructing and Analyzing Component-Based Models of Engineering Systems Having Linear and Nonlinear Regions