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In this article, we introduce a methodology for improving the accuracy and efficiency of reduced order models (ROMs) constructed using the least‐squares Petrov–Galerkin (LSPG) projection method through the introduction of preconditioning. Unlike prior related work, which focuses on preconditioning the linear systems arising within the ROM numerical solution procedure to improve linear solver performance, our approach leverages a preconditioning matrix directly within the minimization problem underlying the LSPG formulation. Applying preconditioning in this way has the potential to improve ROM accuracy for several reasons. First, preconditioning the LSPG formulation changes the norm defining the residual minimization, which can improve the residual‐based stability constant bounding the ROM solution's error. The incorporation of a preconditioner into the LSPG formulation can have the additional effect of scaling the components of the residual being minimized to make them roughly of the same magnitude, which can be beneficial when applying the LSPG method to problems with disparate scales (e.g., dimensional equations, multi‐physics problems). Importantly, we demonstrate that an "ideal preconditioned" LSPG ROM (a ROM in which the preconditioner is the inverse of the Jacobian of its corresponding full order model) emulates projection of the full order model solution increment onto the reduced basis. This quantity defines a lower bound on the error of a ROM solution for a given reduced basis. By designing preconditioners that approximate the Jacobian inverse—as is common in designing preconditioners for solving linear systems—it is possible to obtain a ROM whose error approaches this lower bound. The proposed approach is evaluated on several mechanical and thermo‐mechanical problems implemented within the Albany HPC code and run in the predictive regime, with prediction across material parameter space. We demonstrate numerically that the introduction of simple Jacobi, Gauss‐Seidel, and ILU preconditioners into the proper orthogonal decomposition/LSPG formulation reduces significantly the ROM solution error, the reduced Jacobian condition number, the number of nonlinear iterations required to reach convergence, and the wall time (thereby improving efficiency). Moreover, our numerical results reveal that the introduction of preconditioning can deliver a robust and accurate solution for test cases in which the unpreconditioned LSPG method fails to converge. [ABSTRACT FROM AUTHOR]

This article presents a new reduced order model (ROM) for the angular discretization of the Boltzmann transport equation. The angular ROM is built over a partitioning of the space‐angle phase‐space, by generating independent, optimized angular basis function sets for each partition. The advantage is that each basis function set is optimized to represent the neutron flux distribution in a particular partition of space and angle, rather than being optimized for the entire domain. This serves to reduce the total number of basis functions required, and therefore the solve time. Two numerical examples are presented to demonstrate the efficacy of the methods—a dog‐leg duct problem, involving particle streaming, and the Watanabe–Maynard problem, which includes significant particle scattering. In both cases, it is shown that the method reduces the angular flux error, for a given basis size or solve time, by around an order of magnitude in comparison to other, similar ROM methods. An adaptive version of the method is also presented, whereby the number of basis functions within each space‐angle partition can vary independently. It is shown to potentially provide further significant reductions in error. [ABSTRACT FROM AUTHOR]

The early detection of terrorist threat objects, such as guns and knives, through improved metal detection, has the potential to reduce the number of attacks and improve public safety and security. To achieve this, there is considerable potential to use the fields applied and measured by a metal detector to discriminate between different shapes and different metals since, hidden within the field perturbation, is object characterization information. The magnetic polarizability tensor (MPT) offers an economical characterization of metallic objects and its spectral signature provides additional object characterization information. The MPT spectral signature can be determined from measurements of the induced voltage over a range of frequencies in a metal signature for a hidden object. With classification in mind, it can also be computed in advance for different threat and non‐threat objects. In this article, we evaluate the performance of probabilistic and non‐probabilistic machine learning algorithms, trained using a dictionary of computed MPT spectral signatures, to classify objects for metal detection. We discuss the importance of using appropriate features and selecting an appropriate algorithm depending on the classification problem being solved, and we present numerical results for a range of practically motivated metal detection classification problems. [ABSTRACT FROM AUTHOR]

The Inertia Relief (IR) technique is widely used by industry and produces equilibrated loads allowing to analyze unconstrained systems without resorting to the more expensive full dynamic analysis. The main goal of this work is to develop a computational framework for the solution of unconstrained parametric structural problems with IR and the Proper Generalized Decomposition (PGD) method. First, the IR method is formulated in a parametric setting for both material and geometric parameters. A reduced order model using the encapsulated PGD suite is then developed to solve the parametric IR problem, circumventing the so‐called curse of dimensionality. With just one offline computation, the proposed PGD‐IR scheme provides a computational vademecum that contains all the possible solutions for a predefined range of the parameters. The proposed approach is nonintrusive and it is therefore possible to be integrated with commercial finite element (FE) packages. The applicability and potential of the developed technique is shown using a three‐dimensional test case and a more complex industrial test case. The first example is used to highlight the numerical properties of the scheme, whereas the second example demonstrates the potential in a more complex setting and it shows the possibility to integrate the proposed framework within a commercial FE package. In addition, the last example shows the possibility to use the generalized solution in a multi‐objective optimization setting. [ABSTRACT FROM AUTHOR]

The early detection of terrorist threat objects, such as guns and knives, through improved metal detection, has the potential to reduce the number of attacks and improve public safety and security. To achieve this, there is considerable potential to use the fields applied and measured by a metal detector to discriminate between different shapes and different metals since, hidden within the field perturbation, is object characterisation information. The magnetic polarizability tensor (MPT) offers an economical characterisation of metallic objects that can be computed for different threat and non‐threat objects and has an established theoretical background, which shows that the induced voltage is a function of the hidden object's MPT coefficients. In this article, we describe the additional characterisation information that measurements of the induced voltage over a range of frequencies offer compared with measurements at a single frequency. We call such object characterisations its MPT spectral signature. Then, we present a series of alternative rotational invariants for the purpose of classifying hidden objects using MPT spectral signatures. Finally, we include examples of computed MPT spectral signature characterisations of realistic threat and non‐threat objects that can be used to train machine learning algorithms for classification purposes. [ABSTRACT FROM AUTHOR]

The identification of hidden conducting permeable objects from measurements of the perturbed magnetic field taken over a range of low frequencies is important in metal detection. Applications include identifying threat items in security screening at transport hubs, location of unexploded ordnance, and antipersonnel landmines in areas of former conflict, searching for items of archeological significance and recycling of valuable metals. The solution of the inverse problem, or more generally locating and classifying objects, has attracted considerable attention recently using polarizability tensors. The magnetic polarizability tensor (MPT) provides a characterization of a conducting permeable object using a small number of coefficients, has an explicit formula for the calculation of their coefficients, and a well understood frequency behavior, which we call its spectral signature. However, to compute such signatures, and build a library of them for object classification, requires the repeated solution of a transmission problem, which is typically accomplished approximately using a finite element discretization. To reduce the computational cost, we propose an efficient reduced order model (ROM) that further reduces the problem using a proper orthogonal decomposition for the rapid computation of MPT spectral signatures. Our ROM benefits from a posteriori error estimates of the accuracy of the predicted MPT coefficients with respect to those obtained with finite element solutions. These estimates can be computed cheaply during the online stage of the ROM allowing the ROM prediction to be certified. To further increase the efficiency of the computation of the MPT spectral signature, we provide scaling results, which enable an immediate calculation of the signature under changes in the object size or conductivity. We illustrate our approach by application to a range of homogenous and inhomogeneous conducting permeable objects. [ABSTRACT FROM AUTHOR]

Summary: The design of a new magnetic resonance imaging (MRI) scanner requires multiple numerical simulations of the same magneto‐mechanical problem for varying model parameters, such as frequency and electric conductivity, in order to ensure that the vibrations, noise, and heat dissipation are minimized. The high computational cost required for these repeated simulations leads to a bottleneck in the design process due to an increased design time and, thus, a higher cost. To alleviate these issues, the application of reduced order modeling techniques, which are able to find a general solution to high‐dimensional parametric problems in a very efficient manner, is considered. Building on the established proper orthogonal decomposition technique available in the literature, the main novelty of this work is an efficient implementation for the solution of 3D magneto‐mechanical problems in the context of challenging MRI configurations. This methodology provides a general solution for varying parameters of interest. The accuracy and efficiency of the method are proven by applying it to challenging MRI configurations and comparing with the full‐order solution. [ABSTRACT FROM AUTHOR]

Summary: Multiquery problems such as uncertainty quantification (UQ), optimization of a dynamical system require solving a differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. This issue can be addressed by using a cheaper reduced order model (ROM) instead. However, the ROM entails error in the solution due to approximation in a lower dimensional subspace. Moreover, the ROM lacks robustness over a wide range of parameter values. To address these issues, first, an upper bound on the norm of the state transition matrix is derived. This bound, along with the residual in the governing equation, are then used to develop an error estimator for general nonlinear dynamical systems. Furthermore, this error estimator is used in conjunction with the modified greedy search algorithm proposed by Hossain and Ghosh (Int J Numer Methods Eng, 2018;116(12‐13): 741‐758) to adaptively construct a robust proper orthogonal decomposition‐based ROM. This adaptive ROM is subsequently deployed for UQ by invoking it in a statistical simulation. Two numerical studies: (i) viscous Burgers' equation and (ii) beam on nonlinear Winkler foundation, showed an improved accuracy of the error estimator compared to the current literature. A significant computational speed‐up in UQ is achieved. [ABSTRACT FROM AUTHOR]

Summary: A reduced order model designed by means of a variational multiscale stabilized formulation has been applied successfully to fluid‐structure interaction problems in a strongly coupled partitioned solution scheme. Details of the formulation and the implementation both for the interaction problem and for the reduced models, for both the off‐line and on‐line phases, are shown. Results are obtained for cases in which both domains are reduced at the same time. Numerical results are presented for a semistationary and a fully transient case. [ABSTRACT FROM AUTHOR]

Summary: Defects such as gas pores can be formed and trapped in the fusion zone during laser welding. These defects can significantly affect the mechanical reliability of the welded joint. Current nondestructive inspection technologies are able to detect micro‐voids in a mass production context. Finite element analysis can therefore be used to assess the lifetime of an observed component via image‐based modeling. Unfortunately, running a simulation per component entails a huge and generally unaffordable computational cost. In addition, voids do not admit a parametric modeling. In this paper, a numerical method is proposed to study the impact of defects on the mechanical response of a welded joint. It is based on model order reduction techniques that decrease the computational cost of each simulation related to an image‐based modeling. To tackle the reduction of nonparametric defects, a multiscale construction of the reduced basis is proposed, although no scale separation is assumed when computing the mechanical response of the structure. Some empirical modes are representing the structure behavior and other empirical modes are related to the defect‐induced local fluctuations. They are then assembled to simulate a defective joint. Assets and limitations of the proposed method are explored through a simplified two‐dimensional (2D) problem. For the sake of reproducibility, this 2D problem is fully parametric. Finally, a realistic three‐dimensional (3D) industrial case is presented, where voids geometries have been measured via computed tomography. This 3D problem being nonparametric, fluctuation modes must be computed on the fly, once the computed tomography has been performed. [ABSTRACT FROM AUTHOR]

Summary: Reduced order models (ROMs) are becoming increasingly useful for saving computational cost in response prediction of vibrating systems. In a number of applications such as uncertainty quantification, ROMs require robustness over a wide variation of parameters. Accordingly, often they are classified as local and global, based on their performance in the parametric domain. Availability of an error bound of a ROM helps in achieving this robustness, mainly by allowing adaptivity. In this work, for a linear random dynamical system, first, an a posteriori error bound is developed based on the residual in the governing differential equation. Next, based on this error bound, two adaptive methods are proposed for building robust ROMs, that is, one for local, and another for global. While both methods are based on a greedy search approach, a modification is proposed in the training stage of the global ROM for accelerated convergence. These methods are then applied to an uncertainty quantification problem in a statistical simulation framework, and accordingly, two algorithms are developed. A detailed numerical study on vibration of a bladed disk assembly is conducted to study the accuracy and efficiency of the proposed error bound and adaptive ROMs. It is found that these adaptive ROMs provide a considerable speed‐up in estimating the probability of failure. [ABSTRACT FROM AUTHOR]

Reduced order models are useful for accelerating simulations in many-query contexts, such as optimization, uncertainty quantification, and sensitivity analysis. However, offline training of reduced order models (ROMs) can have prohibitively expensive memory and floating-point operation costs in high-performance computing applications, where memory per core is limited. To overcome this limitation for proper orthogonal decomposition, we propose a novel adaptive selection method for snapshots in time that limits offline training costs by selecting snapshots according an error control mechanism similar to that found in adaptive time-stepping ordinary differential equation solvers. The error estimator used in this work is related to theory bounding the approximation error in time of proper orthogonal decomposition-based ROMs, and memory usage is minimized by computing the singular value decomposition using a single-pass incremental algorithm. Results for a viscous Burgers' test problem demonstrate convergence in the limit as the algorithm error tolerances go to zero; in this limit, the full-order model is recovered to within discretization error. A parallel version of the resulting method can be used on supercomputers to generate proper orthogonal decomposition-based ROMs, or as a subroutine within hyperreduction algorithms that require taking snapshots in time, or within greedy algorithms for sampling parameter space. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The optimization of a vibro-acoustic problem is of main importance for passengers' comfort in transportation vehicles in terms of interior noise. Engineers use numerical tools to predict the response of this coupled problem, but it may lead to a prohibitive computational time. Based on FEM, this work aims at reducing the computational time. The first idea is to keep the same mesh of the acoustic cavity for all the structure configurations and to enrich the pressure approximation by using the extended FEM (XFEM). The enrichment is based on a Heaviside function completed at the structure tip by a continuous ramp function. The second idea is to build reduced basis. The structure basis is composed of its eigenmodes, whereas a modal synthesis method with a fixed interface is used to build the fluid basis. The interface DOFs are the enriched DOF of the XFEM, whereas the internal domain corresponds to the acoustic cavity with no structure. These two combined ideas enable to minimize the computational time in the study of the influence of the structure positions in an acoustic cavity. The method is implemented for shell structures embedded in a 3D acoustic domain. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]