A method for interpolating on manifolds structural dynamics reduced‐order models

A rigorous method for interpolating a set of parameterized linear structural dynamics reduced‐order models (ROMs) is presented. By design, this method does not operate on the underlying set of parameterized full‐order models. Hence, it is amenable to an online real‐time implementation. It is based on mapping appropriately the ROM data onto a tangent space to the manifold of symmetric positive‐definite matrices, interpolating the mapped data in this space and mapping back the result to the aforementioned manifold. Algorithms for computing the forward and backward mappings are offered for the case where the ROMs are derived from a general Galerkin projection method and the case where they are constructed from modal reduction. The proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input. In all cases, good accuracy is demonstrated at real‐time processing speeds. Copyright © 2009 John Wiley & Sons, Ltd.     

The global modal parameterization for non‐linear model‐order reduction in flexible multibody dynamics

In flexible multibody dynamics, advanced modelling methods lead to high‐order non‐linear differential‐algebraic equations (DAEs). The development of model reduction techniques is motivated by control design problems, for which compact ordinary differential equations (ODEs) in closed‐form are desirable. In a linear framework, reduction techniques classically rely on a projection of the dynamics onto a linear subspace. In flexible multibody dynamics, we propose to project the dynamics onto a submanifold of the configuration space, which allows to eliminate the non‐linear holonomic constraints and to preserve the Lagrangian structure. The construction of this submanifold follows from the definition of a global modal parameterization (GMP): the motion of the assembled mechanism is described in terms of rigid and flexible modes, which are configuration‐dependent. The numerical reduction procedure is presented, and an approximation strategy is also implemented in order to build a closed‐form expression of the reduced model in the configuration space. Numerical and experimental results illustrate the relevance of this approach. Copyright © 2006 John Wiley & Sons, Ltd.     

Galerkin reduced‐order modeling scheme for time‐dependent randomly parametrized linear partial differential equations

In this paper, we consider the problem of constructing reduced‐order models of a class of time‐dependent randomly parametrized linear partial differential equations. Our objective is to efficiently construct a reduced basis approximation of the solution as a function of the spatial coordinates, parameter space, and time. The proposed approach involves decomposing the solution in terms of undetermined spatial and parametrized temporal basis functions. The unknown basis functions in the decomposition are estimated using an alternating iterative Galerkin projection scheme. Numerical studies on the time‐dependent randomly parametrized diffusion equation are presented to demonstrate that the proposed approach provides good accuracy at significantly lower computational cost compared with polynomial chaos‐based Galerkin projection schemes. Comparison studies are also made against Nouy's generalized spectral decomposition scheme to demonstrate that the proposed approach provides a number of computational advantages. Copyright © 2012 John Wiley & Sons, Ltd.     

Development and application of reduced‐order modeling procedures for subsurface flow simulation

The optimization of subsurface flow processes is important for many applications, including oil field operations and the geological storage of carbon dioxide. These optimizations are very demanding computationally due to the large number of flow simulations that must be performed and the typically large dimension of the simulation models. In this work, reduced‐order modeling (ROM) techniques are applied to reduce the simulation time of complex large‐scale subsurface flow models. The procedures all entail proper orthogonal decomposition (POD), in which a high‐fidelity training simulation is run, solution snapshots are stored, and an eigen‐decomposition (SVD) is performed on the resulting data matrix. Additional recently developed ROM techniques are also implemented, including a snapshot clustering procedure and a missing point estimation technique to eliminate rows from the POD basis matrix. The implementation of the ROM procedures into a general‐purpose research simulator is described. Extensive flow simulations involving water injection into a geologically complex 3D oil reservoir model containing 60 000 grid blocks are presented. The various ROM techniques are assessed in terms of their ability to reproduce high‐fidelity simulation results for different well schedules and also in terms of the computational speedups they provide. The numerical solutions demonstrate that the ROM procedures can accurately reproduce the reference simulations and can provide speedups of up to an order of magnitude when compared with a high‐fidelity model simulated using an optimized solver. Copyright © 2008 John Wiley & Sons, Ltd.     

Linear random vibration by stochastic reduced‐order models

A practical method is developed for calculating statistics of the states of linear dynamic systems with deterministic properties subjected to non‐Gaussian noise and systems with uncertain properties subjected to Gaussian and non‐Gaussian noise. These classes of problems are relevant as most systems have uncertain properties, physical noise is rarely Gaussian, and the classical theory of linear random vibration applies to deterministic systems and can only deliver the first two moments of a system state if the noise is non‐Gaussian. The method (1) is based on approximate representations of all or some of the random elements in the definition of linear random vibration problems by stochastic reduced‐order models (SROMs), that is, simple random elements having a finite number of outcomes of unequal probabilities, (2) can be used to calculate statistics of a system state beyond its first two moments, and (3) establishes bounds on the discrepancy between exact and SROM‐based solutions of linear random vibration problems. The implementation of the method has required to integrate existing and new numerical algorithms. Examples are presented to illustrate the application of the proposed method and assess its accuracy. Copyright © 2009 John Wiley & Sons, Ltd.     

Model reduction for LPV systems based on approximate modal decomposition

The paper presents a novel model order reduction technique for large‐scale linear parameter‐varying (LPV) systems. The approach is based on decoupling the original dynamics into smaller dimensional LPV subsystems that can be independently reduced by parameter‐varying reduction methods. The decomposition starts with the construction of a modal transformation that separates the modal subsystems. Hierarchical clustering is applied then to collect the dynamically similar modal subsystems into larger groups. The resulting parameter‐varying subsystems are then independently reduced. This approach substantially differs from most of the previously proposed LPV model reduction techniques, since it performs manipulations on the LPV model itself, instead of on a set of linear time‐invariant models defined at fixed scheduling parameter values. Therefore, the interpolation, which is often a challenging part in reduction techniques, is inherently solved. The applicability of the developed algorithm is thoroughly investigated and demonstrated by numerical case studies.     

Stabilization of projection‐based reduced‐order models

A rigorous method for stabilizing projection‐based linear reduced‐order models without significantly affecting their accuracy is proposed. Unlike alternative approaches, this method is computationally efficient. It requires primarily the solution of a small‐scale convex optimization problem. Furthermore, it is nonintrusive in the sense that it operates directly on readily available reduced‐order operators. These can be precomputed using any data compression technique including balanced truncation, balanced proper orthogonal decomposition, proper orthogonal decomposition, or moment matching. The proposed method is illustrated with three applications: the stabilization of the reduction of the Computational Fluid Dynamics‐based model of a linearized unsteady supersonic flow, the reduction of a Computational Structural Dynamics system, and the stabilization of the reduction of a coupled Computational Fluid Dynamics–Computational Structural Dynamics model of a linearized aeroelastic system in the transonic flow regime. Copyright © 2012 John Wiley & Sons, Ltd.     

Subspace recycling accelerates the parametric macro‐modeling of MEMS

A fast computational technique that speeds up the process of parametric macro‐model extraction is proposed. An efficient starting point is the technique of parametric model order reduction (PMOR). The key step in PMOR is the computation of a projection matrix V, which requires the computation of multiple moment matrices of the underlying system. In turn, for each moment matrix, a linear system with multiple right‐hand sides has to be solved. Usually, a considerable number of linear systems must be solved when the system includes more than two free parameters. If the original system is of very large size, the linear solution step is computationally expensive. In this paper, the subspace recycling algorithm outer generalized conjugate residual method combined with generalized minimal residual method with deflated restarting (GCRO‐DR), is considered as a basis to solve the sequence of linear systems. In particular, two more efficient recycling algorithms, G‐DRvar1 and G‐DRvar2, are proposed. Theoretical analysis and simulation results show that both the GCRO‐DR method and its variants G‐DRvar1 and G‐DRvar2 are very efficient when compared with the standard solvers. Furthermore, the presented algorithms overcome the bottleneck of a recently proposed subspace recycling method the modified Krylov recycling generalized minimal residual method. From these subspace recycling algorithms, a PMOR process for macro‐model extraction can be significantly accelerated. Copyright © 2013 John Wiley & Sons, Ltd.     

A POD reduced‐order model for eigenvalue problems with application to reactor physics

A reduced‐order model based on proper orthogonal decomposition (POD) has been presented and applied to solving eigenvalue problems. The model is constructed via the method of snapshots, which is based upon the singular value decomposition of a matrix containing the characteristics of a solution as it evolves through time. Part of the novelty of this work is in how this snapshot data are generated, and this is through the recasting of eigenvalue problem, which is time independent, into a time‐dependent form. Instances of time‐dependent eigenfunction solutions are therefore used to construct the snapshot matrix. The reduced order model's capabilities in efficiently resolving eigenvalue problems that typically become computationally expensive (using standard full model discretisations) has been demonstrated. Although the approach can be adapted to most general eigenvalue problems, the examples presented here are based on calculating dominant eigenvalues in reactor physics applications. The approach is shown to reconstruct both the eigenvalues and eigenfunctions accurately using a significantly reduced number of unknowns in comparison with ‘full’ models based on finite element discretisations. The novelty of this paper therefore includes a new approach to generating snapshots, POD's application to large‐scale eigenvalue calculations, and reduced‐order model's application in reactor physics.Copyright © 2013 John Wiley & Sons, Ltd.     

Hybrid hyper‐reduced modeling for contact mechanics problems

The model reduction of mechanical problems involving contact remains an important issue in computational solid mechanics. In this article, we propose an extension of the hyper‐reduction method based on a reduced integration domain to frictionless contact problems written by a mixed formulation. As the potential contact zone is naturally reduced through the reduced mesh involved in hyper‐reduced equations, the dual reduced basis is chosen as the restriction of the dual full‐order model basis. We then obtain a hybrid hyper‐reduced model combining empirical modes for primal variables with finite element approximation for dual variables. If necessary, the inf‐sup condition of this hybrid saddle‐point problem can be enforced by extending the hybrid approximation to the primal variables. This leads to a hybrid hyper‐reduced/full‐order model strategy. This way, a better approximation on the potential contact zone is further obtained. A posttreatment dedicated to the reconstruction of the contact forces on the whole domain is introduced. In order to optimize the offline construction of the primal reduced basis, an efficient error indicator is coupled to a greedy sampling algorithm. The proposed hybrid hyper‐reduction strategy is successfully applied to a 1‐dimensional static obstacle problem with a 2‐dimensional parameter space and to a 3‐dimensional contact problem between two linearly elastic bodies. The numerical results show the efficiency of the reduction technique, especially the good approximation of the contact forces compared with other methods.     

On the use of modal derivatives for nonlinear model order reduction

Modal derivative is an approach to compute a reduced basis for model order reduction of large‐scale nonlinear systems that typically stem from the discretization of partial differential equations. In this way, a complex nonlinear simulation model can be integrated into an optimization problem or the design of a controller, based on the resulting small‐scale state‐space model. We investigate the approximation properties of modal derivatives analytically and thus lay a theoretical foundation of their use in model order reduction, which has been missing so far. Concentrating on the application field of structural mechanics and structural dynamics, we show that the concept of modal derivatives can also be applied as nonlinear extension of the Craig–Bampton family of methods for substructuring. We furthermore generalize the approach from a pure projection scheme to a novel reduced‐order modeling method that replaces all nonlinear terms by quadratic expressions in the reduced state variables. This complexity reduction leads to a frequency‐preserving nonlinear quadratic state‐space model. Numerical examples with carefully chosen nonlinear model problems and three‐dimensional nonlinear elasticity confirm the analytical properties of the modal derivative reduction and show the potential of the proposed novel complexity reduction methods, along with the current limitations. Copyright © 2016 John Wiley & Sons, Ltd.     

A new unified theory underlying time dependent linear first‐order systems: a prelude to algorithms by design

A new unified theory underlying the theoretical design of linear computational algorithms in the context of time dependent first‐order systems is presented. Providing for the first time new perspectives and fresh ideas, and unlike various formulations existing in the literature, the present unified theory involves the following considerations: (i) it leads to new avenues for designing new computational algorithms to foster the notion of algorithms by design and recovering existing algorithms in the literature, (ii) describes a theory for the evolution of time operators via a unified mathematical framework, and (iii) places into context and explains/contrasts future new developments including existing designs and the various relationships among the different classes of algorithms in the literature such as linear multi‐step methods, sub‐stepping methods, Runge–Kutta type methods, higher‐order time accurate methods, etc. Subsequently, it provides design criteria and guidelines for contrasting and evaluating time dependent computational algorithms. The linear computational algorithms in the context of first‐order systems are classified as distinctly pertaining to Type 1, Type 2, and Type 3 classifications of time discretized operators. Such a distinct classification, provides for the first time, new avenues for designing new computational algorithms not existing in the literature and recovering existing algorithms of arbitrary order of time accuracy including an overall assessment of their stability and other algorithmic attributes. Consequently, it enables the evaluation and provides the relationships of computational algorithms for time dependent problems via a standardized measure based on computational effort and memory usage in terms of the resulting number of equation systems and the corresponding number of system solves. A generalized stability and accuracy limitation barrier theorem underlies the generic designs of computational algorithms with arbitrary order of accuracy and establishes guidelines which cannot be circumvented. In summary, unlike the traditional approaches and classical school of thought customarily employed in the theoretical development of computational algorithms, the unified theory underlying time dependent first‐order systems serves as a viable avenue to foster the notion of algorithms by design. Copyright © 2004 John Wiley & Sons, Ltd.     

An interpolation‐based fast multipole method for higher‐order boundary elements on parametric surfaces

In this article, a black‐box higher‐order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed. Such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis. Due to the exact surface representation, the rate of convergence of higher‐order methods is not limited by approximation errors of the surface. An element‐wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying Galerkin method. By performing the interpolation for the fast multipole method directly on the reference domain, the cost complexity in the polynomial degree is reduced by one order. This gain is independent of the application of either ‐ or ‐matrices. In fact, several simplifications in the construction of ‐matrices are pointed out, which are a by‐product of the surface representation. Extensive numerical examples are provided in order to quantify and qualify the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Fast time‐domain simulation for large‐order linear time‐invariant state space systems

Time‐domain simulation is essential for both analysis and design of complex systems. Unfortunately, high model fidelity leads to large system size and bandwidths, often causing excessive computation and memory saturation. In response we develop an efficient scheme for large‐order linear time‐invariant systems. First, the A matrix is block diagonalized. Then, subsystems of manageable dimensions and bandwidth are formed, allowing multiple sampling rates. Each subsystem is then discretized using a O(n_s) scheme, where n_s is the number of states. Subsequently, a sparse matrix O(n_s) discrete‐time system solver is employed to compute the history of the state and output. Finally, the response of the original system is obtained by superposition. In practical engineering applications, closing feedback loops and cascading filters can hinder the efficient use of the simulation scheme. Solutions to these problems are addressed in the paper. The simulation scheme, implemented as a MATLAB function fastlsim, is benchmarked against the standard LTI system simulator lsim and is shown to be superior for medium to large systems. The algorithm scales close to O(n) for a set of benchmarked systems. Simulation of a high‐fidelity model of (n_s ≈ 2200) the Space Interferometry Mission spacecraft illustrates real world application of the method. Copyright © 2005 John Wiley & Sons, Ltd.     

Transient analysis of electro‐osmotic transport by a reduced‐order modelling approach

Transient behaviour of electro‐osmotic transport in typical electrokinetic channels is studied in this paper. The time needed for the electro‐osmotic flow to reach steady‐state exhibits multiple time scales depending on whether the flow is governed by either a viscous force, electrokinetic force or by a combination of both. When an intersection is present in the electrokinetic channel, such as in a cross or a T‐channel, the flow in the main channel and in the intersection gets to steady‐state at different times. A weighted Karhunen–Loève (KL) decomposition method is proposed in this paper to generate the global basis function for reduced‐order simulation. The key idea in a weighted KL approach is that, instead of minimizing a least‐squares measure of ‘error’ between the linear subspace spanned by the basis functions and the observation space, we minimize the weighted ‘error’ between the two spaces. The global basis functions in a weighted KL approach can be generated by computing the singular value decomposition (SVD) of the matrix containing the weighted snapshots. We show that the weighted KL decomposition based reduced‐order model is computationally more efficient and can capture the multiple time scales encountered in electro‐osmotic transport much more effectively compared to the classical KL decomposition based reduced‐order model. Copyright © 2003 John Wiley & Sons, Ltd.     

Efficient hyper‐reduced‐order model (HROM) for thermal analysis in the moving frame

The hyper‐reduced‐order model (HROM) is proposed for the thermal calculation with a constant moving thermal load. Firstly, the constant velocity transient process is simplified to a steady‐state process in the moving frame. Secondly, the control volume is determined by the temperature rate, and the thermal equilibrium equation in the moving frame is derived by introducing an advective term containing the loading velocity. Thirdly, the HROM is performed on the control volume with a moving frame formulation. This HROM has been applied to the thermal loading on brick and ring disk specimens with a CPU gain of the order of 7 (10⁷). In addition, two strategies are proposed for the HROM to improve its precision. Moreover, the high efficiency and high accuracy are kept for the parametric studies on thermal conductivity and amplitude of heat flux based on the developed HROM. Copyright © 2016 John Wiley & Sons, Ltd.     

Padé approximants and the modal connection: Towards increased robustness for fast parametric sweeps

To increase the robustness of a Padé‐based approximation of parametric solutions to finite element problems, an a priori estimate of the poles is proposed. The resulting original approach is shown to allow for a straightforward, efficient, subsequent Padé‐based expansion of the solution vector components, overcoming some of the current convergence and robustness limitations. In particular, this enables for the intervals of approximation to be chosen a priori in direct connection with a given choice of Padé approximants. The choice of these approximants, as shown in the present work, is theoretically supported by the Montessus de Ballore theorem, concerning the convergence of a series of approximants with fixed denominator degrees. Key features and originality of the proposed approach are (1) a component‐wise expansion which allows to specifically target subsets of the solution field and (2) the a priori, simultaneous choice of the Padé approximants and their associated interval of convergence for an effective and more robust approximation. An academic acoustic case study, a structural‐acoustic application, and a larger acoustic problem are presented to demonstrate the potential of the approach proposed.