An adaptive scheme for a class of interpolatory model reduction methods for frequency response problems

Frequency sweep problems arise in many structural dynamic, acoustic, and structural acoustic applications. In each case, they incur the evaluation of a frequency response function for a typically large number of frequencies. Because each function evaluation requires the solution of an often large‐scale system of equations, frequency sweep problems are computationally intensive. Interpolatory model order reduction is a powerful tool for reducing their cost. However, the performance of this tool depends on the location and number of the interpolation frequency points. It also depends on the number of consecutive frequency derivatives of the response function that are matched at each frequency point. So far, these two choices have been made in the literature in a heuristic manner. In contrast, this paper proposes an automatic adaptive strategy based on monitoring the Euclidean norm of the relative residual associated with the function to be evaluated over the frequency band of interest. More specifically, the number of interpolation points and the number of matched frequency derivatives are adaptively increased until the global Euclidean norm of the relative residual is reduced below a user‐specified tolerance. The robustness, accuracy, and computational efficiency of this adaptive strategy are highlighted with the solution of several frequency sweep problems associated with large‐scale structural dynamic, acoustic, and structural acoustic finite element models. Copyright © 2012 John Wiley & Sons, Ltd.     

Broadband model order reduction of polynomial matrix equations using single‐point well‐conditioned asymptotic waveform evaluation: derivations and theory

To effect a model order reduction (MORe) process on a system which has a polynomial matrix equation dependence on the MORe parameter, researchers generally take one of two approaches. The first is to linearize the system by introducing extra degrees of freedom and then to solve the resulting expanded, linear system with a method such as Lanczos or Arnoldi. The second approach is to work directly with the polynomial system and use a technique such as asymptotic waveform evaluation (AWE). Of course, each approach has advantages and disadvantages. In this paper, a new technique will be presented which has some desirable characteristics from both approaches and which is able to circumvent simultaneously some of their disadvantages. It can be shown that both the Arnoldi and the AWE methods are special cases of this new technique. Finally, numerical results will show the viability of the new method, which will be called the well‐conditioned asymptotic waveform evaluation (WCAWE) method. Copyright © 2003 John Wiley & Sons, Ltd.     

结构模型与控制器降阶的主动控制试验研究

土木工程结构模型的自由度数目较大，从而导致主动控制和半主动控制实施过程中时滞效应十分突出，本文采用均衡截断技术研究了结构和控制器降阶的两种方法。第一种方法是首先采用均衡实现技术得到内部均衡系统，再根据Hankel矩阵奇异值的大小对结构模型进行降阶．对降阶的结构系统设计低阶的线性二次调节（LQ）和H∞控制器；第二种方法是对原结构模型设计LQ和H∞控制器，再采用均衡实现技术对控制器进行降阶，得到低阶控制器模型。通过大量的仿真和地震模拟振动台试验，研究了所建立方法的可行性和有效性。     

Randomized low-rank approximation methods for projection-based model order reduction of large nonlinear dynamical problems

Projection-based nonlinear model order reduction (MOR) methods typically make use of a reduced basis to approximate high-dimensional quantities. However, the most popular methods for computing V , eg, through a singular value decomposition of an m × n snapshot matrix, have asymptotic time complexities of and do not scale well as m and n increase. This is problematic for large dynamical problems with many snapshots, eg, in case of explicit integration. In this work, we propose the use of randomized methods for reduced basis computation and nonlinear MOR, which have an asymptotic complexity of only or . We evaluate the suitability of randomized algorithms for nonlinear MOR and compare them to other strategies that have been proposed to mitigate the demanding computing times incurred by large nonlinear models. We analyze the computational complexities of traditional, iterative, incremental, and randomized algorithms and compare the computing times and accuracies for numerical examples. The results indicate that randomized methods exhibit an extremely high level of accuracy in practice, while generally being faster than any other analyzed approach. We conclude that randomized methods are highly suitable for the reduction of large nonlinear problems.     

A multilevel projection‐based model order reduction framework for nonlinear dynamic multiscale problems in structural and solid mechanics

A reduction/hyper reduction framework is presented for dramatically accelerating the solution of nonlinear dynamic multiscale problems in structural and solid mechanics. At each scale, the dimensionality of the governing equations is reduced using the method of snapshots for proper orthogonal decomposition, and computational efficiency is achieved for the evaluation of the nonlinear reduced‐order terms using a carefully designed configuration of the energy conserving sampling and weighting method. Periodic boundary conditions at the microscales are treated as linear multipoint constraints and reduced via projection onto the span of a basis formed from the singular value decomposition of Lagrange multiplier snapshots. Most importantly, information is efficiently transmitted between the scales without incurring high‐dimensional operations. In this proposed proper orthogonal decomposition–energy conserving sampling and weighting nonlinear model reduction framework, training is performed in two steps. First, a microscale hyper reduced‐order model is constructed in situ, or using a mesh coarsening strategy, in order to achieve significant speedups even in non‐parametric settings. Next, a classical offline–online training approach is performed to build a parametric hyper reduced‐order macroscale model, which completes the construction of a fully hyper reduced‐order parametric multiscale model capable of fast and accurate multiscale simulations. A notable feature of this computational framework is the minimization, at the macroscale level, of the cost of the offline training using the in situ or coarsely trained hyper reduced‐order microscale model to accelerate snapshot acquisition. The effectiveness of the proposed hyper reduction framework at accelerating the solution of nonlinear dynamic multiscale problems is demonstrated for two problems in structural and solid mechanics. Speedup factors as high as five orders of magnitude are shown to be achievable. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

Fixed-precision randomized low-rank approximation methods for nonlinear model order reduction of large systems

Many model order reduction (MOR) methods employ a reduced basis to approximate the state variables. For nonlinear models, V is often computed using the snapshot method. The associated low-rank approximation of the snapshot matrix can become very costly as m,n grow larger. Widely used conventional singular value decomposition methods have an asymptotic time complexity of , which often makes them impractical for the reduction of large models with many snapshots. Different methods have been suggested to mitigate this problem, including iterative and incremental approaches. More recently, the use of fast and accurate randomized methods was proposed. However, most work so far has focused on fixed-rank approximations, where rank k is assumed to be known a priori. In case of nonlinear MOR, stating a bound on the precision is usually more appropriate. We extend existing research on randomized fixed-precision algorithms and propose a new heuristic for accelerating reduced basis computation by predicting the rank. Theoretical analysis and numerical results show a good performance of the new algorithms, which can be used for computing a reduced basis from large snapshot matrices, up to a given precision ε.     

A finite element approach combining a reduced‐order system,Padé approximants,and an adaptive frequency windowing for fast multi‐frequency solution of poro‐acoustic problems

In this work, a solution strategy is investigated for the resolution of multi‐frequency structural‐acoustic problems including 3D modeling of poroelastic materials. The finite element method is used, together with a combination of a modal‐based reduction of the poroelastic domain and a Padé‐based reconstruction approach. It thus takes advantage of the reduced‐size of the problem while further improving the computational efficiency by limiting the number of frequency resolutions of the full‐sized problem. An adaptive procedure is proposed for the discretization of the frequency range into frequency intervals of reconstructed solution. The validation is presented on a 3D poro‐acoustic example. Copyright © 2013 John Wiley & Sons, Ltd.     

Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics

A priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework. The proper generalized decomposition method is used to construct a quasi‐optimal separated representation of the random solution at some frequency samples. At each frequency, an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions. An extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure, which appears to be accurate on the whole frequency band under study and for all values of input random parameters. This strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties. This reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest. Because the dynamic response is highly nonlinear with respect to the input random parameters, a second level of separation of variables is introduced for the representation of functions of multiple random parameters, thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension. Copyright © 2011 John Wiley & Sons, Ltd.     

A ‘nodeless’ dual superelement formulation for structural and multibody dynamics application to reduction of contact problems

A ‘nodeless’ superelement formulation based on dual‐component mode synthesis is proposed, in which the superelement dynamic behavior is described in terms of modal intensities playing the role of intrinsic variables. A computational scheme is proposed to build an orthogonal set of static modes so that the system matrices can have a diagonal or nearly diagonal form, providing thus high computational efficiency for application in the context of structural dynamics as well as flexible multibody dynamics. Connection to adjacent components is expressed through kinematic relationships between intrinsic variables and local displacements. The efficiency of the method is demonstrated on a simple example involving multiple unilateral contact. Copyright © 2015 John Wiley & Sons, Ltd.     

Using Krylov‐Padé model order reduction for accelerating design optimization of structures and vibrations in the frequency domain

In many engineering problems, the behavior of dynamical systems depends on physical parameters. In design optimization, these parameters are determined so that an objective function is minimized. For applications in vibrations and structures, the objective function depends on the frequency response function over a given frequency range, and we optimize it in the parameter space. Because of the large size of the system, numerical optimization is expensive. In this paper, we propose the combination of Quasi‐Newton type line search optimization methods and Krylov‐Padé type algebraic model order reduction techniques to speed up numerical optimization of dynamical systems. We prove that Krylov‐Padé type model order reduction allows for fast evaluation of the objective function and its gradient, thanks to the moment matching property for both the objective function and the derivatives towards the parameters. We show that reduced models for the frequency alone lead to significant speed ups. In addition, we show that reduced models valid for both the frequency range and a line in the parameter space can further reduce the optimization time. Copyright © 2012 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.     

Local/global model order reduction strategy for the simulation of quasi‐brittle fracture

This paper proposes a novel technique to reduce the computational burden associated with the simulation of localized failure. The proposed methodology affords the simulation of damage initiation and propagation while concentrating the computational effort where it is most needed, that is, in the localization zones. To do so, a local/global technique is devised where the global (slave) problem (far from the zones undergoing severe damage and cracking) is solved for in a reduced space computed by the classical proper orthogonal decomposition while the local (master) degrees of freedom (associated with the part of the structure where most of the damage is taking place) are fully resolved. Both domains are coupled through a local/global technique. This method circumvents the difficulties associated with model order reduction for the simulation of highly nonlinear mechanical failure and offers an alternative or complementary approach to the development of multiscale fracture simulators. Copyright © 2011 John Wiley & Sons, Ltd.     

Dimensional model reduction in non‐linear finite element dynamics of solids and structures

A general approach to the dimensional reduction of non‐linear finite element models of solid dynamics is presented. For the Newmark implicit time‐discretization, the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments. To deal with this, it is shown how the problem can be formulated in an approximation (Ritz) basis of much smaller dimension. Similarly, the explicit Newmark algorithm can be also written in a reduced‐dimension basis, and the computation time savings in that case follow from an increase in the stable time step length. In addition, the empirical eigenvectors are proposed as the basis in which to expand the incremental problem. This basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense. Because of this ‘global’ time viewpoint, the basis need not be updated as with reduced bases computed from a linearization of the full finite element model. If the dynamics of a finite element model is expressed in terms of a small number of basis vectors, the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved. At the same time, numerical experiments indicate that by using reduced models, substantial savings can be achieved even in the pre‐asymptotic range. Furthermore, the algorithm parallelizes very efficiently. The method we present is expected to become a useful tool in applications requiring a large number of repeated non‐linear solid dynamics simulations, such as convergence studies, design optimization, and design of controllers of mechanical systems. Copyright © 2001 John Wiley & Sons, Ltd.     

An electrical circuit theoretical method for time‐ and frequency‐ domain solutions of the structural mechanics problems

Shrinking device dimensions in integrated circuit technology made integrated circuits with millions of components a reality. As a result of this advance, electrical circuit simulators that can handle very large number of components have emerged. These programs use new circuit simulation techniques and can find solutions accurately and quickly. In this paper, we apply these techniques to structural mechanics problems by adopting electrical circuit equivalents. We first apply finite element formulation to the mechanical problem. The obtained sets of equations are treated as if they are sets of equations of an equivalent electrical circuit which consists of linear circuit elements such as capacitors, inductors and controlled sources. The equivalent circuit is obtained in the form of a circuit netlist and solved using a general purpose electrical circuit simulator. Several examples showing the advantages of the circuit simulation techniques are demonstrated. Asymptotic waveform evaluation technique which is widely used for simulation of large electrical circuits is also studied for the same examples and the speed‐up advantage is shown. Copyright © 1999 John Wiley & Sons, Ltd.     

Frequency‐domain analysis of time‐integration methods for semidiscrete finite element equations—part II: Hyperbolic and parabolic–hyperbolic problems

Time‐integration methods for semidiscrete finite element equations of hyperbolic and parabolic– hyperbolic types are analysed in the frequency domain. The discrete‐time transfer functions of six popular methods are derived, and subsequently the forced response characteristics of single modes are studied in the frequency domain. Three characteristic numbers are derived which eliminate the parameter dependence of the frequency responses. Frequency responses and L₂‐norms of the phase and magnitude errors are calculated, and comparisons are given of the methods. As shown; the frequency‐domain analysis explains all time‐domain properties of the methods, and gives more insight into the subject. Copyright © 2001 John Wiley & Sons, Ltd.