Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy‐based mesh sampling and weighting for computational efficiency

A rigorous computational framework for the dimensional reduction of discrete, high‐fidelity, nonlinear, finite element structural dynamics models is presented. It is based on the pre‐computation of solution snapshots, their compression into a reduced‐order basis, and the Galerkin projection of the given discrete high‐dimensional model onto this basis. To this effect, this framework distinguishes between vector‐valued displacements and manifold‐valued finite rotations. To minimize computational complexity, it also differentiates between the cases of constant and configuration‐dependent mass matrices. Like most projection‐based nonlinear model reduction methods, however, its computational efficiency hinges not only on the ability of the constructed reduced‐order basis to capture the dominant features of the solution of interest but also on the ability of this framework to compute fast and accurate approximations of the projection onto a subspace of tangent matrices and/or force vectors. The computation of the latter approximations is often referred to in the literature as hyper reduction. Hence, this paper also presents the energy‐conserving sampling and weighting (ECSW) hyper reduction method for discrete (or semi‐discrete), nonlinear, finite element structural dynamics models. Based on mesh sampling and the principle of virtual work, ECSW is natural for finite element computations and preserves an important energetic aspect of the high‐dimensional finite element model to be reduced. Equipped with this hyper reduction procedure, the aforementioned Galerkin projection framework is first demonstrated for several academic but challenging problems. Then, its potential for the effective solution of real problems is highlighted with the realistic simulation of the transient response of a vehicle to an underbody blast event. For this problem, the proposed nonlinear model reduction framework reduces the CPU time required by a typical high‐dimensional model by up to four orders of magnitude while maintaining a good level of accuracy. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Accelerating iterative solution methods using reduced‐order models as solution predictors

We propose the use of reduced‐order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large‐scale numerical simulation. The acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps. The algorithm basically consists of two projection steps: (1) projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions, and (2) solving the governing equations in this reduced space and projecting the solution back on the original, high dimensional one. We applied the algorithm to numerical models for simulation of two‐phase flow through heterogeneous porous media. In particular we considered implicit‐pressure explicit‐saturation (IMPES) schemes and investigated the scope to accelerate the iterative solution of the pressure equation, which is by far the most time‐consuming part of any IMPES scheme. We achieved a substantial reduction in the number of iterations and an associated acceleration of the solution. Our largest test problem involved 93 500 variables, in which case we obtained a maximum reduction in computing time of 67%. The method is particularly attractive for problems with time‐varying parameters or source terms. Copyright © 2006 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.     

Local proper generalized decomposition

One of the main difficulties that a reduced‐order method could face is the poor separability of the solution. This problem is common to both a posteriori model order reduction (proper orthogonal decomposition, reduced basis) and a priori [proper generalized decomposition (PGD)] model order reduction. Early approaches to solve it include the construction of local reduced‐order models in the framework of POD. We present here an extension of local models in a PGD—and thus, a priori—context. Three different strategies are introduced to estimate the size of the different patches or regions in the solution manifold where PGD is applied. As will be noticed, no gluing or special technique is needed to deal with the resulting set of local reduced‐order models, in contrast to most proper orthogonal decomposition local approximations. The resulting method can be seen as a sort of a priori manifold learning or nonlinear dimensionality reduction technique. Examples are shown that demonstrate pros and cons of each strategy for different problems.     

Accelerated mesh sampling for the hyper reduction of nonlinear computational models

In nonlinear model order reduction, hyper reduction designates the process of approximating a projection‐based reduced‐order operator on a reduced mesh, using a numerical algorithm whose computational complexity scales with the small size of the projection‐based reduced‐order model. Usually, the reduced mesh is constructed by sampling the large‐scale mesh associated with the high‐dimensional model underlying the projection‐based reduced‐order model. The sampling process itself is governed by the minimization of the size of the reduced mesh for which the hyper reduction method of interest delivers the desired accuracy for a chosen set of training reduced‐order quantities. Because such a construction procedure is combinatorially hard, its key objective function is conveniently substituted with a convex approximation. Nevertheless, for large‐scale meshes, the resulting mesh sampling procedure remains computationally intensive. In this paper, three different convex approximations that promote sparsity in the solution are considered for constructing reduced meshes that are suitable for hyper reduction and paired with appropriate active set algorithms for solving the resulting minimization problems. These algorithms are equipped with carefully designed parallel computational kernels in order to accelerate the overall process of mesh sampling for hyper reduction, and therefore achieve practicality for realistic, large‐scale, nonlinear structural dynamics problems. Conclusions are also offered as to what algorithm is most suitable for constructing a reduced mesh for the purpose of hyper reduction. Copyright © 2016 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.     

Projection‐based model reduction for contact problems

To be feasible for computationally intensive applications such as parametric studies, optimization, and control design, large‐scale finite element analysis requires model order reduction. This is particularly true in nonlinear settings that tend to dramatically increase computational complexity. Although significant progress has been achieved in the development of computational approaches for the reduction of nonlinear computational mechanics models, addressing the issue of contact remains a major hurdle. To this effect, this paper introduces a projection‐based model reduction approach for both static and dynamic contact problems. It features the application of a non‐negative matrix factorization scheme to the construction of a positive reduced‐order basis for the contact forces, and a greedy sampling algorithm coupled with an error indicator for achieving robustness with respect to model parameter variations. The proposed approach is successfully demonstrated for the reduction of several two‐dimensional, simple, but representative contact and self contact computational models. Copyright © 2015 John Wiley & Sons, Ltd.     

A concurrent model reduction approach on spatial and random domains for the solution of stochastic PDEs

A methodology is introduced for rapid reduced‐order solution of stochastic partial differential equations. On the random domain, a generalized polynomial chaos expansion (GPCE) is used to generate a reduced subspace. GPCE involves expansion of the random variable as a linear combination of basis functions defined using orthogonal polynomials from the Askey series. A proper orthogonal decomposition (POD) approach coupled with the method of snapshots is used to generate a reduced solution space from the space spanned by the finite element basis functions on the spatial domain. POD methods have been extremely popular in fluid mechanics applications and have subsequently been applied to other interesting areas. They have been shown to be capable of representing complicated phenomena with a handful of degrees of freedom. This concurrent model reduction on the random and spatial domains is applied to stochastic partial differential equations (PDEs) in natural convection processes involving randomness in the porosity of the medium and the Rayleigh number. The results indicate that owing to the multiplicative nature of the concurrent model reduction, extremely large computational gains are realized without significant loss of accuracy. Copyright © 2005 John Wiley & Sons, Ltd.     

A local multiple proper generalized decomposition based on the partition of unity

It is well known that model order reduction techniques that project the solution of the problem at hand onto a low-dimensional subspace present difficulties when this solution lies on a nonlinear manifold. To overcome these difficulties (notably, an undesirable increase in the number of required modes in the solution), several solutions have been suggested. Among them, we can cite the use of nonlinear dimensionality reduction techniques or, alternatively, the employ of linear local reduced order approaches. These last approaches usually present the difficulty of ensuring continuity between these local models. Here, a new method is presented, which ensures this continuity by resorting to the paradigm of the partition of unity while employing proper generalized decompositions at each local patch.     

A reduced order model‐based preconditioner for the efficient solution of transient diffusion equations

This paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive‐definite linear systems arising from the numerical discretization of transient parabolic and self‐adjoint partial differential equations. The preconditioners are obtained by nesting appropriate projections of reduced‐order models into the classical iteration of the preconditioned conjugate gradient (PCG). The main idea is to employ the reduced‐order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases. This approach is particularly appealing for transient systems where the full‐model solution has to be computed at each time step. In these cases, the natural reduced space is the one generated by full‐model solutions at previous time steps. When increasing the size of the projection space, the proposed methodology highly reduces the system conditioning number and the number of PCG iterations at every time step. The cost of the application of the preconditioner linearly increases with the size of the projection basis, and a trade‐off must be found to effectively reduce the PCG computational cost. The quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.     

The finite element square reduced (FE2R) method with GPU acceleration: towards three‐dimensional two‐scale simulations

The FE² method is a renown computational multiscale simulation technique for solid materials with fine‐scale microstructure. It allows for the accurate prediction of the mechanical behavior of structures made of heterogeneous materials with nonlinear material behavior. However, the FE² method leads to excessive CPU time and storage requirements, even for academic two‐dimensional problems. In order to allow for realistic three‐dimensional two‐scale simulations, a significant reduction of the CPU and memory usage is required. For this purpose, the authors have recently proposed a reduced basis homogenization scheme based on a mixed incremental variational principle. The approach exploits the potential structure of generalized standard materials. Thereby, important speed‐ups and memory savings can be achieved. Using high‐performance GPUs, the reduced‐basis method can be further accelerated. In the present contribution, our previous works are combined and extended to form the FE²‐reduced method: the FE^2R. The FE^2R can be used to simulate three‐dimensional structural problems with consideration of the nonlinearity and microstructure of the underlying material at acceptable computational cost. Thereby, it allows for a new level of complexity in nonlinear multiscale simulations. Numerical examples illustrate the capabilities of the chosen approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Skeletal reduction of boundary value problems

Boundary value problems posed over thin solids are amenable to a dimensional reduction in that one or more spatial variables may be eliminated from the governing equation, resulting in significant computational gains with minimal loss in accuracy. Extant dimensional reduction techniques rely on representing the solid as a hypothetical mid‐surface plus a possibly varying thickness. Such techniques are however hard to automate since the mid‐surface is often ill‐defined and ambiguous. We propose here a skeletal representation based dimensional reduction of boundary value problems. The proposed technique has the computational advantages of mid‐surface reduction, but can be easily automated. A systematic methodology is presented for reducing boundary value problems to lower‐dimensional problems over the skeleton of a solid. The theoretical properties of the proposed method are derived, and supported by representative numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.     

An adaptive model order reduction with Quasi‐Newton method for nonlinear dynamical problems

Model Order Reduction (MOR) methods are extremely useful to reduce processing time, even nowadays, when parallel processing is possible in any personal computer. This work describes a method that combines Proper Orthogonal Decomposition (POD) and Ritz vectors to achieve an efficient Galerkin projection, which changes during nonlinear solving (online analysis). It is supported by a new adaptive strategy, which analyzes the error and the convergence rate for nonlinear dynamical problems. This model order reduction is assisted by a secant formulation which is updated by the Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) formula to accelerate convergence in the reduced space, and a tangent formulation when correction of the reduced space is needed. Furthermore, this research shows that this adaptive strategy permits correction of the reduced model at low cost and small error. Copyright © 2015 John Wiley & Sons, Ltd.     

Parametrized reduced order modeling for cracked solids

A parametrized reduced order modeling methodology for cracked two dimensional solids is presented, where the parameters correspond to geometric properties of the crack, such as location and size. The method follows the offline-online paradigm, where in the offline, training phase, solutions are obtained for a set of parameter values, corresponding to specific crack configurations and a basis for a lower dimensional solution space is created. Then in the online phase, this basis is used to obtain solutions for configurations that do not lie in the training set. The use of the same basis for different crack geometries is rendered possible by defining a reference configuration and employing mesh morphing to map the reference to different target configurations. To enable the application to complex geometries, a mesh morphing technique is introduced, based on inverse distance weighting, which increases computational efficiency and allows for special treatment of boundaries. Applications in linear elastic fracture mechanics are considered, with the extended finite element method being used to represent discontinuous and asymptotic fields.     

Time‐space PGD for the rapid solution of 3D nonlinear parametrized problems in the many‐query context

This work deals with the question of the resolution of nonlinear problems for many different configurations in order to build a ‘virtual chart’ of solutions. The targeted problems are three‐dimensional structures driven by Chaboche‐type elastic‐viscoplastic constitutive laws. In this context, parametric analysis can lead to highly expensive computations when using a direct treatment. As an alternative, we present a technique based on the use of the time‐space proper generalized decomposition in the framework of the LATIN method. To speed up the calculations in the parametrized context, we use the fact that at each iteration of the LATIN method, an approximation over the entire time‐space domain is available. Then, a global reduced‐order basis is generated, reused and eventually enriched, by treating, one‐by‐one, all the various parameter sets. The novelty of the current paper is to develop a strategy that uses the reduced‐order basis for any new set of parameters as an initialization for the iterative procedure. The reduced‐order basis, which has been built for a set of parameters, is reused to build a first approximation of the solution for another set of parameters. An error indicator allows adding new functions to the basis only if necessary. The gain of this strategy for studying the influence of material or loading variability reaches the order of 25 in the industrial examples that are presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Hierarchical model reduction for incompressible fluids in pipes

Hierarchical model reduction is intended to solve efficiently partial differential equations in domains with a geometrically dominant direction. In many engineering applications, these problems are often reduced to 1‐dimensional differential systems. This guarantees computational efficiency yet dumps local accuracy as nonaxial dynamics are dropped. Hierarchical model reduction recovers the secondary components of the dynamics of interest with a combination of different discretization techniques, following up a natural separation of variables. The dominant direction is generally solved by the finite element method or isogeometric analysis to guarantee flexibility, while the transverse components are solved by spectral methods, to guarantee a small number of degrees of freedom. By judiciously selecting the number of transverse modes, the method has been proven to improve significantly the accuracy of purely 1‐dimensional solvers, with great computational efficiency. A Cartesian framework has been used so far both in slab domains and cylindrical pipes (including arteries) mapped to Cartesian reference domains. In this paper, we investigate the alternative use of a polar coordinates system for the transverse dynamics in circular or elliptical pipes. This seems a natural choice for applications like computational hemodynamics. In spite of this, the selection of a basis function set for the transverse dynamics is troublesome. As pointed out in the literature—even for simple elliptical problems—there is no “best” basis available. In this paper, we perform an extensive investigation of hierarchical model reduction in polar coordinates to discuss different possible choices for the transverse basis, pointing out pros and cons of the polar coordinate system.     

Hyper‐reduction of mechanical models involving internal variables

We propose to improve the efficiency of the computation of the reduced‐state variables related to a given reduced basis. This basis is supposed to be built by using the snapshot proper orthogonal decomposition (POD) model reduction method. In the framework of non‐linear mechanical problems involving internal variables, the local integration of the constitutive laws can dramatically limit the computational savings provided by the reduction of the order of the model. This drawback is due to the fact that, using a Galerkin formulation, the size of the reduced basis has no effect on the complexity of the constitutive equations. In this paper we show how a reduced‐basis approximation and a Petrov–Galerkin formulation enable to reduce the computational effort related to the internal variables. The key concept is a reduced integration domain where the integration of the constitutive equations is performed. The local computations being not made over the entire domain, we extrapolate the computed internal variable over the full domain using POD vectors related to the internal variables. This paper shows the improvement of the computational saving obtained by the hyper‐reduction of the elasto‐plastic model of a simple structure. Copyright © 2008 John Wiley & Sons, Ltd.