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.     

Nonlinear model order reduction based on local reduced‐order bases

A new approach for the dimensional reduction via projection of nonlinear computational models based on the concept of local reduced‐order bases is presented. It is particularly suited for problems characterized by different physical regimes, parameter variations, or moving features such as discontinuities and fronts. Instead of approximating the solution of interest in a fixed lower‐dimensional subspace of global basis vectors, the proposed model order reduction method approximates this solution in a lower‐dimensional subspace generated by most appropriate local basis vectors. To this effect, the solution space is partitioned into subregions, and a local reduced‐order basis is constructed and assigned to each subregion offline. During the incremental solution online of the reduced problem, a local basis is chosen according to the subregion of the solution space where the current high‐dimensional solution lies. This is achievable in real time because the computational complexity of the selection algorithm scales with the dimension of the lower‐dimensional solution space. Because it is also applicable to the process of hyper reduction, the proposed method for nonlinear model order reduction is computationally efficient. Its potential for achieving large speedups while maintaining good accuracy is demonstrated for two nonlinear computational fluid and fluid‐structure‐electric interaction problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Proper orthogonal decomposition‐based model order reduction via radial basis functions for molecular dynamics systems

Model order reduction for molecular dynamics (MD) systems exhibits intrinsic complexities because of the highly nonlinear and nonlocal multi‐atomic interactions in high dimensions. In the present work, we introduce a proper orthogonal decomposition‐based method in conjunction with the radial basis function (RBF) approximation of the nonlinear and nonlocal potential energies and inter‐atomic forces for MD systems. This approach avoids coordinate transformation between the physical and reduced‐order coordinates, and allows the potentials and inter‐atomic forces to be calculated directly in the reduced‐order space. The RBF‐approximated potential energies and inter‐atomic forces in the reduced‐order space are discretized on the basis of the Smolyak sparse grid algorithm to further enhance the effectiveness of the proposed method. The good approximation properties of RBFs in interpolating scattered data make them ideal candidates for the reduced‐order approximation of MD inter‐atomic force calculations. The proposed approach is validated by performing the reduced‐order simulations of DNA molecules under various external loadings. Copyright © 2013 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.     

Adaptive training of local reduced bases for unsteady incompressible Navier–Stokes flows

This report presents a numerical study of reduced‐order representations for simulating incompressible Navier–Stokes flows over a range of physical parameters. The reduced‐order representations combine ideas of approximation for nonlinear terms, of local bases, and of least‐squares residual minimization. To construct the local bases, temporal snapshots for different physical configurations are collected automatically until an error indicator is reduced below a user‐specified tolerance. An adaptive time‐integration scheme is also employed to accelerate the generation of snapshots as well as the simulations with the reduced‐order representations. The accuracy and efficiency of the different representations is compared with examples with parameter sweeps. Copyright © 2015 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.     

A multiscale large time increment/FAS algorithm with time‐space model reduction for frictional contact problems

A multiscale strategy using model reduction for frictional contact computation is presented. This new approach aims to improve computation time of finite element simulations involving frictional contact between linear and elastic bodies. This strategy is based on a combination between the LATIN (LArge Time INcrement) method and the FAS multigrid solver. The LATIN method is an iterative solver operating on the whole time‐space domain. Applying an a posteriori analysis on solutions of different frictional contact problems shows a great potential as far as reducibility for frictional contact problems is concerned. Time‐space vectors forming the so‐called reduced basis depict particular scales of the problem. It becomes easy to make analogies with multigrid method to take full advantage of multiscale information. Copyright © 2013 John Wiley & Sons, Ltd.     

Limited‐memory adaptive snapshot selection for proper orthogonal decomposition

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.     

A non‐intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition

In this paper, a non‐intrusive stochastic model reduction scheme is developed for polynomial chaos representation using proper orthogonal decomposition. The main idea is to extract the optimal orthogonal basis via inexpensive calculations on a coarse mesh and then use them for the fine‐scale analysis. To validate the developed reduced‐order model, the method is implemented to: (1) the stochastic steady‐state heat diffusion in a square slab; (2) the incompressible, two‐dimensional laminar boundary‐layer over a flat plate with uncertainties in free‐stream velocity and physical properties; and (3) the highly nonlinear Ackley function with uncertain coefficients. For the heat diffusion problem, the thermal conductivity of the slab is assumed to be a stochastic field with known exponential covariance function and approximated via the Karhunen–Loève expansion. In all three test cases, the input random parameters are assumed to be uniformly distributed, and a polynomial chaos expansion is found using the regression method. The Sobol's quasi‐random sequence is used to generate the sample points. The numerical results of the three test cases show that the non‐intrusive model reduction scheme is able to produce satisfactory results for the statistical quantities of interest. It is found that the developed non‐intrusive model reduction scheme is computationally more efficient than the classical polynomial chaos expansion for uncertainty quantification of stochastic problems. The performance of the developed scheme becomes more apparent for the problems with larger stochastic dimensions and those requiring higher polynomial order for the stochastic discretization. Copyright © 2015 John Wiley & Sons, Ltd.     

On the limitations of low-rank approximations in contact mechanics problems

Typical strategies for reducing the computational cost of contact mechanics models use low-rank approximations. The underlying hypothesis is the existence of a low-dimensional subspace for the displacement field and a non-negative low-dimensional subcone for the contact pressure. However, given the local nature of contact, it seems natural to wonder whether low-rank approximations are a good fit for contact mechanics or not. In this article, we investigate some of their limitations and provide numerical evidence showing that contact pressure is linearly inseparable in many practical cases. To this end, we consider various mechanical problems involving nonadhesive frictionless contacts and analyze the performance of the low-rank models in terms of three different criteria, namely, compactness, generalization, and specificity.     

Non‐linear model reduction for uncertainty quantification in large‐scale inverse problems

We present a model reduction approach to the solution of large‐scale statistical inverse problems in a Bayesian inference setting. A key to the model reduction is an efficient representation of the non‐linear terms in the reduced model. To achieve this, we present a formulation that employs masked projection of the discrete equations; that is, we compute an approximation of the non‐linear term using a select subset of interpolation points. Further, through this formulation we show similarities among the existing techniques of gappy proper orthogonal decomposition, missing point estimation, and empirical interpolation via coefficient‐function approximation. The resulting model reduction methodology is applied to a highly non‐linear combustion problem governed by an advection–diffusion‐reaction partial differential equation (PDE). Our reduced model is used as a surrogate for a finite element discretization of the non‐linear PDE within the Markov chain Monte Carlo sampling employed by the Bayesian inference approach. In two spatial dimensions, we show that this approach yields accurate results while reducing the computational cost by several orders of magnitude. For the full three‐dimensional problem, a forward solve using a reduced model that has high fidelity over the input parameter space is more than two million times faster than the full‐order finite element model, making tractable the solution of the statistical inverse problem that would otherwise require many years of CPU time. Copyright © 2009 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.     

A semi‐implicit time‐stepping model for frictional compliant contact problems

In this paper, we formulate a semi‐implicit time‐stepping model for multibody mechanical systems with frictional, distributed compliant contacts. Employing a polyhedral pyramid model for the friction law and a distributed, linear, viscoelastic model for the contact, we obtain mixed linear complementarity formulations for the discrete‐time, compliant contact problem. We establish the existence and finite multiplicity of solutions, demonstrating that such solutions can be computed by Lemke's algorithm. In addition, we obtain limiting results of the model as the contact stiffness tends to infinity. The limit analysis elucidates the convergence of the dynamic models with compliance to the corresponding dynamic models with rigid contacts within the computational time‐stepping framework. Finally, we report numerical simulation results with an example of a planar mechanical system with a frictional contact that is modelled using a distributed, linear viscoelastic model and Coulomb's frictional law, verifying empirically that the solution trajectories converge to those obtained by the more traditional rigid‐body dynamic model. Copyright © 2004 John Wiley Sons, Ltd.     

Reduced basis method for managed pressure drilling based on a model with local nonlinearities

To circumvent restrictions of conventional drilling methods, such as slow control actions and inability to drill depleted reservoirs, a drilling method called managed pressure drilling (MPD) has been developed. In MPD, single-phase flow processes can be modeled as a feedback interconnection of a high-order linear system and a low-order nonlinear system. These nonlinearities appear locally both inside and at the boundaries of the computational domain. To obtain a fast simulation platform for real-time purposes (eg, online model-based controller implementation), model order reduction is required for MPD. However, the local nonlinearities render applying model order reduction techniques challenging. In this study, a new approach is proposed to deal with such nonlinearities within the reduced basis (RB) context and it is successfully tested on a model for MPD. Contrary to the classical RB technique, the proposed approach not only does not generate nonphysical spikes at the locations of these local nonlinearities but also yields high speedup factors. The obtained reduced-order model can be used for efficient online simulation and controller design for drilling systems with MPD.     

Non‐singular boundary integral formulations for plane interior potential problems

In this article, a non‐singular formulation of the boundary integral equation is developed to solve smooth and non‐smooth interior potential problems in two dimensions. The subtracting and adding‐back technique is used to regularize the singularity of Green's function and to simplify the calculation of the normal derivative of Green's function. After that, a global numerical integration is directly applied at the boundary, and those integration points are also taken as collocation points to simplify the algorithm of computation. The result indicates that this simple method gives the convergence speed of order N ^?3 in the smooth boundary cases for both Dirichlet and mix‐type problems. For the non‐smooth cases, the convergence speed drops at O(N ^?1/2) for the Dirichlet problems. Copyright © 2001 John Wiley & Sons, Ltd.     

POD‐based model reduction with empirical interpolation applied to nonlinear elasticity

The constantly rising demands on finite element simulations yield numerical models with increasing number of degrees‐of‐freedom. Due to nonlinearity, be it in the material model or of geometrical nature, the computational effort increases even further. For these reasons, it is today still not possible to run such complex simulations in real time parallel to, for example, an experiment or an application. Model reduction techniques such as the proper orthogonal decomposition method have been developed to reduce the computational effort while maintaining high accuracy. Nonetheless, this approach shows a limited reduction in computational time for nonlinear problems. Therefore, the aim of this paper is to overcome this limitation by using an additional empirical interpolation. The concept of the so‐called discrete empirical interpolation method is translated to problems of solid mechanics with soft nonlinear elasticity and large deformations. The key point of the presented method is a further reduction of the nonlinear term by an empirical interpolation based on a small number of interpolation indices. The method is implemented into the finite element method in two different ways, and it is extended by using different solution strategies including a numerical as well as a quasi‐Newton tangent. The new method is successfully applied to two numerical examples concerning hyperelastic as well as viscoelastic material behavior. Using the extended discrete empirical interpolation method combined with a quasi‐Newton tangent enables reductions in computational time of factor 10 with respect to the proper orthogonal decomposition method without empirical interpolation. Negligibly, orders of error can be reached. Copyright © 2015 John Wiley & Sons, Ltd.     

Duality‐based adaptive model reduction for one‐way coupled thermoelastic problems

In this paper, we derive a discrete a posteriori error estimate for a thermoelastic model problem discretized using a reduced finite element method. The problem is one‐way coupled in the sense that heat transfer affects elastic deformation but not vice versa. A reduced model is constructed using component mode synthesis in each of the heat transfer and linear elastic finite element solvers. The error estimate bounds the difference between the reduced and the standard finite element solution in terms of discrete residuals and corresponding dual weights. A main feature with the estimate is that it automatically gives a quantitative measure of the propagation of error between the solvers with respect to a certain computational goal. The analytical results are accompanied by a numerical example. Copyright © 2012 John Wiley & Sons, Ltd.     

Improving the efficiency of large scale topology optimization through on‐the‐fly reduced order model construction

Topology optimization of large scale structures is computationally expensive, notably because of the cost of solving the equilibrium equations at each iteration. Reduced order models by projection, also known as reduced basis models, have been proposed in the past for alleviating this cost. We propose here a new method for coupling reduced basis models with topology optimization to improve the efficiency of topology optimization of large scale structures. The novel approach is based on constructing the reduced basis on the fly, using previously calculated solutions of the equilibrium equations. The reduced basis is thus adaptively constructed and enriched, based on the convergence behavior of the topology optimization. A direct approach and an approach with adjusted sensitivities are described, and their algorithms provided. The approaches are tested and compared on various 2D and 3D minimum compliance topology optimization benchmark problems. Computational cost savings by up to a factor of 12 are demonstrated using the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.     

Hessian‐based model reduction for large‐scale systems with initial‐condition inputs

Reduced‐order models that are able to approximate output quantities of interest of high‐fidelity computational models over a wide range of input parameters play an important role in making tractable large‐scale optimal design, optimal control, and inverse problem applications. We consider the problem of determining a reduced model of an initial value problem that spans all important initial conditions, and pose the task of determining appropriate training sets for reduced‐basis construction as a sequence of optimization problems. We show that, under certain assumptions, these optimization problems have an explicit solution in the form of an eigenvalue problem, yielding an efficient model reduction algorithm that scales well to systems with states of high dimension. Furthermore, tight upper bounds are given for the error in the outputs of the reduced models. The reduction methodology is demonstrated for a large‐scale contaminant transport problem. Copyright © 2007 John Wiley & Sons, Ltd.