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 nonintrusive proper generalized decomposition scheme with application in biomechanics

Proper generalized decomposition (PGD) is often used for multiquery and fast‐response simulations. It is a powerful tool alleviating the curse of dimensionality affecting multiparametric partial differential equations. Most implementations of PGD are intrusive extensions based on in‐house developed FE solvers. In this work, we propose a nonintrusive PGD scheme using off‐the‐shelf FE codes (such as certified commercial software) as an external solver. The scheme is implemented and monitored by in‐house flow‐control codes. A typical implementation is provided with downloadable codes. Moreover, a novel parametric separation strategy for the PGD resolution is presented. The parametric space is split into two‐ or three‐dimensional subspaces, to allow PGD technique solving problems with constrained parametric spaces, achieving higher convergence ratio. Numerical examples are provided. In particular, a practical example in biomechanics is included, with potential application to patient‐specific simulation.     

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.     

Proper generalized decomposition of time-multiscale models

Models encountered in computational mechanics could involve many time scales. When these time scales cannot be separated, one must solve the evolution model in the entire time interval by using the finest time step that the model implies. In some cases, the solution procedure becomes cumbersome because of the extremely large number of time steps needed for integrating the evolution model in the whole time interval. In this paper, we considered an alternative approach that lies in separating the time axis (one-dimensional in nature) in a multidimensional time space. Then, for circumventing the resulting curse of dimensionality, the proper generalized decomposition was applied allowing a fast solution with significant computing time savings with respect to a standard incremental integration. Copyright © 2011 John Wiley & Sons, Ltd.     

Error estimation for proper generalized decomposition solutions: A dual approach

The proper generalized decomposition is a well-established reduced order method, used to efficiently obtain approximate solutions of multi-dimensional problems in a procedure that controls the effects of the “curse of dimensionality.” The question of assessing the quality of the solutions obtained and adapting the approximations assumed, for example, the finite element meshes used, so that the best result is obtained at minimal cost, remains a relevant challenge. This article deals with finite element solutions for solid mechanics problems, using the error obtained from a dual analysis, the difference between complementary solutions, to bound the error in the solutions and to drive an optimal adaptivity process, which obtains meshes with errors significantly lower than those obtained using a uniform refinement.     

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.     

Coupling finite element and reliability analysis through proper generalized decomposition model reduction

The FEM is the main tool used for structural analysis. When the design of the mechanical system involves uncertain parameters, a coupling of the FEM with reliability analysis algorithms allows to compute the failure probability of the system. However, this coupling leads to successive finite element analysis of parametric models involving high computational effort. Over the past years, model reduction techniques have been developed in order to reduce the computational requirements in the numerical simulation of complex models. The objective of this work is to propose an efficient methodology to compute the failure probability for a multi‐material elastic structure, where the Young moduli are considered as uncertain variables. A proper generalized decomposition algorithm is developed to compute the solution of parametric multi‐material model. This parametrized solution is used in conjunction with a first‐order reliability method to compute the failure probability of the structure. Applications to multilayered structures in two‐dimensional plane elasticity are presented.Copyright © 2013 John Wiley & Sons, Ltd.     

Model order reduction for hyperelastic materials

In this paper, we develop a novel algorithm for the dimensional reduction of the models of hyperelastic solids undergoing large strains. Unlike standard proper orthogonal decomposition methods, the proposed algorithm minimizes the use of the Newton algorithms in the search of non‐linear equilibrium paths of elastic bodies. The proposed technique is based upon two main ingredients. On one side, the use of classic proper orthogonal decomposition techniques, that extract the most valuable information from pre‐computed, complete models. This information is used to build global shape functions in a Ritz‐like framework. On the other hand, to reduce the use of Newton procedures, an asymptotic expansion is made for some variables of interest. This expansion shows the interesting feature of possessing one unique tangent operator for all the terms of the expansion, thus minimizing the updating of the tangent stiffness matrix of the problem. The paper is completed with some numerical examples in order to show the performance of the technique in the framework of hyperelastic (Kirchhoff–Saint Venant and neo‐Hookean) solids. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiscale proper generalized decomposition based on the partition of unity

Solutions of partial differential equations could exhibit a multiscale behavior. Standard discretization techniques are constraints to mesh up to the finest scale to predict accurately the response of the system. The proposed methodology is based on the standard proper generalized decomposition rationale; thus, the PDE is transformed into a nonlinear system that iterates between microscale and macroscale states, where the time coordinate could be viewed as a 2D time, representing the microtime and macrotime scales. The macroscale effects are taken into account because of an FEM-based macrodiscretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the domain. The proposed methodology can be seen as an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors.     

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.     

The proper generalized decomposition for the simulation of delamination using cohesive zone model

The use of cohesive zone models is an efficient way to treat the damage especially when the crack path is known a priori. It is the case in the modeling of delamination in composite laminates. However, the simulations using cohesive zone models are expensive in a computational point of view. When using implicit time integration or when solving static problems, the non‐linearity related to the cohesive model requires many iteration before reaching convergence. In explicit approaches, an important number of iterations are also needed because of the time step stability condition. In this article, a new approach based on a separated representation of the solution is proposed. The proper generalized decomposition is used to build the solution. This technique coupled with a cohesive zone model allows a significant reduction of the computational cost. The results approximated with the proper generalized decomposition are very close the ones obtained using the classical finite element approach. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

采用基于KPOD的模型降阶方法提高油藏模拟速度的研究

应用场型了型降阶技术进行了传统油藏模拟器运算速度的研究。考虑到该技术采用本征正交分解(POD)方法能够加速模拟器运算,但对系统的输入参!较敏感,降低模拟的精度和效率,采用了基于Krylov子空间和POD的KPOD方法,以便利用Arnoldi的矩匹配性质和POD的数据泛化能力减少POD方法对模型输入参数的依赖。油藏模拟实例■,KPOD方法在计算速度和精度上都优于POD方法,验证了该方法的有效性和实用性。     

Proper Generalized Decomposition model reduction in the Bayesian framework for solving inverse heat transfer problems

In this paper, the proper generalized decomposition (PGD) is used for model reduction in the solution of an inverse heat conduction problem within the Bayesian framework. Two PGD reduced order models are proposed and the approximation Error model (AEM) is applied to account for the errors between the complete and the reduced models. For the first PGD model, the direct problem solution is computed considering a separate representation of each coordinate of the problem during the process of solving the inverse problem. On the other hand, the second PGD model is based on a generalized solution integrating the unknown parameter as one of the coordinates of the decomposition. For the second PGD model, the reduced solution of the direct problem is computed before the inverse problem within the parameter space provided by the prior information about the parameters, which is required to be proper. These two reduced models are evaluated in terms of accuracy and reduction of the computational time on a transient three-dimensional two region inverse heat transfer problem. In fact, both reduced models result on substantial reduction of the computational time required for the solution of the inverse problem, and provide accurate estimates for the unknown parameter due to the application of the approximation error model approach.     

Streamline upwind/Petrov–Galerkin‐based stabilization of proper generalized decompositions for high‐dimensional advection–diffusion equations

This work is a first attempt to address efficient stabilizations of high dimensional advection–diffusion models encountered in computational physics. When addressing multidimensional models, the use of mesh‐based discretization fails because the exponential increase of the number of degrees of freedom related to a multidimensional mesh or grid, and alternative discretization strategies are needed. Separated representations involved in the so‐called proper generalized decomposition method are an efficient alternative as proven in our former works; however, the issue related to efficient stabilizations of multidimensional advection–diffusion equations has never been addressed to our knowledge. Thus, this work is aimed at extending some well‐experienced stabilization strategies widely used in the solution of 1D, 2D, or 3D advection–diffusion models to models defined in high‐dimensional spaces, sometimes involving tens of coordinates.Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

A real time procedure for affinely dependent parametric model order reduction using interpolation on Grassmann manifolds

Model order reduction helps to reduce the computational time in dealing with large dynamical systems, for example, during simulation, control, optimization. In many cases, the considered model depends on parameters; Model order reduction techniques are, therefore, preferred to symbolically preserve this dependence or to be adaptive to the change of the model caused by the variation in the values of the parameters. In this paper, we first present the application of the interpolation technique on Grassmann manifolds to this problem. We then improve the method for the models whose system matrices depend affinely on parameters by considerably reducing the computational complexity on the basis of analyzing the structure of sums of singular value decompositions and decomposing the whole procedure into offline and online stages. A numerical example is shown to illustrate the method as well as to prove its effectiveness. Copyright © 2012 John Wiley & Sons, Ltd.