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.     

Comparison between the modal identification method and the POD‐Galerkin method for model reduction in nonlinear diffusive systems

This paper presents a comparison between the modal identification method (MIM) and the proper orthogonal decomposition‐Galerkin (POD‐G) method for model reduction. An example of application on a nonlinear diffusive system is used to illustrate the study. The study shows that in both methods, the state formulation of the nonlinear diffusive equation may be similar. However, the ideas behind both methods are completely different. The considered example shows that, for both methods, reducing the order up to 99.5% gives enough accuracy to simulate the dynamic of the original system. It is also seen in this example that the reduced model given through the MIM are slightly faster and more accurate than the ones given through the POD‐G method. Copyright © 2006 John Wiley & Sons, Ltd.     

A modified Newton‐type Koiter‐Newton method for tracing the geometrically nonlinear response of structures

The Koiter‐Newton (KN) method is a combination of local multimode polynomial approximations inspired by Koiter's initial postbuckling theory and global corrections using the standard Newton method. In the original formulation, the local polynomial approximation, called a reduced‐order model, is used to make significantly more accurate predictions compared to the standard linear prediction used in conjunction with Newton method. The correction to the exact equilibrium path relied exclusively on Newton‐Raphson method using the full model. In this paper, we proposed a modified Newton‐type KN method to trace the geometrically nonlinear response of structures. The developed predictor‐corrector strategy is applied to each predicted solution of the reduced‐order model. The reduced‐order model can be used also in the correction phase, and the exact full nonlinear model is applied only to calculate force residuals. Remainder terms in both the displacement expansion and the reduced‐order model are well considered and constantly updated during correction. The same augmented finite element model system is used for both the construction of the reduced‐order model and the iterations for correction. Hence, the developed method can be seen as a particular modified Newton method with a constant iteration matrix over the single KN step. This significantly reduces the computational cost of the method. As a side product, the method has better error control, leading to more robust step size adaptation strategies. Numerical results demonstrate the effectiveness of the method in treating nonlinear buckling problems.     

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.     

Adaptive POD‐based low‐dimensional modeling supported by residual estimates

An adaptive low‐dimensional model is considered to simulate time‐dependent dynamics in nonlinear dissipative systems governed by PDEs. The method combines an inexpensive POD‐based Galerkin system with short runs of a standard numerical solver that provides the snapshots necessary to first construct and then update the POD modes. Switching between the numerical solver and the Galerkin system is decided ‘on the fly’ by monitoring (i) a truncation error estimate and (ii) a residual estimate. The latter estimate is used to control the mode truncation instability and highly improves former adaptive strategies that detected this instability by monitoring consistency with a second instrumental Galerkin system based on a larger number of POD modes. The most computationally expensive run of the numerical solver occurs at the outset, when the whole set of POD modes is calculated. This step is improved by using mode libraries, which may either be generic or result from former applications of the method. The outcome is a flexible, robust, computationally inexpensive procedure that adapts itself to the local dynamics by using the faster Galerkin system for the majority of the time and few, on demand, short runs of a numerical solver. The method is illustrated considering the complex Ginzburg–Landau equation in one and two space dimensions. Copyright © 2015 John Wiley & Sons, Ltd.     

Remarks on the efficiency of POD for model reduction in non‐linear dynamics of continuous elastic systems

The first objective of this paper is to analyse the efficiency of the reduced models constructed using the proper orthogonal decomposition (POD)‐basis and the LIN‐basis in non‐linear dynamics for continuous elastic systems. The POD‐basis is the Hilbertian basis constructed with the POD method while the LIN‐basis is the Hilbertian basis derived from the generalized continuous eigenvalue problem associated with the underlying conservative part of the continuous elastic system and usually called the eigenmodes of vibration. The efficiency of the POD‐basis or the LIN‐basis is related to the rate of convergence in the frequency domain of the solution constructed with the reduced model with respect to its dimension. A basis will be more efficient than another if the reduced‐order solution of the Galerkin projection converges to the solution of the dynamical system more rapidly than the reduced‐order solution of the other. As a second objective of this paper, we present the usual results concerning the POD method using a continuous formulation, with respect to both time and space variables, and then deriving the numerical approximations. Such a presentation allows convergence discussions to be treated. Six examples in non‐linear elastodynamics problems are presented in order to analyse the efficiency of the POD‐basis and the LIN‐basis. It is concluded that the POD‐basis is not more efficient than the LIN‐basis for the examples treated in non‐linear elastodynamics. Copyright © 2007 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.     

Real‐time direct integration of reduced solid dynamics equations

A new method is developed here for the real‐time integration of the equations of solid dynamics based on the use of proper orthogonal decomposition (POD)–proper generalized decomposition (PGD) approaches and direct time integration. The method is based upon the formulation of solid dynamics equations as a parametric problem, depending on their initial conditions. A sort of black‐box integrator that takes the resulting displacement field of the current time step as input and (via POD) provides the result for the subsequent time step at feedback rates on the order of 1 kHz is obtained. To avoid the so‐called curse of dimensionality produced by the large amount of parameters in the formulation (one per degree of freedom of the full model), a combined POD–PGD strategy is implemented. Examples that show the promising results of this technique are included. Copyright © 2014 John Wiley & Sons, Ltd.     

Global and local POD models for the prediction of compressible flows with DG methods

Proper orthogonal decomposition (POD) allows to compress information by identifying the most energetic modes obtained from a database of snapshots. In this work, POD is used to predict the behavior of compressible flows by means of global and local approaches, which exploit some features of a discontinuous Galerkin spatial discretization. The presented global approach requires the definition of high‐order and low‐order POD bases, which are built from a database of high‐fidelity simulations. Predictions are obtained by performing a cheap low‐order simulation whose solution is projected on the low‐order basis. The projection coefficients are then used for the reconstruction with the high‐order basis. However, the nonlinear behavior related to the advection term of the governing equations makes the use of global POD bases quite problematic. For this reason, a second approach is presented in which an empirical POD basis is defined in each element of the mesh. This local approach is more intrusive with respect to the global approach but it is able to capture better the nonlinearities related to advection. The two approaches are tested and compared on the inviscid compressible flow around a gas‐turbine cascade and on the compressible turbulent flow around a wind turbine airfoil.     

Generalized finite element method using proper orthogonal decomposition

A methodology is presented for generating enrichment functions in generalized finite element methods (GFEM) using experimental and/or simulated data. The approach is based on the proper orthogonal decomposition (POD) technique, which is used to generate low‐order representations of data that contain general information about the solution of partial differential equations. One of the main challenges in such enriched finite element methods is knowing how to choose, a priori, enrichment functions that capture the nature of the solution of the governing equations. POD produces low‐order subspaces, that are optimal in some norm, for approximating a given data set. For most problems, since the solution error in Galerkin methods is bounded by the error in the best approximation, it is expected that the optimal approximation properties of POD can be exploited to construct efficient enrichment functions. We demonstrate the potential of this approach through three numerical examples. Best‐approximation studies are conducted that reveal the advantages of using POD modes as enrichment functions in GFEM over a conventional POD basis. Copyright © 2009 John Wiley & Sons, Ltd.     

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L² inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well‐posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well‐posed and stable far‐field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty‐like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd.     

A low‐cost,goal‐oriented ‘compact proper orthogonal decomposition’ basis for model reduction of static systems

A novel model reduction technique for static systems is presented. The method is developed using a goal‐oriented framework, and it extends the concept of snapshots for proper orthogonal decomposition (POD) to include (sensitivity) derivatives of the state with respect to system input parameters. The resulting reduced‐order model generates accurate approximations due to its goal‐oriented construction and the explicit ‘training’ of the model for parameter changes. The model is less computationally expensive to construct than typical POD approaches, since efficient multiple right‐hand side solvers can be used to compute the sensitivity derivatives. The effectiveness of the method is demonstrated on a parameterized aerospace structure problem. Copyright © 2010 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.     

A mixed tetrahedral element with nodal rotations for large‐displacement analysis of inelastic structures

A novel mixed four‐node tetrahedral finite element, equipped with nodal rotational degrees of freedom, is presented. Its formulation is based on a Hu–Washizu‐type functional, suitable to the treatment of material nonlinearities. Rotation and skew‐symmetric stress fields are assumed as independent variables, the latter entering the functional to impose rotational compatibility and suppress spurious modes. Exploiting the choice of equal interpolation for strain and symmetric stress fields, a robust element state determination procedure, requiring no element‐level iteration, is proposed. The mixed element stability is assessed by means of an original and effective numerical test. The extension of the present formulation to geometric nonlinear problems is achieved through a polar decomposition‐based corotational framework. After validation in both material and geometric nonlinear context, the element performances are investigated in demanding simulations involving complex shape memory alloy structures. Supported by the comparison with available linear and quadratic tetrahedrons and hexahedrons, the numerical results prove accuracy, robustness, and effectiveness of the proposed formulation. 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.     

Multivariate predictions of local reduced‐order‐model errors and dimensions

This paper introduces multivariate input‐output models to predict the errors and bases dimensions of local parametric Proper Orthogonal Decomposition reduced‐order models. We refer to these mappings as the multivariate predictions of local reduced‐order model characteristics (MP‐LROM) models. We use Gaussian processes and artificial neural networks to construct approximations of these multivariate mappings. Numerical results with a viscous Burgers model illustrate the performance and potential of the machine learning‐based regression MP‐LROM models to approximate the characteristics of parametric local reduced‐order models. The predicted reduced‐order models errors are compared against the multifidelity correction and reduced‐order model error surrogates methods predictions, whereas the predicted reduced‐order dimensions are tested against the standard method based on the spectrum of snapshots matrix. Since the MP‐LROM models incorporate more features and elements to construct the probabilistic mappings, they achieve more accurate results. However, for high‐dimensional parametric spaces, the MP‐LROM models might suffer from the curse of dimensionality. Scalability challenges of MP‐LROM models and the feasible ways of addressing them are also discussed in this study.     

Higher‐order accurate time‐step‐integration algorithms by post‐integration techniques

In this paper, a framework to construct higher‐order‐accurate time‐step‐integration algorithms based on the post‐integration techniques is presented. The prescribed initial conditions are naturally incorporated in the formulations and can be strongly or weakly enforced. The algorithmic parameters are chosen such that unconditionally A‐stable higher‐order‐accurate time‐step‐integration algorithms with controllable numerical dissipation can be constructed for linear problems. Besides, it is shown that the order of accuracy for non‐linear problems is maintained through the relationship between the present formulation and the Runge–Kutta method. The second‐order differential equations are also considered. Numerical examples are given to illustrate the validity of the present formulation. Copyright © 2001 John Wiley & Sons, Ltd.     

基于特征正交分解的三维壁板非线性颤振分析

基于von Karman 大变形理论及活塞理论建立超音速流中壁板的气动弹性方程。采用特征正交分解法 (POD)结合向伽辽金法(Galerkin)的映射这样一种半解析法建立降阶模型(ROM)求解三维壁板的非线性气动弹性问题,并与传统的Galerkin法对比。发现并证明了POD数值模态与伽辽金法简谐基函数之间转换矩阵的正交性,从而简化了POD降阶模型的建立过程。通过数值算例考察了POD法的准确性、收敛性及高效性。结果表明POD降阶模型能够以更少的模态,更高的计算效率达到与Galerkin法同样的精度。以长宽比4为例,POD法以2个模态,3s的时间计算了壁板的振动响应;而Galerkin法需要16个模态,900s的时间。     

Cost reduction techniques for the design of non‐linear flapping wing structures

This work details a computational framework for gradient‐based optimization of a non‐linear flapping wing structure with a large number of design variables, where analytical sensitivities of the unsteady finite element system are computed using the adjoint method. Two techniques are used to reduce the large computational cost of this structural design process. The first projects the finite element system onto a reduced basis of POD modes. The second uses a monolithic time formulation with spectral elements, and can be used to compute only the desired time‐periodic response. Results are given in terms of the trade‐off between accuracy and computational efficiency of these methods for both system response and adjoint computations, for a variety of mesh/time step refinements, degrees of non‐linearity (i.e. weakly or strongly non‐linear), and harmonic content. The work concludes with the structural design of a flapping wing: the elastic deformation at the wingtip is minimized through the flapping stroke by varying the thickness of each finite element. Significant improvements in computational cost are obtained at little expense to the accuracy of the results obtained via design optimization. Published in 2011 by John Wiley & Sons, Ltd.