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.     

POD–ISAT: An efficient POD‐based surrogate approach with adaptive tabulation and fidelity regions for parametrized steady‐state PDE discrete solutions

A combination of proper orthogonal decomposition (POD) analysis and in situ adaptive tabulation (ISAT) is proposed for the representation of parameter‐dependent solutions of coupled partial differential equation problems. POD is used for the low‐order representation of the spatial fields and ISAT for the local representation of the solution in the design parameter space. The accuracy of the method is easily controlled by free threshold parameters that can be adjusted according to user needs. The method is tested on a coupled fluid‐thermal problem: the design of a simplified aircraft air control system. It is successfully compared with the standard POD; although the POD is inaccurate in certain areas of the design parameters space, the POD–ISAT method achieves accuracy thanks to trust regions based on residuals of the fluid‐thermal problem. The presented POD–ISAT approach provides flexibility, robustness and tunable accuracy to represent solutions of parametrized partial differential equations.Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

A multiscale mass scaling approach for explicit time integration using proper orthogonal decomposition

One of the main computational issues with explicit dynamics simulations is the significant reduction of the critical time step as the spatial resolution of the finite element mesh increases. In this work, a selective mass scaling approach is presented that can significantly reduce the computational cost in explicit dynamic simulations, while maintaining accuracy. The proposed method is based on a multiscale decomposition approach that separates the dynamics of the system into low (coarse scales) and high frequencies (fine scales). Here, the critical time step is increased by selectively applying mass scaling on the fine scale component only. In problems where the response is dominated by the coarse (low frequency) scales, significant increases in the stable time step can be realized. In this work, we use the proper orthogonal decomposition (POD) method to build the coarse scale space. The main idea behind POD is to obtain an optimal low‐dimensional orthogonal basis for representing an ensemble of high‐dimensional data. In our proposed method, the POD space is generated with snapshots of the solution obtained from early times of the full‐scale simulation. The example problems addressed in this work show significant improvements in computational time, without heavily compromising the accuracy of the results. Copyright © 2013 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.     

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POD降阶方法的复合材料曲壁板颤振响应特性研究

建立了三维复合材料曲壁板的气动弹性有限元方程,将本征正交分解方法(POD)应用于三维复合材料曲壁板的非线性颤振响应降阶分析中,通过POD方法构造三维复合材料曲壁板颤振响应的POD模态,然后将系统的运动方程变换到POD模态坐标下,通过数值积分方法计算三维复合材料曲壁板的颤振响应,与传统的模态缩减法计算结果相比,结果很好的吻合,且大大节省了计算时间。     

A new predictor–corrector approach for the numerical integration of coupled electromechanical equations

In this paper, a new approach for the numerical solution of coupled electromechanical problems is presented. The structure of the considered problem consists of the low‐frequency integral formulation of the Maxwell equations coupled with Newton–Euler rigid‐body dynamic equations. Two different integration schemes based on the predictor–corrector approach are presented and discussed. In the first method, the electrical equation is integrated with an implicit single‐step time marching algorithm, while the mechanical dynamics is studied by a predictor–corrector scheme. The predictor uses the forward Euler method, while the corrector is based on the trapezoidal rule. The second method is based on the use of two interleaved predictor–corrector schemes: one for the electrical equations and the other for the mechanical ones. Both the presented methods have been validated by comparison with experimental data (when available) and with results obtained by other numerical formulations; in problems characterized by low speeds, both schemes produce accurate results, with similar computation times. When high speeds are involved, the first scheme needs shorter time steps (i.e., longer computation times) in order to achieve the same accuracy of the second one. A brief discussion on extending the algorithm for simulating deformable bodies is also presented. An example of application to a two‐degree‐of‐freedom levitating device based on permanent magnets is finally reported. Copyright © 2015 John Wiley & Sons, Ltd.     

Seamless coupling of molecular dynamics and material point method via smoothed molecular dynamics

A concurrent multiscale method coupling molecular dynamics (MD) and continuum‐based material point method (MPM) is proposed. Seamless coupling is realized by utilizing smoothed molecular dynamics (SMD) method. One set of background mesh is used in SMD method. Atomic equations of motion are assembled onto mesh nodes, and atomic variables are updated with nodal increments. SMD allows much larger time step size than MD critical time step size but keeps nice global accuracy. SMD is similar to MD except for the mapping process between background mesh nodes and atoms. SMD and MPM share the feature using the background mesh to solve momentum equations and to update variables. So bridging MD and MPM via SMD is straightforward and concise. A recently proposed transition scheme based on frequency decomposition is adopted to suppress phonon reflection at MD‐SMD interface. The nodal equations in SMD–MPM interface region have contributions from both atoms and material points, which ensure the consistency between SMD region and MPM region. A multiple‐time‐step scheme is adopted for high efficiency. Numerical examples including wave propagation, bending, and crack propagation validate the proposed method, and the results show nice accuracy. The computational cost is greatly saved compared with pure MD computation. Copyright © 2017 John Wiley & Sons, Ltd.     

Velocity‐based formulations for standard and quasi‐incompressible hypoelastic‐plastic solids

We present three velocity‐based updated Lagrangian formulations for standard and quasi‐incompressible hypoelastic‐plastic solids. Three low‐order finite elements are derived and tested for non‐linear solid mechanics problems. The so‐called V‐element is based on a standard velocity approach, while a mixed velocity–pressure formulation is used for the VP and the VPS elements. The two‐field problem is solved via a two‐step Gauss–Seidel partitioned iterative scheme. First, the momentum equations are solved in terms of velocity increments, as for the V‐element. Then, the constitutive relation for the pressure is solved using the updated velocities obtained at the previous step. For the VPS‐element, the formulation is stabilized using the finite calculus method in order to solve problems involving quasi‐incompressible materials. All the solid elements are validated by solving two‐dimensional and three‐dimensional benchmark problems in statics as in dynamics. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

A GPU‐based preconditioned Newton‐Krylov solver for flexible multibody dynamics

This paper describes an approach to numerically approximate the time evolution of multibody systems with flexible (compliant) components. Its salient attribute is that at each time step, both the formulation of the system equations of motion and their numerical solution are carried out using parallel computing on graphics processing unit cards. The equations of motion are obtained using the absolute nodal coordinate formulation, yet any other multibody dynamics formalism would fit equally well the overall solution strategy outlined herein. The implicit numerical integration method adopted relies on a Newton–Krylov methodology and a parallel direct sparse solver to precondition the underlying linear system. The proposed approach, implemented in a software infrastructure available under an open‐source BSD‐3 license, leads to improvements in overall simulation times of up to one order of magnitude when compared with matrix‐free parallel solution approaches that do not use preconditioning. Copyright © 2015 John Wiley & Sons, Ltd.     

ODDLS: A new unstructured mesh finite element method for the analysis of free surface flow problems

This paper introduces a new stabilized finite element method based on the finite calculus (Comput. Methods Appl. Mech. Eng. 1998; 151 :233–267) and arbitrary Lagrangian–Eulerian techniques (Comput. Methods Appl. Mech. Eng. 1998; 155 :235–249) for the solution to free surface problems. The main innovation of this method is the application of an overlapping domain decomposition concept in the statement of the problem. The aim is to increase the accuracy in the capture of the free surface as well as in the resolution of the governing equations in the interface between the two fluids. Free surface capturing is based on the solution to a level set equation. The Navier–Stokes equations are solved using an iterative monolithic predictor–corrector algorithm (Encyclopedia of Computational Mechanics. Wiley: New York, 2004), where the correction step is based on imposing the divergence‐free condition in the velocity field by means of the solution to a scalar equation for the pressure. Examples of application of the ODDLS formulation (for overlapping domain decomposition level set) to the analysis of different free surface flow problems are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Novel partitioned time integration methods for DAE systems based on L‐stable linearly implicit algorithms

Real‐time applications based on the principle of Dynamic Substructuring require integration methods that can deal with constraints without exceeding an a priori fixed number of steps. For these applications, first we introduce novel partitioned algorithms able to solve DAEs arising from transient structural dynamics. In particular, the spatial domain is partitioned into a set of disconnected subdomains and continuity conditions of acceleration at the interface are modeled using a dual Schur formulation. Interface equations along with subdomain equations lead to a system of DAEs for which both staggered and parallel procedures are developed. Moreover under the framework of projection methods, also a parallel partitioned method is conceived. The proposed partitioned algorithms enable a Rosenbrock‐based linearly implicit LSRT2 method, to be strongly coupled with different time steps in each subdomain. Thus, user‐defined algorithmic damping and subcycling strategies are allowed. Secondly, the paper presents the convergence analysis of the novel schemes for linear single‐Degree‐of‐Freedom (DoF) systems. The algorithms are generally A‐stable and preserve the accuracy order as the original monolithic method. Successively, these results are validated via simulations on single‐ and three‐DoFs systems. Finally, the insight gained from previous analyses is confirmed by means of numerical experiments on a coupled spring–pendulum system. Copyright © 2011 John Wiley & Sons, Ltd.     

An explicit discontinuous Galerkin method for non‐linear solid dynamics: Formulation,parallel implementation and scalability properties

An explicit‐dynamics spatially discontinuous Galerkin (DG) formulation for non‐linear solid dynamics is proposed and implemented for parallel computation. DG methods have particular appeal in problems involving complex material response, e.g. non‐local behavior and failure, as, even in the presence of discontinuities, they provide a rigorous means of ensuring both consistency and stability. In the proposed method, these are guaranteed: the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation. The semi‐discrete system of ordinary differential equations is integrated in time using a conventional second‐order central‐difference explicit scheme. A stability criterion for the time integration algorithm, accounting for the influence of the DG discretization stability, is derived for the equivalent linearized system. This approach naturally lends itself to efficient parallel implementation. The resulting DG computational framework is implemented in three dimensions via specialized interface elements. The versatility, robustness and scalability of the overall computational approach are all demonstrated in problems involving stress‐wave propagation and large plastic deformations. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical finite element formulation of the Schapery non‐linear viscoelastic material model

This study presents a numerical integration method for the non‐linear viscoelastic behaviour of isotropic materials and structures. The Schapery's three‐dimensional (3D) non‐linear viscoelastic material model is integrated within a displacement‐based finite element (FE) environment. The deviatoric and volumetric responses are decoupled and the strain vector is decomposed into instantaneous and hereditary parts. The hereditary strains are updated at the end of each time increment using a recursive formulation. The constitutive equations are expressed in an incremental form for each time step, assuming a constant incremental strain rate. A new iterative procedure with predictor–corrector type steps is combined with the recursive integration method. A general polynomial form for the parameters of the non‐linear Schapery model is proposed. The consistent algorithmic tangent stiffness matrix is realized and used to enhance convergence and help achieve a correct convergent state. Verifications of the proposed numerical formulation are performed and compared with a previous work using experimental data for a glassy amorphous polymer PMMA. Copyright © 2003 John Wiley & Sons, Ltd.     

Model reduction for linear and nonlinear magneto‐quasistatic equations

We consider model reduction for magneto‐quasistatic field equations in the vector potential formulation. A finite element discretization of such equations leads to large‐scale differential‐algebraic equations of special structure. For model reduction of linear systems, we employ a balanced truncation approach, whereas nonlinear systems are reduced using a proper orthogonal decomposition method combined with a discrete empirical interpolation technique. We will exploit the special block structure of the underlying problem to improve the performance of the model reduction algorithms. Furthermore, we discuss an efficient evaluation of the Jacobi matrix required in nonlinear time integration of the reduced models. Copyright © 2017 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.     

Proper generalized decomposition of a geometrically parametrized heat problem with geophysical applications

The solution of a steady thermal multiphase problem is assumed to be dependent on a set of parameters describing the geometry of the domain, the internal interfaces and the material properties. These parameters are considered as new independent variables. The problem is therefore stated in a multidimensional setup. The proper generalized decomposition (PGD) provides an approximation scheme especially well suited to preclude dramatically increasing the computational complexity with the number of dimensions. The PGD strategy is reviewed for the standard case dealing only with material parameters. Then, the ideas presented in [Ammar et al., “Parametric solutions involving geometry: A step towards efficient shape optimization.” Comput. Methods Appl. Mech. Eng., 2014; 268 :178–193] to deal with parameters describing the domain geometry are adapted to a more general case including parametrization of the location of internal interfaces. Finally, the formulation is extended to combine the two types of parameters. The proposed strategy is used to solve a problem in applied geophysics studying the temperature field in a cross section of the Earth crust subsurface. The resulting problem is in a 10-dimensional space, but the PGD solution provides a fairly accurate approximation (error ≤1%) using less that 150 terms in the PGD expansion. Copyright © 2015 John Wiley & Sons, Ltd.     

Conservation properties of a time FE method. Part I: time‐stepping schemes for N‐body problems

In the present paper one‐step implicit integration algorithms for the N‐body problem are developed. The time‐stepping schemes are based on a Petrov–Galerkin finite element method applied to the Hamiltonian formulation of the N‐body problem. The approach furnishes algorithmic energy conservation in a natural way. The proposed time finite element method facilitates a systematic implementation of a family of time‐stepping schemes. A particular algorithm is specified by the associated quadrature rule employed for the evaluation of time integrals. The influence of various standard as well as non‐standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements. Copyright © 2000 John Wiley & Sons, Ltd.