Selective proper orthogonal decomposition model reduction for forming simulations

Robot-based incremental sheet metal forming is a cost-effective and flexible method for prototype and low batch size production. The simulation of such processes is very challenging and elaborate from the computational point of view. To reduce the computational effort model reduction techniques such as proper orthogonal decomposition (POD) can be applied. But the reduction of highly non-linear models in solid mechanics for example forming simulation still leads to problems of efficiency and accuracy. Therefore, the aim of this paper is to present an alternative way to use POD for forming processes. The presented selective POD (SPOD) method is used to split the model into two domains depending on the degree of plastic strain. Only the domain with approximately linear elastic behavior will be reduced by using POD. Utilizing the SPOD method for the example of forming a horizontal flute reduces the computational time up to around 30 per cent. High accuracy with approximation errors smaller than one per mill is achieved. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Artificial viscosity proper orthogonal decomposition

We introduce improved reduced-order models for turbulent flows. These models are inspired from successful methodologies used in large eddy simulation, such as artificial viscosity, applied to standard models created by proper orthogonal decomposition of flows coupled with Galerkin projection. As a first step in the analysis and testing of our new methodology, we use the Burgers equation with a small diffusion parameter. We present a thorough numerical analysis for the time discretization of the new models. We then test these models in two problems displaying shock-like phenomena. Of course, since the Burgers equation does not model turbulence, we next need to test our new models in realistic turbulent flow settings. This is the subject of a forthcoming report.     

Galerkin proper orthogonal decomposition methods for parabolic problems

Summary. In this work error estimates for Galerkin proper orthogonal decomposition (POD) methods for linear and certain non-linear parabolic systems are proved. The resulting error bounds depend on the number of POD basis functions and on the time discretization. Numerical examples are included. Received September 29, 1999 / Revised version received August 21, 2000 / Published online May 4, 2001     

Gradient-enhanced surrogate modeling based on proper orthogonal decomposition

A new method for enhanced surrogate modeling of complex systems by exploiting gradient information is presented. The technique combines the proper orthogonal decomposition (POD) and interpolation methods capable of fitting both sampled input values and sampled derivative information like Kriging (aka spatial Gaussian processes). In contrast to existing POD-based interpolation approaches, the gradient-enhanced method takes both snapshots and partial derivatives of snapshots of the associated full-order model (FOM) as an input. It is proved that the resulting predictor reproduces these inputs exactly up to the standard POD truncation error. Hence, the enhanced predictor can be considered as (approximately) first-order accurate at the snapshot locations. The technique applies to all fields of application, where derivative information can be obtained efficiently, for example via solving associated primal or adjoint equations. This includes, but is not limited to Computational Fluid Dynamics (CFD). The method is demonstrated for an academic test case exhibiting the main features of reduced-order modeling of partial differential equations.     

A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation

In this article, a reduced optimizing finite difference scheme (FDS) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) for Burgers equation is presented. Also the error estimates between the usual finite difference solution and the POD solution of reduced optimizing FDS are analyzed. It is shown by considering the results obtained for numerical simulations of cavity flows that the error between the POD solution of reduced optimizing FDS and the solution of the usual FDS is consistent with theoretical results. Moreover, it is also shown that the reduced optimizing FDS is feasible and efficient.     

Finite element formulation based on proper orthogonal decomposition for parabolic equations

A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for parabolic equations so that it is reduced into a POD FE formulation with lower dimensions and enough high accuracy. The errors between the reduced POD FE solution and the usual FE solution are analyzed. It is shown by numerical examples that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD method. This work was supported by National Natural Science Foundation of China (Grant Nos. 10871022, 10771065, and 60573158) and Natural Science Foundation of Hebei Province (Grant No. A2007001027)     

An error estimate of the proper orthogonal decomposition in model reduction and data compression

A kind of the proper orthogonal decomposition (POD) is used for data compression of rugged surface and reduction of the Navier–Stokes equations. An error estimate of the POD in model reduction and data compression is discussed. The numerical examples show that the error between the POD approximate solution and reference solution is consistent with theoretical results, and also show that the proposed algorithm is feasible and efficient. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Model reduction of porous-media problems using proper orthogonal decomposition

In the context of finite-element simulations of porous media, computing time and numerical effort is an important issue because the number of degrees of freedom of such coupled problems can become very large. Following this, model reduction plays an important role. A broad variety of materials exhibit a porous microstructure. In order to evaluate the overall response of these materials, a macroscopic continuum-mechanical modelling approach is used. Therefore, the complex inner structure of porous media is regarded in a multi-phasic and multi-component manner by means of the well-founded Theory of Porous Media (TPM). The mechanical behaviour of porous media is solved using the Finite-Element Method (FEM). The basic idea of model reduction is to transform a high dimensional system, in terms of the system's degrees of freedom, to a low dimensional subspace to minimise the computational effort while maintaining the accuracy of the solution. The method of proper orthogonal decomposition (POD) can be seen as a method to approximate a given data set with a low dimensional subspace. Furthermore, the POD method is independent of the type of the model and can be used for nonlinear systems as well as for systems of second order. In several applications, such as consolidation problems of partially saturated soils, commonly occurring motion sequences can be found, which can be used as typical “snapshots” of the system. Therefore, the application of the POD method to the simulation of porous media is discussed in the present contribution. Investigated computations of a biphasic standard problem show that the POD method reduces the numerical effort to solve the linearised system of equations in each iteration step. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identification of muscle determinant for different gait speeds by proper orthogonal decomposition

In this investigation, we explore the use of the proper orthogonal decomposition (POD) on electromyography (EMG) of the leg muscles, with the objective of elucidating the muscles behavior for each gait speeds. Four different gait speeds were used in the study: slow, comfortable, fast and fastest walking. The electromyographic data used in this study were recorded from 12 bilateral lower limb muscles: tensor fascia latae (TFL), semitendinosus (ST), vastus medialis (VM), rectus femoris (RF), tibialis anterior (TA) and medial gastrocnemius (MG). Results indicate that the primary muscles for these gait speeds were bilateral TFL which act alternately to accept weight bearing on each leg. The same muscles also show another dominant function when working simultaneously in pelvic stabilization. In addition, we found that when the speed of walking increases, the number of muscle determinants for gait has been added. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations

The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.     

Insensitizing controls for a large-scale ocean circulation model

We consider here a linear quasi-geostrophic ocean model. We look for controls insensitizing (resp. ε-insensitizing) an observation function of the state. The existence of such controls is equivalent to a null controllability property (resp. an approximate controllability property) for a cascade Stokes-like system. Under reasonable assumptions on the spatial domains where the observation and the control are performed, we are able to prove these properties. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Variational multiscale proper orthogonal decomposition: Navier‐stokes equations

We develop a variational multiscale proper orthogonal decomposition (POD) reduced‐order model (ROM) for turbulent incompressible Navier‐Stokes equations. Under two assumptions on the underlying finite element approximation and the generation of the POD basis, the error analysis of the full discretization of the ROM is presented. All error contributions are considered: the spatial discretization error (due to the finite element discretization), the temporal discretization error (due to the backward Euler method), and the POD truncation error. Numerical tests for a three‐dimensional turbulent flow past a cylinder at Reynolds number show the improved physical accuracy of the new model over the standard Galerkin and mixing‐length POD ROMs. The high computational efficiency of the new model is also showcased. Finally, the theoretical error estimates are confirmed by numerical simulations of a two‐dimensional Navier‐Stokes problem. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 641–663, 2014     

Error estimates for abstract linear–quadratic optimal control problems using proper orthogonal decomposition

In this paper we investigate POD discretizations of abstract linear–quadratic optimal control problems with control constraints. We apply the discrete technique developed by Hinze (Comput. Optim. Appl. 30:45–61, 2005) and prove error estimates for the corresponding discrete controls, where we combine error estimates for the state and the adjoint system from Kunisch and Volkwein (Numer. Math. 90:117–148, 2001; SIAM J. Numer. Anal. 40:492–515, 2002). Finally, we present numerical examples that illustrate the theoretical results.     

A decomposition approach for a very large scale optimal diversity management problem

This paper focuses on the solution of the optimal diversity management problem formulated as a p-Median problem. The problem is solved for very large scale real instances arising in the car industry and defined on a graph with several tens of thousands of nodes and with several millions of arcs. The particularity is that the graph can consist of several non connected components. This property is used to decompose the problem into a series of p-Median subproblems of a smaller dimension. We use a greedy heuristic and a Lagrangian heuristic for each subproblem. The solution of the whole problem is obtained by solving a suitable assignment problem using a Branch-and-Bound algorithm.Received: June 2004 / Revised version: December 2004MSC classification: 49M29, 90C06, 90C27, 90C60All correspondence to: Antonio SforzaIgor Vasilev: Support for this author was provided by NATO grant CBP.NR.RIG.911258.     

Stepwise decomposition approaches for large scale cell formation problems

Cell formation is one of the major steps in cellular manufacturing system (CMS) design. In this paper, two stepwise decomposition approaches are proposed to solve large scale industrial problems. Both of them analyze the part-machine relations, decompose the original system to several large subsystems and then use an optimal solution technique to solve each. Several results are proved to show the conditions under which optimal solutions are obtained.     

Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations

In this paper, some reduced finite difference schemes based on a proper orthogonal decomposition (POD) technique for parabolic equations are derived. Also the error estimates between the POD approximate solutions of the reduced finite difference schemes and the exact solutions for parabolic equations are established. It is shown by considering the results of two numerical examples that the numerical results are consistent with theoretical conclusions. Moreover, it is also shown that the POD reduced finite difference schemes are feasible and efficient.     

Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions

A technique is presented for interpolating unsteady solutions to parameterised fluid flow problems, using a combination of proper orthogonal decomposition and radial basis functions. The technique is validated by considering simulations involving three dimensional unsteady compressible inviscid flow over an oscillating ONERA M6 wing. It is demonstrated that the approach can result in a large reduction in the cpu time required to find solutions at new parameter values, without a significant loss in accuracy.     

A large scale interface multi-fluid model for simulating multiphase flows

The scope of the Eulerian Multiphase (EMP) model in STAR-CCM+ is extended to simulate multi-scale two-phase flows using Large Scale Interface (LSI) model. The LSI model provides a criteria based on local phase-distribution to distinguish between regimes characterized by small and large scale interfaces. An appropriate closure for conserved variable is specified for each regime, weighted sum of which forms the closure for the interaction between the phases. The LSI model also allows to model surface tension effects in the vicinity of large scale interfaces as well. The large scale interface is treated as a moving wall using a turbulence damping procedure near the interface. This extended multifluid methodology implemented in STAR-CCM+® Software is validated using several standard two-phase flow problems.     

A proper generalized decomposition approach for optical flow estimation

This paper introduces the use of the proper generalized decomposition (PGD) method for the optical flow (OF) problem in a classical framework of Sobolev spaces, ie, optical flow methods including a robust energy for the data fidelity term together with a quadratic penalizer for the regularization term. A mathematical study of PGD methods is first presented for general regularization problems in the framework of (Hilbert) Sobolev spaces, and their convergence is then illustrated on OF computation. The convergence study is divided in two parts: (a) the weak convergence based on the Brézis-Lieb decomposition and (b) the strong convergence based on a growth result on the sequence of descent directions. A practical PGD-based OF implementation is then proposed and evaluated on freely available OF data sets. The proposed PGD-based OF approach outperforms the corresponding non-PGD implementation in terms of both accuracy and computation time for images containing a weak level of information, namely, low image resolution and/or low signal-to-noise ratio (SNR).