Proper Generalized Decomposition Method Research Articles

A numerical B-spline Galerkin method with proper generalized decomposition for reduced order modeling of partial differential equations

A numerical study is presented on reduced order modeling of low and high-dimensional partial differential equations using a new B-spline Galerkin proper generalized decomposition (PGD) method. Galerkin-PGD schemes based on B-splines are numerically investigated for various benchmark problems, including the Poisson equation, linear/nonlinear advection–diffusion equations, and the Navier–Stokes equations. The treatment of inhomogeneous boundary conditions, non-separable terms and nonlinearities is numerically investigated. Numerical errors for high-fidelity B-spline solutions and reduced order solutions are provided to demonstrate the effectiveness and accuracy of the proposed method. Computational costs are reported to illustrate the efficiency of the developed approaches.

Real-time simulation of the transplanted tooth using model order reduction.

The biomechanics of transplanted teeth remain poorly understood due to a lack of models. In this context, finite element (FE) analysis has been used to evaluate the influence of occlusal morphology and root form on the biomechanical behavior of the transplanted tooth, but the construction of a FE model is extremely time-consuming. Model order reduction (MOR) techniques have been used in the medical field to reduce computing time, and the present study aimed to develop a reduced model of a transplanted tooth using the higher-order proper generalized decomposition method. The FE model of a previous study was used to learn von Mises root stress, and axial and lateral forces were used to simulate different occlusions between 75 and 175N. The error of the reduced model varied between 0.1% and 5.9% according to the subdomain, and was the highest for the highest lateral forces. The time for the FE simulation varied between 2.3 and 7.2h. In comparison, the reduced model was built in 17s and interpolation of new results took approximately 2.10-2s. The use of MOR reduced the time for delivering the root stresses by a mean 5.9h. The biomechanical behavior of a transplanted tooth simulated by FE models was accurately captured with a significant decrease of computing time. Future studies could include using jaw tracking devices for clinical use and the development of more realistic real-time simulations of tooth autotransplantation surgery.

A Reduced Model Based on Proper Generalized Decomposition for the Fast Analysis of IGBT Power Modules Lifetime

Abstract A reduced weakly-coupled thermo-mechanical model based on the proper generalized decomposition method was developed for the numerical analysis of power modules. The employed model reduction method enabled us to obtain, in a preliminary offline phase, the solution of the thermo-mechanical problem over a large range of design parameters, with much time saving compared to a classical (brute force) multi-resolution finite element method. In an online postprocessing phase, the power module lifetime, modeled with a strain-life law, was then computed in a straightforward manner by rapidly evaluating the solution for any value of the parameters. Sensitivity analysis was conducted to select parameters values leading to acceptable module lifetimes with respect to given criteria. A robust design study was also performed to illustrate the performance of the proposed approach.

Forced vibration analysis of composite beams with piezoelectric layers based on the variable separation method

This paper proposes a new strategy based on the proper generalized decomposition method to solve the forced vibration problem for bi-dimensional piezoelectric composite beams. The approach considers a classical harmonic space-frequency description of the dynamic problem and redefines it by approximating the displacement and electric potential fields as a sum of separated functions of x (beam axis coordinate), z (thickness coordinate) and ω (load frequency). The methodology consists in an iterative algorithm that solves three one-dimensional linear problems at each iteration. The result is a 2D solution in frequency domain with 1D computational complexity. Several numerical tests with wide range of slenderness ratios are considered in order to assess the validity of the method. Moreover, a study of different combinations of boundary conditions is carried out. The results in terms of modal parameters and frequency response functions are analysed and compared with exact elasticity solution and finite element simulations.

A mixed formulation of proper generalized decomposition for solving the Allen-Cahn and Cahn-Hilliard equations

In this paper, we propose a new implementation strategy for solving the Allen-Cahn and Cahn-Hilliard equations with the proper generalized decomposition (PGD) method for parametric studies. As is common to all PGD methods, it is not necessary to do essentially repeating computations; instead, solutions for a range of parameters can be obtained in a single computation. The proposed implementation strategy includes a mixed formulation and a new data structure. The mixed formulation is applied with the staggered method which does not require the first variation of residual with the derivative of the bulk free energy and thereby avoid repeated matrix reassembling in the Newton-Raphson procedure. In addition, the proposed data structure stores pointwise values of the solution functions instead of the PGD representation for the residual of the Newton method, which is efficient for nonlinear time-dependent problems. Overall, the computational efficiency is significantly improved compared with most traditional PGD formulations and implementations.

Nonintrusive uncertainty quantification for automotive crash problems with VPS/Pamcrash

Uncertainty Quantification (UQ) is a key discipline for computational modeling of complex systems, enhancing reliability of engineering simulations. In crashworthiness, having an accurate assessment of the behavior of the model uncertainty allows reducing the number of prototypes and associated costs. Carrying out UQ in this framework is especially challenging because it requires highly expensive simulations. In this context, surrogate models (metamodels) allow drastically reducing the computational cost of Monte Carlo process. Different techniques to describe the metamodel are considered, Ordinary Kriging, Polynomial Response Surfaces and a novel strategy (based on Proper Generalized Decomposition) denoted by Separated Response Surface (SRS). A large number of uncertain input parameters may jeopardize the efficiency of the metamodels. Thus, previous to define a metamodel, kernel Principal Component Analysis (kPCA) is found to be effective to simplify the model outcome description. A benchmark crash test is used to show the efficiency of combining metamodels with kPCA. • Nonintrusive combination of data techniques and surrogates for parametric analysis, design and uncertainty quantification. • Application to VPS/Pamcrash finite elements crashworthiness simulations for automotive industry. • A novel surrogate model, the Separated Response Surface (SRS), based on Proper Generalized Decomposition method. • Application on a benchmark crash model of interest for automotive industry (Volkswagen Group).

Parametric PGD model used with orthogonal polynomials to assess efficiently the building's envelope thermal performance

ABSTRACT Estimating the temperature field of a building envelope could be a time-consuming task. The use of a reduced-order method is then proposed: the Proper Generalized Decomposition method. The solution of the transient heat equation is then re-written as a function of its parameters: the boundary conditions, the initial condition, etc. To avoid a tremendous number of parameters, the initial condition is parameterized. This is usually done by using the Proper Orthogonal Decomposition method to provide an optimal basis. Building this basis requires data and a learning strategy. As an alternative, the use of orthogonal polynomials (Chebyshev, Legendre) is here proposed. Highlights Chebyshev and Legendre polynomials are used to approximate the initial condition Performance of Chebyshev and Legendre polynomials are compared to the POD basis Each basis combined with the PGD model is compared to laboratory measurements The influence of four different parameters on the accuracy of the basis is studied For each approximation basis, CPU calculation times are evaluated and compared

Crack and wrinkle evaluation of press forming analysis using reduced order model by PGD

While the calculation model has been refined and scaled up for the purpose of improving the prediction accuracy of computer simulation, the method of obtaining results while suppressing the calculation cost is also attracting a lot of attention in the market. In recent years, cases have been reported in which nonlinear problems and dynamic problems were tackled using Proper Orthogonal Decomposition (POD) or Proper Generalized Decomposition (PGD). In this study, we focus on the PGD method and confirm its applicability to dynamic nonlinear problems. The validity and practicality of the method will be verified by taking up the press forming analysis with material nonlinearity and contact definition as the subject, and comparing the approximate solution derived by the PGD framework with the calculation result of commercial software.

Bayesian data assimilation with Transport Map sampling and PGD model order reduction

The motivation of this work is to address real-time sequential inference of parameters with a full Bayesian formulation. The Transport Maps method allows to determine a coupling between a reference density and the posterior density. Here, the contribution is to use the Proper Generalized Decomposition method (PGD) to reduce the evaluation cost of the multi-parametric numerical model and to speed up the minimization algorithm in the Transport Map method by calculating the models derivatives with reference to the parameters in a straightforward manner with non-analytical models. A numerical example highlights the performance of the method.

Non-Uniform Rational Basis Spline hyper-surfaces for metamodelling

This study presents an original metamodelling technique based on Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces. The proposed approach is able to fit general non-convex sets of target points (TPs) by extending the NURBS formalism to the N-dimensional (N-D) case. The shape of such a hyper-surface is tuned by several parameters: the number of control points (CPs), their coordinates and the related weights, the degrees of the blending functions and the knot-vector components defined along each direction. The goal of the proposed strategy is to automatically determine (i.e. without the user’s intervention) the full set of parameters defining the NURBS hyper-surface approximating a given set of TPs, without considering simplifying hypotheses. To this purpose, the problem is formulated as a constrained nonlinear programming problem (CNLPP) wherein the optimization variables are all the parameters tuning the shape of the NURBS hyper-surface. Nevertheless, when the number of CPs and the degrees of the basis functions are included among the design variables, the resulting problem is defined over a space having a changing dimension. This problem is solved by means of an original genetic algorithm able to determine, simultaneously, the optimum value of both the design space size (related to the integer variables of the NURBS hyper-surface) and the NURBS hyper-surface continuous parameters. The effectiveness of the proposed approach is shown by means of two meaningful test cases. In addition, the proposed method has been applied to a benchmark taken from the literature and the results have been compared to those provided by the Proper Generalized Decomposition method.

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).

A mixed PGD-a priori time basis strategy for the simulation of cyclic transient thermal behavior

The knowledge of the service life of polymers under cyclic loading, widely used in industrial applications, is required and usually based on the use of methods necessitating an accurate prediction of the stabilized cycle. This implies a large computation time using the Finite Element Method (FEM) since it requires a large number of cycles for polymers. To alleviate this difficulty, a model order reduction method can be used. In this paper, a mixed strategy is investigated. Through the Proper Generalized Decomposition Method (PGD) framework, this strategy combines the Fast Fourier Transform (FFT) to create a priori time basis and the FEM to compute the related spatial modes. The method is applied to 3D thermal problems under cyclic loadings. The robustness of the proposed strategy is discussed for various boundary conditions, multi-times, and different cyclic loadings. A large time saving is obtained proving the interest of this alternative strategy to deal with fatigue simulations.

Fast solution of elliptic harbor agitation problems under frequency-direction input spectra by model order reduction and NURBS-enhanced FEM

Many harbor applications are based on the solution of linear elliptic agitation problems for many spectral conditions. One of the main goals consists in computing the linear combination of numerous simulations of the harbor agitation problem, using monochromatic waves of different spectral components (i.e. frequency and incoming wave direction). In practice, the standard strategy selects the number of wave components according to a prescribed discretization of the 2D input spectra. The main issue relies on some quantities of interest that are very sensitive to the level of refinement of the spectra, such as the significant wave height at every mesh point or the identification of resonance modes induced by long wave scattering. In many cases, achieving enough quality in these quantities may impose numerous simulations and, consequently, non-practical computer costs. This can drastically limit the final accuracy of results. To overcome this situation, here a new strategy is proposed to efficiently solve a large number of harbor agitation problems derived from dense discretizations of the 2D input spectra. The strategy is based on the combination of two different numerical approaches. Firstly, each required monochromatic simulation is solved via high order NURBS (non-uniform rational B-splines) enhanced finite elements (NEFEM). More precisely, NEFEM captures the exact harbor geometry using large mesh elements that produce accurate solutions and significant savings on the system size, particularly in long wave cases. Secondly, a model order reduction technique is used to approximate the original elliptic harbor model by a so-called surrogate model. The main advantage is that, once the surrogate model is constructed, it can be rapidly evaluated to provide simulations for any value of the spectral components within a range of interest, and without the need of solving any new harbor agitation problem (as the standard strategy does). Thus, this enables the possibility of using any desired discretization of the 2D input spectra with no additional computer cost. The construction of the surrogate model is performed using the proper generalized decomposition method with a novel incremental computation along the frequency dimension. The proposed strategy is discussed, and its superior performance with respect to standard strategies is demonstrated, on two harbor agitation examples with several applications.

Forced vibration analysis of composite beams based on the variable separation method

This paper proposes a new strategy based on the proper generalized decomposition method to solve the forced vibration problem in two-dimensional (2D) layered beams. The approach redefines the problem by approximating the displacement field as a sum of separated functions of x, z, and ω and proposes an iterative algorithm that solves three 1D problems at each iteration. Several numerical tests with different slenderness ratios and boundary conditions are considered to validate the method. The results in terms of modal parameters and frequency response functions are analyzed and compared with exact elasticity solution and finite element simulations with very good agreements.

On a Goal-Oriented Version of the Proper Generalized Decomposition Method

In this paper, we introduce, analyze, and numerically illustrate a goal-oriented version of the Proper Generalized Decomposition method. The objective is to derive a reduced-order formulation such that the accuracy in given quantities of interest is increased when compared to a standard Proper Generalized Decomposition method. Traditional goal-oriented methods usually compute the solution of an adjoint problem following the calculation of the primal solution for error estimation and adaptation. In the present work, we propose to solve the adjoint problem first, based on a reduced approach, in order to extract estimates of the quantities of interest and use this information to constrain the reduced primal problem. The resulting reduced-order constrained solution is thus capable of delivering more accurate estimates of the quantities of interest. The performance of the proposed approach is illustrated on several numerical examples.

Characterization of the proper generalized decomposition method for fixed-source diffusion problems

Reduced-order models can greatly decrease the computational time required to solve a large numerical problem. Proper Generalized Decomposition (PGD) is a relatively new solution method that builds the reduced-order basis on the fly and thus does not require an expensive learning phase. By generating a reduced-order basis in the form of a finite sum of separable functions, a multi-dimensional partial differential equation can be transformed into a set of coupled one-dimensional partial differential equations, which greatly reduces the overall degrees of freedom. To date, some preliminary work has been done to implement PGD as a neutronics solver. The PGD method has been implemented to solve one-group, two-dimensional neutron diffusion eigenvalue and time-dependent problems. The work done so far has demonstrated the viability of PGD neutronics solvers, but more research is needed to warrant the deployment of PGD for practical analysis applications. The purpose of this work is to characterize the performance of the PGD solution method over a broad range of problem specifications and demonstrate the application of several proposed methodological improvements. One-group and two-group derivations of the separated PGD equations are provided along with detailed descriptions of the solution algorithms. Several practical details are discussed, including a new algorithm for selecting the inner solver tolerances in order to avoid over-solving. PGD is found to be a robust solver, with performance only moderately influenced by the scattering ratio or whether the system is absorption dominated or diffusion dominated. Heterogeneous systems with arbitrary cross sections can be solved with PGD, although performance is affected by the separability of the cross sections. By direct comparison with a standard finite element solver, substantial time savings are observed in agreement with the theoretical linear scaling behavior with respect to mesh refinement for both one-group and two-group problems.

A mixed POD–PGD approach to parametric thermal impervious soil modeling: Application to canyon streets

Numerical simulation is a powerful tool for assessing the causes of an Urban Heat Island (UHI) effect or quantifying the impact of mitigation solutions on local climatic conditions. However, the numerical cost associated with such a tool is quite significant at the scale of an entire district. Today, the main challenge consists of achieving both a proper representation of the physical phenomena and a critical reduction in the numerical costs of running simulations. This paper presents a combined parametric urban soil model that accurately reproduces thermal heat flux exchanges between the soil and the urban environment with a reduced computational time. For this purpose, the use of a combination of two reduced-order methods is proposed herein: the Proper Orthogonal Decomposition method, and the Proper Generalized Decomposition method. The developed model is applied to two case studies in order to establish a practical evaluation: an open area independent of the influences of the surrounding surface, and a theoretical urban scene with two canyon streets. The error due to the model reduction remains below 0.2 °C on the mean surface temperature for a reduced computational cost of 80%. Compared to in situ measurements the error remains bellow 1.24 °C at the surface.

A fast and robust sub-optimal control approach using reduced order model adaptation techniques

Classical adjoint-based optimization approach for the optimal control of partial differential equations is known to require a large amount of CPU time and memory storage. In this article, in order to reduce these requirements, a posteriori and a priori model order reduction techniques such as POD (Proper Orthogonal Decomposition) and PGD (Proper Generalized Decomposition) are used. As a matter of fact, these techniques allows a fast access to the temporal dynamics of a solution approximated in a suitable subspace of low dimension, spanned by a set of basis functions that form a reduced basis. The costly high fidelity model is then projected onto this basis and results in a system of ordinary differential equations which can be solved in quasi-real time. A disadvantage of considering a fixed POD basis in a suboptimal control loop, is basically the dependence of such bases on a posteriori information coming from high fidelity simulations. Therefore, a non robustness of the POD basis can be expected for certain perturbations in the original parameter for which it was built. As a result, update the reduced bases with respect to each variation in the control parameter using the POD method is still costly. To get over this difficulty, we equip the usual reduced optimal control algorithm with an intermediate basis adaptation step. The first proposed approach consists in adapting the reduced basis for a new control parameter by interpolating over a set of POD bases previously computed for a range of control parameters. To achieve that, an interpolation technique based on properties of the tangent subspace of the Grassmann manifold (ITSGM) is considered. The second approach is the PGD method, which by nature, enrich a space time decomposition trying to enhance the approximation by learning from its own errors. Relaying on this property, this method is employed in the control loop as a basis corrector, in such a way the given spatial basis is adapted for the new control parameter by performing just few enrichments. These two approaches are applied in the sub-control of the two dimensional non-linear reaction-diffusion equations and Burgers equations.

Adaptive sparse grid based HOPGD: Toward a nonintrusive strategy for constructing space‐time welding computational vademecum

SummarySimulation‐based engineering usually needs the construction of computational vademecum to take into account the multiparametric aspect. One example concerns the optimization and inverse identification problems encountered in welding processes. This paper presents a nonintrusive a posteriori strategy for constructing quasi‐optimal space‐time computational vademecum using the higher‐order proper generalized decomposition method. Contrary to conventional tensor decomposition methods, based on full grids (eg, parallel factor analysis/higher‐order singular value decomposition), the proposed method is adapted to sparse grids, which allows an efficient adaptive sampling in the multidimensional parameter space. In addition, a residual‐based accelerator is proposed to accelerate the higher‐order proper generalized decomposition procedure for the optimal aspect of computational vademecum. Based on a simplified welding model, different examples of computational vademecum of dimension up to 6, taking into account both geometry and material parameters, are presented. These vademecums lead to real‐time parametric solutions and can serve as handbook for engineers to deal with optimization, identification, or other problems related to repetitive task.

Accelerating isogeometric boundary element analysis for 3‐dimensional elastostatics problems through black‐box fast multipole method with proper generalized decomposition

SummaryThe isogeometric approach to computational engineering analysis makes use of nonuniform rational B‐splines to discretise both the geometry and the analysis field variables, giving a higher‐fidelity geometric description and leading to improved convergence properties of the solution over conventional piecewise polynomial descriptions. Because of its boundary‐only modelling, with no requirement for a volumetric nonuniform rational B‐spline geometric definition, the boundary element method is an ideal choice for isogeometric analysis of solids in 3D. An isogeometric boundary element analysis (IGABEM) algorithm is presented for the solution of such problems in elasticity and is accelerated using the black‐box fast multipole method (bbFMM). The bbFMM scheme is of O(n) complexity, giving a general kernel‐independent separation that can be easily integrated into existing conventional IGABEM codes with little modification. In the bbFMM scheme, an important process of obtaining a low‐rank approximation of moment‐to‐local operators has been hitherto based on singular value decomposition, which can be very time consuming for large 3D problems, and this motivates the present work. We introduce the proper generalized decomposition method as an alternative approach, and this is demonstrated to enhance efficiency in comparison with schemes that rely on the singular value decomposition. In the worst case, a factor of approximately 2 performance gain is achieved. Numerical examples show the performance gains that are achievable in comparison to standard IGABEM solutions and demonstrate that solution accuracy is not affected. The results illustrate the potential of this numerical technique for solving arbitrary large scale elastostatics problems directly from computer‐aided design models.