Proper Generalized Decomposition Research Articles

Metamodel of Parametric Geometric Magnetostatic Problem Based on PGD and RBF Approaches

In order to reduce the computational time induced by solving a finite element (FE) model for a magnetostatic problem with varying geometric parameters, a parametric geometric metamodel is defined using the proper generalized decomposition (PGD) approach. The mesh deformation associated with the geometric variation is implemented using the radial basis functions (RBFs) interpolation method. The proposed approach is applied to a variable inductance and then to a permanent magnet synchronous machine (PMSM). The results show that the PGD metamodel can accurately approximate the FE solution for the case of parametric geometric problems.

Proper Generalized Decomposition using Taylor expansion for non-linear diffusion equations

For a physical problem described by a parameterized mathematical model, different configurations of the problem require computing the solution over a range of parameters in order to study the phenomenon when parameters change. In other words, it is a process of looking for a continuum of solutions of the equation, relative to these parameters, in order to find the ones that fit the experimental data. However, solving a direct problem for each parametric configuration will generate a cascade of direct problems, which will cost a huge amount of time, especially when we deal with non-linear equations. Therefore, the parametric solution is a suitable alternative strategy to compute the solution of the equation. In this paper, we will use the Proper Generalized Decomposition (PGD) method to solve non-linear diffusion equations and produce parametric solutions. To treat the non-linear functions, we will not use the Discrete Empirical Interpolation Methods (DEIM), which has proven their utility, but the non-linear terms will be replaced by their Taylor series expansion up to an order m. This will produce a new model, which we call here the ”developed equation” and therefore the PGD is applied on. Polynomial equations appear for each tensor element computation. While space and time tensor elements’ equations are to be solved using Finite Elements Methods (FEM) and Borel–Padé–Laplace (BPL) integrator respectively, Newton solver is used for tensors relative to the parameters’ equations. Here, rational polynomial functions arise for parametric tensor elements, which are known to extrapolate solutions. Numerical simulations are done for a non-linear diffusion equation with exponential diffusion coefficient as first trial, and with a magnetic diffusion coefficient as a second one.

Hybrid Polynomial Chaos Expansion and proper generalized decomposition approach for uncertainty quantification problems in the frame of elasticity

Polynomial Chaos Expansion (PCE) is a powerful and flexible metamodeling technique, but suffers from the fact that the number of required training samples grows exponentially with the dimensionality of the problem. Recently, proper generalized decomposition (PGD) has become a popular way to address this challenge since its complexity grows linearly with the dimension of the problem. In this work, we present a new technique called PGD–PCE, which combines the advantages of both methods. The algorithm is based on separate representations and constructed by orthonormal polynomial functions. We test the proposed approach on simple toy problems and engineering finite element problems. The results show that PGD–PCE performs well in terms of accuracy and computational efficiency when dealing with large problems. • We propagate uncertainties in mechanical problems. • We develop a simple to implement and non-intrusive method. • We show that accuracy is preserved while cost drastically decrease. • We illustrate the capabilities of the approach on various numerical tests.

PGD reduced-order modeling for structural dynamics applications

We propose in this paper a Proper Generalized Decomposition (PGD) approach for the solution of problems in linear elastodynamics. The novelty of the work lies in the development of weak formulations of the PGD problems based on the Lagrangian and Hamiltonian Mechanics, the main objective being to devise numerical methods that are numerically stable and energy conservative. We show that the methodology allows one to consider the Galerkin-based version of the PGD and numerically demonstrate that the PGD solver based on the Hamiltonian formulation offers better stability and energy conservation properties than the Lagrangian formulation. The performance of the two formulations is illustrated and compared on several numerical examples describing the dynamical behavior of a one-dimensional bar.

A proper generalized decomposition-based harmonic balance method with arc-length continuation for nonlinear frequency response analysis

The objective of this study is to incorporate the proper generalized decomposition (PGD) into the harmonic balance method (HBM) for an efficient numerical continuation of nonlinear frequency response. PGD utilizes the low-dimensional subspaces of the HBM response, approximating the solution as a low-rank separated representation of spatial and harmonic components. The progressive Galerkin approach is employed to formulate subproblems for each component, and the solution is built on the fly without prior assumption or computation of nonlinear response characteristics. During the numerical continuation, the spatial modes obtained at the previous computation are utilized as a reduced basis, and arc-length constraint is applied to follow the proper path of the frequency response curve. Numerical studies demonstrate that the proposed method allows significant computational savings compared to HBM, while accurately reflecting the nonlinear behavior.

Reduced-order modeling via proper generalized decomposition for uncertainty quantification of frequency response functions

This paper presents a new reduced-order modeling methodology for frequency response analysis of linear dynamical systems with parametric uncertainty. The proposed framework consists of offline and online stages in the computation process. The offline stage introduces a variant of proper generalization decomposition (PGD), which approximates the solution of the full-order model (FOM) as a low-rank separated representation. A key feature of the proposed PGD is the collocation representation equipped with the Dirac-delta function, which handles non-smooth characteristics of frequency responses point-wisely. This strategy replaces direct computations of FOM on a large number of samples and frequencies with iterative solving of the subproblems formulated by the progressive Galerkin approach. In the online stage, the PGD modes acquired from the offline stage are used to generate a reduced-order model (ROM). Solutions for unsampled points are evaluated through ROM, which is further utilized to estimate the statistics of the frequency response function. Numerical examples demonstrate that the proposed methodology accelerates computational efficiency while maintaining accuracy over the frequency range including resonance.

Proper generalized decomposition solutions for composite laminates parametrized with fibre orientations

Composite materials are gaining popularity as an alternative to classical materials in many different applications. Moreover, their design is even more flexible due to the potential of additive manufacturing. Thus, one can produce a tailored composite laminate with the optimal values of some design parameters providing the desired mechanical performance. In this context, having a parametric numerical model for the mechanical response of the composite laminate is essential to compute the optimal parameters. In the present paper, the design parameters under consideration are the angles describing the orientation of the reinforcement fibers in different layers or patches of the composite laminates. We obtain a generalized solution using Proper Generalized Decomposition (PGD) which is adopted to provide solutions with explicit parametric dependence. The Tsai-Wu failure criterion is used to estimate first ply failure. In this context, Tsai-Wu criterion is used as the objective function for the optimization of the fibre orientations in the laminate. The PGD solution provides also sensitivities for a gradient-based optimization algorithm. The potentiality and efficiency of the presented approach is demonstrated through some numerical tests.

Consideration of Rotational Motion in the Proper Generalized Decomposition by a Sliding Interface Technique

The simulation of rotating electrical machines, particularly in the development process, involves many parameters, such as the current excitation, permanent magnet materials, and the relative rotor position. To reduce the computational effort related to a large number of degrees of freedom (DOFs), the proper generalized decomposition (PGD) can be employed. Although the mentioned reduction technique is able to efficiently cope with parameter variations, it is limited to conformal meshes. However, the consideration of moving parts in the model for the simulation of rotating electrical machines is a necessity to characterize the machine’s behavior. For this purpose, the sliding interface technique, based on Lagrange multipliers (LMs) can be utilized. In this contribution, the sliding interface technique, which imposes no restrictions on the finite element discretization of the interface between stator and rotor, is combined with the PGD to lift the restriction of the latter to conformal meshes while keeping the symmetry and positive definiteness of the system without the creation of additional elements. This approach enables the parametric simulation of rotating electrical machines by the PGD.

Bayesian inference for random field parameters with a goal-oriented quality control of the PGD forward model’s accuracy

Numerical models built as virtual-twins of a real structure (digital-twins) are considered the future of monitoring systems. Their setup requires the estimation of unknown parameters, which are not directly measurable. Stochastic model identification is then essential, which can be computationally costly and even unfeasible when it comes to real applications. Efficient surrogate models, such as reduced-order method, can be used to overcome this limitation and provide real time model identification. Since their numerical accuracy influences the identification process, the optimal surrogate not only has to be computationally efficient, but also accurate with respect to the identified parameters. This work aims at automatically controlling the Proper Generalized Decomposition (PGD) surrogate’s numerical accuracy for parameter identification. For this purpose, a sequence of Bayesian model identification problems, in which the surrogate’s accuracy is iteratively increased, is solved with a variational Bayesian inference procedure. The effect of the numerical accuracy on the resulting posteriors probability density functions is analyzed through two metrics, the Bayes Factor (BF) and a criterion based on the Kullback-Leibler (KL) divergence. The approach is demonstrated by a simple test example and by two structural problems. The latter aims to identify spatially distributed damage, modeled with a PGD surrogate extended for log-normal random fields, in two different structures: a truss with synthetic data and a small, reinforced bridge with real measurement data. For all examples, the evolution of the KL-based and BF criteria for increased accuracy is shown and their convergence indicates when model refinement no longer affects the identification results.

Nonintrusive parametric NVH study of a vehicle body structure

A reduced order model technique is presented to perform the parametric Noise, Vibration and Harshness (NVH) study of a vehicle body-in-white (BIW) structure characterized by material and shape design variables. The ultimate goal is to develop a methodology which allows to efficiently explore the variation in the design space of the BIW static and dynamic global stiffnesses, such that the NVH performance can be evaluated already in the preliminary phase of the development process. The proposed technique is based on the proper generalized decomposition (PGD) method. The obtained PGD solution presents an explicit dependency on the introduced design variables, which allows to obtain solutions in 0.1 milliseconds and therefore opens the door to fast optimization studies and real-time visualizations of the results in a pre-defined range of parameters. The method is nonintrusive, such that an interaction with commercial software is possible. A parametrized finite element (FE) model of the BIW is built by means of the ANSA CAE preprocessor software, which allows to account for material and geometric parameters. A comparison between the parametric NVH solutions and the full-order FE simulations is performed using the MSC-Nastran software, to validate the accuracy of the proposed method. In addition, an optimization study is presented to find the optimal materials and shape properties with respect to the NVH performance. Finally, in order to support the designers in the decision-making process, a graphical interface app is developed which allows to visualize in real-time how changes in the design variables affect pre-defined quantities of interest.

On a wave-based reduced order model for transient effects computation including mid frequencies

The wave-based computational method named Variational Theory of Complex Rays (VTCR) is revisited in order to improve its robustness and its efficiency to solve transient dynamic problems including medium frequencies. The first problem addressed in the paper concerns the ill-conditioning of wave-based computational approaches. We introduce the concept of “effective portrait” which provides a new framework to the VTCR with a strong reduction of the conditioning number. The second issue deals with the computation over a large frequency bandwidth for which the Proper Generalised Decomposition (PGD) has been retained. A thorough analysis of the efficiency and the limitations of the PGD-VTCR is given. The last issue concerns the synergy between the PGD-VTCR and the exponential window method for solving transient problems.

Domain decomposition involving subdomain separable space representations for solving parametric problems in complex geometries

A domain decomposition technique combined with an enhanced geometry mapping based on the use of NURBS is considered for solving parametrized models in complex geometries (non simply connected) within the so-called proper generalized decomposition (PGD) framework, enabling the expression of the solution in each subdomain in a fully separated form, involving both the space and the model parameters. NURBS allow a compact and powerful domain mapping into a fully separated reference geometry, while the PGD allows recovering an affine structure of the problem in the reference domain, facilitating the use of the standard PGD solver for computing the parametric solution in each subdomain first, and then by enforcing the interface transmission conditions, in the whole domain.

On the High-Resolution Discretization of the Maxwell Equations in a Composite Tape and the Heating Effects Induced by the Dielectric Losses

Electromagnetic field propagation inside composite materials represents a challenge where fiber-scale simulation remains intractable using classical simulation methods. The present work proposes an original 3D simulation with a mesh resolution fine enough to resolve the fiber scale, thanks to the use of Proper Generalized Decomposition (PGD)-based space decomposition, which avoids the necessity of considering homogenized properties and considers the richest description of the involved physics from the solution of the Maxwell equations. This high-resolution simulation enables comparing the electromagnetic field propagation in a composite part, depending on the considered frequency and the fiber’s/wave polarization’s relative orientation. The electromagnetic fields are then post-processed to identify the heat generation terms and- the resulting induced thermal field. The results prove the ability of the PGD-based discretization to attain extremely high levels of resolution, the equivalent of 1010 finite-element degrees of freedom. The obtained results show an enhanced wave penetration when the electric field polarization coincides with the fiber orientation. On the contrary, when the electric field is polarized along the normal to the fiber orientation, both the penetration and the associated heating reduce significantly, compromising the use of homogenized models, rendering them unable to reproduce the observed behaviors.

HiDeNN-TD: Reduced-order hierarchical deep learning neural networks

This paper presents a tensor decomposition (TD) based reduced-order model of the hierarchical deep-learning neural networks (HiDeNN). The proposed HiDeNN-TD method keeps advantages of both HiDeNN and TD methods. The automatic mesh adaptivity makes the HiDeNN-TD more accurate than the finite element method (FEM) and conventional proper generalized decomposition (PGD) and TD, using a fraction of the FEM degrees of freedom. This work focuses on the theoretical foundation of the method. Hence, the accuracy and convergence of the method have been studied theoretically and numerically, with a comparison to different methods, including FEM, PGD, TD, HiDeNN and Deep Neural Networks. In addition, we have theoretically shown that the PGD/TD converges to FEM at increasing modes, and the PGD/TD solution error is a summation of the mesh discretization error and the mode reduction error. The proposed HiDeNN-TD shows a high accuracy with orders of magnitude fewer degrees of freedom than FEM, and hence a high potential to achieve fast computations with a high level of accuracy for large-size engineering and scientific problems. As a trade-off between accuracy and efficiency, we propose a highly efficient solution strategy called HiDeNN-PGD. Although the solution is less accurate than HiDeNN-TD, HiDeNN-PGD still provides a higher accuracy than PGD/TD and FEM with only a small amount of additional cost to PGD.

Nonintrusive parametric solutions in structural dynamics

A nonintrusive reduced order method able to solve a parametric modal analysis is proposed in this work. The main objective is being able to efficiently identify how a variation of user-defined parameters affects the dynamic response of the structure in terms of fundamental natural frequencies and corresponding mode shapes. A parametric version of the inverse power method (IPM) is presented by using the proper generalised decomposition (PGD) rationale. The proposed approach utilises the so-called encapsulated PGD toolbox and includes a new algorithm for computing the square root of a parametric object. With only one offline computation, the proposed PGD-IPM approach provides an analytical parametric expression of the smallest (in magnitude) eigenvalue (or natural frequency) and corresponding eigenvector (mode shape), which contains all the possible solutions for every combination of the parameters within pre-defined ranges. A Lagrange multiplier deflation technique is introduced in order to compute subsequent eigenpairs, which is also valid to overcome the stiffness matrix singularity in the case of a free-free structure. The proposed approach is nonintrusive and it is therefore possible to be integrated with commercial finite element (FE) packages. Two numerical examples are shown to underline the properties of the technique. The first example includes one material and one geometric parameter. The second example shows a more realistic industrial example, where the nonintrusivity of the approach is demonstrated by employing a commercial FE package for assembling the FE matrices. Finally, a multi-objective optimisation study is performed proving that the developed method could significantly assist the decision-making during the preliminary phase of a new design process.

Proper Generalized Decomposition for Parametric Study and Material Distribution Design of Multi-Directional Functionally Graded Plates Based on 3D Elasticity Solution.

The use of mesh-based numerical methods for a 3D elasticity solution of thick plates involves high computational costs. This particularly limits parametric studies and material distribution design problems because they need a large number of independent simulations to evaluate the effects of material distribution and optimization. In this context, in the current work, the Proper Generalized Decomposition (PGD) technique is adopted to overcome this difficulty and solve the 3D elasticity problems in a high-dimensional parametric space. PGD is an a priori model order reduction technique that reduces the solution of 3D partial differential equations into a set of 1D ordinary differential equations, which can be solved easily. Moreover, PGD makes it possible to perform parametric solutions in a unified and efficient manner. In the present work, some examples of a parametric elasticity solution and material distribution design of multi-directional FGM composite thick plates are presented after some validation case studies to show the applicability of PGD in such problems.

Parametric solutions of turbulent incompressible flows in OpenFOAM via the proper generalised decomposition

An a priori reduced order method based on the proper generalised decomposition (PGD) is proposed to compute parametric solutions involving turbulent incompressible flows of interest in an industrial context, using OpenFOAM. The PGD framework is applied for the first time to the incompressible Navier-Stokes equations in the turbulent regime, to compute a generalised solution for velocity, pressure and turbulent viscosity, explicitly depending on the design parameters of the problem. In order to simulate flows of industrial interest, a minimally intrusive implementation based on OpenFOAM SIMPLE algorithm applied to the Reynolds-averaged Navier-Stokes equations with the Spalart-Allmaras turbulence model is devised. The resulting PGD strategy is applied to parametric flow control problems and achieves both qualitative and quantitative agreement with the full order OpenFOAM solution for convection-dominated fully-developed turbulent incompressible flows, with Reynolds number up to one million.

Reduced-order modeling of neutron transport separated in energy by Proper Generalized Decomposition with applications to nuclear reactor physics

In this article, we formulate and implement an a priori Reduced-Order Model (ROM) of neutron transport separated in energy by Proper Generalized Decomposition (PGD). Numerical experiments characterize efficacy compared to the full-order model in approximating the neutron flux, multigroup cross sections, and coarse-group flux. Problems consider representative light water reactor pins of UO2 or Mixed Oxide fuel with CASMO-70, XMAS-172, and SHEM-361 energy meshes. We demonstrate the ROM can estimate the angular flux within roughly 0.36% L2 error—or 0.05% with update of the energy basis—given 50 modes, or enrichment iterations, even for the finest mesh, SHEM-361. The corresponding scalar flux errors round to 0.2% and 0.01%. Further, collapsing from SHEM-361 to (an aligned analogue of) CASMO-70, we find few (1–3) modes yield multigroup cross sections comparable to those of an infinite medium approximation, while 30 produces values practically indistinguishable from those of the full-order model, as assessed by the error of the coarse-group solution. This holds true with both Consistent-P and Inconsistent-P transport corrections. Comparing the coarse-group solutions, we observe the L2 error of the ROM with 10 modes (0.5%) is comparable to that introduced by cross section condensation. Given additional (20, 30) modes, the ROM is able to converge below this threshold (to 0.11–0.16%, 0.06–0.09%, respectively). Ultimately, we expect this PGD ROM may achieve considerable savings in computing multigroup cross sections. Moreover, the ROM provides an alternative means of approximation to cross section condensation, preferable in that it introduces neither a loss of resolution nor irrecoverable error.

A stabilized mixed space–time Proper Generalized Decomposition for the Navier–Stokes equations

In this paper, a reduced order model based on the Proper Generalized Decomposition (PGD) is applied to the numerical modeling of the incompressible Navier–Stokes equations. Despite of all the advances achieved in numerical analysis in the last decades, the simulation of large-scale, real-time and parametric problems remain a challenge, especially in the Computational Fluid Dynamics (CFD) context. This can be associated to several aspects such as the solution of large algebraic systems, the governing equations nonlinearities and numerical instabilities, computational limitations, among others. This work focuses on providing low cost solution for incompressible flows with low Reynolds numbers by means of a PGD reduced basis built under a space–time decomposition employing a mixed-stabilized approach for two-dimensional flows. The methodology is evaluated with a set of numerical examples, confirming its robustness and precision, as well as a reduction in the number of linear systems to be solved and the run time compared to the standard stabilized finite element approach.

A proper generalized decomposition based Padé approximant for stochastic frequency response analysis

AbstractThis article presents a proper generalized decomposition (PGD) based Padé approximant for efficient analysis of the stochastic frequency response. Due to the high nonlinearity of the stochastic response with respect to the input uncertainties, the classical stochastic Galerkin (SG) method utilizing polynomial chaos exhibits slow convergence near the resonance. Furthermore, the dimension of the SG method is the product of deterministic and stochastic approximation spaces, and hence resolution over a banded frequency range is computationally expensive or even prohibitive. In this study, to tackle these problems, the PGD first generates the solution of stochastic frequency equations as a separated representation of deterministic and stochastic components. For the banded frequency range computations, the deterministic vectors are exploited as a reduced basis in conjunction with singular value decomposition. Subsequently, the Padé approximant is applied based on the PGD solution, and the stochastic frequency response is represented by a rational function. Through various numerical studies, it is demonstrated that the proposed framework improves not only the accuracy in the vicinity of resonance but also the computational efficiency, compared with the SG method.