Proper Generalized Decomposition Research Articles

An efficient PGD solver for structural dynamics applications

We propose in this paper a Proper Generalized Decomposition (PGD) solver for reduced-order modeling of linear elastodynamic problems. It primarily focuses on enhancing the computational efficiency of a previously introduced PGD solver based on the Hamiltonian formalism. The novelty of this work lies in the implementation of a solver that is halfway between Modal Decomposition and the conventional PGD framework, so as to accelerate the fixed-point iteration algorithm. Additional procedures such that Aitken’s delta-squared process and mode-orthogonalization are incorporated to ensure convergence and stability of the algorithm. Numerical results regarding the ROM accuracy, time complexity, and scalability are provided to demonstrate the performance of the new solver when applied to dynamic simulation of a three-dimensional structure.

On the benefits of a multiscale domain decomposition method to model-order reduction for frictional contact problems

In this paper, the efficiency of a multiscale strategy based on a domain decomposition method (DDM) for model-order reduction of time-dependent frictional contact problems is presented. The proposed strategy relies on the LArge Time INcrement (LATIN) nonlinear solver combined with model reduction based on the Proper Generalized Decomposition (PGD). The LATIN presents a robust treatment of contact conditions, sharing similarities with augmented Lagrangian approaches, and naturally leads to a mixed DDM. In addition, the global space–time formulation of the method allows PGD-based model reduction to be used during computations, creating and enriching on-the-fly reduced bases per substructure to better track sliding fronts and propagative phenomena. The introduction of a multiscale strategy in the LATIN framework is consistent with the physics of contact problems, in which phenomena with different wavelengths interact: local solutions at contact interfaces presents high gradient effects with a short wavelength compared to the characteristic length of the structure. By taking advantage of this, the coarse scale problem of the strategy enables to capture efficiently the behavior of the problem at the structural level, focusing then on capturing the local contact variations at the contact interfaces. The most important features of the approach are shown comprehensively on a simple one-dimensional frictional contact problem. Then, its robustness and effectiveness are tested on a two-dimensional multibody frictional contact problem with more complex loadings. Guidelines are also given for choosing the parameters of the strategy, in particular those driving the construction of the reduced basis.

Skin and Proximity Effect Calculation of a System of Rectangular Conductors Using the Proper Generalized Decomposition Technique

This paper presents the application of a numerical approach known as proper generalized decomposition (PGD) to calculate the per-unit length (PUL) ac resistance of rectangular conductors. PGD has been successfully used in areas such as fluid mechanics and biomedical applications. It solves a partial differential equation (PDE) by decomposing the answer into a set of unknown one-dimensional (1D) functions in an iterative approach until it reaches a predetermined convergence. In this paper, a frequency-dependent meshing scheme is employed in the PGD technique at each frequency to properly take skin and proximity effects into account. One of the main advantages of PGD over traditional numerical approaches such as finite element or finite difference methods is that it confines the answers within a set of one-dimensional functions, which require fewer computational resources. Different examples of single and multiple rectangular conductors are considered to study skin and proximity effects. The PGD results are compared with those obtained using a commercial finite element method (FEM) software to verify the accuracy of the model. This approach can be used in applications such as white box modeling of transformers, EMC analysis, hairpin winding design used in electric vehicles, and busbar simulation.

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.

PGD in thermal transient problems with a moving heat source: A sensitivity study on factors affecting accuracy and efficiency

AbstractThermal transient problems, essential for modeling applications like welding and additive metal manufacturing, are characterized by a dynamic evolution of temperature. Accurately simulating these phenomena is often computationally expensive, thus limiting their applications, for example for model parameter estimation or online process control. Model order reduction, a solution to preserve the accuracy while reducing the computation time, is explored. This article addresses challenges in developing reduced order models using the proper generalized decomposition (PGD) for transient thermal problems with a specific treatment of the moving heat source within the reduced model. Factors affecting accuracy, convergence, and computational cost, such as discretization methods (finite element and finite difference), a dimensionless formulation, the size of the heat source, and the inclusion of material parameters as additional PGD variables are examined across progressively complex examples. The results demonstrate the influence of these factors on the PGD model's performance and emphasize the importance of their consideration when implementing such models. For thermal example, it is demonstrated that a PGD model with a finite difference discretization in time, a dimensionless representation, a mapping for a moving heat source, and a spatial domain non‐separation yields the best approximation to the full order model.

Proper generalized decomposition-based iterative enrichment process combined with shooting method for steady-state forced response analysis of nonlinear dynamical systems

This paper presents a novel framework combining proper generalized decomposition (PGD) with the shooting method to determine the steady-state response of nonlinear dynamical systems upon a general periodic input. The proposed PGD approximates the response as a low-rank separated representation of the spatial and temporal dimensions. The Galerkin projection is employed to formulate the subproblem for each dimension, then the fixed-point iteration is applied. The subproblem for the spatial vector can be regarded as computing a set of reduced-order basis vectors, and the shooting problem projected onto the subspace spanned by these basis vectors is defined to obtain the temporal coefficients. From this procedure, the proposed framework replaces the complex nonlinear time integration of the full-order model with the series of solving simple iterative subproblems. The proposed framework is validated through two descriptive numerical examples considering the conventional linear normal mode method for comparison. The results show that the proposed shooting method based on PGD can accurately capture nonlinear characteristics within 10 modes, whereas linear modes cannot easily approximate these behaviors. In terms of computational efficiency, the proposed method enables CPU time savings of about one order of magnitude compared with the conventional shooting methods.

Efficient bead-on-plate weld model for parameter estimation towards effective wire arc additive manufacturing simulation

AbstractDespite the advances in hardware and software techniques, standard numerical methods fail in providing real-time simulations, especially for complex processes such as additive manufacturing applications. A real-time simulation enables process control through the combination of process monitoring and automated feedback, which increases the flexibility and quality of a process. Typically, before producing a whole additive manufacturing structure, a simplified experiment in the form of a bead-on-plate experiment is performed to get a first insight into the process and to set parameters suitably. In this work, a reduced order model for the transient thermal problem of the bead-on-plate weld simulation is developed, allowing an efficient model calibration and control of the process. The proposed approach applies the proper generalized decomposition (PGD) method, a popular model order reduction technique, to decrease the computational effort of each model evaluation required multiple times in parameter estimation, control, and optimization. The welding torch is modeled by a moving heat source, which leads to difficulties separating space and time, a key ingredient in PGD simulations. A novel approach for separating space and time is applied and extended to 3D problems allowing the derivation of an efficient separated representation of the temperature. The results are verified against a standard finite element model showing excellent agreement. The reduced order model is also leveraged in a Bayesian model parameter estimation setup, speeding up calibrations and ultimately leading to an optimized real-time simulation approach for welding experiment using synthetic as well as real measurement data.

Multi-temporal decomposition for elastoplastic ratcheting solids

This paper presents a multi-temporal formulation for simulating elastoplastic solids under cyclic loading. We leverage the proper generalized decomposition (PGD) to decompose the displacements into multiple time scales, separating the spatial and intra-cyclic dependence from the inter-cyclic variation. In contrast with the standard incremental approach, which solves the (non-linear and computationally intensive) mechanical balance equations at every time step, the proposed PGD approach allows the mechanical balance equations to be solved exclusively for the small-time intra-cyclic response, while the large-time inter-cyclic response is described by simple scalar algebraic equations. Numerical simulations exhibiting complex cyclic responses, including a 2D problem and an application to a monopile foundation, demonstrate that PGD solutions with a limited number of space–time degrees of freedom may be obtained numerically, only requiring a few modes to accurately capture the reference response.

An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition

A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection–diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods.

Empowering PGD-based parametric analysis with Optimal Transport

The Proper Generalized Decomposition (PGD) is a Model Order Reduction framework that has been proposed to be able to do parametric analysis of physical problems. These parameters may include material properties, boundary conditions, etc. With this framework most of the computation may done in an off-line stage allowing to perform real time simulation in a variety of situations. Nevertheless, this scheme may lose its efficiency where the domain itself is also considered as “a parameter”. Optimal transport techniques, on the other hand, have demonstrated exceptional performance in interpolating different types of fields described over geometrical domains with varying shapes. Hence trying allying both techniques is quite natural. The core idea is that PGD handles the parametric solution, while the optimal transport-based methodology transports the solution for a family of domains defined by geometrical parameters such as lengths, radii, thicknesses, etc. In this first attempt the associated methodology is proposed and apply in simple 1D and 2D cases showing interesting performances.

Fast simulation of 3D elastic response for wheel–rail contact loading using Proper Generalized Decomposition

To increase computational efficiency, we adopt Proper Generalized Decomposition (PGD) to solve a reduced-order problem of the displacement field for a three-dimensional rail head exposed to different contact scenarios. The three-dimensional solid rail head is modeled as a two-dimensional cross-section, with the coordinate along the rail being treated as a parameter in the PGD approximation. A novel feature is that this allows us to solve the full three-dimensional model with a nearly two-dimensional computational effort. Additionally, we incorporate the distributed contact load predicted from dynamic vehicle–track simulations as extra coordinates in the PGD formulation, using a semi-Hertzian contact model. The problem is formulated in two ways; one general ansatz which considers the treatment of numerous parameters, some of which exhibit a linear influence, and a linear ansatz where multiple PGD solutions are solved for. In particular, situations where certain parameters become invariant are handled. We assess the accuracy and efficiency of the proposed strategy through a series of verification examples. It is shown that the PGD solution converges toward the FE solution with reduced computational cost. Furthermore, solving for the PGD approximation based on load parameterization in an offline stage allows expedient handling of the wheel–rail contact problem online.

A real-time approach to structural seismic response analysis

This paper presents a real-time seismic response analysis method based on the Proper Orthogonal Decomposition (POD) and Proper Generalized Decomposition (PGD) theories. Initially, the damping matrix and ground motion load are incorporated using the parametric form of the solid dynamic equation. By leveraging the POD method, the computational challenge arising from the high number of parameters in the equation (one parameter for each degree of freedom of the entire model) is overcome, enabling the calculation of complex computational models. Subsequently, the proposed method's effectiveness, stability, and accuracy are demonstrated through the analysis of various seismic waves characterized by different magnitudes, durations, and frequencies. Furthermore, the method successfully achieves real-time response feedback of the structure to arbitrary seismic waves. Importantly, this approach exhibits significant potential for engineering applications.

Application of PGD separation of space to create a reduced-order model of a lithium-ion cell structure

Lithium-ion cells can be considered a laminate of thin plies comprising the anode, separator, and cathode. Lithium-ion cells are vulnerable toward out-of-plane loading. When simulating such structures under out-of-plane mechanical loads, subordinate approaches such as shells or plates are sub-optimal because they are blind toward out-of-plane strains and stresses. On the other hand, the use of solid elements leads to limitations in terms of computational efficiency independent of the time integration method. In this paper, the bottlenecks of both (implicit and explicit) methods are discussed, and an alternative approach is shown. Proper generalized decomposition (PGD) is used for this purpose. This computational method makes it possible to divide the problem into the characteristic in-plane and out-of-plane behaviors. The separation of space achieved with this method is demonstrated on a static linearized problem of a lithium-ion cell structure. The results are compared with conventional solution approaches. Moreover, an in-plane/out-of-plane separated representation is also built using proper orthogonal decomposition (POD). This simply serves to compare the in-plane and out-of-plane behaviors estimated by the PGD and does not allow computational advantages relative to conventional techniques. Finally, the time savings and the resulting deviations are discussed.

PGD based meta modelling of a lithium-ion battery for real time prediction

Despite the existence of computationally efficient tools, the effort for parametric investigations is currently high in industry. In this paper, within the context of Li-Ion batteries, an efficient meta-modelling approach based on the Proper Generalized Decomposition (PGD) is considered. From a suitable design of experiments, a parametric model is trained and then exploited to predict, in real time, the system response to a specific parameter combination. In particular, two different methods are considered, the sparse PGD (sPGD) and the anchored-ANOVA based one (ANOVA-PGD). As a use case for the method the dynamic indentation test of a commercial lithium-ion pouch cell with a cylindrical impactor is selected. The cell model considers a homogenised macroscopic structure suitably calibrated for explicit finite element simulations. Four parameters concerning the impactor are varied, both non-geometric (mass and initial velocity) and geometric (diameter and orientation). The study focuses on multi-dimensional outputs, such as curves and contour plots. Inspired by earlier studies, the sPGD is used to predict the force-displacement curves. As a further development, the impactor kinetic energy curve and the displacement contours are both predicted using its recently developed variant ANOVA-PGD. Moreover, a novel curve alignment technique based on the Gappy Proper Orthogonal Decomposition (Gappy-POD) is suggested here. The meta-model is compared to the results of an FE simulation and the resulting deviations are then discussed.

Proper Generalized Decomposition stereocorrelation to measure kinematic fields for high speed impact on laminated glass

Stereocorrelation (SC) is a powerful tool for measuring 3D surface deformations in mechanical tests. However, it may suffer from convergence issues for large displacements or speckle changes. A global space–time formulation is proposed herein using Proper Generalized Decomposition (PGD) as a regularization strategy. The entire image sequence is considered as a whole, and the kinematics is sought as the sum of modes, each of which is a function of space multiplied by a function of time (smoothed over high frequencies if needed). A discriminative benchmark case is chosen to validate this approach: an impact test on glass laminated with PVB layers. This case is difficult for many reasons, particularly because of severe contrast changes due to light reflections occurring at several places. It is shown that the new formulation outperforms classical approaches in robustness and measurement accuracy and provides results within a much shorter processing time.

A hybrid enrichment strategy for the efficient construction of reduced order models by the proper generalized decomposition

PurposeThe calculation of electromagnetic fields can involve many degrees of freedom (DOFs) to achieve accurate results. The DOFs are directly related to the computational effort of the simulation. The effort is decreased by using the proper generalized decomposition (PGD) and proper orthogonalized decomposition (POD). The purpose of this study is to combine the advantages of both methods. Therefore, a hybrid enrichment strategy is proposed and applied to different electromagnetic formulations.Design/methodology/approachThe POD is an a-priori method, which exploits the solution space by decomposing reference solutions of the field problem. The disadvantage of this method is given by the unknown number of solutions necessary to reconstruct an accurate field representation. The PGD is an a-priori approach, which does not rely on reference solutions, but require much more computational effort than the POD. A hybrid enrichment strategy is proposed, based on building a small POD model and using it as a starting point of the PGD enrichment process.FindingsThe hybrid enrichment process is able to accurately approximate the reference system with a smaller computational effort compared to POD and PGD models. The hybrid enrichment process can be combined with the magneto-dynamic T-Ω formulation and the magnetic vector potential formulation to solve eddy current or non-linear problems.Originality/valueThe PGD enrichment process is improved by exploiting a POD. A linear eddy current problem and a non-linear electrical machine simulation are analyzed in terms of accuracy and computational effort. Further the PGD-AV formulation is derived and compared to the PGD-T-Ω reduced order model.

Reduced-order modeling of neutron transport eigenvalue problems separated in energy by Proper Generalized Decomposition

In this article, we develop and validate an a priori Reduced-Order Model (ROM) of neutron transport separated in energy by Proper Generalized Decomposition (PGD) as applied to the k-eigenvalue problem. To do so, we devise a novel PGD algorithm for eigenvalue problems, in which the update step is solved as an eigenproblem. Further, we demonstrate this problem can be efficiently solved by a recursive LU factorization which exploits the matrix structure imposed by Progressive PGD. Numerical experiments validate Galerkin and Petrov-Galerkin (Minimax) ROMs, as compared to the full-order model, in approximating the fine- and coarse-group neutron flux, the fission source, and multiplication factor (k-eigenvalue). These benchmarks consider representative light water reactor pins of UO2 or Mixed Oxide fuel with CASMO-70, XMAS-172, and SHEM-361 energy meshes. In all cases, the ROM achieves an L2 error of the angular flux less than 0.1% given fifty modes, or enrichment iterations. At the same number of modes, the eigenvalue error is found to be less than 2×10−4 and 2×10−5 for the Galerkin and Minimax ROMs respectively. Meanwhile, the fine-group ROM surpasses the accuracy of the coarse-group full-order model—comparing both to the fine-group full-order model—in estimating the angular and scalar coarse-group fluxes and k-eigenvalue between roughly ten and twenty modes. The computational cost of the PGD ROM is comparable to that of the otherwise-identical fixed-source ROM presented in previous work. Altogether, we expect this PGD ROM may achieve considerable computational savings in modeling fine-group reactor physics. Moreover, it offers an alternative means of approximation to cross section condensation, preferable in that it requires no reference solution or loss of resolution, yet may achieve superior precision for a comparable computational effort.

Regularized regressions for parametric models based on separated representations

Regressions created from experimental or simulated data enable the construction of metamodels, widely used in a variety of engineering applications. Many engineering problems involve multi-parametric physics whose corresponding multi-parametric solutions can be viewed as a sort of computational vademecum that, once computed offline, can be then used in a variety of real-time engineering applications including optimization, inverse analysis, uncertainty propagation or simulation based control. Sometimes, these multi-parametric problems can be solved by using advanced model order reduction—MOR-techniques. However, solving these multi-parametric problems can be very costly. In that case, one possibility consists in solving the problem for a sample of the parametric values and creating a regression from all the computed solutions. The solution for any choice of the parameters is then inferred from the prediction of the regression model. However, addressing high-dimensionality at the low data limit, ensuring accuracy and avoiding overfitting constitutes a difficult challenge. The present paper aims at proposing and discussing different advanced regressions based on the proper generalized decomposition (PGD) enabling the just referred features. In particular, new PGD strategies are developed adding different regularizations to the s-PGD method. In addition, the ANOVA-based PGD is proposed to ally them.

Parametric Analysis of Thick FGM Plates Based on 3D Thermo-Elasticity Theory: A Proper Generalized Decomposition Approach.

In the present work, the general and well-known model reduction technique, PGD (Proper Generalized Decomposition), is used for parametric analysis of thermo-elasticity of FGMs (Functionally Graded Materials). The FGMs have important applications in space technologies, especially when a part undergoes an extreme thermal environment. In the present work, material gradation is considered in one, two and three directions, and 3D heat transfer and theory of elasticity equations are solved to have an accurate temperature field and be able to consider all shear deformations. A parametric analysis of FGM materials is especially useful in material design and optimization. In the PGD technique, the field variables are separated to a set of univariate functions, and the high-dimensional governing equations reduce to a set of one-dimensional problems. Due to the curse of dimensionality, solving a high-dimensional parametric problem is considerably more computationally intensive than solving a set of one-dimensional problems. Therefore, the PGD makes it possible to handle high-dimensional problems efficiently. In the present work, some sample examples in 4D and 5D computational spaces are solved, and the results are presented.

A phase field model for the solid-state sintering with parametric proper generalized decomposition

Solid state sintering is a common process for additive manufacturing (AM), which is often simulated with the phase field method. However, high computational cost limits the application and prediction capacity of numerical models of sintering. In this paper, we apply the proper generalized decomposition (PGD) technique, an instance of Model Order Reduction (MOR), to solid-state sintering problems in three dimensions, with a focus on the influence of temperature. The proposed model is an effective way to simplify computational models and avoid repeated modeling, which shows superiority in lowering the computational cost and improving the computational efficiency for solid-state sintering simulations comparing to traditional approaches.