Proper Generalized Decomposition Research Articles

Proper Generalized Decomposition Applied on a Rotating Electrical Machine

The proper generalized decomposition (PGD) is a model order reduction method which allows to reduce the computational time of a numerical problem by seeking for a separated representation of the solution. The PGD has been already applied to study an electrical machine but at standstill without accounting the motion of the rotor. In this paper, we propose a method to account for the rotation in the PGD approach in order to build an efficient metamodel of an electrical machine. Then, the machine metamodel will be coupled to its electrical and mechanical environment in order to obtain accurate results with an acceptable computational time on a full simulation.

Inclusion of frequency‐dependent parameters in power transmission lines simulation using harmonic analysis and proper generalized decomposition

AbstractThis study presents the application of proper generalized decomposition on transmission line models involving frequency‐dependent parameters. Frequency dependence of parameters can be due to a number of reasons including phenomena such as “ground return effects,” “proximity effects,” and “skin effects.” In the current study, simplified methods of skin effects based on Bessel functions are used to showcase the method. Although we implement our study using a specific model for skin effects to demonstrate the effectiveness of the proposed method to accommodate frequency‐dependent effects in proper generalized decomposition, the present work is not meant to discuss the merits of any particular skin effects model. The method can easily accommodate other effects, which induces frequency dependence in the transmission line parameters. In time‐domain modeling, the parameters are assumed constant, and these models prove inefficient when incorporating these parameters as function of frequency. Therefore, a frequency‐domain simulation is implemented using harmonic analysis. Proper generalized decomposition (PGD) presents a separated representation and provides a quick and accurate solution for such problems.

A reduced model to simulate the damage in composite laminates under low velocity impact

This article presents an efficient numerical strategy to simulate the damage in composite laminates under low velocity impact. The proposed method is based on a separated representation of the solution in the context of the Proper Generalized Decomposition (PGD). This representation leads to an important reduction of the number of degrees of freedom. In addition to the PGD, the main ingredients of the model are the following: (a) cohesive zone models (CZM) to represent the delamination and the matrix cracking, (b) a modified nonlinear Hertzian contact law to calculate the impact force, (c) the implicit Newmark integration scheme to compute the evolution of the solution during the impact. The method is applied to simulate an impact on a laminated plate. The results are similar to the solution obtained with a classical finite element simulation. The shape of the delaminated area is found to be coherent with some experimental results from the literature.

Proper generalized decomposition solutions within a domain decomposition strategy

SummaryDomain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in‐plane–out‐of‐plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user‐defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution.

Proper Generalized Decomposition (PGD) for the numerical simulation of polycrystalline aggregates under cyclic loading

The numerical modelling of the behaviour of materials at the microstructural scale has been greatly developed over the last two decades. Unfortunately, conventional resolution methods cannot simulate polycrystalline aggregates beyond tens of loading cycles, and they do not remain quantitative due to the plasticity behaviour. This work presents the development of a numerical solver for the resolution of the Finite Element modelling of polycrystalline aggregates subjected to cyclic mechanical loading. The method is based on two concepts. The first one consists in maintaining a constant stiffness matrix. The second uses a time/space model reduction method. In order to analyse the applicability and the performance of the use of a space–time separated representation, the simulations are carried out on a three-dimensional polycrystalline aggregate under cyclic loading. Different numbers of elements per grain and two time increments per cycle are investigated. The results show a significant CPU time saving while maintaining good precision. Moreover, increasing the number of elements and the number of time increments per cycle, the model reduction method is faster than the standard solver.

Explicit parametric solutions of lattice structures with proper generalized decomposition (PGD)

Architectured materials (or metamaterials) are constituted by a unit-cell with a complex structural design repeated periodically forming a bulk material with emergent mechanical properties. One may obtain specific macro-scale (or bulk) properties in the resulting architectured material by properly designing the unit-cell. Typically, this is stated as an optimal design problem in which the parameters describing the shape and mechanical properties of the unit-cell are selected in order to produce the desired bulk characteristics. This is especially pertinent due to the ease manufacturing of these complex structures with 3D printers. The proper generalized decomposition provides explicit parametic solutions of parametric PDEs. Here, the same ideas are used to obtain parametric solutions of the algebraic equations arising from lattice structural models. Once the explicit parametric solution is available, the optimal design problem is a simple post-process. The same strategy is applied in the numerical illustrations, first to a unit-cell (and then homogenized with periodicity conditions), and in a second phase to the complete structure of a lattice material specimen.

Proper Generalized Decomposition Method for Solving Fisher-Type Equation and Heat Equation

A model reduction technique—the Proper Generalized Decomposition (PGD) for solving time dependent and multidimensional parameters is reviewed and applied to both the Fisher-type equations and the heat equation. Space-time discretization and separated representation technique for obtaining fast convergence computation while maintaining real time is detailed. Three situations of the Fisher-type equation are solved by the PGD and the results show a perfect agreement with the exact solutions. The source term of the heat equation is given a Huxley source and the thermal diffusivity is taken to be linearly dependent of the spatial parameter. The results show how the Fisher-type equation finds application to the heat equation and that the PGD method allows a perfect representation of the temperature distribution defined in a 5-D tensorial product space and time.

Sensitivity thermal analysis in the laser-assisted tape placement process

Nowadays, the production of large pieces made of thermoplastic composites is an industrial challenging issue as there are yet several difficulties associated to their processing. The laser-assisted tape placement (LATP) process is an automated manufacturing technique to produce long-fiber reinforced thermoplastic matrix composites. In this process, a tape is placed and progressively welded on the substrate. The main aim of the present work is to solve an almost state of the art thermal model by using an efficient numerical technique, the so-called Proper Generalized Decomposition (PGD) that considers parameters (geometrical and material) as model extra-coordinates. Within the PGD rationale the parametric temperature field is expressed in a separated form, as a finite sum of functional products, where each term depends on a single coordinate (space, time or each one of the parameters considered as extra-coordinates). Such a separated representation allows the explicit expression of the sensitivity fields, from the temperature derivative with respect to each parameter. These sensitivity fields represent a very valuable methodology to analyze and establish the influence of the critical input parameters on the thermal response, and therefore, for performing process optimization and control, as well as for evaluating the effect of parameters variability on the thermal response.

A New Algorithm of Proper Generalized Decomposition for Parametric Symmetric Elliptic Problems

We introduce a new algorithm of proper generalized decomposition (PGD) for parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optima...

Multi-parametric space-time computational vademecum for parametric studies: Application to real time welding simulations

Real time simulations of welding processes remain intractable despite the impressive increasing computing power. This paper presents the case of a thermo-elasto-plastic problem with located moving heat loading. A novel non-intrusive a posteriori reduced order strategy for building multiparametric computational vademecum dedicated to real-time simulations of nonlinear thermo-mechanical problems is proposed. The high order proper generalized decomposition (HOPGD) is used to seek separated representation of solutions with some precomputed snapshots. Furthermore, a relaxation method is successfully applied to accelerate this procedure. The accuracy of the constructed computational vademecum is controlled by a localized multigrid selection method that allows an automatic selection of snapshots in the areas of interest of the parameter space. Examples of multiparametric computational vademecum taking into account some material parameters will be shown in this paper.

On the Existence of a Progressive Variational Vademecum based on the Proper Generalized Decomposition for a Class of Elliptic Parameterized Problems

In this study, we present the mathematical analysis needed to explain the convergence of a progressive variational vademecum based on the proper generalized decomposition (PGD). The PGD is a novel technique that was developed recently for solving problems with high dimensions, and it also provides new approaches for obtaining the solutions of elliptic and parabolic problems via the abstract separation of variables method. This new scenario requires a mathematical framework in order to justify its application to the solution of numerical problems and the PGD can help in the change to this paradigm. The main aim of this study is to provide a mathematical environment for defining the notion of progressive variational vademecum. We prove the convergence of this iterative procedure and we also provide the first order optimality conditions in order to construct the numerical approximations of the parameterized solutions. In particular, we illustrate this methodology based on a robot path planning problem. This is one of the common tasks when designing the trajectory or path of a mobile robot. The construction of a progressive variational vademecum provides a novel methodology for computing all the possible paths from any start and goal positions derived from a harmonic potential field in a predefined map.

Generalized parametric solutions in Stokes flow

Design optimization and uncertainty quantification, among other applications of industrial interest, require fast or multiple queries of some parametric model. The Proper Generalized Decomposition (PGD) provides a separable solution, a computational vademecum explicitly dependent on the parameters, efficiently computed with a greedy algorithm combined with an alternated directions scheme and compactly stored. This strategy has been successfully employed in many problems in computational mechanics. The application to problems with saddle point structure raises some difficulties requiring further attention. This article proposes a PGD formulation of the Stokes problem. Various possibilities of the separated forms of the PGD solutions are discussed and analyzed, selecting the more viable option. The efficacy of the proposed methodology is demonstrated in numerical examples for both Stokes and Brinkman models.

A multiscale separated representation to compute the mechanical behavior of composites with periodic microstructure

The requirements for advanced numerical computations are very high when studying the multiscale behavior of heterogeneous structures such as composites. For the description of local phenomena taking place on the microscopic scale, the computation must involve a fine discretization of the structure. This condition leads to problems with a high number of degrees of freedom that lead to prohibitive computational costs when using classical numerical methods such as the finite element method (FEM). To overcome these problems, this paper presents a new domain decomposition method based on the proper generalized decomposition (PGD) to predict the behavior of periodic composite structures. Several numerical tests are presented. The PGD results are compared with those obtained using the classical finite element method. A very good agreement is observed.

Immersed boundary method combined with proper generalized decomposition for simulation of a flexible filament in a viscous incompressible flow

In this paper, a combination of the Proper Generalized Decomposition (PGD) with the Immersed Boundary method (IBM) for solving fluid-filament interaction problem is proposed. In this combination, a forcing term constructed by the IBM is introduced to Navier-Stokes equations to handle the influence of the filament on the fluid flow. The PGD is applied to solve the Poission's equation to find the fluid pressure distribution for each time step. The numerical results are compared with those by previous publications to illustrate the robustness and effectiveness of the proposed method.

Local proper generalized decomposition

SummaryOne of the main difficulties that a reduced‐order method could face is the poor separability of the solution. This problem is common to both a posteriori model order reduction (proper orthogonal decomposition, reduced basis) and a priori [proper generalized decomposition (PGD)] model order reduction. Early approaches to solve it include the construction of local reduced‐order models in the framework of POD. We present here an extension of local models in a PGD—and thus, a priori—context. Three different strategies are introduced to estimate the size of the different patches or regions in the solution manifold where PGD is applied. As will be noticed, no gluing or special technique is needed to deal with the resulting set of local reduced‐order models, in contrast to most proper orthogonal decomposition local approximations.The resulting method can be seen as a sort of a priori manifold learning or nonlinear dimensionality reduction technique. Examples are shown that demonstrate pros and cons of each strategy for different problems.

3-D IC Interconnect Parasitic Capacitance Extraction With a Reformulated PGD Algorithm

Proper generalized decomposition (PGD) is a recently developed model-order reduction method based on the use of variable-separated representations. In this paper, space variable-separated PGD is applied on the 3-D capacitance extraction of interconnects in integrated circuits to reduce its computational complexity. To make the PGD solver feasible, the complex boundary conditions are simplified by a characteristic function technique. 3-D singular-value decomposition of the coefficient functions is avoided by using a reformulated PGD algorithm, and therefore, the proposed method can effectively deal with problems with inhomogeneous dielectric layers and dummy fills. Numerical examples are given to verify the method.

The Reference Point Method, a “hyperreduction” technique: Application to PGD-based nonlinear model reduction

A new approximation technique, called Reference Point Method (RPM), is proposed in order to reduce the computational complexity of algebraic operations for constructing reduced-order models in the case of time dependent and/or parametrized nonlinear partial differential equations. Even though model reduction techniques enableone to decrease the dimension of the initial problem in the sense that far fewer degrees of freedom are needed to represent the solution, the complexity of evaluating the nonlinear terms and assembling the low dimensional operator associated with the reduced-order model still scales with the size of the original high-dimensional model. This point can be critical, especially when the reduced-order basis changes throughout the solution strategy as it is the case for model reduction techniques based on Proper Generalized Decomposition (PGD). Based on the concept of spatial, parameter/time reference points and influence patches, the RPM defines a compressed version of the data from which an approximate low-rank separated representation by patch of the operators can be constructed by explicit formulas at low-cost without resorting to SVD-based techniques. An application of the RPM to PGD-based model reduction for a nonlinear parametrized elliptic PDE previously studied by other authors with reduced-basis method and EIM is proposed. It is shown that computational complexity to construct the reduced-order model can be divided in practice by one order of magnitude compared with the classical PGD approach.

Bayesian and variational Bayesian approaches for flows in heterogeneous random media

In this paper, we study porous media flows in heterogeneous stochastic media. We propose an efficient forward simulation technique that is tailored for variational Bayesian inversion. As a starting point, the proposed forward simulation technique decomposes the solution into the sum of separable functions (with respect to randomness and the space), where each term is calculated based on a variational approach. This is similar to Proper Generalized Decomposition (PGD). Next, we apply a multiscale technique to solve for each term (as in [1]) and, further, decompose the random function into 1D fields. As a result, our proposed method provides an approximation hierarchy for the solution as we increase the number of terms in the expansion and, also, increase the spatial resolution of each term. We use the hierarchical solution distributions in a variational Bayesian approximation to perform uncertainty quantification in the inverse problem. We conduct a detailed numerical study to explore the performance of the proposed uncertainty quantification technique and show the theoretical posterior concentration.

Monitoring a PGD solver for parametric power flow problems with goal‐oriented error assessment

SummaryThe parametric analysis of electric grids requires carrying out a large number of power flow computations. The different parameters describe loading conditions and grid properties. In this framework, the proper generalized decomposition (PGD) provides a numerical solution explicitly accounting for the parametric dependence. Once the PGD solution is available, exploring the multidimensional parametric space is computationally inexpensive. The aim of this paper is to provide tools to monitor the error associated with this significant computational gain and to guarantee the quality of the PGD solution. In this case, the PGD algorithm consists in three nested loops that correspond to (1) iterating algebraic solver, (2) number of terms in the separable greedy expansion, and (3) the alternated directions for each term. In the proposed approach, the three loops are controlled by stopping criteria based on residual goal‐oriented error estimates. This allows one for using only the computational resources necessary to achieve the accuracy prescribed by the end‐user. The paper discusses how to compute the goal‐oriented error estimates. This requires linearizing the error equation and the quantity of interest to derive an efficient error representation based on an adjoint problem. The efficiency of the proposed approach is demonstrated on benchmark problems. Copyright © 2016 John Wiley & Sons, Ltd.

An innovative method for the design of high energy performance building envelopes

In this paper, an innovative method to minimise energy losses through building envelopes is presented, using the Proper Generalised Decomposition (PGD), written in terms of space x, time t, thermal diffusivity α and envelope thickness L. The physical phenomenon is solved at once, contrarily to classical numerical methods that cannot create a parameter dependent model. First, the PGD solution is validated with an analytical solution to prove its accuracy. Then a complex case study of a multi-layer wall submitted to transient boundary conditions is investigated. The parametric solution is computed as a function of the space and time coordinates, as well as the thermal insulation thickness and the load material thermal diffusivity. Physical behaviour and conduction loads are analysed for 76 values of thermal insulation thickness and 100 types of load material properties. Furthermore, the reduced computational cost of the PGD is highlighted. The method computes the solution 100 times faster than standard numerical approaches. In addition, the PGD solution has a low storage cost, providing interesting development of parametric solutions for real-time applications of energy management in buildings.