A separated representation involving multiple time scales within the Proper Generalized Decomposition framework

Angelo Pasquale,Jean-Louis Duval,Francisco Chinesta,Simona Perotto,Antonio Falcó,Elías Cueto,Amine Ammar

doi:10.1186/s40323-021-00211-7

Abstract

Solutions of partial differential equations can exhibit multiple time scales. Standard discretization techniques are constrained to capture the finest scale to accurately predict the response of the system. In this paper, we provide an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors. The proposed methodology is based on a time-separated representation within the standard Proper Generalized Decomposition, where the time coordinate is transformed into a multi-dimensional time through new separated coordinates, each representing one scale, while continuity is ensured in the scale coupling. For instance, when considering two different time scales, the governing Partial Differential Equation is commuted into a nonlinear system that iterates between the so-called microtime and macrotime, so that the time coordinate can be viewed as a 2D time. The macroscale effects are taken into account by means of a finite element-based macro-discretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the time domain. The resulting separated representation allows us a very fine time discretization without impacting the computational efficiency. The proposed formulation is explored and numerically verified on thermal and elastodynamic problems.

Highlights

Many engineering problems are defined in very large time intervals and, at the same time, the response must encompass the different time scales present in the model
When considering two different time scales, the governing Partial Differential Equation is commuted into a nonlinear system that iterates between the so-called microtime and macrotime, so that the time coordinate can be viewed as a 2D time
This paper aims at computing the time evolution of the unknown fields involved by a multiscale Partial Differential Equation (PDE), taking into account all the information that it involves, i.e., covering all the time scales

Summary

Introduction

Many engineering problems are defined in very large time intervals (e.g., when dealing with fatigue, aging, dynamics with loadings involving multiple characteristic times) and, at the same time, the response must encompass the different time scales present in the model. In order to apply the new PGD formulation based on a two-scale time discretization to problem (10), we have to recast the differential operators in (20) into a multiscale framework, by adding a further temporal independent variable This leads us to deal with a framework characterized by two dimensions in time and one dimension in space. A11 = A21 = INx , A31 = −k DNx , Fig. 3 Sparsity pattern of discrete PGD operators associated with the separated representation (21), for Nx = 101 and Nt = 1000 whereas the two-scale time discretization leads us to adopt the modified tensorial decomposition in (16), ENt = A21 ⊗ A31 + A22 ⊗ A32, with A12 = Em+1 the matrix characterizing the implicit Euler discretization at the microscale; A13 = A33 = IM the discrete identity matrix associated with the subintervals i+1; A22 = Cm+1 the correction matrix, sized according to the microscale, with a unique non-null element in the upper-right corner; A23 = LM the lower shift matrix, sized according to the macroscale, with the first subdiagonal of elements all equal to one; A32 = Im+1 the discrete identity matrix associated with the microscale.

Conclusions

Methods