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A reduced simulation applied to viscoelastic fatigue of polymers using a time multi-scale approach based on Partition of Unity method

Sebastian Rodriguez, Angelo Pasquale, Jad Mounayer, Diego Canales, Marianne Beringhier, Chady Ghnatios, Amine Ammar, Francisco Chinesta

TL;DR

This work addresses the high computational cost of simulating viscoelastic polymers with many internal variables across a broad relaxation-time spectrum. It introduces a space-time PGD framework augmented with a temporal multi-scale strategy based on Partition of Unity to efficiently resolve multiple time scales, including a transient-to-fatigue transition. By coupling a fixed-point PGD expansion for displacement and internal variables with low-rank approximations and PU-enriched time functions, the paper demonstrates feasibility on a 1D bar under cyclic loading, including scenarios with up to 50 internal variables. Although the multi-scale approach incurs additional setup costs and may be less efficient for short horizons, it shows promise for long-time fatigue problems and paves the way for extensions to 3D and nonlinear viscoelasticity, with richer macro temporal bases to further reduce the mode count.

Abstract

The simulation of viscoelastic time-evolution problems described by a large number of internal variables and with a large spectrum of relaxation times requires high computational resources for their resolution. Furthermore, the internal variables evolution is described by a set of linear differential equations which involves many time scales. In this context, the use of a space-time PGD approximation is proposed here to boost their resolution, where the temporal functions are constructed following a multi-scale strategy along with the Partition of Unity method, in order to catch each dynamic efficiently. The feasibility and the robustness of the method are discussed in the case of a polymer in a non-equilibrium state under cyclic loading.

A reduced simulation applied to viscoelastic fatigue of polymers using a time multi-scale approach based on Partition of Unity method

TL;DR

This work addresses the high computational cost of simulating viscoelastic polymers with many internal variables across a broad relaxation-time spectrum. It introduces a space-time PGD framework augmented with a temporal multi-scale strategy based on Partition of Unity to efficiently resolve multiple time scales, including a transient-to-fatigue transition. By coupling a fixed-point PGD expansion for displacement and internal variables with low-rank approximations and PU-enriched time functions, the paper demonstrates feasibility on a 1D bar under cyclic loading, including scenarios with up to 50 internal variables. Although the multi-scale approach incurs additional setup costs and may be less efficient for short horizons, it shows promise for long-time fatigue problems and paves the way for extensions to 3D and nonlinear viscoelasticity, with richer macro temporal bases to further reduce the mode count.

Abstract

The simulation of viscoelastic time-evolution problems described by a large number of internal variables and with a large spectrum of relaxation times requires high computational resources for their resolution. Furthermore, the internal variables evolution is described by a set of linear differential equations which involves many time scales. In this context, the use of a space-time PGD approximation is proposed here to boost their resolution, where the temporal functions are constructed following a multi-scale strategy along with the Partition of Unity method, in order to catch each dynamic efficiently. The feasibility and the robustness of the method are discussed in the case of a polymer in a non-equilibrium state under cyclic loading.
Paper Structure (13 sections, 20 equations, 16 figures)

This paper contains 13 sections, 20 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Viscoelastic model
  3. PGD model-reduction method applied to the viscoelastic problem
  4. Computation of the low-rank approximation of displacement
  5. Convergence criteria for the construction of the low-rank decomposition
  6. Computation of the low-rank approximation of internal variables
  7. Convergence criteria for the construction of the low-rank decomposition
  8. Determination of temporal PGD functions as a multi-scale approximation throw Partition of Unity method
  9. Dealing with transient behavior in the time function to be approximated
  10. Numerical examples
  11. One internal variable
  12. 50 internal variables
  13. Conclusions and perspectives

Figures (16)

  • Figure 1: Reference problem.
  • Figure 2: Illustration of multi-scale discretization and used shape functions for the macro problem.
  • Figure 3: Illustration of overlapping of microscales.
  • Figure 4: Reference signal and multi-scale approximation.
  • Figure 5:
  • ...and 11 more figures