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The LATIN-PGD methodology to nonlinear dynamics and quasi-brittle materials for future earthquake engineering applications

Sebastian Rodriguez, Pierre-Etienne Charbonnel, Pierre Ladevèze, David Néron

TL;DR

The paper develops a LATIN-PGD framework integrated with Time-Discontinuous Galerkin (TDGM) for efficient, low-frequency nonlinear dynamics in quasi-brittle concrete with isotropic damage. By reformulating the problem into Gamma/Ad spaces and enriching space-time PGD modes at each global step, the method achieves accurate results with reduced online cost, as demonstrated on a 3D bending beam and benchmarked against standard Newmark/Quasi-Newton solvers. The combination of PD-based model order reduction and TDGM enables handling of many temporal degrees of freedom and supports parametric/uncertainty analyses relevant to seismic risk assessment. The work lays the groundwork for scalable, uncertainty-aware earthquake engineering simulations with reusable PGD bases and efficient temporal computations.

Abstract

This paper presents a first implementation of the LArge Time INcrement (LATIN) method along with the model reduction technique called Proper Generalized Decomposition (PGD) for solving nonlinear low-frequency dynamics problems when dealing with a quasi-brittle isotropic damage constitutive relations. The present paper uses the Time-Discontinuous Galerkin Method (TDGM) for computing the temporal contributions of the space-time separate-variables solution of the LATIN-PGD approach, which offers several advantages when considering a high number of DOFs in time. The efficiency of the method is tested for the case of a 3D bending beam, where results and benchmarks comparing LATIN-PGD to classical time-incremental Newmark/Quasi-Newton nonlinear solver are presented. This work represents a first step towards taking into account uncertainties and carrying out more complex parametric studies imposed by seismic risk assessment.

The LATIN-PGD methodology to nonlinear dynamics and quasi-brittle materials for future earthquake engineering applications

TL;DR

The paper develops a LATIN-PGD framework integrated with Time-Discontinuous Galerkin (TDGM) for efficient, low-frequency nonlinear dynamics in quasi-brittle concrete with isotropic damage. By reformulating the problem into Gamma/Ad spaces and enriching space-time PGD modes at each global step, the method achieves accurate results with reduced online cost, as demonstrated on a 3D bending beam and benchmarked against standard Newmark/Quasi-Newton solvers. The combination of PD-based model order reduction and TDGM enables handling of many temporal degrees of freedom and supports parametric/uncertainty analyses relevant to seismic risk assessment. The work lays the groundwork for scalable, uncertainty-aware earthquake engineering simulations with reusable PGD bases and efficient temporal computations.

Abstract

This paper presents a first implementation of the LArge Time INcrement (LATIN) method along with the model reduction technique called Proper Generalized Decomposition (PGD) for solving nonlinear low-frequency dynamics problems when dealing with a quasi-brittle isotropic damage constitutive relations. The present paper uses the Time-Discontinuous Galerkin Method (TDGM) for computing the temporal contributions of the space-time separate-variables solution of the LATIN-PGD approach, which offers several advantages when considering a high number of DOFs in time. The efficiency of the method is tested for the case of a 3D bending beam, where results and benchmarks comparing LATIN-PGD to classical time-incremental Newmark/Quasi-Newton nonlinear solver are presented. This work represents a first step towards taking into account uncertainties and carrying out more complex parametric studies imposed by seismic risk assessment.
Paper Structure (20 sections, 56 equations, 13 figures, 4 tables, 1 algorithm)

This paper contains 20 sections, 56 equations, 13 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Equations of the reference problem
  3. The reference problem
  4. Isotropic damage model for concrete materials
  5. Quasi-brittle behavior in tension
  6. Progressive micro-crack re-closure
  7. Uniaxial response of the concrete medium -- Comparison with test results
  8. The LATIN-PGD solver
  9. Reformulation of the reference problem
  10. The LATIN method: main ingredients
  11. Convergence and stop criterion
  12. Implementation details
  13. Local Stage
  14. Global linear stage: equilibrium and compatibility equations
  15. Enrichment step: addition of a new rank-one space-time PGD mode to the existing decomposition
  16. ...and 5 more sections

Figures (13)

  • Figure 1: The mechanical domain under study.
  • Figure 2: Handling unilateral effect in concrete RVE -- uniaxial illustration with only one micro-crack.
  • Figure 3: Mechanical response of the model at a Gauss point to a uni-axial tension-compression loading.
  • Figure 4: Uni-axial test results on double-notched concrete specimen described in nouailletas2015experimental. (Left) Imposed displacement in the vertical direction, two Crack Mouth Opening Displacement (CMOD) sensors are measuring the displacement between the two faces of each notch. (Right) Stress vs. strain plot in the vertical direction; the "normal strain" on abscissa is an averaged value computed from the two CMOD sensors.
  • Figure 5: Iterative strategy with search directions $\mathit{G}$ and $\mathit{A}$.
  • ...and 8 more figures