A Robust and Versatile Parallel FFT-Based Mechanical Solver for General Non-Periodic and Periodic Boundary Conditions

Jul 16, 2026

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General boundary conditions are integrated into a fast Fourier transform framework for linear and non-linear mechanical problems using small or finite transformation formulations. For distributed-memory parallel computing, a unified scheme combines periodic and non-periodic (Dirichlet or Neumann) boundary conditions. Non-periodic constraints are handled via discrete trigonometric transforms leveraging symmetry in fluctuation displacement and stress fields. A displacement-based fixed-point solver with convergence acceleration and discrete Green operators built from finite-difference discrete differential operators yields efficient parallel boundary-value problem solutions and robust performance.

LanguageEnglish

CategoryResearch & Report

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Pages40

Reading Time1 h 41 min

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1. Introduction

1.1. State of the art

A robust and versatile parallel FFT-based mechanical solver for general non-periodic and periodic boundary conditions Yaovi Armand AMOUZOU-ADOUNa , Lionel GÉLÉBARTa,∗, Cédric FLAGEULb , Yushan WANGc a Université Paris-Saclay, CEA, SRMA, Gif-sur-Yvette, 91191, France b Curiosity Group, Pprime Institute, CNRS - University of Poitiers-ENSMA, France c Université Paris-Saclay, UVSQ, CNRS, CEA, Maison de la Simulation, 91191, Gif-sur-Yvette,France Abstract General boundary conditions are implemented within a fast Fourier transform framework for linear and non-linear mechanical problems using small or finite transformation formulations. In the context of parallel computing (distributed memory), we present a framework that enables the combination of periodic and non-periodic (Dirichlet or Neumann) boundary conditions. Taking advantage of the link between non-periodic boundary conditions and the symmetries of the relevant components of the fluctuation displacement and stress fields, discrete trigonometric transforms are employed to adapt the classical Moulinec–Suquet fast Fourier transform approach. The present study employs an original displacement-based fixed-point algorithm in combination with a convergence acceleration method in order to solve boundary value problems. Finite difference approaches are used to build the discrete Green operators associated with a pre-conditioner (reference material), whose choice depends on the loading type and the small or finite transformation frameworks. The newly developed double tetrahedron scheme is employed to investigate non-periodic problems. Outcomes are compared to those of the classical hexahedral scheme. The robustness and computational efficiency of the presented parallel solver is demonstrated through numerical experiments of non-trivial loading scenarios (tension, bending, normal-mixed loading, torsion-bending), complex and densely discretized microstructures and diverse behavior laws (elasticity, isotropic plasticity, crystal plasticity), within small and finite transformation frameworks. Keywords: Non-periodic boundary conditions; Discrete Trigonometric and Fourier transforms; Discrete Green operators; Small and Finite transformations; Crystal plasticity; Massively parallel simulations. 1. Introduction 1.1. State of the art Accurate comparisons between experimental results and simulations requires robust and efficient numerical methods capable of handling large 3D grids. Since the seminal work of Moulinec and Suquet (1994, 1998), fast Fourier transform (FFT)-based solvers establish themselves as efficient alternatives to standard finite element (FE) codes, in describing the macroscopic properties of heterogeneous media, as well as the distribution of local fields (e.g., strain, stress) . For an extensive overview of classical FFT-based solvers and their applications, we refer the reader to the review articles (Schneider, 2021 ; Lucarini et al., 2022 ; Gierden et al., 2022) . FFT-based solvers exploit the application of a discrete Green operator in Fourier space to solve the governing equations efficiently. The initial proposition of Moulinec and Suquet (1998) can be regarded as a collocation method, starting from the continuous equation and using a truncation of Fourier series to solve the so-called Lippman-Schwinger equation for an auxiliary problem defined on a homogeneous material. Another possibility arXiv :2607 .05929v 1 [ cs .CE] 7 Jul 2026 ∗ Corresponding author Email addresses: [yaovi.amouzou-adoun@cea.fr](yaovi.amouzou-adoun@cea.fr) (Yaovi Armand AMOUZOU-ADOUN), [lionel.gelebart@cea.fr](lionel.gelebart@cea.fr) (Lionel GÉLÉBART), [cedric.flageul@univ-poitiers.fr](cedric.flageul@univ-poitiers.fr) (Cédric FLAGEUL), [yushan.wang@cea.fr](yushan.wang@cea.fr) (Yushan WANG) consists of using Galerkin variational formulations (Brisard and Dormieux, 2012 ; Vondˇrejc et al., 2014 ; Bignonnet et al., 2026) to build the discrete Green operator. Moreover, using

FAQs

How does the proposed framework handle both periodic and non-periodic boundary conditions in FFT-based mechanics?

It implements general boundary conditions within an FFT framework and uses discrete trigonometric transforms to adapt the classical Moulinec–Suquet approach for combining periodic and non-periodic (Dirichlet or Neumann) conditions.

What numerical strategy is used to solve mechanical boundary value problems?

The method uses a displacement-based fixed-point algorithm together with a convergence acceleration technique to improve solution efficiency.

How are discrete Green operators constructed and why does the choice of reference material matter?

Finite difference approaches are used to build discrete Green operators associated with a preconditioner (reference material), and its choice depends on the loading type and whether small or finite transformation frameworks are used.

Jul 16, 2026

Upload by Oliver