An implicit FFT-based method for wave propagation in elastic heterogeneous media
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Achieving geometric accuracy in FFT-based micromechanical models using conformal grid
Owing to its efficiency, simplicity and robustness, the FFT-based method has become the standard for computation of mechanical fields in a heterogeneous periodic unit cell. One of the main disadvantages of the FFT-based method is the inaccurate representation of the initial microstructure on a regular grid of voxels, which can be alleviated through the use of distorted initial grids. Here, in this paper, a method for generation of distorted initial grids conforming to the microstructural features (e.g. straight/curved boundaries) is proposed. The method determines the positions of the grid nodes in the initial configuration by solving a system of springs connecting the nodes. Microstructures consisting of layers, Voronoi tessellation and circular/spherical inclusions are considered, and mechanical fields simulated using the FFT-based method. It is found that distorted initial grids, conforming to the microstructural features, lead to more accurate mechanical fields in comparison to the corresponding non-distorted initial grid solution. The effect of initial grid distortion on the convergence of the FFT-based method is analyzed and discussed.
36 MATERIALS SCIENCE↗
A Green’s function fast multipole method for computation of micromechanical fields in heterogeneous materials
Computation of micromechanical fields in heterogeneous materials is usually performed using either the finite element method or the Green’s function method based on FFTs. The finite element method allows for accurate discretization and for non-periodic boundary conditions but is computationally expensive. On the other hand, the FFT-based method is computationally efficient but requires discretization on a regular grid of hexahedral voxels. In this paper, a Green’s function method allowing for accurate discretization using tetrahedral elements and for non-periodic boundary conditions is proposed. The convolution is computed using the fast multipole method, which provides good accuracy even for low-order expansion due to the fast decay of interactions between elements. The proposed Green’s function fast multipole method is verified by comparison with analytical and FFT-based solutions. Furthermore, the computational time is analyzed and compared to the FFT-based method for non-periodic convolution. Finally, effective properties of an elastic polycrystalline microstructure containing thin intergranular cracks are computed and analyzed.
36 MATERIALS SCIENCE↗
MOSAIC: An Effective FFT-based Numerical Method to Assess Aging Properties of Concrete
As the nuclear fleet in the United States ages and subsequent license renewal applications grow, the prediction of concrete durability at extended operation becomes more important. To address this issue, a Fast-Fourier Transform (FFT) method is utilized to simulate aging-related degradation of concrete within the Microstructure Oriented Scientific Analysis of Irradiated Concrete (MOSAIC) software. MOSAIC utilizes compositional phase maps to simulate damage from radiation-induced volumetric expansion (RIVE), applied force, creep, and thermal expansion. This compositional detail allows each mineral in the microstructure to be assigned specific material properties, allowing the simulation to be as accurate and representative as possible. The principal goal of MOSAIC is to simulate the effects of nonlinear aging mechanisms occurring in nuclear concrete on the macroscopic mechanical properties, using only the aggregate microstructure compositional information as a starting point. Here, several realistic example simulations are shown to demonstrate the utility and uniqueness of the MOSAIC software.
36 MATERIALS SCIENCE↗
Extended FFT-based micromechanical formulation to consider general non-periodic boundary conditions
Here, this paper presents a new approach for applying non-periodic boundary conditions in the context of FFT-based methods to solve micromechanical problems in heterogeneous solids. The domain of the original problem is extended to satisfy the periodicity requirements at the boundary of the extended domain. The velocity constraint on the boundary of the original domain is replaced by a corresponding constraint on the velocity gradient in the extended volume, and a two-level augmented Lagrangian method is used to enforce the constraint. The proposed method is implemented as an extension of the large-strain elasto-viscoplastic FFT-based (LS-EVPFFT) model of Zecevic et al. (2022). The proposed method is verified in the cases of fully imposed velocity boundary conditions and mixed velocity/traction-free boundary conditions. The accuracy and convergence of the method are studied next, followed by applications to bending and indentation of polycrystals that illustrate the extended capabilities of the proposed formulation.
36 MATERIALS SCIENCE↗
An FFT-based micromechanical model for gradient enhanced brittle fracture
Damage models incorporated within FFT-based micromechanical methods have received much attention recently because of the need to better understand and predict brittle and ductile fracture. An important aspect of a damage model is non-local regularization, which removes the mesh dependence of the predictions that otherwise become physically unacceptable upon grid refinement. In this work, the Helmholtz-type equation for non-local gradient regularization of a damage model on a distorted grid is solved using an FFT-based approach. Further, the resulting system of equations is solved using the Jacobi iterative method. The model is applied to simulate brittle fracture of an intermetallic. The influence of the time and space discretization, the length-scale parameter, and intermetallic crystallographic orientation on crack evolution is studied.
36 MATERIALS SCIENCE↗
Approximation of periodic Green's operator in real space using numerical integration and its use in fast Fourier transform-based micromechanical models
In this work, we propose an expression for the periodic first derivative of Green's function in real space. The proposed expression allows an alternative way of computing the periodic Green's operator based on periodically summing the free-space Green's operator in terms of an appropriate quadrature rule. We provide computational examples, which show the accuracy of the proposed approach, together with reduced spurious oscillations in the solution fields.
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Generalized grain boundary constitutive description implemented in a strain-gradient large-strain FFT-based formulation: Application to nano-metallic laminates
Herein this paper presents a general treatment of grain boundary constitutive behavior in the context of strain-gradient (SG) plasticity, and its numerical implementation in a large-strain (LS) elasto-viscoplastic (EVP) fast Fourier transform (FFT)-based micromechanical model. Two novel grain boundary constitutive equations are proposed, allowing for more accurate description of the Burgers vector flow at the grain boundary. The capabilities of the generalized SG-LS-EVPFFT formulation are illustrated for the case of kink-band formation during layer-parallel compression of nano-metallic laminates (NMLs), requiring consideration of the interaction between dislocations and interfaces.
36 MATERIALS SCIENCE↗
New large-strain FFT-based formulation and its application to model strain localization in nano-metallic laminates and other strongly anisotropic crystalline materials
This paper presents a new robust large-strain (LS) elasto-viscoplastic (EVP) formulation based on Fast Fourier Transforms (FFTs) for the prediction of the micro-mechanical response and microstructure evolution of polycrystalline and multiphase materials, with emphasis on the effect of strong crystallographic and/or morphologic anisotropy on localization of plastic deformation. In this work, the novel LS-EVPFFT formulation allows treatment of complex initial geometries and large deformations considering three grids of material points: a regular grid in the reference configuration, where FFTs can be performed; an irregular grid in the initial configuration, created by applying a stress-free displacement field to the reference regular grid; and an irregular grid in the current configuration, undergoing large strains and rotations as the material is loaded. Further numerical stability of the new formulation also required the use of a novel expression for the discrete modified Green’s operator, which reduces spurious field oscillations. After presenting and validating the new formulation by comparison with preexisting implementations and analytical solutions, LS-EVPFFT is applied to the prediction of slip and kink bands formation in polycrystalline columnar ice, and kink bands in single crystal zinc wires, showing good agreement with classic experiments. Finally, the model is used to study kink band formation during compression of Cu–Nb nano-metallic laminates (NMLs), in which accurate treatment of the complex geometry associated with the tortuosity of interfaces and large deformations become critical, showing consistency with corresponding micropillar experiments.
36 MATERIALS SCIENCE↗
FFT-based solver for higher-order and multi-phase-field fracture models applied to strongly anisotropic brittle materials
This paper presents the application of a fast Fourier transform (FFT) based method to solve two phase field models designed to simulate crack growth of strongly anisotropic materials in the brittle regime. By leveraging the ability of the FFT-based solver to generate solutions with higher-order and global continuities, we design two simple algorithms to capture the complex fracture patterns (e.g. sawtooth, and curved crack growth) common in materials with strongly anisotropic surface energy via the multi-phase-field and high-order phase-field frameworks. A staggered operator-split solver is used where both the balance of linear momentum and the phase field governing equations are formulated in the periodic domain. The unit phase field of the initial failure region is prescribed by the penalty method to alleviate the sharp material contrast between the initial failure region and the base material. The discrete frequency vectors are generalized to estimate the second and fourth order gradients such that the Gibbs effect near shape interfaces or jump conditions can be suppressed. Furthermore, a preconditioner is adopted to improve the convergence rate of the iterative linear solver. Three numerical experiments are used to systematically compare the performance of the FFT-based method in the multi-phase-field and high-order phase-field frameworks.
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BM3DORNL
BM3DORNL is a high-performance, open-source library for removing streak and ring artifacts from computed-tomography (CT) data, developed for neutron imaging at Oak Ridge National Laboratory's Spallation Neutron Source (VENUS beamline) and applicable to X-ray CT as well. Ring artifacts — concentric rings in reconstructed slices caused by detector pixel-to-pixel response non-uniformities — appear as vertical streaks in the sinogram and degrade both image quality and quantitative analysis. BM3DORNL operates in the sinogram domain using an adaptation of the BM3D (block-matching and 3D collaborative filtering) algorithm (Dabov et al., 2007). It provides a dedicated streak-removal mode, a true multi-scale BM3D variant (after Mäkinen et al., 2021) that suppresses wide streaks single-scale methods miss, and an alternative Fourier–SVD method (~2.6× faster) combining FFT-based energy detection with rank-1 SVD. The computationally intensive core is implemented in Rust with parallel (Rayon) block matching, integral-image pre-screening, and optimized transforms, and is exposed through a simple Python API (with an optional GUI) so it integrates directly into existing tomography reconstruction pipelines. It processes both 2D sinograms and 3D sinogram stacks, is pip-installable for Linux and macOS, and is documented at https://bm3dornl.readthedocs.io.
Zhang, Chen [Oak Ridge National Laboratory (ORNL),↗
An FFT-based approach for Bloch wave analysis: application to polycrystals
A method based on the Fast Fourier Transform is proposed to obtain the dispersion relation of acoustic waves in heterogeneous periodic media with arbitrary microstructures. The microstructure is explicitly considered using a voxelized Representative Volume Element (RVE). The dispersion diagram is obtained solving an eigenvalue problem for Bloch waves in Fourier space. To this aim, two linear operators representing stiffness and mass are defined through the use of differential operators in Fourier space. The smallest eigenvalues are obtained using the implicitly restarted Lanczos and the subspace iteration methods, and the required inverse of the stiffness operator is done using the conjugate gradient with a preconditioner. The method is used to study the propagation of acoustic waves in elastic polycrystals, showing the strong effect of crystal anistropy and polycrystaline texture on the propagation. It is shown that the method combines the simplicity of classical Fourier series analysis with the versatility of Finite Elements to account for complex geometries proving an efficient and general approach which allows the use of large RVEs in 3D.
97 MATHEMATICS AND COMPUTING↗
Weighted FFT estimators for 1D and 3D correlations of the Lyman- α forest
Correlations in the Lyman-α (Lyα) forest, both as a function of line of sight separation (1D) and 3D separation, provide a unique window to the distribution of matter at redshifts not accessible by current galaxy surveys. While optimal quadratic estimators have been used to measure 1D correlations, they are computationally expensive and difficult to extend to 3D analyses. On the other hand, estimators based on the Fast Fourier Transform (FFT) are significantly faster, but are affected by missing data in the spectra (masked pixels) and so far have not used pixel weights to reduce the uncertainties in the measurement. In this publication we describe how to compute the window matrix that enables forward-modelling the impact of masked pixels and weights on the FFT-based estimators. Here, we use Gaussian and hydrodynamical simulations with artificially masked pixels to validate the method on the measurement of 1D correlations. Finally, we show that the formalism can be extended to model the impact on 3D correlations, in particular on the cross-spectrum, the correlation of 1D Fourier modes as a function of transverse separation. This work will enable more precise clustering measurements with the Lyα forest dataset recently collected by the Dark Energy Spectroscopic Instrument (DESI).
Lokken, Martine [Univ. Autonoma de Barcelona (Spai↗
Enhancing photoionization rate calculations in low-temperature plasmas using spectral methods
Photoionization plays a central role in the development of streamer discharges and other non-equilibrium plasma phenomena. It creates seed electrons, which are essential for positive streamer propagation, allowing the ionization front to move forward. Because of this, accurate modeling of photoionization is very important for predicting streamer behavior and plasma evolution. The photoionization process in air (N 2 – O 2 mixture) is often described by the Zheleznyak model (1982). This model is usually solved through Helmholtz-type equations that approximate the Zheleznyak photoionization model (Zheleznyak et al. 1982) as Partial Differential Equations (PDEs). Conventional numerical methods, such as the Finite Difference Method (FDM) or Finite Volume Method (FVM), are widely used to solve these equations. Although they are prevalent, the computational cost of these methods due to their need for matrix operations and iterative solver is demanding. To address this challenge, this work develops a spectral solver based on the Fast Fourier Transform (FFT) combined with Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST) to calculate the photoionization rate efficiently in an axisymmetric cylindrical domain. This method naturally satisfies the boundary conditions used in the model and converts the PDE into algebraic ones in spectral space. Thus, avoids the need for iterative matrix solvers. When compared with FDM results, it is demonstrated that the new solver not only maintains accuracy, but also reduces the computational cost, showing a performance increase of approximately 100 compared to FDM over a wide range of problem sizes. The method is parallelized using Message Passing Interface (MPI) and has been integrated into a fluid plasma model for streamer simulation. Here, this FFT-based approach provides a fast and reliable alternative for calculating photoionization in fluid models, helping large-scale plasma simulations run faster and efficiently, and allows higher-resolution simulation without extra computational cost.
Axisymmetric system↗
Imposing equilibrium on experimental 3-D stress fields using Hodge decomposition and FFT-based optimization
Here, we present a methodology to impose micromechanical constraints, i.e. stress equilibrium at grain and sub-grain scale, to an arbitrary (non-equilibrated) voxelized stress field obtained, for example, by means of synchrotron X-ray diffraction techniques. The method consists in finding the equilibrated stress field closest (in L 2 -norm sense) to the measured non-equilibrated stress field, via the solution of an optimization problem. The extraction of the divergence-free (equilibrated) part of a general (non-equilibrated) field is performed using the Hodge decomposition of a symmetric matrix field, which is the generalization of the Helmholtz decomposition of a vector field into the sum of an irrotational field and a solenoidal field. The combination of: a) the Euler–Lagrange equations that solve the optimization problem, and b) the Hodge decomposition, gives a differential expression that contains the bi-harmonic operator and two times the curl operator acting on the experimental stress field. These high-order derivatives can be efficiently performed in Fourier space. The method is applied to filter the non-equilibrated parts of a synthetic piecewise constant stress fields with a known ground truth, and stress fields in Gum Metal, a beta-Ti-based alloy measured in-situ using Diffraction Contrast Tomography (DCT). In both cases, the largest corrections were obtained near grain boundaries.
36 MATERIALS SCIENCE↗
Microstructural Simulation of Ion-Irradiated Natural Rocks and Minerals
This report details the contribution of Oak Ridge National Laboratory to the Nuclear Energy University Program (NEUP) project, "Rapid Characterization of Concrete Mineralogy using Multi-Scale Tools" led by the University of Illinois at Urbana-Champaign. To support the subsequent license renewal (SLR) of the US nuclear power plants (NPPs) fleet, the performance of concrete-forming aggregates against neutron irradiation needs to be assessed. Ion irradiation is proposed as a rapid, cost-effective surrogate method to full neutron irradiation testing. To complement the characterization of ion-irradiated rock specimens, ORNL has run finite-elements and fast-Fourier transform (FFT-based) simulations using the codes MARS and Microstructure-Oriented Scientific Analysis of Irradiated Concrete (MOSAIC). The main conclusion of this analysis is that the apparent post–ion-irradiation step-height underestimates the accumulated free radiation-induced volumetric expansion (RIVE) in the ion-implanted depth by about 15% at full amorphization and about 25% toward the beginning of the ion irradiation experiment. The discrepancy is explained by the fact that the step height is proportional to the sum of the RIVE (i.e., one third of the RIVE) and the irradiation-assisted plastic strains in the vertical direction (lower than two thirds of the RIVE). Because of the large lateral compressive stresses caused by the restraining effect of the unirradiated substrate, the stress field in the mineral grains and at the grain boundary (GB) in the ion-implanted layer is much different than the comparable stress field occurring during neutron irradiation. Hence, the mismatch RIVE causing cracks in the rock-forming minerals leads to different cracking patterns. Ion irradiation continues to be used as a rapid technique to assess the RIVE potential of rocks.
11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
Fast and Scalable FFT-Based GPU-Accelerated Algorithms for Block-Triangular Toeplitz Matrices with Application to Linear Inverse Problems Governed by Autonomous Dynamical Systems
In this work, we present an efficient and scalable algorithm for performing matrix-vector multiplications (matvecs) for block Toeplitz matrices. Such matrices, which are shift-invariant with respect to their blocks, arise in the context of solving inverse problems governed by autonomous systems, and time-invariant systems in particular. In this article, we consider inverse problems that infer unknown parameters from observational data of a linear time-invariant dynamical system given in the form of partial differential equations (PDEs). Matrix-free Newton-conjugate-gradient methods are often the gold standard for solving these inverse problems, but they require numerous actions of the Hessian on a vector. Matrix-free adjoint-based Hessian matvecs require solution of a pair of linearized forward/adjoint PDE solves per Hessian action, which may be prohibitive for large-scale inverse problems. Time invariance of the forward PDE problem leads to a block Toeplitz structure of the discretized parameter-to-observable (p2o) map defining the mapping from inputs (parameters) to outputs (observables) of the PDEs. This block Toeplitz structure enables us to exploit two key properties: (1) compact storage of the p2o map and its adjoint, and (2) efficient fast Fourier transform–based Hessian matvecs. The proposed algorithm is mapped onto large multi-GPU clusters and achieves more than 80% of peak bandwidth on NVIDIA A100 GPUs. Excellent weak scaling is shown for up to 48 A100 GPUs. For the targeted problems, the implementation executes Hessian matvecs within fractions of a second, which is orders of magnitude faster than can be achieved by conventional matrix-free Hessian matvecs via forward/adjoint PDE solves.
97 MATHEMATICS AND COMPUTING↗
A FFT-based mesoscale continuum dislocation mechanics with defect energy: Applications to composites and polycrystals
A crystal plasticity elastoviscoplastic FFT (fast Fourier transform) formulation with a mesoscale continuum field dislocation mechanics model is presented, which incorporates a defect energy density that depends on GND densities and an associated material length scale. This allows to thermodynamically derive internal length scale dependent intra-crystalline backstress and Peach–Koehler force acting on GND densities. The model considers GND density evolution through a filtered numerical spectral approach, which is coupled with stress equilibrium through the elastoviscoplastic FFT algorithm. The discrete Fourier transform (DFT) method together with finite difference (FD) schemes is applied to solve both the backstress tensor and the Fourier–Green operator. Numerical results are first reported for two-phase laminate composites with plastic single crystal channels and elastic precipitates for shear loadings. Channel size effects are simulated and analyzed on the overall and local hardening behaviors during monotonous loadings. In addition, the evolutions of GND densities and the role of their associated backstress on size effects are examined during reversible shear loading. In a second part, the role of the defect energy internal length scale on polycrystal’s hardening during tension–compression is discussed. The results are compared to those obtained using FFT-based continuum field dislocation mechanics without defect energy.
36 MATERIALS SCIENCE↗