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DOE OSTI2021

Fourier Neural Networks as Function Approximators and Differential Equation Solvers

We present a Fourier neural network (FNN) that can be mapped directly to the Fourier decomposition. The choice of activation and loss function yields results that replicate a Fourier series expansion closely while preserving a straightforward architecture with a single hidden layer. The simplicity of this network architecture facilitates the integration with any other higher-complexity networks, at a data pre- or postprocessing stage. We validate this FNN on naturally periodic smooth functions and on piecewise continuous periodic functions. We showcase the use of this FNN for modeling or solving partial differential equations with periodic boundary conditions. The main advantages of the current approach are the validity of the solution outside the training region, interpretability of the trained model, and simplicity of use.

Fourier decomposition↗

DOE OSTI2021

Finite-size correction for slab supercell calculations of materials with spontaneous polarization

Abstract The repeated slab approach has become a de facto standard to accurately describe surface properties of materials by density functional theory calculations with periodic boundary conditions. For materials exhibiting spontaneous polarization, we show that the conventional scheme of passivation with pseudo hydrogen is unable to realize a charge-neutral surface. The presence of a net surface charge induces via Gauss’s law a macroscopic electric field through the slab and results in poor size convergence with respect to the thickness of the slab. We propose a modified passivation method that accounts for the effect of spontaneous polarization, describes the correct bulk limits and boosts convergence with respect to slab thickness. The robustness, reliability, and superior convergence of energetics and electronic structure achieved by the proposed method are demonstrated using the example of polar ZnO surfaces.

Chemistry↗

DOE OSTI2021

Higher-order particle representation for particle-in-cell simulations

In this paper we present an alternative approach to the representation of simulation particles for unstructured electrostatic and electromagnetic PIC simulations. In our modified PIC algorithm we represent particles as having a smooth shape function limited by some specified finite radius, r 0 . A unique feature of our approach is the representation of this shape by surrounding simulation particles with a set of virtual particles with delta shape, with fixed offsets and weights derived from Gaussian quadrature rules and the value of r 0 . As the virtual particles are purely computational, they provide the additional benefit of increasing the arithmetic intensity of traditionally memory bound particle kernels. The modified algorithm is implemented within Sandia National Laboratories' unstructured EMPIRE-PIC code, for electrostatic and electromagnetic simulations, using periodic boundary conditions. We show results for a representative set of benchmark problems, including electron orbit, a transverse electromagnetic wave propagating through a plasma, numerical heating, and a plasma slab expansion. In this work, good error reduction across all of the chosen problems is achieved as the particles are made progressively smoother, with the optimal particle radius appearing to be problem-dependent.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1993

On the spline-based wavelet differentiation matrix

The differentiation matrix for a spline-based wavelet basis is constructed. Given an n-th order spline basis it is proved that the differentiation matrix is accurate of order 2n + 2 when periodic boundary conditions are assumed. This high accuracy, or superconvergence, is lost when the boundary conditions are no longer periodic. Furthermore, it is shown that spline-based bases generate a class of compact finite difference schemes.

Jameson, Leland↗

DOE OSTI2021

Verification of Boundary Conditions in MCNP

To verify in MCNP that arrays formed with reflective and periodic boundary conditions produce a neutron multiplication factor (keff) that is equal to or higher than an array of the same size with no boundary conditions

36 MATERIALS SCIENCE↗

DOE OSTI2023

A methodology to generate crystal-based molecular structures for atomistic simulations

Abstract We propose a systematic method to construct crystal-based molecular structures often needed as input for computational chemistry studies. These structures include crystal ‘slabs’ with periodic boundary conditions (PBCs) and non-periodic solids such as Wulff structures. We also introduce a method to build crystal slabs with orthogonal PBC vectors. These methods are integrated into our code, Los Alamos Crystal Cut ( LCC ), which is open source and thus fully available to the community. Examples showing the use of these methods are given throughout the manuscript.

42 ENGINEERING↗

DOE OSTI2024

An unstructured body-of-revolution electromagnetic particle-in-cell algorithm with radial perfectly matched layers and dual polarizations

A novel electromagnetic particle-in-cell algorithm has been developed for fully kinetic plasma simulations on unstructured (irregular) meshes in complex body-of-revolution geometries. The algorithm, implemented in the BORPIC++ code, utilizes a set of field scalings and a coordinate mapping, reducing the Maxwell field problem in a cylindrical system to a Cartesian finite element Maxwell solver in the meridian plane. The latter obviates the cylindrical coordinate singularity in the symmetry axis. The choice of an unstructured finite element discretization enhances the geometrical flexibility of the BORPIC++ solver compared to the more traditional finite difference solvers. Symmetries in Maxwell’s equations are explored to decompose the problem into two dual polarization states with isomorphic representations that enable code reuse. The particle-in-cell scatter and gather steps preserve charge conservation at the discrete level. Our previous algorithm (BORPIC+) discretized the E and B field components of TE Φ and TM Φ polarizations on the finite element (primal) mesh. Here, we employ a new field-update scheme. Using the same finite element (primal) mesh, this scheme advances two sets of field components independently: (1) E and B of TE Φ polarized fields, (E z , E ρ , B Φ ) and (2) D and H of TM Φ polarized fields, (D Φ , H z , H ρ ). Since these field updates are not explicitly coupled, the new field solver obviates the coordinate singularity, which otherwise arises at the cylindrical symmetric axis, ρ = 0 when defining the discrete Hodge matrices (generalized finite element mass matrices). Here, a cylindrical perfectly matched layer is implemented as a boundary condition in the radial direction to simulate open space problems, with periodic boundary conditions in the axial direction. We investigate effects of charged particles moving next to the cylindrical perfectly matched layer. We model azimuthal currents arising from rotational motion of charged rings, which produce TMΦ polarized fields. Several numerical examples are provided to illustrate the first application of the algorithm.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

Coordinate transformation methodology for simulating quasistatic elastoplastic solids

Molecular dynamics simulations frequently employ periodic boundary conditions where the positions of the periodic images are manipulated in order to apply deformation to the material sample. For example, Lees-Edwards conditions use moving periodic images to apply simple shear. Here, we examine the problem of precisely comparing this type of simulation to continuum solid mechanics. We employ a hypo-elastoplastic mechanical model, and develop a projection method to enforce quasi-static equilibrium. We introduce a simulation framework that uses a fixed Cartesian computational grid on a reference domain, and imposes deformation via a time-dependent coordinate transformation to the physical domain. As a test case for our method, we consider the evolution of shear bands in a bulk metallic glass using the shear transformation zone theory of amorphous plasticity. We examine the growth of shear bands in simple shear and pure shear conditions as a function of the initial preparation of the bulk metallic glass.

74 ATOMIC AND MOLECULAR PHYSICS↗

NASA NTRS1996

Periodic Time-Domain Nonlocal Nonreflecting Boundary Conditions for Duct Acoustics

Periodic time-domain boundary conditions are formulated for direct numerical simulation of acoustic waves in ducts without flow. Well-developed frequency-domain boundary conditions are transformed into the time domain. The formulation is presented here in one space dimension and time; however, this formulation has an advantage in that its extension to variable-area, higher dimensional, and acoustically treated ducts is rigorous and straightforward. The boundary condition simulates a nonreflecting wave field in an infinite uniform duct and is implemented by impulse-response operators that are applied at the boundary of the computational domain. These operators are generated by convolution integrals of the corresponding frequency-domain operators. The acoustic solution is obtained by advancing the Euler equations to a periodic state with the MacCormack scheme. The MacCormack scheme utilizes the boundary condition to limit the computational space and preserve the radiation boundary condition. The success of the boundary condition is attributed to the fact that it is nonreflecting to periodic acoustic waves. In addition, transient waves can pass rapidly out of the solution domain. The boundary condition is tested for a pure tone and a multitone source in a linear setting. The effects of various initial conditions are assessed. Computational solutions with the boundary condition are consistent with the known solutions for nonreflecting wave fields in an infinite uniform duct.

Watson, Willie R.↗

DOE OSTI2023

A Formulation of the Many-Body Expansion (MBE) for Periodic Systems: Application to Several Ice Phases

A new formulation of the many-body expansion (MBE) for periodic systems, based on their inherent translational symmetry, is introduced and applied to 7 ice polymorphs. This new formulation is built via a hierarchical procedure connecting gas phase unit cells over finite supercells to infinite solids. For finite supercells, it is demonstrated that this method successfully recovers the energetics while reducing the scaling of the calculation of the many-body terms by a factor of ~N, where N is the system size. Furthermore, the proposed framework delineates a straightforward way to obtain properties of macroscopic systems in the limit of an infinite cell. For periodic systems, the success of this approach is demonstrated by showing that the lattice energies computed (up to the 4-th order in the MBE) reproduce the lattice energies obtained using periodic boundary conditions with an Ewald summation for 7 polymorphs of ice (Ih, II, VIII, IX, XIII, XIV, XV). This development makes it possible to quantify, for the first time, the many-body contributions to the lattice energy of various ice polymorphs. The many-body (three-body and higher) interactions were found to vary significantly among the 7 ice phases, amounting between 7-24% of the total lattice energies. So, this development opens the door for obtaining insights into solid-state properties, while leveraging the computational benefits of the MBE.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS2024

Filament Wound Composite Analysis Using the NASA Multiscale Analysis Tool (NASMAT) and Finite Element Analysis

Fiber reinforced composite materials, owing to their tailorable thermomechanical and functional properties, allow one to produce a structure that is stronger, stiffer, and lighter than its metal counterpart while performing the same function, yielding a more efficient structure. This not only allows for the improvement of current technologies like aircraft structures, but also enables new technologies like gaseous hydrogen storage for mobility applications, which are otherwise impractical when manufactured using traditional metals due to weight and space restrictions or material embrittlement. However, the use of composites imposes greater design and manufacturing challenges on an engineer, since they are heterogenous, having a distinct structure across multiple length scale, behave generally anisotropically at the structural level and require complex manufacturing and processing methods. Capturing this complex behavior requires detailed numerical simulations, including the modeling of microstructural features like undulations, voids, and fiber alignment. In this paper, multiple repeating unit cells (RUCs), representing filament wound composites, are developed (via a script provided in the Appendix) and analyzed. The refinement of these RUCs is varied, and the analyses are performed using both the Abaqus finite element software and the NASA Multiscale Analysis Tool (NASMAT). A study is undertaken to compare the predicted effective elastic properties of the wound RUC to a laminate representation of the wound RUC, which neglects the undulations. Additionally, two different sets of periodic boundary conditions (PBCs) have been examined. One approximates the real boundary conditions using a standard approach and the other represents the PBCs exactly through the use of an offset. Lastly, a comparison of the local elastic stress fields is made among the models and approaches. Since wound structures are often approximated as laminated structures, it is important to understand the degree to which this assumption is valid, namely by first comparing the elastic constants and local elastic fields. This will provide, on the one hand, information concerning the bulk mechanical behavior and, on the other hand, insights concerning local load distributions and likely damage initiation sites.

Composites↗

DOE OSTI2023

DFT-based mesoscopic interactions of N adatoms on Cu(100)

The first-principles density functional theory (DFT) method is used to directly calculate the interactions of N atoms of various configurations on Cu(100) surfaces. This allows us to derive a mesoscopic substratemediated interaction model. We find that, for these Cu(100)-supported two-dimensional N clusters, oscillating substrate-mediated interactions obtained from an elastic eigenvector approach including a mesoscopic interacting range (called the mesoscopic model) have reasonable agreement with the DFT results. The DFT periodic boundary conditions are properly mapped into the mesoscopic model. The model is used to analyze the energies of c(2×2) square islands independent of their sizes. Previous experimental studies have shown that, in the low coverage region, N atoms agglomerate to isolated square islands of about 5 nm × 5 nm in size with a c(2×2) structure. With increasing coverage, those islands form regular patterns but do not touch below about 0.4 monolayers. From previous experiments and calculations, the square island pattern shows significant elastic effects which are now confirmed.

74 ATOMIC AND MOLECULAR PHYSICS↗

NASA NTRS1993

Numerical modeling of the interstellar medium in galactic disks

We have been developing detailed hydrodynamic models of the global interstellar medium in the hope of understanding the mass and volume occupied by various phases, as well as their structure and kinematics. In our model, the gas is modeled by one fluid while representative Pop 1 stars are modeled by a second fluid. The two fluids are coupled in that the gas forms into stars at a rate given by a Schmidt law while stellar mass loss returns matter into the gas phase (on a time scale of 100 Myr). Also, the stars heat the gas through stellar winds and the gas cools through optically thin radiation. The time behavior of these two fluids is studied in two spatial dimensions with the Eulerian finite difference numerical hydrodynamic code Zen. The two spatial dimensions are along the plane of a disk (x, total length of 2 kpc) and perpendicular to the disk (z, total height of +/- 15 kpc) and a galactic gravitational field in the z direction, typical of that at the solar circle, is imposed upon the simulation; self-gravity and rotation are absent. For the boundary conditions, outflow is permitted at the top and bottom of the grid (z = +/- 15 kpc) while periodic boundary conditions are imposed upon left and right sides of the grid. As initial conditions, we assumed a gaseous distribution like that seen for the H1 by earlier researchers, although the results are insensitive to the initial conditions. We have run simulations in which the heating due to stars, parameterized as a stellar wind velocity, a, is varied from low (a = 150 km/s), to intermediate (a = 300 km/s), to high (a = 600 km/s). Since the intermediate case is roughly equivalent to the Galactic energy injection rate from supernovae, this summary will concentrate on results from this simulation.

Rosen, A.↗

NASA NTRS1994

Optimizations on Designing High-Resolution Finite-Difference Schemes

We describe a general optimization procedure for both maximizing the resolution characteristics of existing finite differencing schemes as well as designing finite difference schemes that will meet the error tolerance requirements of numerical solutions. The procedure is based on an optimization process. This is a generalization of the compact scheme introduced by Lele in which the resolution is improved for single, one-dimensional spatial derivative, whereas in the present approach the complete scheme, after spatial and temporal discretizations, is optimized on a range of parameters of the scheme and the governing equations. The approach is to linearize and Fourier analyze the discretized equations to check the resolving power of the scheme for various wave number ranges in the solution and optimize the resolution to satisfy the requirements of the problem. This represents a constrained nonlinear optimization problem which can be solved to obtain the nodal weights of discretization. An objective function is defined in the parametric space of wave numbers, Courant number, Mach number and other quantities of interest. Typical criterion for defining the objective function include the maximization of the resolution of high wave numbers for acoustic and electromagnetic wave propagations and turbulence calculations. The procedure is being tested on off-design conditions of non-uniform mesh, non-periodic boundary conditions, and non-constant wave speeds for scalar and system of equations. This includes the solution of wave equations and Euler equations using a conventional scheme with and without optimization and the design of an optimum scheme for the specified error tolerance.

Liu, Yen↗

NASA NTRSDate not supplied

Process Modeling of Woven Textiles

Woven polymer matrix composites offer numerous advantages over traditional unidirectional laminates. Processing conditions affect residual stress that builds up during manufacturing. This work proposes the analysis of three curing parameters on the local residual stress distribution of a plain weave textile architecture, including chemical shrinkage, the temperature cool down, and the evolution of matrix properties as a function of curing. Virtual curing is applied to the inter-tow matrix material of the plain weave Repeating Unit Cell (RUC) modeled in TexGen. Curing is implemented through user-written subroutines within the commercial software suite Abaqus. Results from process modeling are compared with a linear-elastic thermal cool down in Abaqus. A boundary conditions study is also conducted to compare the residual stress build up predicted by Flat(FBCs) and Periodic Boundary Conditions (PBCs). Preliminary results indicate that a thermal cool down over-estimates the end-of-cure stresses in the matrix. Process modeling is conducted with and without chemical shrinkage, then compared with a thermal cool down. Results show that 11.32% of the final stress state is due to chemical shrinkage. Material property evolution during curing also has a non-negligible effect on the end stress.

woven composites↗

NASA NTRS1974

Partition function for a two dimensional plasma in the random phase approximation

The partition function for a two-dimensional plasma is evaluated within the random phase approximation. The periodic boundary conditions are fully taken into account by including the periodic image interactions. In the guiding-center limit, the negative temperature threshold energy is evaluated, and a value different from previous calculations results. When an identical random phase evaluated, and a value different from previous calculations results. When an identical random phase evaluation is applied to the finite gyroradius plasma, the Salzberg-Prager-May equation of state is recovered.

Seyler, C. E., Jr.↗

NASA NTRS1974

Partition function for a two-dimensional plasma in the random-phase approximation

The partition function for a two-dimensional plasma is evaluated within the random-phase approximation. The periodic boundary conditions are taken fully into account by including the periodic image interactions. In the guiding-center limit, the 'negative temperature' threshold energy is evaluated, and a value different from previous calculations results. When an identical random-phase evaluation is applied to the finite-gyroradius plasma, the Salzberg-Prager-May equation of state is recovered.

Seyler, C. E., Jr.↗

DOE OSTI2021

Verification of Boundary Conditions in MCNP

MCNP (Monte Carlo N-Particle) is code that is used to model the transport of different particles. To verify in MCNP6.1 that arrays formed with reflective and periodic boundary conditions produce a neutron multiplication factor (keff) that is equal to or higher than an array of the same size with no boundary conditions.

36 MATERIALS SCIENCE↗