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Measuring the Burgers vector of dislocations with dark-field X-ray microscopy
The subsurface dynamics of dislocations are essential to many properties of bulk crystalline materials. However, it is challenging to characterize a bulk crystal by conventional transmission electron microscopy (TEM) due to the limited penetration depth of electrons. A novel X-ray imaging technique – dark-field X-ray microscopy (DFXM) – was developed to image hierarchical dislocation structures in bulk crystals. While today's DFXM can effectively map the line structures of dislocations, it is still challenging to quantify the Burgers vectors, the key characterization governing the dislocation behaviors. Here, we extend the 'invisibility criterion' formalism from the TEM theory to the geometrical-optics model of DFXM and demonstrate the consistency between DFXM and dark-field TEM using multi-diffraction-peak imaging for a single edge dislocation. Due to the practical difficulty of multi-peak DFXM experiments, we further study how the Burgers vector effect is encoded for a single-peak DFXM experiment. Using the geometrical-optics DFXM simulation, we explore the asymmetry of rocking tilt scans at different rolling tilts and develop a new method to characterize the Burgers vector. The conclusions of this study advance our understanding of the use of DFXM in characterizing individual dislocations, enabling the connection from bulk DFXM imaging to dislocation mechanics.
Differentiable physics-enabled closure modeling for Burgers’ turbulence
Abstract Data-driven turbulence modeling is experiencing a surge in interest following algorithmic and hardware developments in the data sciences. We discuss an approach using the differentiable physics paradigm that combines known physics with machine learning to develop closure models for Burgers’ turbulence. We consider the one-dimensional Burgers system as a prototypical test problem for modeling the unresolved terms in advection-dominated turbulence problems. We train a series of models that incorporate varying degrees of physical assumptions on an a posteriori loss function to test the efficacy of models across a range of system parameters, including viscosity, time, and grid resolution. We find that constraining models with inductive biases in the form of partial differential equations that contain known physics or existing closure approaches produces highly data-efficient, accurate, and generalizable models, outperforming state-of-the-art baselines. Addition of structure in the form of physics information also brings a level of interpretability to the models, potentially offering a stepping stone to the future of closure modeling.
A scalable exponential-DG approach for nonlinear conservation laws: With application to Burger and Euler equations
In this work, we propose an Exponential DG framework for partial differential equations. We decompose 7 governing equations into linear and nonlinear parts to which we apply the discontinuous Galerkin 8 (DG) spatial discretization. In particular, we construct the linear part using Jacobian that effectively 9 capture stiff characteristics in the system. The former is integrated analytically, whereas the latter 10 is approximated. This approach i) is stable with a large Courant number (Cr > 1); ii) supports 11 high-order solutions both in time and space; iii) is computationally favorable compared to IMEX 12 DG methods with no preconditioner; iv) becomes comparable to explicit RKDG methods on uniform 13 mesh and beneficial on non-uniform grid for Euler equations; v) is scalable in a modern massively 14 parallel computing architecture due to its explicit nature of exponential time integrators and com15 pact communication stencil of DG method. Numerical results demonstrate the performance of our 16 proposed methods through various examples. We also discuss the stability and convergence analysis 17 for our exponential DG scheme in the context of Burgers equation.
Synchronization of dust acoustic waves in a forced Korteweg–de Vries–Burgers model
The synchronization of dust acoustic waves to an external periodic source is studied in the framework of a driven Korteweg–de Vries–Burgers equation that takes into account the appropriate nonlinear and dispersive nature of low-frequency waves in a dusty plasma medium. For a spatiotemporally varying source term, the system is shown to demonstrate harmonic (1:1) and superharmonic (1:2) synchronized states. Here, the existence domains of these states are delineated in the form of Arnold tongue diagrams in the parametric space of the forcing amplitude and forcing frequency and their resemblance to some past experimental results is discussed.
Solution of Burgers' Equation using the Finite Volume Method and Runge-Kutta 4 Time-Stepping under Various Boundary Conditions
Burgers’ equation is a 1D partial differential equation (PDE) developed as a model to understand fluid flow.
A Tensor Network-Based Quantum Algorithm for the Nonlinear 1D Burgers' Equation
In this work, we implement a tensor network-based quantum algorithm to solve unsteady, nonlinear partial differential equations (PDEs). The challenge lies in how to effectively represent, encode, process, and evolve the nonlinear system of PDEs on quantum computers. We will discuss the new techniques using the compressible 1-dimensional (1D) Burgers' equation as an example, because it represents the fundamental nonlinear feature and yet removes certain complexity in physics, allowing us to focus on the design of quantum algorithms. Previous attempts to solve nonlinear PDEs in quantum computation have often involved storing multiple copies of solutions or employing linearizations. Neither is practical due to exponential scaling with evolution time or insufficient solution accuracy. Our framework is based on matrix product states (MPSs) and matrix product operators (MPOs). For example, the velocity field is represented by MPS, whereas the linear and nonlinear spatial differential terms of the velocity field are processed by MPOs. Our primary focus herein is to verify and validate the various tensor network components of the algorithm using solutions obtained by the classical algorithms on high performance computing (HPC) architectures. We use a classical time marching method to demonstrate the functionality of the tensor network operations to model the PDE and their robustness with the time evolution of the system. Our classical simulation results demonstrate the utility of tensor network-based operations in modeling nonlinear PDEs and highlight the necessity as well as potential advantages of using quantum simulations for these techniques.
Sculpting crystals one Burgers vector at a time: Toward colloidal lattice robot swarms
Plastic deformation of crystalline materials with isotropic particle attractions proceeds by the creation and migration of dislocations under the influence of external forces. If dislocations are produced and migrated under the action of local forces, then material shape change can occur without the application of surface forces. We investigate how particles with variable diameters can be embedded in colloidal monolayers to produce dislocations on demand. Additionally, we find in simulation that when embedded clusters of variable diameter particles are taken through multiple cycles of swelling and shrinking, large cumulative plastic slip is produced by the creation and biased motion of dislocation pairs in the solid for embedded clusters of particular geometries. In this way, dislocations emitted by these clusters (biased “dislocation emitters”) can be used to reshape colloidal matter. Our results are also applicable to larger-scale swarms of robotic particles that organize into dense ordered two-dimensional (2D) arrangements. We conclude with a discussion of how dislocations fulfill for colloids the role sought by “metamodules” in lattice robotics research and show how successive applications of shear as a unit operation can produce shape change through slicing and swirling.
Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping Methods
In this work, we propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.
Automated extraction of interfacial dislocations and disconnections from atomistic data
Here we introduce Interfacial Line Defect Analysis (ILDA), a method for identifying and extracting interfacial dislocations and disconnections with little-to-no input from the user on the nature of the interface. By simply providing the orientations and coherency strains for the crystals in a coherent reference state of the interface, ILDA provides exact Burgers vectors. Alternatively, these orientations and strains can be estimated using local atomic deformation gradients, making ILDA fully automated and providing estimated Burgers vectors. ILDA also determines the step height associated with each defect line segment, in case the associated defect is a disconnection. The heart of the method is the identification of atoms residing at co-incidence sites between the two crystals and the construction of a surface mesh connecting these sites that is used to compose Burgers circuits and insert defect line segments. We demonstrate the performance of ILDA in two test cases: a twist grain boundary and a phase boundary.
Weak-form latent space dynamics identification
Recent work in data-driven modeling has demonstrated that a weak formulation of model equations enhances the noise robustness of a wide range of computational methods. In this paper, we demonstrate the power of the weak form to enhance the LaSDI (Latent Space Dynamics Identification) algorithm, a recently developed data-driven reduced order modeling technique. We introduce a weak form-based version WLaSDI (Weak-form Latent Space Dynamics Identification). WLaSDI first compresses data, then projects onto the test functions and learns the local latent space models. Notably, WLaSDI demonstrates significantly enhanced robustness to noise. With WLaSDI, the local latent space is obtained using weak-form equation learning techniques. Compared to the standard sparse identification of nonlinear dynamics (SINDy) used in LaSDI, the variance reduction of the weak form guarantees a robust and precise latent space recovery, hence allowing for a fast, robust, and accurate simulation. We demonstrate the efficacy of WLaSDI vs. LaSDI on several common benchmark examples including viscid and inviscid Burgers', radial advection, and heat conduction. For instance, in the case of 1D inviscid Burgers' simulations with the addition of up to 100% Gaussian white noise, the relative error remains consistently below 6% for WLaSDI, while it can exceed 10,000% for LaSDI. Similarly, for radial advection simulations, the relative errors stay below 15% for WLaSDI, in stark contrast to the potential errors of up to 10,000% with LaSDI. Moreover, speedups of several orders of magnitude can be obtained with WLaSDI. For example applying WLaSDI to 1D Burgers' yields a 140X speedup compared to the corresponding full order model.
A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder
Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, e.g., any advection-dominated flow phenomena such as in traffic flow, atmospheric flows, and air flow over vehicles, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed a fast and accurate physics-informed neural network ROM, namely nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models. The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 1D and 2D Burgers' equations. A speedup of up to 2.6 for 1D Burgers' and a speedup of 11.7 for 2D Burgers' equations are achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique. Lastly, a posteriori error bounds for the NM-ROMs are derived that take account of the hyper-reduced operators.
Dislocation content of grain boundary phase junctions and its relation to grain boundary excess properties
In this work, we analyze the dislocation content of grain boundary (GB) phase junctions, i.e., line defects separating two different GB phases coexisting on the same GB plane. While regular GB disconnections have been characterized for a variety of interfaces, GB phase junctions formed by GBs with different structures and different numbers of excess atoms have not been previously studied. We apply a general Burgers circuit analysis to calculate the Burgers vectors b of junctions in two Σ5 Cu boundaries previously simulated with molecular dynamics. The Burgers vectors of these junctions cannot be described by the displacement shift complete (DSC) lattice alone. We show that, in general, the normal component of b is not equal to the difference in the GB excess volumes but contains another contribution from the numbers of GB atoms per unit area ΔN* required to transform one GB phase into another. In the boundaries studied, the latter component dominates and even changes the sign of b. We derive expressions for the normal and tangential components of b in terms of the DSC lattice vectors and the non-DSC part due to ΔN* and additional GB excess properties, including excess volume and shears. These expressions provide a connection between GB phase transformations driven by the GB free energy difference and the motion of GB junctions under applied normal and shear stresses. The proposed analysis quantifies b and therefore makes it possible to calculate the elastic part of the energy of these defects, evaluate their contribution to the nucleation barrier during GB phase transformations, and treat elastic interactions with other defects.
Weak-Form Latent Space Dynamics Identification
This software showcases the enhanced capabilities of the Latent Space Dynamics Identification (LaSDI) algorithm through the application of the weak form, resulting in WLaSDI. WLaSDI first compresses the data, then projects it onto test functions, and subsequently learns the local latent space models. Notably, WLaSDI demonstrates significantly improved robustness to noise. Using weak-form equation learning techniques, WLaSDI achieves local latent space modeling. Compared to the standard sparse identification of nonlinear dynamics (SINDy) used in LaSDI, the variance reduction of the weak form ensures robust and precise latent space recovery, enabling fast, robust, and accurate simulations. We demonstrate the efficacy of WLaSDI against LaSDI using several common benchmark examples, including viscid and inviscid Burgers', radial advection, and heat conduction. For instance, in 1D inviscid Burgers' simulations with up to 100% Gaussian white noise, WLaSDI maintains relative errors consistently below 6%, whereas LaSDI errors can exceed 10,000%. Similarly, in radial advection simulations, WLaSDI keeps relative errors below 16%, compared to potential errors of up to 10,000% with LaSDI. Additionally, WLaSDI achieves significant speedups, such as a 140X speedup in 1D Burgers' simulations compared to the corresponding full order model.
Nonlinear manifold reduced order model
Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, e.g., any advection-dominated flow phenomena such as in traffic flow, atmospheric flows, and air flow over vehicles, a lowdimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed a fast and accurate physics-informed neural network ROM, namely nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our software takes advantage of the existing numerical methods that are used to solve the corresponding full order models. The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 1D and 2D Burgers' equations. A speedup of up to 2.6 for 1D Burgers' and a speedup of 11.7 for 2D Burgers' equations are achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique.
Nonlinear manifold-based reduced order model
Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, e.g., any advection-dominated flow phenomena such as in traffic flow, atmospheric flows, and air flow over vehicles, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed a fast and accurate physicsinformed neural network ROM, namely nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LSROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models. The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 1D and 2D Burgers’ equations. A speedup of up to 2.6 for 1D Burgers’ and a speedup of 11.7 for 2D Burgers’ equations are achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique. Finally, a posteriori error bounds for the NM-ROMs are derived that take account of the hyper-reduced operators.
Dislocation‐Driven Formation of Oriented Macroperiodic Metastructures of Curved Single Crystal Lattices in Glass
Abstract Single crystals fabricated in glass by localized heating can develop uniquely deformed lattices stabilized by the surrounding amorphous medium. The development of lattice curvature appears to be intrinsic to the crystal growth process in some systems, while the result of the locally changing crystallography in others. In this work, a model laser‐fabricated rotating lattice Sb 2 S 3 crystal grown in stoichiometric glass is used to demonstrate fabrication of novel macroperiodic metastructures that utilize intrinsic lattice curvature superimposed with subtle crystallographic influences. The limited availability of slip systems drives the lattice curvature magnitude to vary with crystal growth direction, maximizing for lattices aligned with the predominant Burgers vector along with corresponding increases in dislocation density. Misaligned lattice orientations form smaller secondary lattice curvatures arising from misaligned Burgers vectors with further elastic contributions. Over extended crystal growth, these secondary components align the lattice to rotate about either the <001> or <010> crystal axes forming repeating metastructures of lattice orientation with periodicity 20–160 microns in length. The mechanistic approach used in this work may be expanded to other systems with known slip systems to better understand and design macroperiodic metastructures.
< c+a > dislocations in {101 - 2} twins in Mg: A kinematic and energetic requirement
Consideration of published TEM and molecular dynamics studies of interactions between matrix dislocations and {101¯2} twin boundaries (TB) in hexagonal close packed metals inspires a hypothesis regarding the formation of dislocations with < c+a > Burgers vectors inside the twins. It was previously reported that when basal < a > dislocations from the matrix are swept by an advancing TB, they are converted into partial dislocations bounding basal I 1 stacking faults (SFs) within the twin. In the present work, we show that this dislocation configuration near TB renders it energetically favorable for a pair of partial dislocations to constrict into a dislocation within the twin. New in situ TEM observations confirm this hypothesis by revealing long < c+a > dislocation dipoles terminated by the TB at one end and by I1 SFs within the twin at the other end. Finally, the resulting near-screw < c+a > dislocation dipoles are observed to be glissile, enabling self-annihilation in some circumstances. Thus, rather than requiring a high energy dislocation pile-up to press two < a > dislocations through a TB into a single < c+a > dislocation, it is demonstrated that < c+a > dislocations naturally form, within {101¯2} twins in Mg, as TB migrates into a matrix, which inevitably contains dislocations with < a > Burgers vectors. Furthermore, these observations help to explain how the twins themselves may subsequently deform and why they may be particularly vulnerable to plastic, shear localization.