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Below its Jahn--Teller transition temperature, $T_\mathrm{JT}$, NaNiO$_2$ has a monoclinic layered structure consisting of alternating layers of edge-sharing NaO$_6$ and Jahn-Teller-distorted NiO$_6$ octahedra. Above $T_\mathrm{JT}$ where NaNiO$_2$ is rhombohedral, diffraction measurements show the absence of a cooperative Jahn-Teller distortion, accompanied by an increase in the unit cell volume. Using neutron total scattering, solid-state Nuclear Magnetic Resonance (NMR), and extended X-ray absorption fine structure (EXAFS) experiments as local probes of the structure we find direct evidence for a displacive, as opposed to order-disorder Jahn-Teller transition at $T_\mathrm{JT}$. This is supported by \textit{ab initio} molecular dynamics (AIMD) simulations. To our knowledge this study is the first to show a displacive Jahn-Teller transition in any material using direct observations with local probe techniques., Comment: Under review. For SI, email the authors

Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a background grid. Interpolation-based immersed boundary methods augment existing finite element software to non-invasively implement immersed boundary capabilities through extraction. Extraction interpolates the background basis as a linear combination of Lagrange polynomials defined on a foreground mesh, creating an interpolated basis that can be easily integrated by existing methods. This work extends the interpolation-based immersed boundary method to multi-material and multi-physics problems. Beginning from level-set descriptions of domain geometries, Heaviside enrichment is implemented to accommodate discontinuities in state variable fields across material interfaces. Adaptive refinement with truncated hierarchical B-splines is used to both improve interface geometry representations and resolve large solution gradients near interfaces. Multi-physics problems typically involve coupled fields where each field has unique discretization requirements. This work presents a novel discretization method for coupled problems through the application of extraction, using a single foreground mesh for all fields. Numerical examples illustrate optimal convergence rates for this method in both 2D and 3D, for heat conduction, linear elasticity, and a coupled thermo-mechanical problem. The utility of this method is demonstrated through image-based analysis of a composite sample, where in addition to circumventing typical meshing difficulties, this method reduces the required degrees of freedom compared to classical boundary-fitted finite element methods.

We introduce stabilized spline collocation schemes for the numerical solution of nonlinear, hyperbolic conservation laws. A nonlinear, residual-based viscosity stabilization is combined with a projection stabilization-inspired linear operator to stabilize the scheme in the presence of shocks and prevent the propagation of spurious, small-scale oscillations. Due to the nature of collocation schemes, these methods possess the possibility for greatly reduced computational cost of high-order discretizations. Numerical results for the linear advection, Burgers, Buckley-Leverett, and Euler equations show that the scheme is robust in the presence of shocks while maintaining high-order accuracy on smooth problems.

Recent years have seen many successful applications of machine learning (ML) to facilitate fluid dynamic computations. As simulations grow, generating new training datasets for traditional offline learning creates I/O and storage bottlenecks. Additionally, performing inference at runtime requires non-trivial coupling of ML framework libraries with simulation codes. This work offers a solution to both limitations by simplifying this coupling and enabling in situ training and inference workflows on heterogeneous clusters. Leveraging SmartSim, the presented framework deploys a database to store data and ML models in memory, thus circumventing the file system. On the Polaris supercomputer, we demonstrate perfect scaling efficiency to the full machine size of the data transfer and inference costs thanks to a novel co-located deployment of the database. Moreover, we train an autoencoder in situ from a turbulent flow simulation, showing that the framework overhead is negligible relative to a solver time step and training epoch.

Direct numerical simulation (DNS) of a turbulent boundary layer over the Gaussian (Boeing) bump is performed. This boundary layer exhibits a series of adverse and favorable pressure gradients and convex and concave curvature effects before separating. These effects on turbulent boundary layers are characterized and compared to a lower Reynolds number flow over the same geometry. The momentum budgets are analyzed in the streamline-aligned coordinate system upstream of the separation region. These momentum budgets allow the simplification of equations to facilitate an integral analysis. Integral analysis-based scalings for Reynolds stresses in the inner and outer regions of the boundary layer are also formulated. These proposed scalings exhibit a better collapse of Reynolds stress profiles compared to friction velocity scaling and Zagarola-Smits scaling in the strong favorable pressure gradient region and in the mild adverse pressure region that precedes it in this flow.

In this work we adapt classical residual-based stabilization techniques to the spline collocation setting. Inspired by the Streamline-Upwind-Petrov-Galerkin and Pressure-Stabilizing-Petrov-Galerkin methods, our stabilized collocation schemes address spurious oscillations that can arise from advection and pressure instabilities. Numerical examples for the advection-diffusion equation, Stokes equations, and incompressible Navier-Stokes equations show the effectiveness of the proposed stabilized schemes while maintaining the high-order convergence rates and accuracy of standard isogeometric collocation on smooth problems.

We develop two isogeometric divergence-conforming collocation schemes for incompressible flow. The first is based on the standard, velocity-pressure formulation of the Navier-Stokes equations, while the second is based on the rotational form and includes the vorticity as an unknown in addition to the velocity and pressure. We describe the process of discretizing each unknown using B-splines that conform to a discrete de Rham complex and collocating each governing equation at the Greville abcissae corresponding to each discrete space. Results on complex domains are obtained by mapping the equations back to a parametric domain using structure-preserving transformations. Numerical results show the promise of the method, including accelerated convergence rates of the three field, vorticity-velocity-pressure scheme when compared to the two field, velocity-pressure scheme.

We present a new approach for constructing data-driven subgrid stress models for large eddy simulation of turbulent flows using anisotropic grids. The key to our approach is a Galilean, rotationally, reflectionally and unit invariant model form that also embeds filter anisotropy in such a way that an important subgrid stress identity is satisfied. We use this model form to train a data-driven subgrid stress model using only a small amount of anisotropically filtered DNS data and a simple and inexpensive neural network architecture. A priori and a posteriori tests indicate that the trained data-driven model generalizes well to filter anisotropy ratios, Reynolds numbers and flow physics outside the training dataset.

This paper presents an immersed, isogeometric finite element framework to predict the response of multi-material, multi-physics problems with complex geometries using locally refined discretizations. To circumvent the need to generate conformal meshes, this work uses an eXtended Finite Element Method (XFEM) to discretize the governing equations on non-conforming, embedding meshes. A flexible approach to create truncated hierarchical B-splines discretizations is presented. This approach enables the refinement of each state variable field individually to meet field-specific accuracy requirements. To obtain an immersed geometry representation that is consistent across all hierarchically refined B-spline discretizations, the geometry is immersed into a single mesh, the XFEM background mesh, which is constructed from the union of all hierarchical B-spline meshes. An extraction operator is introduced to represent the truncated hierarchical B-spline bases in terms of Lagrange shape functions on the XFEM background mesh without loss of accuracy. The truncated hierarchical B-spline bases are enriched using a generalized Heaviside enrichment strategy to accommodate small geometric features and multi-material problems. The governing equations are augmented by a formulation of the face-oriented ghost stabilization enhanced for locally refined B-spline bases. We present examples for two- and three-dimensional linear elastic and thermo-elastic problems. The numerical results validate the accuracy of our framework. The results also demonstrate the applicability of the proposed framework to large, geometrically complex problems.

We introduce a new paradigm for immersed finite element and isogeometric methods based on interpolating function spaces from an unfitted background mesh into Lagrange finite element spaces defined on a foreground mesh that captures the domain geometry but is otherwise subject to minimal constraints on element quality or connectivity. This is a generalization of the concept of Lagrange extraction from the isogeometric analysis literature and also related to certain variants of the finite cell and material point methods. Crucially, the interpolation may be approximate without sacrificing high-order convergence rates, which distinguishes the present method from existing finite cell, CutFEM, and immersogeometric approaches. The interpolation paradigm also permits non-invasive reuse of existing finite element software for immersed analysis. We analyze the properties of the interpolation-based immersed paradigm for a model problem and implement it on top of the open-source FEniCS finite element software, to apply it to a variety of problems in fluid, solid, and structural mechanics where we demonstrate high-order accuracy and applicability to practical geometries like trimmed spline patches.

Nitsche's method is a well-established approach for weak enforcement of boundary conditions for partial differential equations (PDEs). It has many desirable properties, including the preservation of variational consistency and the fact that it yields symmetric, positive-definite discrete linear systems that are not overly ill-conditioned. In recent years, the method has gained in popularity in a number of areas, including isogeometric analysis, immersed methods, and contact mechanics. However, arriving at a formulation based on Nitsche's method can be a mathematically arduous process, especially for high-order PDEs. Fortunately, the derivation is conceptually straightforward in the context of variational problems. To facilitate the process, we devised an abstract framework for constructing Nitsche's method for these types of problems in [J. Benzaken, J. A. Evans, S. McCormick, and R. Tamstorf, Nitsche's method for linear Kirchhoff-Love shells: Formulation, error analysis, and verification, Comput. Methods Appl. Mech. Eng., 374 (2021), p. 113544]. The goal of this paper is to elucidate the process through a sequence of didactic examples. First, we show the derivation of Nitsche's method for Poisson's equation to gain an intuition for the various steps. Next, we present the abstract framework and then revisit the derivation for Poisson's equation to use the framework and add mathematical rigor. In the process, we extend our derivation to cover the vector-valued setting. Armed with a basic recipe, we then show how to handle a higher-order problem by considering the vector-valued biharmonic equation and the linearized Kirchhoff-Love plate. In the end, the hope is that the reader will be able to apply Nitsche's method to any problem that arises from variational principles., Comment: 43 pages, 2 figures

We show that the second-order accurate generalized-$\alpha$ method on a uniform temporal mesh may be viewed as an implicit midpoint method on a shifted temporal mesh. With this insight, we demonstrate generalized-$\alpha$ time integration of a finite element spatial discretization of a conservation law system results in a fully-discrete method admitting discrete balance laws when (i) the time integration is second-order accurate, (ii) a uniform temporal mesh is employed, (iii) the spatial discretization is conservative, and (iv) conservation variables are discretized.

We consider a stabilization method for divergence-conforming B-spline discretizations of the incompressible Navier--Stokes problem wherein jumps in high-order normal derivatives of the velocity field are penalized across interior mesh facets. We prove that this method is pressure robust, consistent, and energy stable, and we show how to select the stabilization parameter appearing in the method so that excessive numerical dissipation is avoided in both the cross-wind direction and in the diffusion-dominated regime. We examine the efficacy of the method using a suite of numerical experiments, and we find the method yields optimal $\textbf{L}^2$ and $\textbf{H}^1$ convergence rates for the velocity field, eliminates spurious small-scale structures that pollute Galerkin approximations, and is effective as an Implicit Large Eddy Simulation (ILES) methodology.

Structural subgrid stress models for large eddy simulation often allow for backscatter of energy from unresolved to resolved turbulent scales, but excessive model backscatter can eventually result in numerical instability. A commonly employed strategy to overcome this issue is to set predicted subgrid stresses to zero in regions of model backscatter. This clipping procedure improves the stability of structural models, however, at the cost of reduced correlation between the predicted subgrid stresses and the exact subgrid stresses. In this article, we propose an alternative strategy that removes model backscatter from model predictions through the solution of a constrained minimization problem. This procedure, which we refer to as optimal clipping, results in a parameter-free mixed model, and it yields predicted subgrid stresses in higher correlation with the exact subgrid stresses as compared with those attained with the traditional clipping procedure. We perform a series of a priori and a posteriori tests to investigate the impact of applying the traditional and optimal clipping procedures to Clark's gradient subgrid stress model, and we observe that optimal clipping leads to a significant improvement in model predictions as compared to the traditional clipping procedure.

We introduce a new hybridized discontinuous Galerkin method for the incompressible magnetohydrodynamics equations. If particular velocity, pressure, magnetic field, and magnetic pressure spaces are employed for both element and trace solution fields, we arrive at an energy stable method which returns pointwise divergence-free velocity fields and magnetic fields and properly balances linear momentum. We discretize in time using a second-order-in-time generalized-$\alpha$ method, and we present a block iterative method for solving the resulting nonlinear system of equations at each time step. We numerically examine the effectiveness of our method using a manufactured solution and observe our method yields optimal convergence rates in the $L_2$ norm for the velocity field, pressure field, magnetic field, and magnetic pressure field. We further find our method is pressure robust. We then apply our method to a selection of benchmark problems and numerically confirm our method is energy stable.

A recent paper [J. A. Evans, D. Kamensky, Y. Bazilevs, "Variational multiscale modeling with discretely divergence-free subscales", Computers & Mathematics with Applications, 80 (2020) 2517-2537] introduced a novel stabilized finite element method for the incompressible Navier-Stokes equations, which combined residual-based stabilization of advection, energetic stability, and satisfaction of a discrete incompressibility condition. However, the convergence analysis and numerical tests of the cited work were subject to the restrictive assumption of a divergence-conforming choice of velocity and pressure spaces, where the pressure space must contain the divergence of every velocity function. The present work extends the convergence analysis to arbitrary inf-sup-stable velocity-pressure pairs (while maintaining robustness in the advection-dominated regime) and demonstrates the convergence of the method numerically, using both the traditional and isogeometric Taylor-Hood elements.