We present the three-dimensional version of the Elliptical Parcel-In-Cell (EPIC) method for the simulation of fluid flows and analogous continuum systems. The method represents a flow using a space-filling set of ellipsoidal parcels, which move, rotate and deform in the flow field. Additionally, parcels may carry any number of attributes, such as vorticity, density, temperature, etc, which generally evolve in time on the moving parcels. An underlying grid is used for efficiency in computing the velocity field from the interpolated vorticity field, and in obtaining parcel attribute tendencies. Mixing is enabled by permitting parcels to split when excessively deformed, and by merging very small parcels with the nearest other parcel. Several tests are provided which illustrate the behaviour of the method and demonstrate its effectiveness in modelling complex, buoyancy-driven turbulent fluid flows. The results are compared with a large eddy simulation (LES) and a direct numerical simulation (DNS) model.

Geometric conduction blocks stop cardiac electric propagation due to the shape or conductivity properties of the domain. The blocks are considered to cause many abnormal cardiac electric propagations, leading to cardiac electrophysiological pathologies, such as cardiac fibrillation and arrhythmia. Locating such multidimensional conduction blocks is challenging, particularly in a complex domain with a complex shape and strong anisotropy, such as the heart. To address this problem, we propose a novel mathematical model of the geometric conduction block using the relative acceleration adopted from space-time physics. An efficient numerical scheme for the mathematical model is also proposed to predict the unidirectional conduction block effectively, even in a complex domain. The relative acceleration in the cardiac electric propagation corresponds to the sink-source relationship between the excited (after repolarization) and excitable (before depolarization) cardiac cells, representing the geometric growth rate of the volume of metric balls. The trajectory is constructed from the wavefront of diffusion-reaction equations by aligning orthonormal basis vectors along the gradient of the action potential. Relative acceleration is computed along the propagational direction from the connection 1-form of the basis vectors. The proposed mathematical model and numerical scheme are applied to demonstrate geometric conduction blocks in two-dimensional (2D) simple curved domains with strong anisotropy.

Smoothing is a specialized form of Bayesian inference for state-space models that characterizes the posterior distribution of a collection of states given an associated sequence of observations. Ramgraber et al. [38] proposes a general framework for transport-based ensemble smoothing, which includes linear Kalman-type smoothers as special cases. Here, we build on this foundation to realize and demonstrate nonlinear backward ensemble transport smoothers. We discuss parameterization and regularization of the associated transport maps, and then examine the performance of these smoothers for nonlinear and chaotic dynamical systems that exhibit non-Gaussian behavior. In these settings, our nonlinear transport smoothers yield lower estimation error than conventional linear smoothers and state-of-the-art iterative ensemble Kalman smoothers, for comparable numbers of model evaluations.

Smoothers are algorithms for Bayesian time series re-analysis. Most operational smoothers rely either on affine Kalman-type transformations or on sequential importance sampling. These strategies occupy opposite ends of a spectrum that trades computational efficiency and scalability for statistical generality and consistency: non-Gaussianity renders affine Kalman updates inconsistent with the true Bayesian solution, while the ensemble size required for successful importance sampling can be prohibitive. This paper revisits the smoothing problem from the perspective of measure transport, which offers the prospect of consistent prior-to-posterior transformations for Bayesian inference. We leverage this capacity by proposing a general ensemble framework for transport-based smoothing. Within this framework, we derive a comprehensive set of smoothing recursions based on nonlinear transport maps and detail how they exploit the structure of state-space models in fully non-Gaussian settings. We also describe how many standard Kalman-type smoothing algorithms emerge as special cases of our framework. A companion paper [35] explores the implementation of nonlinear ensemble transport smoothers in greater depth.

We present an extension to the well-known peel off optimization for Monte Carlo radiative transfer simulations. The classical method is only applicable when the distance between the detector and the peel off event is much bigger than the size of the detector. We use two alternatives to the classical method and calculate the peel off intensity via a subdivision and an integration method. We compare their performance for a realistic scenario and derive guidelines for a general treatment. This allows for precise peel off calculations at any distance to the detector surface.

A fast and reliable geometry optimization algorithm is presented that optimizes atomic positions and lattice vectors simultaneously. Using a series of benchmarks, it is shown that the method presented in this paper outperforms in most cases the standard optimization methods implemented in popular codes such as Quantum ESPRESSO and VASP. To motivate the variable cell shape optimization method presented in here, the eigenvalues of the lattice Hessian matrix are investigated thoroughly. It is shown that they change depending on the shape of the cell and the number of particles inside the cell. For certain cell shapes the resulting condition number of the lattice matrix can grow quadratically with respect to the number of particles. By a coordinate transformation, which can be applied to all variable cell shape optimization methods, the undesirable conditioning of the lattice Hessian matrix is eliminated.

We present an algorithm for compressing the radiosity view factor model commonly used in radiation heat transfer and computer graphics. We use a format inspired by the hierarchical off-diagonal low rank format, where elements are recursively partitioned using a quadtree or octree and blocks are compressed using a sparse singular value decomposition—the hierarchical matrix is assembled using dynamic programming. The motivating application is time-dependent thermal modeling on vast planetary surfaces, with a focus on permanently shadowed craters which receive energy through indirect irradiance. In this setting, shape models are comprised of a large number of triangular facets which conform to a rough surface. At each time step, a quadratic number of triangle-to-triangle scattered fluxes must be summed; that is, as the sun moves through the sky, we must solve the same view factor system of equations for a potentially unlimited number of time-varying righthand sides. We first conduct numerical experiments with a synthetic spherical cap-shaped crater, where the equilibrium temperature is analytically available. We also test our implementation with triangle meshes of planetary surfaces derived from digital elevation models recovered by orbiting spacecraft. Our results indicate that the compressed view factor matrix can be assembled in quadratic time, which is comparable to the time it takes to assemble the full view matrix itself. Memory requirements during assembly are reduced by a large factor. Finally, for a range of compression tolerances, the size of the compressed view factor matrix and the speed of the resulting matrix vector product both scale linearly (as opposed to quadratically for the full matrix), resulting in orders of magnitude savings in processing time and memory space.

Herein we describe a new approach to modelling inviscid two-dimensional stratified flows in a general domain. The approach makes use of a conformal map of the domain to a rectangle. In this transformed domain, the equations of motion are largely unaltered, and in particular Laplace's equation remains unchanged. This enables one to construct exact solutions to Laplace's equation and thereby enforce all boundary conditions.An example is provided for two-dimensional flow under the Boussinesq approximation, though the approach is much more general (albeit restricted to two-dimensions). This example is motivated by flow under a weir in a tidal estuary. Here, we discuss how to use a dynamically-evolving conformal map to model changes in the water height on either side of the weir, though the example presented keeps these heights fixed due to limitations in the computational speed to generate the conformal map.The numerical approach makes use of contour advection, wherein material buoyancy contours are advected conservatively by the local fluid velocity, while a dual contour-grid representation is used for the vorticity in order to account for vorticity generation from horizontal buoyancy gradients. This generation is accurately estimated by using the buoyancy contours directly, rather than a gridded version of the buoyancy field. The result is a highly-accurate, efficient numerical method with extremely low levels of numerical damping.

We present a multi-physics model for the approximation of the coupled system formed by the heat equation and the Navier-Stokes equations with solidification and free surfaces. The computational domain is the union of two overlapping regions: a larger domain to account for thermal effects, and a smaller region to account for the fluid flow. Temperature-dependent surface effects are accounted for via surface tension and Marangoni forces. The volume-of-fluid approach is used to track the free surfaces between the metal (liquid or solidified) and the ambient air. The numerical method incorporates all the physical phenomena within an operator splitting strategy. The discretization relies on a two-grid approach that uses an unstructured finite element mesh for diffusion phenomena and a structured Cartesian grid for advection phenomena. The model is validated through numerical experiments, the main application being laser melting and polishing.

We propose a physics-driven stretching function for direct numerical simulation (DNS) of compressible turbulent wall-bounded flows, which blends uniform near-wall spacing with uniform resolution in terms of semi-local Kolmogorov units in the outer wall layer. Given target Mach number, Reynolds number and wall temperature, our procedure yields a well-defined prescription for the number of grid points and their distribution which guarantee at the same time numerical accuracy and judicious exploitation of computational resources. DNS of high-speed turbulent boundary layers are used to evaluate the quality of the proposed stretching function, which show that one can achieve identical results as with general-purpose stretching functions, however with substantially higher efficiency. A Python script is provided to facilitate implementation of the proposed grid stretching.