Multi-material flows involve boundaries between two non-mixing fluids, seen in scenarios such as underwater detonations, oil-water separation in petroleum extraction, lava-water interactions during volcanic eruptions, and gas-liquid interfaces in bubble column reactors. Accurate algorithms for these dynamic interfaces are crucial in computational fluid dynamics. While various methods exist, moment-of-fluid interface reconstruction stands out because of its efficiency and accuracy. This technique uses non-linear optimization to derive a shape that matches not only the volume, but also the first and even the second and third order moments of the fluid. However, the main limitation of the moment-of-fluid methods is their dependence on, and lack of, a good initial guess, which is critical for their robust convergence. We derive a highly accurate initial guess using the iso-contour of the polynomial that exactly matches the higher-order moments. This allows a vertex-wise optimization and with it the reconstruction of many interface shapes that have not been possible in the past, such as polygons with multiple connected components and holes.

We present our results by using a machine learning (ML) approach for the solution of the Riemann problem for the Euler equations of fluid dynamics. The Riemann problem is an initial-value problem with piecewise-constant initial data and it represents a mathematical model of the shock tube. The solution of the Riemann problem is the building block for many numerical algorithms in computational fluid dynamics, such as finite-volume or discontinuous Galerkin methods. Therefore, a fast and accurate approximation of the solution of the Riemann problem and construction of the associated numerical fluxes is of crucial importance. The exact solution of the shock tube problem is fully described by the intermediate pressure and mathematically reduces to finding a solution of a nonlinear equation. Prior to delving into the complexities of ML for the Riemann problem, we consider a much simpler formulation, yet very informative, problem of learning roots of quadratic equations based on their coefficients. We compare two approaches: (i) Gaussian process (GP) regressions, and (ii) neural network (NN) approximations. Among these approaches, NNs prove to be more robust and efficient, although GP can be appreciably more accurate (about 30% ). We then use our experience with the quadratic equation to apply the GP and NN approaches to learn the exact solution of the Riemann problem from the initial data or coefficients of the gas equation of state (EOS). We compare GP and NN approximations in both regression and classification analysis and discuss the potential benefits and drawbacks of the ML approach.

To accurately model flows with shock waves using staggered-grid Lagrangian hydrodynamics, the artificial viscosity has to be introduced to convert kinetic energy into internal energy, thereby increasing the entropy across shocks. Determining the appropriate strength of the artificial viscosity is an art and strongly depends on the particular problem and experience of the researcher. The objective of this study is to pose the problem of finding the appropriate strength of the artificial viscosity as an optimization problem and solve this problem using machine learning (ML) tools, specifically using surrogate models based on Gaussian Process regression (GPR) and Bayesian analysis. We describe the optimization method and discuss various practical details of its implementation. The shock-containing problems for which we apply this method all have been implemented in the LANL code FLAG (Burton in Connectivity structures and differencing techniques for staggered-grid free-Lagrange hydrodynamics, Tech. Rep. UCRL-JC-110555, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1992, in Consistent finite-volume discretization of hydrodynamic conservation laws for unstructured grids, Tech. Rep. CRL-JC-118788, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1994, Multidimensional discretization of conservation laws for unstructured polyhedral grids, Tech. Rep. UCRL-JC-118306, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1994, in FLAG, a multi-dimensional, multiple mesh, adaptive free-Lagrange, hydrodynamics code. In: NECDC, 1992). First, we apply ML to find optimal values to isolated shock problems of different strengths. Second, we apply ML to optimize the viscosity for a one-dimensional (1D) propagating detonation problem based on Zel’dovich-von Neumann-Doring (ZND) (Fickett and Davis in Detonation: theory and experiment. Dover books on physics. Dover Publications, Mineola, 2000) detonation theory using a reactive burn model. We compare results for default (currently used values in FLAG) and optimized values of the artificial viscosity for these problems demonstrating the potential for significant improvement in the accuracy of computations.

A conservative data transfer (remap) between two meshes is an important step of arbitrary Lagrangian-Eulerian (ALE) hydrodynamics simulations. High-order numerical methods for ALE simulations require both high-order (curvilinear) meshes and high-order remap algorithms. We develop a conservative and bounds-preserving method for accurate remapping of discrete fields on generalized polygonal meshes with curvilinear edges. The properties of the proposed method are studied theoretically and numerically for various (smooth and non-smooth) mesh deformations and discrete fields that represent smooth and discontinuous functions.

We explore an intersection-based remap method between meshes consisting of isoparametric elements. We present algorithms for the case of serendipity isoparametric elements (QUAD8 elements) and piece-wise constant (cell-centered) discrete fields. We demonstrate convergence properties of this remap method with a few numerical experiments.

We present a new moment-of-fluid (MOF2) interface reconstruction method. It uses the zeroth, first, and second moments of the fragment of material inside a cell of the mesh to reconstruct a convex material polygon or a union of convex polygons that approximate the respective material fragment. The new method requires information about the material moments only for the cell under consideration. The MOF2 method allows to exactly reproduce several convex shapes: corners, filaments, and some concave shapes: cell-complements to corners and filaments.Interface reconstruction is formulated as a local (for each cell), non-linear, equality constrained optimization problem, which does not require additional communication and allows for an efficient parallel implementation.We present an extensive set of test problems, both for interface reconstruction on a single cell, and for reconstruction of a variety of shapes on a variety of meshes.We describe how to perform two-material advection using the MOF2 method and present the results for the classical advection tests.We also show the examples of material interface remapping needed in the framework of multi-material arbitrary Lagrangian-Eulerian methods, and give a brief description of a procedure that can be used to update the material moments on the Lagrangian stage of those methods.(c) 2023 Elsevier Inc. All rights reserved.

We present a new adaptive moment-of-fluid (A-MOF) interface reconstruction method. It uses the zeroth, first, and second moments of the fragment of material inside a cell of the mesh to construct a shape that approximates the respective material fragment. The new method requires information about the material moments only for the cell under consideration. The adaptive method chooses between shapes obtained by the intersection of the cell with one half-plane, two half-planes, or a circle. The A-MOF method allows to exactly reproduce several convex shapes: corners, filaments, and their concave cell-complements; as well as pieces of the circles and its cell-compliments. Interface reconstruction is formulated as a local (for each cell), non-linear, equality constrained optimization problem, which does not require additional communication and allows for an efficient parallel implementation. We present an extensive set of test problems, both for interface reconstruction on a single cell, and for reconstruction of a variety of shapes on the entire mesh. (c) 2023 Elsevier Inc. All rights reserved.

Remapping is a conservative interpolation of a discretized intensive quantity between two meshes. In this article, we propose a novel multi-material flux remapping method that avoids the geometric computation of mesh-mesh intersections needed for an accurate intersection based remap. The flux remap is applicable to scalar quantities such as material density describing the multi-material flow between meshes with the same connectivity but small mesh displacements. The method is described for two-and three-dimensional polygonal/polyhedral meshes as it is implemented in Portage.(1) Another open source library, Tangram,(2) is used to calculate material interfaces in cells containing more than one material. Performance and accuracy of the flux remap are discussed with respect to Arbitrary Lagrangian-Eulerian simulations and compared to an accurate intersection based remap. In particular, cyclic remapping shows that the accuracy of the flux remap is limited to first order on material boundaries while maintaining second order accuracy in pure material regions. Published by Elsevier Inc.

data, the predicted time-dependent mass accumulation on a downstream sensor rises too sharply at early times and too slowly at late times because the SSVD overestimates the amount of mass stored in the fastest particles and underestimates the mass stored in the slowest particles. The functional form of the resulting m(t) is inconsistent with the available time-dependent data; numerical simulations and analytic studies agree on this point. Simulated mass tallies are highly sensitive to radial expansion of the ejecta cloud. It is not clear if the same effect is present in the experimental data but if so, depending on the degree, this may challenge the model’s compatibility with tin coupon data. The current implementation of the model in FLAG is sensitive to the detailed interaction between kinematics (hydrodynamic methods) and thermodynamics (material models); this sensitivity prohibits certain physics modeling choices. The appendices contain an extensive analytic study of piezoelectric ejecta mass measurements, along with test problems, excerpted from a longer work (LA-UR-17-21218).

In this work, we propose an interpolation or remapping algorithm of material-dependent fields on polyhedral meshes where any source or target cell contains only one material. It is conservative and it preserves sharp material boundaries on the target mesh, even if the source and target regions delineating the same material are slightly misaligned. If those material regions are aligned, then the algorithm is also linearity-preserving and bounds-preserving. For a given material, it consists of a conservative field reconstruction on a target mesh part from a source mesh part associated with that material, followed by a repair step in case of misaligned boundaries. No assumption is made regarding the topology of the input meshes

One of the important features of the arbitrary Lagrangian-Eulerian (ALE) hydrodynamics methods is the appearance of mesh cells containing fractional amounts of more than one material. To resolve sub-scale material dynamics, interfaces between materials inside a cell have to be calculated from material moments. The moment-of-fluid (MOF) method is one of the most accurate interface reconstruction methods. Unlike other methods, it is local and does not require data from neighboring cells which makes it amenable for emerging computer architectures. We analyze this method for axisymmetric problems. Our analysis closes a few gaps in its theoretical justification and provides useful insight on its properties and practical implementation. The theoretical conclusions are confirmed with numerical experiments.

Remap aims to accurately interpolate numerical fields from a source to a target mesh.It is an important component in arbitrary Lagrangian-Eulerian simulations in which the mesh evolves between time steps following the fluid displacement, as well as for accurately transferring numerical fields between codes.In this work, we present a new remap method that is conservative, in the sense that the integral of each field on the domain is preserved during the process, while minimizing field perturbations and avoiding diffusion on discontinuous fields.

We introduce new intersection-distribution-based remapping tools for indirect staggered arbitrary Lagrangian-Eulerian (ALE) simulations of multi-material shock hydrodynamics on arbitrary meshes. In addition to conserving momentum and total energy, the three-stage remapper proposed in this work preserves non-negativity of the internal energy. At the first stage, we construct slope-limited piecewise-linear reconstructions of all conserved quantities on zones of the source mesh and perform intersection-based remap to obtain bound-preserving zonal quantities on the target mesh. At the second stage, we define bound-preserving nodal quantities of the staggered ALE discretization as convex combinations of corner quantities. The nodal internal energy is corrected in a way which keeps it non-negative, while providing exact conservation of total energy. At the final stage, we distribute the non-negative nodal internal energy to corners, zones and materials using non-negative weights. Proofs of positivity preservation are provided for each stage. This work is a natural extension of our paper [14] in which a similar intersection-distribution-based remapping procedure was employed. The original version used a nodal kinetic energy fix which did not provably ensure positivity preservation for the zonal internal energy after the final distribution stage. The new algorithm cures this potential drawback by using 'coordinated' limiters for piecewise-linear reconstructions, remapping the internal energy to nodes and correcting it before redistribution. The effectiveness of the new nodal fix is illustrated by numerical examples. (c) 2021 Elsevier Inc. All rights reserved.

3 The role of Ristra at LANL post-ATDM 88Nodes

A high-order accurate reconstructed discontinuous Galerkin (rDG) method is developed for solving two-dimensional hydrodynamic problems in cell-centered updated Lagrangian formulation. This method is the Lagrangian limit of the unsplit rDG-ALE formulation, and is obtained by assuming the equality of the grid velocity to the fluid velocity only at cell boundaries. The conservative variables and the Taylor basis defined on the time-dependent moving mesh, provide the piece-wise polynomial expansion in the updated Lagrangian formulation. A multi-directional nodal Riemann solver is implemented for computing the grid velocity at the vertices and the numerical flux at the cell boundaries. A characteristic limiting procedure is extended from the primitive variable version to the conservative variable version, and its performance is compared with the limiter on physical variables. A number of benchmark test cases are conducted to assess the accuracy, robustness, and non-oscillatory property of the DG(P0), DG(P1) and rDG(P1P2) methods. The numerical experiments demonstrate that the developed rDG method is able to attain the designed order of accuracy and the characteristic limiting procedure outperforms the limiter on physical variables in terms of the monotonicity and symmetry preservation for shock problems. (C) 2020 Elsevier Ltd. All rights reserved.

We present a multi-material cell model (closure model) for demanding arbitrary Lagrangian-Eulerian (ALE) simulations of fluids and solids. It is based on the interface-aware sub-scale dynamics (IASSD) approach which utilizes the exact material interface geometry within the computational cell to calculate internal material interactions. Our formulation of the closure model also aims to improve the accuracy in low-speed impact events. Voids are used to represent ambient vacuum and internal free boundaries of the distinct materials. Void regions can close and open at contact surfaces, allowing a transition from contact physics to free motion in vacuum. The coupling of void closure and opening with a new formulation of the IASSD model for solids is tested on several one- and two-dimensional numerical examples, ranging from gas expansion in vacuum to planar and round object impacts at various speeds. (C) 2020 Published by Elsevier Ltd.

We present a new distribution-based remapping of the nodal mass and momentum between arbitrary source (Lagrangian) and arbitrary target (rezoned) meshes for indirect staggered arbitrary Lagrangian-Eulerian hydrodynamics. The method is based on the following ideas: define cell-centered momentum and mass on the source mesh; conservatively remap those cell-centered quantities from source to target meshes; and use local constrained optimization for each cell of the target mesh to conservatively distribute cell mass and momentum to the nodes of the cell. This new method is efficient, conservative, accurate and bounds preserving. Published by Elsevier Ltd.

To accurately model inviscid flow with shock waves using staggered-grid Lagrangian hydrodynamics, artificial viscosity is introduced to convert kinetic energy into internal energy, thereby providing a mechanism to generate the required entropy increase across shocks. In this paper, we propose a new method for constructing an adaptive, artificial viscosity in the context of one-dimensional, staggered-grid Lagrangian hydrodynamics. Our adaptive, artificial viscosity is defined in terms of two parameters that depend on density locally in a neighborhood around a shock, and hence, vary cell-by-cell. Our methodology is based on building a reference set of pre-computed optimal, globally constant artificial viscosity coefficients for a family of isolated shock test problems. For arbitrary flows, the evaluation of the unknown coefficients is automated by first estimating shock intensity locally, and second computing the corresponding adaptive parameter value via interpolation over the data from the pre-computed reference set of optimal, globally constant coefficient values. To illustrate the performance of our new approach, we compare our results against two existing methods. The first method is a limiter-based approach, which relies on estimating velocity gradients of the flow, and the second method is utilized in several commercial codes. We demonstrate that our new adaptive methodology produces more accurate results for a variety of tests with propagating shock waves, as well as for the aforementioned family of isolated shock problems. (C) 2020 Elsevier Ltd. All rights reserved.

We present a new intersection-distribution-based remapping method between arbitrary polygonal meshes for indirect staggered multi-material arbitrary Lagrangian-Eulerian hydrodynamics. All cell-centered material quantities are conservatively remapped using intersections between the Lagrangian (old, source) mesh and the rezoned (new, target) mesh. The new nodal masses are obtained by conservative distribution of all material masses in each new cell to the cell's corners and then collecting those corner masses at new nodes. This distribution is done using a local constrained optimization approach for each cell in the new mesh. In order to remap nodal momentum we first define cell-centered momentum for each cell in the old mesh, conservatively remap this to the new mesh and then conservatively distribute the new zonal momentum to each cell's bounding nodes, again using local constrained optimization. Our method also conserves total energy by applying a new nodal kinetic energy correction that relies on a process similar to that used for remapping nodal mass and momentum. Cell-centered kinetic energy is computed, conservatively remapped and then distributed to nodes. The discrepancy between this conservatively remapped and actual nodal kinetic energy is then conservatively distributed to the internal energies of the materials in the cells surrounding each node. Unlike conventional cell-based corrections of this type, this new nodal kinetic energy correction has not been observed to drive material internal energy negative in any of our testing. Unlike flux based remapping, our new intersection-distribution method can be applied to remapping between source and target meshes that are arbitrarily different, which provides superior flexibility in the rezoning strategy. Our method is accurate, essentially conservative and essentially bounds preserving.