We present an energy conservative, quadrature based model reduction framework for the compressible Euler equations of Lagrangian hydrodynamics. Building on a finite element discretization of the governing equations, we develop reduced models using data based reduced basis functions and the empirical quadrature procedure (EQP). We introduce a strongly energy conservative variant of EQP that enforces exact energy conservation in the reduction process. Numerical experiments for four benchmark problems – Sedov blast, Gresho vortex, triple point and Taylor-Green vortex – demonstrate that the numerical implementation of our proposed method conserves total energy to near machine precision, while maintaining accuracy comparable to the basic EQP formulation.

The MFEM (Modular Finite Element Methods) library is a high-performance C++ library for finite element discretizations. MFEM supports numerous types of finite element methods and is the discretization engine powering many computational physics and engineering applications across a number of domains. This paper describes some of the recent research and development in MFEM, focusing on performance portability across leadership-class supercomputing facilities, including exascale supercomputers, as well as new capabilities and functionality, enabling a wider range of applications. Much of this work was undertaken as part of the Department of Energy's Exascale Computing Project (ECP) in collaboration with the Center for Efficient Exascale Discretizations (CEED).

While projection-based reduced order models can reduce the dimension of full order solutions, the resulting reduced models may still contain terms that scale with the full order dimension. Hyper-reduction techniques are sampling-based methods that further reduce this computational complexity by approximating such terms with a much smaller dimension. The goal of this work is to introduce a points selection algorithm developed by Shin and Xiu [SIAM J. Sci. Comput., 38 (2016), pp. A385--A411], as a hyper-reduction method. The selection algorithm is originally proposed as a stochastic collocation method for uncertainty quantification. Since the algorithm aims at maximizing a quantity S that measures both the column orthogonality and the determinant, we refer to the algorithm as S-OPT. Numerical examples are provided to demonstrate the performance of S-OPT and to compare its performance with an over-sampled Discrete Empirical Interpolation (DEIM) algorithm. We found that using the S-OPT algorithm is shown to predict the full order solutions with higher accuracy for a given number of indices.

The Rayleigh-Taylor instability is a classical hydrodynamic instability of great interest in various disciplines of science and engineering, including astrophysics, atmospheric sciences and climate, geophysics, and fusion energy. Analytical methods cannot be applied to explain the long-time behavior of the Rayleigh-Taylor instability, and therefore, numerical simulation of the full problem is required. However, in order to capture the growth of amplitude of perturbations accurately, both the spatial and temporal discretizations need to be extremely fine for traditional numerical methods, and long-time simulation may become prohibitively expensive. In this paper, we propose efficient reduced order model techniques to accelerate the simulation of the Rayleigh-Taylor instability in compressible gas dynamics. We introduce a general framework for decomposing the solution manifold to construct the temporal domain partition and temporally-local reduced order model construction with varying Atwood number. We propose two practical approaches in this framework, namely decomposition by physical time and by penetration distance. Numerical results are presented to examine the performance of the proposed approaches.

As a mathematical model of high-speed flow and shock wave propagation in a complex multimaterial setting, Lagrangian hydrodynamics is characterized by moving meshes, advection-dominated solutions, and moving shock fronts with sharp gradients. These challenges hinder the existing projection-based model reduction schemes from being practical. We develop several variations of projection-based reduced order model techniques for Lagrangian hydrodynamics by introducing three different reduced bases for position, velocity, and energy fields. A time-windowing approach is also developed to address the challenge imposed by the advection-dominated solutions. Lagrangian hydrodynamics is formulated as a nonlinear problem, which requires a proper hyper-reduction technique. Therefore, we apply the over-sampling DEIM and SNS approaches to reduce the complexity due to the nonlinear terms. Finally, we also present both a posteriori and a priori error bounds associated with our reduced order model. We compare the performance of the spatial and time-windowing reduced order modeling approaches in terms of accuracy and speed-up with respect to the corresponding full order model for several numerical examples, namely Sedov blast, Gresho vortices, Taylor-Green vortices, and triple-point problems.

We present an extension of a linearized Coulomb collision operator, previously used in several Eulerian kinetic codes for like-species collisions and unlike-species collisions in the case where the backgrounds about which the linearization is made all are in collisional equilibrium, to the situation of interpenetrating plasma streams. In the latter case, the backgrounds cannot be taken to be in equilibrium and a significant generalization is required. Our development is targeted toward the Eulerian kinetic plasma code LOKI, which evolves the Vlasov–Poisson or Vlasov–Maxwell system in a Cartesian “2 + 2-dimensional” phase space. The extended operator has been implemented in a test code, and results of both quantitative verification and qualitative “realizability” tests are presented.

Abstract Hydraulic fracturing is a stimulation technique used in unconventional reservoirs to generate sufficient reservoir contact for viable production. During stimulation, proppant is pumped into the system to keep fractures open when, at a later stage, injection ceases and fluid is recovered as fractures close. Thus, controlling proppant distribution in the fracture is important to improve well productivity. Several factors influence proppant placement and settling such as proppant size, proppant density and carrier-fluid rheology. However, accurately predicting, and consequently optimizing, proppant placement in complex fracture systems remains an industry challenge. The objective of this study is to use advanced modeling techniques of proppant transport inside complex fracture systems using a 3D simulator that couples geomechan ics, fracture mechanics, fluid behavior and proppant transport to assess the optimization of stimulation treatments. In this study proppant transport models incorporate the interaction between hydraulic fractures (HF) and natural fractures (NF) of various sizes and orientations. Treatments were simulated in systematic variations of proppant size, proppant density and fluid viscosity. Fluid rheology, proppant size and density are all major parameters affecting the proppant settling rate. In more complex systems, proppant placement is strongly influenced by the location, size, and orientation of natural fractures. Based on the size of proppant and viscosity of the injected fluid, proppant may or may not enter the natural fractures.

Summary In this study, we implemented and tested a new processing-based broadband solution for mitigating F-K transform artifacts for receiver deghosting in a marine environment. The FK transform has traditionally been used for flat cable (constant depth) deghosting and often times tailored to meet the slanted (variable depth) cable criteria. Recently, the usage of τ − p do- main deterministic deghost operator has been more prominent with slant cable deghosting. Irrespective of the type of transform or deghost operator used, a windowed process is essential due to the time and offset varying character of the ghost. This use of a windowed process usually results in poor reconstruction of deghosted signals and artifacts beyond the control of the transform(s) itself. The windowing in time and offset produces edgy effects which can be clearly seen in the difference plots. Our method, using a non-windowing approach, demonstrates a better representation of the deghosted signals without the artifacts caused by the boundary of the windows. This method has also been well-tested for both the flat and slant cable receiver deghosting workflows in synthetic and field data examples.

We develop a family of expanded mixed multiscale finite element methods (MsFEMs) and their hybridizations for second-order elliptic equations. This formulation expands the standard mixed multiscale finite element formulation in the sense that four unknowns (hybrid formulation) are solved simultaneously: pressure, gradient of pressure, velocity, and Lagrange multipliers. We use multiscale basis functions for both the velocity and the gradient of pressure. In the expanded mixed MsFEM framework, we consider both separable and nonseparable spatial scales. Specifically, we analyze the methods in three categories: periodic separable scales, G-convergent separable scales, and a continuum of scales. When there is no scale separation, using some global information can significantly improve the accuracy of the expanded mixed MsFEMs. We present a rigorous convergence analysis of these methods that includes both conforming and nonconforming formulations. Numerical results are presented for various multiscale models of flow in porous media with shale barriers that illustrate the efficacy of the proposed family of expanded mixed MsFEMs.

This paper presents and analyzes a negative-norm least-squares finite element discretization method for the dimension-reduced time-harmonic Maxwell equations in the case of axial symmetry. The reduced equations are expressed in cylindrical coordinates, and the analysis consequently involves weighted Sobolev spaces based on the degenerate radial weighting. The main theoretical results established in this work include existence and uniqueness of the continuous and discrete formulations and error estimates for simple finite element functions. Numerical experiments confirm the error estimates and efficiency of the method for piecewise constant coefficients.

In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple elliptic equation for the amplitude. This is true for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution, we can expand the solution in a Fourier series. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled nonlinear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach. In this paper we look at nonlinear, time-harmonic potential problems as simple model problems. We construct and analyze almost optimal solvers for the Jacobi systems arising from the Newton linearization of the large-scale coupled nonlinear system that one has to solve instead of performing the expensive time-integration procedure.

The paper is devoted to fast iterative solvers for frequency-domain finite element equations approximating linear and nonlinear parabolic initial boundary value problems with time-harmonic excitations. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple linear elliptic system for the amplitudes belonging to the sine- and to the cosine-excitation or a large nonlinear elliptic system for the Fourier coefficients in the linear and nonlinear case, respectively. The fast solution of the corresponding linear and nonlinear system of finite element equations is crucial for the competitiveness of this method.

Consider the space of two-dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable. This space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates. We prove that if the original three-dimensional domain is convex, then the multigrid V-cycle applied to the inner product in this space converges, provided certain modern smoothers are used. For the convergence analysis, we first prove several intermediate results, e.g., the approximation properties of a commuting projector in weighted norms, and a superconvergence estimate for a dual mixed method in weighted spaces. The uniformity of the multigrid convergence rate with respect to mesh size is then established theoretically and illustrated through numerical experiments.

We present new finite element methods for Helmholtz and Maxwell equations for general three-dimensional polyhedral meshes, based on domain decomposition with boundary elements on the surfaces of the polyhedral volume elements. The methods use the lowest-order polynomial spaces and produce sparse, symmetric linear systems despite the use of boundary elements. Moreover, piecewise constant coefficients are admissible. The resulting approximation on the element surfaces can be extended throughout the domain via representation formulas. Numerical experiments confirm that the convergence behavior on tetrahedral meshes is comparable to that of standard finite element methods, and equally good performance is attained on more general meshes.

We derive and analyze new boundary element (BE) based finite element discretizations of potential-type, Helmholtz and Maxwell equations on arbitrary polygonal and polyhedral meshes. The starting point of this discretization technique is the symmetric BE Domain Decomposition Method (DDM), where the subdomains are the finite elements. This can be interpreted as a local Trefftz method that uses PDE-harmonic basis functions. This discretization technique leads to largescale sparse linear systems of algebraic equations which can efficiently be solved by Algebraic Multigrid (AMG) methods or AMG preconditioned conjugate gradient methods in the case of the potential equation and by Krylov subspace iterative methods in general.