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• Hyperviscosity for ALE hydrodynamics with arbitrary order on unstructured meshes. • Robust shock capturing on moving high-order meshes with high-order convergence. • A method for computing hypervisocisty operator in an efficient, matrix-free manner. • Reduced numerical dissipation and enhanced resolution of complex vortical flow. The numerical approximation of compressible hydrodynamics is at the core of high-energy density (HED) multiphysics simulations as shocks are the driving force in experiments like inertial confinement fusion (ICF). In this work, we describe our extension of the hyperviscosity technique, originally developed for shock treatment in finite difference simulations, for use in arbitrarily high-order finite element methods for Lagrangian hydrodynamics. Hyperviscosity enables shock capturing while preserving the high-order properties of the underlying discretization away from the shock region. Specifically, we compute a high-order term based on a product of the mesh length scale to a high power scaled by a hyper-Laplacian operator applied to a scalar field. We then form the total artificial viscosity by taking a non-linear blend of this term and a traditional artificial viscosity term. We also present a matrix-free formulation for computing the finite element based hyper-Laplacian operator. Such matrix-free methods have superior performance characteristics compared to traditional full matrix assembly approaches and offer advantages for GPU based HPC hardware. We demonstrate the numerical convergence of our method and its application to complex, multi-material ALE simulations on high-order (curved) meshes. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present an extension of our general high-order curvilinear finite element approach for solving the Euler equations in a Lagrangian frame [1] to the case of axisymmetric problems. The numerical approximation of these equations is important in a number of applications of compressible shock hydrodynamics and the reduction of 3D problems with axial symmetry to 2D computations provides a significant computational advantage. Unlike traditional staggered-grid hydrodynamics (SGH) methods, which use the so-called “area-weighting” scheme, we formulate our semi-discrete axisymmetric conservation laws directly in 3D and reduce them to a 2D variational form in a meridian cut of the original domain. This approach is a natural extension of the high-order curvilinear finite element framework we have developed for 2D and 3D problems in Cartesian geometry, leading to a rescaled momentum conservation equation which includes new radial terms in the pressure gradient and artificial viscosity forces. We show that this approach exactly conserves energy and we demonstrate via computational examples that it also excels at preserving symmetry in problems with symmetric initial conditions. The results also illustrate that our computational method does not produce spurious symmetry breaking near the axis of rotation, as is the case with many area-weighted approaches. [Copyright &y& Elsevier]

• Mesh adaptivity is used to control numerical dissipation in ALE hydrodynamics. • Algebraic problem formulation applicable to curved meshes in any dimension. • Adaptivity to discrete simulation features is done through node movement. • General target construction methods can adapt all geometric properties. • Adaptive ALE triggers can improve both accuracy and simulation time. In this paper we propose tools for high-order mesh optimization and demonstrate their benefits in the context of multi-material Arbitrary Lagrangian-Eulerian (ALE) compressible shock hydrodynamic applications. The mesh optimization process is driven by information provided by the simulation which uses the optimized mesh, such as shock positions, material regions, known error estimates, etc. These simulation features are usually represented discretely, for instance, as finite element functions on the Lagrangian mesh. The discrete nature of the input is critical for the practical applicability of the algorithms we propose and distinguishes this work from approaches that strictly require analytical information. Our methods are based on node movement through a high-order extension of the Target-Matrix Optimization Paradigm (TMOP) of [1]. The proposed formulation is fully algebraic and relies only on local Jacobian matrices, so it is applicable to all types of mesh elements, in 2D and 3D, and any order of the mesh. We discuss the notions of constructing adaptive target matrices and obtaining their derivatives, reconstructing discrete data in intermediate meshes, node limiting that enables improvement of global mesh quality while preserving space-dependent local mesh features, and appropriate normalization of the objective function. The adaptivity methods are combined with automatic ALE triggers that can provide robustness of the mesh evolution and avoid excessive remap procedures. The benefits of the new high-order TMOP technology are illustrated on several simulations performed in the high-order ALE application BLAST [2]. [ABSTRACT FROM AUTHOR]

• Matrix-free methods for transport and remap problems based on residual distribution. • High-order Bernstein finite element discretizations. • Theoretical foundation that guarantees reasonable, CFL-like time step restrictions. • Comparison of new monolithic limiters to Flux-Corrected-Transport. • Optimal convergence rates for smooth solutions achieved by nodal smoothness indicators. This paper is focused on the aspects of limiting in residual distribution (RD) schemes for high-order finite element approximations to advection problems. Both continuous and discontinuous Galerkin methods are considered in this work. Discrete maximum principles are enforced using algebraic manipulations of element contributions to the global nonlinear system. The required modifications can be carried out without calculating the element matrices and assembling their global counterparts. The components of element vectors associated with the standard Galerkin discretization are manipulated directly using localized subcell weights to achieve optimal accuracy. Low-order nonlinear RD schemes of this kind were originally developed to calculate local extremum diminishing predictors for flux-corrected transport (FCT) algorithms. In the present paper, we incorporate limiters directly into the residual distribution procedure, which makes it applicable to stationary problems and leads to well-posed nonlinear discrete problems. To circumvent the second-order accuracy barrier, the correction factors of monolithic limiting approaches and FCT schemes are adjusted using smoothness sensors based on second derivatives. The convergence behavior of presented methods is illustrated by numerical studies for two-dimensional test problems. [ABSTRACT FROM AUTHOR]