Robert N. Rieben

High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics

The numerical approximation of the Euler equations of gas dynamics in a movingLagrangian frame is at the heart of many multiphysics simulation algorithms. In this paper, we present a general framework for high-order Lagrangian discretization of these compressible shock hydrodynamics equations using curvilinear finite elements. This method is an extension of the approach outlined in [Dobrev et al., Internat. J. Numer. Methods Fluids, 65 (2010), pp. 1295--1310] and can be formulated for any finite dimensional approximation of the kinematic and thermodynamic fields, including generic finite elements on two- and three-dimensional meshes with triangular, quadrilateral, tetrahedral, or hexahedral zones. We discretize the kinematic variables of position and velocity using a continuous high-order basis function expansion of arbitrary polynomial degree which is obtained via a corresponding high-order parametric mapping from a standard reference element. This enables the use of curvilinear zone geometry, higher-ord...

FEMSTER: An object-oriented class library of high-order discrete differential forms

FEMSTER is a modular finite element class library for solving three-dimensional problems arising in electromagnetism. The library was designed using a modern geometrical approach based on differential forms (or p-forms) and can be used for high-order spatial discretizations of well-known H(div)- and H(curl)-conforming finite element methods. The software consists of a set of abstract interfaces and concrete classes, providing a framework in which the user is able to add new schemes by reusing the existing classes or by incorporating new user-defined data types.

A tensor artificial viscosity using a finite element approach

We derive a tensor artificial viscosity suitable for use in a 2D or 3D unstructured arbitrary Lagrangian-Eulerian (ALE) hydrodynamics code. This work is similar in nature to that of Campbell and Shashkov [1]; however, our approach is based on a finite element discretization that is fundamentally different from the mimetic finite difference framework. The finite element point of view leads to novel insights as well as improved numerical results. We begin with a generalized tensor version of the Von Neumann-Richtmyer artificial viscosity, then convert it to a variational formulation and apply a Galerkin discretization process using high order Gaussian quadrature to obtain a generalized nodal force term and corresponding zonal heating (or shock entropy) term. This technique is modular and is therefore suitable for coupling to a traditional staggered grid discretization of the momentum and energy conservation laws; however, we motivate the use of such finite element approaches for discretizing each term in the Euler equations. We review the key properties that any artificial viscosity must possess and use these to formulate specific constraints on the total artificial viscosity force term as well as the artificial viscosity coefficient. We also show, that under certain simplifying assumptions, the two-dimensional scheme from [1] can be viewed as an under-integrated version of our finite element method. This equivalence holds on general distorted quadrilateral grids. Finally, we present computational results on some standard shock hydro test problems, as well as some more challenging problems, indicating the advantages of the new approach with respect to symmetry preservation for shock wave propagation over general grids.

A generalized mass lumping technique for vector finite-element solutions of the time-dependent Maxwell equations

Time-domain finite-element solutions of Maxwells equations require the solution of a sparse linear system involving the mass matrix at every time step. This process represents the bulk of the computational effort in time-dependent simulations. As such, mass lumping techniques in which the mass matrix is reduced to a diagonal or block-diagonal matrix are very desirable. In this paper, we present a special set of high order 1-form (also known as curl-conforming) basis functions and reduced order integration rules that, together, allow for a dramatic reduction in the number of nonzero entries in a vector finite element mass matrix. The method is derived from the Nedelec curl-conforming polynomial spaces and is valid for arbitrary order hexahedral basis functions for finite-element solutions to the second-order wave equation for the electric (or magnetic) field intensity. We present a numerical eigenvalue convergence analysis of the method and quantify its accuracy and performance via a series of computational experiments.

A Step towards Energy Efficient Computing: Redesigning a Hydrodynamic Application on CPU-GPU

Power and energy consumption are becoming an increasing concern in high performance computing. Compared to multi-core CPUs, GPUs have a much better performance per watt. In this paper we discuss efforts to redesign the most computation intensive parts of BLAST, an application that solves the equations for compressible hydrodynamics with high order finite elements, using GPUs BLAST, Dobrev. In order to exploit the hardware parallelism of GPUs and achieve high performance, we implemented custom linear algebra kernels. We intensively optimized our CUDA kernels by exploiting the memory hierarchy, which exceed the vendors library routines substantially in performance. We proposed an auto tuning technique to adapt our CUDA kernels to the orders of the finite element method. Compared to a previous base implementation, our redesign and optimization lowered the energy consumption of the GPU in two aspects: 60% less time to solution and 10% less power required. Compared to the CPU-only solution, our GPU accelerated BLAST obtained a 2.5× overall speedup and 1.42× energy efficiency (green up) using 4th order (Q_4) finite elements, and a 1.9× speedup and 1.27× green up using 2nd order (Q2) finite elements.

FEMSTER: an object oriented class library of discrete differential forms

The FEMSTER finite element class library described in this paper is unique in several aspects. First, it is based upon the language of differential forms. This language provides a unified description of a great variety of PDEs and thus leads us directly to a concise and abstract interface to finite element methods. This language also unifies the seemingly disparate Lagrange, H(curl) and H(div) basis functions that are used in computational electromagnetics. Secondly, FEMSTER utilizes higher-order elements, bases, and integration rules. Higher-order elements are important for accurate modeling of curved surfaces. The use of higher-order basis functions reduces the demands put upon mesh generation, e.g. a billion element mesh is no longer required for a numerically converged solution. The FEMSTER class library is ideally suited for researchers who wish to experiment with unstructured-grid, higher-order solution of Poissons equation, the Helmholtz equation, Maxwell equations, and related PDEs that employ the standard gradient, curl, and divergence operators.

An arbitrary Lagrangian–Eulerian discretization of MHD on 3D unstructured grids

Abstract We present an arbitrary Lagrangian–Eulerian (ALE) discretization of the equations of resistive magnetohydrodynamics (MHD) on unstructured hexahedral grids. The method is formulated using an operator-split approach with three distinct phases: electromagnetic diffusion, Lagrangian motion , and Eulerian advection . The resistive magnetic induction equation is discretized using a compatible mixed finite element method with a second order accurate implicit time differencing scheme which preserves the divergence-free nature of the magnetic field. At each discrete time step, electromagnetic force and heat terms are calculated and coupled to the hydrodynamic equations to compute the Lagrangian motion of the conducting materials. By virtue of the compatible discretization method used, the invariants of Lagrangian MHD motion are preserved in a discrete sense. When the Lagrangian motion of the mesh causes significant distortion, that distortion is corrected with a relaxation of the mesh, followed by a second order monotonic remap of the electromagnetic state variables. The remap is equivalent to Eulerian advection of the magnetic flux density with a fictitious mesh relaxation velocity. The magnetic advection is performed using a novel variant of constrained transport (CT) that is valid for unstructured hexahedral grids with arbitrary mesh velocities. The advection method maintains the divergence-free nature of the magnetic field and is second order accurate in regions where the solution is sufficiently smooth. For regions in which the magnetic field is discontinuous (e.g. MHD shocks) the method is limited using a novel variant of algebraic flux correction (AFC) which is local extremum diminishing (LED) and divergence preserving. Finally, we verify each stage of the discretization via a set of numerical experiments.

Verification of high-order mixed finite-element solution of transient magnetic diffusion problems

We develop and present high-order mixed finite-element discretizations of the time-dependent electromagnetic diffusion equations for solving eddy-current problems on three-dimensional unstructured grids. The discretizations are based on high-order H(Grad), H(Curl), and H(Div) conforming finite-element spaces combined with an implicit and unconditionally stable generalized Crank-Nicholson time differencing method. We develop three separate electromagnetic diffusion formulations, namely the E (electric field), H(magnetic field), and the A-/spl phi/ (potential) formulations. For each formulation, we also provide a consistent procedure for computing the secondary variables J(current flux density) and B(magnetic flux density), as these fields are required for the computation of electromagnetic force and heating terms. We verify the error convergence properties of each formulation via a series of numerical experiments on canonical problems with known analytic solutions. The key result is that the different formulations are equally accurate, even for the secondary variables J and B, and hence the choice of which formulation to use depends mostly on relevance of the natural and essential boundary conditions to the problem of interest. In addition, we highlight issues with numerical verification of finite-element methods that can lead to false conclusions on the accuracy of the methods.

Arbitrary Lagrangian-Eulerian methods for modeling high-speed compressible multimaterial flows

This paper reviews recent developments in Arbitrary Lagrangian Eulerian (ALE) methods for modeling high speed compressible multimaterial flows in complex geometry on general polygonal meshes. We only consider the indirect ALE approach which consists of three key stages: a Lagrangian stage, in which the solution and the computational mesh are updated; a rezoning stage, in which the nodes of the computational mesh are moved to improve grid quality; and a remapping stage, in which the Lagrangian solution is transferred to the rezoned mesh.

Development and Application of Compatible Discretizations of Maxwell's Equations

We present the development and application of compatible finite element discretizations of electromagnetics problems derived from the time dependent, full wave Maxwell equations. We review the H(curl)-conforming finite element method, using the concepts and notations of differential forms as a theoretical framework. We chose this approach because it can handle complex geometries, it is free of spurious modes, it is numerically stable without the need for filtering or artificial diffusion, it correctly models the discontinuity of fields across material boundaries, and it can be very high order. Higher-order H(curl) and H(div) conforming basis functions are not unique and we have designed an extensible C++ framework that supports a variety of specific instantiations of these such as standard interpolatory bases, spectral bases, hierarchical bases, and semi-orthogonal bases. Virtually any electromagnetics problem that can be cast in the language of differential forms can be solved using our framework. For time dependent problems a method-of-lines scheme is used where the Galerkin method reduces the PDE to a semi-discrete system of ODE’s, which are then integrated in time using finite difference methods. For time integration of wave equations we employ the unconditionally stable implicit Newmark-Beta method, as well as the high order energy conserving explicit Maxwell Symplectic method; for diffusion equations, we employ a generalized Crank-Nicholson method. We conclude with computational examples from resonant cavity problems, time-dependent wave propagation problems, and transient eddy current problems, all obtained using the authors massively parallel computational electromagnetics code EMSolve.