Veselin Dobrev

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...

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.

High-performance Tensor Contractions for GPUs

We present a computational framework for high-performance tensor contractions on GPUs. High-performance is difficult to obtain using existing libraries, especially for many independent contractions where each contraction is very small, e.g., sub-vector/warp in size. However, using our framework to batch contractions plus application-specifics, we demonstrate close to peak performance results. In particular, to accelerate large scale tensor-formulated high-order finite element method (FEM) simulations, which is the main focus and motivation for this work, we represent contractions as tensor index reordering plus matrix-matrix multiplications (GEMMs). This is a key factor to achieve algorithmically many-fold acceleration (vs. not using it) due to possible reuse of data loaded in fast memory. In addition to using this context knowledge, we design tensor data-structures, tensor algebra interfaces, and new tensor contraction algorithms and implementations to achieve 90+% of a theoretically derived peak on GPUs. On a K40c GPU for contractions resulting in GEMMs on square matrices of size 8 for example, we are 2.8 faster than CUBLAS, and 8.5 faster than MKL on 16 cores of Intel Xeon E5-2670 (Sandy Bridge) 2.60GHz CPUs. Finally, we apply autotuning and code generation techniques to simplify tuning and provide an architecture-aware, user-friendly interface.

Preconditioning of Symmetric Interior Penalty Discontinuous Galerkin FEM for Elliptic Problems

This is a further development of [9] regarding multilevel preconditioning for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. We assume that the mesh on the finest level is a results of a geometrically refined fixed coarse mesh. The preconditioner is a multilevel method that uses a sequence of finite element spaces of either continuous or piecewise constant functions. The spaces are nested, but due to the penalty term in the DG method the corresponding forms are not inherited. For the continuous finite element spaces we show that the variable V-cycle provides an optimal preconditioner for the DG system. The piece-wise constant functions do not have approximation property so in order to control the energy growth of the inter-level transfer operator we apply W–cycle MG. Finally, we present a number of numerical experiments that support the theoretical findings.

Surface Reconstruction and Image Enhancement via

A surface reconstruction technique based on minimization of the total variation of the gradient is introduced. Convergence of the method is established, and an interior-point algorithm solving the associated linear programming problem is introduced. The reconstruction algorithm is illustrated on various test cases including natural and urban terrain data, and enhancement of low-resolution or aliased images.

L^1

Abstract In this work we present a FCT- like Maximum-Principle Preserving (MPP) method to solve the transport equation. We use high-order polynomial spaces; in particular, we consider up to 5th order spaces in two and three dimensions and 23rd order spaces in one dimension. The method combines the concepts of positive basis functions for discontinuous Galerkin finite element spatial discretization, locally defined solution bounds, element-based flux correction, and non-linear local mass redistribution. We consider a simple 1D problem with non-smooth initial data to explain and understand the behavior of different parts of the method. Convergence tests in space indicate that high-order accuracy is achieved. Numerical results from several benchmarks in two and three dimensions are also reported.

-Minimization

We show how a scalable preconditioner for the primal discontinuous Petrov--Galerkin (DPG) method can be developed using existing algebraic multigrid (AMG) preconditioning techniques. The stability ...We show how a scalable preconditioner for the primal discontinuous Petrov-Galerkin (DPG) method can be developed using existing algebraic multigrid (AMG) preconditioning techniques. The stability of the DPG method gives a norm equivalence which allows us to exploit existing AMG algorithms and software. We show how these algebraic preconditioners can be applied directly to a Schur complement system of interface unknowns arising from the DPG method. To the best of our knowledge, this is the first massively scalable algebraic preconditioner for DPG problems.

High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation

We present a new approach for multi-material arbitrary Lagrangian--Eulerian (ALE) hydrodynamics simulations based on high-order finite elements posed on high-order curvilinear meshes. The method builds on and extends our previous work in the Lagrangian [V. A. Dobrev, T. V. Kolev, and R. N. Rieben, SIAM J. Sci. Comput., 34 (2012), pp. B606--B641] and remap [R. W. Anderson et al., Internat. J. Numer. Methods Fluids, 77 (2015), pp. 249--273] phases of ALE, and depends critically on a functional perspective that enables subzonal physics and material modeling [V. A. Dobrev et al., Internat. J. Numer. Methods Fluids, 82 (2016), pp. 689--706]. Curvilinear mesh relaxation is based on node movement, which is determined through the solution of an elliptic equation. The remap phase is posed in terms of advecting state variables between two meshes over a fictitious time interval. The resulting advection equation is solved by a discontinuous Galerkin (DG) formulation, combined with a customized Flux Corrected Transpor...

A Scalable Preconditioner for a Primal Discontinuous Petrov--Galerkin Method

Abstract We present a new predictor-corrector approach to enforcing local maximum principles in piecewise-linear finite element schemes for the compressible Euler equations. The new element-based limiting strategy is suitable for continuous and discontinuous Galerkin methods alike. In contrast to synchronized limiting techniques for systems of conservation laws, we constrain the density, momentum, and total energy in a sequential manner which guarantees positivity preservation for the pressure and internal energy. After the density limiting step, the total energy and momentum gradients are adjusted to incorporate the irreversible effect of density changes. Antidiffusive corrections to bounds-compatible low-order approximations are limited to satisfy inequality constraints for the specific total and kinetic energy. An accuracy-preserving smoothness indicator is introduced to gradually adjust lower bounds for the element-based correction factors. The employed smoothness criterion is based on a Hessian determinant test for the density. A numerical study is performed for test problems with smooth and discontinuous solutions.

High-Order Multi-Material ALE Hydrodynamics

The traditional way to numerically solve time-dependent problems is to sequentially march through time, solving for one time step and then the next. The parallelism in this approach is limited to the spatial dimension, which is quickly exhausted, causing the gain from using more processors to solve a problem to diminish. One approach to overcome this barrier is to use methods that are parallel in time. These methods have the potential to achieve dramatically better performance compared to time-stepping approaches, but achieving this performance requires carefully choosing the amount of parallelism devoted to space versus the amount devoted to time. Here, we present a performance model that, for a multigrid-in-time solver, makes the decision on when to switch to parallel-in-time and on how much parallelism to devote to space vs. time. In our experiments, the model selects the best parallel configuration in most of our test cases and a configuration close to the best one in all other cases.