Tzanio Kolev

Veselin Dobrev, Patrick Knupp, Tzanio Kolev et al.

We describe a framework for controlling and improving the quality of high-order finite element meshes based on extensions of the Target-Matrix Optimization Paradigm (TMOP) of Knupp. This approach allows high-order applications to have a very precise control over local mesh quality, while still improving the mesh globally. We address the adaption of various TMOP components to the settings of general isoparametric element mappings, including the mesh quality metric in 2D and 3D, the selection of sample points and the solution of the resulting mesh optimization problem. We also investigate additional practical concerns, such as tangential relaxation and restricting the deviation from the original mesh. The benefits of the new high-order TMOP algorithms are illustrated on a number of test problems and examples from a high-order arbitrary Eulerian-Lagrangian (ALE) application. Our implementation is freely available in an open-source library form.

Veselin A. Dobrev, Tzanio V. Kolev, Chak S. Lee et al.

We propose an unified algebraic approach for static condensation and hybridization, two popular techniques in finite element discretizations. The algebraic approach is supported by the construction of scalable solvers for problems involving H(div)-spaces discretized by conforming (Raviart-Thomas) elements of arbitrary order. We illustrate through numerical experiments the relative performance of the two (in some sense dual) techniques in comparison with a state-of-the-art parallel solver, ADS, available in the software hypre and MFEM. The superior performance of the hybridization technique is clearly demonstrated with increased benefit for higher order elements.

Jiachen Yang, Tarik Dzanic, Brenden Petersen et al.

Large-scale finite element simulations of complex physical systems governed by partial differential equations (PDE) crucially depend on adaptive mesh refinement (AMR) to allocate computational budget to regions where higher resolution is required. Existing scalable AMR methods make heuristic refinement decisions based on instantaneous error estimation and thus do not aim for long-term optimality over an entire simulation. We propose a novel formulation of AMR as a Markov decision process and apply deep reinforcement learning (RL) to train refinement policies directly from simulation. AMR poses a new problem for RL as both the state dimension and available action set changes at every step, which we solve by proposing new policy architectures with differing generality and inductive bias. The model sizes of these policy architectures are independent of the mesh size and hence can be deployed on larger simulations than those used at train time. We demonstrate in comprehensive experiments on static function estimation and time-dependent equations that RL policies can be trained on problems without using ground truth solutions, are competitive with a widely-used error estimator, and generalize to larger, more complex, and unseen test problems.

Jakub Červený, Veselin Dobrev, Tzanio Kolev

We propose a general algorithm for non-conforming adaptive mesh refinement (AMR) of unstructured meshes in high-order finite element codes. Our focus is on h-refinement with a fixed polynomial order. The algorithm handles triangular, quadrilateral, hexahedral and prismatic meshes of arbitrarily high order curvature, for any order finite element space in the de Rham sequence. We present a flexible data structure for meshes with hanging nodes and a general procedure to construct the conforming interpolation operator, both in serial and in parallel. The algorithm and data structure allow anisotropic refinement of tensor product elements in 2D and 3D, and support unlimited refinement ratios of adjacent elements. We report numerical experiments verifying the correctness of the algorithms, and perform a parallel scaling study to show that we can adapt meshes containing billions of elements and run efficiently on 393,000 parallel tasks. Finally, we illustrate the integration of dynamic AMR into a high-order Lagrangian hydrodynamics solver.

Tzanio V. Kolev, Jinchao Xu, Yunrong Zhu

In this paper, we extend some of the multilevel convergence results obtained by Xu and Zhu in [Xu and Zhu, M3AS 2008], to the case of second order linear reaction-diffusion equations. Specifically, we consider the multilevel preconditioners for solving the linear systems arising from the linear finite element approximation of the problem, where both diffusion and reaction coefficients are piecewise-constant functions. We discuss in detail the influence of both the discontinuous reaction and diffusion coefficients to the performance of the classical BPX and multigrid V-cycle preconditioners.