A matrix-free macro-element variant of the hybridized discontinuous Galerkin method
V. Badrkhani, R.R. Hiemstra, M.L. Mika, and D. Schillinger
February 18, 2023;
preprint
We investigate a macro-element variant of the hybridized discontinuous Galerkin (HDG) method, using patches of
standard simplicial elements that can have non-matching interfaces. Coupled via the HDG technique, our method
enables local refinement by uniform simplicial subdivision of each macro-element. By enforcing one spatial
discretization for all macro-elements, we arrive at local problems per macro-element that are embarrassingly
parallel, yet well balanced. Therefore, our macro-element variant scales efficiently to n-node clusters and
can be tailored to available hardware by adjusting the local problem size to the capacity of a single node,
while still using moderate polynomial orders such as quadratics or cubics. Increasing the local problem size
means simultaneously decreasing, in relative terms, the global problem size, hence effectively limiting the
proliferation of degrees of freedom. The global problem is solved via a matrix-free iterative technique that
also heavily relies on macro-element local operations.
A comparison of matrix-free isogeometric Galerkin and collocation methods for Karhunen–Loève expansion
M.L. Mika, R.R. Hiemstra, T.J.R. Hughes and D. Schillinger
March 13, 2022;
Current Trends and Open Problems in Computational Mechanics
Numerical computation of the Karhunen–Loève expansion is computationally challenging in terms of both memory
requirements and computing time. We compare two state-of-the-art methods that claim to efficiently solve for the
K–L expansion: (1) the matrix-free isogeometric Galerkin method using interpolation based quadrature proposed
by the authors in [1] and (2) our new matrix-free implementation of the isogeometric collocation method proposed in [2].
Two three-dimensional benchmark problems indicate that the Galerkin method performs significantly better for smooth
covariance kernels, while the collocation method performs slightly better for rough covariance kernels.
A matrix-free isogeometric Galerkin method for Karhunen-Loève
approximation of random fields using tensor product splines, tensor
contraction and interpolation based quadrature
M.L. Mika, T.J.R. Hughes, D. Schillinger, P. Wriggers and R.R. Hiemstra
February 9, 2021;
Computer Methods in Applied Mechanics and Engineering
The Karhunen-Loève series expansion decomposes a Gaussian stochastic process
into an infinite series of pairwise uncorrelated random variables and pairwise
L2 orthogonal functions. The computational complexity of standard Galerkin finite
element formation and assembly techniques, as well as memory requirements of
direct solution techniques become quickly computationally intractable with
increasing polynomial degree and the number of elements. In this work we
present a novel matrix-free solution strategy that scales optimally with
the problem size, is indepedent of the polynomial degree and is emberassingly
parallel. A high-order three-dimensional benchmark
illustrates exceptional computational performance combined with high
accuracy and robustness.
