666 research outputs found
On a general implementation of - and -adaptive curl-conforming finite elements
Edge (or N\'ed\'elec) finite elements are theoretically sound and widely used
by the computational electromagnetics community. However, its implementation,
specially for high order methods, is not trivial, since it involves many
technicalities that are not properly described in the literature. To fill this
gap, we provide a comprehensive description of a general implementation of edge
elements of first kind within the scientific software project FEMPAR. We cover
into detail how to implement arbitrary order (i.e., -adaptive) elements on
hexahedral and tetrahedral meshes. First, we set the three classical
ingredients of the finite element definition by Ciarlet, both in the reference
and the physical space: cell topologies, polynomial spaces and moments. With
these ingredients, shape functions are automatically implemented by defining a
judiciously chosen polynomial pre-basis that spans the local finite element
space combined with a change of basis to automatically obtain a canonical basis
with respect to the moments at hand. Next, we discuss global finite element
spaces putting emphasis on the construction of global shape functions through
oriented meshes, appropriate geometrical mappings, and equivalence classes of
moments, in order to preserve the inter-element continuity of tangential
components of the magnetic field. Finally, we extend the proposed methodology
to generate global curl-conforming spaces on non-conforming hierarchically
refined (i.e., -adaptive) meshes with arbitrary order finite elements.
Numerical results include experimental convergence rates to test the proposed
implementation
On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections
In this work, we analyze the scalability of inexact two-level balancing domain decomposition by constraints (BDDC) preconditioners for Krylov subspace iterative solvers, when using a highly scalable asynchronous parallel implementation where fine and coarse correction computations are overlapped in time. This way, the coarse-grid problem can be fully overlapped by fine-grid computations (which are embarrassingly parallel) in a wide range of cases. Further, we consider inexact solvers to reduce the computational cost/complexity and memory consumption of coarse and local problems and boost the scalability of the solver. Out of our numerical experimentation, we conclude that the BDDC preconditioner is quite insensitive to inexact solvers. In particular, one cycle of algebraic multigrid (AMG) is enough to attain algorithmic scalability. Further, the clear reduction of computing time and memory requirements of inexact solvers compared to sparse direct ones makes possible to scale far beyond state-of-the-art BDDC implementations. Excellent weak scalability results have been obtained with the proposed inexact/overlapped implementation of the two-level BDDC preconditioner, up to 93,312 cores and 20 billion unknowns on JUQUEEN. Further, we have also applied the proposed setting to unstructured meshes and partitions for the pressure Poisson solver in the backward-facing step benchmark domain
Distributed-memory parallelization of the aggregated unfitted finite element method
The aggregated unfitted finite element method (AgFEM) is a methodology
recently introduced in order to address conditioning and stability problems
associated with embedded, unfitted, or extended finite element methods. The
method is based on removal of basis functions associated with badly cut cells
by introducing carefully designed constraints, which results in well-posed
systems of linear algebraic equations, while preserving the optimal
approximation order of the underlying finite element spaces. The specific goal
of this work is to present the implementation and performance of the method on
distributed-memory platforms aiming at the efficient solution of large-scale
problems. In particular, we show that, by considering AgFEM, the resulting
systems of linear algebraic equations can be effectively solved using standard
algebraic multigrid preconditioners. This is in contrast with previous works
that consider highly customized preconditioners in order to allow one the usage
of iterative solvers in combination with unfitted techniques. Another novelty
with respect to the methods available in the literature is the problem sizes
that can be handled with the proposed approach. While most of previous
references discussing linear solvers for unfitted methods are based on serial
non-scalable algorithms, we propose a parallel distributed-memory method able
to efficiently solve problems at large scales. This is demonstrated by means of
a weak scaling test defined on complex 3D domains up to 300M degrees of freedom
and one billion cells on 16K CPU cores in the Marenostrum-IV platform. The
parallel implementation of the AgFEM method is available in the large-scale
finite element package FEMPAR
Enhanced balancing Neumann-Neumann preconditioning in computational fluid and solid mechanics
In this work, we propose an enhanced implementation of balancing Neumann-Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo-inverses appearing in local solvers. It is based on a kernel identication strategy that allows us to eciently compute the action of the pseudo-inverse via local indenite solvers. We further show how, identifying a minimum set of degrees of freedom to be xed, an equivalent denite system can be solved instead, even in the elastic case. On the other hand, we propose a simple modication of the preconditioned conjugate gradient (PCG) algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN PCG algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large-scale distributed-memory platforms. The enhanced BNN method is a competitive preconditioner for three-dimensional Poisson and elasticity problems, and outperforms the BDDC method in many cases
Scalable solvers for complex electromagnetics problems
In this work, we present scalable balancing domain decomposition by
constraints methods for linear systems arising from arbitrary order edge finite
element discretizations of multi-material and heterogeneous 3D problems. In
order to enforce the continuity across subdomains of the method, we use a
partition of the interface objects (edges and faces) into sub-objects
determined by the variation of the physical coefficients of the problem. For
multi-material problems, a constant coefficient condition is enough to define
this sub-partition of the objects. For arbitrarily heterogeneous problems, a
relaxed version of the method is defined, where we only require that the
maximal contrast of the physical coefficient in each object is smaller than a
predefined threshold. Besides, the addition of perturbation terms to the
preconditioner is empirically shown to be effective in order to deal with the
case where the two coefficients of the model problem jump simultaneously across
the interface. The new method, in contrast to existing approaches for problems
in curl-conforming spaces does not require spectral information whilst
providing robustness with regard to coefficient jumps and heterogeneous
materials. A detailed set of numerical experiments, which includes the
application of the preconditioner to 3D realistic cases, shows excellent weak
scalability properties of the implementation of the proposed algorithms
A highly scalable parallel implementation of balancing domain decomposition by constraints
In this work we propose a novel parallelization approach of two-level balancing domain decomposition by constraints preconditioning based on overlapping of fine-grid and coarse-grid duties in time. The global set of MPI tasks is split into those that have fine-grid duties and those that have coarse-grid duties, and the different computations and communications in the algorithm are then re-scheduled and mapped in such a way that the maximum degree of overlapping is achieved while preserving data dependencies among them. In many ranges of interest, the extra cost associated to the coarse-grid problem can be fully masked by fine-grid related computations (which are embarrassingly parallel). Apart from discussing code implementation details, the paper also presents a comprehensive set of numerical experiments, that includes weak scalability analyses, with structured and unstructured meshes, and exact and inexact solvers for the 3D Poisson and linear elasticity problems on a pair of state-of-the-art multicore-based distributed-memory machines. This experimental study reveals remarkable weak scalability in the solution of problems with thousands of millions of unknowns on several tens of thousands of computational cores
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