637 research outputs found
Preconditioning for radial basis function partition of unity methods
Meshfree radial basis function (RBF) methods are of interest for solving partial differential equations due to attractive convergence properties, flexibility with respect to geometry, and ease of implementation. For global RBF methods, the computational cost grows rapidly with dimension and problem size, so localised approaches, such as partition of unity or stencil based RBF methods, are currently being developed. An RBF partition of unity method (RBF--PUM) approximates functions through a combination of local RBF approximations. The linear systems that arise are locally unstructured, but with a global structure due to the partitioning of the domain. Due to the sparsity of the matrices, for large scale problems, iterative solution methods are needed both for computational reasons and to reduce memory requirements. In this paper we implement and test different algebraic preconditioning strategies based on the structure of the matrix in combination with incomplete factorisations. We compare their performance for different orderings and problem settings and find that a no-fill incomplete factorisation of the central band of the original discretisation matrix provides a robust and efficient preconditioner
A trivariate interpolation algorithm using a cube-partition searching procedure
In this paper we propose a fast algorithm for trivariate interpolation, which
is based on the partition of unity method for constructing a global interpolant
by blending local radial basis function interpolants and using locally
supported weight functions. The partition of unity algorithm is efficiently
implemented and optimized by connecting the method with an effective
cube-partition searching procedure. More precisely, we construct a cube
structure, which partitions the domain and strictly depends on the size of its
subdomains, so that the new searching procedure and, accordingly, the resulting
algorithm enable us to efficiently deal with a large number of nodes.
Complexity analysis and numerical experiments show high efficiency and accuracy
of the proposed interpolation algorithm
Adaptive meshless refinement schemes for RBF-PUM collocation
In this paper we present an adaptive discretization technique for solving
elliptic partial differential equations via a collocation radial basis function
partition of unity method. In particular, we propose a new adaptive scheme
based on the construction of an error indicator and a refinement algorithm,
which used together turn out to be ad-hoc strategies within this framework. The
performance of the adaptive meshless refinement scheme is assessed by numerical
tests
An extended finite element method with smooth nodal stress
The enrichment formulation of double-interpolation finite element method
(DFEM) is developed in this paper. DFEM is first proposed by Zheng \emph{et al}
(2011) and it requires two stages of interpolation to construct the trial
function. The first stage of interpolation is the same as the standard finite
element interpolation. Then the interpolation is reproduced by an additional
procedure using the nodal values and nodal gradients which are derived from the
first stage as interpolants. The re-constructed trial functions are now able to
produce continuous nodal gradients, smooth nodal stress without post-processing
and higher order basis without increasing the total degrees of freedom. Several
benchmark numerical examples are performed to investigate accuracy and
efficiency of DFEM and enriched DFEM. When compared with standard FEM,
super-convergence rate and better accuracy are obtained by DFEM. For the
numerical simulation of crack propagation, better accuracy is obtained in the
evaluation of displacement norm, energy norm and the stress intensity factor
Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations
The iterative diagonalization of a sequence of large ill-conditioned
generalized eigenvalue problems is a computational bottleneck in quantum
mechanical methods employing a nonorthogonal basis for {\em ab initio}
electronic structure calculations. We propose a hybrid preconditioning scheme
to effectively combine global and locally accelerated preconditioners for rapid
iterative diagonalization of such eigenvalue problems. In partition-of-unity
finite-element (PUFE) pseudopotential density-functional calculations,
employing a nonorthogonal basis, we show that the hybrid preconditioned block
steepest descent method is a cost-effective eigensolver, outperforming current
state-of-the-art global preconditioning schemes, and comparably efficient for
the ill-conditioned generalized eigenvalue problems produced by PUFE as the
locally optimal block preconditioned conjugate-gradient method for the
well-conditioned standard eigenvalue problems produced by planewave methods
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