2,705 research outputs found
Average Interpolation Under the Maximum Angle Condition
Interpolation error estimates needed in common finite element applications
using simplicial meshes typically impose restrictions on the both the
smoothness of the interpolated functions and the shape of the simplices. While
the simplest theory can be generalized to admit less smooth functions (e.g.,
functions in H^1(\Omega) rather than H^2(\Omega)) and more general shapes
(e.g., the maximum angle condition rather than the minimum angle condition),
existing theory does not allow these extensions to be performed simultaneously.
By localizing over a well-shaped auxiliary spatial partition, error estimates
are established under minimal function smoothness and mesh regularity. This
construction is especially important in two cases: L^p(\Omega) estimates for
data in W^{1,p}(\Omega) hold for meshes without any restrictions on simplex
shape, and W^{1,p}(\Omega) estimates for data in W^{2,p}(\Omega) hold under a
generalization of the maximum angle condition which previously required p>2 for
standard Lagrange interpolation
Macro-element interpolation on tensor product meshes
A general theory for obtaining anisotropic interpolation error estimates for
macro-element interpolation is developed revealing general construction
principles. We apply this theory to interpolation operators on a macro type of
biquadratic finite elements on rectangle grids which can be viewed as a
rectangular version of the Powell-Sabin element. This theory also shows
how interpolation on the Bogner-Fox-Schmidt finite element space (or higher
order generalizations) can be analyzed in a unified framework. Moreover we
discuss a modification of Scott-Zhang type giving optimal error estimates under
the regularity required without imposing quasi uniformity on the family of
macro-element meshes used. We introduce and analyze an anisotropic
macro-element interpolation operator, which is the tensor product of
one-dimensional macro interpolation and Lagrange interpolation.
These results are used to approximate the solution of a singularly perturbed
reaction-diffusion problem on a Shishkin mesh that features highly anisotropic
elements. Hereby we obtain an approximation whose normal derivative is
continuous along certain edges of the mesh, enabling a more sophisticated
analysis of a continuous interior penalty method in another paper
Superconvergence Using Pointwise Interpolation in Convection-Diffusion Problems
Considering a singularly perturbed convection-diffusion problem, we present
an analysis for a superconvergence result using pointwise interpolation of
Gau{\ss}-Lobatto type for higher-order streamline diffusion FEM.
We show a useful connection between two different types of interpolation,
namely a vertex-edge-cell interpolant and a pointwise interpolant. Moreover,
different postprocessing operators are analysed and applied to model problems.Comment: 19 page
Galerkin projection of discrete fields via supermesh construction
Interpolation of discrete FIelds arises frequently in computational physics.
This thesis focuses on the novel implementation and analysis of Galerkin
projection, an interpolation technique with three principal advantages over
its competitors: it is optimally accurate in the L2 norm, it is conservative,
and it is well-defined in the case of spaces of discontinuous functions.
While these desirable properties have been known for some time, the implementation
of Galerkin projection is challenging; this thesis reports the first
successful general implementation.
A thorough review of the history, development and current frontiers of
adaptive remeshing is given. Adaptive remeshing is the primary motivation
for the development of Galerkin projection, as its use necessitates the interpolation
of discrete fields. The Galerkin projection is discussed and the
geometric concept necessary for its implementation, the supermesh, is introduced.
The efficient local construction of the supermesh of two meshes
by the intersection of the elements of the input meshes is then described.
Next, the element-element association problem of identifying which elements
from the input meshes intersect is analysed. With efficient algorithms for
its construction in hand, applications of supermeshing other than Galerkin
projections are discussed, focusing on the computation of diagnostics of simulations
which employ adaptive remeshing. Examples demonstrating the effectiveness
and efficiency of the presented algorithms are given throughout.
The thesis closes with some conclusions and possibilities for future work
Spectral tensor-train decomposition
The accurate approximation of high-dimensional functions is an essential task
in uncertainty quantification and many other fields. We propose a new function
approximation scheme based on a spectral extension of the tensor-train (TT)
decomposition. We first define a functional version of the TT decomposition and
analyze its properties. We obtain results on the convergence of the
decomposition, revealing links between the regularity of the function, the
dimension of the input space, and the TT ranks. We also show that the
regularity of the target function is preserved by the univariate functions
(i.e., the "cores") comprising the functional TT decomposition. This result
motivates an approximation scheme employing polynomial approximations of the
cores. For functions with appropriate regularity, the resulting
\textit{spectral tensor-train decomposition} combines the favorable
dimension-scaling of the TT decomposition with the spectral convergence rate of
polynomial approximations, yielding efficient and accurate surrogates for
high-dimensional functions. To construct these decompositions, we use the
sampling algorithm \texttt{TT-DMRG-cross} to obtain the TT decomposition of
tensors resulting from suitable discretizations of the target function. We
assess the performance of the method on a range of numerical examples: a
modifed set of Genz functions with dimension up to , and functions with
mixed Fourier modes or with local features. We observe significant improvements
in performance over an anisotropic adaptive Smolyak approach. The method is
also used to approximate the solution of an elliptic PDE with random input
data. The open source software and examples presented in this work are
available online.Comment: 33 pages, 19 figure
Topology optimization of multiple anisotropic materials, with application to self-assembling diblock copolymers
We propose a solution strategy for a multimaterial minimum compliance
topology optimization problem, which consists in finding the optimal allocation
of a finite number of candidate (possibly anisotropic) materials inside a
reference domain, with the aim of maximizing the stiffness of the body. As a
relevant and novel application we consider the optimization of self-assembled
structures obtained by means of diblock copolymers. Such polymers are a class
of self-assembling materials which spontaneously synthesize periodic
microstructures at the nanoscale, whose anisotropic features can be exploited
to build structures with optimal elastic response, resembling biological
tissues exhibiting microstructures, such as bones and wood. For this purpose we
present a new generalization of the classical Optimality Criteria algorithm to
encompass a wider class of problems, where multiple candidate materials are
considered, the orientation of the anisotropic materials is optimized, and the
elastic properties of the materials are assumed to depend on a scalar
parameter, which is optimized simultaneously to the material allocation and
orientation. Well-posedness of the optimization problem and well-definition of
the presented algorithm are narrowly treated and proved. The capabilities of
the proposed method are assessed through several numerical tests
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