6,964 research outputs found
Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization
We introduce a new variational method for the numerical homogenization of
divergence form elliptic, parabolic and hyperbolic equations with arbitrary
rough () coefficients. Our method does not rely on concepts of
ergodicity or scale-separation but on compactness properties of the solution
space and a new variational approach to homogenization. The approximation space
is generated by an interpolation basis (over scattered points forming a mesh of
resolution ) minimizing the norm of the source terms; its
(pre-)computation involves minimizing quadratic (cell)
problems on (super-)localized sub-domains of size .
The resulting localized linear systems remain sparse and banded. The resulting
interpolation basis functions are biharmonic for , and polyharmonic
for , for the operator -\diiv(a\nabla \cdot) and can be seen as a
generalization of polyharmonic splines to differential operators with arbitrary
rough coefficients. The accuracy of the method ( in energy norm
and independent from aspect ratios of the mesh formed by the scattered points)
is established via the introduction of a new class of higher-order Poincar\'{e}
inequalities. The method bypasses (pre-)computations on the full domain and
naturally generalizes to time dependent problems, it also provides a natural
solution to the inverse problem of recovering the solution of a divergence form
elliptic equation from a finite number of point measurements.Comment: ESAIM: Mathematical Modelling and Numerical Analysis. Special issue
(2013
Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods
In this paper, we discuss a general multiscale model reduction framework
based on multiscale finite element methods. We give a brief overview of related
multiscale methods. Due to page limitations, the overview focuses on a few
related methods and is not intended to be comprehensive. We present a general
adaptive multiscale model reduction framework, the Generalized Multiscale
Finite Element Method. Besides the method's basic outline, we discuss some
important ingredients needed for the method's success. We also discuss several
applications. The proposed method allows performing local model reduction in
the presence of high contrast and no scale separation
Computational homogenization of fibrous piezoelectric materials
Flexible piezoelectric devices made of polymeric materials are widely used
for micro- and nano-electro-mechanical systems. In particular, numerous recent
applications concern energy harvesting. Due to the importance of computational
modeling to understand the influence that microscale geometry and constitutive
variables exert on the macroscopic behavior, a numerical approach is developed
here for multiscale and multiphysics modeling of thin piezoelectric sheets made
of aligned arrays of polymeric nanofibers, manufactured by electrospinning. At
the microscale, the representative volume element consists in piezoelectric
polymeric nanofibers, assumed to feature a piezoelastic behavior and subjected
to electromechanical contact constraints. The latter are incorporated into the
virtual work equations by formulating suitable electric, mechanical and
coupling potentials and the constraints are enforced by using the penalty
method. From the solution of the micro-scale boundary value problem, a suitable
scale transition procedure leads to identifying the performance of a
macroscopic thin piezoelectric shell element.Comment: 22 pages, 13 figure
Nonlinear nonlocal multicontinua upscaling framework and its applications
In this paper, we discuss multiscale methods for nonlinear problems. The main
idea of these approaches is to use local constraints and solve problems in
oversampled regions for constructing macroscopic equations. These techniques
are intended for problems without scale separation and high contrast, which
often occur in applications. For linear problems, the local solutions with
constraints are used as basis functions. This technique is called Constraint
Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM).
GMsFEM identifies macroscopic quantities based on rigorous analysis. In
corresponding upscaling methods, the multiscale basis functions are selected
such that the degrees of freedom have physical meanings, such as averages of
the solution on each continuum.
This paper extends the linear concepts to nonlinear problems, where the local
problems are nonlinear. The main concept consists of: (1) identifying
macroscopic quantities; (2) constructing appropriate oversampled local problems
with coarse-grid constraints; (3) formulating macroscopic equations. We
consider two types of approaches. In the first approach, the solutions of local
problems are used as basis functions (in a linear fashion) to solve nonlinear
problems. This approach is simple to implement; however, it lacks the nonlinear
interpolation, which we present in our second approach. In this approach, the
local solutions are used as a nonlinear forward map from local averages
(constraints) of the solution in oversampling region. This local fine-grid
solution is further used to formulate the coarse-grid problem. Both approaches
are discussed on several examples and applied to single-phase and two-phase
flow problems, which are challenging because of convection-dominated nature of
the concentration equation
Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems
The paper addresses a numerical method for solving second order elliptic
partial differential equations that describe fields inside heterogeneous media.
The scope is general and treats the case of rough coefficients, i.e.
coefficients with values in . This class of coefficients
includes as examples media with micro-structure as well as media with multiple
non-separated length scales. The approach taken here is based on the the
generalized finite element method (GFEM) introduced in \cite{107}, and
elaborated in \cite{102}, \cite{103} and \cite{104}. The GFEM is constructed by
partitioning the computational domain into to a collection of
preselected subsets and constructing finite dimensional
approximation spaces over each subset using local information. The
notion of the Kolmogorov -width is used to identify the optimal local
approximation spaces. These spaces deliver local approximations with errors
that decay almost exponentially with the degrees of freedom in the
energy norm over . The local spaces are used within the
GFEM scheme to produce a finite dimensional subspace of
which is then employed in the Galerkin method. It is shown that the error in
the Galerkin approximation decays in the energy norm almost exponentially
(i.e., super-algebraicly) with respect to the degrees of freedom . When
length scales "`separate" and the microstructure is sufficiently fine with
respect to the length scale of the domain it is shown that
homogenization theory can be used to construct local approximation spaces with
exponentially decreasing error in the pre-asymtotic regime.Comment: 30 pages, 6 figures, updated references, sections 3 and 4 typos
corrected, minor text revision, results unchange
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