287 research outputs found
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
Contraction and optimality properties of an adaptive Legendre-Galerkin method: the multi-dimensional case
We analyze the theoretical properties of an adaptive Legendre-Galerkin method
in the multidimensional case. After the recent investigations for
Fourier-Galerkin methods in a periodic box and for Legendre-Galerkin methods in
the one dimensional setting, the present study represents a further step
towards a mathematically rigorous understanding of adaptive spectral/
discretizations of elliptic boundary-value problems. The main contribution of
the paper is a careful construction of a multidimensional Riesz basis in ,
based on a quasi-orthonormalization procedure. This allows us to design an
adaptive algorithm, to prove its convergence by a contraction argument, and to
discuss its optimality properties (in the sense of non-linear approximation
theory) in certain sparsity classes of Gevrey type
Optimal control in ink-jet printing via instantaneous control
This paper concerns the optimal control of a free surface flow with moving
contact line, inspired by an application in ink-jet printing. Surface tension,
contact angle and wall friction are taken into account by means of the
generalized Navier boundary condition. The time-dependent differential system
is discretized by an arbitrary Lagrangian-Eulerian finite element method, and a
control problem is addressed by an instantaneous control approach, based on the
time discretization of the flow equations. The resulting control procedure is
computationally highly efficient and its assessment by numerical tests show its
effectiveness in deadening the natural oscillations that occur inside the
nozzle and reducing significantly the duration of the transient preceding the
attainment of the equilibrium configuration
On the decay of the inverse of matrices that are sum of Kronecker products
Decay patterns of matrix inverses have recently attracted considerable
interest, due to their relevance in numerical analysis, and in applications
requiring matrix function approximations. In this paper we analyze the decay
pattern of the inverse of banded matrices in the form where is tridiagonal, symmetric and positive definite, is
the identity matrix, and stands for the Kronecker product. It is well
known that the inverses of banded matrices exhibit an exponential decay pattern
away from the main diagonal. However, the entries in show a
non-monotonic decay, which is not caught by classical bounds. By using an
alternative expression for , we derive computable upper bounds that
closely capture the actual behavior of its entries. We also show that similar
estimates can be obtained when has a larger bandwidth, or when the sum of
Kronecker products involves two different matrices. Numerical experiments
illustrating the new bounds are also reported
Adaptive Fourier-Galerkin Methods
We study the performance of adaptive Fourier-Galerkin methods in a periodic
box in with dimension . These methods offer unlimited
approximation power only restricted by solution and data regularity. They are
of intrinsic interest but are also a first step towards understanding
adaptivity for the -FEM. We examine two nonlinear approximation classes,
one classical corresponding to algebraic decay of Fourier coefficients and
another associated with exponential decay. We study the sparsity classes of the
residual and show that they are the same as the solution for the algebraic
class but not for the exponential one. This possible sparsity degradation for
the exponential class can be compensated with coarsening, which we discuss in
detail. We present several adaptive Fourier algorithms, and prove their
contraction and optimal cardinality properties.Comment: 48 page
A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems
We propose and analyze a two-level method for mimetic finite difference
approximations of second order elliptic boundary value problems. We prove that
the two-level algorithm is uniformly convergent, i.e., the number of iterations
needed to achieve convergence is uniformly bounded independently of the
characteristic size of the underling partition. We also show that the resulting
scheme provides a uniform preconditioner with respect to the number of degrees
of freedom. Numerical results that validate the theory are also presented
An optimal adaptive Fictitious Domain Method
We consider a Fictitious Domain formulation of an elliptic partial
differential equation and approximate the resulting saddle-point system using
an inexact preconditioned Uzawa iterative algorithm. Each iteration entails the
approximation of an elliptic problems performed using adaptive finite element
methods. We prove that the overall method converges with the best possible rate
and illustrate numerically our theoretical findings
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