2,200 research outputs found
Homogenization of cohesive fracture in masonry structures
We derive a homogenized mechanical model of a masonry-type structure
constituted by a periodic assemblage of blocks with interposed mortar joints.
The energy functionals in the model under investigation consist in (i) a linear
elastic contribution within the blocks, (ii) a Barenblatt's cohesive
contribution at contact surfaces between blocks and (iii) a suitable unilateral
condition on the strain across contact surfaces, and are governed by a small
parameter representing the typical ratio between the length of the blocks and
the dimension of the structure. Using the terminology of Gamma-convergence and
within the functional setting supplied by the functions of bounded deformation,
we analyze the asymptotic behavior of such energy functionals when the
parameter tends to zero, and derive a simple homogenization formula for the
limit energy. Furthermore, we highlight the main mathematical and mechanical
properties of the homogenized energy, including its non-standard growth
conditions under tension or compression. The key point in the limit process is
the definition of macroscopic tensile and compressive stresses, which are
determined by the unilateral conditions on contact surfaces and the geometry of
the blocks
Quantitative Homogenization of Elliptic PDE with Random Oscillatory Boundary Data
We study the averaging behavior of nonlinear uniformly elliptic partial
differential equations with random Dirichlet or Neumann boundary data
oscillating on a small scale. Under conditions on the operator, the data and
the random media leading to concentration of measure, we prove an almost sure
and local uniform homogenization result with a rate of convergence in
probability
A Tensor Analogy of Yuan's Theorem of the Alternative and Polynomial Optimization with Sign structure
Yuan's theorem of the alternative is an important theoretical tool in
optimization, which provides a checkable certificate for the infeasibility of a
strict inequality system involving two homogeneous quadratic functions. In this
paper, we provide a tractable extension of Yuan's theorem of the alternative to
the symmetric tensor setting. As an application, we establish that the optimal
value of a class of nonconvex polynomial optimization problems with suitable
sign structure (or more explicitly, with essentially non-positive coefficients)
can be computed by a related convex conic programming problem, and the optimal
solution of these nonconvex polynomial optimization problems can be recovered
from the corresponding solution of the convex conic programming problem.
Moreover, we obtain that this class of nonconvex polynomial optimization
problems enjoy exact sum-of-squares relaxation, and so, can be solved via a
single semidefinite programming problem.Comment: acceted by Journal of Optimization Theory and its application, UNSW
preprint, 22 page
The Hessian Riemannian flow and Newton's method for Effective Hamiltonians and Mather measures
Effective Hamiltonians arise in several problems, including homogenization of
Hamilton--Jacobi equations, nonlinear control systems, Hamiltonian dynamics,
and Aubry--Mather theory. In Aubry--Mather theory, related objects, Mather
measures, are also of great importance. Here, we combine ideas from mean-field
games with the Hessian Riemannian flow to compute effective Hamiltonians and
Mather measures simultaneously. We prove the convergence of the Hessian
Riemannian flow in the continuous setting. For the discrete case, we give both
the existence and the convergence of the Hessian Riemannian flow. In addition,
we explore a variant of Newton's method that greatly improves the performance
of the Hessian Riemannian flow. In our numerical experiments, we see that our
algorithms preserve the non-negativity of Mather measures and are more stable
than {related} methods in problems that are close to singular. Furthermore, our
method also provides a way to approximate stationary MFGs.Comment: 24 page
Convergence Rates in L^2 for Elliptic Homogenization Problems
We study rates of convergence of solutions in L^2 and H^{1/2} for a family of
elliptic systems {L_\epsilon} with rapidly oscillating oscillating coefficients
in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a
consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov
eigenvalues of {L_\epsilon}. Most of our results, which rely on the recently
established uniform estimates for the L^2 Dirichlet and Neumann problems in
\cite{12,13}, are new even for smooth domains.Comment: 25 page
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