353 research outputs found
Concurrent Multiscale Computing of Deformation Microstructure by Relaxation and Local Enrichment with Application to Single-Crystal Plasticity
This paper is concerned with the effective modeling of deformation microstructures within a concurrent multiscale computing framework. We present a rigorous formulation of concurrent multiscale computing based on relaxation; we establish the connection between concurrent multiscale computing and enhanced-strain elements; and we illustrate the approach in an important area of application, namely, single-crystal plasticity, for which the explicit relaxation of the problem is derived analytically. This example demonstrates the vast effect of microstructure formation on the macroscopic behavior of the sample, e.g., on the force/travel curve of a rigid indentor. Thus, whereas the unrelaxed model results in an overly stiff response, the relaxed model exhibits a proper limit load, as expected. Our numerical examples additionally illustrate that ad hoc element enhancements, e.g., based on polynomial, trigonometric, or similar representations, are unlikely to result in any significant relaxation in general
Stochastic homogenization of nonconvex unbounded integral functionals with convex growth
We consider the well-travelled problem of homogenization of random integral
functionals. When the integrand has standard growth conditions, the qualitative
theory is well-understood. When it comes to unbounded functionals, that is,
when the domain of the integrand is not the whole space and may depend on the
space-variable, there is no satisfactory theory. In this contribution we
develop a complete qualitative stochastic homogenization theory for nonconvex
unbounded functionals with convex growth. We first prove that if the integrand
is convex and has -growth from below (with , the dimension), then it
admits homogenization regardless of growth conditions from above. This result,
that crucially relies on the existence and sublinearity at infinity of
correctors, is also new in the periodic case. In the case of nonconvex
integrands, we prove that a similar homogenization result holds provided the
nonconvex integrand admits a two-sided estimate by a convex integrand (the
domain of which may depend on the space-variable) that itself admits
homogenization. This result is of interest to the rigorous derivation of rubber
elasticity from polymer physics, which involves the stochastic homogenization
of such unbounded functionals.Comment: 64 pages, 2 figure
Well-Behavior, Well-Posedness and Nonsmooth Analysis
AMS subject classification: 90C30, 90C33.We survey the relationships between well-posedness and well-behavior. The latter
notion means that any critical sequence (xn) of a lower semicontinuous function
f on a Banach space is minimizing. Here âcriticalâ means that the remoteness of
the subdifferential âf(xn) of f at xn (i.e. the distance of 0 to âf(xn)) converges
to 0. The objective function f is not supposed to be convex or smooth and the
subdifferential â is not necessarily the usual Fenchel subdifferential. We are thus
led to deal with conditions ensuring that a growth property of the subdifferential
(or the derivative) of a function implies a growth property of the function itself.
Both qualitative questions and quantitative results are considered
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
Auxiliary problem principles for equilibria
The auxiliary problem principle allows solving a given equilibrium problem (EP) through an equivalent auxiliary problem with better properties. The paper investigates two families of auxiliary EPs: the classical auxiliary problems, in which a regularizing term is added to the equilibrium bifunction, and the regularized Minty EPs. The conditions that ensure the equivalence of a given EP with each of these auxiliary problems are investigated exploiting parametric definitions of different kinds of convexity and monotonicity. This analysis leads to extending some known results for variational inequalities and linear EPs to the general case together with new equivalences. Stationarity and convexity properties of gap functions are investigated as well in this framework. Moreover, both new results on the existence of a unique solution and new error bounds based on gap functions with good convexity properties are obtained under weak quasimonotonicity or weak concavity assumptions
Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane
We complete the complexity classification by degree of minimizing a
polynomial over the integer points in a polyhedron in . Previous
work shows that optimizing a quadratic polynomial over the integer points in a
polyhedral region in can be done in polynomial time, while
optimizing a quartic polynomial in the same type of region is NP-hard. We close
the gap by showing that this problem can be solved in polynomial time for cubic
polynomials.
Furthermore, we show that the problem of minimizing a homogeneous polynomial
of any fixed degree over the integer points in a bounded polyhedron in
is solvable in polynomial time. We show that this holds for
polynomials that can be translated into homogeneous polynomials, even when the
translation vector is unknown. We demonstrate that such problems in the
unbounded case can have smallest optimal solutions of exponential size in the
size of the input, thus requiring a compact representation of solutions for a
general polynomial time algorithm for the unbounded case
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