1,414 research outputs found
Complete damage in linear elastic materials - Modeling, weak formulation and existence results
In this work, we introduce a degenerating PDE system with a time-depending
domain for complete damage processes under time-varying Dirichlet boundary
conditions. The evolution of the system is described by a doubly nonlinear
differential inclusion for the damage process and a quasi-static balance
equation for the displacement field which are strongly nonlinearly coupled. In
our proposed model, the material may completely disintegrate which is
indispensable for a realistic modeling of damage processes in elastic
materials. Complete damage theories lead to several mathematical problems since
for instance coercivity properties of the free energy are lost and, therefore,
several difficulties arise. For the introduced complete damage model, we
propose a classical formulation and a corresponding suitable weak formulation
in an -framework. The main aim is to prove existence of weak solutions for
the introduced degenerating model. In addition, we show that the classical
differential inclusion can be regained from the notion of weak solutions under
certain regularity assumptions which is a novelty in the theory of complete
damage models of this type
One-dimensional parametric determining form for the two-dimensional Navier-Stokes equations
The evolution of a determining form for the 2D Navier-Stokes equations (NSE),
which is an ODE on a space of trajectories is completely described. It is
proved that at every stage of its evolution, the solution is a convex
combination of the initial trajectory and the fixed steady state, with a
dynamical convexity parameter , which will be called the characteristic
determining parameter. That is, we show a remarkable separation of variables
formula for the solution of the determining form. Moreover, for a given initial
trajectory, the dynamics of the infinite-dimensional determining form are
equivalent to those of the characteristic determining parameter which
is governed by a one-dimensional ODE. %for the parameter specifying the
position on the line segment. This one-dimensional ODE is used to show that if
the solution to the determining form converges to the fixed state it does so no
faster than , otherwise it converges to a projection
of some other trajectory in the global attractor of the NSE, but no faster than
, as , where is the
evolutionary variable in determining form. The one-dimensional ODE also
exploited in computations which suggest that the one-sided convergence rate
estimates are in fact achieved. The ODE is then modified to accelerate the
convergence to an exponential rate. Remarkably, it is shown that the zeros of
the scalar function that governs the dynamics of , which are called
characteristic determining values, identify in a unique fashion the
trajectories in the global attractor of the 2D NSE. Furthermore, the
one-dimensional characteristic determining form enables us to find
unanticipated geometric features of the global attractor, a subject of future
research
Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods
The convex feasibility problem (CFP) is at the core of the modeling of many
problems in various areas of science. Subgradient projection methods are
important tools for solving the CFP because they enable the use of subgradient
calculations instead of orthogonal projections onto the individual sets of the
problem. Working in a real Hilbert space, we show that the sequential
subgradient projection method is perturbation resilient. By this we mean that
under appropriate conditions the sequence generated by the method converges
weakly, and sometimes also strongly, to a point in the intersection of the
given subsets of the feasibility problem, despite certain perturbations which
are allowed in each iterative step. Unlike previous works on solving the convex
feasibility problem, the involved functions, which induce the feasibility
problem's subsets, need not be convex. Instead, we allow them to belong to a
wider and richer class of functions satisfying a weaker condition, that we call
"zero-convexity". This class, which is introduced and discussed here, holds a
promise to solve optimization problems in various areas, especially in
non-smooth and non-convex optimization. The relevance of this study to
approximate minimization and to the recent superiorization methodology for
constrained optimization is explained.Comment: Mathematical Programming Series A, accepted for publicatio
Numerical analysis of parabolic p-Laplacian: Approximation of trajectories
The long time numerical approximation of the parabolic p-Laplacian problem with a time-independent forcing term and sufficiently smooth initial data is studied. Convergence and stability results which are uniform for t is an element of [0, infinity) are established in the L-2, W-1,W-p norms for the backward Euler and the Crank-Nicholson schemes with the finite element method (FEM). This result extends the existing uniform convergence results for exponentially contractive semigroups generated by some semilinear systems to nonexponentially contractive semigroups generated by some quasilinear systems
- ā¦