2,548 research outputs found
Analytical Solutions to General Anti-Plane Shear Problems In Finite Elasticity
This paper presents a pure complementary energy variational method for
solving anti-plane shear problem in finite elasticity. Based on the canonical
duality-triality theory developed by the author, the nonlinear/nonconex partial
differential equation for the large deformation problem is converted into an
algebraic equation in dual space, which can, in principle, be solved to obtain
a complete set of stress solutions. Therefore, a general analytical solution
form of the deformation is obtained subjected to a compatibility condition.
Applications are illustrated by examples with both convex and nonconvex stored
strain energies governed by quadratic-exponential and power-law material
models, respectively. Results show that the nonconvex variational problem could
have multiple solutions at each material point, the complementary gap function
and the triality theory can be used to identify both global and local extremal
solutions, while the popular (poly-, quasi-, and rank-one) convexities provide
only local minimal criteria, the Legendre-Hadamard condition does not guarantee
uniqueness of solutions. This paper demonstrates again that the pure
complementary energy principle and the triality theory play important roles in
finite deformation theory and nonconvex analysis.Comment: 23 pages, 4 figures. Mathematics and Mechanics of Solids, 201
Passing to the limit in maximal slope curves: from a regularized Perona-Malik equation to the total variation flow
We prove that solutions of a mildly regularized Perona-Malik equation
converge, in a slow time scale, to solutions of the total variation flow. The
convergence result is global-in-time, and holds true in any space dimension.
The proof is based on the general principle that "the limit of gradient-flows
is the gradient-flow of the limit". To this end, we exploit a general result
relating the Gamma-limit of a sequence of functionals to the limit of the
corresponding maximal slope curves.Comment: 19 page
Global attractors for gradient flows in metric spaces
We develop the long-time analysis for gradient flow equations in metric
spaces. In particular, we consider two notions of solutions for metric gradient
flows, namely energy and generalized solutions. While the former concept
coincides with the notion of curves of maximal slope, we introduce the latter
to include limits of time-incremental approximations constructed via the
Minimizing Movements approach.
For both notions of solutions we prove the existence of the global attractor.
Since the evolutionary problems we consider may lack uniqueness, we rely on the
theory of generalized semiflows introduced by J.M. Ball. The notions of
generalized and energy solutions are quite flexible and can be used to address
gradient flows in a variety of contexts, ranging from Banach spaces to
Wasserstein spaces of probability measures.
We present applications of our abstract results by proving the existence of
the global attractor for the energy solutions both of abstract doubly nonlinear
evolution equations in reflexive Banach spaces, and of a class of evolution
equations in Wasserstein spaces, as well as for the generalized solutions of
some phase-change evolutions driven by mean curvature
Convergence of the one-dimensional Cahn-Hilliard equation
We consider the Cahn-Hilliard equation in one space dimension with scaling a
small parameter \epsilon and a non-convex potential W. In the limit \espilon
\to 0, under the assumption that the initial data are energetically
well-prepared, we show the convergence to a Stefan problem. The proof is based
on variational methods and exploits the gradient flow structure of the
Cahn-Hilliard equation.Comment: 23 page
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
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