778 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
International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book
The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions.
This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more
Optimal control of the sweeping process over polyhedral controlled sets
The paper addresses a new class of optimal control problems governed by the
dissipative and discontinuous differential inclusion of the sweeping/Moreau
process while using controls to determine the best shape of moving convex
polyhedra in order to optimize the given Bolza-type functional, which depends
on control and state variables as well as their velocities. Besides the highly
non-Lipschitzian nature of the unbounded differential inclusion of the
controlled sweeping process, the optimal control problems under consideration
contain intrinsic state constraints of the inequality and equality types. All
of this creates serious challenges for deriving necessary optimality
conditions. We develop here the method of discrete approximations and combine
it with advanced tools of first-order and second-order variational analysis and
generalized differentiation. This approach allows us to establish constructive
necessary optimality conditions for local minimizers of the controlled sweeping
process expressed entirely in terms of the problem data under fairly
unrestrictive assumptions. As a by-product of the developed approach, we prove
the strong -convergence of optimal solutions of discrete
approximations to a given local minimizer of the continuous-time system and
derive necessary optimality conditions for the discrete counterparts. The
established necessary optimality conditions for the sweeping process are
illustrated by several examples
Sublabel-Accurate Relaxation of Nonconvex Energies
We propose a novel spatially continuous framework for convex relaxations
based on functional lifting. Our method can be interpreted as a
sublabel-accurate solution to multilabel problems. We show that previously
proposed functional lifting methods optimize an energy which is linear between
two labels and hence require (often infinitely) many labels for a faithful
approximation. In contrast, the proposed formulation is based on a piecewise
convex approximation and therefore needs far fewer labels. In comparison to
recent MRF-based approaches, our method is formulated in a spatially continuous
setting and shows less grid bias. Moreover, in a local sense, our formulation
is the tightest possible convex relaxation. It is easy to implement and allows
an efficient primal-dual optimization on GPUs. We show the effectiveness of our
approach on several computer vision problems
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