10,268 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
Global Solutions to Nonconvex Optimization of 4th-Order Polynomial and Log-Sum-Exp Functions
This paper presents a canonical dual approach for solving a nonconvex global
optimization problem governed by a sum of fourth-order polynomial and a
log-sum-exp function. Such a problem arises extensively in engineering and
sciences. Based on the canonical duality-triality theory, this nonconvex
problem is transformed to an equivalent dual problem, which can be solved
easily under certain conditions. We proved that both global minimizer and the
biggest local extrema of the primal problem can be obtained analytically from
the canonical dual solutions. As two special cases, a quartic polynomial
minimization and a minimax problem are discussed. Existence conditions are
derived, which can be used to classify easy and relative hard instances.
Applications are illustrated by several nonconvex and nonsmooth examples
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