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David Y. Gao

Publications and source records attributed to David Y. Gao.

9 recordsLinked to original sources
arXiv2018

On SDP Method for Solving Canonical Dual Problem in Post Buckling of Large Deformed Elastic Beam

This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

math.OC↗
arXiv2015

Remarks on Analytic Solutions in Nonlinear Elasticity and Anti-Plane Shear Problem

This paper revisits a well-studied anti-plane shear deformation problem formulated by Knowles in 1976 and analytical solutions in general nonlinear elasticity proposed by Gao since 1998. Based on minimum potential principle, a well-determined fully nonlinear system is obtained for isochoric deformation, which admits non-trivial states of finite anti-plane shear without ellipticity constraint. By using canonical duality theory, a complete set of analytical solutions are obtained for 3-D finite deformation problems governed by generalized neo-Hookean model. Both global and local extremal solutions to the nonconvex variational problem are identified by a triality theory. Connection between challenges in nonconvex analysis and NP-hard problems in computational science is revealed. It is proved that the ellipticity condition for general fully nonlinear boundary value problems depends not only on differential operators, but also sensitively on the external force field. The homogenous hyper-elasticity for general anti-plane shear deformation must be governed by the generalized neo-Hookean model. Knowles' over-determined system is simply due to a pseudo-Lagrange multiplier and two extra equilibrium conditions in the plane. The constitutive condition in his theorems is naturally satisfied with $b = λ/2$. His ellipticity condition is neither necessary nor sufficient for general homogeneous materials to admit nontrivial states of anti-plane shear.

math-ph↗
arXiv2013

Canonical duality for solving general nonconvex constrained problems

This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem (without duality gap)can be obtained in a unified form with global optimality conditions provided. While the popular augmented Lagrangian method may produce more difficult nonconvex problems due to the nonlinearity of constraints.

math.OC↗
arXiv2013

Global Solutions to Large-Scale Spherical Constrained Quadratic Minimization via Canonical Dual Approach

This paper presents global optimal solutions to a nonconvex quadratic minimization problem over a sphere constraint. The problem is well-known as a trust region subproblem and has been studied extensively for decades. The main challenge is the so called 'hard case', i.e., the problem has multiple solutions on the boundary of the sphere. By canonical duality theory, this challenging problem is able to reformed as an one-dimensional canonical dual problem without duality gap. Sufficient and necessary conditions are obtained by the triality theory, which can be used to identify whether the problem is hard case or not. A perturbation method and the associated algorithms are proposed to solve this hard case problem. Theoretical results and methods are verified by large-size examples.

math.OC↗
arXiv2013

Post-buckling Solutions of Hyper-elastic Beam by Canonical Dual Finite Element Method

Post buckling problem of a large deformed beam is analyzed using canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy functional is reformulated in a mixed complementary energy form with both displacement and stress fields, and a pure complementary energy is explicitly formulated in finite dimensional space. Based on the canonical duality theory and the associated triality theorem, a primal-dual algorithm is proposed, which can be used to find all possible solutions of this nonconvex post-buckling problem. Numerical results show that the global maximum of the pure-complementary energy leads to a stable buckled configuration of the beam. While the local extrema of the pure-complementary energy present unstable deformation states, especially. We discovered that the unstable buckled state is very sensitive to the number of total elements and the external loads. Theoretical results are verified through numerical examples and some interesting phenomena in post-bifurcation of this large deformed beam are observed.

cs.CE↗
arXiv2012

Canonical Duality Approach for Nonlinear Dynamical Systems

This paper presents a canonical dual approach for solving a nonlinear population growth problem governed by the well-known logistic equation. Using the finite difference and least squares methods, the nonlinear differential equation is first formulated as a nonconvex optimization problem with unknown parameters. We then prove that by the canonical duality theory, this nonconvex problem is equivalent to a concave maximization problem over a convex feasible space, which can be solved easily to obtain global optimal solution to this challenging problem. Several illustrative examples are presented.

nlin.CD↗
arXiv2012

On the Triality Theory in Global Optimization

Triality theory is proved for a general unconstrained global optimization problem. The method adopted is simple but mathematically rigorous. Results show that if the primal problem and its canonical dual have the same dimension, the triality theory holds strongly in the tri-duality form as it was originally proposed. Otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a super-symmetrical form as it was expected. Additionally, a complementary weak saddle min-max duality theorem is discovered. Therefore, an open problem on this statement left in 2003 is solved completely. This theory can be used to identify not only the global minimum, but also the largest local minimum, maximum, and saddle points. Application is illustrated. Some fundamental concepts in optimization and remaining challenging problems in canonical duality theory are discussed.

math.OC↗
arXiv2011

A novel canonical dual computational approach for prion AGAAAAGA amyloid fibril molecular modeling

Many experimental studies have shown that the prion AGAAAAGA palindrome hydrophobic region (113-120) has amyloid fibril forming properties and plays an important role in prion diseases. However, due to the unstable, noncrystalline and insoluble nature of the amyloid fibril, to date structural information on AGAAAAGA region (113-120) has been very limited. This region falls just within the N-terminal unstructured region PrP (1-123) of prion proteins. Traditional X-ray crystallography and nuclear magnetic resonance (NMR) spectroscopy experimental methods cannot be used to get its structural information. Under this background, this paper introduces a novel approach of the canonical dual theory to address the 3D atomic-resolution structure of prion AGAAAAGA amyloid fibrils. The novel and powerful canonical dual computational approach introduced in this paper is for the molecular modeling of prion AGAAAAGA amyloid fibrils, and that the optimal atomic-resolution structures of prion AGAAAAGA amyloid fibils presented in this paper are useful for the drive to find treatments for prion diseases in the field of medicinal chemistry. Overall, this paper presents an important method and provides useful information for treatments of prion diseases. Overall, this paper could be of interest to the general readership of Theoretical Biology.

q-bio.BM↗