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30102 Applied Mathematics
10199 Other Mathematical Sciences
10905 Civil Engineering
1Canonical dual finite element method
1Canonical duality
1Canonical duality theory
1Complex systems
1Global optimizatio
1Multiple solutions
1Non-convex variational problem
1Nonconvex optimization
1Nonconvex variational problem
1Nonlinear analysis
1Nonlinear beam model
1Post buckling
1Regularity
1Semiregularity
1Semitransversality
1Subregularity

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Post-buckling solutions of hyper-elastic beam by canonical dual finite element method

- Cai, Kun, Gao, David, Qin, Qing

**Authors:**Cai, Kun , Gao, David , Qin, Qing**Date:**2014**Type:**Text , Journal article**Relation:**Mathematics and Mechanics of Solids Vol. 19, no. 6 (2014), p. 659-671**Full Text:**false**Reviewed:****Description:**The post-buckling problem of a large deformed beam is analyzed using the 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 non-convex 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. 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.

Canonical duality theory and triality for solving general global optimization problems in complex systems

- Morales-Silva, Daniel, Gao, David

**Authors:**Morales-Silva, Daniel , Gao, David**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics and Mechanics of Complex Systems Vol. 3, no. 2 (2015), p. 139-161**Full Text:****Reviewed:****Description:**General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples. © 2015 Mathematical Sciences Publishers.

**Authors:**Morales-Silva, Daniel , Gao, David**Date:**2015**Type:**Text , Journal article**Relation:**Mathematics and Mechanics of Complex Systems Vol. 3, no. 2 (2015), p. 139-161**Full Text:****Reviewed:****Description:**General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples. © 2015 Mathematical Sciences Publishers.

Geometric and metric characterizations of transversality properties

- Bui, Hoa, Cuong, Nguyen, Kruger, Alexander

**Authors:**Bui, Hoa , Cuong, Nguyen , Kruger, Alexander**Date:**2020**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 48, no. 2 (2020), p. 277-297**Full Text:****Reviewed:****Description:**This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. We clarify quantitative relations between several geometric and metric characterizations of the transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings. We expose all the parameters involved in the definitions and characterizations and establish relations between them. This allows us to classify the quantitative geometric and metric characterizations of transversality and regularity, and subdivide them into two groups with complete exact equivalences between the parameters within each group and clear relations between the values of the parameters in different groups. © 2020, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.

**Authors:**Bui, Hoa , Cuong, Nguyen , Kruger, Alexander**Date:**2020**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 48, no. 2 (2020), p. 277-297**Full Text:****Reviewed:****Description:**This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. We clarify quantitative relations between several geometric and metric characterizations of the transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings. We expose all the parameters involved in the definitions and characterizations and establish relations between them. This allows us to classify the quantitative geometric and metric characterizations of transversality and regularity, and subdivide them into two groups with complete exact equivalences between the parameters within each group and clear relations between the values of the parameters in different groups. © 2020, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.

On the extrema of a nonconvex functional with double-well potential in 1D

**Authors:**Gao, David , Lu, Xioajun**Date:**2016**Type:**Text , Journal article**Relation:**Zeitschrift fur Angewandte Mathematik und Physik Vol. 67, no. 3 (2016), p. 1-7**Full Text:**false**Reviewed:****Description:**This paper mainly investigates the extrema of a nonconvex functional with double-well potential in 1D through the approach of nonlinear differential equations. Based on the canonical duality method, the corresponding Euler–Lagrange equation with Neumann boundary condition can be converted into a cubic dual algebraic equation, which will help find the local extrema for the primal problem. © 2016, Springer International Publishing.

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