20101 Pure Mathematics
2Canonical duality theory
10102 Applied Mathematics
11502 Banking, Finance and Investment
13D topology
1Buckling
1Canonical dual finite element method
1Canonical penalty-duality
1Elasticity theory
1Finite element method
1Finite element methods
1Global optimization
1Interpolation
1Large deformation
1Nonconvex functional
1Nonconvex mechanics
1Nonlinear Gao beam
1Numerical solutions
1Optimisation problem
1Post buckling

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Canonical dual finite element method for solving nonconvex mechanics and topology optimization

**Authors:**Ali, Elaf**Date:**2018**Type:**Text , Thesis , PhD**Full Text:****Description:**Canonical duality theory (CDT) is a newly developed, potentially powerful methodological theory which can transfer general multi-scale nonconvex/discrete problems in Rn to a unified convex dual problem in continuous space Rm with m n and without a duality gap. The associated triality theory provides extremality criteria for both global and local optimal solutions, which can be used to develop powerful algorithms for solving general nonconvex variational problems. This thesis, first, presents a detailed study of large deformation problems in 2-D structural system. Based on the canonical duality theory, a canonical dual finite element method is applied to find a global minimization to the general nonconvex optimization problem using a new primal-dual semi-definite programming algorithm. Applications are illustrated by numerical examples with different structural designs and different external loads. Next, a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam is investigated. 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 the canonical dual finite element method, a new primal-dual semi-definite programming algorithm is presented, which can be used to obtain all possible post-buckled solutions. In order to verify the triality theory, mixed meshes of different dual stress interpolation are applied to obtain the closed dimensions between discretized displacement and discretized stress. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential 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 the external load, thickness of the beam, numerical precision, and the size of finite elements. Finally, a mathematically rigorous and computationally powerful method for solving 3-D topology optimization problems is demonstrated. This method is based on CDT developed by Gao in nonconvex mechanics and global optimization. It shows that the so-called NP-hard Knapsack problem in topology optimization can be solved deterministically in polynomial-time via a canonical penalty-duality (CPD) method to obtain precise global optimal 0-1 density distribution at each volume evolution. The relation between this CPD method and Gao's pure complementary energy principle is revealed for the first time. A CPD algorithm is proposed for 3-D topology optimization of linear elastic structures. Its novelty is demonstrated by benchmark problems. Results show that without using any artificial technique, the CPD method can provide mechanically sound optimal design, also it is much more powerful than the well-known BESO and SIMP methods. Finally, computational complexity and conceptual/mathematical mistakes in topology optimization modeling and popular methods are explicitly addressed.**Description:**Doctor of Philosophy**Description:**Canonical duality theory (CDT) is a newly developed, potentially powerful method- ological theory which can transfer general multi-scale nonconvex/discrete problems in Rn to a unified convex dual problem in continuous space Rm with m

**Authors:**Ali, Elaf**Date:**2018**Type:**Text , Thesis , PhD**Full Text:****Description:**Canonical duality theory (CDT) is a newly developed, potentially powerful methodological theory which can transfer general multi-scale nonconvex/discrete problems in Rn to a unified convex dual problem in continuous space Rm with m n and without a duality gap. The associated triality theory provides extremality criteria for both global and local optimal solutions, which can be used to develop powerful algorithms for solving general nonconvex variational problems. This thesis, first, presents a detailed study of large deformation problems in 2-D structural system. Based on the canonical duality theory, a canonical dual finite element method is applied to find a global minimization to the general nonconvex optimization problem using a new primal-dual semi-definite programming algorithm. Applications are illustrated by numerical examples with different structural designs and different external loads. Next, a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam is investigated. 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 the canonical dual finite element method, a new primal-dual semi-definite programming algorithm is presented, which can be used to obtain all possible post-buckled solutions. In order to verify the triality theory, mixed meshes of different dual stress interpolation are applied to obtain the closed dimensions between discretized displacement and discretized stress. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential 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 the external load, thickness of the beam, numerical precision, and the size of finite elements. Finally, a mathematically rigorous and computationally powerful method for solving 3-D topology optimization problems is demonstrated. This method is based on CDT developed by Gao in nonconvex mechanics and global optimization. It shows that the so-called NP-hard Knapsack problem in topology optimization can be solved deterministically in polynomial-time via a canonical penalty-duality (CPD) method to obtain precise global optimal 0-1 density distribution at each volume evolution. The relation between this CPD method and Gao's pure complementary energy principle is revealed for the first time. A CPD algorithm is proposed for 3-D topology optimization of linear elastic structures. Its novelty is demonstrated by benchmark problems. Results show that without using any artificial technique, the CPD method can provide mechanically sound optimal design, also it is much more powerful than the well-known BESO and SIMP methods. Finally, computational complexity and conceptual/mathematical mistakes in topology optimization modeling and popular methods are explicitly addressed.**Description:**Doctor of Philosophy**Description:**Canonical duality theory (CDT) is a newly developed, potentially powerful method- ological theory which can transfer general multi-scale nonconvex/discrete problems in Rn to a unified convex dual problem in continuous space Rm with m

**Authors:**Ali, Elaf**Date:**2020**Type:**Text , Journal article , Article**Relation:**Bulletin of the Australian Mathematical Society Vol. 101, no. 1 (Feb 2020), p. 172-173**Full Text:**false**Reviewed:**

Canonical finite element method for solving nonconvex variational problems to post buckling beam problem

**Authors:**Ali, Elaf , Gao, David**Date:**2016**Type:**Text , Conference proceedings**Relation:**2nd International Conference on Numerical Computations : Theory and Algorithms, NUMTA 2016; Pizzo Calabro, Italy; 19th-25th June 2016; published in AIP Proceedings of the 2nd International Conference "Numerical Computations: Theory and Algorithms Vol. 1776, p. 1-4**Full Text:****Reviewed:****Description:**The goal of this paper is to solve the post buckling phenomena of a large deformed elastic beam by a canonical dual mixed finite element method (CD-FEM). The total potential energy of this beam is a nonconvex functional which can be used to model both pre-and post-buckling problems. Different types of dual stress interpolations are used in order to verify the triality theory. Applications are illustrated with different boundary conditions and external loads by using semi-definite programming (SDP) algorithm. The results show that the global minimum of the total potential energy is stable buckled configuration, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. While the local minimum is unstable buckled configuration and very sensitive to both stress interpolations and the external loads.

**Authors:**Ali, Elaf , Gao, David**Date:**2016**Type:**Text , Conference proceedings**Relation:**2nd International Conference on Numerical Computations : Theory and Algorithms, NUMTA 2016; Pizzo Calabro, Italy; 19th-25th June 2016; published in AIP Proceedings of the 2nd International Conference "Numerical Computations: Theory and Algorithms Vol. 1776, p. 1-4**Full Text:****Reviewed:****Description:**The goal of this paper is to solve the post buckling phenomena of a large deformed elastic beam by a canonical dual mixed finite element method (CD-FEM). The total potential energy of this beam is a nonconvex functional which can be used to model both pre-and post-buckling problems. Different types of dual stress interpolations are used in order to verify the triality theory. Applications are illustrated with different boundary conditions and external loads by using semi-definite programming (SDP) algorithm. The results show that the global minimum of the total potential energy is stable buckled configuration, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. While the local minimum is unstable buckled configuration and very sensitive to both stress interpolations and the external loads.

On SPD method for solving canonical dual problem in post buckling of large deformed elastic beam

**Authors:**Ali, Elaf , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Communications in Mathematical Sciences Vol. 16, no. 5 (2018), p. 1225-1240**Full Text:****Reviewed:****Description:**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.

**Authors:**Ali, Elaf , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Communications in Mathematical Sciences Vol. 16, no. 5 (2018), p. 1225-1240**Full Text:****Reviewed:****Description:**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.

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