Analysis And Design Of Bi-modulus Structures And Investigation On Symmetry Problems In Structural Optimization

Along with the development of computational methods in engineering science and equipment, e.g., mathematical programming, finite element method, computer hardware, etc. structural optimization technology has taken an important part in the innovative design platform. In order to simplify theoretical analysis and save numerical cost, symmetry reduction technique is usually adopted for symmetric structural optimization problems, which can only obtain symmetric designs. Recently, it is shown that the global optimal solution of a symmetric structural optimization problem can be highly asymmetric, which implies that improper symmetry reduction would lead to the loss of global optima. Then there arises a fundamental question "Under what cases, there must be a symmetric global optimal solution for a symmetric structural optimization problem?" On the other hand, there are ubiquitous materials and structures with asymmetric properties under tension and compression in nature and engineering, for instance, concrete, rubber, alloy, nacre, cable membrane structure, etc. And bi-modulus constitutive relation can well simulate the mechanical responses of some aforementioned materials and structures. Due to the difficulties induced by the non-smooth property of the constitutive relation, there’s no consistent variational principles and bounding theorems, mature computational framework and topology optimization techniques for bi-modulus materials, which severely restrict the applications of bi-modulus elastic theory in engineering. Besides, investigation on the fundamental theory and computational framework for non-smooth materials is also of theoretical significance.Based on above analysis, through convex analysis (corresponding basic knowledges are presented in Chapter 2), this thesis devotes to investigate both the symmetry problem of structural optimization problems and the variational principles, bounding theorem, efficient computational framework and topology optimization problems of bi-modulus materials. And the main contents and results can be summarized as follows:Studies on symmetry property of global optimal solution of structural optimization problems. With use of group theory, the mathematical description of symmetric deterministic structural optimization problems is provided firstly, then the symmetry theorem is established and demonstrated, which reveals that quasi-convexity and continuity are the sufficient conditions to guarantee the symmetry of global optimal solution, and also two counter examples are presented to show asymmetric solutions may exist for structural optimization problems which do not possess those two conditions. After that, symmetry theorem is generalized to multi-objective and multi-load cases. For structural optimization considering uncertainties, both non-probabilistic robust optimization and probabilistic reliability based design optimization are investigated, respectively. Alternative types of symmetry are defined for both the two kinds of structural optimization problems, and the corresponding symmetry theorems are also demonstrated. Those results generalize the former realization about symmetry property of global optimal solutions, and also present theoretical foundation for symmetry reduction in structural optimization problems.Unified variational principles and bounding theorems for bi-modulus materials. Both the partitioned constitutive relations and strain energy densities are embedded in some semi-definite programming with introducing semi-definite variables. Then the minimum potential/minimum complementary energy principles, Hellinger-Reissner variational principles, Hu-Washizu variational principles and Hashin-Shtrikman variational principles are established. By choosing appropriate trail fields, effective bounds of composites containing bi-modulus components are obtained with use of those variational principles, and the attainability of theoretical bounds is also discussed. It is illustrated that the bounds of effective property would be related to the boundary conditions, through the concrete expressions of Voigt-Reuss bounds of the effective stiffness matrices of the bi-modulus composites, because of the non-smooth property of the bi-modulus constitutive relations. According to those results, traditional variational principles and bounding theorems for linear case are extended to non-smooth bi-linear case.Efficient computational framework for analysis of bi-modulus structures and its applications. Under the assumption of small deformation, both the linearity and non-linearity of bi-modulus elastic problems are demonstrated with use of the unified variational principles at first. Through analysis of Ambartsumyan constitutive model, the underlying non-convergence reason is pointed out, and the tangent constitutive matrix and complemented constitutive matrix are derived in order to establish alternative Newton-Raphson algorithms which have asymptotically quadratic rate of convergence. After that, the algorithms are also generalized to 3D case and finite deformation with small deformation and large rotation case. To promote the engineering application of those algorithms, Umat codes are also developed for ABAQUS. Besides, the successful applications of approximate prediction of the wrinkled region of tensile membrane and explanation in unusual phenomena of cell mechanosensing process, show the potential value of bi-modulus elastic theory.Studies on topology optimization involving bi-modulus materials. The sensitivity results of minimum compliance design of bi-modulus structure are derived strictly in SIMP approach at first. With use of the developed computational algorithms, the solution procedure for solving topology optimization of bi-modulus structures are presented. Taking the bi-material case for example, topology optimization of multi-phase bi-modulus materials is also solved efficiently. Numerical results show that, the structure under pure tension or pure compression states and strut-and-tie design can be obtained as the results of minimum compliance design of bi-modulus structures. Finally, the symmetry property of global optimal solution of bi-modulus truss is investigated. It is found that the symmetry results established for linear elastic truss also apply to bi-modulus truss structure, and the anti-symmetric property (i.e., structural optimization problem with anti-symmetric external load, symmetric ground structure, mass distribution, displacement constraints, etc, possesses symmetric global optima), however, is not valid yet due to the asymmetry of bi-modulus constitutive relation.