Summary and concluding remarks

The basics of topology optimization and reliability-based design optimization are described in the first chapter. The topology optimization approach is introduced in comparison with the classical size and shape optimization approaches. The well-known Solid Isotropic Material with Penalization (SIMP) model is briefly reviewed. Additionally, the reliability-based design

optimization problem is introduced.

Chapter 2 presents the formulations of the topology optimization for minimum compliance. The formulations of the integration of the stiffness matrix are discussed. Next, the

multiresolution topology optimization approach is proposed by using different discretizations. The MTOP elements are introduced for both 2D (Q4/n25) and 3D (B8/n125) problems, and several other element types such as hexagonal and tetrahedral using MTOP are also discussed. Chapter 3 further develops the MTOP approach in Chapter 2 by reducing the number of design variables in the MTOP approach in Chapter 2. The adaptive multiresolution topology

optimization is introduced in this chapter. Also, a ratio to measure the resolution and efficiency of a model is proposed. The approaches proposed in Chapters 2 and 3 are demonstrated by

numerous numerical examples to show the features of the approaches over the conventional element-based approach.

In Chapter 4, the reliability-based design optimization approaches in the literature are

reviewed. The formulations of the reliability index approach (RIA) and the performance measure approach (MPA) are compared. Also, the double-loop and single-loop approaches are discussed. Next, the component and system reliability-based design optimization problems

(CRBDO/SRBDO) are described. The matrix-based system reliability (MSR) method is briefly reviewed and further developed for integration with the single-loop approach. Finally, the

SRBDO/MSR procedure is proposed and demonstrated with numerical examples and verified by Monte Carlo simulation.

The RBDO approach is applied to topology optimization problem, so-called reliability-based topology optimization (RBTO) in Chapter 5. The single-loop approach is derived using the first- order reliability method (FORM) which is not accurate when the limit-state functions are highly nonlinear. Hence, the SORM-based RBTO formulations are proposed in this chapter to improve the accuracy. The RBTO frame work is integrated with the MTOP approach above to enhance the efficiency. Numerical examples are presented to demonstrate the SORM-based RBTO over the FORM-based approach.

The major contributions of this study are summarized as follows:

• A multiresolution topology optimization (MTOP) approach is proposed based on the discretizations of the density, design variable and displacement fields with distinct resolutions. The MTOP approach is first developed using the same mesh for design variables and density. MTOP elements including Q4/n25, B8/n125 are introduced in comparison to the conventional Q4/U, B8/U elements. The MTOP approach enables us to obtain high-resolution topology design with a relatively low computational cost. The MTOP approach is demonstrated by numerous two- and three-dimensional topology optimization problems. The MTOP approach developed in this study has been successfully applied in designing the optimal shape of bone replacement

structures to improve the current practice of the craniofacial reconstruction (Sutradhar et al., 2010).

• The MTOP approach is further developed by employing fully distinct meshes for density, design variable and displacement. In comparison to the first development of MTOP approach which uses the same mesh for design variables and density, the second development of MTOP approach employs different resolutions for design variable and density fields. Specifically, a relatively fine mesh for element density, a moderately fine mesh for design variables, and a relatively coarse mesh for finite elements are employed. This improvement further reduces the number of design variables in comparison to the original MTOP approach. New iMTOP elements including Q4/n25/d4, B8/n125/d27 are introduced.

• An adaptive multiresolution topology optimization scheme is proposed in which the MTOP or iMTOP elements are used only when and where needed. This scheme allows us to further reduce the number of density elements and the number of design variables.

• “iMTOP ratio” is introduced as a measurement of the resolution and efficiency model. It is based on the ratio of the number of finite elements, the number of density elements, and number of design variables.

• A single-loop system reliability-based design optimization approach using matrix- based system reliability method is introduced. The SRBDO/MSR approach is applicable to general system events including link-set, cut-set systems under dependence between component events.

• The reliability-based topology optimization (RBTO) problems are investigated in both component and system constraint levels. Its efficiency is enhanced by employing the proposed MTOP approach and the single-loop approach.

• To improve the accuracy of the RBTO problem for component and system levels, the second-order reliability method (SORM) is employed to enhance the accuracy of the

probability computation. The SORM-based CRBTO and SRBTO provide more accurate results than the FORM-based approaches.

• MTOP approach is further developed to include the pattern symmetry and repetition constraint to apply for practical design. These constraints are often employed in the concept design of structures such as buildings, bridges.

• The proposed approaches are demonstrated by numerical examples for the structural system of a building core in both DTO and RBTO.

• The accuracy of the developed RBDO and RBTO algorithms are confirmed by Monte Carlo simulations of the failure probabilities of the optimal designs and topologies.