Continuum Structural Topology Optimization Approach Based On Bone Remodeling Theory

Due to the limitation of the available resources, the efficiency of using the resources has to be improved for the sustainable development. Structural optimization is an effective method to design the best structural performances. With the growth of computer power, the structural optimization methods have been developed greatly. In the last several decades, structural optimization is used widely in engineering fields, such as architectural engineering, mechanical engineering, chemical engineering, aerospace and spaceflight engineering. In structural optimization field, topology optimization is a hot topic and several approaches has been developed, which has become an effective approach in the initial or concept design phase to shorten the design time for a product. For topology optimization of a complicated structure with large sum of design variables, it is difficult to solve the problem efficiently by any method. With the consideration of such aspects, a new bionics approach is proposed in this research based on bone remodeling theory and the fabric tensor theory, whose major idea is to consider a structure as a piece of 'bone'. The optimization for structure is equivalent to the 'bone' remodeling following Wolff's law which states bone microstructure and local stiffness tend to align with the stress principal directions to adapt to the mechanical environments. As the 'bone' reaches the equilibrium state of remodeling, the final optimal structure is obtained. The thesis is orgnized as follows:In chapter 1, a review on continuum structural topology optimization and the bone remodeling theories is provided, which is composed of the history and advance of structural optimization, some major approaches for continuum structural topology optimization, and the methods of updating the design variables. The bionics optimization approaches and some bone remodeling theories are introduced particularly.In chapter 2, the fabric tensor theory for describing the microstructure of heterogeneous material is introduced, which consist of the measurement of the fabric tensor of a heterogeneous material (i.e. the Mean Intercept Length (MIL) approach), the micro-finite element model of three-dimensional porous media, and the theories on the relation between the fabric tensor and the elastic constitutive model of porous media.In chapter 3, the fixed reference strain interval approach to solve continuum structural topology optimization is developed, in which fabric tensor is introduced as the design variables of porous media in design domain. To obtain the relative density of a porous material point having an imposed fabric tensor-elastic tensor relation, Stiffness Equivalent Condensation (SEC) method is proposed, by which the relative density is expressed as the function of the invariables of its fabric tensor. The update rule of the design variables, called as growth law, is established as: 1) during the iteration process of the optimization of a structure, the eigenvectors of the stress tensor at any material point in the present step are those of the fabric tensor used in the next step based on Wolff's law. 2) the increments of the eigenvalues of the fabric tensor are dependable with the principal strains and the interval of reference strain corresponding to the dead zone in bone mechanics. When the eigenpairs of the fabric tensors need to update, the process is called as anisotropic growth. And when all the fabric tensors are proportional to the second order identity tensor in the simulation, the process is called as isotropic growth. Numerical examples demonstrate the significance of the algorithm parameters, such as the interval of reference strain, increments of the eigenvalues of the fabric tensors and initial material distribution parameter.In chapter 4, several numerical examples are provided to illustrate the applicability of the proposed bionics approach, like the isotropic and anisotropic growthes of two-dimensional structures, the isotropic growth of the three-dimensional structures and plates. As an application in biomechanics, the density distribution of proximal femur is predicted. Compared with Stanford models in the isotropic and anisotropic cases, the proposed method is validated.In chapter 5, the floating reference strain interval approach extends the fixed reference strain interval approach to solve structural topology optimization problems with the constraints, like structural volume constraint, displacement constraint(s) and/or stress constraint. In the floating reference strain interval approach, the fabric tensor is changed, when anyone of the absolute values of the principal strains at a material point is out of the current reference interval. In optimization, the interval of reference strain is changed and the update rule of the interval depends on the active constaint of an optimization problem. Numerical results demonstrate the validity of the method developed.In chapter 6, the double floating reference strain intervals approach is proposed to optimize structures with Different Tension and Compression Properties (DTCPs), in which there are two kinds of differences: 1) the difference on the tensile and compressive properties of material, such as the material can only resist tension (e.g. cable, membrane) or only resist compression (e.g. brick wall), or the compressive and tensile elasticity of the material are not identical. 2) two different floating reference strain intervals in tensile and compressive states to control the growth of material point are introduced in the present method. To solve the topology optimization of the strucutures with DTCPs with traditional approach, the efficiency of computation is too low. To overcome the difficulty, the structure with DTCPs is replaced with the structure with isotropic porous media and the growth of the porous media is controlled by two different intervals of reference strain in terms of the local stress strate. The equivalence of these two differences is verified with the theoretical analysis and in numerical simulations, which reveals that the different TCPs lead to different topologies even if the same objective and the character constraint functions are used in the same initial configuration. The more practical results can be obtained if the differences as mentioned above are considered in structural optimization.In chapter 7, the floating interval of reference strain energy density (SED) approach is proposed, which provide a floating interval of SED to control the update of design variables and whose validity is verified with some numerical caculations. The mesh sensitivity of the plate subjected to uniform pressure is studied, which is useful to guide the practical engineering with this method.Finally, the main contributions of the dissertation are concluded and some further aspects of the work are previewed.