Configurational optimization for optimal topological and fracture-resistant designs of solids
The objective of this dissertation is to develop a novel topological design approach that enables insertion of finite-sized heterogeneities into homogeneous solid domain based on arbitrary design objective to improve both load-carrying capacity and fracture resistance of structures. To this end, a configurational optimization problem for determining the optimal location, orientation, and shape of a finite-sized heterogeneity inserted into a homogeneous solid subdomain is first proposed. Then, the material derivative (termed the configurational derivative) of arbitrary design objectives to arbitrary design modifications of the subdomain is derived and is later simplified to obtain the sensitivities of the objective corresponding to translation, rotation, and scaling of the heterogeneity. In addition, the equivalent path-independent integral forms of the translation, rotation, and scaling sensitivities are derived based on the notion of conservation rules in solids. Next, the configurational derivative is specialized for insertion of an infinitesimal heterogeneity. That is, the change in an arbitrary objective corresponding to the insertion of any infinitesimal heterogeneity (i.e., the topological derivative) is obtained. This sensitivity is shown to correspond to a self-similar expansion of the heterogeneity with respect to its centroid, and may be thought of as the generalized topological derivative for an arbitrary-shaped heterogeneity. For the two objectives that are of relevance to structures, namely the structural compliance and the total potential energy, the expansion sensitivities for an ellipsoidal heterogeneity are analytically derived using the concepts of eigenstrain and Eshelby's tensor of micromechanics. Moreover, for the total potential energy objective, the configurational derivative yields the path-independent J-, L-, and M-integrals that are used to characterize the energy release rates corresponding to unit translation, rotation, and expansion of small crack(s) in an infinite body. The configurational derivative is used to demonstrate the optimal topological design of structures by sequentially inserting new finite-sized heterogeneities of regular shapes into a structure followed by the optimization of their configurations. This approach successfully avoids the difficult-to-manufacture skeletal structures resulting from traditional topology optimization approaches. Further, by relating the M-integral to Rice's J-integral, the average energy release rate over the entire crack front of an arbitrary planar crack is derived. The average energy release rate is then proposed as the criterion for fracture-resistant design of structures. Using this average energy release criterion, a systematic fracture-resistant design approach that naturally unifies the assessment of risk as well as the mitigation strategy relying on the configurational optimization is proposed. Finally, the configurational optimization approach is used to perform optimal topological design of structure under pad in wirebonded chips by sequentially inserting copper lines/vias into the homogeneous dielectric layers. In order to overcome the constraints due to fabrication procedure and design rules ensuring electrical connectivity, a specialized configurational optimization exploiting the generalized topological derivative is developed to achieve the design goal. The optimal metal stack solutions corresponding to various copper volume fractions, dielectric layer configurations, as well as metal layout constraints are presented. (Abstract shortened by UMI.)
Subbarayan, Purdue University.
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