Hierarchical partition of unity constructions for multiscale design optimization of heterogeneous objects

Mahidhar Rayasam, Purdue University

Abstract

Designs in Nature demonstrate intricate multi-scale forms that optimally fit their function. Developing means for discovering multiscale design solutions, driven by simple objectives inspired by Nature, is the goal of this thesis. Such multiscale design frameworks would enable the design of high performance heterogeneous materials and structures whose applications are many, and are spread across multitude of domains. Multiscale design paradigms that can achieve the above stated objective do not exist in the literature. In addition, challenges such as remeshing, material heterogeneity representation in CAD systems, and integrated topology-shape design need to be addressed as part of such a design framework. Thus, the specific aims of this thesis are to develop design methodologies, computational techniques, and simulation tools that go beyond analysis to enable efficient multiscale design. A unified representational paradigm for design and analysis that is inspired by the set theoretic Boolean operations of the constructive solid geometry (CSG) procedure of computer-aided design (CAD) is developed. The notion of a primitive design state is defined. A behavior state is defined that is associated with a primitive design state, and that is determined through a global analysis problem. A global multi-level design problem is defined to determine the primitive design states. Boolean operations on the fields belonging to primitive design and behavior spaces are defined, and it is shown that the compositions of these fields amount to a hierarchical partition of unity constructions. Non-uniform rational B-Splines (NURBS) are used to discretize the geometry, material, and behavioral fields. It is shown that, this leads to recursive partition of unity constructions. A heterogeneous material modeling framework is defined as a generalization of classical topology optimization. Analytical formulations for optimal domain connectivity modifications are developed, and integrated with freeform shape modeling techniques. The developed methodology is implemented into an existing symbolic, meshless computational framework (jNURBS) and is demonstrated on several classes of problems. As a conclusion, a lofty goal of this thesis it to make a small step towards replicating hierarchical multiscale structures such as Euplectella sp.

Degree

Ph.D.

Advisors

Subbarayan, Purdue University.

Subject Area

Mechanical engineering

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