Topological-derivative-based design of stiff fiber-reinforced structures with optimally oriented continuous fibers
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RESEARCH PAPER
Topological-derivative-based design of stiff fiber-reinforced structures with optimally oriented continuous fibers Akshay Desai1 · Mihir Mogra1 · Saketh Sridhara1 · Kiran Kumar2 · Gundavarapu Sesha2 · G. K. Ananthasuresh1 Received: 2 April 2020 / Revised: 2 August 2020 / Accepted: 10 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We use topological derivatives to obtain fiber-reinforced structural designs with non-periodic continuous fibers optimally arranged in specific patterns. The distribution of anisotropic fiber material within isotropic matrix material is determined for given volume fractions of void and material as well as fiber and matrix simultaneously, for maximum stiffness. In this threephase material distribution approach, we generate a Pareto surface of stiffness and two volume fractions by adjusting the level-set plane in the topological sensitivity field. For this, we utilize topological derivatives for interchanging (i) isotropic material and void; (ii) fiber material and void; and (iii) isotropic and fiber materials, during iterative optimization. While the isotropic topological derivative is well known, the latter two required modification of the anisotropic topological derivative. Furthermore, we used the polar form of the topological derivative to determine the optimal orientation of the fiber at every point. Thus, in the discretized finite element model, we get element-wise optimal fiber orientation in the portions where fiber is present. Using these discrete sets of orientations, we extract continuous fibers as streamlines of the vector field. We show that continuous fibers are aligned with the principal stress directions as first reported by Pedersen. Three categories of examples are presented: (i) embedding fiber everywhere in the isotropic matrix without voids; (ii) selectively embedding fiber for a given volume fraction of the fiber without voids; and (iii) including voids for given volume fractions of fiber and matrix materials. We also present an example with multiple load cases. Additionally, in view of practical implementation of laying up or 3D-printing of fibers within the matrix material, we simplify the dense arrangement of fibers by evenly spacing them while retaining their specific patterns. Keywords Topological derivatives · Fiber-reinforced structural design · Non-periodic continuous fibers · Pareto surface · Anisotropic topological derivative · Polar form · Multiple load-cases
1 Introduction Topology optimization, with its beginning in homogenization-based parameterization (Bendsoe and Kikuchi 1988), is a well-established method for designing structures made of a single isotropic material using power-law material interpolation techniques (e.g., Bendsoe and Sigmund 2003; Stolpe and Svanberg 2001). Optimality criteria method (Bendsoe and Sigmund 2003), method of moving asymptotes (Svanberg 1987), level-set method
Responsible Editor: Juli´an Andr´es Norato G. K. Ananthasuresh
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