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This work uses multi-material topology optimization (MMTO) to maximize the average torque of a 3-phase permanent magnet synchronous machine (PMSM). Eight materials are considered in the stator: air, soft magnetic steel, three electric phases, and their three returns. To address the challenge of designing a 3-phase PMSM stator, a generalized density-based framework is used. The proposed methodology places the prescribed material candidates on the vertices of a convex polytope, interpolates material properties using Wachspress shape functions, and defines Cartesian coordinates inside polytopes as design variables. A rational function is used as penalization to ensure convergence towards meaningful structures, without the use of a filtering process. The influences of different polytopes and penalization parameters are investigated. The results indicate that a hexagonal-based diamond polytope is a better choice than the classical orthogonal domains for this MMTO problem. In addition, the proposed methodology yields high-performance designs for 3-phase PMSM stators by implementing a continuation method on the electric load angle.

We introduce a framework for modeling dynamic fracture problems using cohesive polygonal finite elements. Random polygonal meshes provide a robust, efficient method for generating an unbiased network of fracture surfaces. Further, these meshes have more facets per element than standard triangle or quadrilateral meshes, providing more possible facets per element to insert cohesive surfaces. This property of polygonal meshes is advantageous for the modeling of pervasive fracture. We use both Wachspress and maximum entropy shape functions to form a finite element basis over the polygons. Fracture surfaces are captured through dynamically inserted cohesive zone elements at facets between the polygons in the mesh. Contact is enforced through a penalty method that is applied to both closed cohesive surfaces and general interpenetration of two polygonal elements. Several numerical examples are presented that illustrate the capabilities of the method and demonstrate convergence of solutions.

Traditionally, standard Lagrangian-type finite elements, such as linear quads and triangles, have been the elements of choice in the field of topology optimization. However, finite element meshes with these conventional elements exhibit the well-known “checkerboard” pathology in the iterative solution of topology optimization problems. A feasible alternative to eliminate such long-standing problem consists of using hexagonal (honeycomb) elements with Wachspress-type shape functions. The features of the hexagonal mesh include two-node connections (i.e. two elements are either not connected or connected by two nodes), and three edge-based symmetry lines per element. In contrast, quads can display one-node connections, which can lead to checkerboard; and only have two edge-based symmetry lines. In addition, Wachspress rational shape functions satisfy the partition of unity condition and lead to conforming finite element approximations. We explore the Wachspress-type hexagonal elements and present their implementation using three approaches for topology optimization: element-based, continuous approximation of material distribution, and minimum length-scale through projection functions. Examples are presented that demonstrate the advantages of the proposed element in achieving checkerboard-free solutions and avoiding spurious fine-scale patterns from the design optimization process.