A critical step in topology optimization (TO) is finding sensitivities. Manual derivation and implementation of the sensitivities can be quite laborious and error-prone, especially for non-trivial objectives, constraints and material models. An alternate approach is to utilize automatic differentiation (AD). While AD has been around for decades, and has also been applied in TO, wider adoption has largely been absent. In this educational paper, we aim to reintroduce AD for TO, and make it easily accessible through illustrative codes. In particular, we employ JAX, a high-performance Python library for automatically computing sensitivities from a user defined TO problem. The resulting framework, referred to here as AuTO, is illustrated through several examples in compliance minimization, compliant mechanism design and microstructural design.

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A fundamental requirement in standard finite element method (FEM) over four‐node quadrilateral meshes is that every element must be convex, else the results can be erroneous. A mesh containing concave element is said to be tangled, and tangling can occur, for example, during: mesh generation, mesh morphing, shape optimization, and/or large deformation simulation. The objective of this article is to introduce a tangled finite element method (TFEM) for handling concave elements in four‐node quadrilateral meshes. TFEM extends standard FEM through two concepts. First, the ambiguity of the field in the tangled region is resolved through a careful definition, and this naturally leads to certain correction terms in the FEM stiffness matrix. Second, an equality condition is imposed on the field at re‐entrant nodes of the concave elements. When the correction terms and equality conditions are included, we demonstrate that one can achieve accurate results, and optimal convergence, even over severely tangled meshes. The theoretical properties of the proposed TFEM are established, and the implementation, that requires minimal changes to standard FEM, is discussed in detail. Several numerical experiments are carried out to illustrate the robustness of the proposed method.

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