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  1. Abstract

    Extracting quadrilateral layouts from surface triangulations is an important step in texture mapping, semi-structured quadrilateral meshing for traditional analysis and spline reconstruction for isogeometric analysis. Current methods struggle to yield high-quality layouts with appropriate connectivity between singular nodes (known as “extraordinary points” for spline representations) without resorting to either mixed-integer optimization or manual constraint prescription. The first of these is computationally expensive and comes with no guarantees, while the second is laborious and error-prone. In this work, we rigorously characterize curves in a quadrilateral layout up to homotopy type and use this information to quickly define high-quality connectivity constraints between singular nodes. The mathematical theory is accompanied by appropriate computational algorithms. The efficacy of the proposed method is demonstrated in generating quadrilateral layouts on the United States Army’s DEVCOM Generic Hull vehicle and parts of a bilinear quadrilateral finite element mesh (with some linear triangles) of a 1996 Dodge Neon.

  2. Abstract

    Over 50% of the energy from power plants, vehicles, oil refining, and steel or glass making process is released to the atmosphere as waste heat. As an attempt to deal with the growing energy crisis, the solid-state thermoelectric generator (TEG), which converts the waste heat into electricity using Seebeck phenomenon, has gained increasing popularity. Since the figures of merit of the thermoelectric materials are temperature dependent, it is not feasible to achieve high efficiency of the thermoelectric conversion using only one single thermoelectric material in a wide temperature range. To address this challenge, this paper proposes a method based on topology optimization to optimize the layouts of functional graded TEGs consisting of multiple materials. The objective of the optimization problem is to maximize the output power and conversion efficiency as well. The proposed method is implemented using the Solid Isotropic Material with Penalization (SIMP) method. The proposed method can make the most of the potential of different thermoelectric materials by distributing each material into its optimal working temperature interval. Instead of dummy materials, both the P and N-type electric conductors are optimally distributed with two different practical thermoelectric materials, namely Bi2Te3 & PbTe for P-type, and Bi2Te3 & CoSb3more »for N-type respectively, with the yielding conversion efficiency around 12.5% in the temperature range Tc = 25°C and Th = 400°C. In the 2.5D computational simulation, however, the conversion efficiency shows a significant drop. This could be attributed to the mismatch of the external load and internal TEG resistance as well as the grey region from the topology optimization results as discussed in this paper.

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  3. Abstract

    Topology optimization has been proved to be an automatic, efficient and powerful tool for structural designs. In recent years, the focus of structural topology optimization has evolved from mono-scale, single material structural designs to hierarchical multimaterial structural designs. In this research, the multi-material structural design is carried out in a concurrent parametric level set framework so that the structural topologies in the macroscale and the corresponding material properties in mesoscale can be optimized simultaneously. The constructed cardinal basis function (CBF) is utilized to parameterize the level set function. With CBF, the upper and lower bounds of the design variables can be identified explicitly, compared with the trial and error approach when the radial basis function (RBF) is used. In the macroscale, the ‘color’ level set is employed to model the multiple material phases, where different materials are represented using combined level set functions like mixing colors from primary colors. At the end of this optimization, the optimal material properties for different constructing materials will be identified. By using those optimal values as targets, a second structural topology optimization is carried out to determine the exact mesoscale metamaterial structural layout. In both the macroscale and the mesoscale structural topology optimization,more »an energy functional is utilized to regularize the level set function to be a distance-regularized level set function, where the level set function is maintained as a signed distance function along the design boundary and kept flat elsewhere. The signed distance slopes can ensure a steady and accurate material property interpolation from the level set model to the physical model. The flat surfaces can make it easier for the level set function to penetrate its zero level to create new holes. After obtaining both the macroscale structural layouts and the mesoscale metamaterial layouts, the hierarchical multimaterial structure is finalized via a local-shape-preserving conformal mapping to preserve the designed material properties. Unlike the conventional conformal mapping using the Ricci flow method where only four control points are utilized, in this research, a multi-control-point conformal mapping is utilized to be more flexible and adaptive in handling complex geometries. The conformally mapped multi-material hierarchical structure models can be directly used for additive manufacturing, concluding the entire process of designing, mapping, and manufacturing.

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  4. Free, publicly-accessible full text available August 1, 2023
  5. Abstract With specific fold patterns, a 2D flat origami can be converted into a complex 3D structure under an external driving force. Origami inspires the engineering design of many self-assembled and re-configurable devices. This work aims to apply the level set-based topology optimization to the generative design of origami structures. The origami mechanism is simulated using thin shell models where the deformation on the surface and the deformation in the normal direction can be simplified and well captured. Moreover, the fold pattern is implicitly represented by the boundaries of the level set function. The folding topology is optimized by minimizing a new multiobjective function that balances kinematic performance with structural stiffness and geometric requirements. Besides regular straight folds, our proposed model can mimic crease patterns with curved folds. With the folding curves implicitly represented, the curvature flow is utilized to control the complexity of the folds generated. The performance of the proposed method is demonstrated by the computer generation and physical validation of two thin shell origami designs.
    Free, publicly-accessible full text available August 1, 2023
  6. This work proposes a rigorous and practical algorithm for quad-mesh generation based the Abel-Jacobi theory of algebraic \textcolor{red}{curves}. We prove sufficient and necessary conditions for a flat metric with cone singularities to be compatible with a quad-mesh, in terms of the deck-transformation, then develop an algorithm based on the theorem. The algorithm has two stages: first, a meromorphic quartic differential is generated to induce a T-mesh; second, the edge lengths of the T-mesh are adjusted by solving a linear system to satisfy the deck transformation condition, which produces a quad-mesh. In the first stage, the algorithm pipeline can be summarized as follows: calculate the homology group; compute the holomorphic differential group; construct the period matrix of the surface and Jacobi variety; calculate the Abel-Jacobi map for a given divisor; optimize the divisor to satisfy the Abel-Jacobi condition by integer programming; compute \textcolor{red}{a} flat Riemannian metric with cone singularities at the divisor by Ricci flow; \textcolor{red}{isometrically} immerse the surface punctured at the divisor onto the complex plane and pull back the canonical holomorphic differential to the surface to obtain the meromorphic quartic differential; construct a motorcycle graph to generate a T-Mesh. In the second stage, the deck transformation constraints are formulatedmore »as a linear equation system of the edge lengths of the T-mesh. The solution provides a flat metric with integral deck transformations, which leads to the final quad-mesh. The proposed method is rigorous and practical. The T-mesh and quad-mesh results can be applied for constructing Splines directly. The efficiency and efficacy of the proposed algorithm are demonstrated by experimental results on surfaces with complicated topologies and geometries.« less
    Free, publicly-accessible full text available April 1, 2023