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Abstract This work presents a method for the topology optimization of welded frame structures to minimize the manufacturing cost. The structures considered here consist of assemblies of geometric primitives such as bars and plates that are common in welded frame construction. A geometry projection technique is used to map the primitives onto a continuous density field that is subsequently used to interpolate material properties. As in density-based topology optimization techniques, the ensuing ersatz material is used to perform the structural analysis on a fixed mesh, thereby circumventing the need for re-meshing upon design changes. The distinct advantage of the representation by geometric primitives is the ease of computation of the manufacturing cost in terms of the design parameters, while the geometry projection facilitates the analysis within a continuous design region. The proposed method is demonstrated via the manufacturing-cost-minimization subject to a displacement constraint of 2D bar, 3D bar, and plate structures.
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Algebraic Representations for Volumetric Frame Fields
Field-guided parameterization methods have proven effective for quad meshing of surfaces; these methods compute smoothcross fieldsto guide the meshing process and then integrate the fields to construct a discrete mesh. A key challenge in extending these methods to three dimensions, however, isrepresentationof field values. Whereas cross fields can be represented by tangent vector fields that form a linear space, the 3D analog—an octahedral frame field—takes values in a nonlinear manifold. In this work, we describe the space of octahedral frames in the language of differential and algebraic geometry. With this understanding, we develop geometry-aware tools for optimization of octahedral fields, namely geodesic stepping and exact projection via semidefinite relaxation. Our algebraic approach not only provides an elegant and mathematically sound description of the space of octahedral frames but also suggests a generalization to frames whose three axes scale independently, better capturing the singular behavior we expect to see in volumetric frame fields. These newodeco frames, so called as they are represented by orthogonally decomposable tensors, also admit a semidefinite program–based projection operator. Our description of the spaces of octahedral and odeco frames suggests computing frame fields via manifold-based optimization algorithms; we show that these algorithms efficiently produce high-quality fields while maintaining stability and smoothness.
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- Award ID(s):
- 1838071
- PAR ID:
- 10603638
- Publisher / Repository:
- Association for Computing Machinery (ACM)
- Date Published:
- Journal Name:
- ACM Transactions on Graphics
- Volume:
- 39
- Issue:
- 2
- ISSN:
- 0730-0301
- Format(s):
- Medium: X Size: p. 1-17
- Size(s):
- p. 1-17
- Sponsoring Org:
- National Science Foundation
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