Aligning feature-mapping designs to given density fields, first and second order

Abstract

Feature-mapping methods represent structural designs by explicit geometric primitives on fixed analysis grids, an interpretable and parametric alternative to the density fields of density-based topology optimization (SIMP). In contrast to SIMP, however, feature-mapping optimization is sensitive to the initial design. We present a gradient-based approach to align a feature-mapping configuration to a given (pseudo) density field, e.g., to initialize a subsequent feature-mapping optimization. The staged approach is based on a least-squares tracking formulation, preceded by a variant that only rewards alignment and does not penalize mismatch. Without an underlying finite element simulation, iterations are cheap, but isolated features receive little sensitivity information. As a remedy, we propose an asymmetric transition function based on automatically parametrized Bézier curves. We demonstrate the approach for a 2D cantilever and the five-bar design; for the latter, we show optional feature minimization based on the feature scaling variable of the geometry projection method. The approach works well with first-order optimizers. The absence of a state problem and the small number of variables, however, also make a second-order formulation attractive. To the best of the authors' knowledge, we present the first Hessian formulation for feature mapping and report the behavior of first- and second-order optimizers on our benchmark problems. For completeness, we also give the exact Hessian of the state-based compliance, which requires one additional solution of the FEM system per feature variable.

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BibTeX RIS

BibTeX

@misc{wein2026aligning,
  title = {{Aligning feature-mapping designs to given density fields, first and second order}},
  author = {Fabian Wein and Patrick Jung and Arash Moradian and Michael Stingl},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.39472},
  eprint = {2609.39472},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.39472},
}

RIS

TY  - UNPB
TI  - Aligning feature-mapping designs to given density fields, first and second order
AU  - Fabian Wein
AU  - Patrick Jung
AU  - Arash Moradian
AU  - Michael Stingl
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.39472
UR  - https://arxiv.org/abs/2609.39472
AB  - Feature-mapping methods represent structural designs by explicit geometric primitives on fixed analysis grids, an interpretable and parametric alternative to the density fields of density-based topology optimization (SIMP). In contrast to SIMP, however, feature-mapping optimization is sensitive to the initial design. We present a gradient-based approach to align a feature-mapping configuration to a given (pseudo) density field, e.g., to initialize a subsequent feature-mapping optimization. The staged approach is based on a least-squares tracking formulation, preceded by a variant that only rewards alignment and does not penalize mismatch. Without an underlying finite element simulation, iterations are cheap, but isolated features receive little sensitivity information. As a remedy, we propose an asymmetric transition function based on automatically parametrized Bézier curves. We demonstrate the approach for a 2D cantilever and the five-bar design; for the latter, we show optional feature minimization based on the feature scaling variable of the geometry projection method. The approach works well with first-order optimizers. The absence of a state problem and the small number of variables, however, also make a second-order formulation attractive. To the best of the authors' knowledge, we present the first Hessian formulation for feature mapping and report the behavior of first- and second-order optimizers on our benchmark problems. For completeness, we also give the exact Hessian of the state-based compliance, which requires one additional solution of the FEM system per feature variable.
ER  -

Abstract

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