Generation of Tubular and Membranous Shape Textures with Curvature Functionals

Song, Anna

doi:10.1007/s10851-021-01049-9

Cited by 3 publications

(4 citation statements)

References 97 publications

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“…Detailed mathematical proof of phase‐field approximation as well as implementation details can be found in ref. [33].…”

Section: Resultsmentioning

confidence: 99%

“…We adopt the curvature‐based phase‐field framework of Song, [ 33 ] which makes it possible to unify topologies with spherical, tubular, and membranous features as well as their combinations with seamless transition. Briefly, for a given surface

false

, let us consider an energy functional F based on the surface curvatures as

false

where the integrand

$f \left[\right.

…”

Section: Resultsmentioning

confidence: 99%

“…Despite a wide variety of explorations ranging from TPMS to spinodoid metamaterials, a more general and unified topology description of smooth porous microstructures remains to be investigated. In this direction, Song [ 33 ] recently proposed a unifying phase‐field framework to generate tubular and membranous topologies with diverse curvature profiles. Similar to the canonical spinodal decomposition model, [ 34 ] topologies within this framework are obtained as optimizers (subject to constant volume restriction) of energy functionals parameterized based on principal curvatures of the phase‐field interface.…”

Section: Introductionmentioning

confidence: 99%

“…To this end, we enable instant identification of microstructural topologies with precise targeted curvature profiles and unlock new possibilities in design of metamaterials and porous microstructures. This is achieved by strategically using a dual deep‐neural‐network (NN) setup [ 22,47,55 ] in tandem with a phase‐field framework driven by curvature‐based energetics [ 33 ] that bypasses both the aforementioned key challenges—namely, the computational bottleneck of phase‐field methods and the ill‐posedness in inverse design.…”

Section: Introductionmentioning

confidence: 99%

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Inverse Designing Surface Curvatures by Deep Learning

Guo,

Sharma,

Kumar

2024

Advanced Intelligent Systems

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Smooth and curved microstructural topologies found in nature—from soap films to trabecular bone—have inspired several mimetic design spaces for architected metamaterials and bio‐scaffolds. However, the design approaches so far are ad hoc, raising the challenge: how to systematically and efficiently inverse design such artificial microstructures with targeted topological features? Herein, surface curvature is explored as a design modality and a deep learning framework is presented to produce topologies with as‐desired curvature profiles. The inverse design framework can generalize to diverse topological features such as tubular, membranous, and particulate features. Moreover, successful generalization beyond both the design and data space is demonstrated by inverse designing topologies that mimic the curvature profile of trabecular bone, spinodoid topologies, and periodic nodal surfaces for application in bio‐scaffolds and implants. Lastly, curvature and mechanics are bridged by showing how topological curvature can be designed to promote mechanically beneficial stretching‐dominated deformation over bending‐dominated deformation.

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“…Detailed mathematical proof of phase‐field approximation as well as implementation details can be found in ref. [33].…”

Section: Resultsmentioning

confidence: 99%

false

, let us consider an energy functional F based on the surface curvatures as

false

where the integrand

$f \left[\right.

…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Inverse Designing Surface Curvatures by Deep Learning

Guo,

Sharma,

Kumar

2024

Advanced Intelligent Systems

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Designing Mixed-Category Stochastic Microstructures by Deep Generative Model-Based and Curvature Functional-Based Methods

Xu,

Naghavi Khanghah,

2023

Journal of Mechanical Design

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Bridging the gaps among various categories of stochastic microstructures remains a challenge in the design representation of microstructural materials. Each microstructure category requires certain unique mathematical and statistical methods to define the design space (design representation). The design representation methods are usually incompatible between two different categories of stochastic microstructures. The common practice of pre-selecting the microstructure category and the associated design representation method before conducting rigorous computational design restricts the design freedom and hinders the discovery of innovative microstructure designs. To overcome this issue, this paper proposes and compares two novel methods, the deep generative modeling-based method and the curvature functional-based method, to understand their pros and cons in designing mixed-category stochastic microstructures for desired properties. For the deep generative modeling-based method, the Variational Autoencoder is employed to generate an unstructured latent space as the design space. For the curvature functional-based method, the microstructure geometry is represented by curvature functionals, of which the functional parameters are employed as the microstructure design variables. Regressors of the microstructure design variables-property relationship are trained for microstructure design optimization. A comparative study is conducted to understand the relative merits of these two methods in terms of computational cost, continuous transition, design scalability, design diversity, dimensionality of the design space, interpretability of the statistical equivalency, and design performance.

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Willmore-type variational problem for foliated hypersurfaces

Rovenski

2024

era

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<abstract><p>After Thomas James Willmore, many authors were looking for an immersion of a manifold in Euclidean space or Riemannian manifold, which is the critical point of functionals whose integrands depend on the mean curvature or the norm of the second fundamental form. We study a new Willmore-type variational problem for a foliated hypersurface in Euclidean space. Its general version is the Reilly-type functional, where the integrand depends on elementary symmetric functions of the eigenvalues of the restriction on the leaves of the second fundamental form. We find the 1st and 2nd variations of such functionals and show the conformal invariance of some of them. For a critical hypersurface with a transversally harmonic foliation, we derive the Euler-Lagrange equation and give examples with low-dimensional foliations. We present critical hypersurfaces of revolution and show that they are local minima for special variations of immersion.</p></abstract>

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Generation of Tubular and Membranous Shape Textures with Curvature Functionals

Cited by 3 publications

References 97 publications

Inverse Designing Surface Curvatures by Deep Learning

Inverse Designing Surface Curvatures by Deep Learning

Designing Mixed-Category Stochastic Microstructures by Deep Generative Model-Based and Curvature Functional-Based Methods

Willmore-type variational problem for foliated hypersurfaces

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