Jiong Chen
ABSTRACT
The fundamental solutions (Green’s functions) of linear elasticity for an infinite and isotropic media are ubiquitous in interactive graphics applications that cannot afford the computational costs of volumetric meshing and finite-element simulation. For instance, the recent work of de Goes and James [2017] leveraged these Green’s functions to formulate sculpting tools capturing in real-time broad and physically-plausible deformations more intuitively and realistically than traditional editing brushes. In this paper, we extend this family of Green’s functions by exploiting the anisotropic behavior of general linear elastic materials, where the relationship between stress and strain in the material depends on its orientation. While this more general framework prevents the existence of analytical expressions for its fundamental solutions, we show that a finite sum of spherical harmonics can be used to decompose a Green’s function, which can be further factorized into directional, radial, and material-dependent terms. From such a decoupling, we show how to numerically derive sculpting brushes to generate anisotropic deformation and finely control their falloff profiles in real-time.
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- Josephine Ainley, Sandra Durkin, Rafael Embid, Priya Boindala, and Ricardo Cortez. 2008. The method of images for regularized Stokeslets. J. Comput. Phys. 227, 9 (2008), 4600–4616.Google Scholar
Digital Library
- Alexis Angelidis, Geoff Wyvill, and Marie-Paule Cani. 2004. Sweepers: swept user-defined tools for modeling by deformation. In Proceedings of Shape Modeling Applications. 63–73.Google Scholar
- David M. Barnett. 1972. The precise evaluation of derivatives of the anisotropic elastic Green’s functions. Physica Status Solidi (b) 49, 2 (1972), 741–748.Google ScholarCross Ref
- Boost. 2021. Boost C++ Libraries. http://www.boost.org/.Google Scholar
- John Burkardt. 2020. Quadrature Rules for the Unit Sphere. https://people.sc.fsu.edu/~jburkardt/cpp_src/sphere_lebedev_rule/sphere_lebedev_rule.htmlGoogle Scholar
- Ricardo Cortez, Lisa Fauci, and Alexei Medovikov. 2005. The method of regularized Stokeslets in three dimensions: analysis, validation, and application to helical swimming. Physics of Fluids 17, 3 (2005), 031504.Google ScholarCross Ref
- Fernando de Goes and Doug L James. 2017. Regularized Kelvinlets: sculpting brushes based on fundamental solutions of elasticity. ACM Trans. Graph. 36, 4 (2017), 1–11.Google ScholarDigital Library
- Fernando de Goes and Doug L James. 2018. Dynamic Kelvinlets: secondary motions based on fundamental solutions of elastodynamics. ACM Trans. Graph. 37, 4 (2018), 1–10.Google ScholarDigital Library
- Fernando de Goes and Doug L. James. 2019. Sharp Kelvinlets: Elastic Deformations with Cusps and Localized Falloffs. In Digital Production Symposium. Article 2.Google ScholarDigital Library
- Brian Gough. 2009. GNU Scientific Library Reference Manual(third ed.). Network Theory Ltd.Google Scholar
- Loukas Grafakos. 2008. Classical fourier analysis(second ed.). Springer.Google Scholar
- Robin Green. 2003. Spherical harmonic lighting: The gritty details. In Archives of the game developers conference, Vol. 56. 4.Google Scholar
- Doug L James, Jernej Barbič, and Dinesh K Pai. 2006. Precomputed acoustic transfer: output-sensitive, accurate sound generation for geometrically complex vibration sources. ACM Trans. Graph. 25, 3 (2006), 987–995.Google ScholarDigital Library
- Doug L James and Dinesh K Pai. 1999. Artdefo: accurate real time deformable objects. In Proceedings of the Conference on Computer Graphics and Interactive Techniques. 65–72.Google ScholarDigital Library
- Doug L James and Dinesh K Pai. 2003. Multiresolution Green’s function methods for interactive simulation of large-scale elastostatic objects. ACM Trans. Graph. 22, 1 (2003), 47–82.Google ScholarDigital Library
- Lily Kharevych, Patrick Mullen, Houman Owhadi, and Mathieu Desbrun. 2009. Numerical coarsening of inhomogeneous elastic materials. ACM Trans. Graph. 28, 3 (2009), 1–8.Google ScholarDigital Library
- Yaron Lipman, David Levin, and Daniel Cohen-Or. 2008. Green coordinates. ACM Trans. Graph. 27, 3 (2008), 1–10.Google ScholarDigital Library
- Toshio Mura and N Kinoshita. 1972. The precise evaluation of derivatives of the anisotropic elastic Green’s functions. Physica Status Solidi (b) 49, 2 (1972), 741–748.Google ScholarCross Ref
- Roger G. Newton. 2002. Scattering Theory of Waves and Particles. Springer-Verlag.Google Scholar
- Rohan Sawhney and Keenan Crane. 2020. Monte Carlo geometry processing: A grid-free approach to PDE-based methods on volumetric domains. ACM Trans. Graph. 39, 4 (2020).Google ScholarDigital Library
- Camille Schreck, Christian Hafner, and Chris Wojtan. 2019. Fundamental solutions for water wave animation. ACM Trans. Graph. 38, 4 (2019), 1–14.Google ScholarDigital Library
- Wolfram Research, Inc.2021. Mathematica (version 13.0.0). https://www.wolfram.com/mathematicaGoogle Scholar
- Longtao Xie, Chuanzeng Zhang, Jan Sladek, and Vladimir Sladek. 2016. Unified analytical expressions of the three-dimensional fundamental solutions and their derivatives for linear elastic anisotropic materials. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, 2186 (2016), 20150272.Google ScholarCross Ref
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