Quasi-Newton Exploration of Implicitly Constrained Manifolds
A family of methods for the efficient update of second order approximations of a constraint manifold is proposed in this thesis. The concept of such a constraint manifold corresponds to an abstract space prescribed by implicit nonlinear constraints, which can be a set of objects satisfying certain d...
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| Language: | en |
| Published: |
2012
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| Online Access: | Tang, C. (2011). Quasi-Newton Exploration of Implicitly Constrained Manifolds. KAUST Research Repository. https://doi.org/10.25781/KAUST-C94C8 http://hdl.handle.net/10754/209387 |
| Summary: | A family of methods for the efficient update of second order approximations of a
constraint manifold is proposed in this thesis. The concept of such a constraint manifold
corresponds to an abstract space prescribed by implicit nonlinear constraints,
which can be a set of objects satisfying certain desired properties. This concept has
a variety of applications, and it has been successfully introduced to fabrication-aware
architectural design as a shape space consisting of all the implementable designs.
The local approximation of such a manifold can be first order, in the tangent space,
or second order, in the osculating surface, with higher precision. For a nonlinearly
constrained manifold with rather high dimension and codimension, the computation
of second order approximants (osculants) is time consuming. In this thesis, a type
of so-called quasi-Newton manifold exploration methods which approximate the new
osculants by updating the ones of a neighbor point by 1st-order information is introduced.
The procedures are discussed in detail and the examples implemented to
visually verify the methods are illustrated. |
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