Presented at IASS 2018 @ MIT

1. Optimized Quad Gridshell from Stress Field and
Curvature Field
Lorenzo Greco, www.parametricism.co.uk, AKT II
Paper number
204
lorenzogreco@gmail.com

2. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 2
Court Queen Elizabeth II - British Museum, London
Yas Marina Hotel - Abu Dhabi
Cour Visconti – Louvre, Paris
Neumünster Abbey – Luxemburg

3. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 3
In architecture there might be more constraints compared with computer graphics, for example:
● Dictated position for some nodes
● Positioning of irregular vertices
● Strict boundary conditions for aesthetic and structural reasons
● Localized anisotropy to enhance details or sharp features
● Regular quad shape reduces glass cut-offs and is generally more aesthetically pleasant
● Planarity or single curvature at most, greatly reduces costs

4. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 4
Top-down quadrangulation
It becomes hard when it puts restraints to the quality
and size of individual quads and to the overall mesh
topology, whilst minimizing the deviation from it.
Bottom-up quadrangulation
The bottom-up approach is the easiest to
achieve. It consists of recreating the given
surface using translational, rotational, and
scaled surfaces, or any combination of them.
● Developable
● Planar quad
● Hard to match the initial surface
● The quad topology is strictly linked to
how the mesh has been rationalized.

5. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 5
Workflow showing the steps to
generate a planar quad mesh

6. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 6
Initial surface

7. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 7
Principal stress directions (black) principal curvature (red) directions of the freeform continuous shell

8. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 8
Selecting a subset of
vectors to be used to
generate a global
smooth vector field

9. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 9
Comparison between fields: (black)
principal stress, (red) principal
curvature,
(white) smooth vector field from n-
PolyVector field

10. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 10
Comparison between fields: (black) principal stress, (red) principal curvature, (green) naïf
point-wise interpolation, and (white) smooth vector field from n-PolyVector field

11. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 11
Conjugated vector field from principal stress and curvature fields

12. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 12
Orthogonal map derived from frame field global parameterization

13. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 13
The vectors chosen to inform the grid
directionality are in the black rectangles.
The resulting gridshell (red and green)
follows those directions with good
accuracy whilst still being smooth across
the whole domain.

14. Lorenzo Greco, parametricism.co.uk, lorenzogreco@gmail.com 14
Capybara Dodo