As regular readers of this blog will know, I’m passionate about the use of relaxation and force-based methods for optimizing geometry in a very interactive way.

There is a great variety of form-finding that can be done by assigning physical forces as interactions between sets of particles. However, in my investigations so far, the topology of these interactions has usually remained fixed. So the overall shape changes dramatically, but the number of points -and the underlying network of which point interacts with which- remains the same throughout the simulation.

I have written about topology optimization before, in my post on self organizing structures, but the videos above use a fundamentally different approach than any of the examples shown there, as this new work is based on purely local interactions, which reconfigure their connectivity as the geometry changes.

I did experiment a while back with a different technique for dynamic remeshing based on repelling/colliding particles or spheres (see this video, skip to around 1:27 for the mesh generation), but this was limited by a number of factors – although the spheres only exert force on each other when they are closer than a certain distance, they still need to be constantly checked against all the other spheres to see if they are within this distance. Checking every sphere against every other one means the number of interactions increases as the square of the number of particles, which becomes slow for even modest meshes of a few thousand vertices.

There are various ways of significantly reducing this complexity and cutting down the number of distance calculations to be performed at each iteration (such as partitioning the space into cubic cells, and only checking for interactions between particles in the same or neighbouring cells).  However, there still remains the question of how to remove or add spheres if they get too close or too spread out. Then even when a fairly equal distribution of points across the surface has been achieved, turning this into a clean mesh can still be tricky.

Instead, using the mesh connectivity itself from the very beginning as the network of which vertices interact with which, and updating it iteratively and locally based on its changing geometry, solves all of these problems at once. This connectivity update can be done by using repetitions and combinations of just 3 essential moves :

edge flip

edge collapse

edge split

(or alternatively, one can use edge flip, vertex insertion and vertex removal)

vertex insertion/removal

I’m always using a triangulated mesh here. In the 2nd and 3rd videos at the start, the underlying physics is still working with triangles, but I’ve shown the dual of this, which exchanges the roles of faces and vertices, turning a mesh of mostly triangles into one of mostly hexagons:

Notice that the irregular vertices of the triangle mesh (ones which have 5 or 7 connected edges instead of the regular 6) correspond to pentagons and heptagons in the dual. Finding the right number and placement of these irregular vertices is an important part of making a good mesh. There is a close relationship between these irregular vertices and the overall topology and curvature of the mesh. Euler’s polyhedron theorem gives us a precise relationship between the number of faces/edges/vertices, and the genus of the mesh (F-E+V=2-2g). This is linked to Descarte’s polyhedron theorem, which relates the total angle defect at the vertices to the genus, and is itself a discrete version of the continuous Gauss-Bonnet theorem which describes a similar relationship for the integral of Gaussian curvature in the smooth case.

Keeping a good quality triangular mesh (close to even sized equilateral elements, with no obtuse, skinny or degenerate ones) can be very useful for many other types of optimization and simulation, as well as an advantage for fabrication if it is to be built as a physical structure.

I’m thinking I could also vary the remeshing rules depending on whether the priority is regularity of geometry, or regularity of connectivity. For example, on a sphere, a geodesic dome has a small number of irregular vertices (this is sometimes referred to as a semi-regular mesh), but a fair amount of variation in edge length, whereas solutions to the Thomson or Tammes problems reduce the variation of distance, but have larger numbers of irregular vertices.

If the triangles all have identical edge lengths, then all of the angle defect is concentrated at the irregular vertices, whereas if the edges can vary slightly, the angle defect can be spread out across the mesh. As mentioned above, by Descarte’s theorem, the total angle defect is fixed, but if the number of vertices is increased it can be divided between more of them. Subdividing each triangle (using a smoothing scheme such as Loop subdivision) does not alter the configuration of irregular vertices, but reduces the angle defect at each vertex (and in the smooth limit it approaches zero as the number of vertices approaches infinity).

The remeshing can also be based on other criteria than just equal triangles. Reducing triangle size in high curvature areas is an obvious one, but I have a hunch there could also be some interesting ways of basing it not just on geometry, but on stresses, and using it for structural optimization.

I find remeshing fascinating because the same geometric rules and operations have relevance across so many different disciplines and at a variety of scales.

For example, in the carbon nanostructures graphene and nanotubes, which are hexagonal arrangements of atoms, there is a much studied crystallographic defect which occurs called the Stone-Wales defect – where instead of all hexagons we get 2 pentagons and 2 heptagons. It has important implications for the material’s mechanical and electrical properties. If we look again at our edge-flip move on a regular triangular mesh, and its effect on the dual, we see that it is exactly the Stone Wales defect!

There are even hypothetical carbon nanostructures (proposed by Mackay and Terrones)  in the shape of doubly curved triply periodic minimal surfaces known as Schwarzites.

(from the paper Curved nanostructured materials)

As the bond lengths between the Carbon atoms are very rigid, the only way these curved structures can be formed is through the variations in mesh connectivity.

Because such complex geometries are only dependent on the way in which many identical elements are connected, rather than variation in the elements themselves, they can be modelled using simple materials (no laser cutting involved – just clever assembly!):

(by Bih-Yaw Jin from the beaded molecules)

(by Loop.ph)

Going down to even smaller scales, we can even find the use of similar ideas about mesh connectivity in theories of loop quantum gravity, spin foams, and Regge calculus (for example, see Canonical Simplicial Gravity by Dittrich and Höhn, or The Feynman diagramatics for the spin foam models). Pachner moves or bistellar flips generalize the mesh moves described earlier to higher dimensional simplices.

(images from Canonical Simplicial Gravity)

So as well being potentially useful for design, meshes and their properties have profound relevance to our understanding of the nature of space and curvature, because of the way they link the discrete and the continuous.

I’ll be continuing to work on these remeshing tools and their combination with other types of relaxation and optimization. These recent developments are not publicly available for now, but I’ll update here with any news.

Here are a few more references on remeshing:

Dynamic Remeshing and Applications

A Remeshing Approach to Multiresolution Modeling

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also – a general update :  Around a year ago I started working full time for Foster+Partners in the Specialist Modelling Group. This has been a really great experience, working on (and applying Kangaroo on) some big and exciting projects, but core Kangaroo development (and posts here) did necessarily slow down a bit during this time. However, I have recently left F+P and things will be changing ! More news to follow…

soliton is a kind of solitary, stable and localised wave which acts in many ways like a particle. They are useful in describing a diverse range of physical phenomena, and their mathematics is a large and active topic of research.

One way of demonstrating the idea of solitons is the coupled pendulum model: Imagine a series of pendulums attached at even spacings along a horizontal rod, free to rotate only in the plane perpendicular to the rod, and connected to their neighbouring pendulums by torsion springs. If the pendulum at one end is pulled over the top of the rod it introduces a twist into the system. The weight of the pendulums makes them want to hang straight down, and along most of the rod they do, but at this twist they cannot. Because of the pendulums pulling down either side of the twist, it remains localised and cannot spread out. It can move along the rod like a wave, but unlike a normal wave, it cannot simply dissipate and disappear (except by leaving one end of the rod). These twists can be in either direction (sometimes called kink and anti-kink solitons), and interestingly they can pass through each other and emerge unchanged. Curious about this behaviour, I had a go at simulating it in Kangaroo :

[vimeo 35462854]

Another model for solitons can be made with playing cards :

[vimeo 35469549]

(I think I first encountered this model in the book ‘The New Ambidextrous Universe’ by Martin Gardner)

A form of soliton even occurs in traffic jams, and here is yet another model of soliton like behaviour – the Jacob’s ladder toy.

I see a link here with the the idea of bistable structures. A normal elastic object has a single unstressed configuration, and if you bend it away from this shape(provided you do not bend it too far) it will always try and spring back to that one rest configuration. A bistable object on the other hand has 2 different states in which it is at rest, and in between a more highly stressed state. An example of a bistable structure that some readers may remember are the ‘snap bracelets‘ popular during the early 90’s.

These were very similar to metal measuring tape – a strip of thin metal with a slight curvature across its short direction when straight along its length. A normal metal tape measure is not actually bistable, because it is still stressed when in its rolled state, but it is much less stressed in the rolled or straight configurations than in the in between state.

This allows a sharp bend to move along the tape while staying localized, a bit like the pendulum soliton.

[vimeo 35469746]

I’ve recently been exploring the idea of whether it is possible to create something similar, but on a surface, not just a linear element.

Here I am also drawing on the idea of auxetic materials – defined as having a negative Poisson’s ratio. This means that unlike most materials which get thinner in cross section as you stretch them along their length, auxetics surprisingly actually get fatter. This can be achieved by small scale structures within the material behaving as linkages.

Many of the origami textured surfaces I have explored in previous posts actually have auxetic properties. Something I am recently trying to do is combine auxetic and bistable properties in a single sheet material through 3d printing. I’ll update when the test print arrives, but I’m hoping that the transition region between the two different contracted states of the material will behave something like a 2 dimensional version of the unstable region of the tape measure, causing some interesting out of plane buckling.

I’ve barely scratched the surface of the real study of solitons, and what I’ve mentioned here are only really toy models of a highly complex subject. For some more related reading :

Wikipedia article on the Sine-Gordon equation

Many Faces of Solitons

A real version of the pendulum model, and another

and there are also all sorts of exotic Soliton related objects, such as SkyrmionsTwistors Instantons and Hopfions

For more on auxetics, check out Rod Lakes’ page

and for more on bistable structures try Simon Guest’s page

Finally – sorry for such a long hiatus, it’s good to be writing again, more posts coming soon.

There have been some fairly heated posts and discussions in the blogs recently on the subject of some patents relating to architectural geometry held by Evolute, Helmut Pottmann and RFR,  about the commercialisation of academic research, and particularly how all this relates to what I do in Kangaroo, and what people can legally use Kangaroo for.

Firstly, rather than attempting to paraphrase or summarise, I will link to the posts in question so you can read them (and particularly the comments people made in reply) yourself:

Patenting Geometry on Daniel Davis’ Digital Morphogenesis blog

and Evolute’s response

Why is Evolute Patenting Geometry ?

I welcome this debate, and think it is of great interest and importance to everyone involved in the future of computational geometry. There is also a discussion starting on the Grasshopper forum (http://www.grasshopper3d.com/forum/topics/patenting-geometry) and I’d be interested to hear the thoughts of you my readers on the matter. There are also some further comments and discussion on the download site here http://www.food4rhino.com/project/kangaroo#comment-209

To clarify some aspects of my own position :

Several of the geometry optimization features I have developed in Kangaroo have been  informed and inspired by the works of the Polyhedral Surfaces Unit at TU Berlin lead by Alexander Bobenko, and the Geometric Modeling and Industrial Geometry Group at TU Wien lead by Helmut Pottmann. I first encountered some of these works as a student, via a long running interest in minimal surfaces, circle packing, and conformal mapping, which lead me to papers and presentations such as these:

Discrete differential geometry. Consistency as integrability – Bobenko and Suris 2005

Discrete Minimal Surfaces from Quadrilaterals – Bobenko 2007

The focal geometry of circular and conical meshes – Pottmann and Wallner 2006

Geometry of Multi-layer Freeform Structures for Architecture – Pottmann, Liu, Wallner, Bobenko and Wang 2007

Geometry of Architectural Freeform Structures – Pottmann, Schiftner and Wallner 2008

Despite usually not fully understanding most of the technical details of these often beautiful papers I somehow found the results very appealing, and they alerted me to the great potential for the use in design of planar quad meshes, circular and conical meshes and other related geometry.

Another important later influence for me in my recent work has been the paper Physics-based Generative Design – Ramtin Attar, Robert Aish, Jos Stam, Duncan Brinsmead, Alex Tessier, Michael Glueck & Azam Khan 2010, where among other things they describe embedding properties useful for fabrication (such as planar mesh faces and constant offsets) in an interactive model, and iteratively enforcing these constraints.

Following the example of how Grasshopper has made advanced computational methods vastly more accessible to many people, I saw a potential to put some of these geometrical results in the hands of ordinary designers by adding them to my physics engine Kangaroo, and make them intuitive to use and cheaper(free!) than the existing commercially available software for this (Evolute Tools). I don’t claim to replicate all the capabilities of their software, and my work is still under development, but I do think I offer a unique way of interacting with some of these geometric properties, and feel the results so far are very promising.

The method I use is quite simple compared to many other optimization techniques, and in fact I can describe it here:

It works by assigning every vertex  a mass and velocity, and calculating and applying forces for the set of points acted on by each force object (such as the 2 end points for a spring, or the 4 vertices of a quad for a planarization force), adding up all of the forces acting on each particle, applying the corresponding accelerations and repeating, with a damping term to cause the system to settle to an equilibrium. Similar techniques are widely used in computer games and animation, and a good overview can be found in Baraff and Witkin’s Siggraph 97 course notes Physically based modeling.

The force calculations themselves are also quite simple – for example, to force 4 points to lie on a circle they must be equidistant from a 5th point and also planar. In Kangaroo this is achieved by:

-An ‘equalization’ force acting on the lines from these points to a floating 5th point (simply taking the average length of the 4 lines and then treating each one as a Hooke’s law spring with this average as its rest length). When these lines are equal length clearly the 4 points lie on a common sphere.

-A planarization force. If the 4 points are not coplanar, they define a tetrahedron. As the volume of this tetrahedron decreases to zero, the quad becomes planar, so a zero rest-length spring-like force is applied pulling the the top and bottom diagonals towards each other. When the distance between these diagonals is zero the quad is planar.

(That takes care of circularizing the mesh, but where all this gets really interesting for me is when this is combined with other forces to shape the form – Laplacian smoothing, gravity, inflation, etc and point or boundary constraints which can be interactively adjusted by the designer)

This is not to say that the reasons behind why circular meshes have the interesting properties they do is simple, just that the methods of their creation can be. In fact the study of these discretizations of curved surfaces has its roots in some very deep mathematics going back quite some way.

Much of the discussion in the pages linked to at the start centres around the distinction between patenting the use of geometric results vs geometric methods. This can lead to some fascinating abstract philosophical and moral debates, such as whether mathematics and mathematical truths are invented or discovered, and to what extent can or should they, or certain uses of them, be owned. However the debate about the legality of these things also has much more immediately practical and economic implications for those in the business of designing buildings.

I’m not actually going to go into my own personal thoughts and feelings on these matters much here, and I do not have the expertise to contribute meaningfully to any legal debate, but I will certainly be following developments with interest, and am keen to hear your opinions.

Fortunately for me, whatever debate there may be about  who has what right to use certain geometry in their buildings, nobody is trying to stop me developing Kangaroo. In fact I have had some enjoyable and encouraging discussions with the people at Evolute, and even provided them with some meshes which they used to demonstrate some of their new optimization procedures in a recent paper.

Aside from any other discussion I also want to publicly thank Helmut Pottmann and his co-authors for their many great papers over the years. I would not want anyone to get the impression that I am ungrateful for or ignorant of the contribution they have made to the field and to some of my own recent work.

I have tried to give credit to those that have influenced and inspired me in my postings on this blog and elsewhere, and make no secret of the fact that all of my work builds on that of others from many sources, just as those sources themselves build on others. I believe  everyone who contributes to the increase of knowledge with original work is entitled to public credit (and I believe that there is at least general consensus about this, aside from the issue of legal and commercial entitlement).  As this is an informal blog and not an academic publication these acknowledgements have not always followed a standard format, which is why I felt the need to clarify matters.

Finally, on a related note, I want to end with an example of what I enjoy about all this – learning from the work shared by others, applying it in hopefully interesting ways, and sharing my work in turn. So – I am happy to introduce one of the most recent additions to the Kangaroo repertoire of forces – for optimizing a given pair of triangles towards having tangent incircles. Meshes where all pairs of adjacent triangles have this property are referred to as circle packing meshes – the wonderful properties of which are described and illustrated in the paper

Packing Circles and Spheres on surfaces (thesis of Mathias Hobinger 2009)

and also a shorter paper of the same name – Shiftner Hobinger, Wallner and Pottmann 2009

from which I learned of these meshes and took the simple length criteria to optimize for with this force.

[vimeo 28519758]

(more details and documentation for this feature to follow shortly)

Enjoy, join in the discussion, and keep on optimizing!