-
Design of Self-supporting Surfaces
Etienne VougaColumbia Univ. / KAUST
Mathias HobingerEvolute / TU Wien
Johannes WallnerTU Graz / TU Wien
Helmut PottmannKAUST
Figure 1: Left: Surfaces with irregularly placed holes are hard
to realize as masonry, where the mortar between bricks must not be
subject totensile stresses. The surface shown here, surprisingly,
has this property it has been found as the nearest self-supporting
shape from a givenfreeform geometry. The fictitious thrust network
used in our algorithms is superimposed, with edges cross-section
and coloring visualizingthe magnitude of forces (warmer colors
represent higher stresses.) Right: Curvature analysis with respect
to the Airy stress surface tells ushow to remesh shapes by
self-supporting quad meshes with planar faces. This guides
steel/glass constructions with low moments in nodes.
Abstract
Self-supporting masonry is one of the most ancient and
eleganttechniques for building curved shapes. Because of the very
geomet-ric nature of their failure, analyzing and modeling such
strutures ismore a geometry processing problem than one of
classical contin-uum mechanics. This paper uses the thrust network
method of anal-ysis and presents an iterative nonlinear
optimization algorithm forefficiently approximating freeform shapes
by self-supporting ones.The rich geometry of thrust networks leads
us to close connectionsbetween diverse topics in discrete
differential geometry, such asa finite-element discretization of
the Airy stress potential, perfectgraph Laplacians, and computing
admissible loads via curvaturesof polyhedral surfaces. This
geometric viewpoint allows us, in par-ticular, to remesh
self-supporting shapes by self-supporting quadmeshes with planar
faces, and leads to another application of thetheory: steel/glass
constructions with low moments in nodes.
CR Categories: I.3.5 [Computer Graphics]: Computational
Ge-ometry and Object ModelingCurve, surface, solid, and
objectrepresentations;
Keywords: Discrete differential geometry, architectural
geome-try, self-supporting masonry, thrust networks, reciprocal
force dia-grams, discrete Laplacians, isotropic geometry, mean
curvature
Links: DL PDF
1 Introduction
Vaulted masonry structures are among the simplest and at the
sametime most elegant solutions for creating curved shapes in
buildingconstruction. For this reason they have been an object of
inter-est since antiquity; large, non-convex examples of such
structuresinclude gothic cathedrals. They continue to be an active
topic ofresearch today.
Our paper is concerned with a combined geometry+statics
analysisof self-supporting masonry and with tools for the
interactive mod-eling of freeform self-supporting structures. Here
self-supportingmeans that the structure, considered as an
arrangement of blocks(bricks, stones), holds together by itself,
with additional supportpresent only during construction. This
analysis is based on the fol-lowing assumptions, which follow the
classic [Heyman 1966]:
Assumption 1: Masonry has no tensile strength, but the
individualbuilding blocks do not slip against each other (because
of frictionor mortar). On the other hand, their compressive
strength is suffi-ciently high so that failure of the structure is
by a sudden change ingeometry and not by material failure.
Assumption 2 (The Safe Theorem): If a system of forces can
befound which is in equilibrium with the load on the structure
andwhich is contained within the masonry envelope then the
structurewill carry the loads, although the actual forces present
may not bethose postulated by that system.
Our approach is twofold: We first give an overview of the
con-tinuous case of a smooth surface under stress, which turns out
tobe governed locally by the Airy stress function. This
mathemat-ical model is called a membrane in the engineering
literature andhas been applied to the analysis of masonry before.
The surfaceis self-supporting if and only if stresses are entirely
compressive.For computational purposes, stresses are discretized as
a fictitiousthrust network [Block and Ochsendorf 2007] contained in
the ma-sonry structure; this network is a system of forces in
equilibriumwith the structures deadload. It can be interpreted as a
finite ele-ment discretization of the continuous case, and it turns
out to havevery interesting geometry, with the Airy stress function
becoming a
-
polyhedral surface directly related to a reciprocal force
diagram.
While previous work in architectural geometry was mostly
con-cerned with aspects of rationalization and purely geometric
side-conditions which occur in freeform architecture, the focus of
thispaper is design with statics constraints. In particular, our
contribu-tions are the following:
Contributions. We present an optimization algorithm, basedon the
theory of thrust networks and Airy potentials, for
efficientlyfinding a self-supporting surface near a given arbitrary
referencesurface (3), and build a tool for interactive design of
self-support-ing surfaces based on this algorithm (4). Freeform
masonry isbased on such surfaces.
The discrete stress Laplacian derived from a thrust networkwith
compressive forces is a so-called perfect one (2.2). We use itto
argue why our discretizations are faithful to the continuous
case.
We connect the physics of self-supporting surfaces with the
ge-ometry of isotropic 3-space, and express the equations
governingself-supporting surfaces in terms of curvatures (2.3) and
(2.4).Likewise we establish a connection between the stress
Laplacianand mean curvatures of polyhedral surfaces. This
theoretical partof the paper is a contribution to Discrete
Differential Geometry.
We use the geometric knowledge we have gathered to find
par-ticularly nice families of self-supporting surfaces, especially
planarquadrilateral representations of thrust networks (5). This
leads tosteel/glass structures with low bending and torsion
moments.
Related Work. Unsupported masonry has been an active topic
ofresearch in the engineering community. The foundations for
themodern approach were laid by Jacques Heyman [1966] and
areavailable as the textbook [Heyman 1995]. The theory of
recipro-cal force diagrams in the planar case was studied by J.
Maxwell;a unifying view on polyhedral surfaces, compressive forces
andcorresponding convex force diagrams is presented by [Ash et
al.1988]. F. Fraternali [2002; 2010] established a connection
betweenthe continuous theory of stresses in membranes and the
discretetheory of forces in thrust networks, by interpreting the
latter as anon-conforming finite element discretization of the
former.
Several authors have studied the problem of finding discrete
com-pressive force networks contained within the boundary of
masonrystructures; previous work in this area includes [ODwyer
1998] and[Andreu et al. 2007]. Fraternali [2010] proposed solving
for thestructures discrete stress surface, and examining its convex
hull tostudy the structures stability and susceptibility to
cracking. Thisapproach works well for analyzing existing
structures, where theboundary tractions can be measured and the
stress surface is knownto be close to convex, but is not an ideal
design tool since in suchsettings the boundary tractions are
unknown, and where replacing anon-convex intial stress surface by
its convex hull can cause large,uncontrolled global changes to the
surface being designed.
Philippe Blocks seminal thesis introduced Thrust Network
Analy-sis, which pioneered the use of thrust networks and their
reciprocaldiagrams for efficient and practical design of
self-supporting ma-sonry structures. By first seeking a reciprocal
diagram of the topview, guaranteeing equilibrium of horizontal
forces, then solvingfor the heights that balance the vertical
loads, Thrust Network Anal-ysis linearizes the form-finding
problem. For a thorough overviewof this methodology, see e.g.
[Block and Ochsendorf 2007; Block2009]. Recent work by Block and
coauthors extends this methodin the case where the reciprocal
diagram is not unique; for differentchoices of reciprocal diagram,
the optimal heights can be found us-ing the method of least squares
[Van Mele and Block 2011], and the
search for the best such reciprocal diagram can be automated
usinga genetic algorithm [Block and Lachauer 2011].
Other approaches to the design of self-supporting structures
includemodeling these structures as damped particle-spring systems
(dy-namic relaxation methods) [Kilian and Ochsendorf 2005;
Barnes2009], and mirroring the rich tradition in architecture of
designingself-supporting surfaces using hanging chain or membrane
models(for instance by Frei Otto, Antoni Gaudi, and Heinz Isler)
[Hey-man 1998; Kotnik and Weinstock 2012]. Force density meth-ods
[Linkwitz and Schek 1971] linearize the form-finding prob-lem by
solving for static equilibrium with respect to position vari-ables,
given prescribed prestresses in the form of axial force den-sities
[Grundig et al. 2000]. Alternatively, masonry structures canbe
represented by networks of rigid blocks [Livesley 1992],
whoseconditions on the structural feasibility were incorporated
into pro-cedural modeling of buildings [Whiting et al. 2009].
Algorithmic and mathematical methods relevant to this paper
arework on the geometry of PQ meshes [Liu et al. 2006],
discretecurvatures for such meshes [Pottmann et al. 2007; Bobenko
et al.2010], in particular curvatures in isotropic geometry
[Pottmann andLiu 2007]. Schiftner and Balzer [2010] discuss
approximating areference surface by a quad mesh with planar faces,
whose layoutis guided by statics properties of that surface.
2 Self-supporting Surfaces
This section contains the theoretical basis of the paper. We
be-gin in 2.1 and 2.2 with a review of the continuous theoryof
self-supporting surfaces, and their discretization as thrust
net-works [Block and Ochsendorf 2007]. This discrete model and
itsassociated equilibrium equations form the groundwork of our
opti-mization algorithm (4) for designing self-supporting surfaces.
In2.3, 2.4, and 2.5 we draw connections between the theory
ofself-supporting surfaces, curvature measures in isotropic
geome-try, and discrete Laplace-Beltrami operators. These insights
lead tosome observations on existence of convergence of discrete
approx-imations to smooth self-supporting surfaces, and are
important forthe later discussions of planar quad remeshing and
special classesof self-supporting surfaces (5).
2.1 The Continuous Theory
We model masonry as a surface given by a height field s(x, y)
de-fined in some planar domain . We assume a vertical load densityF
(x, y) over the top view usually F represents the structuresown
weight. By definition this surface is self-supporting if and onlyif
there exists a negative semidefinite (compressive) stress tensor
over the surface whose stresses are in equilibrium with the
actingforces. Rewriting the equilibrium equations in plane
coordinates(x, y), we have that such a stress tensor exists if and
only if thereexists a field M(x, y) = g1 det g of 2 2 symmetric
positivesemidefinite matrices satisfying
div(Ms) = F, divM = 0, (1)
where g is the induced metric`1+s2x sxsysxsy 1+s2y
and the divergence
operator div`u(x,y)v(x,y
= ux+vy is understood to act on the columns
of a matrix (see e.g. [Fraternali 2010], [Giaquinta and Giusti
1985]).
The condition divM = 0 says that M is locally the Hessian of
areal-valued function (the Airy stress potential) [Green and
Zerna2002]: With the notation
M =`m11 m12m12 m22
cM = ` m22 m12m12 m11
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S
viAiFiAiFiAiFiAiFiAiFiAiFiAiFiAiFiAiFiAiFiAiFiAiFiAiFiAiFiAiFiAiFiAiFi
S
S
wijeij
eij
Figure 2: A thrust network S with dangling edges indicating
ex-ternal forces (left). This network together with compressive
forceswhich balance vertical loads AiFi projects onto a planar mesh
S with equilibrium compressive forces wijeij in its edges.
Rotatingforces by 90 leads to the reciprocal force diagram S
(right).
it is clear that divM = 0 is an integrability condition for cM ,
solocally there is a potential with
cM = 2, i.e., M = d2.If the domain is simply connected, this
relation holds globally.Positive semidefiniteness of M (or
equivalently of cM ) character-izes convexity of the Airy potential
. The Airy function enterscomputations only by way of its
derivatives, so global existence isnot an issue.
Remark: Stresses at boundary points depend on the way the
sur-face is anchored: A fixed anchor means no condition, but a
freeboundary with outer normal vector n means Ms,n = 0.
Stress Laplacian. Note that divM = 0 yields div(Ms) =tr(M2s),
which we like to call s. The operator is sym-metric. It is elliptic
(as a Laplace operator should be) if and only ifM is positive
definite, i.e., is strictly convex. The balance condi-tion (1) may
be written as s = F.
2.2 Discrete Theory: Thrust Networks
We discretize a self-supporting surface by a mesh S = (V,E, F
)(see Figure 2). Loads are again vertical, and following Block
[2007]we discretize them as force densities Fi associated with
vertices vi.The load acting on this vertex is then given by FiAi,
whereAi is anarea of influence (using a prime to indicate
projection onto the xyplane, Ai is the area of the Voronoi cell of
vi w.r.t. V
). We assumethat stresses are carried by the edges of the mesh:
the force exertedon the vertex vi by the edge connecting vi,vj is
given by
wij(vj vi), where wij = wji 0.The weights wij in these equations
can be interpreted as axialforce densities along the edges. The
nonnegativity of the individ-ual weights wij expresses the
compressive nature of forces. Thebalance conditions at vertices
then read as follows: With vi =(xi, yi, si) we haveX
jiwij(xj xi) =
Xji
wij(yj yi) = 0, (2)Xji
wij(sj si) = AiFi. (3)
A mesh equipped with edge weights in this way is a discrete
thrustnetwork [Block 2009]. Invoking the safe theorem, we can state
thata masonry structure is self-supporting, if we can find a thrust
net-work with compressive forces which is entirely contained within
thestructure. In other words, for a given surface the vi are known
and
wk
vkS =
= S
fk
Figure 3: Airy stress potential and its polar dual .
projectsonto the same planar mesh as S does, while projects ontothe
reciprocal force diagram. A primal face fk lies in the planez = x +
y + the corresponding dual vertex iswk = (, ,).
the wij are unknown; the surface is self-supporting whenever a
so-lution {wij} to the above equations exist. Finding a
self-supportingsurface near one that is not amounts to solving for
a simultaneoussolution in vi andwij ; we describe one efficient
approach for doingso in 3.
Reciprocal Diagram. Equations (2) have a geometric
interpreta-tion: with edge vectors
eij = vj vi = (xj , yj) (xi, yi),
Equation (2) asserts that vectors wijeij form a closed cycle.
Rotat-ing them by 90 degrees, we see that likewise
eij = wijJeij , with J =
`0 11 0
,
form a closed cycle (see Figure 2). If the mesh S is simply
con-nected, there exists an entire reciprocal diagram S which is
acombinatorial dual of S, and which has edge vectors eij [Blockand
Ochsendorf 2007]. Its vertices are denoted by vi .
Remark: If S is a Delaunay triangulation, then the
correspondingVoronoi diagram is an example of a reciprocal
diagram.
Polyhedral Stress Potential. We can go further and construct
aconvex polyhedral Airy stress potential surface with verticeswi =
(xi, yi, i) combinatorially equivalent to S by requiring thata
primal face of lies in the plane z = x+ y + if and only if(, ) is
the corresponding dual vertex of S (see Figure 3). Ob-viously this
condition determines up to vertical translation. Forexistence see
[Ash et al. 1988]. The inverse procedure constructsa reciprocal
diagram from . This procedure works also if forcesare not
compressive: we can construct an Airy mesh which hasplanar faces,
but it will no longer be a convex polyhedron.
The vertices of can be interpolated by a piecewise-linear
function(x, y). It is easy to see that the derivative of (x, y)
jumps by theamount eij = wijeij when crossing over the edge eij at
rightangle, with unit speed. This identifies as the Airy polyhedron
in-troduced by [Fraternali et al. 2002] as a finite element
discretizationof the continuous Airy function (see also [Fraternali
2010]).
If the mesh is not simply connected, the reciprocal diagram
andthe Airy polyhedron exist only locally. Our computations do
notrequire global existence.
Polarity. Polarity with respect to the Maxwell paraboloid z
=12(x2+y2) maps the plane z = x+y+ to the point (, ,).
Thus, applying polarity to and projecting the result into the
xyplane reconstructs the reciprocal diagram = S (see Fig. 3).
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Discrete Stress Laplacian. The weights wij may be used to
de-fine a graph Laplacian which on vertex-based functions acts
as
s(vi) =X
jiwij(sj si).
This operator is a perfect discrete Laplacian in the sense of
[War-detzky et al. 2007], since it is symmetric by construction,
Equa-tion (2) implies linear precision for the planar top view mesh
S (i.e., f = 0 if f is a linear function), and wij 0
ensuressemidefiniteness and a maximum principle for -harmonic
func-tions. Equation (3) can be written as s = AF .
Note that is well defined even when the underlying meshes arenot
simply connected.
2.3 Surfaces in Isotropic Geometry
It is worthwhile to reconsider the basics of self-supporting
surfacesin the language of dual-isotropic geometry, which takes
place in R3with the z axis as a distinguished vertical direction.
The basic ele-ments of this geometry are planes, having equation z
= f(x, y) =x + y + . The gradient vector f = (, ) determines
theplane up to translation. A plane tangent to the graph of the
functions(x, y) has gradient vectors.There is the notion of
parallel points: (x, y, z) (x, y, z) x = x, y = y.
Remark: The Maxwell paraboloid is considered the unit sphere
ofisotropic geometry, and the geometric quantities considered
aboveare assigned specific meanings: The forces eij = wijeij
aredihedral angles of the Airy polyhedron , and also lengths
ofedges of . We do not use this terminology in the sequel.
Curvatures. Generally speaking, in the differential geometry
ofsurfaces one considers the Gauss map from a surface S to a
con-vex unit sphere by requiring that corresponding points have
par-allel tangent planes. Subsequently mean curvature Hrel and
Gaus-sian curvature Krel relative to are computed from the
derivatived. Classically is the ordinary unit sphere x2 + y2 + z2 =
1, sothat maps each point to its unit normal vector.
In our setting, parallelity is a property of points rather than
planes,and the Gauss map goes the other way, mapping the
tangentplanes of the unit sphere z = (x, y) to the corresponding
tan-gent plane of the surface z = s(x, y). If we know which pointa
plane is attached to, then the Gauss map is determined by theplanes
gradient. So we simply write
7 s.By moving along a curve u(t) = (x(t), y(t)) in the
parameterdomain we get the first variation of tangent planes: d
dt|u(t) =
(2)u. This yields the derivative (2)u d7 (2s)u, for allu, and
the matrix of d is found as (2)1(2s). By definition,curvatures of
the surface s relative to are found as
Krels = det(d) =det2sdet2 ,
Hrels =12
tr(d) = 12
tr
M
det22s
=s
2 det2 .
The Maxwell paraboloid 0(x, y) = 12 (x2 + y2) is the
canonical
unit sphere of isotropic geometry, with Hessian E2. Curvatures
rel-ative to 0 are not called relative and are denoted by the
symbolsH,K instead of Hrel,Krel. The observation
= tr(M2) = tr(d22) = 2 det2
together with the formulas above implies
Ks = det2s, K = det2 = Hrels = s2K
=s
.
Relation to Self-supporting Surfaces. Summarizing the for-mulas
above, we rewrite the balance condition (1) as
2KHrels = s = F. (4)
Let us draw some conclusions:
Since Hrel = 1 we see that the load F = 2K is admissiblefor the
stress surface (x, y), which is hereby shown as self-supporting.
The quotient of loads yields Hrels = F/F.
If the stress surface coincides with the Maxwell paraboloid,then
constant loads characterize constant mean curvaturesurfaces,
because we get K = 1 and Hs = F/2.
If s1, s2 have the same stress potential , then Hrels1s2 =Hrels1
Hrels2 = 0, so s1 s2 is a (relative) minimal surface.
2.4 Meshes in Isotropic Geometry
A general theory of curvatures of polyhedral surfaces with
respectto a polyhedral unit sphere was proposed by [Pottmann et al.
2007;Bobenko et al. 2010], and its dual complement in isotropic
geome-try was elaborated on in [Pottmann and Liu 2007]. As
illustrated byFigure 4, the mean curvature of a self-supporting
surface S relativeto its discrete Airy stress potential is
associated with the vertices ofS. It is computed from areas and
mixed areas of faces in the polarpolyhedra S and :
Hrel(vi) =Ai(S,)Ai(,)
, where
Ai(S,) = 14
Xk:fk1-ring(vi)
det(vk ,wk+1) + det(w
k ,v
k+1).
The prime denotes the projection into the xy plane, and
summationis over those dual vertices which are adjacent to vi.
Replacing vkby wk yields Ai(,) =
12
Pdet(wk ,w
k+1).
viS
w0v0
w1
v1
w2
v2
w3v3
f0
fS0
f1
fS1
f2
fS2
f3
fS3
S
Figure 4: Mean curvature of a vertex vi of S: Correspondingedges
of the polar duals S, are parallel, and mean curvatureaccording to
[Pottmann et al. 2007] is computed from the verticespolar to faces
adjacent to vi. For valence 4 vertices the case ofzero mean
curvature shown here is characterized by parallelity
ofnon-corresponding diagonals of corresponding quads in S,.
Proposition. If is the Airy surface of a thrust network S,
thenthe mean curvature of S relative to is computable as
Hrel(vi) =
Pji wij(sj si)Pji wij(j i)
=s
vi
. (5)
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||||||||||||||||||||||||||||||||||||
impossible feature
(a) (b)
S
(c)
S = (d) (view from below)
Figure 5: The top of the Lilium Tower (a) cannot stand as a
masonry structure, because its central part is concave. Our
algorithm findsa nearby self-supporting mesh (b) without this
impossible feature. (c) shows the corresponding Airy mesh and
reciprocal force diagramS . (d) The user can edit the original
surface, such as by specifying that the center of the surface is
supported by a vertical pillar, and theself-supporting network
adjusts accordingly.
Proof. It is sufficient to show 2Ai(S,) = Pji wij(sj si).For
that, consider edges e1, . . . , en emanating from vi. The
dualcycles in and S without loss of generality are given by
ver-tices (v1 , . . . ,vn ) and (w1 , . . . ,wn ), respectively.
The latterhas edges wj+1 wj = wijJej (indices modulo n).Without
loss of generality vi = 0, so the vertex vj by constructionequals
the gradient of the linear function x 7 vj ,x defined bythe
properties ej1 7 sj1 si, ej 7 sj si. Correspondingedge vectors vj+1
vj and wj+1 wj are parallel, becausevj+1vj , ej = (sj si) (sj si) =
0. Expand 2Ai(S,):
12
Pdet(wj ,v
j+1) + det(v
j ,w
j+1)
= 12
Pdet(wj wj+1,vj+1) + det(vj ,wj+1 wj )
= 12
Pdet(wijJej ,vj+1) + det(vj , wijJej)
=P
det(vj , wijJej) =
Pwijvj , ej =
Pwij(sj si).
Here we have used det(a, Jb) = a,b.
In order to discretize (4), we also need a discrete Gaussian
curva-ture, usually defined as a quotient of areas which correspond
underthe Gauss mapping. We define
K(vi) =Ai(,)
Ai,
where Ai is the Voronoi area of vertex vi in the projected mesh
S used in (3).
Remark: If the faces of the thrust network S are not planar, the
sim-ple trick of introducing additional edges with zero forces in
themmakes them planar, and the theory is applicable. In the
interest ofspace, we refrain from elaborating further.
Discrete Balance Equation. The discrete version of the
balanceequation (4) reads as follows:
Theorem. A simply-connected mesh S with vertices vi =(xi, yi,
si) can be put into static equilibrium with vertical nodalforces
AiFi if and only if there exists a combinatorially equivalentmesh
with planar faces and vertices (xi, yi, i), such that cur-vatures
of S relative to obey
2K(vi)Hrel(vi) = Fi (6)
at every interior vertex and every free boundary vertex vi. S
canbe put into compressive static equilibrium if and only if there
existsa convex such .
Proof. The relation between equilibrium forces wijeij in S
andthe polyhedral stress potential has been discussed above, andso
has the equivalence wij 0 convex (see e.g.[Ash et al. 1988] for a
survey of this and related results). It re-mains to show that
Equations (2) and (6) are equivalent. This isthe case because the
proposition above implies 2K(vi)Hrel(vi) =2Ai(,)
Ai
Ai(,S)Ai(,)
= 1Ai
(Pji wij(sj si)) = 1AiAiFi.
2.5 Convergence.
When considering discrete thrust networks as discretizations
ofcontinuous self-supporting surfaces, the following question is
im-portant: For a given smooth surface s(x, y) with stress
potential ,does there exist a polyhedral surface S in equilibrium
approximat-ing s(x, y), whose top view is a given planar mesh S ?
We restrictour attention to triangle meshes, where planarity of the
faces of thediscrete stress surface is not an issue. Equivalently,
we ask:
Does S have a reciprocal diagram whose corresponding
Airypolyhedron approximates the continuous Airy potential ?(if the
surfaces involved are not simply connected, these ob-jects are
defined locally).
Does S possess a perfect discrete Laplace-Beltrami opera-tor in
the sense of Wardetzky et al. [2007] whose weightsare the edge
length scalars of such a reciprocal diagram?
From [Wardetzky et al. 2007] we know that perfect Laplacians
ex-ist only on regular triangulations which are projections of
convexpolyhedra. On the other hand, previous sections show how to
ap-propriately re-triangulate: Let be a triangle mesh convex hull
ofthe vertices (xi, yi, (xi, yi)), where (xi, yi) are vertices of S
.Then its polar dual projects onto a reciprocal diagram with
pos-itive edge weights, so has positive weights, and the
vertices(xi, yi, si) of S can be found by solving the discrete
Poisson prob-lem (s)i = AiFi.
We expect, but we dont prove, that the discrete approximatesits
continuous counterpart for reasonable sampling (after all it
isdirectly derived from (x, y)). This implies that solving the
dis-crete Poisson equation leads to a mesh approximating its
continu-ous counterpart s(x, y), and we have convergence as the
samplingdensity increases. A rigorous analysis is a topic for
future research.
3 Thrust Networks from Reference Meshes
Consider now the problem of taking a given reference mesh, sayR,
and finding a combinatorially equivalent mesh S in static
equi-librium approximating R. The loads on S include
user-prescribedloads as well as the dead load caused by the meshs
own weight.Conceptually, finding S amounts to minimizing some
formulation
-
Figure 6: A freeform surface (left) needs adjustments around
theentrance arch and between the two pillars in order to be
self-supporting; our algorithm finds the nearby surface in
equilibrium(right) that incorporates these changes.
of distance between R and S, subject to constraints (2), (3),
andwij 0. For any choice of distance this minimization will be
anonlinear, non-convex, inequality-constrained variational
problem.Our experience with black-box solvers [Wachter and Biegler
2006]is that they perform well for surfaces without complex
geometry orfor polishing reference meshes close to self-supporting,
but fail toconverge in reasonable time for more complicated shapes
such asthe one of Fig. 1, left. We therefore propose the following
special-ized, staggered linearization for solving the optimization
problem:
0. Start with an initial guess S = R.1. Estimate the self-load
on the vertices of S, using their current
positions.2. Fixing S, locally fit an associated stress surface
.3. Alter positions vi to improve the fit.4. Repeat from Step 1
until convergence.
Remark: This staggered approach shares several advantages
ofsolving the full nonlinear problem: a nearby self-supporting
sur-face is found given only a suggested reference shape, without
need-ing to single one of the many possible top view reciprocal
diagramsor needing to specify boundary tractions these are found
automat-ically during optimization. Although providing an initial
top viewgraph with good combinatorics remains important, by not
fixing thetop view our approach allows the thrust network to slide
both ver-tically and tangentially to the ground, essential to
finding faithfulthrust networks for surfaces with free boundary
conditions.
Step 1: Estimating Self-Load. The dead load due to the sur-faces
own weight depends not only on the top view of S, but alsoon the
surface area of its faces. To avoid adding nonlinearity tothe
algorithm, we estimate the load coefficients Fi at the beginningof
each iteration, and assume they remain constant until the
nextiteration. We estimate the load AiFi associated with each
vertexby calculating its Voronoi surface area on each of its
incident faces(note that this surface area is distinct from Ai, the
vertexs Voronoiarea on the top view), and then multiplying by a
user-specified sur-face density .
Step 2: Fit a Stress Surface. In this step, we fix S and try
tofit a stress surface subordinate to the top view S of the
primalmesh. We do so by searching for dihedral angles between the
facesof which minimize, in the least-squares sense, the error in
forceequilibrium (6) and local integrability of . Doing so is
equivalentto minimizing the squared residuals of Equations (3) and
(2), withthe positions held fixed. We define the equilibrium
energy
E =X
i
00AiFi
X
jiwij(vj vi)
2, (7)where i runs through interior and free boundary vertices,
and solve
minwij E, s.t. 0 wij wmax. (8)
Fig. Vertices Edges Time (s) Iterations Max. Rel. Error5b 1201
3504 21.6 9 4.2 1055d 1200 3500 26.5 10 8.5 1056 1535 2976 17.0 21
2.7 1058 752 2165 8.0 9 5.8 10511 2358 4302 19.5 9 3.0 10416 527
998 5.7 25 2.4 105
Table 1: Numerical details about our examples. We show the
clocktime needed by an Intel Xeon 2.3GHz desktop PC with 4 GB of
RAMto find a self-supporting thrust network and associated stress
sur-face from the examples reference mesh; we also give the number
ofouter iterations of the four steps in (3). The maximum relative
er-ror is the dimensionless quantity maxi AiFi(001)
Pji wij(vj
vi)/AiFi (the maximum is taken over interior vertices vi).
Here wmax is an optional maximum weight we are willing to
as-sign (to limit the amount of stress in the surface). This
convex,sparse, box-constrained least-squares problem [Friedlander
2007]always has a solution. If the objective is 0 at this solution,
S is self-supporting we are done. Otherwise, S is not
self-supporting andits vertices must be moved.
Step 3: Alter Positions. In the previous step we fit as best
aspossible a stress surface to S. There are two possible kinds
oferror with this fit: the faces around a vertex (equivalently, the
recip-rocal diagram) might not close up; and the resulting stress
forcesmight not be exactly in equilibrium with the loads. These
errorscan be decreased by modifying the top view and heights of S,
re-spectively. It is possible to simply solve for new vertex
positionsthat put S in static equilibrium, since Equations (2) and
(3) withwij fixed form a square linear system that is typically
nonsingular.
While this approach would yield a self-supporting S, this mesh
isoften far from the reference mesh R, since any local errors in
thestress surface from Step 2 amplify into global errors in S. We
pro-pose instead to look for new positions that decrease the
imbalancein the stresses and loads, while also penalizing drift
away from thereference mesh:
minv E + X
i
ni,vi v0i
2 + v v0P2,where v0i is the position of the i-th vertex at the
start of this stepof the optimization, ni is the starting vertex
normal (computed asthe average of the incident face normals), v0P
is the projection of v
0
onto the reference mesh, and > are penalty coefficients that
aredecreased proportionally to the decrease in E at every iteration
ofSteps 13. = 1 and = 0.1 worked well as initial values of
theseparameters for the examples shown in this paper. The second
termallows S to slide over itself (if doing so improves
equilibrium) butpenalizes drift in the normal direction. The third
term, weaker thanthe second, regularizes the optimization by
preventing large driftaway from the reference surface or excessive
tangential sliding.
Implementation Details. Solving the weighted
least-squaresproblem of Step 3 amounts to solving a sparse,
symmetric linearsystem. While the MINRES algorithm [Paige and
Saunders 1975]is likely the most robust algorithm for solving this
system, in prac-tice we have observed that the method of conjugate
gradients workswell despite the potential ill-conditioning of the
objective matrix.
Limitations. This algorithm is not guaranteed to always
con-verge; this fact is not surprising from the physics of the
problem
-
(a) (b) (c)
Figure 7: Interactive Edits. (a) shows aself-supporting thrust
network in fact itis a Koebe mesh as mentioned in 5. In(b) boundary
conditions are defined andpillars have been added. Cutting alongthe
highlighted edge and optimizing forthe self-supporting property
results in themesh shown in (c).
(a) (b) (c) (d)
Figure 8: Destruction sequence. We simulate removing small parts
of masonry (their location is shown by a yellow ball) and the
falling offof further pieces which are no longer supported after
removal. For this example, removing a certain small number of
single bricks does notaffect stability (a,b). Removal of material
at a certain point (yellow ball in (b)) will cause a greater part
of the structure to collapse, as seenin (c). (d) shows the result
after one more removal (all images show the respective thrust
networks, not the reference surface).
(if the boundary of the reference mesh encloses too large of a
re-gion, wmax is set too low, and the density of the surface too
high,a thrust network in equilibrium simply does not exist the
vault istoo ambitious and cannot be built to stand; pillars are
needed.)
We can, however, make a few remarks. Only Step 1 can increasethe
equilibrium energy E of Equation (7). Step 2 always decreasesit,
and Step 3 does as well as 0. Moreover, as 0 and 0, Step 3
approaches a linear system with as many equationsas unknowns; if
this system has full rank, its solution sets E = 0.These facts
suggest that the algorithm should generally convergeto a thrust
network in equilibrium, provided that Step 1 does notincrease the
loads by too much at every iteration, and this is indeedwhat we
observe in practice. One case where this assumption isguaranteed to
hold is if the thickness of the surface is allowed tofreely vary,
so that it can be chosen so that the surface has uniformdensity
over the top view.
We have observed several situations where our algorithm has
diffi-culty converging to a high-quality solution, even though the
under-lying optimization problem is feasible:
Vertices with high valence (such as can occur at irregu-lar
vertices of triangle meshes) often become local maxima(bumps) after
optimization. In the worst case, the algorithmcan stall atE > 0
due to the linear system in Step 3 becomingsingular and infeasible.
This failure occurs, for instance, whenan interior vertex has
height zi lower than all of its neighbors,and Step 2 assigns all
incident edges to that vertex a weight ofzero: clearly no amount of
moving the vertex or its neighborscan bring the vertex into
equilibrium. We avoid such degen-erate configurations by bounding
weights slightly away fromzero in (8), trading increased robustness
for slight smoothingof the resulting surface. We also havent
noticed these arti-facts in lower-valence meshes such as quad
meshes.
The algorithm does not usually converge if the reference meshhas
a self-intersecting top view (i.e., isnt a height field), al-though
it can occasionally correct slight overhangs (by de-forming the top
view so that the network mesh is a heightfield).
The algorithm may not converge if the deformation needed to
make the refence mesh self-supporting is too large. Specify-ing
a reference mesh insufficiently supported by fixed bound-ary
vertices is the most common such situation for instance,marking the
bases of the pillars in Fig. 7 as free instead offixed boundaries
causes our algorithm to fail, even though a(drastic)
self-supporting deformation of that surface does ex-ist.
4 Results
Interactive Design of Self-Supporting Surfaces. The
opti-mization algorithm described in the previous section forms the
ba-sis of an interactive design tool for self-supporting surfaces.
Usersmanipulate a mesh representing a reference surface, and the
com-puter searches for a nearby thrust network in equilibrium (see
e.g.Figure 7). Features of the design tool include:
Handle-based 3D editing of the reference mesh using Lapla-cian
coordinates [Lipman et al. 2004; Sorkine et al. 2003] toextrude
vaults, insert pillars, and apply other deformations tothe
reference mesh. Handle-based adjustments of the heights,keeping the
top view fixed, and deformation of the top view,keeping the heights
fixed, are also supported. The thrust net-work adjusts
interactively to fit the deformed positions, givingthe usual visual
feedback about the effects of edits on whetheror not the surface
can stand.
Specification of boundary conditions. Points of contact be-tween
the reference surface and the ground or environmentare specified by
pinning vertices of the surface, specifyingthat the thrust network
must coincide with the reference meshat this point, and relaxing
the condition that forces must be inequilibrium there.
Interactive adjustment of surface density, external loads,
andmaximum permissible stress per edge, with visual feedbackof how
these parameters affect the fitted thrust network.
Upsampling of the thrust network through Catmull-Clark
sub-division and polishing of the resulting refined thrust
networkusing optimization (3).
Visualization of the stress surface dual to the thrust
networkand corresponding reciprocal diagram.
-
Figure 9: Stability Test. Left: Coloring and cross-section of
edgesvisualize the magnitude of forces in a thrust network which is
inequlibrium with this domes dead load. Right: When an
additionalload is applied, there exists a corresponding compressive
thrust net-work which is still contained in the masonry hull of the
originaldome. This implies stability of the dome under that load.
14 m
shell thickness 0.1 m, = 2, 500 kg/m311,000 kg
Figure 10: Stability test similar to Figure 9, but with a shell
thickness of 1 m, in order to better visualize the way the thrust
network starts toleave the masonry hull as the load increases.
Additional loads are 0 kg, 5,000 kg, 10,000 kg, and 20,000 kg,
resp., from left to right.
Examples. Top of the Lilium Tower: Consider the top portion
ofthe steel-glass exterior surface of the Lilium Tower (being built
inWarszaw, see Fig. 5). What is if we had wanted to build this
sur-face out of masonry instead? This surface contains a concave
partwith local minimum in its interior and so cannot possibly be
self-supporting without modification. Given this surface as a
referencemesh, our algorithm constructs a nearby thrust network in
equilib-rium without the impossible feature. The user can then
explore howediting the reference mesh adding a pillar, for example
affectsthe thrust network and its deviation from the reference
surface.
Example: Freeform Structure with Two Pillars. Suppose an
archi-tects experience and intuition has permitted the design of a
nearlyself-supporting freeform surface (Figure 6). Our algorithm
revealsthose edits needed to make the structure sound principally
aroundthe entrance arch, and the area between the two pillars.
(a) (b)
Figure 11: A mesh with holes (a) requires large deformations
toboth the top view and heights to render it self-supporting
(b)
Example: Interactive Editing. Figure 7 shows an example of the
de-sign and optimization workflow. Starting with a mesh, we first
addpillars in the center and clean up the outer boundary (by fixing
it tothe floor). A subsequent cut needs a further round of
optimization.This surface is neither convex nor simply connected,
and exhibits amix of boundary conditions, none of which cause our
algorithm anydifficulty; it always finds a self-supporting thrust
network near thedesigned reference mesh. The user is free to make
edits to the refer-ence mesh, and the thrust network adapts to
these edits, providingthe user feedback on whether these designs
are physically realiz-able [we refer to the accompanying video for
interactive buildingand editing of freeform self-supporting
shapes].
Example: Destruction Sequence. In Figure 8 we simulate
removingparts of masonry and the falling off of further pieces
which are nolonger supported after removal. This is done by
deleting the 1-neighborhood of a vertex and solving for a new
thrust network in
compressive equilibrium close to the original reference surface.
Wedelete those parts of the network which deviate too much and
areno longer contained in the masonry hull, and iterate.
Example: Swiss Cheese. Cutting holes in a self-supporting
surfaceinterrupts force flow lines and causes dramatic global
changes tothe surface stresses, often to the point that the surface
is no longerin equilibrium. Whether a given surface with many such
holes canstand is far from obvious. Figure 11a shows such an
implausibleand unstable surface; our optimization finds a nearby,
equally im-plausible but stable surface without difficulty (Figs.
1, left and 11b).
Example: Stability Test: See Figures 9, 10 for a series of
imageswhich visualize the effect of additional loads on a thrust
network.
Example: Structural Glass. See Figure 16 for details on a
self-supporting surface which is realized not as masonry, but as a
steel/glass construction with glass as a structural element.
5 Special Self-Supporting Surfaces
PQ Meshes. Meshes with planar faces are of particular interestin
architecture, so in this section we discuss how to remesh a
giventhrust network in equilibrium such that it becomes a quad
meshwith planar faces (again in equilibrium). If this mesh is
realizedas a steel-glass construction, it is self-supporting in its
beams alone,with no forces exerted on the glass (this is the usual
manner of usingglass). The beams constitute a self-supporting
structure which is inperfect force equilibrium (without moments in
the nodes) if onlythe deadload is applied. (In such constructions,
the restriction thatthe internal forces are compressive does not
apply.)
((a) ((b) ((c)
Figure 12: Directly enforcing planarity of the faces of even
avery simple self-supporting quad-mesh vault (a) results in a
sur-face far removed from the original design (b). Starting instead
froma remeshing of the surface with edges following relative
principalcurvature directions yields a self-supporting, PQ mesh far
morefaithful to the original (c).
Taking an arbitrary non-planar quad mesh and attempting naive,
si-
-
Figure 13: Planar quad remeshing ofthe surface of Fig. 5. (a)
Relative prin-cipal directions, found from eigenvec-tors of (2)12s.
(b) Quad meshguided by principal directions is al-most planar and
almost self-support-ing. (c) Small changes achieve
bothproperties.
(a) (b) (c)
Figure 14: Planar quad remeshing ofthe surface of Figure 6.
Left: Relativeprincipal directions. Right: The re-sult of
optimization is a self-support-ing PQ mesh, which guides a
moment-free steel/glass construction (interiorview, see also Fig.
1).
multaneous enforcement of planarity and static equilibrium
eitherby staggering a planarity optimization step every outer
iteration, oradding a planarity penalty term to the position update
does notyield good results, as shown in Figure 12. Indeed, as we
will seelater in this section, such a planar perturbation of a
thrust networkis not expected to generally exist.
Consider a planar quad mesh S with vertices vij = (xij , yij ,
sij)which approximates a given continuous surface s(x, y). It is
knownthat S must approximately follow a network of conjugate curves
inthe surface (see e.g. [Liu et al. 2006]). We can derive this
conditionin an elementary way as follows: Using a Taylor expansion,
wecompute the volume of the convex hull of the quadrilateral vij
,vi+1,j , vi+1,j+1, vi,j+1, assuming the vertices lie exactly on
thesurface s(x, y). This results in
vol = 16
det(a1,a2) `(a1)
T 2sa2
+ ,where a1 =
`xi+1,jxijyi+1,jyij
, a2 =
`xi,j+1xijyi,j+1yij
,
and the dots indicate higher order terms. We see that planarity
re-quires (a1)T 2sa2 = 0. In addition to the mesh S
approximatingthe surface s(x, y), the corresponding polyhedral Airy
surface must approximate (x, y); thus we get the conditions
(a1)T 2s a2 = (a1)T 2 a2 = 0.
a1,a2 are therefore eigenvectors of (2)12s. In view of 2.3,a1,a2
indicate the principal directions of the surface s(x, y) rela-tive
to (x, y).
In the discrete case, where s, are not given as continuous
surfaces,but are represented by a mesh in equilibrium and its Airy
mesh, weuse the techniques of Schiftner [2007] and Cohen-Steiner
and Mor-van [2003] to approximate the Hessians 2s, 2, compute
prin-cipal directions as eigenvectors of (2)12s, and
subsequentlyfind meshes S, approximating s, which follow those
directions.Global optimization can now polish S, to a valid thrust
networkwith discrete stress potential, where before it failed: we
do so bytaking the planarity energy
Pf (2pi f )2, where the sum runs
over faces and f is the sum of the interior angles of face f ,
lin-earizing it at every iteration, and adding it to the objective
functionof the position update (Step 3). Convexity of ensures that
S isself-supporting.
Note that for each , the relative principal curvature directions
givethe unique curve network along which a planar quad
discretization
of a self-supporting surface is possible. Other networks wont
work(see Figure 12). Figures 13 and 14 further illustrate the
result ofapplying this procedure to self-supporting surfaces.
Remark: When remeshing a given shape by planar quad meshes,
weknow that the circular and conical properties require that the
meshfollows the ordinary, Euclidean principal curvature directions
[Liuet al. 2006]. It is remarkable that the self-supporting
property in asimilar manner requires us to follow certain relative
principal direc-tions. Practictioners observations regarding the
beneficial staticsproperties of principal directions can be
explained by this analogy,because the relative principal directions
are close to the Euclideanones, if the stress distribution is
uniform and s is small.
Koenigs Meshes. Given a self-supporting thrust network S
withstress surface , we ask the question: Which vertical
perturbationS+R is self-supporting, with the same loads as S? As to
notation,all involved meshes S,R, have the same top view, and
arithmeticoperations refer to the respective z coordinates si, ri,
i of vertices.
The condition of equal loads then is expressed as (s+r) = sin
terms of Laplacians or as HrelS = H
relS+R in terms of mean cur-
vature, and is equivalent to r = 0, i.e., HrelR = 0. So R is
aminimal surface relative to . While in the triangle mesh case
thereare enough degrees of freedom for nontrivial solutions, the
case ofplanar quad meshes is more intricate: Polar polyhedraR,
haveto be Christoffel duals of each other [Pottmann and Liu 2007],
as il-lustrated by Figure 4. Unfortunately not all quad meshes have
sucha dual; the condition is that the mesh is Koenigs, i.e., the
derivedmesh formed by the intersection points of diagonals of faces
againhas planar faces [Bobenko and Suris 2008].
Koebe meshes. An interesting special case occurs if is aKoebe
mesh of isotropic geometry, i.e., a PQ mesh whose edgestouch the
Maxwell paraboloid. Since approximates the Maxwellparaboloid, we
get 2K(vi)Hrel(vi) 1 and consequently isself-supporting for unit
load. Applying the Christoffel dual con-struction described above
yields a minimal mesh R, and meshes + R which are self-supporting
for unit load (see Figure 15).
6 Conclusion and Future Work
Conclusion. This paper builds on relations between statics
andgeometry, some of which have been known for a long time, and
-
+ R
R
Figure 15: Left: A Koebe mesh is self-supporting for unit
deadload. An family of self-supporting meshes with the same top
view isdefined by S = + R, where R is chosen as s Christoffel-dual.
The right hand image shows a different example of the
sameconnectivity.
connects them with newer methods of discrete differential
geome-try, such as discrete Laplace operators and curvatures of
polyhedralsurfaces. We were able to find efficient ways of modeling
self-supporting freeform shapes, and provide architects and
engineerswith an interactive tool for evaluating the statics of
freeform ge-ometries. The self-supporting property of a shape is
directly rele-vant for freeform masonry. The actual thrust networks
we use forcomputation are relevant e.g. for steel constructions,
where equilib-rium of deadload forces implies absence of moments.
This theoryand accompanying algorithms thus constitute a new
contribution toarchitectural geometry, connecting statics and
geometric design.
Acknowledgments. This work was very much inspired byPhilippe
Blocks plenary lecture at the 2011 Symposium on Geom-etry
Processing in Lausanne. Several illustrations (the
destructionsequence of Fig. 8 and the maximum load example of Fig.
9) havereal-world analogues on his web page at ETH Zurich, see
[Block2011] and [Davis et al. 2011].
We would also like to thank Florin Isvoranu and Alexander
Schift-ner for their help and input during preparation of this
paper; the de-velopers of the OpenMesh, Ipopt, Eigen, and BCLS
software pack-ages used in our implementation; and Miklos Bergou,
Eitan Grin-spun, Danny Kaufman, and Niloy Mitra for their advice
and sugges-tions on early drafts of the paper. This work was funded
in part bythe NSF (CMMI-11-29917, IIS-11-17257, IIS-10-48948,
IIS-09-16129, CCF-06-43268) and generous gifts from Adobe, ATI,
Au-todesk, mental images, NVIDIA, Side Effects Software, the
WaltDisney Company, and Weta Digital. The work of Helmut
Pottmannand Mathias Hobinger was supported by the Austrian Science
Fund(FWF) through grants No. P9206-N12 and P23735-N13 and the
Eu-ropean Communitys 7th Framework Programme under grant agree-ment
230520 (ARC).
FutureWork. There are several directions of future research.
Oneis to incorporate non-manifold meshes, which occur naturally
whene.g. supporting walls are introduced. It is also obvious that
non-ver-tical loads, e.g. wind load, play a role. The surfaces
produced byour algorithm are the solutions of discrete elliptic
boundary valueproblems and so tend to be smooth; another avenue for
future workis modification of the discretization to allow for
self-supporting sur-faces with sharp features. There are also some
directions to pur-sue in improving the algorithms, for instance
adaptive remeshing inproblem areas. Probably the interesting
connections between stat-
ics and geometry are not yet exhausted: on the one hand we
ex-pect that interesting new geometry arises from questions of
statics,on the other hand we would like to propose the
geometrization ofproblems as a general solution paradigm.
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