Global Minimization of Vertex Height Differences for Freeform Architectural Design 661

original outside form of the structure, as designed by the

architect. Triangular mesh elements do not require pla-

narization because their geometry is always planar. The

planarity of an element in a selected mesh should be

executed to the level that still allows the assembly of

closure elements.

Problem Formulation

Figure 4 shows a CAD model, designed according to

planar and conical pre-optimized mesh. Figure 4 shows

the support structure only; composed of joint cylinders

and I beams. In every single joint cylinder (or joint box

[8]) is positioned according to a maximum vertex height

distance (,;maxij ). Therefore the top flange of the corre-

sponding I beam is leveled with the top edge of the joint

cylinder. All the other I beams, having smaller vertex

height distances, must be positioned higher, producing

additional internal loading in the joint cylinder and joint

cylinder-beam connection problems. The idea is to pro-

vide minimal possible joint height differences (,),

between top flange of the I beams in every single joint

cylinder, which significantly reduces additional forces

and moments in a joint cylinder (Figure 7). An post-

optimization algorithm was created to do the task for all

the joints in the structure. In chapter 4, a graphical analy-

sis of joint height differences is made for the entire mesh

of the sample free form structure. Minimization of these

differences in particular joint significantly reduces addi-

tional forces and moments in a joint cylinders.

h

Δij

h

2. Related Work

Not many papers cover our ideas presented in the

introduction. Multi-layer architecture, including planar

and conical meshes was discussed by Pottmann et al. [2,

5,9,10]. Although visually appealing P-hex meshes that

were also extended to meshes with parallel edges. P-hex

geometry inherits similar problems with a physical rea-

lization of vertex. The elimination of edge offset dif-

ferences can be achieved with Koebe polyhedra [11], but

this brings very restrictive geometry, which cannot

approximate arbitrary shapes. Still this is a promising

approach for glass structures with no closure layer

provided. Such surfaces can also be part of the freeform

structure that can be included in mesh optimization as a

rigid body. Pottmann et al. [10] also suggest to appro-

ximate beam offsets with fairness functional during

vertex perturbation. In their optimization they neglect the

physical realization of the vertex junction and concen-

trate on optimization to achieve an approximately con-

stant offset from a theoretical point of view. So far, we

are unaware of any architectural project that should use

the present geometry processing ideas, as it seems that

solutions need to be solved in detail in CAD (Figure 4)

before the realization is possible. In addition, a stress

analysis of such structure is required, which is not a

trivial task as adequate stucture computations are yet to

be determined.

3. Post-Optimization Method

3.1. The Joint Connection Differencesat Cylinder

In this section we briefly present the geometric algorithm

to locate the intersection points between a ray and a

cylinder following the Cychosz and Waggenspack [12]

algorithm.

We are only interested in the intersection point in

between the vertex cylinder and the beam. Connecting

height (

Figure 5) from vertex origin and beam-

cylinder intersection in

do

is calculated by projecting

vector in

H

o to vertex normal (see Figure 6). After

some algebra calculations we finally obtain

v

in

in in

H

dH H

vo

ovo (1)

Beam offset distance d can be regarded as a function

of many parameters, face normals, inclination and di

hedral angles. This function is obviously nonlinear and

so is the post-optimization procedure that generally mini-

mizes differences between each beam d in each vertex of

the structure. Solving such global post-optimization

problem can easily lead to local optimums. Especially

when structures are large. Many algorithms from com-

putational geometry try to avoid such situations by

defining local operators rather than solving the problem

globally. In this paper we follow such ideas by intro-

ducing two competitive algorithms with local impact and

compare their results.

3.2. Vertex Element Differences

In order to generate a CAD model, it is necessary to

specify the points where the support structures and the

joint elements are positioned. The beams are displaced at

the fixing points in the joint cylinder. The calculation of

the beam positions, displaced from the reference points,

is shown below.

Figure 6 shows a cross-section of the cylindrical joint,

beams with insulation, spacers and outside closure metal.

The outside closure sheet normals 1 and 2 are

joined at the vertex reference point (

Figure 6).

n n

o

,1ij e

hh r

1

1

e

e

nn er

nn

(2)

where

12

12

enn

nnn and

e

rr

e

e

en

en

.

Vertex positioning. For each vertex of the selected

i

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