Global Minimization of Vertex Height Differences for Freeform Architectural Design

doi:10.4236/jsea.2012.59077

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Journal of Software Engineering and Applications, 2012, 5, 659-663

http://dx.doi.org/10.4236/jsea.2012.59077 Published Online September 2012 (http://www.SciRP.org/journal/jsea)

659

Global Minimization of Vertex Height Differences for

Freeform Architectural Design*

Simon Kulovec, Leon Kos, Jože Duhovnik

Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia.

Email: [email protected]

Received June 15th, 2012; revised July 20th, 2012; accepted August 3rd, 2012

ABSTRACT

Architectural design is leading us in the direction of structures with free and irregular forms. As a consequence of this

the connection between the design’s intent and its fabrication represents a challenge when creating a support structure

that is geometrically viable and which needs to possess certain aesthetic, fabricational, thermal and strength require-

ments. To ensure the contacts of the edges of the neighboring insulation panels along their thicknesses, the edges must

be cut at different angles, which causes differences in the vertex heights and, furthermore, differences in the positions of

the inner metal sheets of the insulation panels. The main goal of the presented research is the development of a

post-optimization procedure, by which the minimum joint-height differences will be achieved for all the joints, taking

into account all the free-form surfaces of the individual architectural design. To compensate for the residual height dif-

ferences the use of spacers of different thicknesses is proposed. The paper considers the global minimization of the

joint-height differences for a sample free-form architectural design that is meshed with a quad-dominant mesh.

Keywords: Planar Mesh; Support Structure; Post-Optimization; Height-Difference Function

1. Introduction

In the physical realization of a project the arbitrary

shapes designed by the architect must satisfy a number of

requirements that limit the possible realizable solutions

for the final “free forms”. Those are frequently used in

“sculptured” designs, like museums and towers that are

intended to be city landmarks. Such designs seek their

representation through shape, while applying cost-effec-

tive materials and making use of the whole of the avail-

able area.

The freeform surfaces of the structures can be de-

scribed by different types of meshes providing the plana-

rity of each of the mesh surface elements, i.e., the faces.

The existing freeform structures mainly consist of train-

gular meshes, where the condition of planarity is satisfied

automatically. We chose to use quadrilateral meshes [1]

(Figure 1), because they are cost effective, but also geo-

metrically complex. The condition of planarity is satis-

fied by an optimization algorithm based on Sequential

Quadratic Optimization [2].

Freeform façades consist of relatively thick, planar

(Figure 2), panel elements and of a corresponding sup-

port structure. The outer surfaces of the panel elements

coincide with the mesh faces (Figure 3). To ensure the

contact of the edges (Figure 3) of the neighboring panel

elements along their thicknesses, their edges must be cut

at different angles, which causes differences in the edge

heights and furthermore differences in the positions of

the inner metal sheets of the insulation panels.

In common support structures (Figure 4) where the

beams have the same height, the position of the beams

must compensate for the above-mentioned difference in

the positions of the outer metal sheets. For this reason

relatively large joint-height differences (Figures 5-

7) are necessary.

Δij

The differences in the positions of the beams ensure a

Figure 1. Double-curved freeform quad-dominant mesh.

*Post-optimization of height differences of Freeform Structures.

Global Minimization of Vertex Height Differences for Freeform Architectural Design

660

Figure 2. Planar optimized mesh.

Figure 3. Mesh elements: vertex, edge and face.

Figure 4. CAD model of a support structure of planar mesh

structure.

constant distance between the outer metal sheet and the

beams of the structure. Producing, as a result, the differ-

ent vertex height distances (Figure 6).

,ij

The aim of this research is to develop a post-optimi-

zation procedure by which the minimum joint-height

differences ,,;max,;min

can be achieved for

all the joints, taking into account all the faces of the con-

sidered free-form architectural design. It should be pointed

out that the angles

Δij ijij

hh h

,1i



and ,2i



in all the joints are

already optimized [3,4] (Figures 6 and 7) and therefore

not subject to change.

Freeform structures require planarity for each closure

metal panel (insulation panel element) [5-7]. Planarity is

necessary for the building of the structure, particularly in

cases when the structure is covered with non-deformable

elements (e.g., glass). We are trying to make planar ele-

ments for the selected structure, while still keeping the

Figure 5. Geometry of the cylinder-beam intersection prob-

lem.

Figure 6. Initial vertex cross-section with equal spacer thick-

ness (h2 + di,2).

Figure 7. Post-optimized vertex cross-section with variable

spacer thickness (Δhi,z < Δhi,2 → di,z < di,2).

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.

Δij

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

is calculated by projecting

vector in



o to vertex normal (see Figure 6). After

some algebra calculations we finally obtain



in in

dH H



 

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

,1ij e

hh r









nn er

nn 

(2)

where







enn

nnn and



rr

Vertex positioning. For each vertex of the selected

Global Minimization of Vertex Height Differences for Freeform Architectural Design

662

mesh, the vertex height distance is determined:

,ij

1,i



, ;max1,11,

max, ,, ,

ij n

hhhdhd

(3)

where the index represents the vertices, and

the index represents the beams of a given

mesh.

1i

m1j

3.3. Definition of the Height-Difference Function

The height-difference function is created by taking into

account all the vertex height differences in the selected

mesh.



,,, 1,

ijijiji j

Fhh hh h









1 (4)

where and .

1i

i1jm

The index represents the vertices; the index de-

notes the I beams.

The first part of the height-difference function repre-

sents the vertex height differences ,ij

(see Figure 6),

while the second part of the height-difference function

represents the connection of the current vertex with the

neighboring vertices. The height-difference function com-

pares the vertex height differences of the support struc-

ture on both sides of the beam.

4. Results

The resulting shift of the beams introduces different dis-

tances between the outside closure metal of the panel and

the individual beam (see ,1i and ,2i in Figure 7).

These differences are compensated for by the spacers of

adapted thicknesses (Figures 6 and 7). After optimiza-

tion the thickness ,1i is no longer a constant for the

whole of the façade, but can be different for each beam,

(see

Figure 7).

d d

d

,2,1,2 ,1i

Figure 8 shows the convergence of the height-differ-

ence function (Equation (4) that describes the problem of

joint-height differences , (Figures 6 and 7), which

we would like to minimize. The height-difference func-

tion is composed of the sum of the vertex height differ-

ences (,ij

) for each vertex in a given mesh. The con-

straint, determining the interval by which each beam can

move, is [−5, 30] mm.

ΔΔ

hh d

Δij

In ten iterations, the height-difference function reaches

a maximum drop, followed by reaching almost the top

convergence after 15 iterations and not changing signifi-

cantly up to the end.

A comparison between the individual vertex height

distances for the beams before and after optimization is

shown as a standard deviation in Figure 9. It shows the

standard deviation of the vertex height differences ,ij

(Figure 6) of the beams for all the vertices of the mesh.

It can be seen that the deviation of the vertex height dif-

ferences (

Figures 6 and 7) is reduced.

,ij

Figure 8. Height-difference function [mm] convergence.

Figure 9 shows that the standard deviation of the ver-

tex height distances from the vertex improves in almost

all the vertices after optimization. On the vertices with

only three beams, which are mostly on beam vertices, it

is not possible to achieve a quasi-ideal situation.

It can be concluded that the distribution of the joint-

height distances (,

Δij) on the mesh is improved after

the post-optimization procedure. This post-optimization

procedure results in more constant joint-height distances

on the mesh. We believe that this optimized mesh is: 1)

cost effective, and 2) simpler to construct.

5. Conclusions

The aim of the post-optimization procedure is to provide

the minimum joint-height differences ,2,1i

(Fig-

ure 7). The focus is on quadrilateral meshes because they

are cost effective. An analysis of the initial and opti-

mized meshes was made in order to ascertain whether the

structure is improved after the optimization and whether

the mesh keeps its original shape.

ΔΔ

hh

The first part is represented by the post-optimization

algorithm. It provides, globally, the minimum joint-

height differences for the beams in the individual vertices

of a given mesh (Figure 1). The optimization tries to

provide the minimum joint-height distances (,),

which provides the strength stability (the additional

forces and momentums in the joint cylinder are minimal)

of the construction. The height-difference function for

the optimization is so structured that it keeps the original

shape of any given mesh (Equation (4)).

Δij

The second part involves a mesh analysis before and

after the optimization. In a quasi-ideal situation the stan-

dard deviation of the vertices would be zero (Figure 8),

which would mean that all the vertex’s beams have the

minimum joint-height differences. We believe that this

optimized mesh is: 1) cost effective, and 2) simpler to

Global Minimization of Vertex Height Differences for Freeform Architectural Design

663

050100 150 200 250 300 350 400 450 500 550

σ[mm]

Vertex no.

Figure 9. Standard deviation of vertex height distances before and after post-optimization.

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with Constraints,” 2nd Edition, 2004.

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[7] O. Schenk and K. Gärtner, “Solvingunsymmetric Sparse

Systems of Linear Equations with Pardiso,” Journal of

Future Generation Computer Systems, Vol. 20, No. 4,

2004, pp. 475-487. doi:10.1016/j.future.2003.07.011

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