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Building and Environment 42 (2007) 1632–1643

Optimization of the form of a building on an oval base

Mariusz AdamskiÃ

Department of Heat Engineering, Technical University, PL-15-351 Bia! ystok, ul. Wiejska 45E, Poland

Received 13 January 2006; accepted 15 February 2006

Abstract

The aim of this work is to analyze the possibility of optimizing an abstract, symmetrical with respect to the north–south axis form of a

building with vertical walls and windows, and constant volume and height. The external south partitions of the building are walls withwhole windowpanes. The heat losses through walls, floors, roof and the gain of solar radiation through transparent partitions with

respect to their direct correlation with the shape of building form are next taken into consideration. The gain of solar energy for the north

part of the building have been disregarded.

The optimization problem of the form of the building on an oval base is examined by variational methods taking into account two

opposite criteria:

minimum construction costs including the cost of materials and erection,

minimum seasonal demand for heat energy.

The obtained solution is composed of the semicircular bounding of the northern part of the building and a curve of the southern part

described by a parametrical function. The design is also compared with the buildings on square and circular bases.

The calculations have been performed using specially prepared computer program and illustrated by a numerical example.r 2006 Elsevier Ltd. All rights reserved.

Keywords: Optimization; Building; Heat losses; Heat gains; Solar radiation

1. Introduction

Optimization problems of the shape of buildings with

regard to the energy consumption were the subject of many

publications. To papers discussing unicriterial optimization

of building shape belong, among others to [1–4]. In Fokin’s

paper [1] the shape of a building of given volume was

optimized, taking the minimium heat losses as the criterion.Spherical shape was obtained as the result. With an

additional restraint introduced that the building be a

rectangular prism, a cubic shape was obtained. The

problem of determining optimum dimensions of a building

on a rectangular plan with the criterion of minimum heat

power requirements per 1 m3 was solved by Gadomski in

[2]. Heat gains due to insolation were not taken into

account. The geometry of building shape was analyzed in

paper [3]. The concept of geometric compactness was

introduced as the ratio of the area of external walls with

respect to the volume of the building. Buildings of various

shapes were composed using four equal cubes of side a,

obtaining the geometric compactness coefficients between

4/a to 14,1/a. In [4] a building of prismatic shape was

optimized, taking into account heat gains due to insolationthrough transparent and opaque partitions. On the basis of

the criterion of the minimum energy demand for heating,

optimum ratio of the dimensions of sides of the building

and the optimum number of storeys were determined. In

[5], the Authors analyzed the influence of shape coefficient

of the building and that of the climatic zone on the

seasonal energy demand index.

The examples of application of multi-criteria optimiza-

tion methods in the solution of architectural problems

can be found in [6–9]. The problems of optimization of

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energy-saving buildings were also presented comprehen-

sively by Owczarek [10]. He presented a model of solar

radiation heat gains across windows. In his paper the

thermal resistance of the outer walls was fixed geometry of

building shape, linear dimensions of the hexagonal building

plan at the imposed angles and wall azimuths. Bi-criterial

problem of the optimization of building shape on

hexagonal plan was also solved for the criteria of minimum

cost of materials and minimum yearly heating energy cost.

In papers [11–14] the multi-criteria optimization problem

was presented for buildings of a given volume andoctagonal in plan. Decision variables were: building height,

wall lengths, wall azimuths, window sizes and thermal

resistance of individual external partitions Minimum

construction costs and minimum the seasonal heating cost

were assumed as optimization criteria.

The papers related to the optimization of form analyze

buildings with vertical walls. There are two groups of tasks:

optimization of building form on a polygonal base, which

is presented in numerous publications and optimization of

building form on an oval base. The optimization problem

of the form of a building with an arbitrary base [12,15] has

been solved by variational methods. The criterion of

minimum building cost and minimum annual heating cost

are assumed for optimization. The decision variables of the

problem are the curves describing the base of the building.

The obtained solution is composed of a circular segment

bounding the northern part of the building and a curve

described by a sixth degree polynomial bounding its

southern part. It was assumed that the ratio of the area

of the windows to that a wall element is a trinomial square.

The aim of the work [16] was to analyze the possibility of

optimizing an abstract, symmetrical with respect to the

north–south axis form of a building with vertical walls and

windows, and constant volume and height. The external

south partitions of the building are walls with whole

windowpanes (Fig. 1). The heat losses through walls,

floors, roof and the gains of solar radiation through

transparent partitions with respect to their direct correla-

tion with the shape of building form are next taken into

consideration. The gains of solar energy for the south part

of the building have been calculated using the following

approximation:

E ¼ E aoð Þ ¼ y1cos2 aoð Þ þ y2sin2 aoð Þ. (1.1)

The maximum error of approximation can be found for the

azimuth of 901 (east or west elevation). The gains of solarenergy for the north part of the building have been

disregarded.

The optimization problem of the form of the building on

an oval base is examined by variational methods taking

into account two opposite criteria:

minimum construction costs including the cost of

materials and erection,

minimum seasonal demand for heat energy.

The obtained solution is composed of the semicircular

bounding of the northern part of the building and a curveof the southern part described by a complicated logarith-

mic function. In order to compare the building on a square

base and a building on a circular base have also been

considered.

The obtained solutions have been considerably influ-

enced by

the model of solar energy gains,

the applied sort of external partitions.

The effect of the physical environment on a building was

the subject of numerous works. The problems connected

ARTICLE IN PRESS

Nomenclature

A heat transfer area, area of ceiling or floor, m2,

Ac ¼ A f

c specific heat at constant pressure, J/(kg K)

cc, ce unit cost of heat energy, of electrical energy,respectively, z"/MJ

dl length element of a wall or of a window, m

E the annual heat losses, MJ/year

F cost, z"

h building height, m

k thermal conductivity coefficient, W/(m2K)

S the area, m2

SD annual number of degree-days [K Â day]

t temperature, 1C

te, ti mean temperature of the external air, of heated

rooms, oC

tce, t fe mean temperature of the air of unheated garret,

cellar, 1C

V volume of the building m3

Greek symbols

a angle

r mean density of air, kg/m3

jo1 coefficient for temperature of the air of

unheated garret and cellar, where: jc ¼ti Àtce

ti Àte; j f ¼

ti Àt fe

ti Àte;

Subscripts

c, o, f, w a ceiling, a window, a floor, a wall, respectively

e, i external, interior

ce, fe unheated garret, cellar

M. Adamski / Building and Environment 42 (2007) 1632–1643 1633



with the optimization of the form of a building are related

to the following, most important, quantities:

SD—annual number of degree-days [K Â day],

y1, y2 —average values of the total amount of solar

radiation falling during the heating season on the south,

east or west vertical plane [MJ/m2], y1 —south elevation,

y2 —east or west elevation.

A model is developed to optimize an abstract, symmetrical

with respect to the north-south axis form of building with

vertical walls and windows, and constant volume V and

height h (Fig. 2). The south external partitions of the

building are walls of whole windowpanes (Fig. 1). The heat

losses through walls, floors, roof and the gains of solarradiation through transparent partitions with respect to

their direct correlation with the shape of building form are

taken into consideration.

2. Components of thermal balance of a building

The thermal balance of a building is composed of [16]:

heat losses through walls, floors and ceilings,

heat losses through transparent partitions,

heat losses due to infiltration of air through external

partitions,

and

gain of solar radiation through transparent partitions,

heat gains emitted by lighting installations, household

equipment and human bodies,

heat losses or gains recovered from the ventilation air.

Each component plays a different role in the general

thermal balance of a building. The heat losses through

walls, floors, roof and transparent partitions and the gain

of solar radiation through transparent partitions with

respect to their direct correlation with the shape of building

form are next taken into consideration.

2.1. Annual heat losses through the walls, ceilings and floors

The annual heat losses E through an element h dl 2 of a

wall E w and through an element h dl 2 of a window E o, a

ceiling E c and floor E f were found from the formulae:

E w ¼ 3600 Á 24 Á k w Á SD Á h dl 2 Â 10À6

¼ 0:0864 Á k w Á SD Á h dl 2, ð2:1Þ

E o ¼ 0:0864 Á k o Á SD Á h dl 1, (2.2)

E c ¼ 0:0864 Á jc Á k c Á SD Á Ac, (2.3)

E f ¼ 0:0864 Á j f Á k f Á SD Á A f . (2.4)

The annual heat loss due to ventilation E V can be

expressed by the formula

E V ¼ 3600 Á 24 Á SD Á c Á r Áni Á V

3600Â 10À6

¼ 24 Á 1011 Á 1; 25 Á SD Á ni Á V Â 10À6

¼ 0; 03033 Á SD Á ni Á V , ð2:5Þ

where ni is a number of air changes per hour, usually

ni ¼ 12

2, but in the energy-saving buildings it can beni ¼ 0:5.

2.2. The heat gain by solar radiation

The gain of direct solar radiation through transparent

partitions is examined. It is assumed that the momentary

gain due to solar radiation E S

[kW] through a vertical

window, the azimuth of which is aw and the area A, is

y

S awð Þ ¼ AJ cos aS À awð Þ, (2.6)

where the symbols J [kW/m2] and aS denote the intensity

and the azimuth of solar radiation.

ARTICLE IN PRESS

Fig. 1. The whole windowpane structure wall of security, tempered glass.Fig. 2. View of oval-shaped building and symbols used.

M. Adamski / Building and Environment 42 (2007) 1632–16431634



It should be pointed out that the difference aS Àaw does

not depend on the axis we measure the azimuth from. In

further calculations azimuths will be determined with

regard to plus semi-axis OY. The value of azimuth angle

at noon is zero, whereas its value at 6 pm is p/2. The

vertical window azimuth has been determined with regard

to the normal , that is ao ¼ 0 for the south elevation andao ¼ p=2 for the east elevation (Fig. 1). The function (2.6)

can be assumed as follows:

y

S aoð Þ ¼ AJ cos aS ð Þ cos awð Þ þ sin aS ð Þ sin awð Þ½ . (2.7)

The heat gain due to solar radiation for the examined

partition in the heating season have been obtained by

integrating the above equation:

y aoð Þ ¼

Z heatingseason

y

S awð Þdt

¼ Z heatingseason

AJ tð Þ cos aS tð Þð Þ cos awð Þ½

þ sin aS tð Þð Þ sin awð Þ dt. ð2:8Þ

Hence we obtain

y1 ¼ y 0ð Þ ¼

Z heatingseason

J tð Þ cos aS tð Þð Þ dt (2.9)

and

y2 ¼ y p=2À Á

¼

Z heatingseason

J tð Þ sin aS tð Þð Þ dt (2.10)

for A ¼ 1 m2.

The issue has been thoroughly examined in paper [13]. In

particular the seasonal solar radiation energy for vertical

planes has been established for Warsaw. Different numbers

of the heating season days have been analyzed. The

variation of seasonal radiation has been calculated using

the cosine Fourier series including its four first elements as

follows:

y awð Þ ¼ a0 þ a1 cos awð Þ

þ a2 cos 2awð Þ þ a3 cos 3awð Þ. ð2:11Þ

For instance for south plane we obtain

y1 ¼ y 0ð Þ ¼ a0 þ a1 þ a2 þ a3 (2.12)

and for east or west elevation we obtain

y2 ¼ y p=2À Á

¼ a0 À a2. (2.13)

Use of this model of the heat gain by solar radiation

require four parameters. The estimated values a1, a2, a3, a4

are included in [13]. From the evenness function cosine it

appears that:

y awð Þ ¼ y Àawð Þ ¼ y awjj . (2.14)

The function (2.11) has been used for the azimuths of

vertical wall from 0 to p/2 as an approximation of the

results of calculations included in paper [13].

3. Formulation of the optimization problem

The subject of our considerations is a building with

vertical walls, constant volume V and height h. The base of

the building has the shape shown in Fig. 2. It was assumed

that the building is symmetrical with respect to the

north–south axis. The walls of the south part of the buildingare partitions with whole windowpanes. The gain in solar

energy for the north part has been disregarded, external

partitions of this part of the building can be transparent or

not. In the considered problem a constant value of overall

heat-transfer coefficient k w for the whole outer partition in

the north part of the building has been assumed.

The aim of the present considerations is to determine the

form of the building using two opposite criteria:

minimum building cost,

minimum seasonal cost of heating.

The function expressing the construction cost is defined

as follows:

F 1 ¼ 2

Z A1

B 1

co Á h dl 1 þ 2

Z A1

B 2

cw Á h dl 2 þV

hcc þ

V

hc f þ D1,

(3.1)

where cw, co, cc and c f are the construction costs of wall,

window, ceiling and floor, respectively, z"/m2, V volume of

the building m3 and D1 other costs independent of the

decision variables, z".

The function expressing the seasonal heating cost is

described in the following way:

F 2 ¼ 0:0864 Á SD Á 2ce Á

Z A1

B 1

k o Á h dl 1 þ

Z A1

B 1

k w Á h dl 2

&

þ jck c þ j f k f

Á

V

2hþ

f V Á V

2

'

À 2ce Á

Z A1

B 1

y awð Þ Á h dl 1 þ D2, ð3:2Þ

where ce indicates the price of thermal energy used for

heating the building, z"/MJ, dl 1, dl 2 are the lengths of

elements of the curves y1 and y2, D2 other costs

independent of the decision variables, z". The definition

of parameter f V follows from (2.5): f V ¼ 0:35104 Á ni . (3.3)

Because

dl 1;2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ y021;2

q dx, (3.4)

dl 1 Á cos aw ¼ dx, (3.5a)

dl 1 Á sin aw ¼ d y1 (3.5b)

and

tan aw

ð Þ ¼ y0

1 ¼

d y1dx

(3.6a)

ARTICLE IN PRESS




or

aw ¼ arctan y01

¼ arctand y1

dx

, (3.6b)

where d y140 and y01 ¼ d y1=dx ¼ tan awð Þ for west eleva-

tion the south part of the building or d y1o0 and y01 ¼

À tan awð Þ for east elevation the south part of the building,we obtain:

F 1 ¼ 2

Z xa

0

co Á h Á

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y02

1

q dx

þ 2

Z xa

0

cw Á h Á

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y02

1

q dx þ

V

hcc þ

V

hc f þ D1,

ð3:7Þ

F 2 ¼ 0:0864 Á SD Á ce Á 2

Z xa

0

k oh


1

q dx

&

þZ xa

0k wh


q dx þ jck c þ j f k f

V

2h þf V V

2'

À 2ce

Z A1

B 1

y arctan y01

À Áh


1

q dx þ D2. ð3:8Þ

The decision variables of the problem are the functions

y1(x), y2(x). The following conditions have been assumed:

1. y1(x) and y2(x) are continuous functions of C 2 class

within the range (0, xa),

y1 xð Þ; y2 xð Þ 2 C 2 0; xað Þ. (3.9)

2. The functions y1(x) and y2(x) are zero at the point, the

abscissa of which is xa:

y1ðxaÞ ¼ 0, (3.10)

y2ðxaÞ ¼ 0, (3.11)

3. The oval base is free of nibs, that means

y01ð0Þ ¼ 0, (3.12)

y

0

2ð0Þ ¼ 0, (3.13)

limx!xÀ

a

y01 ¼ À1, (3.14)

limx!xÀ

a

y02 ¼ þ1, (3.15)

4. The functions y1(x) and y2(x) bound a region of the area

V/h, that is

2 Á Z xa

0

y1 xð Þ À y2 xð ÞÀ Á dx ¼ V =h. (3.16)

5. The form of the building is symmetrical in relation to

OY axis (Fig. 2).

The set of nonpredominant valuation can be determined

by the method of weight coefficients. The minimum of the

objective function is sought:

F ¼ lF 1 þ ð1 À lÞF 2, (3.17)

where lA[0, 1], conditions (3.9)–(3.16) being satisfied. In

the present work the weight coefficient l can be subjected

to the modified number N of utilization time (years) of the

building as follows:

l ¼1

N þ 1. (3.18)

The modified number of utilization years is the number

of years multiplied by a coefficient expressing the rate of

interest and inflation. If the value of the coefficient l is

zero, this means disregarding the costs of building

materials and construction. The same effect is producedby assuming the time of utilization of the building to tend

to infinity (N -N).

The assumption of l ¼ 1 corresponds to the utilization

costs being disregarded, that is to the assumption that

N ¼ 0. Both cases are not interesting, therefore it is not the

entire set of compromises corresponding to 0plp1 that

will be determined, but its part

1

2XlX

1

101, (3.19)

corresponding to the utilization period of 1–100 years:

1pN p100. (3.20)

4. Solution of the optimization problem

Eqs. (3.7), (3.8) and (3.17) leads to

F ¼ l Á co þ 1 À lð Þ Á 0:0864 Á SD Á ce Á k oð Þ Á 2h Á

Z xa

0

ffiffiffiffiffiffiffiffiffiffiffi ffiffiffi1 þ y02

1

q dx

þ l Á cw þ 1 À lð Þ Á 0:0864 Á SD Á ce Á k wð Þ Á 2h Á

Z xa

0

ffiffiffiffiffiffiffiffiffiffiffi ffiffiffi1 þ y02

2

q dx

þ l Á cc þ c f À Á þ 1 À lð Þ Á 0:0864 Á SD Á ce Á jck c þ j f k f ÁV

hþ l Á D1 þ 1 À lð Þ Á 0:0864 Á SD Á ce Á f V Á V þ 1 À lð Þ Á D2

À 2ce 1 À lð Þ Á h Á

Z xa

0

y arctan y01

À Á ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y02

2

q dx. ð4:1Þ

Taking into consideration (3.16) we obtain the Lagrange

Function [17]:

F ¼ Ao

Z xa

0


1

q dx þ Aw

Z xa

0


2

q dx þ C

þ D À E

Z xa

0

y arctan y01

À Á ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y02

1

q dx

þ 2l1 Á Z xa

0

y1 xð Þ À y2 xð ÞÀ Á dx À l1 Á V =h, ð4:2Þ

ARTICLE IN PRESS




where

Ao ¼ l Á co þ 1 À lð Þ Á 0:0864SDcek oð Þ Á 2h,

Aw ¼ l Á cw þ 1 À lð Þ Á 0:0864SDcek wð Þ Á 2h,

C ¼ l Á cc þ c f

À Áþ 1 À lð Þ Á 0:0864SDce jck c þ j f k f

Á V =h,

D ¼ l Á D1 þ 1 À lð Þ Á 0:0864SDce f V V þ 1 À lð Þ Á D2,

E ¼ 2ce 1 À lð Þ Á h.

This is an isoperimetric problem of the variational

calculus. Conditions (3.9)–(3.15) enable us to determine the

integration constants and the constant l1.

Functional (4.2) reaches its extreme value, if Euler’s

equations:

f y01 y0

1 y00

1 þ f y1 y01 y0

1 þ f xy01

À f y1¼ 0, (4.3)

f y02 y0

2 y00

2 þ f y2 y02 y0

2 þ f xy02

À f y2¼ 0. (4.4)

are satisfied. The symbol f z denoting the integrand qF =qz,

that is

d Ao À E y arctan y01

À ÁÀ ÁÀ ÁÁ

y01


1q À E


1

q Á

dy arctan y01

À ÁÀ Ád y0

1

0B@

1CA

d y01

Â y001 À 2l1 ¼ 0, ð4:5Þ

Aw Á1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ y022

q 3y00

2 þ 2l1 ¼ 0. (4.6)

Because y001 ¼ d y0

1=dx and taking into consideration (4.5)

we obtain

d Ao À E y arctan y01À ÁÀ ÁÀ Á Á

y01 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ y021

q 0B@

ÀE


1

q Á

dy arctan y01

À ÁÀ Ád y0

1

1CA ¼ 2l1 dx ð4:7Þ

and on integrating

Ao À E y arctan y01

À ÁÀ ÁÀ ÁÁ


1 þ y021

q À E ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y02

1q Ády arctan y0

1À ÁÀ Ád y

01

¼ 2l1x þ C 1, ð4:8Þ

where

dy arctan y01

À ÁÀ Ád y0

1

¼ À 4a2 þ

a1 þ 9a3 þ a1 À 3a3ð Þ y021 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ y021

q 0B@ 1CAÁ

y01

1 þ y021

À Á2 .

ð4:9Þ

From condition (3.12) we find the integration constant

C1. Because y01ð0Þ ¼ 0, therefore dy arctan 0ð Þð Þ

d y0

1 ¼ 0 and

0 ¼ 2l1 Á 0 þ C 1, (4.10)

or


À ÁÀ ÁÀ ÁÁ y

0

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y02

1

q À E


1

q Á

dy arctan y01

À ÁÀ Ád y0

1

¼ 2l1x. ð4:11Þ

The left part of the obtained equation is the

function depended on y01, This is designated as f y0

1

À Á.

The value of x can be easily calculated from the following

equation:

x ¼ 12l1

f y01À Á

¼ 12l1

Ao À E y arctan y01À ÁÀ ÁÀ Á

Á y

0

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y02

1

q 0B@

ÀE


1

q Á

dy arctan y01

À ÁÀ Ád y0

1

1CA, ð4:12Þ

i.e. for assumed value of the directional coefficient y01

tangent to the curve y1 we know the abscissa of the point of

tangency. The ordinate of the point of tangency of the

looked curve y1 we find from

y1 xð Þ ¼

Z x

Àxa

y01 dx (4.13a)

or

y1 xð Þ ¼

Z xa

x

y01 dx: (4.13b)

Furthermore, the application of (3.10) yield:

y1 xað Þ ¼ Z xa

Àxa

y01 dx ¼ 0. (4.14)

ARTICLE IN PRESS




We calculateR

y01 dx on integrating by parts and taking

into consideration (4.12) and we find

y1 xð Þ ¼

Z y0

1 dx ¼ y01x À

Z d y0

1

dxxdx ¼ y0

1x À

Z xd y0

1

¼1

2l1

y01 f y0

1À Á À Z f y01À Ád y0

1 . ð4:15Þ

Because

Z f y0

1

À Ád y0

1 ¼

Z Ao À E y arctan y0

1

À ÁÀ ÁÀ ÁÁ


1 þ y021

q 264

ÀE


1

q Á

dy arctan y01

À ÁÀ Ád y0

1

375d y0

1

¼ Ao À E y arctan y01À ÁÀ ÁÀ Á Á ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y02

1q , ð4:16Þ

hence

y1 ¼

Z y0

1dx ¼1

2l1

y01 f y0

1

À Á

À Ao À E y arctan y01

À ÁÀ ÁÀ ÁÁ


1

q þ C 2, ð4:17Þ

or

y1 ¼1

2l1

Ao À E y arctan y01À ÁÀ ÁÀ Á

ÁÀ1


1q

0

B@ÀE Á y0

1 Á


1

q Á

dy arctan y01

À ÁÀ Ád y0

1

1CA þ C 2. ð4:18Þ

From (3.10) y1ðxaÞ ¼ 0 and from (3.14) limx!xÀa

y01 ¼

À1 and regarding from (4.9) the following condition:

lim y0

1!À1

y01 Á


1

q Á

dy arctan y01

À ÁÀ Ád y0

1

¼ lim y0

1

!À1 y0

1 Á ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y021q Á 4a2 þ

a1 þ 9a3 þ a1 À 3a3ð Þ y021 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ y021

q 0B@

1CA

ÂÀ y0

1

1 þ y021

À Á2¼ À a1 À 3a3ð Þ, ð4:19Þ

we find the integration constant C 2:

0 ¼ 0 À E Á À a1 À 3a3ð Þð Þ þ C 2,

or C 2 ¼ ÀE Á a1 À 3a3ð Þ. Hence (4.18) takes the form

y1 ¼1

2l1

Ao À E y arctan y01À ÁÀ ÁÀ Á Á

À1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y

02

1q

0B@

ÀE Á y01 Á


1

q Á

dy arctan y01

À ÁÀ Ád y0

1

1CA À E Á a1 À 3a3ð Þ,

ð4:20Þ

i.e. for assumed value of the directional coefficient

y01 tangent to the curve y1 we know the ordinate of

the point of tangency. Eqs. (4.12) and (4.20) show

the curve delimiting the south part of the building base

as follows:

x ¼1

2l1


À ÁÀ ÁÀ ÁÁ


1 þ y021

q À E


1

q Á

dy arctan y01

À ÁÀ Ád y0

1

,

y1 ¼1

2l1

Ao À E y arctan y01À ÁÀ ÁÀ Á Á

À1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y021

q 0B@

ÀE Á y01 Á


1

q Á

dy arctan y01

À ÁÀ Ád y0

1

1CA À E Á a1 À 3a3ð Þ,

where y01 is parameter, which takes optional real values.

Taking into consideration (3.5a), (3.5b) and (3.6a), (3.6b)

we obtain

x awð Þ ¼1

2l1

Ao À E y awð Þð Þ Á sin aw À E Ády awð Þ

daw

cos aw

,

y1 awð Þ ¼ À12l1

Ao À E y awð Þð Þ Á cos aw þ E Á dy awð Þdaw

sin aw

À E Á a1 À 3a3ð Þ. ð4:21Þ

Above show the curve delimiting the south part of the

building base. The angle aw takes real values from section

Àp=2; p=2

.

On integrating (4.6) we obtain

y02 xð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ y022

q ¼À2l1 Á x

Aw

,

hence

j y02 xð Þj ¼

x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAw=2l1

À Á2À x2

q . (4.22)

The expression (À2l1x/Aw) is the sine of the inclination

angle of the tangent at the point (x, y2(x)) to the OX -axis.

Because y02ðxÞ40 for x x 2 0; xað Þ, we have

y02 xð Þ ¼

x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAw=2l1

À Á2À x2

q . (4.23)

Hence, on integrating, we obtain

y2 xð Þ ¼ À ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Aw=2l1À Á2

À x2q

þ C R2 . (4.24)

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From the condition (3.11) y2ðxaÞ ¼ 0 we find the

constant C R2:

0 ¼ À

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAw

2l1

2

À x2a

s þ C R2

hence

C R2¼


2l1

2

À x2a

s .

From condition (3.15) limx!xÀa

y02 ¼ þ1 we find

Aw=ð2l1Þ ¼ xa and

y2 xð Þ ¼ À


2l1

2

À x2

s ¼ À

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

a À x2

q . (4.25)

It is seen that

y22 xð Þ þ x2 ¼ x2

a. (4.26)

This is an equation of a circle with its centre at the pointO2(0, 0) and the radius R2 ¼ Aw=2=l1 ¼ xa.

The area of the semicircle bounded by the OX -axis, is,

for y2(xa)o0,

S 2 ¼ 2

Z xa

0

y2 xað Þ dx ¼

p

2x2

a ¼p

2

Aw

2l1

2

. (4.27)

Parameter l1 is determined from (3.16), which takes the

form

2 Á

Z p=2

0

Àx awð Þ Á y01 awð Þ daw þ

p

2x2

a ¼ V =h. (4.28)

From (4.21) we obtain

x0 awð Þ ¼dx

daw

¼cos aw

2l1

Ao À E y awð Þ À E d2y awð Þ

da2w

,

y01 awð Þ ¼

À sin aw

2l1

Ao À E y awð Þ À E d2y awð Þ

da2w

, ð4:29Þ

and for angle aw ¼ p=2 we find co-ordinates of the point A1

(xa, 0) (Fig. 2).

x p=2À Á

¼1

2l1

Ao À E y p=2À ÁÀ Á

,

y1 p=2

À Á¼ 0. ð4:30Þ

The integral

I 1 ¼ 4l21

Z p=2

0

Àx awð Þ Á y01 awð Þd aw

¼

Z p=2

0

Ao À E y awð Þð Þ Á sin aw À E Ády awð Þ

daw

cos aw

Â Ao À E y awð Þ À E d2y awð Þ

da2w

sin aw daw ð4:31Þ

was numerical calculated. Then from (4.28) and (4.30) we

obtain

2l1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI 1 þp

4Ao À E y p=2À ÁÀ Á2 2h

V r . (4.32)

To determine the functions F1, F2 and F we must

calculate the integrals

I 2 ¼

Z xa

0


1

q dx,

I 3 ¼ Z xa

0

y arctan y01À ÁÀ Á ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ y02

1q dx,

I 4 ¼

Z xa

0


2

q dx.

The first two have been converted to the following forms:

I 2 ¼

Z p=2

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidx

daw

2

þd y1

daw

2s

daw,

I 3 ¼

Z p=2

0

y awð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidx

daw

2

þd y1

daw

2s

daw

and have been determined using numerical procedures.

Whereas the integral I 4 has been presented by the use of

elementary functions in the following way:Z xa

0


2

q dx ¼

p

2

As

2l1

¼ pxa. (4.33)

The values of functions F 1, F 2 and F have been

determined using the specially prepared computer

program.

5. Numerical example

The present calculations have been performed using

specially prepared computer program. The data assumed

for computation were as follows:

cw ¼ 145.29 z"/m2, co ¼ 522.46 z"/m2,

cc ¼ 225.47 z"/m2, c f ¼ 261.18 z"/m2,

ce ¼ 0,10 z"/MJ,

k w ¼ 0.386 W/m2/K, k o ¼ 1.4 W/m2/K,

k c ¼ 0.306 W/m2/K, k f ¼ 0.324 W/m2/K,

jc ¼ 1.0, j f ¼ 1.0,

SD ¼ 4000 K day, h ¼ 1 m,

S ¼ 2.00 m2.

The values of the parameter l used for computation were

l ¼ 1=2; 1=11; 1=26; 1=51 and 1=101,

which corresponds to the modified utilization times of the

building, expressed in years

N ¼ 1; 10; 25; 50; 100.

The results of the calculations for the building on an oval

base have been included in Table 1. The obtained solutions

are very much alike. The points of the set of nonpredo-

minant evaluations which have been determined are shown

in Fig. 3 and the corresponding forms of the bases of the

buildings – in Fig. 4.

In order to compare evaluation the functions F 1, F 2 and

F have been determined for the above data, for the building

ARTICLE IN PRESS




ARTICLE IN PRESS

Table 1

The results of the optimization computation of the form of a building on an oval base

l[dimensionless] 1/2 1/11 1/26 1/51 1/101

N [a] 1 10 25 50 100

xa [m] 0.79897 0.80501 0.80925 0.81205 0.81405

aw ¼ p=36 . . . x, y1 [m]

0 0.00 0.79 0.00 0.78 0.00 0.76 0.00 0.75 0.00 0.75

1 0.07 0.79 0.07 0.77 0.08 0.76 0.08 0.75 0.08 0.74

2 0.14 0.78 0.15 0.76 0.15 0.75 0.15 0.74 0.16 0.73

3 0.21 0.77 0.22 0.75 0.22 0.73 0.23 0.72 0.23 0.72

4 0.28 0.75 0.29 0.72 0.30 0.71 0.30 0.70 0.30 0.69

5 0.34 0.72 0.35 0.70 0.36 0.68 0.37 0.67 0.37 0.67

6 0.40 0.69 0.42 0.66 0.43 0.65 0.43 0.64 0.44 0.63

7 0.46 0.65 0.48 0.63 0.49 0.61 0.49 0.60 0.50 0.59

8 0.52 0.61 0.53 0.58 0.54 0.57 0.55 0.56 0.55 0.55

9 0.57 0.56 0.58 0.54 0.59 0.52 0.60 0.51 0.60 0.51

10 0.61 0.51 0.63 0.49 0.64 0.47 0.64 0.46 0.65 0.46

11 0.66 0.45 0.67 0.44 0.68 0.42 0.68 0.41 0.69 0.41

12 0.69 0.40 0.70 0.38 0.71 0.37 0.72 0.36 0.72 0.35

13 0.72 0.33 0.73 0.32 0.74 0.31 0.74 0.30 0.75 0.30

14 0.75 0.27 0.76 0.26 0.77 0.25 0.77 0.25 0.77 0.24

15 0.77 0.21 0.78 0.20 0.78 0.19 0.79 0.19 0.79 0.18

16 0.79 0.14 0.79 0.13 0.80 0.13 0.80 0.13 0.80 0.12

17 0.80 0.07 0.80 0.07 0.81 0.06 0.81 0.06 0.81 0.06

18 ¼ pX2 0.80 0.00 0.81 0.00 0.81 0.00 0.81 0.00 0.81 0.00

S1 [m2] (% S ) 0.997 (49.9) 0.982 (49.1) 0.971 (48.6) 0.964 (48.2) 0.959 (48.0)

S2 [m2] 1.003 1.018 1.029 1.036 1.041

S [m2] 2.000 2.000 2.000 2.000 2.000

Heat losses [kWh/a] 550.4 548.62 547.42 546.65 546.1

Heat gains [kWh/a] 152.52 152.24 152.06 151.95 151.9

F 1 [z"] 2645.8 2638.9 2634.3 2631.3 2629.2

F 2 [z"/a] 143.2 142.7 142.3 142.0 141.9

F [z"/a] 1 394.5 369.6 238.2 190.9 166.6

141.8

142

142.2

142.4

142.6

142.8

143

143.2

143.4

143.6

2625 2630 2635 2640 2645 2650F2 [z /a]

F 1 [ z ]

circle F1 = 2647.1 z

F2 = 143.3

A (N=1)

F = 1394.5 z /a

B (N=10)

F = 369.6 zl/a

C (N=25)

F = 238.2 z /a

D (N=50)

F = 190.9 z /a

E (N=100) = perfect evaluation F1 = 2 629.2 z

F = 166.6 z /a F2 = 141.9 z /a

l

l

l l

l

l

l

l

l

Fig. 3. The set of nonpredominant evaluations with indication of square, circle and perfect evaluation.




on a square base and a building on a circular base. The

results of calculations are included in Tables 2 and 3.

6. Conclusions

The polyoptimization of the form of a building on an oval

base has been solved by variational method. The obtained

solution is composed of the semicircular bounding of the

northern part of the building and a curve of the southern

part described by a parametrical function.

The ratio of the area of the southern part of the building

S1 to the area of the building S decreases with the number

N , which determines the modified utilization time of the

building. In the present problem they are S1=S ¼ 49:9%

for N ¼ 1 and S1=S ¼ 48:0% for N ¼ 100 (Fig. 5).

Values of the eccentricity parameters of the southern

part of the building S1, defined as y1(0)/xa, are shown in

Fig. 6. If the number N increase, then eccentricity

parameter y1(0)/xa decrease.

The examined forms on an oval base are better than the

form of the building on a circular base (Fig. 3) and much

better than the form of the building on a square base.

Values F 1, F 2 for the building on a square base are out

the Fig. 3.

ARTICLE IN PRESS

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

x [m]

y 1

[ m ]

N=1N=10

N=25

N=50

N=100

Fig. 4. The forms of the preferred bases of a building for various values N .

Table 2

The results of the computation of the form of a building on a square base

l [dimensionless] 1/2 1/11 1/26 1/51 1/101

N [a] 1 10 25 50 100

S1 ¼ 50% S [m2]

Heat losses [kWh/a] 605.9

Heat gains [kWh/a] 163.5

F 1 ¼ 2862.0 z"

F 2 ¼ 159.3 z"/a

F [z"/a] 1510.6 405.0 263.2 212.3 186.0

Table 3

The results of the computation of the form of a building on a circular base

l[dimensionless] 1/2 1/11 1/26 1/51 1/101

N [a] 1 10 25 50 100

S1 ¼ 50% S [m2]

Heat losses [kWh/a] 550.7

Heat gains [kWh/a] 152.6

F 1 ¼ 2647.1 z"

F 2 ¼ 143.3 z"/a

F [z"/a] 1395.2 371.0 239.6 192.4 168.1




The cylinder-shaped form of the building involves

higher operating costs and the investment costs in

comparison with the analyzed forms of an oval shape.

The investment and operating costs F of the building on

a circular base related to one year are higher for the

buildings on a circular base. Optimization of the form of building should be a

standard tool of designers at an early stage of the

architectural project.

References

[1] Fokin KF. Stroitelnaja tieplotechnika ograzdajuszczych czastiej

zdanij, Moskwa 1934.

[2] Gadomski J. Analysis of influence of the architectural concepts on

heat loss amounts in buildings and some predictions of this fields.

Warszawa: Institute of Buildings; 1987.

[3] Menkhoff H, Blum A, Trykowski M, Wente E, Zapke W.

Energetische Bauen. Energiewirtschaftliche Aspekte zur Planung

und Gestaltung von Wohngeba ¨ uden, 04.086/1983, Schriftenreihe

‘‘Bau- und Wohnforschung’’ des Bundesministers fu ¨ r Raumordnung,

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[4] Petzold K. Zum Einfluss non Form und Grosse der Geba ¨ ude auf den

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PAN and KN PZITB, (6) Krako ´ w-Krynica 1995;139–146.

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conservation buildings, Scientific Bull. WSI, Zielona Gora 1989;88.

[7] Markus TA. Cost-benefit analysis in building design. Problems and

solutions. Journal of Architectural Research 1976;5(3).

[8] Radford AD, Gero JS. Trade off diagrams for the integrated design

of the physical environment in buildings. Buildings and Environment

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[9] Radford AD, Gero JS, Murthy NS. Designing optimal multi-

functional surface materials, ACMBSM. The University of New-

castle; 1982.

[10] Owczarek S. Optymization of the shape of energy-savings buildings

on the plan of polygon, Studia z zakresu inzynierii, No. 30, IPPT

PAN, Warszawa; 1993.

ARTICLE IN PRESS

0.47

0.475

0.48

0.485

0.49

0.495

0.5

1 10 25 50 100

N [years]

S1 / S

Fig. 5. The ratio of the area of the southern part of the building S1 to the area of the building S as the function the modified number N of utilization time

of the building.

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1 10 25 50 100

N [years]

y1(0)/xa

Fig. 6. The eccentricity parameter y1(0)/xa as the function of the modified number N of utilization time of the building.




[11] Adamski M, Marks W. Multicriteria optimization of shapes and

structures of external walls of energy conservation buildings.

Archives of Civil Engineering 1993;39(1):77–91.

[12] Adamski M. Zagadnienia optymalizacji wielokryterialnej budynkow

energooszczednych, Praca doktorska, IPPT PAN, Warszawa, 1993.

[13] Marks W, Owczarek S, [pod red.], Optymalizacja wielokryterialna

budynko ´ w energooszczednych, Studia z zakresu inzynierii, nr 46,

IPPT, KILW PAN, Warszawa, 1999.

[14] Jedrzejuk H, Marks W. Discrete polyoptimization of energy-saving

buildings. Archives of Civil Engineering 2002;48(3):331–47.

[15] Adamski M. Optimization of the form of a building with an arbitrary

base. Engineering Trans 1994;42(4):359–76.

[16] Adamski M. Polyoptimization of the form of a building on an oval

base. Archives of Civil Engineering 2003;XLIX(4):511–30.

[17] Findeisen W, Szymanowski J, Wierzbicki A, Teoria i metody

obliczeniowe optymalizacji. PWN, Warszawa, 1977.

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