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# Optimizing Damper Connectors for Adjacent Buildings

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Many theoretical and experimental studies have used heuristic methods to investigate the dynamic behaviour of the passive coupling of adjacent structures. However, few papers have used optimization techniques with guaranteed convergence in order to increase the efficiency of the passive coupling of adjacent structures. In this paper, the combined problem of optimal arrangement and mechanical properties of dampers placed between two adjacent buildings is considered. A new bi-level optimization approach is presented. The outer-loop of the approach optimizes damper configuration and is solved using the inserting dampers'' method, which was recently shown to be a very effective heuristic method. Under the assumption that the dampers have varying damper coefficients, the inner-loop finds the optimal damper coefficients by solving an $n$-dimensional optimization problem, where derivative information of the objective function is not available. Three different non-gradient methods are compared for solving the inner loop: a genetic algorithm (GA), the mesh adaptive direct search (MADS) algorithm, and the robust approximate gradient sampling (RAGS) algorithm. It is shown that by exploiting this new bi-level problem formulation, modern derivative free optimization techniques with guaranteed convergence (such as MADS and RAGS) can be used. The results indicate a great increase in the efficiency of the retrofitting system, as well as the existence of a threshold on the number of dampers inserted with respect to the efficiency of the retrofitting system.
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Optimizing Damper Connectors for Adjacent Buildings
K. Bigdeli, W. Hare, J. Nutini and S. Tesfamariam
October 16, 2015
Abstract
Many theoretical and experimental studies have used heuristic methods to investi-
gate the dynamic behaviour of the passive coupling of adjacent structures. However,
few papers have used optimization techniques with guaranteed convergence in order
to increase the eﬃciency of the passive coupling of adjacent structures. In this paper,
the combined problem of optimal arrangement and mechanical properties of dampers
placed between two adjacent buildings is considered. A new bi-level optimization ap-
proach is presented. The outer-loop of the approach optimizes damper conﬁguration
and is solved using the “inserting dampers” method, which was recently shown to be a
very eﬀective heuristic method. Under the assumption that the dampers have varying
damper coeﬃcients, the inner-loop ﬁnds the optimal damper coeﬃcients by solving
an n-dimensional optimization problem, where derivative information of the objective
function is not available. Three diﬀerent non-gradient methods are compared for solv-
ing the inner loop: a genetic algorithm (GA), the mesh adaptive direct search (MADS)
algorithm, and the robust approximate gradient sampling (RAGS) algorithm. It is
shown that by exploiting this new bi-level problem formulation, modern derivative free
optimization techniques with guaranteed convergence (such as MADS and RAGS) can
be used. The results indicate a great increase in the eﬃciency of the retroﬁtting sys-
tem, as well as the existence of a threshold on the number of dampers inserted with
respect to the eﬃciency of the retroﬁtting system.
Keywords: Seismic retroﬁtting, passive coupling, derivative-free optimization, bi-level
optimization
1 Introduction
Increase in demand for residency and oﬃce buildings, coupled with limited land avail-
ability, has resulted in the construction of high-rise buildings in close proximity. During
an earthquake, such closely spaced buildings are prone to pounding induced damages
[30, 7, 22, 14]. By using damper connectors, the seismic vulnerability of adjacent
buildings can be reduced, and thus, these pounding induced damages can be controlled
[4, 8, 32]. Due to often having limited available resources, decision makers need to be
able to optimize the number and placement of dampers. In this paper, a new problem
formulation and optimization approach are presented to solve the combined problem of
1
ﬁnding the optimal arrangement and optimal mechanical properties of dampers placed
Considerable research has been reported on diﬀerent damping devices, conﬁrm-
ing the eﬃciency of damper connectors in mitigating the vibrations of structures
[33, 36, 8, 28, 9, 25, 38, 39, 35, 37, 25, 34, 35]. The research on optimizing damper
connections between adjacent buildings can be categorized into two categories: the
placement of dampers; and the determination of mechanical properties for dampers
[11, 10]. The mechanical properties of dampers can be further subdivided into the
kind of dampers used and the optimal damper coeﬃcients required (see [10, 26] or
Section 3.1 herein for further details on damper coeﬃcients). This paper focuses on
the combined optimization of both the placement of dampers and the corresponding
damper coeﬃcients. This is set as a novel bi-level optimization problem of ﬁnding
the optimal conﬁguration of dampers and the corresponding optimal damper coeﬃ-
cients. The conﬁguration of dampers is a discrete optimization problem, whereas, the
determination of optimal damper coeﬃcients is a continuous optimization problem.
Several studies have presented results for optimizing damper coeﬃcients [32, 40, 5,
6, 27]. Xu et al. [32] considered multiple uniform dampers throughout the buildings,
connecting every adjacent ﬂoor of the multiple degree-of-freedom (MDOF) structures.
Zhu and Xu [40] presented an analytical closed form solution for the damper coeﬃcients
of a ﬂuid damper connecting two single degree-of-freedom (SDOF) structures. Basili
and De Angelis [5, 6] studied optimal mechanical properties of nonlinear hysteretic
dampers connecting SDOF and MODF structures. They presented explicit equations
relating dissipation energy, relative displacement and relative acceleration of the me-
chanical properties of the dampers. Patel and Jangid [27] assumed that the optimal
damper coeﬃcients were functions of the relative velocity between the structures; the
damper coeﬃcients increased from a small value for the base ﬂoor to a large value for
the top ﬂoor.
The discrete optimization problem of damper positioning has also been the topic
of several recent studies [35, 26, 10]. For example, Yang and Lu [35] constructed
a series of experiments to show that one can eliminate half of the dampers without
compromising eﬃciency. However, a method to determine the optimal arrangement of
the dampers was not included. Bigdeli et al. [10] presented an optimization algorithm
to ﬁnd the optimal placement of dampers with a limited number of available dampers.
All dampers were assumed to be identical with the same mechanical properties. The
results of Bigdeli et al. [10] also demonstrated that if all damper coeﬃcients are assumed
to be equal, then increasing the number of dampers does not necessarily increase the
dynamic stability of a structure. Moreover, under these conditions, increasing the
number of dampers may actually exacerbate the dynamic behaviour of the buildings.
This is in agreement with other studies [26].
To the authors’ knowledge, the only research that studies the combined damper
location and coeﬃcient selection problem is Ok et al. [26]. Ok et al. [26] examined a
multi-objective optimization method using genetic algorithm (GA) for a set of coupled
MDOF structures connected to each other by magneto-rheological (MR) dampers. The
number of dampers and the voltage for the MR dampers installed at each ﬂoor were
assumed as design parameters. Since a bounded domain for voltage was assumed,
they allowed each ﬂoor to have more than one damper if the result was more eﬀective.
Unlike this paper, Ok et al. used a single-level optimization framework, and thus,
2
used a heuristic (speciﬁcally genetic algorithm) to seek optimality. In this paper,
a novel bi-level framework is considered to optimize the arrangement and mechanical
properties of dampers placed between two adjacent buildings. The objective function is
set to minimize the maximum inter-story drift over all possible damper conﬁgurations.
This is a common objective in seismic retroﬁtting literature [23, 10, 21], however,
it should be noted that other objective functions have been used [35, 26, 17]. The
optimization techniques and bi-level framework used herein are independent of the
objective function, so could be readily applied to optimize other aspects of seismic
retroﬁtting. Such exploration is left for future research.
The bi-level approach presented in this paper uses an outer-loop that seeks to opti-
mize the combinatorial problem of where to place dampers, and an inner-loop that seeks
to solve the continuous problem of determining the optimal design for each damper.
This bi-level approach presents one key advantage over a single level approach, namely,
the inner-loop can be solved using modern derivative-free optimization (DFO) software,
and therefore some assurance of (local) optimality can be attained. The approach is
tested on a suite of 150 test-cases. Results support the eﬀectiveness of the approach and
demonstrate the importance of using high quality DFO software (opposed to heuristic
methods). Results further demonstrate that there exists a threshold on the number
of dampers inserted with respect to the eﬃciency of the retroﬁtting system. That is,
maximal eﬃciency can be achieved using a limited number of dampers, provided that
the internal damper designs are fully optimized.
The remainder of this paper is organized as follows. In Section 2, the physical
model and the optimization problem are presented. In Section 3, a discussion of each
level of the bi-level problem is presented, as are the details of the optimization methods
used in this paper. In Section 4, numerical results for 150 test problems are presented.
Some conclusions are provided in Section 5.
2 Physical Model & Optimization
Figure 1 provides an illustrative model of two adjacent buildings connected by dampers.
In this paper, buildings are modelled as 1-dimensional systems assuming the center
of rigidity and the center of mass of adjacent ﬂoors are in the same plane with the
viscous dampers (with variable damper coeﬃcients) are connected at the ﬂoor level.
Each building is modelled as a multi degree-of-freedom (MDOF) system consisting
of lumped masses (the mass of each ﬂoor), linear springs (stiﬀness of the columns),
and linear viscous dampers [32, 10]. The ground motion and dynamic response of
the buildings are assumed to be unidirectional. Both buildings are assumed to be
symmetric in plane (i.e., their centers of mass located in the same plane), and as a
result, the eﬀect of torsional vibrations is not considered. Equal building heights are
not required, and consequently, the maximum number of dampers that can be placed
is controlled by the shortest building.
Buildings 1 and 2 have n+mand nstories and are connected by nddampers.
Thus, the dynamic model for both structures is a 2n+mdegree-of-freedom system.
Let X(t)RN, where N= 2n+m, be the vector of displacements of each ﬂoor at
time t. The governing equation of the system can be expressed as
M¨
X(t)+(C+Cd)˙
X(t) + KX(t) = M E g(t),(1)
3
Building1
Building 2
c
n1
c
n+1,1
c
n+m,1
c
n2
n-1,2
c
12
c
n-2,2
c
32
k
12
k
22
k
32
k
n-2,2
k
n-1,2
k
n2
k
11
k
21
k
31
k
n-2,1
k
n-1,1
k
n1
k
n+1
,1
k
n+m,1
m
11
m
21
m
n-2,1
m
n-3,1
m
n-1,1
m
n1
m
n+m-1,1
m
n+m,1
m
12
m
22
m
n-2,2
m
n-3,2
m
n-1,2
m
n2
Figure 1: Model of adjacent buildings with damper connectors.
where matrices MRN×N,CRN×Nand KRN×Nare generated by the given
mass, damping, and stiﬀness factors of the buildings, respectively. The vector ERN
is a vector of ones, and the function g:RRis the ground acceleration during
the earthquake. The matrix CdRN×Nis constructed using the damper coeﬃcients
and the locations of the nddampers. Let cdRn,cd0, be the vector of damper
coeﬃcients for each ﬂoor, where any ﬂoor ithat is without a damper has coeﬃcient
cd,i = 0. The matrix Cdtakes the form
Cd=
diag(cd) zero(n, m)diag(cd)
zero(m, n) zero(m, m) zero(m, n)
diag(cd) zero(n, m) diag(cd)
,(2)
4
where diag(x) is the diagonal matrix whose entries coincide with the vector x, and
zero(a, b) is an a×bzero matrix.
Note that equation (1) describes the motion in the time domain. Considering the
spectral density of the ground excitation, the equation of motion can be written in the
frequency domain as
eiωt M ω2X(ω)+(C+Cd)iωX(ω) + KX(ω)=M EqSg(ω)eiωt ,(3)
where the response of the building is given by
X(ω) = Mω2+ (C+Cd)+K1×M EqSg(ω).(4)
In this paper, a Kanai-Tajimi ﬁltered white noise function is used for the spectral
density function of ground acceleration:
Sg(ω) =
1+4ζ2
gω
ωg2
1ω
ωg22
+ 4ζ2
gω
ωg2S0,(5)
where ωg,ζgand S0represent dynamics characteristics and the intensity of the earth-
quake. These parameters are chosen based on geological characteristics.
For a given vector of damper coeﬃcients, cd, a numerical approximation of the
standard deviation of the displacement response for the ith ﬂoor of building bis possible,
and is given by
σib =Z+
−∞
kxib(ω)k21
2
,(6)
where xib(ω) is the component of X(ω) corresponding to ﬂoor iof building b. The
value σib is used to calculate the inter-story drift for each ﬂoor, which is denoted by
fib =σib σ(i1)b2, where σ0bis deﬁned as 0. This in turn deﬁnes the maximum
inter-story drift as
F= max max
i=1,...,n+m(σi1σ(i1)1)2,max
i=1,...,n (σi2σ(i1)2)2.(7)
The objective is to determine the optimal conﬁguration of dampers and damper coef-
ﬁcients in order to minimize the maximum inter-story drift.
As an analytic solution to equation (6) is unavailable, a numerical approximation
of σib is required. To do this, upper and lower limits of ±20rad/s are imposed, and
a trapezoidal rule approximation is applied to the integral in equation (6) with a step
size of 0.02. (Previous studies show that the eﬀect of frequencies greater than 20rad/s
on the response of the structure is negligible [32]).
In summary, to place nddamper connectors between two buildings of heights n+m
and n, an optimization problem of the following form is considered:
min
cdRn
+
F(cd)s.t. cd,j = 0 for at least nndvalues of j, (8)
where
F(cd) = max fib(cd) : i= 1, . . . , n +m, b = 1; i= 1, . . . , n, b = 2(9)
5
and each fib is numerically approximated using computer simulation. The bi-level
optimization approach presented in the next section is designed for this problem for-
mulation: the outer-loop seeks to optimize the combinatorial problem of which values
of cd,j are 0, and the inner-loop seeks to solve the continuous problem of determining
the optimal values of the damper coeﬃcients.
3 A Bi-level Approach
As discussed in the previous section, the optimization problem in this paper is con-
sidered as a bi-level optimization problem with an inner continuous optimization algo-
rithm and an outer discrete optimization algorithm. The inner-loop uses a non-gradient
based method to ﬁnd an optimal set of damper coeﬃcients for a ﬁxed conﬁguration
and a ﬁxed number of dampers. Based on the research in [10], the outer-loop uses a
heuristic optimization algorithm, which seeks the optimal conﬁguration of dampers. A
schematic outline of the presented bi-level approach is given in Figure 2.
Beginning at initialization, the algorithm obtains the mechanical properties of the
two adjacent buildings under consideration. The buildings are initially considered
as two unconnected structures, and the iteration count kis set to 0. From here,
the algorithm initializes the ﬁrst iteration of the outer-loop, which optimizes damper
location.
The heuristic algorithm used for the outer-loop (as schematically shown in full
in Figure 2) is the inserting dampers method from [10]. In [10], this method was
shown to be the most eﬀective among others at ﬁnding the optimal conﬁguration of
dampers. In addition, this approach has the advantage that the optimization of nd
dampers automatically provides the solutions to the optimization of 1,2, ..., and nd1
dampers.
During the ﬁrst iteration, the algorithm checks all possible locations, i.e., all ﬂoors
i= 1,...n, to put the ﬁrst damper by determining the optimal damper coeﬃcients
(inner-loop, dashed line in Figure 2) for each location. Following each inner-loop
iteration, the resulting objective value (minimum over the maximum inter-story drifts
for the current ﬁxed damper conﬁguration) is compared to the best objective value for
the current ﬂoor. In the ﬁrst iteration, this step consists of initializing the objective
value for each ﬂoor. For subsequent iterations, if placing the damper on the current
ﬂoor in the current conﬁguration produces a smaller objective function value than the
current best, then the stored best objective function value is updated accordingly for
that ﬂoor. The damper is then removed from the current ﬂoor, and a damper is placed
on the next (consecutive) ﬂoor. This continues until the top ﬂoor is reached.
Once all the ﬂoors have been cycled through (condition i<nis not satisﬁed),
the algorithm computes an element of the argmin over the resulting set of nobjective
function values. (Recall, the argmin, or the argument of the minimum, is the set of all
minimizers for an optimization problem.) The solution to this problem, j, indicates
that the placement of a damper on ﬂoor jresults in the minimal objective value for the
current overall damper conﬁguration. Thus, a ‘permanent’ damper is inserted at ﬂoor
j. From here, the iteration count is increased and this iterative procedure is repeated,
starting again at ﬂoor 1, until all available dampers are inserted into the structure. (In
order to aid the inner-loop, when possible the solution to the previous iteration is used
as a warm-start for the solver used in the inner loop (see Section 3.1)). If a ‘permanent’
6
Start
Get:
Mechanical Properties of
two buildings, m, n
i=1
Determining optimal damping
coefficients, and put
Obj(i)= Best objective function
value
damper in i th floor
Initialize with .
(Unconnected buildings)
i=i+1
End
Add a damper in j th
floor, and set
Yes
Yes
No
in i th floor?
Add a damper in i th floor
No
Obj(i) =
No
Yes
0k
nk
1kk
})({minarg },...,2,1{ iObjj ni
ni
Figure 2: Schematic of the bi-level optimization problem: outer-loop in full, inner-loop (dashed
box) in brief.
damper has been inserted in a previous iteration at the current ﬂoor, then when that
ﬂoor is reached in the cycle, the algorithm assigns an objective function value of inﬁnity
for this ﬂoor. It then proceeds onto the next step: either continuing onto the next ﬂoor,
or computing the argmin over the objective values. This ensures that any ﬂoor with a
‘permanent’ damper will not be selected again for damper insertion when calculating
the argmin over the objective function values. This iterative process continues until
the maximum number of allowable dampers have been placed.
In the next section, details of the methods used to solve for the optimal damper
coeﬃcients in the inner-loop are presented.
3.1 Damper Coeﬃcients Optimization
The main purpose of the inner-loop of the optimization algorithm (the dashed box in
Figure 2) is to ﬁnd optimal damper coeﬃcients for a ﬁxed conﬁguration of damper
7
connectors. As derivative information of the objective function is not available for the
optimization problem, the inner-loop requires the use of a non-gradient based method.
Many non-gradient optimization options exist. These can be widely split into
two categories, heuristic optimization methods and derivative-free optimization (DFO)
methods. Here, DFO refers to methods that are mathematically derived and studied
to provide (theoretical) proof of convergence to (local) minimizers, whereas heuristic
methods are any other non-gradient based methods that do not ﬁt this deﬁnition. For
a thorough introduction into several well-known DFO frameworks, see [20, 15].
Due to their versatility, heuristic optimization methods are widely used in structural
engineering. One such method is the genetic algorithm (GA) [26]. However, heuristic
methods do not guarantee convergence to (locally) optimal solutions. As such, there
has been a recent increase in the use of derivative-free optimization techniques that
guarantee optimality. In this work, the mesh adaptive direct search (MADS) algorithm
[3] is examined, as implemented in MATLAB’s global optimization toolbox; as well as a
novel robust approximate gradient sampling (RAGS) algorithm [19] that is speciﬁcally
designed for ﬁnite minimax problems [19]. The stochastic based GA is used as a
baseline comparison to previous studies that have principally employed this method.
A detailed description of each algorithm is given in Section 3. Greater detail on each
of these methods is provided next.
3.1.1 Genetic Algorithm
The genetic algorithm (GA) is a popular heuristic search method. It has been argued to
be a reasonably eﬃcient method, particularly in engineering applications ([2, 29, 18, 26]
and references therein). A simple GA consists of several steps, including the generation
of initial points, selection, competition and reproduction [31]. A brief description of
the genetic algorithm follows.
procedure GeneticAlgorithm
begin
Initialize and evaluate random population P(t);
while stopping conditions not satisﬁed do
begin
Mutate and crossover P(t) to yield C(t);
Evaluate C(t);
Select P(t+ 1) from C(t) and elite individuals from P(t);
end
end
In this paper, the GA is used in a standard form that is included in the MATLAB
global optimization toolbox [24]. All parameter values and settings are the MATLAB
default choices, future research may explore alternate parameter selections.
The mesh adaptive direct search (MADS) method [3] is a sub-category of pattern search
(PS) methods. A brief description of a general pattern search method follows.
8
procedure PatternSearch
begin
Initialize x0and a set of directions D;
while stopping conditions not satisﬁed do
begin
Search for a point with f(x)< f(xk) (optional);
Poll points from {xk+αkd:dDk(∈ D)};
if f(xk+αkdk)< f(xk)
Stop polling;
xk+1 xk+αkdk;
else
xk+1 xk;
Update mesh parameter αk;
end
end
end
Speciﬁc to MADS, randomly rotated bases are used in each iteration to provide a more
robust convergence. In this paper, a MATLAB interface is employed with the version
of MADS that is implemented in the NOMAD project [1]. All parameter values and
settings are the MADS default choices, future research may explore alternate parameter
selections.
The robust approximate gradient sampling algorithm (RAGS algorithm) is a derivative-
free optimization algorithm that exploits the smooth substructure of the ﬁnite minimax
problem,
min
xF(x) where F(x) = max{fi:i= 1, . . . , N }.
The general concept of the RAGS algorithm relies on the deﬁnition of the active set of
a ﬁnite max function fat a point ¯x,
Ax) = {i:Fx) = fix)}.
Loosely speaking, the subdiﬀerential of Fat a point ¯xis the set of all possible gradients.
For a ﬁnite max function, the subdiﬀerential, as shown in [13], is given by
∂f ( ¯x) = conv{∇fix)}iAx),(10)
and the direction of steepest descent can be deﬁned via Proj(0|∂f (¯x)). Although the
direction of steepest descent is ﬁne, it tends to get stuck on non-diﬀerentiable ridges
of the function.
In 2005, Burke et al. [12] introduced a robust gradient sampling algorithm. This
algorithm uses information from around the current iterate to help minimize along
non-diﬀerentiable ridges of nonsmooth functions. A brief description of the RAGS al-
gorithm follows.
9
procedure RAGS
begin
Initialize x0, search radius ∆0, Armijo-like parameter ηand other parameters;
begin
Generate a set of n+ 1 points;
Use points to generate robust approximate subdiﬀerential Gk
Y;
Set search direction dk
Y= Proj(0|Gk
Y);
if ksmall, but |dk|large
Carry out line search: ﬁnd tk>0 such that f(xk+tkdk)< f(xk)ηtk|dk|2;
Success: update xkand loop;
Failure: decrease accuracy measure and loop;
else if klarge
Decrease ∆kand loop;
else
Terminate;
end
end
end
pling algorithm to a DFO setting. Like MADS, the RAGS algorithm is proven to
converge to a local minimizer. Readers are referred to [19] for further information.
All parameter values and settings are the RAGS default choices, future research may
explore alternate parameter selections.
3.2 Warm-starting
All of the algorithms tested allow the user to input an initial point. Thus, once the
outer-loop has completed at least one iteration, the solution to the past outer-loop
conﬁguration can be used to ‘warm-start’ the inner-loop computation. Speciﬁcally,
the damper coeﬃcients are initialized by setting cj
d,i =cj1
d,i if there was a damper in
position iduring outer-loop j1, and cj
d,i = 0 if there was not a damper in position
iduring outer-loop j1. Two versions of each algorithm are considered, for six
algorithms total: GA using a random initial point for each new inner-loop (denoted
GAr), GA using the warm-start initial point for each new inner-loop (denoted GAw),
initial point (denoted MADSw), RAGS using a random initial point (denoted RAGSr),
and RAGS using the warm-start initial point (denoted RAGSw).
4 Numerical Results
In this section, a summary of results are presented for various numerical problems
for damper-connected structures. The solution times and quality measures of the
previously presented non-gradient based methods are compared.
10
4.1 Test Problems
In order to compare the presented methods, three diﬀerent sets of mechanical properties
and ﬁve diﬀerent sets of heights for the two buildings are considered. It should be noted
that the mechanical properties are taken from previous studies in the ﬁeld and are
reasonable examples of buildings requiring seismic retroﬁtting [32] [10]. However, the
building heights are artiﬁcial, and selected to be representative of a variety of scenarios.
These are provided in Tables 1 and 2.
Table 1: Mechanical Properties
Building aBuilding b
ma (kg) ka (N/m) ca (N.s/m) mb (kg) kb (N/m) cb (N.s/m)
Set I 1.29E+06 4.00E+09 1.00E+05 1.29E+06 2.00E+09 1.00E+05
Set II 2.60E+06 1.20E+10 2.40E+06 1.60E+06 1.20E+10 2.40E+06
Set III 4.80E+06 1.60E+10 1.20E+06 4.00E+06 2.30E+10 1.20E+06
Table 2: Building Heights
Case fafb
1 10 10
2 10 20
3 20 10
4 10 40
5 40 10
In Table 2, faand fbrepresent the number of ﬂoors for buildings aand b, re-
spectively. For all numerical examples, to generate the ground excitation spectrum,
the following values are used for the ground acceleration parameters in equation (5):
parameter values are the same as those used in [10] and [32].) For each of the 3 sets of
mechanical properties, the number of dampers changes from 1 to 10. Therefore, incor-
porating all 5 building height combinations, a total of 150 test problems are generated,
representing a wide range of situations. Each problem is solved via a combination of
the inserting dampers method and a non-gradient based method (either GA, MADS
or RAGS). Optimal arrangements and damper coeﬃcients, as well as corresponding
objective function values are determined.
As an example output, for Building Height 1, Material Set I, using 4 dampers,
the optimal conﬁgurations represented in vector form, as solved via GA, MADS, and
RAGS were as follows:
GA: [0,0,2.4331,0.4821,1.5187,0,0,0,0,0.2146] 107,
RAGS: [0,0,2.4188,0.1135,1.6505,0,0,0,0,0.2099] 107,
where a bold zero (0) denotes a ﬂoor that has no damper. Notice that, while the
damper coeﬃcients diﬀer, they are similar, and all methods resulted in dampers on
11
ﬂoors 3, 4, 5, and 10. Also note that the damper coeﬃcients diﬀer from ﬂoor to ﬂoor,
emphasizing the need for multi-variable optimization.
4.2 Solution Time and Quality
Tables 3 to 8 in Appendix A show the number of function calls required and the
optimal objective function values obtained using various methods. For the sake of
brevity, optimal design variables, including conﬁgurations of dampers and damper
coeﬃcients, are not included. The full data is available upon request by contacting the
corresponding author.
Note that in Tables 3 to 8, instead of reporting actual solution times in seconds, the
number of performed simulations is reported. It is worth noting that each simulation
takes approximately 2 seconds, regardless of the details or dimension of the problem.
Clearly, the rate of convergence is a crucial factor for any optimization algorithm.
As a ﬁrst comparison of convergence rates, in Figures 3 and 4, for Building Height 1,
Material Set I, and the case when all adjacent ﬂoors are connected, the objective value
(a) and the minimum objective value (b) for each function call are plotted. Figure
3 displays the three algorithms using random initial points and Figure 4 displays the
three algorithms using warm-start initial points. For brevity, other ﬁgures displaying
similar results are omitted.
200 400 600 800 1000 1200
100
101
102
Function Calls
Objective Value
GAr
RAGSr
(a) Objective value at each function call.
200 400 600 800 1000 1200
10−0.29
10−0.27
10−0.25
10−0.23
10−0.21
Function Calls
Minimum Objective Value
GAr
RAGSr
(b) Minimum objective value at each func-
tion call.
Figure 3: Objective values for Material Set I and Building Heights 1 when algorithms use random
initial points.
In Figure 3(a), it can be seen that, GA is a stochastic based method. In particular,
it evaluates the objective function at a wide range of points, resulting in a large range
of function values, even after convergence is essentially established. For the MADS
algorithm, a similar variation in objective value range is seen, with multiple spikes in
the objective value as the number of function calls increases. Looking closely, it can
be seen that RAGS converges with minimal objective value variation. This is because
RAGS lacks any global search heuristics.
Examining Figures 3 and 4, one notes that all three algorithms do fairly well on
these problems. In Figure 3, RAGS outperforms the other methods, while in Figure 4,
GA does extremely well. However, it should be noted that these ﬁgures only represent
1 out of 150 test problems. In order to investigate the overall performance of the
presented methods, a performance proﬁle is used [16].
12
0 200 400 600 800 1000 1200
100
101
102
103
Function Calls
Objective Value
GAw
RAGSw
(a) Objective value at each function call.
200 400 600 800 1000 1200
100.101
100.104
100.107
100.11
100.113
100.116
Function Calls
Minimum Objective Value
GAw
RAGSw
(b) Minimum objective value at each func-
tion call.
Figure 4: Objective values for Material Set I and Building Heights 1 when algorithms use warm-
start initial points.
Performance proﬁles are designed to graphically compare both the speed and the
robustness of algorithms across a test set. This is done by plotting, for each algorithm,
the percentage of problems that are solved within a factor of the best solve time. For
a more detailed description of performance proﬁles, see [16].
To calculate the performance proﬁle, a deﬁnition of when a method “solves” a
speciﬁc problem is required. In this paper, a method is considered as a “failed method”
if the diﬀerence between the objective value obtained using the method in question and
the best objective value obtained by any of the methods for that problem exceeds the
deﬁned allowable tolerance. Performance proﬁles for the presented methods are plotted
in Figure 5 for allowable tolerances of 5% and 1%, respectively.
0 50 100 150 200 250 300 350
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
τ
ρ(τ)
GAw
RAGSw
GAr
RAGSr
(a) Maximum tolerance of 5%
0 50 100 150 200 250 300 350
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
τ
ρ(τ)
GAw
RAGSw
GAr
RAGSr
(b) Maximum tolerance of 1%
Figure 5: Performance proﬁles for GAw, MADSw, RAGSw, GAr, MADSr, and RAGSr.
In Figure 5(a), it is shown that for 5% tolerance the maximum accuracy is obtained
the problems, and takes only a fraction of the time portion to achieve this. In Figure
5(b), it is shown that for 1% tolerance, MADSwnot only provides maximum accuracy,
but also uses the least solving time. Overall, it appears that the MADS algorithm
using a warm-start procedure is well suited to solve these problems.
13
An interesting note occurs in comparing the warm-start with random initial points.
Warm-starting seems to give a small positive boost to RAGS, particularly in rate of
convergence. Conversely, RAGSractually outperforms RAGSwin ﬁnal accuracy. This
is likely because, unlike MADS and GA, RAGS has no embedded heuristics to break
out of local minimizers. This suggests that the warm-start locations, while good, are
local minimizers of the next problem. Finally, without warm-starting GA performs
quite poorly.
4.3 Number of Dampers
In this section, the eﬀects of the number of dampers on the eﬃciency of the retroﬁtting
system for one of the test problems are presented. As a multi-objective optimization
study, fewer dampers and increased eﬃciency of the system are desired. One of the key
results in this section (and this research) is that, if a proper optimization is applied,
then there is no need to place dampers on every story. In fact, one can get optimal
results by placing dampers on only a fraction of the total number of stories.
To help visualize this result, Figure 6 plots the number of dampers used against the
optimal maximum inter-story drift achieved for Building Heights I and Material Sets I,
II and III. (Plots for other Building Heights look similar, and are available by contacting
the corresponding author.) As expected, the objective value generally decreases as the
number of dampers are increased. What is surprising is how rapidly the objective value
decreases. For Material Set I, Figure 6(a), optimal values are obtained using just 5 or
6 dampers for every algorithm except GAw. Similar trends occur in Figures 6(b) and
6(c).
points. As expected, the use of warm-start initial points means that the objective value
never increases when the number of dampers is increased. When random initial points
are employed, this trend is not present, and indeed GArdoes notably worse using 10
dampers than using just 5 or 6. On the other hand, examining Figures 6(b) and 6(c),
it is seen that RAGSrstuck in a local minimizer that requires just 3 dampers. So,
warm-starting appears valuable, but only if the algorithm includes some heuristic to
break free of local minimizers.
Figure 6 inspires us to consider how many dampers are required by each of the
algorithms to ﬁnd an optimal solution for a single building. For a ﬁxed Mechanical
Property and Building Height, the optimal solution is taken to be the overall lowest
value found by all six algorithms given any number of dampers. This yields 15 test
problems (3 sets of Mechanical Properties and 5 sets of Building Heights). The results
are represented in Figure 7.
In Figure 7(a), a histogram of the number of dampers used in the exact optimal
solution for a ﬁxed Mechanical Property and Building Height is provided. While Fig-
ure 7(b), provides a histogram of the minimum number of dampers used in order to
minimize the objective within 1% of the optimal solution. Examining Figure 7(a), no-
tice that only one problem requires 10 dampers to achieve the minimum value (this is
Material Set III, Building Height 4). More interestingly, in Figure 7(b) it is seen that
the vast majority of problems are solved with less than 6 dampers, and many require
just 2 or 3 dampers to achieve a solution within 1% of the optimal solution. It is worth
noting that the problem that requires 9 dampers to achieve a solution within 1% of
the optimal solution is also Material Set III, Building Height 4.
14
1 2 3 4 5 6 7 8 9 10
8
8.5
9
9.5
10
10.5 x 10−7
Number of Dampers
Objective Value
GAw
RAGSw
GAr
RAGSr
(a) Material Set I
1 2 3 4 5 6 7 8 9 10
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9 x 10−7
Number of Dampers
Objective Value
GAw
RAGSw
GAr
RAGSr
(b) Material Set II
1 2 3 4 5 6 7 8 9 10
4.5
4.6
4.7
4.8
4.9
5
5.1
5.2
5.3
5.4
5.5 x 10−7
Number of Dampers
Objective Value
GAw
RAGSw
GAr
RAGSr
(c) Material Set III
Figure 6: Objective value for an increasing number of inserted dampers using Building Heights I.
4.4 Damper Conﬁguration
In Subsection 4.3, it is found that for most building combinations, the number of
dampers required to solve problems within a tolerance of the optimal solution is less
than half of the maximum number of dampers that could be inserted. It is particu-
larly interesting that several problems can be solved (with 1% tolerance) using just 2
dampers. A natural question at this point is, at which ﬂoors are dampers most com-
monly inserted? Figure 8 examines this question, speciﬁcally looking at the cases when
1 damper, 2 dampers, and 5 dampers are inserted.
Figure 8 plots a histogram of the optimal damper locations for the 15 buildings.
Notice that if only one damper is used, then by far the most common location is on the
tenth ﬂoor (as high as possible in the problem). Examining Figures 8(b) and 8(c), the
pattern becomes less apparent. The most popular location is always the tenth ﬂoor,
but as more dampers are added, the locations become more scattered. This emphasizes
the importance of optimization and considering each building conﬁguration uniquely.
5 Conclusion
This paper presents a comprehensive optimization problem formulation and proce-
dure that can be used to ﬁnd the optimal conﬁguration and mechanical properties of
15
1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
Number of Dampers
Number of Problems Solved
(a) Exact Minimal Solution
1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
Number of Dampers
Number of Problems Solved
(b) Tolerance of 1%
Figure 7: Number of dampers required to solve problems within a tolerance of the optimal solution.
1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Floor
Number of Dampers
(a) 1 Damper
1 2 3 4 5 6 7 8 9 10
0
2
4
6
8
10
12
Floor
Number of Dampers
(b) 2 Dampers
1 2 3 4 5 6 7 8 9 10
0
2
4
6
8
10
12
14
Floor
Number of Dampers
(c) 5 Dampers
Figure 8: Histograms of aggregate optimal damper locations for the 15 test buildings.
dampers for connected structures. In particular, two adjacent buildings are considered
as lumped mass models connected to each other using discrete viscous dampers. A
pseudo excitation formula is used to generate an earthquake load in a frequency do-
main. Assuming a linear behaviour of the buildings (linear springs and linear viscous
dampers), the dynamic response of the whole system is found. Using the dynamic
response of the system, the desired objective function, i.e., the maximum inter-story
drift, is calculated.
The optimization procedure consists of two parts including discrete and continuous
optimizations. An outer-loop (discrete optimization algorithm) ﬁnds the best conﬁgu-
ration of a limited number of dampers between two buildings; an inner-loop (continuous
optimization algorithm) ﬁnds the optimal damper coeﬃcients of the dampers. Three
diﬀerent algorithms (GA, MADS and RAGS) for the continuous optimization problem
are considered, each using a random initial point and a warm-start initial point. In
order to compare speed and robustness of these non-gradient based methods, 150 test
problems were generated and solved via these three methods. Results showed that
MADS using a warm-start initial point is quite fast and robust.
Furthermore, the eﬃciency of the retroﬁtting system with respect to the number of
dampers used was investigated. In [10], it is shown that when assuming equal damper
coeﬃcients, increasing the number of dampers may exacerbate the dynamic behaviour
of the buildings. When the assumption of equal damper coeﬃcients is removed, it
was observed that although increasing the number of dampers no longer exacerbates
the dynamic behaviour of the system, there is nonetheless a threshold after which
16
increasing the number of dampers provides little beneﬁt to the system. Using 15 test
problems, it was found that in most cases, the optimal behaviour of a seismic retroﬁt
can be achieved within 1% using 4 or less dampers, and only one problem required
more than 6 dampers. This represents a signiﬁcant saving in material and overall cost
of retroﬁtting.
Finally, it is worth mentioning that a very similar bi-level optimization procedure
as presented can be followed for diﬀerent types of damper connectors, such as MR
dampers, friction dampers and so on. In these cases, the only element that changes
is the simulation core of the problem; the same discrete and continuous optimization
algorithms can be used. Furthermore, it should be clear that, while this paper focused
on minimizing the maximum inter-story drift, the techniques within this paper can
easily be adapted to any objective function.
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19
6 Appendix A
Table 3: Number of function calls and objective values found for Material Set I for algorithms
Number of function calls Objective value
1 1 380 272 203 1.02E-06 1.01E-06 1.01E-06
2 1030 930 837 9.27E-07 8.92E-07 8.88E-07
3 1870 1849 1455 8.89E-07 8.73E-07 8.66E-07
4 2850 3014 2297 8.59E-07 8.36E-07 8.40E-07
5 3900 4445 3056 8.30E-07 8.21E-07 8.20E-07
6 4950 6224 3892 8.23E-07 8.23E-07 8.04E-07
7 5930 8192 4746 8.21E-07 8.22E-07 8.02E-07
8 6770 10282 5600 8.40E-07 8.25E-07 8.04E-07
9 7400 11327 6212 8.38E-07 8.22E-07 8.07E-07
10 350 695 552 8.53E-07 8.41E-07 8.09E-07
2 1 380 279 170 4.36E-06 4.33E-06 4.33E-06
2 1100 873 571 1.79E-06 1.72E-06 1.72E-06
3 2000 1883 1661 1.90E-06 1.72E-06 1.72E-06
4 2980 3680 2965 1.87E-06 1.72E-06 1.71E-06
5 4055 5995 4360 1.76E-06 1.72E-06 1.73E-06
6 5105 8999 6376 1.77E-06 1.72E-06 1.73E-06
7 6155 13127 9192 1.84E-06 1.71E-06 1.73E-06
8 6995 16589 11644 1.88E-06 1.71E-06 1.74E-06
9 7625 18710 13511 1.89E-06 1.73E-06 1.74E-06
10 350 1511 1208 1.96E-06 1.71E-06 1.77E-06
3 1 370 266 164 1.67E-06 1.59E-06 1.59E-06
2 1020 729 724 1.62E-06 1.53E-06 1.52E-06
3 1860 1367 1719 1.56E-06 1.52E-06 1.51E-06
4 2840 2030 2814 1.54E-06 1.51E-06 1.51E-06
5 3890 2933 3911 1.54E-06 1.52E-06 1.52E-06
6 4940 4133 5228 1.55E-06 1.52E-06 1.52E-06
7 5920 5780 6443 1.54E-06 1.52E-06 1.52E-06
8 6760 8014 7794 1.54E-06 1.52E-06 1.53E-06
9 7390 9667 8845 1.54E-06 1.54E-06 1.53E-06
10 350 1812 513 1.57E-06 1.52E-06 1.54E-06
4 1 405 322 154 8.24E-06 8.21E-06 8.19E-06
2 1065 1045 640 5.87E-06 5.58E-06 5.56E-06
3 1920 2190 1247 5.57E-06 5.48E-06 5.47E-06
4 2940 3962 1846 5.57E-06 5.45E-06 5.44E-06
5 3990 6221 2598 5.47E-06 5.44E-06 5.43E-06
6 5040 9050 3402 5.46E-06 5.43E-06 5.42E-06
7 6020 11796 4167 5.46E-06 5.42E-06 5.42E-06
8 6860 14535 4941 5.44E-06 5.42E-06 5.43E-06
9 7490 17575 5666 5.53E-06 5.42E-06 5.45E-06
10 350 1933 487 5.66E-06 5.43E-06 5.46E-06
5 1 385 312 153 5.96E-06 5.94E-06 5.94E-06
2 1045 1010 756 5.86E-06 5.86E-06 5.87E-06
3 1930 2097 1766 5.94E-06 5.86E-06 5.87E-06
4 2930 4345 3424 5.89E-06 5.86E-06 5.87E-06
5 3980 7131 5166 6.17E-06 5.86E-06 5.88E-06
6 5030 11821 7045 5.96E-06 5.86E-06 5.89E-06
7 6010 16077 8844 6.22E-06 5.86E-06 5.89E-06
8 6850 21024 10533 5.97E-06 5.86E-06 5.91E-06
9 7480 25831 11646 6.14E-06 5.86E-06 5.92E-06
10 350 3545 820 6.40E-06 5.86E-06 5.92E-06
20
Table 4: Number of function calls and objective values found for Material Set I for algorithms
Number of function calls Objective value
1 1 1040 35 19 1.01E-06 1.01E-06 1.01E-06
2 1060 68 32 8.88E-07 8.96E-07 1.00E-06
3 1040 110 242 8.70E-07 8.70E-07 8.71E-07
4 1060 48 71 8.61E-07 8.68E-07 8.67E-07
5 1040 326 44 8.52E-07 8.54E-07 8.65E-07
6 1040 534 84 8.48E-07 8.50E-07 8.63E-07
7 1040 160 113 8.46E-07 8.49E-07 8.61E-07
8 1040 524 686 8.20E-07 8.47E-07 8.37E-07
9 1040 205 144 8.02E-07 8.49E-07 8.35E-07
10 1040 92 281 8.04E-07 8.49E-07 8.30E-07
2 1 1040 26 23 4.33E-06 4.33E-06 4.33E-06
2 1040 76 151 1.72E-06 1.73E-06 1.72E-06
3 1040 35 51 1.72E-06 1.72E-06 1.72E-06
4 1040 95 45 1.72E-06 1.72E-06 1.72E-06
5 1040 317 40 1.72E-06 1.72E-06 1.72E-06
6 1060 182 57 1.72E-06 1.72E-06 1.72E-06
7 1040 254 78 1.72E-06 1.72E-06 1.73E-06
8 1040 1188 143 1.72E-06 1.72E-06 1.74E-06
9 1040 391 383 1.72E-06 1.72E-06 1.74E-06
10 1040 351 214 1.72E-06 1.72E-06 1.74E-06
3 1 1040 31 31 1.59E-06 1.59E-06 1.59E-06
2 1040 60 16 1.53E-06 1.53E-06 1.58E-06
3 1040 85 32 1.52E-06 1.52E-06 1.58E-06
4 1040 107 41 1.52E-06 1.52E-06 1.57E-06
5 1040 256 50 1.52E-06 1.52E-06 1.57E-06
6 1040 133 76 1.52E-06 1.52E-06 1.57E-06
7 1040 204 56 1.51E-06 1.52E-06 1.56E-06
8 1040 179 97 1.51E-06 1.52E-06 1.56E-06
9 1040 447 123 1.51E-06 1.52E-06 1.56E-06
10 1041 261 127 1.51E-06 1.52E-06 1.56E-06
4 1 1041 32 19 8.19E-06 8.21E-06 8.19E-06
2 1041 66 70 5.56E-06 5.60E-06 5.56E-06
3 1041 123 33 5.47E-06 5.49E-06 5.55E-06
4 1041 168 41 5.45E-06 5.45E-06 5.54E-06
5 1041 242 62 5.43E-06 5.43E-06 5.54E-06
6 1041 176 39 5.42E-06 5.42E-06 5.53E-06
7 1041 133 60 5.42E-06 5.42E-06 5.52E-06
8 1041 595 64 5.41E-06 5.42E-06 5.52E-06
9 1041 133 51 5.42E-06 5.42E-06 5.53E-06
10 1041 184 106 5.42E-06 5.42E-06 5.54E-06
5 1 1041 31 23 5.94E-06 5.94E-06 5.94E-06
2 1041 54 28 5.86E-06 5.86E-06 5.89E-06
3 1041 252 44 5.86E-06 5.86E-06 5.87E-06
4 1041 301 42 5.86E-06 5.86E-06 5.87E-06
5 1041 202 43 5.86E-06 5.86E-06 5.87E-06
6 1041 196 39 5.86E-06 5.86E-06 5.87E-06
7 1041 59 76 5.86E-06 5.86E-06 5.87E-06
8 1041 98 73 5.86E-06 5.86E-06 5.88E-06
9 1041 789 84 5.86E-06 5.86E-06 5.90E-06
10 1041 1153 113 5.86E-06 5.86E-06 5.91E-06
21
Table 5: Number of function calls and objective values found for Material Set II for algorithms
Number of function calls Objective value
1 1 425 298 153 4.81E-07 4.69E-07 4.68E-07
2 1055 871 575 4.65E-07 4.49E-07 4.33E-07
3 1895 1884 1421 4.46E-07 4.27E-07 4.21E-07
4 2875 3406 2264 4.33E-07 4.17E-07 4.12E-07
5 3925 5119 3236 4.20E-07 4.09E-07 4.03E-07
6 4975 6565 4496 4.14E-07 4.05E-07 4.03E-07
7 5955 8001 5680 4.18E-07 4.03E-07 4.02E-07
8 6795 10055 6503 4.16E-07 4.03E-07 4.02E-07
9 7425 11942 7259 4.21E-07 4.03E-07 4.02E-07
10 350 930 460 4.26E-07 4.12E-07 4.04E-07
2 1 355 324 196 2.43E-07 2.43E-07 2.43E-07
2 1025 1053 550 2.41E-07 2.37E-07 2.36E-07
3 1895 2203 1036 2.38E-07 2.37E-07 2.36E-07
4 2875 3460 1429 2.37E-07 2.36E-07 2.36E-07
5 3925 4481 1894 2.37E-07 2.37E-07 2.37E-07
6 4975 5281 2263 2.37E-07 2.38E-07 2.37E-07
7 5955 6159 2810 2.37E-07 2.36E-07 2.37E-07
8 6795 7178 3288 2.37E-07 2.38E-07 2.38E-07
9 7425 7632 3780 2.37E-07 2.40E-07 2.38E-07
10 350 172 307 2.42E-07 2.41E-07 2.38E-07
3 1 410 308 123 5.35E-07 4.99E-07 4.99E-07
2 1060 1036 611 4.68E-07 3.32E-07 3.22E-07
3 1945 2130 1740 3.82E-07 3.24E-07 3.19E-07
4 2945 3689 3284 3.50E-07 3.21E-07 3.19E-07
5 4020 5121 4374 3.42E-07 3.20E-07 3.18E-07
6 5070 7196 5776 3.61E-07 3.13E-07 3.17E-07
7 6050 9158 7933 3.33E-07 3.11E-07 3.21E-07
8 6930 10932 10490 3.35E-07 3.15E-07 3.16E-07
9 7560 12724 12279 3.41E-07 3.20E-07 3.15E-07
10 350 1401 1196 3.54E-07 3.24E-07 3.13E-07
4 1 390 299 409 3.89E-06 3.86E-06 3.86E-06
2 1060 954 851 3.50E-06 2.74E-06 2.74E-06
3 1945 1889 1663 2.74E-06 2.68E-06 2.72E-06
4 2945 3383 2903 2.75E-06 2.66E-06 2.70E-06
5 3995 5003 4040 2.77E-06 2.64E-06 2.71E-06
6 5075 7171 5382 2.73E-06 2.62E-06 2.70E-06
7 6055 8987 6510 2.75E-06 2.62E-06 2.71E-06
8 6895 10931 7559 2.79E-06 2.61E-06 2.70E-06
9 7525 12645 8550 2.77E-06 2.61E-06 2.71E-06
10 350 981 573 2.78E-06 2.61E-06 2.71E-06
5 1 415 299 199 2.06E-06 1.93E-06 1.93E-06
2 1085 945 699 1.65E-06 1.24E-06 1.24E-06
3 1940 2047 1424 1.52E-06 1.23E-06 1.23E-06
4 2940 3455 2284 1.31E-06 1.22E-06 1.22E-06
5 3990 6590 3236 1.22E-06 1.21E-06 1.22E-06
6 5040 9516 4253 1.25E-06 1.21E-06 1.22E-06
7 6020 12164 5278 1.26E-06 1.21E-06 1.22E-06
8 6860 15396 6126 1.25E-06 1.21E-06 1.22E-06
9 7490 18878 6842 1.27E-06 1.21E-06 1.22E-06
10 350 1422 421 1.26E-06 1.21E-06 1.23E-06
22
Table 6: Number of function calls and objective values found for Material Set II for algorithms
Number of function calls Objective value
1 1 1041 25 18 4.68E-07 4.69E-07 4.68E-07
2 1421 55 34 4.33E-07 4.35E-07 4.65E-07
3 1041 85 320 4.20E-07 4.22E-07 4.56E-07
4 1141 222 41 4.12E-07 4.13E-07 4.56E-07
5 1041 264 45 4.11E-07 4.12E-07 4.55E-07
6 1041 76 50 4.11E-07 4.11E-07 4.54E-07
7 1041 256 55 4.10E-07 4.11E-07 4.54E-07
8 1041 270 80 4.04E-07 4.11E-07 4.53E-07
9 1041 129 84 4.04E-07 4.11E-07 4.53E-07
10 1041 218 87 4.04E-07 4.11E-07 4.53E-07
2 1 1041 31 14 2.43E-07 2.43E-07 2.43E-07
2 1181 93 13 2.39E-07 2.38E-07 2.43E-07
3 1041 106 36 2.36E-07 2.37E-07 2.42E-07
4 1041 289 33 2.36E-07 2.37E-07 2.42E-07
5 1041 232 46 2.36E-07 2.37E-07 2.42E-07
6 1041 141 56 2.36E-07 2.37E-07 2.41E-07
7 1041 106 71 2.36E-07 2.37E-07 2.41E-07
8 1041 226 69 2.36E-07 2.37E-07 2.41E-07
9 1041 565 97 2.36E-07 2.37E-07 2.41E-07
10 1041 260 92 2.36E-07 2.37E-07 2.41E-07
3 1 1041 37 15 4.99E-07 4.99E-07 4.99E-07
2 1381 68 19 3.21E-07 3.27E-07 4.97E-07
3 1041 64 35 3.21E-07 3.18E-07 4.95E-07
4 1041 121 32 3.21E-07 3.15E-07 4.94E-07
5 1041 223 73 3.21E-07 3.15E-07 4.93E-07
6 1041 92 66 3.21E-07 3.15E-07 4.91E-07
7 1041 192 92 3.21E-07 3.15E-07 4.90E-07
8 1041 1430 53 3.21E-07 3.13E-07 4.90E-07
9 1041 1232 942 3.21E-07 3.13E-07 3.43E-07
10 1041 451 324 3.21E-07 3.13E-07 3.40E-07
4 1 1041 32 26 3.86E-06 3.86E-06 3.86E-06
2 1041 73 100 2.74E-06 2.74E-06 2.74E-06
3 1041 89 23 2.74E-06 2.66E-06 2.74E-06
4 1241 208 38 2.68E-06 2.63E-06 2.74E-06
5 1041 70 30 2.67E-06 2.62E-06 2.74E-06
6 1041 157 34 2.67E-06 2.61E-06 2.74E-06
7 1041 92 30 2.67E-06 2.61E-06 2.74E-06
8 1041 219 44 2.67E-06 2.61E-06 2.74E-06
9 1041 108 63 2.61E-06 2.61E-06 2.74E-06
10 1041 308 50 2.61E-06 2.61E-06 2.74E-06
5 1 1041 19 18 1.93E-06 1.93E-06 1.93E-06
2 1041 80 96 1.24E-06 1.24E-06 1.24E-06
3 1141 171 33 1.22E-06 1.22E-06 1.24E-06
4 1161 210 30 1.22E-06 1.21E-06 1.24E-06
5 1041 136 43 1.22E-06 1.21E-06 1.24E-06
6 1041 280 34 1.21E-06 1.21E-06 1.24E-06
7 1041 285 68 1.21E-06 1.21E-06 1.24E-06
8 1041 338 58 1.21E-06 1.21E-06 1.24E-06
9 1041 103 80 1.21E-06 1.21E-06 1.24E-06
10 1041 169 92 1.21E-06 1.21E-06 1.24E-06
23
Table 7: Number of function calls and objective values found for Material Set III for algorithms
Number of function calls Objective value
1 1 385 293 159 5.39E-07 5.42E-07 5.37E-07
2 1015 1158 635 5.22E-07 5.16E-07 5.16E-07
3 1855 2408 1631 5.03E-07 4.81E-07 4.79E-07
4 2835 3891 2823 5.04E-07 4.80E-07 4.68E-07
5 3885 5849 4165 4.84E-07 4.74E-07 4.60E-07
6 4935 8353 5533 4.83E-07 4.61E-07 4.62E-07
7 5915 10105 7017 4.84E-07 4.68E-07 4.59E-07
8 6755 12987 7926 4.70E-07 4.62E-07 4.63E-07
9 7385 14166 9089 4.72E-07 4.68E-07 4.59E-07
10 350 1365 610 4.87E-07 4.65E-07 4.59E-07
2 1 395 273 117 4.70E-07 4.64E-07 4.62E-07
2 1055 1127 302 4.41E-07 4.40E-07 4.54E-07
3 1910 2559 836 4.39E-07 4.40E-07 4.50E-07
4 2890 4558 1562 4.40E-07 4.37E-07 4.43E-07
5 3940 6421 3539 4.38E-07 4.35E-07 4.40E-07
6 4990 8294 5246 4.38E-07 4.36E-07 4.39E-07
7 5970 10386 6726 4.38E-07 4.38E-07 4.39E-07
8 6810 13105 8304 4.40E-07 4.40E-07 4.38E-07
9 7440 15972 9417 4.40E-07 4.41E-07 4.38E-07
10 350 1038 715 4.50E-07 4.51E-07 4.38E-07
3 1 415 290 156 8.98E-07 8.41E-07 8.41E-07
2 1075 946 702 8.09E-07 4.97E-07 4.91E-07
3 1945 1930 1494 6.49E-07 4.94E-07 4.91E-07
4 2985 3352 2573 5.04E-07 4.93E-07 4.91E-07
5 4110 5335 3581 5.28E-07 4.85E-07 5.00E-07
6 5220 7326 5248 4.97E-07 4.95E-07 5.08E-07
7 6270 8942 7197 5.11E-07 4.85E-07 5.09E-07
8 7190 11827 9596 5.04E-07 4.90E-07 5.10E-07
9 7820 13554 13366 5.14E-07 4.92E-07 5.09E-07
10 350 1306 955 5.78E-07 4.88E-07 5.11E-07
4 1 425 317 202 8.63E-06 8.64E-06 8.62E-06
2 1075 989 1064 8.65E-06 6.51E-06 8.63E-06
3 1945 2143 2196 6.11E-06 6.10E-06 8.67E-06
4 2965 4028 3572 6.17E-06 5.87E-06 8.70E-06
5 4015 6565 5950 6.32E-06 5.84E-06 8.70E-06
6 5065 10071 8418 8.73E-06 5.80E-06 8.73E-06
7 6150 12257 10836 6.47E-06 5.81E-06 8.77E-06
8 7030 15980 12886 6.35E-06 5.79E-06 8.78E-06
9 7930 18156 14585 6.36E-06 5.72E-06 8.81E-06
10 500 1194 1098 6.19E-06 5.71E-06 8.82E-06
5 1 465 298 284 6.85E-06 6.21E-06 6.19E-06
2 1125 1163 813 5.60E-06 3.20E-06 3.20E-06
3 2070 2332 1586 3.60E-06 2.46E-06 2.47E-06
4 3110 3957 2438 3.40E-06 2.41E-06 2.43E-06
5 4310 6951 3373 2.68E-06 2.37E-06 2.41E-06
6 5420 9313 4481 2.56E-06 2.36E-06 2.39E-06
7 6470 12817 5570 2.41E-06 2.36E-06 2.38E-06
8 7350 16541 6600 2.41E-06 2.35E-06 2.38E-06
9 7980 19290 7276 2.57E-06 2.35E-06 2.38E-06
10 350 2299 367 2.66E-06 2.34E-06 2.39E-06
24
Table 8: Number of function calls and objective values found for Material Set III for algorithms
Number of function calls Objective value
1 1 1041 36 29 5.37E-07 5.42E-07 5.37E-07
2 1041 84 37 5.16E-07 5.17E-07 5.34E-07
3 1961 141 47 4.89E-07 4.80E-07 5.32E-07
4 1041 205 501 4.88E-07 4.74E-07 5.17E-07
5 1041 187 72 4.88E-07 4.69E-07 5.16E-07
6 1041 103 43 4.87E-07 4.69E-07 5.16E-07
7 1041 441 38 4.87E-07 4.68E-07 5.15E-07
8 1041 162 85 4.87E-07 4.68E-07 5.15E-07
9 1041 584 75 4.86E-07 4.68E-07 5.15E-07
10 1041 839 59 4.86E-07 4.68E-07 5.15E-07
2 1 1041 29 15 4.62E-07 4.64E-07 4.62E-07
2 1081 69 32 4.39E-07 4.40E-07 4.62E-07
3 1041 83 27 4.39E-07 4.35E-07 4.62E-07
4 1041 68 38 4.39E-07 4.35E-07 4.61E-07
5 1041 108 48 4.39E-07 4.35E-07 4.61E-07
6 1041 131 80 4.38E-07 4.34E-07 4.61E-07
7 1041 185 68 4.39E-07 4.34E-07 4.61E-07
8 1041 178 78 4.39E-07 4.34E-07 4.60E-07
9 1041 523 71 4.39E-07 4.34E-07 4.60E-07
10 1041 152 96 4.39E-07 4.34E-07 4.60E-07
3 1 1041 29 12 8.41E-07 8.41E-07 8.41E-07
2 1041 53 30 5.01E-07 4.94E-07 8.39E-07
3 1041 65 24 5.01E-07 4.93E-07 8.37E-07
4 1041 56 41 4.95E-07 4.93E-07 8.36E-07
5 1041 90 63 4.88E-07 4.93E-07 8.34E-07
6 1041 55 48 4.87E-07 4.93E-07 8.33E-07
7 1041 300 57 4.87E-07 4.92E-07 8.33E-07
8 1041 199 59 4.87E-07 4.91E-07 8.32E-07
9 1041 416 123 4.87E-07 4.91E-07 8.32E-07
10 1041 291 68 4.87E-07 4.91E-07 8.32E-07
4 1 1041 37 19 8.62E-06 8.64E-06 8.62E-06
2 1041 54 18 8.62E-06 8.62E-06 8.62E-06
3 1041 63 33 8.62E-06 8.62E-06 8.62E-06
4 1041 59 42 8.62E-06 8.62E-06 8.63E-06
5 1041 118 64 8.62E-06 8.62E-06 8.64E-06
6 1041 319 51 8.62E-06 8.62E-06 8.66E-06
7 1041 41 71 8.62E-06 8.62E-06 8.68E-06
8 1041 1360 62 8.62E-06 5.92E-06 8.71E-06
9 1041 1008 75 8.62E-06 5.72E-06 8.73E-06
10 1041 753 81 8.62E-06 5.68E-06 8.76E-06
5 1 1041 26 30 6.19E-06 6.21E-06 6.19E-06
2 1041 74 24 3.20E-06 3.20E-06 6.17E-06
3 1041 160 31 2.46E-06 2.46E-06 6.15E-06
4 1201 162 20 2.42E-06 2.42E-06 6.14E-06
5 1261 286 630 2.37E-06 2.37E-06 2.49E-06
6 1041 281 321 2.37E-06 2.36E-06 2.42E-06
7 1061 456 68 2.35E-06 2.36E-06 2.42E-06
8 1041 459 59 2.34E-06 2.36E-06 2.42E-06
9 1041 204 56 2.34E-06 2.35E-06 2.42E-06
10 1041 640 106 2.34E-06 2.34E-06 2.42E-06
25
... Later, Klein and Healy [2] developed an algorithm for semi-active control and found that to ensure that the system is controllable, the natural frequencies of adjacent buildings should be different. From these works, several researches were developed using the coupling technique, evaluating several control devices in the connection between the buildings [3][4][5][6][7][8][9]. ...
... The mathematical formulation for adjacent coupled buildings with deterministic base excitation is based on studies [7,8,[12][13]. For stochastic excitation with white noise as an input, the formulation is based on studies [14][15][16][17][18]. ...
... Still, considering that the excitation is stationary, is independent of time, so the stationary solution of Eq. (7) can be obtained by solving the Lyapunov matrix equation: (8) When using the PSDF to represent a stationary Gaussian random process , can be used a white noise filter through two linear filters: (9) In which , , and are soil parameters (sub index g) and filter parameters (sub index f). Equation (9) lead to the following PSDF described by Clough and Penzien [15]: (10) Being the intensity of the process. ...
... Previous studies have shown that the vibration of the interconnected adjacent structures can be effectively mitigated by applying the passive energy dissipation devices [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], active [15,20,[24][25][26][27][28][29] and semi-active [19,[30][31][32][33][34][35] control strategies to the connectors. Among these methods, the use of the viscous/visco-elastic dampers has attracted extensive attention because of its easy installation and almost free maintenance. ...
... Recently, after removing the assumption of equal damping coefficients, Bigdel et al. [22] found that increasing the number of dampers no longer exacerbates the dynamic behaviour of the system, by using a newly developed bi-level optimization procedure. Considering two linear elastic shear cantilever beams interconnected by a viscous/visco-elastic damper at the top of the shorter building, Tubaldi [21] derived the approximate expressions for the damping ratios of the first two fundamental vibration modes in the case of low added damping, by utilizing assumed mode method. ...
Article
Full-text available
With consideration of a high-rise coupled building system, a flexible beams-based analytical model is setup to characterize the dynamic behavior of the system. The general motion equation for the two beams interconnected by multiple viscous/visco-elastic dampers is rewritten into a non-dimensional form to identify the minimal set of parameters governing the dynamic characteristics. The corresponding exact solution suitable for arbitrary boundary conditions is presented. Furthermore, the methodology for computing the coefficients of the modal shape function is proposed. As an example, the explicit expression of the modal shape function is derived, provided only one damper is adopted to connect the adjacent buildings. Finally, to validate the proposed methodologies, three case studies are performed, in which the existence of the overdamping and the optimal damping coefficient are revealed. In the case of using one damper in connecting two similar buildings, the estimating equations for the first modal damping ratio are formulated.
... Bigdeli et al. [38] used a bi-level optimization approach to determine the best arrangement and concomitant mechanical properties of the VFD device to connect two adjacent buildings. The arrangement of the dampers is a discrete optimization, while the mechanical properties, stiffness, and damping, are of the continuous type. ...
Article
Due to the large population density in big urban centers, building taller and closer buildings is necessary. The high flexibility of these buildings can lead to excessive vibration problems due to the action of earthquakes and strong winds. Furthermore, there is also the possibility of collision between them (pounding). The structural coupling technique has been extensively studied in recent decades due to its efficiency in attenuating excessive displacements and avoiding pounding between coupled structures. This method connects two or more buildings close together through control devices to attenuate the dynamic responses of both structures. Since 2002, several inerter-based control devices have been developed. Inerter is an element in which the force applied to it is proportional to the relative acceleration between its terminals. In 2017, inerter-based devices were introduced to the connected system to improve the structural coupling technique. Since then, several studies have been conducted to show the advantages of this combined use. Thus, this work illustrated the most recent and relevant developments on structural coupling, inerter devices, and their simultaneous use. So far, this proposal is quite promising, with motivating results, bringing possibilities for real applications with clear advantages about already consolidated devices.
... The use of meta-heuristic methods for multi-objective optimization for efficient vibration control of structures may be found in Refs. [71][72][73]. With this in view, the objectives are set for the present study as follows: ...
Article
This study proposes a shape memory alloy (SMA)-based damped outrigger system for vibration control of tall timber buildings. In this study, the core of the structure is idealized as a cantilever beam, and each floor mass is considered as a discrete mass that acts at the junction between the floor and the core of the structure. The governing equations of motion of the combined system of shear core and outrigger are derived using the Lagrange formulation. The SMA spring is installed between the outrigger beam and the column to dissipate earthquake-induced energy and reduce excessive load demand on the column. Optimal performance of the proposed system requires optimizing the outriggers’ location and tuning the SMA properties in an uncertain environment. Two conflicting objective functions, minimization of acceleration and inter-storey drift ratio, are solved through a multi-objective optimization. Four different multi-objective meta-heuristic optimization algorithms (ant-lion, dragonfly, particle swarm optimization, and non-dominated sorting genetic algorithm II) are considered. Three different tall timber buildings (10-, 15-, and 20-storey) and up to two outrigger beams are considered for optimization. The seismicity of Vancouver, BC is used for the numerical simulation and vulnerability assessment. The optimum outrigger location for a one-outrigger system is found to be approximately 60%, while for a two-outrigger system, the same is obtained as 33% and 68%, respectively. Finally, the fragility analysis is carried out, which shows the superiority of this passive device in terms of minimizing the probability of failure exceeding a given threshold limit, which yields an improvement in the reliability of the structure.
... en, the response attenuation results were compared by employing LQR-RNN and LQR-CVL. Bigdeli et al. [13,14] discussed the number and location of connecting dampers so that the performance of adjacent buildings can be promoted. Gao et al. [15] put forward a dynamic output feedback control method for mitigating structural seismic vibration and obtaining the location of the actuators and sensors, which were placed between adjacent buildings. ...
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Full-text available
This paper presents an overlapping decentralized guaranteed cost hybrid control method for adjacent buildings with uncertain parameters, by combining the guaranteed cost control algorithm with the overlapping decentralized control strategy. The passive dampers are used as link members between the two parallel buildings, and the active control devices are installed between two consecutive floors in two adjacent buildings. The passive coupling dampers modulate the relative responses between the two buildings, and the active control devices modulate the interstory responses of each building. Based on the inclusion principle, a large-scale structure is divided into a set of paired substructures with common parts first. Then, the controller of each pair of substructures is designed by using the guaranteed cost algorithm. After that, the controller of the original system is formed by using the contraction principle. Consequently, the proposed approach is used to prevent pounding damage and achieve the best results in earthquake response reduction of uncertain adjacent buildings when compared with the calculation results obtained by the centralized control strategy. Furthermore, the stability and reliability of the control system are promoted by adopting the overlapping decentralized control strategy.
... To increase the efficiency of viscous fluid dampers, optimal arrangement and mechanical properties of connecting viscous fluid dampers were considered. A number of meta-heuristic methods were used and compared to find out appropriate location of viscous dampers in coupled buildings with different height [114][115][116][117][118][119][120][121][122][123]. The results of all mentioned studies indicate that optimal location and value parameters of viscous dampers play an important role to protect adjacent buildings against pounding and absolutely reduce dynamic responses of structures. ...
Article
Full-text available
This paper presents a state-of-the-art review of significant studies published on coupled buildings. The approach of connecting two adjacent buildings has been effectively applied to reduce structural responses of a coupled building which is subjected to seismic excitation. The coupled building control utilizes two adjacent structures to transfer dynamic force which is applied to each other in order to reduce critical responses. The aim of this article is to review technologies and methods in structural connection of coupled buildings which are classified as (a) rigid (b) passive, (c) semi-active, (d) active, and (e) hybrid. The concepts of these connections are explained and formulated. The comparison, application and feature of connections are tabulated. Finally, some approaches to enhance the performance of connections and draw backs of studies are suggested.
... Apart from the above mentioned advantages, the link in these structures helps to reduce the seismic responses of the adjacent structures and overcome the problem of pounding [2] , occurred in the past major seismic events such as 1985 Mexico City and 1989 Loma Prieta earthquakes. Most of the existing literature addresses this problem with two different approaches, namely: two adjacent multi degree of freedom (MDOF) systems [3][4][5][6] , and two adjacent single degree of freedom (SDOF) systems [7,8] . In the context of seismic energy dissipation through passive control, literature [6] showed that despite having half of the mass of the individually placed TMDs, shared tuned mass damper (STMD) is more effective in reducing the seismic vibrations ( Table 1 ). ...
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This work introduces a passive control device, shape memory alloy tuned mass damper inerter (SMA-TMDI), for the control of linked-single degree of freedom (SDOF) systems subjected to base excitation. The adjacent SDOF systems are connected through a device called inerter with force proportional to the relative acceleration of the individual SDOF oscillators of the linked-SDOF systems. The SMA element of SMA-TMDI dissipates the energy of primary oscillator through the hysteretic phase transformation, while, the mass-amplification effect of the inerter is utilized to reduce displacement of the secondary oscillator of the linked-SDOF systems. The mean square displacement responses of both the oscillators of the linked-SDOF systems subjected to white noise base excitation are derived based on stochastic equivalent linear parameters of the SMA spring through an iterative process. Parametric studies under white noise excitation are conducted and based on the results obtained, a multi-objective optimization is performed considering displacement variances of both the SDOF oscillators as the objective function. Under white noise excitation, the optimal performances of SMA-TMDI and TMDI systems are analyzed in terms of displacement mean square responses and the root mean square control forces transferred to the linked-SDOF systems are also examined. Further, the performance comparison of the SMA-TMDI and TMDI passive control devices are carried out under non-stationary Kanai-Tajimi excitation based on an Ito-Taylor formulation of the mean square stochastic differential equations. Based on the results obtained for both white noise and ground motion base excitation cases, it can be observed that the SMA-TMDI system performs better in comparison to the TMDI system with significantly lesser requirement on total damper mass and inertance.
... There are other types of grey box optimization problems, in which some part of the structure may be exploited. For example, [11] studies min-max functions in the context of seismic retrofitting design, [13] considers a combination of blackbox constraints and constraints with a known functional form, and in [17], the authors study problems with a 1 norm objective function. ...
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