Guided Firework Algorithm (GFWA) Optimization Research on Viscoelastic Damper (VED) Structure Based on Vulnerability Evaluation

Abstract

The vulnerability curve serves as a precise evaluation metric for structural seismic performance and a critical component in earthquake loss assessment. In this study, the orthogonal expansion method for random ground motion generation is integrated with the probability density evolution method (PDEM) to address the dynamic reliability and vulnerability of general Multi-Degree of Freedom (MDOF) nonlinear structures. By employing dynamic reliability as a constraint and vulnerability as an evaluation index, the guided firework algorithm (GFWA) is introduced to optimize the design of viscoelastic damper (VED) structure systems. To validate the proposed methods, several examples are presented, including the generation of artificial waves, the vulnerability analysis of a five-story reinforced concrete (RC) structure, and a comparative study of GFWA and genetic algorithm (GA) optimization for VED parameters to assess the optimization efficiency. The results demonstrate that the proposed vulnerability method achieves satisfactory accuracy and is well suited for evaluating damper structure optimization designs. Furthermore, GFWA outperforms GA significantly in terms of efficiency and feasibility, offering a promising approach for optimization design in architectural structures.

1. Introduction

Safety and economic efficiency are the primary objectives in structural seismic design. Rapid urbanization and economic growth have intensified the need to reduce construction costs while maintaining or improving structural safety. Structural vibration control is an effective method for mitigating seismic responses and cumulative damage, with VED being among the most widely used passive control devices [1,2]. Additionally, the increased computational capabilities and advancements in swarm intelligence algorithms have facilitated structural optimization, enhancing disaster resistance while reducing costs [3,4].

However, integrating damper control with structural optimization presents two significant challenges: (1) Conventional structural failure indices, such as inter-story displacement (or drift), do not fully capture VED performance, especially for structures on sites with varying soil properties. This limitation conflicts with the principles of performance-based earthquake engineering (PBEE), highlighting the need for a more precise structural performance evaluation method in VED structure seismic design. (2) For complex engineering optimization problems, improving optimization efficiency while reducing computational costs remains a critical challenge.

The concept of seismic vulnerability has emerged as a key focus in PBEE. Vulnerability curves establish a one-to-one correspondence between structural failure probability and ground motion intensity across different performance levels, enabling the prediction of damage probabilities under various seismic excitations and providing a precise description of structural seismic performance. These curves have been widely adopted by researchers to evaluate seismic performance and estimate earthquake-induced losses for various structures [5,6,7,8,9,10].

Traditional vulnerability analysis typically involves nonlinear time history analysis of structures subjected to selected earthquake records, with structural response probability distributions often assumed to follow a lognormal distribution. However, this approach overlooks the coupling between the strong randomness of ground motion and structural nonlinearity, as well as the evolving nature of structural response probability density distributions [11].

Recognizing that vulnerability fundamentally revolves around probability and performance, this study proposes generating artificial ground motions to account for the inherent randomness of seismic excitations and conducting deterministic nonlinear time history analysis of an MDOF structure under these artificial waves. This transforms the vulnerability problem into a structural reliability problem. While existing methods such as the Poisson method [12], Monte Carlo simulation [13], and first-order reliability method [14] can address structural reliability, they remain underdeveloped for the dynamic response analysis and reliability assessment of nonlinear structures. In contrast, the probability density evolution method (PDEM), recently developed by Li and Chen based on the probability conservation principle and density evolution theory, offers a more efficient and effective approach for obtaining the probability density function (PDF) and its evolution in the stochastic dynamic response of general structures [15,16,17].

Swarm intelligence algorithms, including the Genetic Algorithm (GA) [18], Particle Swarm Optimization (PSO) [19], Whale Optimization Algorithm (WOA) [20], Swordfish Optimization Algorithm (SFO) [21], and Moth Flame Optimization Algorithm (MFO) [22], have been extensively applied in optimization design due to their distinct advantages. For instance, the Genetic Algorithm excels in global search capabilities, making it particularly effective for large-scale and multi-objective optimization problems. The PSO algorithm, known for its simplicity and rapid convergence, is well suited for continuous optimization tasks. The WOA stands out for its minimal parameter requirements and ease of implementation. Similarly, the SFO algorithm demonstrates fast convergence and is highly effective in solving complex optimization problems. Meanwhile, the MFO algorithm is particularly adept at handling nonlinear optimization challenges.

However, when tackling complex engineering optimization problems characterized by multi-modal minima and multiple objective functions, these algorithms often fall short in terms of both efficiency and accuracy. The GA, for example, is computationally expensive and highly sensitive to parameter settings, which can lead to premature convergence. The WOA tends to exhibit low efficiency and slow convergence rates when applied to high-dimensional problems. The SFO algorithm, despite its rapid convergence, is prone to local optima due to its sensitivity to parameter configurations. Additionally, the MFO algorithm suffers from high computational complexity and inconsistent convergence behavior.

Compared to the aforementioned swarm intelligence algorithms, the firework algorithm (FWA) demonstrates significant advantages in several key areas, including diversity, adaptive search, parallel search, simplicity of implementation, multi-modal problem solving, robustness, and innovation. These strengths enable FWA to effectively address the limitations observed in other algorithms. Initially proposed by Professor Tan in 2010, FWA draws inspiration from the explosive patterns of fireworks in the night sky [23,24]. Over the years, FWA has undergone continuous development, resulting in various improved versions [25,26,27], and has been successfully applied across numerous fields [28,29,30]. Among these variants, GFWA [27] stands out as one of the most efficient members of the FWA family. By incorporating adaptive operators and guide vectors into the original FWA framework, GFWA enhances the algorithm’s performance. The GFWA employs a dynamic population adjustment strategy based on its firework explosion mechanism, thereby eliminating the necessity for explicit selection, crossover, and mutation operations. Furthermore, its adaptive mechanisms optimize the search process by dynamically adjusting parameters according to solution quality. This approach not only simplifies the parameter tuning process but also ensures a balance between search diversity and efficiency, enabling GFWA to achieve superior performance in addressing complex optimization challenges.

Within the framework of PBEE, this study proposes a novel approach to vulnerability assessment by integrating random ground motion generation and the PDEM. In Section 2, the stochastic ground motion method is extended to generate artificial waves that fully reflect ground motion randomness. Additionally, a related random point selection method is introduced. Section 3 presents the dynamic equations governing structures under random excitation and proposes a numerical solution for PDF of structural responses using PDEM. This section also addresses dynamic reliability analysis and vulnerability assessment based on structural reliability theory.

Furthermore, Section 4 applies GFWA to optimize the critical inter-story drift in a VED-equipped structure, establishing an optimization model for this purpose. Section 5 provides numerical examples to validate the proposed methods, including comparative case studies. Finally, Section 6 summarizes the key findings and conclusions drawn from this study.

2. Stochastic Ground Motion Generation

Li and Liu proposed a stochastic process expansion method utilizing a standard orthogonal basis [31,32]. This method begins with the orthogonal decomposition of the ground motion process and employs the principle of energy equivalence. As a result, only a few independent random variables are needed to capture the primary probabilistic characteristics of the ground motion. The power spectral density (PSD) function of the modified Y Hu-X Zhou model is adopted to demonstrate the ground motion generation method and produce reliable artificial waves.

2.1. Stochastic Ground Motion Model

The acceleration PSD function of the modified Y Hu-X Zhou model [33] is defined as

$$S_{\ddot{X}}(\omega )={\displaystyle \frac{{\omega }_g^2{\omega }^2}{{({\omega }_g^2-{\omega }^2)}^2+4{\xi }_g^2{\omega }_g^2{\omega }^2}}{\displaystyle \frac{{\omega }^2}{{\omega }^2+{\gamma }^2}}{S}_0$$

(1)

Here, ${\xi }_g$ and ${\omega }_g$ are, respectively, the damping ratio and the predominant frequency of the site soil; $\omega$ is the seismic dynamic circle frequency; $\gamma$ is the low-frequency reduction, usually taken as 2 rad/s; $S_0$ is the spectral intensity factor.

According to the relation between the displacement PSD function and acceleration PSD function

$$S_X(\omega )={\omega }^{-4}{S}_{\ddot{X}}(\omega )$$

(2)

The displacement PSD function of the modified Hu Y Hu-X Zhou model can be obtained as

$$S_X(\omega )={\displaystyle \frac{{\omega }_g^2}{{({\omega }_g^2-{\omega }^2)}^2+4{\xi }_g^2{\omega }_g^2{\omega }^2}}{\displaystyle \frac{1}{{\omega }^2+{\gamma }^2}}{S}_0$$

(3)

$R_X(\tau )$, the autocorrelative function of the earthquake displacement process, can be obtained according to Wiener–Kintchine theorem [34]

$$\begin{array}{cc}\hfill R_X(\tau )& =2{\displaystyle \underset{0}{\overset{+\infty }{\int }}{S}_X\left(\omega \right)}{e}^{i\omega \tau }d\omega \hfill \\ & =2S_0{\displaystyle \underset{0}{\overset{+\infty }{\int }}\frac{{\omega }_g^2}{{({\omega }_g^2-{\omega }^2)}^2+4{\xi }_g^2{\omega }_g^2{\omega }^2}\frac{1}{{\omega }^2+{\gamma }^2}{e}^{i\omega \tau }d\omega }\hfill \end{array}$$

(4)

Considering that the duration of the actual ground motion is finite, $R_X(\tau )$ is modified as

$$R_X(\tau ,T_s)=(1-\left|\tau \right|/T_s)R_X(\tau ), \left|\tau \right|\le T_s$$

(5)

Here, $T_s$ is the seismic dynamic stationary period.

2.2. Orthogonal Decomposition Based on Hartley Transformation

The Hartley transformation [32] of real-valued function can be expressed as

$$H_x(f)={\displaystyle {\int }_{-\infty }^{\infty }x(t)}cas(2\pi ft)dt$$

(6)

Its inverse transformation can be defined as

$$x(t)={\displaystyle {\int }_{-\infty }^{\infty }{H}_x(f)}cas(2\pi ft)df$$

(7)

where the core function $cas(·)$ can be defined as

$$cas(t)=\sin (t)+\cos (t)$$

(8)

Select the normalized Hartley orthogonal basis as the standard orthogonal basis

$${\varphi }_n(t)={\displaystyle \frac{1}{\sqrt{T_s}}}\mathrm{cas}({\displaystyle \frac{2\pi nt}{T_s}})\hspace{1em}n=0,1,\dots ,\infty$$

(9)

Based on the principle of energy equivalence, the orthogonal expansion formula of the random ground motion acceleration process can be obtained as

$${\ddot{X}}_g(\mathsf{\Theta },t)={\displaystyle \sum _{n=1}^N{\displaystyle \sum _{j=1}^r\sqrt{2{\lambda }_j}{\xi }_j{\phi }_{jk}}}{\varphi }_k(t)=\sqrt{2}{\displaystyle \sum _{j=1}^r\sqrt{{\lambda }_j}{\xi }_j{f}_j(t)}$$

(10a)

$$f_j(t)=-{{\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left(\frac{2\pi n}{{\mathrm{T}}_{\mathrm{s}}}\right)}}^2{\eta }_{n+1}{\phi }_{j,n+1}{\varphi }_n(t)$$

(10b)

Here, ${\lambda }_j$ is the eigenvalue of $R_X$; ${\phi }_{j,n+1}$ is the (n + 1)th element of standard feature vector ${\mathsf{\Phi }}_j$; ${\varphi }_n(t)$ is the standard orthogonal basis; $N$ represents the expansion terms (taken as 500 in this article); $r$ is the truncated terms (taken as 9 in this article); ${\eta }_{n+1} (n=1,2,\cdots ,N)$ is the energy equivalent coefficient, which can be solved; the random vector $\mathsf{\Theta }=\left\{{\xi }_1, {\xi }_2, \cdots , {\xi }_r\right\}$ is composed of a set of independent standard Gauss random variables.

In Equation (11), ${\lambda }_j$ and ${\mathsf{\Phi }}_j$ can be determined by the correlation matrix R of the displacement process

$$R{\mathsf{\Phi }}_j={\lambda }_j{\mathsf{\Phi }}_j$$

(11)

where the expression of R is

$$R={({\rho }_{ij})}_{(N+1)\times (N+1)}$$

(12)

Here, ${\rho }_{ij}$ is the element of R and can be calculated as

$${\rho }_{ij}={\displaystyle \underset{0}{\overset{T_s}{\int }}}{\displaystyle \underset{0}{\overset{T_s}{\int }}{R}_X(t_2-t_1,T_s)}{\varphi }_i(t_1){\varphi }_j(t_2)d{t}_{1}d{t}_2\hspace{1em}i,j=0,1,2,\cdots ,N$$

(13)

2.3. Number Theoretic Point Selection of Random Variable Space

When seismic waves involve multi-dimensional random variables, the selection of discrete points in the random variable space becomes critical. This selection is essential for accurately representing the probability space of the random function and solving the PDF of the structural nonlinear response. Three common strategies for point selection are typically employed: the tangent sphere method [35], the number theoretical method [36], and the cubature points method [37]. The choice of selection method directly impacts computational efficiency. For problems involving more than six random variables, the number theoretical method is recommended and is adopted in this study.

If there are S elements in $\mathsf{\Theta }$, the point of $Z$, which is an S dimensional independent standard normal vector, is selected according to the number theoretical method yielding

$$\begin{array}{cc}{z}_{j,k}={\displaystyle \frac{k{\eta }_j}{N_T}}-\mathrm{int}({\displaystyle \frac{k{\eta }_j}{N_T}}),& (j=1,\dots ,S,\hspace{0.17em}k=1,\dots ,N_T)\end{array}$$

(14)

where $N_T$ is the total number of points, $\left\{N_T,{\eta }_1,\cdots ,{\eta }_S\right\}$ is the integer vector, which is available from Ref. [37], and $\mathrm{int}(·)$ takes the remaining integer after trimming the fractional part.

Translating and sizing the spread points $z_{j,k}$ in the unit hypercube ${\left[0,1\right]}^S$,

$$\begin{array}{cc}{\overline{\theta }}_{j,k}=2(z_{j,k}-0.5)\lambda ,& (j=1,\dots ,S,\hspace{0.33em}k=1,\dots ,N_T)\end{array}$$

(15)

where $\lambda$ is the truncated boundary; the point set ${\overline{\theta }}_{j,k}$ is uniformly scattered in the hypercube ${\left[-\lambda ,\lambda \right]}^S$.

Because the joint PDF of $Z$ is spherically symmetric and monotone, decreasing with the distance between the point and zero point, the effective points can be scattered further from $\overline{\theta }$ according to the following condition

$${\displaystyle \sum _{j=1}^S{\overline{\theta }}_{j,k}^2}\le {(r_0\lambda )}^2$$

(16)

where $r_0$ is the bounded radius coefficient. Note the effective point set as ${\overline{\theta }}_{j,q}(q=1,\cdots ,N_{sel})$.

The assigned probability is given by

$$P_q={\displaystyle {\int }_{V_q}{\phi }_S}(\overline{\theta })d\overline{\theta }\approx {\displaystyle \frac{{(2\lambda )}^S}{n}}{\phi }_S({\overline{\theta }}_{j,q}),\hspace{1em}(q=1,\cdots ,N_{sel})$$

(17)

where $V_q$ is the representative volume of ${\overline{\theta }}_{j,q}$, and ${\phi }_S(\cdot )$ is the joint PDF of $Z$.

3. Dynamic Reliability and Structural Vulnerability

3.1. The Evolutionary Probability Density Function of Dynamic Response

Without loss of generality, the dynamic equation of the nonlinear MDOF system under the stochastic earthquake excitation can be governed by

$$M\ddot{X}+C\dot{X}+K(X)=-MI{\ddot{X}}_g(\mathsf{\Theta },t)$$

(18)

Here, $\ddot{X}$$\dot{X}$$X$ are the $n\times 1$ acceleration, velocity, and displacement vectors, with the overdot denoting the derivation with regard to time; $M$, $C$ are $n\times n$ mass and damping matrix, respectively; $K$ is the restoring force vector; $I$ is the $n\times 1$ unit column vector; ${\ddot{X}}_g(\mathsf{\Theta },t)$ is the ground acceleration (Equation (10)).

Thoughts on PDEM indicate that the solution of Equation (18) exists and relies solely on the random vector $\mathsf{\Theta }$, and the structure displacement physical quality can be expressed as $X(t)=H(\mathsf{\Theta },t)$ [38]. The system composed of $(X(t),\mathsf{\Theta })$ is a probability conserve, and the joint PDF $P_{x\theta }(x,\theta ,t)$ satisfies the generalized density evolution equation as

$${\displaystyle \frac{\partial p_{x\mathsf{\Theta }}(x,\theta ,t)}{\partial t}}+\dot{X}(\theta ,t){\displaystyle \frac{\partial p_{x\mathsf{\Theta }}(x,\theta ,t)}{\partial x_j}}=0$$

(19)

With the initial condition

$$p_{x\mathsf{\Theta }}(x,\theta ,t_0)=p_{x\mathsf{\Theta }}(x,\theta ,t)\delta (X-X_0)$$

(20)

Here, $X_0$ is the initial value of $X$; $\delta (·)$ is the Dirac function.

After Problems (14) and (15) are solved, one can obtain the PDF of $X(t)$ as the marginal distribution of $p_{x\mathsf{\Theta }}(x,\theta ,t)$, i.e.,

$$p_x(x,t)={\displaystyle {\int }_{{\mathrm{\Omega }}_{\theta }}{p}_{x\mathsf{\Theta }}(x,\theta ,t)}d\theta$$

(21)

Here, ${\mathrm{\Omega }}_{\theta }$ is the distribution domain of $\mathsf{\Theta }$.

3.2. Numerical Implementation of $p_x(x,t)$

The numerical solution combining the deterministic dynamic response analysis and the finite difference method includes the following steps:

Step 1.

Take representative points as ${\theta }_q(q=1,2,\dots ,N_{\mathrm{sel}})$, (Equation (16)), and solve the corresponding assigned probability $P_q$ (Equation (17)).

Step 2.

For the prescribed $\theta ={\theta }_q$, solve Equation (13) with a deterministic time integration method to obtain the velocity $\dot{X}({\theta }_q,t_m)$, where $t_m=m\cdot \Delta t$, $\Delta t$ is the time step.

Step 3.

Solve the initial boundary problem defined by Equations (14) and (15) with the finite difference method [38].

3.3. Classification of Failure State Criterion

Structural vulnerability analysis requires defining multi-level ultimate failure states (FSs), which align with structural performance criteria. These criteria are typically quantified by metrics such as top displacement, inter-story drift, or maximum inter-story drift angle exceeding predefined thresholds.

The seismic design code for building structures defines five performance levels (failure states, FSs) for reinforced concrete (RC) frame structures under varying seismic excitations: normal operation (NO), immediate occupancy (IO), moderate damage (MD), life safety (LS), and collapse prevention (CP) [39]. In this study, the maximum inter-story drift angle is adopted as a performance metric for the frame structure. The relationship between structural failure states (FSs) and the corresponding maximum inter-story drift angle limits is summarized in

Table 1. The maximum inter-layer displacement angle limit values of structure failure states.

Structure FS	Discriminant Criterion	Structural Capability Index	Limit Value
NO	$\left|{\theta }_{\max }\right|\le {\theta }_1$	-	-
IO	${\theta }_1<\left|{\theta }_{\max }\right|\le {\theta }_2$	${\theta }_1$	1/550
MD	${\theta }_2<\left|{\theta }_{\max }\right|\le {\theta }_3$	${\theta }_2$	1/400
LS	${\theta }_3<\left|{\theta }_{\max }\right|\le {\theta }_4$	${\theta }_3$	1/250
CP	$\left|{\theta }_{\max }\right|>{\theta }_4$	${\theta }_4$	1/50

.

3.4. Solution of Structural Reliability and Vulnerability

The limit value of the structural capacity index delineates the boundary between the safe and failure regions of the structural response, corresponding to the performance level of the structure. Consider the collapse prevention (CP) level as an example:

$$\left\{\left|\theta \right|\le {\theta }_4\right\}={\mathrm{\Omega }}_s,\left\{\left|\theta \right|>{\theta }_4\right\}={\mathrm{\Omega }}_f$$

(22)

Here, ${\mathrm{\Omega }}_s$ stands for the safety domain of the structure response preventing from the failure of CP, and ${\mathrm{\Omega }}_f$ stands for the failure domain in the structure response of failure of CP. It is easy to know

$${\mathrm{\Omega }}_s\cap {\mathrm{\Omega }}_f=\varnothing ,{\mathrm{\Omega }}_s\cup {\mathrm{\Omega }}_f=\mathrm{\Omega },$$

(23)

$\mathrm{\Omega }$ stands for the overall domain of the structure response.

The limit value of the structural capacity index can be defined as a boundary condition in solving the generalized probability density evolution equation. This implies that if the structural response exceeds the safety boundary, the structure is considered to have failed, and the probability transitions into the failure region without returning to the safe region. In other words, the probability in this region is entirely absorbed. For the first-passage failure problem, the dynamic reliability of the structure can be expressed as

$$F_r(t)={\displaystyle {\int }_{x\in {\mathrm{\Omega }}_s}{p}_x(x,t)dx}\hspace{1em}t\in (0,T_t)$$

(24)

Here, $T_t$ is the total analysis time.

Seismic vulnerability is defined as the probability of a structure reaching a predefined failure state under a specified seismic intensity. By incorporating the uncertainty factors of vulnerability assessment into stochastic artificial waves and defining the structural limit state between safe and failure regions, the vulnerability assessment reduces to a standard failure probability problem. The mathematical formulation can be expressed as

$$F_f(a)=P(FS | A=a)={\displaystyle {\int }_{x\in {\mathrm{\Omega }}_f}{p}_{X|A}[x | a]dr}=1-F_r(t,A=a),\hspace{1em}t=T_t$$

(25)

Here, $F_f$ is the seismic vulnerability of the structure; $P(·)$ is the failure probability operator; $A$ is the ground motion strength factor (in this passage, we choose the ground peak acceleration (PGA)); $p_{X|A}[x | a]$ is the PDF of the structure response when A = a.

At last, by applying the Incremental Dynamic Analysis (IDA) method and curve fitting the computing results of $F_f$, the obtained smooth curves are then the vulnerability curve.

4. VED Structure Optimization

4.1. VED Structure Finite Element Model

The calculation model of VED can be given by the Kelvin model

$$f_d=c_d{\dot{x}}_d+k_d{x}_d$$

(26)

Here, $f_d$ is the resilience of VED; ${\dot{x}}_d,x_d$ are, respectively, the relative velocity and relative displacement of the VED layer; $c_d$ and $k_d$ are the equivalent impedance and stiffness, and they can be obtained as

$$c_d={\displaystyle \frac{G^{″}{A}_d}{\omega h_d}}={\displaystyle \frac{\eta G^{\prime }{A}_d}{\omega h_d}}$$

(27)

$$k_d={\displaystyle \frac{G^{\prime }{A}_d}{h_d}}$$

(28)

Here, $G^{\prime }$ is the shear storage modulus; $G^{″}$ is the shear loss modulus; $\omega$ means the self-vibratory frequency of the main structure; $A_d$ is the total shear area of viscoelastic (VE) material; $h_d$ is the thickness of each VE slab; the damper loss factor is ${\eta }_v={\displaystyle \frac{G^{″}}{G^{\prime }}}$.

After adding the VED devices into the main structure, the dynamic differential equation of the VED structure yields from Formula (13) to

$$M\ddot{X}+C\dot{X}+KX+R_v=-MI{\ddot{X}}_g(\mathsf{\Theta },t)$$

(29a)

$$R_v=A_1\dot{X}+A_{2}X$$

(29b)

Here, $r$ is the nonlinear force matrix contributed by VED; $A_1$ and $A_2$ are the coefficient matrixes of the nonlinear force, and they yield

$$A_1=\left[\begin{array}{cccc}{c}_{d1}+c_{d2}& -c_{d2}& & \\ -c_{d2}& c_{d2}+c_{d3}& -c_{d3}& \\ & -c_{d3}& \ddots & -c_{dn}\\ & & -c_{dn}& c_{dn}\end{array}\right]$$

(30)

$$A_2=\left[\begin{array}{cccc}{k}_{d1}+k_{d2}& -k_{d2}& & \\ -k_{d2}& k_{d2}+k_{d3}& -k_{d3}& \\ & -k_{d3}& \ddots & -k_{dn}\\ & & -k_{dn}& k_{dn}\end{array}\right]$$

(31)

Here, $k_{dn}$ and $c_{dn}$ are the sum of stiffness and impedance of the VED of the nth layer.

4.2. VED Parameter Optimization Model

To reduce the dynamic response of the VED structure under excitation and enhance the reliability of the VED structure to prevent failure, the optimization objective function is defined here as follows: on the premise of satisfying certain constrain conditions, optimize the VED parameters to reach the best shock absorption effect. To simplify the optimization problem, the numerical values of ${\eta }_v$, $A_d$, $h_d$ are fixed, and $G^{\prime }$ is taken as the only design variable; the optimization model is shown as

$$\mathrm{Design} \mathrm{variable}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}Y\in \left[\underset{¯}{G^{\prime }},\overline{G^{\prime }}\right]$$

$$\mathrm{Objective} \mathrm{function}:\hspace{1em}\hspace{1em}\hspace{1em} \min \hspace{1em}\Delta (Y)=\underset{1\le j\le cn}{\max }\left[{\Delta }_j(Y)\right]$$

(32)

$$\mathrm{constrain} \mathrm{conditions}: \mathrm{s}.\mathrm{t}.\hspace{1em}\hspace{1em}\hspace{1em}\underset{¯}{G^{\prime }}\le G^{\prime }\le \overline{G^{\prime }}$$

(33)

$$\varphi (y_i)=\Delta (y_i)(1+c)\hspace{1em}c=0.2,if{F}_r(i)\le 0.9$$

(34)

$$\mathrm{Fitness} \mathrm{function}:\hspace{1em}\hspace{1em}\hspace{1em} f(y_i)={\displaystyle \frac{{\varphi }_i(y_i)}{\varphi {(y)}_{\min }+\varphi {(y)}_{\max }}}$$

(35)

Here, $\Delta (Y)$ is the most disadvantageous inter-layer displacement (MDID) of the VED structure; cn is the total structure layers; $f(y_i)$ is the fitness function composed of $\Delta (y_i)$, the constrain breach factor c, and $F_r$, which illustrates that once the MDID is negative or the reliability fails to reach 0.9, $f_i$ enlarges and the individual $y_i$ will be more likely to be eliminated.

4.3. GFWA

The explosion of a firework can be interpreted as a local search around a specific point in the solution space. When a point $y_i$ is found to be satisfying $\min \Delta (y_i)$, the GFWA can continually set off ‘fireworks’ in potential space searching for better solutions through the iteration of generating sparks from the fireworks and the selection of fireworks among the sparks until one ‘spark’ targets or is fairly near the point $y_i$. The GFWA has a very simple idea and works stably; the operators will be briefly introduced in three parts as follows.

(1) Explosion Operator: Each firework undergoes an explosion process, producing a predefined number of sparks within a controlled amplitude range. The quantity of sparks and the explosion amplitude are determined adaptively based on the objective function’s fitness values. This adaptive mechanism ensures that higher-quality fireworks generate more sparks within reduced explosion amplitudes to enhance local search precision, while lower-quality fireworks produce fewer sparks across broader amplitudes to facilitate global exploration.

For each firework Xi, its explosion sparks’ number is calculated as follows:

$${\lambda }_i=\widehat{\lambda }\cdot {\displaystyle \frac{\underset{j}{\max (f(y_j))-f(y_i)}}{{\displaystyle \sum _j(\underset{j}{\max }(f(y_k))-f(y_j))}}}$$

(36)

where $\widehat{\lambda }$ is a constant parameter, which controls the total number of explosion sparks in one generation, taken as 15 here.

In each generation, the firework with the best fitness is called the core firework (CF):

$${\mathrm{y}}_{CF}=\underset{{\mathrm{X}}_{\mathrm{i}}}{\arg \min }(\mathrm{f}({\mathrm{y}}_{\mathrm{i}}))$$

(37)

The fireworks’ explosion amplitudes (except for the CF’s) are calculated as:

$$A_i=\widehat{A}\cdot {\displaystyle \frac{\underset{j}{f(y_i)-f(y_{CF})}}{{\displaystyle \sum _j(f(y_j)-f(y_{CF}))}}}$$

(38)

where $\widehat{A}$ is a constant parameter, which controls the explosion amplitudes generally, taken as 40 here.

But, for the CF, its explosion amplitude is adjusted according to the search results in the last generation:

$$A_{CF}(i)=\left\{\begin{array}{c}{A}_{CF}(1)\hspace{0.33em}\hspace{0.33em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i=1\\ C_r{A}_{CF}(t-1)\hspace{1em}f(y_{CF}(i))=f(y_{CF}(i-1))\\ C_a{A}_{CF}(i-1)\hspace{1em}f(y_{CF}(t))<f(y_{CF}(i-1))\end{array}\right.$$

(39)

where $A_{CF}(i)$ is the explosion amplitude of the CF in generation i. In the first generation, the CF is the best among all the randomly initialized fireworks, and its amplitude is preset to a constant number, which is usually the diameter of the search space. After that, if in generation i − 1, the algorithm found a better solution than the best in generation i − 2, the amplitude of the CF will be multiplied by an amplification coefficient Ca > 1 (taken as 1.2 here); otherwise, it will be multiplied by a reduction coefficient Cr < 1 (taken as 0.9 here). The best solution in generation i − 1 is always selected into generation i as the CF, so the righthand conditions in Equation (39) indicate whether the best solution found has been improved.

(2) In GFWA, a novel guiding spark mechanism is introduced to improve information utilization. This approach leverages the objective function’s information obtained from explosion sparks to construct a guiding vector with both promising direction and adaptive magnitude. By adding this guiding vector to the firework’s current position, an elite solution, termed the guiding spark, is generated to enhance search efficiency. The guiding vector is defined as

$${\mathrm{\Lambda }}_i={\displaystyle \frac{1}{⌈\sigma {\lambda }_i⌉}}{\displaystyle \sum _{j=1}^{⌈\sigma {\lambda }_i⌉}(s_{ij}-s_{i,{\lambda }_i-j+1})}$$

(40)

Here, the sparks are sorted based on their fitness values in ascending order. ${\lambda }_i$ is the explosion sparks’ number for each $y_i$; $s_{ij}$ is the jth spark generated by $y_i$; $\sigma \le 0.5$, as only the top and bottom $\sigma {\lambda }_i$ sparks’ indexes are needed, taken as 0.2 here.

The position of the guiding spark $G_i$ for firework $y_i$ is determined by Algorithm 1.

Algorithm 1. Generating the Guiding Spark for $y_i$.

Require: $y_i$, $s_{ij}$, $f(s_{ij})$, ${\lambda }_i$ and $\sigma$

1: Sort the sparks by their fitness values $f(s_{ij})$ in ascending order.

2: ${\mathrm{\Lambda }}_i\leftarrow {\displaystyle \frac{1}{\sigma {\lambda }_i}}({\displaystyle \sum _{j=1}^{\sigma {\lambda }_i}{s}_{ij}-{\displaystyle \sum _{j={\lambda }_i-\sigma {\lambda }_i+1}^{{\lambda }_i}{s}_{ij}}})$

3: $G_i\leftarrow y_i+{\mathrm{\Lambda }}_i$

4: return $G_i$

(3) Selection Operator: In each iteration, after generating the guiding sparks, the candidate pool for selecting the next generation of fireworks comprises three types of individuals: current fireworks, explosion sparks, and guiding sparks. The best-performing individual from this pool is first selected as a firework for the next iteration. The remaining fireworks are then chosen uniformly at random from the rest of the candidates.

In brief, the framework of the GFWA is shown in Algorithm 2.

Algorithm 2. Guided Fireworks Algorithm.

1: Randomly initialize $\mu$ fireworks in the potential space.

2: Evaluate the fireworks’ fitness.

3: repeat

4:  Calculate ${\lambda }_i$ according to Equation (36).

5:  Calculate $A_i$ according to Equations (38) and (39).

6:  For each firework, generate ${\lambda }_i$ sparks within the amplitude $A_i$.

7:  For each firework, generate guiding sparks according to Algorithm 1.

8:  Evaluate all the sparks’ fitness.

9:  Keep the best individual as a firework.

10:  Randomly choose other $\mu -1$ fireworks among the rest of individuals.

11: until termination criteria is met.

12: return the position and the fitness of the best individual.

At last, based on the theories of stochastic ground motion generation and PDEM, this study attempts to apply the vulnerability analysis method to the VED structure GFWA optimization.

5. Case Studies

5.1. Example of Stochastic Ground Motion Generation

This example is presented to prove the correctness of the artificial wave generation method. The main parameters of the NTM are as follows: $\gamma =4.0$, $N_T=155093$, ${\mathrm{r}}_0=0.95$, $N_{sel}=394$. Then, 394 artificial waves are generated, where the seismic fortification intensity is taken as 8 degrees, and the epicentral distance is 60 km. The site soil property parameters of kind IV are listed in

Table 2. Site soil-related parameters of seismic oscillation.

Site Category	${\mathit{\omega }}_{\mathit{g}}$ (rad/s)	${\mathit{\xi }}_{\mathit{g}}$	${\mathit{S}}_0$ (cm2/s3)	${\mathit{T}}_{\mathbf{s}}$ (s)
IV	9.67	0.90	185.37	15.66

, and their computation method is taken from [40], which we will not give further discussion.

The power spectral densities (PSDs) and acceleration response spectra of the generated artificial waves are illustrated in Figure 1 and Figure 2, respectively, with selected representative data for comparison. Figure 1 demonstrates the attenuation characteristics of the average energy distribution across frequencies, while Figure 2 highlights the attenuation of the maximum acceleration response with respect to the natural vibration period. Both figures reveal the non-uniform energy distribution of the waves, effectively capturing the stochastic nature of the artificial ground motions. Additionally, Figure 3 presents two randomly selected acceleration time–history curves, which not only exhibit the intensity and frequency attenuation characteristics but also clearly reflect the randomness through variations in peak values and temporal patterns.

5.2. Structure Elastoplastic Time History Analysis and PDFM Solution

A total of 394 artificial ground motion records were applied to a five-story reinforced concrete (RC) frame structure, and nonlinear time history analyses conducted. The stiffness degradation model used in the analysis, proposed by Takeda [41], is illustrated in Figure 4. The relevant structural parameters and elastoplastic constitutive model parameters are summarized in

Table 3. Structure selection.

Story Height	Span	Width	Sections of Columns	Beams	Concrete	Bar
3.9 m	6 m	4.5 m	500 × 500 mm2	$EI\to \infty$	C35	HRB335

and

Table 4. Structure parameters.

Layer	1	2	3	4	5	First Stiffness Reduction Coefficient	Second Stiffness Reduction Coefficient
Mass (×105 kg)	1.00	0.95	0.95	0.95	0.95	0.4	0.1
Stiffness (×107 N/m)	1.92	1.59	1.59	1.59	1.65
$x_c$ (mm)	6.3	4.9	4.2	3.8	2.6
$x_y$ (mm)	21.8	18.9	17.2	15.3	13.6

, respectively. The ground motions were scaled to a peak acceleration of 110 gal, with an excitation time step of 0.02 s.

The life safety (LS) failure state is adopted as the absorbing boundary condition for solving the generalized probability density evolution equation, which evaluates the seismic reliability of the structure. Two failure criteria are considered: (1) a single failure criterion, where structural failure is defined as the response of any single layer exceeding the limit, and (2) a complex failure criterion, where structural failure occurs when the responses of all layers exceed their respective limits [42]. The results are summarized in

Table 5. Structure seismic reliability.

Failure Criterion	Seismic Reliability	Failure Probability
Failure of layer 1	0.5690	0.4310
Failure of layer 2	0.5566	0.4434
Failure of layer 3	0.5758	0.4242
Failure of layer 4	0.5720	0.4280
Failure of layer 5	0.5762	0.4238
Failure of structure system	0.4243	0.5757

. As shown, the structural system reliability calculated under the complex failure criterion is lower, confirming that the reliability under this criterion does not equate to the reliability of the weakest link in the system.

The evolution of the probability density function (PDF) is illustrated in Figure 5, with representative PDFs at selected time instants shown in Figure 6. The results demonstrate that the PDF exhibits non-stationary temporal evolution, including transient multi-modal characteristics. This behavior precludes conventional assumptions of normality or standard parametric distributions for displacement response modeling. In contrast, the probability density evolution method (PDEM) employed in this study enables the precise characterization of stochastic structural responses and direct computation of time-dependent reliability and displacement failure probabilities.

By defining boundary conditions corresponding to distinct performance levels and incrementally increasing artificial wave intensity in 40 gal increments, failure probability datasets were generated. These datasets were subsequently fitted using MATLAB^® R2023b (Version 9.14)’s curve fitting toolbox to derive vulnerability curves (Figure 7). The curves reveal two critical observations: (1) the structure maintains high reliability under frequent seismic events (low intensity), and (2) failure probability escalates nonlinearly with seismic intensity, indicating significantly degraded performance under rare or extreme earthquakes. These findings underscore the necessity of implementing vibration control measures and structural optimization for enhanced seismic resilience.

5.3. VED Structure Optimization

To enhance the seismic performance of the structure, viscous energy dampers (VEDs) are implemented in this study. Each floor of the original structure is equipped with a VED, and the optimal damper parameters are determined through optimization algorithms to minimize the maximum inter-story drift. The guided firework algorithm (GFWA) is employed and compared with a Genetic Algorithm (GA) to evaluate computational efficiency. The optimized VED parameters are summarized in

Table 6. VED parameters.

${\mathit{G}}^{\mathbf{\prime }}$ (N/m2)	${\mathit{\eta }}_{\mathit{v}}$	${\mathit{A}}_{\mathit{d}}$ (m2)	${\mathit{h}}_{\mathit{d}}$ (m)
0~120 × 105	1.4	3 × 10−2	1.3 × 10−2

.

In the subsequent analysis, a constraint is imposed to ensure the structural system reliability for collapse prevention (CP) remains no less than 0.9. Both algorithms are executed 40 times, with a maximum of 40 evaluations per run. To ensure computational efficiency, a termination criterion is applied: the optimization loop terminates automatically if the improvement rate between consecutive generations falls below 0.000005.

The optimization results are summarized in

Table 7. Structural optimization results.

Algorithm	Breaking Generation	Displacement Reliability	The Optimal	Optimization Rate
GA	13	0.5975	15.0900	29.18%
GFWA	15	0.9194	67.6111	66.86%

, which compares the performance of the guided firework algorithm (GFWA) and Genetic Algorithm (GA) across key metrics: termination generation, optimal inter-story drift, structural system reliability, optimized variable values, and computational efficiency. While the GA terminated the loop earlier (at the 13th generation), it achieved limited optimization effectiveness. In contrast, the GFWA demonstrated superior stability and convergence, yielding significantly lower inter-story drift, higher structural system reliability, and markedly improved computational efficiency.

A comparison of the convergence behavior and computational time between the two algorithms is illustrated in Figure 8 and Figure 9, where it is evident that GFWA exhibits a significantly faster convergence rate. However, GA tends to converge to a local optimum and terminates prematurely, limiting its effectiveness. In summary, GFWA demonstrates superior computational efficiency and convergence performance compared to GA.

However, suppressing the displacement response should not come at the expense of increasing other performance indicators, as this would undermine the effectiveness of structural vibration control optimization. To address this, Figure 10 and Figure 11 present a comparison of the peak inter-layer displacement and acceleration response time histories before and after GFWA optimization. The results demonstrate that the VED damping effect achieved through GFWA is highly effective, as both the acceleration and displacement responses are significantly reduced. In contrast, as shown in Figure 12 and Figure 13, while the displacement response is partially reduced using GA, the acceleration response remains largely unaffected. These comparisons highlight two key points: (1) the selection of VED parameters is critical for effective structural shock absorption, and damper parameter optimization plays a vital role in achieving this; (2) for this case study, GFWA demonstrates significantly superior optimization performance compared to GA.

Using the optimized results from GFWA, the vulnerability curve of the structure post-optimization is plotted. A comparison of the vulnerability curves before and after optimization is shown in Figure 14 to assess changes in structural performance. The optimized curves exhibit smoother trends and reduced slopes, suggesting a decrease in the exceedance probabilities for all damage levels. Notably, under high-intensity ground motion excitation, the system’s reliability improves significantly. These findings demonstrate a marked enhancement in the overall structural performance, highlighting the success and effectiveness of the GFWA-based VED parameter optimization approach.

6. Conclusions

Based on the proposed vulnerability evaluation method, this study successfully applies GFWA to optimize a reinforced concrete structure. The following conclusions can be drawn:

(1)

The stochastic ground motion generation method proposed in this study, based on the orthogonal expansion model, effectively incorporates site soil properties and ground motion randomness. This approach eliminates the need for complex wave selection in vulnerability analysis, offering a more reliable framework for assessing structural performance. Additionally, by integrating PDEM, this study introduces a vulnerability method that couples ground motion randomness with structural nonlinearity, providing an exact probabilistic solution for vulnerability assessment from both performance and probability perspectives.

(2)

VED demonstrates excellent energy dissipation and shock absorption capabilities in building structures, but their effectiveness heavily depends on parameter selection. Evaluating VED-based shock absorption control requires a comprehensive assessment of multiple performance indicators, making vulnerability analysis a valuable tool in this context.

(3)

GFWA demonstrates robust optimization capabilities. The experimental results highlight its superiority over GA in terms of optimization efficiency, computational performance, applicability, and global search capability, effectively avoiding local optima. As a result, GFWA offers a highly efficient and practical solution for seismic optimization design in building structures.