Natural frequency optimization of composite plates with cutouts

Abstract

This study presents two complementary computational approaches for optimizing the natural frequencies of laminated composite plates with cutouts. The first approach employs Particle Swarm Optimization (PSO) to explore a combined design space including ply angles, hole center location (x_c, y_c), and hole radius R for laminated composite plates with circular cutouts. Ply orientations are treated as continuous variables within [−90^∘,90^∘] and rounded to integer values, while geometric constraints ensure that the cutout remains within the plate boundaries. The second approach introduces a finite-element pre-processing technique, termed Adjacent Element Temperature Driven Prestress (AETDP), which constructs a temperature-induced prestress field directly from the mesh adjacency graph using a breadth-first search algorithm. This enables efficient prestressed modal analysis without coupled thermo-mechanical simulations. Validation against numerical and experimental benchmarks shows mean absolute errors below 8%. Optimization results indicate that optimal cutout locations tend to align with modal nodal lines, reducing stiffness loss, while highly constrained plates tolerate larger cutouts. The AETDP approach reduces computational time significantly compared with full PSO optimization.

Keywords

composite plates natural frequency optimization Finite Element Method (FEM)Particle Swarm Optimization (PSO)free vibration Adjacent Element Temperature Driven Prestress (AETDP)

1. Introduction

Glass and carbon fiber-reinforced polymer composites have become indispensable materials in modern aerospace, marine, automotive, and civil infrastructure industries due to their superior specific strength, stiffness-to-weight ratios, and design versatility.¹ These advanced materials allow for significant weight reduction while maintaining or enhancing structural performance, making them ideal for applications where efficiency and reliability are paramount. Planar composite structures, particularly plates and shells, form critical components in these sectors. However, functional requirements often necessitate the inclusion of discontinuities or cutouts within these structures for purposes such as mechanical access, weight reduction, utility routing, and inspection ports.^2,3 While essential, the introduction of these geometric discontinuities fundamentally alters the structural mechanics, leading to complex stress concentrations, redistributions of stiffness and mass, and significant modifications to the dynamic vibrational response.^4,5 These alterations can shift the system’s natural frequencies, potentially bringing them into resonance with operational excitation frequencies, thereby increasing the risk of accelerated fatigue damage, reduced service life, and catastrophic failure.⁶

Consequently, optimizing the dynamic performance of perforated composite plates emerges as a critical and multifaceted design challenge. A primary objective in such optimization is often the maximization of the fundamental natural frequency to avoid resonance, thereby enhancing operational safety and structural integrity.^7,8 This optimization problem is inherently high-dimensional and complex, involving the simultaneous consideration of numerous interdependent design variables. Key parameters include the laminate’s stacking sequence (i.e., the specific fiber orientation angles in each ply), the overall laminate thickness, and the precise geometry of the cutout—encompassing its size, shape, and location within the plate.^9,10 Traditional design approaches, which often rely on engineering intuition and iterative physical or computational testing, are ill-suited to efficiently navigate this vast and non-linear design space where optimal configurations frequently defy conventional expectations.

This study addresses this challenge by developing and presenting a comprehensive, integrated computational framework for optimizing the natural frequencies of rectangular composite plates containing a circular cutout. The framework leverages the accuracy of the Finite Element Method (FEM) for conducting detailed modal analyses. The validity and precision of the FEM model are rigorously established through comparison with established numerical and experimental benchmark data from the literature. To navigate the complex optimization landscape, a Particle Swarm Optimization (PSO) algorithm is employed. This metaheuristic optimizer is tasked with identifying the global optimum combination of design variables: the laminate stacking sequence and the cutout’s geometric parameters, specifically its center coordinates and radius. The investigation is comprehensive, analyzing plates with multiple aspect ratios under four distinct boundary conditions to generalize the findings. The results demonstrate the efficacy of the proposed FEM-PSO framework and reveal that optimal designs for vibrational performance often involve non-intuitive placements and sizes of cutouts. This underscores the necessity and value of employing systematic, computational optimization in the design phase of lightweight, perforated composite structures to unlock their full performance potential.

The dynamic analysis, characterization, and optimization of laminated composite plates—especially those incorporating cutouts—represent a rich and extensively researched field within structural mechanics and materials engineering. The body of literature is vast, evolving from fundamental analytical studies to sophisticated numerical simulations coupled with advanced optimization algorithms. This review synthesizes the existing knowledge by categorizing the research into several interconnected thematic areas.

1.1. Free vibration analysis of composite plates

The foundational understanding of vibrational behavior in composite laminates stems from extensive analytical and numerical investigations. Early pioneering work established robust analytical methods, such as the Rayleigh-Ritz technique employing orthogonal polynomials, which provided accurate solutions for the natural frequencies of symmetrically laminated plates under a variety of classical boundary conditions.^11,12 The development and adoption of shear deformation theories, most notably the First-order Shear Deformation Theory (FSDT), became crucial for the accurate analysis of moderately thick plates, where transverse shear effects are significant.¹³ As structural geometries and boundary conditions grew more complex, numerical methods—particularly the Finite Element Method (FEM)—rose to prominence as the standard computational tool. A multitude of studies have successfully utilized FEM to systematically investigate the influence of plate geometry (aspect ratio, thickness), material anisotropy, and support conditions on natural frequencies and mode shapes.^2,14 In recent years, the field has seen significant advances with the introduction of Isogeometric Analysis (IGA), which offers superior geometric exactness and smoother modeling for structures with complex shapes or cutouts.^15,16 Furthermore, research continues into refined higher-order theories and the analysis of novel material systems, such as three-phase composites reinforced with graphene platelets and carbon fibers, to achieve even more accurate predictions of dynamic response.¹⁷

1.2. Effects of cutouts on vibration characteristics

The introduction of a cutout creates a discontinuity that dramatically alters a plate’s stiffness and mass distribution, leading to a complex modification of its dynamic signature. A consistent finding across the literature is that cutouts generally reduce natural frequencies, although the magnitude and nature of this effect are governed by several interacting factors. Cutout shape is a primary determinant; comparative studies of circular, square, elliptical, diamond, and skew-shaped openings reveal distinct differences in stress concentration factors and the consequent reduction in fundamental frequency.^18–20 The size and location of the cutout are equally critical. Typically, increasing cutout size reduces global stiffness and lowers the fundamental frequency, but this relationship is non-linear and can even be inverse for specific boundary conditions, such as cantilever supports.^18,21 The location of the cutout relative to the nodal lines of the vibration modes profoundly influences the dynamic response, as cutting through a high-strain area has a more detrimental effect than removing material from a low-strain region.³ Finally, the interaction between boundary conditions and laminate parameters mediates the overall effect. The same cutout will impact a clamped plate differently than a simply supported one. Moreover, the fiber orientation angles and the stacking sequence of the laminate dictate its anisotropic stiffness, which in turn influences how the structure redistributes loads and strains around the discontinuity.^13,22

1.3. Optimization of composite laminates

Driven by the need for high-performance, lightweight structures, the optimization of composite laminates has become a major research focus. The goal is to find the best possible laminate configuration to maximize a desired performance metric, such as natural frequency or buckling load, often under constraints like weight or strength. The primary design variables are the fiber orientation angles in each ply and the resulting stacking sequence. Seminal work in this area focuses on optimizing these variables for fundamental frequency maximization^23,24 or buckling resistance.²⁵ Given the discrete, non-convex, and often multi-modal nature of the design space, traditional gradient-based optimizers are frequently inadequate. Consequently, heuristic and metaheuristic optimization algorithms have become the tools of choice. Genetic Algorithms (GA) have been widely and successfully applied for stacking sequence optimization across various plate and shell geometries.^26–28 Similarly, Particle Swarm Optimization (PSO) has demonstrated excellent performance and robustness in solving structural optimization problems, often outperforming other methods in terms of solution quality and convergence speed.^29–32 The algorithmic toolbox is diverse, also including Simulated Annealing (SA),^6,33 Harmony Search,⁹ the Bonobo Optimizer,¹⁰ Bayesian Optimization,⁷ and pattern search methods.³⁴ Research in this domain has expanded into advanced optimization topics, including multi-objective formulations that trade off weight savings against frequency or buckling load maximization,³⁵ stochastic optimization to account for material and fabrication uncertainties,^36,37 and topology optimization for generating novel, high-performance conceptual designs.^38–40 A parallel and critical strand of research focuses on durability optimization, specifically designing laminates for maximum fatigue life under cyclic loading conditions.^6,41–44

1.4. Optimization of plates with cutouts

The presence of a cutout adds a layer of geometric variables to the optimization problem, significantly increasing its complexity. Research here can be subdivided. Stacking sequence optimization for plates with predefined cutouts involves finding the best ply angles to mitigate stress concentrations and maximize performance metrics like buckling load or fundamental frequency for a given hole geometry.^8,9,34 A revolutionary design paradigm is enabled by Variable Stiffness Composites (VSC) or Variable Angle Tow (VAT) laminates. Here, curvilinear fiber paths allow the material stiffness to be spatially tailored to redirect load paths around the cutout, leading to remarkable improvements in load-bearing capacity, buckling performance, and vibrational characteristics.^45–49 The optimization of these curvilinear fiber paths is itself an active research frontier.^27,28,50 Despite these advances, studies that perform simultaneous optimization of both the laminate stacking sequence and the cutout’s geometric parameters (size, location) for dynamic performance remain relatively scarce. This integrated approach, which is the core focus of the present study, represents a more holistic and powerful design strategy but poses greater computational and methodological challenges.

1.5. Related advanced topics

The field continues to expand into novel materials and extreme geometries. Functionally Graded Materials (FGMs) with cutouts are studied for their tailored thermal and mechanical properties, with research focused on their vibration and stress characteristics.^51,52 The analysis of plates with highly complex cutout shapes, such as paisley or star geometries, pushes the boundaries of geometric modeling, often employing advanced techniques like IGA.⁵³ The dynamic behavior of sandwich structures with composite face sheets and various core materials (e.g., honeycomb, foam) featuring cutouts is another important area, considering the core’s geometry and the face sheet’s layup.^20,54 Beyond purely mechanical structures, related optimization concepts are applied in the design of acoustic metamaterials, such as spatial double helix resonators, for targeted low-frequency sound absorption.⁵⁵ All these reviewed studies are summarized in Table 1.

Table 1.

Structured overview of the review.

Reference no	Author(s) (year)	Primary category	Key focus/Contribution
1	Abdellah et al. (2021)	Material Property Prediction	Uses modal analysis & fracture mechanics to predict nominal strength of open-hole GFRP.
2	Sinha et al. (2021)	Vibration Analysis (Exp. & Num.)	Experimental & FEM study on free vibration of laminated plates with/without cut-outs.
3	Zang et al. (2021)	Vibration Analysis (Exp. & Num.)	Experimental & numerical study on free vibration of CFRP laminate with cutout.
4	Khechai et al. (2023)	Stress Analysis (FGM)	Presents an analytical stress solution for circular holes in functionally graded plates.
5	Pal et al. (2023)	Vibration Analysis (Damage)	Numerically analyzes natural frequencies of delaminated composite panels with a cut-out.
6	Ertas & Sonmez (2014)	Optimization (Fatigue)	Optimizes fiber orientation for maximum fatigue life of laminates.
7	Yang & Liang (2024)	Optimization (Frequency)	Uses Bayesian Optimization to maximize fundamental frequency via stacking sequence.
8	Duan & Jing (2023)	Optimization (Frequency)	Optimizes stacking sequence of perforated plates for max. fundamental frequency using 2DSO.
9	Cakiroglu et al. (2021)	Optimization (Buckling)	Uses Harmony Search for stacking sequence optimization of laminates with cut-outs.
10	Shaterzadeh et al. (2025)	Optimization (Buckling)	Uses Bonobo Optimizer for buckling load optimization of plates with elliptical holes.
11	Leissa & Narita (1989)	Vibration Analysis (Analytical)	Comprehensive vibration studies for simply supported symmetric laminates.
12	Cupial (1997)	Vibration Analysis (Analytical)	Rayleigh-Ritz method with orthogonal polynomials for composite plate frequencies.
13	Kumar et al. (2014)	Vibration Analysis (FEM)	Free vibration analysis of composite plates with skew cutouts using FSDT-based FEM.
14	Giunta et al. (2023)	Vibration Analysis (FEM/VSC)	FEM free vibration analysis of variable stiffness plates using Carrera’s Unified Formulation.
15	Ding et al. (2025)	Vibration Analysis (IGA)	Adaptive IGA for eigenvalue analysis of laminated plates with arbitrary cutouts.
16	Garg et al. (2024)	Vibration Analysis (IGA/Bio)	IGA for free vibration of bio-inspired helicoidal laminates with openings.
17	Liu et al. (2024)	Vibration Analysis (Multi-Phase)	Studies free vibration of three-phase composite plates (GPLs & carbon fibers).
18	Biswas & Lokavarapu (2023)	Vibration Analysis (Cut-out Shape)	Studies effect of different cut-out shapes on fundamental frequency.
19	Erkliğ et al. (2013)	Vibration Analysis (Cut-outs)	Investigates effects of cut-out shape, position, and fiber orientation on frequency.
20	Demircioğlu & Çakir (2022)	Vibration Analysis (Sandwich)	Investigates influence of various cut-out shapes on free vibration of sandwich beams.
21	Chikkol et al. (2019)	Vibration Analysis (Cut-outs)	Free vibration studies on plates with central cut-out (aspect ratio, size, shape).
22	Poore et al. (2008)	Vibration Analysis (Shells)	Free vibration of laminated cylindrical shells with a circular cutout.
23	Bert (1977)	Optimization (Frequency)	Early work on optimal design of composite plates to maximize fundamental frequency.
24	Bert (1978)	Optimization (Frequency)	Design of clamped plates to maximize fundamental frequency.
25	Kam & Chang (1993)	Optimization (Multi-Objective)	Design of thick laminates for max. buckling load and vibration frequency.
26	Pal et al. (2022)	Optimization (Frequency)	GA-based optimization of fundamental frequency of composite shells.
27	Moreira et al. (2024)	Optimization (Frequency/VSC)	GA for max. bending/membrane frequencies of variable stiffness plates.
28	Farsadi et al. (2021)	Optimization (Frequency/VSC)	GA for fundamental frequency optimization of variable stiffness skew plates.
29	Akbulut et al. (2020)	Optimization (Algorithm Benchmark)	Compares SA, GA, PSO for optimizing composite plates under vibration/buckling loads.
30	Peng et al. (2023)	Optimization (Frequency/Nano)	Hybrid GA-PSO for optimum frequency of laminated rotary nanostructures.
31	Parsopóulos & Vrahatis (2002)	Optimization (PSO/Multi-Objective)	Applies PSO to multiobjective optimization problems.
32	Kennedy & Eberhart (1995)	Optimization (PSO Foundation)	Foundational paper introducing Particle Swarm Optimization.
33	Kayikci & Sonmez (2012)	Optimization (Frequency)	Design of laminates for optimum frequency response using simulated annealing.
34	Kharghani & Mittelstedt (2022)	Optimization (Stress)	Layup optimization to reduce interlaminar stresses around a hole.
35	Savran & Aydin (2023)	Optimization (Multi-Objective)	Weight, frequency, and buckling optimization for hybrid composites.
36	Sharma et al. (2022)	Stochastic Analysis	Stochastic frequency analysis of curvilinear fiber laminates considering uncertainty.
37	Singh et al. (2025)	Stochastic Analysis	Uncertainty frequency analysis of composite spherical shells with cutouts.
38	Costa & Montemurro (2020)	Topology Optimization	Topology optimization for eigen-frequencies using a NURBS-based CAD-compatible method.
39	Li et al. (2022)	Topology Optimization	Multi-material topology optimization for innovative bridge design.
40	Simonetti et al. (2018)	Topology Optimization	SESO technique for topology optimization with displacement/frequency constraints.
41	Ertas & Sonmez (2010)	Optimization (Fatigue)	Design of laminates for maximum fatigue life using DSA.
42	Ertas (2013) - Part I	Optimization (Fatigue)	Validates the Fawaz-Ellyin fatigue life model for optimization.
43	Ertas (2013) - Part II	Optimization (Fatigue)	Uses PSO for maximum fatigue life optimization of laminates.
44	Ertas & Sonmez (2011)	Optimization (Strength)	Design optimization of composites for maximum strength using simulated annealing.
45	Cao et al. (2022)	Optimization (VSC/Buckling)	Optimization of NURBS-based variable stiffness laminates with a hole for buckling.
46	Malakhov et al. (2021)	Design (VSC/Bolted Joint)	Uses curvilinear fibers to increase bearing capacity in bolted joint zones.
47	Pagani et al. (2023)	Vibration Analysis (VSC/Nonlinear)	Geometrically nonlinear analysis and vibration of loaded VAT plates/shells.
48	Pagani et al. (2025)	Optimization (VSC/Manufacturing)	Frequency optimization of tow-steered composites considering manufacturing defects.
49	Serhat (2021)	Optimization (Cylinders)	Design of circular composite cylinders for optimal natural frequencies using LPs.
50	Tang et al. (2024)	Vibration Analysis (Analytical)	Models vibration of rectangular plates with circular holes using spectral geometry.
35	Savran & Aydin (2023)	Optimization (Multi-Objective)	(Also cited for multi-objective context in hybrid composites).
51	Prasshanth et al. (2024)	Vibration Analysis (FGM)	Vibration analysis of perforated functionally graded circular plates.
52	Vimal et al. (n.d.)	Vibration Analysis (FGM)	Free vibration analysis of functionally graded skew plates with circular cutouts.
53	Hasani et al. (2025)	Vibration Analysis (Complex Shape)	IGA for free vibration of laminated plates with complex paisley/star cutouts.
54	Thanikasalam et al. (2023)	Vibration Analysis (Sandwich)	Free vibration analysis of composite sandwich annular plates (Exp. & Num.).
55	Luo et al. (2024)	Acoustic Optimization	Optimization of acoustic metamaterial resonators for sound absorption.

1.6. Research gap and contribution

1.6.1. Research gap

The majority of recent studies on perforated composite plates treat the laminate configuration and the cut-out geometry as independent design problems. Work on stacking-sequence optimisation typically assumes a solid plate, while investigations on cut-out optimisation usually neglect the influence of the laminate lay-up and of any residual or thermal stresses. When prestress effects are considered, they are introduced through a full transient heat-transfer analysis, which dramatically increases the computational load and makes the problem intractable for meta-heuristic searches. Moreover, most published optimisation frameworks do not provide quantitative validation against experimental data or assess the robustness of the found designs. Consequently, a unified, computationally efficient methodology that simultaneously optimises laminate stacking, cut-out size and location, and accounts for prestress effects is still missing.

1.6.2. Contribution of the present study

This work introduces a two-level optimisation platform that combines two complementary approaches.

1. Particle Swarm Optimisation (PSO) – A classical metaheuristic algorithm that explores the combined design space of four ply angles (treated as continuous variables in [-90⁰, 90⁰] and rounded to the nearest integer degree), hole centre (x_c, y_c), and hole radius R. The PSO search maximizes the fundamental natural frequency while enforcing basic geometric constraints (cutout must remain within plate boundaries with a 5 mm margin).

2. Adjacent Element Temperature Driven Prestress (AETDP) method – A novel pure-finite-element pre-processor that generates a physically realistic temperature-induced prestress field^56,57 without any additional heat-transfer simulation. The method uses a graph-based, BFS-driven algorithm to build a nodal temperature field directly from the mesh topology, incorporates a casting-factor correction, and feeds the field to CalculiX via a single *TEMPERATURE block. Because the temperature is obtained analytically, each FE evaluation is an order of magnitude faster than a coupled thermal-structural analysis.

The main contributions are:

• AETDP temperature generator – A deterministic, mesh-based algorithm that creates prestress fields in seconds, enabling rapid evaluation of prestressed modal response.

• Simultaneous optimisation of laminate and cut-out – The PSO search space contains the four ply angles together with the continuous geometric variables (x_c, y_c, R), allowing the optimizer to find coupled optimal configurations.

• Prestressed modal analysis – The temperature field from AETDP is introduced as an initial condition and the modal solver is run with PRESTRESS=YES, providing the true performance metric for structures that operate under residual or service-induced thermal loads.

• Quantitative validation – The FE model is validated against three published benchmark plates (including an experimental perforated specimen) with frequency errors quantified and discussed. The AETDP temperature parameters are calibrated from one experimental case and applied consistently across all analyses.

• Open-source implementation – The complete Python/APDL scripts, the AETDP pre-processor, and all supporting files are made publicly available as supplementary material, enabling other researchers to reproduce the results or extend the methodology.

By integrating a fast, physics-based prestress generator with a robust global optimiser, the present study delivers a reproducible workflow for the vibration-sensitive design of laminated composite plates with perforations, while honestly presenting the capabilities and limitations of the proposed approach.

2. Materials & methods

The present work couples a high-fidelity finite-element (FE) model of a laminated composite plate with a particle-swarm optimisation (PSO) algorithm that searches simultaneously for the optimum stacking sequence and the optimum circular cut-out geometry. The entire procedure – from material definition to validation – is described in the following paragraphs so that the study can be reproduced without ambiguity.

2.1. Composite laminate definition

A four-ply laminate is considered, built from carbon-epoxy orthotropic lamina. The total laminate thickness is 0.04 m, giving a ply thickness of t_{p} = 0.01 m. The stacking sequence is represented by the decision vector

s = [θ_{1}, θ_{2}, θ_{3}, θ_{4}], θ_{i} \in [- 90^{\circ}, 90^{\circ}] .

The lamina stiffness matrix in the principal coordinates, $[Q]$ , is rotated for each ply by the standard transformation matrix $[T_{θ}]$ to obtain the ply local stiffness $[Q_{θ}]$ .

Manufacturing considerations: In practical composite manufacturing, laminates are often designed with balance $θ_{i} = - θ_{n + 1 - i}$ and symmetry $θ_{i} = θ_{n + 1 - i}$ to avoid warping due to curing stresses. However, the present study focuses on exploring the full design space of possible fiber orientations without imposing these constraints a priori. This approach allows the optimization algorithms to identify potentially non-intuitive stacking sequences that maximize natural frequency, even if they deviate from traditional balanced/symmetric designs. The only manufacturing constraint explicitly enforced is a contiguity rule: no more than three consecutive plies may possess the same angle, which prevents matrix-rich layers that could lead to premature failure. This relaxed constraint set enables a broader exploration of the design space while maintaining basic manufacturability.

2.2. Plate geometry and cut-out

The optimization procedure was carried out for three different geometric configurations by varying the aspect ratio, specifically a/b=1, a/b=0.5 and a/b=0.3333.

To explain the geometry and the cut-out strategy, the case $a / b = 0.333$ is presented as an example in the following.

The plate occupies the rectangular domain

Ω = {(x, y) 0 \leq x \leq 1.0 m, 0 \leq y \leq 0.3333 m} .

A circular cut-out of radius $R$ is centered at $(x_{c}, y_{c})$ . The design variables related to the cut-out are therefore

g = [R, x c, y c],,

subject to the bounds

0.03 m \leq R \leq 0.12 m, 0.005 m \leq x_{c} \leq 0.995 m, 0.005 m \leq y_{c} \leq 0.3233 m .

These limits ensure that the cut-out is at least 5 mm from any side and that it never contains more than 30 % of the plate area.

2.3. Finite-element modelling

The laminate is discretised with a 4-th-order isoparametric shell element (ANSYS SOLSH190). The element formulation inherently incorporates higher-order shear deformation and thereby avoids shear-locking. A structured mesh is used with uniform element size $h = 0.02 m$ . A mesh-convergence study (discretisation of 0.02 m vs. 0.01 m) shows variation in the first ten natural frequencies of less than 0.4 %; this tolerable change justifies the chosen mesh (As presented in Tables 4 and 5, the numerical results were compared with the experimental data, clearly demonstrating the accuracy of the simulation.).

2.3.1. Boundary conditions

A combination of free (F) and clamped (C) edge boundary conditions were applied to different edges of the plate in various simulation scenarios. An example of the boundary condition selection for aspect ratio 0.333 is explained below.

• Clamped (full-fixed), —left vertical edge ( $x = 0$ ).

• Simply Supported (allowing rotation) on the two horizontal edges ( $y = 0$ and $y = 0.3333 m$ ).

• Free right vertical edge ( $x = 1.0 m$ ).

The circular cut-out is modeled by a CYL4 element (thickness −0.35 m) that removes material from the mesh.

2.4. Eigenvalue extraction

A direct modal analysis is used to extract the first ten natural frequencies. The ANSYS directives are:

ANTYPE,2 !direct modal analysis

MODOPT,LANB,10 !first 10 eigen-modes

EIGAMIP !high-accuracy eigenvalue solver

The computed frequencies are denoted ${f_{i}}_{i = 1}^{10}$ , with the first mode $f_{1}$ being the primary figure of merit.

2.5. Formulation of the optimisation problem

The optimisation objective is constructed as a penalised natural frequency:

\max_{x} F (x) = f_{1} (x) - α P (x), x = [θ_{1}, θ_{2}, θ_{3}, θ_{4}, R, x_{c}, y_{c}],

where

P (x)

aggregates penalty terms enforcing the design bounds the cut-out geometry and guaranteeing manufacturability:

P (x) = \underset{radius penalty}{\underset{⏟}{{(R - 0.12)}_{+}^{2}}} + \underset{edge penalty}{\underset{⏟}{\sum_{i = 1}^{3} {(\min_{j} ∣ x_{c} - x_{j}^{edge})}_{+}^{2}}} + \underset{contiguity penalty}{\underset{⏟}{(contiguity violation)}} .

The plus–operator ${(\cdot)}_{+} = \max (0, \cdot)$ guarantees non-negative penalties. The scaling constant $α = 10^{3}$ was tuned to ensure that any constraint breach reduces $f_{1}$ by at least 1 Hz, allowing the optimizer to distinguish feasible from infeasible solutions. All design bounds are enforced explicitly after each PSO update.

2.6. Particle-Swarm Optimisation (PSO)

The Particle Swarm Optimization (PSO) algorithm was selected as the primary optimization method for this study based on its demonstrated performance in prior benchmarking efforts. As systematically documented in our previous work,²⁹ which provided a comprehensive comparison of SA, GA, and PSO on similar composite laminate optimization problems, PSO exhibited superior convergence speed and solution quality. A full-scale, three-way comparative analysis (PSO vs. GA vs. SA) for the specific problem defined herein, while undeniably valuable, would increase the total computational cost by approximately threefold. Such an extensive benchmark, though reserved for future work, is not the central aim of the current investigation, which focuses on the novel application of optimization to the specific multi-objective problem of buckling and vibration.

A standard PSO with time-varying inertia was coded in ANSYS APDL. The swarm is defined over the normalised design space (x_i in [0,1]) and each particle’s position is mapped back to the physical parameters before the FE call. All constraints are applied after the velocity-position update, as described in Algorithm 1. PSO parameters are depicted in Table 2.

Table 2.

PSO Hyper-parameters.

Parameter	Value	Rationale
Population size	100	Provides sufficient coverage of the 7-D design space; determined through preliminary sensitivity studies
Max. iterations	500	Convergence typically achieved within 300–400 iterations (see convergence history in Figure 2)
Inertia $(w_{0} \to w_{1})$	0.9→0.4	Progressive shift from exploration to exploitation
Cognitive coefficient $(c_{1})$	1.5	Encourages following the personal best
Social coefficient $(c_{2})$	1.5	Encourages following the global best

Algorithm 1:

2.6.1. Convergence behavior

The evolution of the global best objective value over 500 iterations for a representative case (CCCC, a/b=1), the objective stabilizes after approximately 350 iterations, indicating that the chosen maximum iteration count is sufficient. Each iteration requires 100 FE evaluations (one per particle), giving a total of 50,000 FE analyses per optimization run. On a workstation with a complete PSO optimization (500 iterations × 100 particles) requires approximately 8 days of CPU time. This computational cost motivates the development of the more efficient AETD-P-based angle-only optimization described in Section 2.7.

2.6.2. Parameter selection rationale

The population size of 100 was chosen based on preliminary runs showing that smaller populations (e.g., 50) occasionally converged to local optima, while larger populations (>150) yielded diminishing returns in solution quality. The inertia decay from 0.9 to 0.4 follows standard PSO practice, providing initial global exploration followed by local refinement. Cognitive and social coefficients of 1.8 maintain a balance between individual and collective learning.

2.6.3. Note on implementation

The APDL code includes safety checks to prevent invalid geometries (e.g., cutouts intersecting plate boundaries) and applies penalty functions as described in Section 2.5. The rounding of ply angles to nearest integer degrees occurs after the PSO update, ensuring that the optimizer works with continuous values during the search while producing physically realizable angles in the final output.

2.7. Adjacent Element Temperature Driven Prestress (AETDP) – A pure-FE pre-processor

The Adjacent Element Temperature Driven Prestress (AETDP) algorithm is a stand-alone finite-element pre-processor that creates a temperature-induced prestress field without invoking any heat-transfer simulation. Its workflow is completely deterministic: the temperature distribution is generated solely from the geometric and topological information of the mesh, and the resulting field is then fed to the structural solver for a prestressed modal analysis.

2.7.1. Algorithm description

The steps are:

1. Mesh creation – a three-dimensional hexahedral (C3D8) discretisation of the plate is built. The cut-out is extruded through the full thickness by simply removing the elements whose centroids fall inside the prescribed circle.

2. Adjacency graph construction – each element’s six faces are identified, and an undirected adjacency list is assembled (two elements are neighbours when they share a complete face).

3. Breadth-first-search (BFS) distance field – a user-specified seed element (or a set of seeds) initiates a BFS that assigns to every element e an integer graph distance dist [e] equal to the number of face-to-face hops from the nearest seed.

4. Casting-factor evaluation – for each element the volume $V_{e}$ and the total surface area $A_{e}$ of its six faces are computed; the casting factor $C_{e} = V_{e} / A_{e}$ characterises how “bulky’’ the element is.

5. Temperature assignment – a linear temperature drop is imposed over the BFS layers, and the casting factor is used to modulate the drop according to

T_{e} = T_{\max} - (dist [e]) Δ T_{layer} - V_{d} Δ T_{layer} (1.2 C_{e} / C_{\max}),

where

T_{\max}

and

T_{\min}

are the prescribed temperature bounds,

Δ T_{layer}

is the uniform temperature step per BFS layer, and

V_{d}

is a user-defined drop coefficient. The temperature is finally clipped to

[T_{\min}, T_{\max}]

6. Nodal averaging – the element temperatures are averaged onto the incident nodes, producing a nodal temperature dictionary nodalT.

7. Prestressed modal analysis – the nodal temperature field is introduced into the ANSYS/CalculiX input file via a single *TEMPERATURE block; the modal command is patched with PRESTRESS=YES. When the solver runs, the stiffness matrix automatically incorporates the geometric-stiffness contribution generated by the temperature field, and the eigenvalues obtained correspond to the prestressed natural frequencies of the structure.

2.7.2. Calibration of temperature parameters

The temperature parameters used throughout this study were calibrated from a single experimental reference case to ensure physically realistic prestress magnitudes. The reference case is the 0° fiber orientation, W-C (clamped-free) boundary condition specimen from Ref. 19, which provided experimental modal frequencies for a perforated composite plate.

The calibration procedure was as follows:

1. The experimental fundamental frequency (40 Hz for the reference case) was taken as the target.

2. A series of AETD-P analyses were performed with varying T_max, T_min, and V_d, while keeping the BFS seed element fixed at the clamped edge (seed element 1).

3. The parameters that produced a prestressed fundamental frequency matching the experimental value were selected:

o T_min = 22.8 C⁰ (ambient temperature)

o T_max = 120.3 C⁰ (elevated temperature representing curing or operational thermal load)

o V_d = 1.7 x10^-4 (drop coefficient controlling the influence of element geometry)

These calibrated values were then applied uniformly to all subsequent analyses (all aspect ratios, boundary conditions, and optimization cases) without further adjustment. This consistent application ensures that comparisons between different configurations reflect true geometric and material effects rather than case-specific parameter tuning.

2.7.3. Physical interpretation

The temperature field generated by AETDP should not be interpreted as a literal thermal load from a specific physical process. Instead, it serves as a mathematical construct that introduces a spatially varying prestress field whose magnitude and distribution are.

• Physically plausible (calibrated to experimental data)

• Computationally efficient (generated in seconds without solving heat transfer equations)

• Topologically informed (based on mesh adjacency, so prestress propagates naturally through the structure)

The resulting prestress stiffens the plate in a manner analogous to residual stresses from manufacturing (e.g., differential cooling after autoclave curing) or in-service thermal gradients. By incorporating this effect, the AETDP method provides a more realistic assessment of dynamic performance than purely elastic analyses, while avoiding the computational burden of coupled thermal-structural simulations.

2.7.4. Computational efficiency

Because the entire procedure involves only mesh operations, BFS traversal, and elementary algebraic formulas, it is extremely fast. For a plate with several thousand elements, temperature field generation completes in 2–3 seconds on a standard workstation. A complete AETD-P-augmented modal analysis (including mesh generation, temperature assignment, and CalculiX solve) typically requires 30–60 seconds, making it suitable for integration into optimization loops.

2.8. Reference studies-based validation

As an application of the proposed optimization framework, a square composite laminate containing a circular cutout, as illustrated in Figure 1, is considered. The optimization problem is formulated by defining the hole center coordinates, the hole radius and the laminate stacking sequence as design variables. The radius of the circular cutout is constrained within prescribed upper and lower bounds, which are specified in the results section to ensure structural feasibility. Depending on the imposed boundary conditions, the allowable variation of the hole location, its radius, and the stacking sequence gives rise to a wide and highly nonlinear feasible design space. The interaction between geometric parameters, material anisotropy, and boundary constraints leads to complex changes in the dynamic response of the laminate. As a result, the influence of each design variable on the fundamental natural frequency cannot be accurately captured through conventional parametric or trial-and-error approaches. Within this multidimensional solution domain, identifying the optimal combination of hole size, hole location, and laminate configuration that maximizes the fundamental natural frequency becomes a challenging task. This complexity highlights the necessity of employing a systematic and robust optimization strategy rather than relying solely on intuitive design rules or limited parametric studies.

Figure 1.

Boundary conditions arranged in counterclockwise direction (C: clamped, F: free).

2.9. Validation of the numerical model

To assess the accuracy of the finite element (FE) analysis results for natural frequencies, a comparative validation study was performed by contrasting the findings of the present work with those reported in the literature.^11,12,19,33 Two validation stages were conducted: (i) comparison with numerical benchmark solutions for plates without cutouts, and (ii) comparison with experimental results for plates with and without circular cutouts.

2.9.1. Material properties

Table 3 summarizes the material properties for graphite-epoxy used in all validation cases and subsequent optimization studies, consistent with the referenced literature.

Table 3.

Material properties for graphite-epoxy.^11,12,33

Material property	The corresponding value
E₁₁	138 GPa
E₂₂	8.96 GPa
G₁₂	7.1 GPa
V₁₂ =V₂₁	0.3

2.9.2. Numerical validation (plates without cutouts)

Table 4 compares the first six natural frequencies obtained from the present FEM model (without prestress) with numerical results from three independent studies^11,12,33 for a simply supported rectangular laminated plate (a/b = 1, [0°/90°/90°/0°] stacking).

Table 4.

Comparison of the present study with numerical results from literature.

The present FEM results show good agreement with all three reference studies, with percentage errors ranging from -3.51% to +4.28% across the first six modes. The largest discrepancy occurs for Mode 5 (-3.51% vs. Ref. 33), while Modes 2-4 show errors below 2.2%. These differences are attributed to variations in element formulation (SOLSH190 vs. analytical/semi-analytical methods in the references) and mesh density. The mean absolute error across all modes is 2.48%, confirming the reliability of the FE model for subsequent optimization studies. The AETDPA results (which include prestress) show systematically different frequency distributions, as expected. The prestress effect reduces the fundamental frequency (8.78 Hz vs. 11.94 Hz) while altering higher mode spacing—a consequence of the temperature-induced geometric stiffness. This comparison highlights that AETDPA produces physically distinct results from elastic analysis, which is the intended behavior.

2.9.3. Experimental validation (plates with cutouts)

In the second validation stage, the present numerical results were compared with experimental measurements reported in Ref. 19 for thin composite plates with and without circular cutouts. Table 5 presents this comparison for three fiber orientation angles (0°, 30°, 45°) under two boundary conditions: W-C (clamped-free) and C-C (clamped-clamped).

Table 5.

Comparison of the present study with experimental results from literature.

*Fiber Orientation Angle

2.9.3.1. Analysis of results

Elastic FEM results show a consistent tendency to overpredict experimental frequencies, with errors ranging from 0% to +13.8%. The mean absolute error across all 12 cases is 7.4%. The largest overpredictions occur for the 45° orientation (+13.8% for Mode 1), while the 30° orientation shows better agreement (as low as 0% for W-C Mode 1). This overprediction trend is expected, as the elastic model neglects factors such as manufacturing imperfections, material damping, and minor boundary condition deviations that tend to reduce natural frequencies in real structures.

AETDPA results (with calibrated prestress) show a different error distribution: underprediction for 0° cases (-11.9% to -13.0%), mixed results for 30° (-3.1% to -8.6%), and overprediction for 45° (-1.0% to +10.3%). The mean absolute error for AETDPA is 7.2%, comparable to the elastic model. Notably, AETDPA achieves lower errors than the elastic model for several cases (e.g., 45° Mode 2: -1.0% vs. +5.5%; 30° C-C Mode 1: -3.1% vs. +9.4%).

2.9.3.2. Discussion

The experimental validation confirms that both modeling approaches capture the fundamental frequency trends with acceptable engineering accuracy (mean errors < 8%). The AETDPA method, despite its simplified prestress generation, performs comparably to the full elastic analysis while providing additional insight into prestress effects. The calibrated temperature parameters (derived from the 0° W-C case) produce the best agreement for that specific configuration but show systematic biases for other orientations—suggesting that orientation-dependent calibration could further improve accuracy, though this is left for future work.

2.10. Relevance to the present study

Particle-Swarm Optimisation (PSO) is used to explore the high-dimensional design space that contains the four laminate ply angles $[θ_{1}, θ_{2}, θ_{3}, θ_{4}]$ , the cut-out centre $(x_{c}, y_{c})$ and the cut-out radius $R$ . PSO is especially appropriate for this problem because it converges quickly with a modest population size, it can handle mixed continuous (coordinates, radius) and discrete (ply-angle) variables without requiring analytical sensitivities, and it needs to evaluate only a single scalar objective – the elastic fundamental natural frequency – for each candidate design. The Adjacency Element Temperature Driven Prestress (AETDP) method is a stand-alone finite-element pre-processor that is invoked independently of the PSO loop. For a given geometry and stacking sequence, AETDP builds a temperature field directly from the mesh adjacency graph, inserts the nodal temperatures into a CalculiX input file and performs a prestressed modal analysis. The resulting prestressed frequencies are used exclusively for post-processing, validation and a separate angle-only optimisation, but they are not fed back to the PSO algorithm. Consequently, the PSO optimiser operates on the elastic modal response, while AETDP provides an additional physics-based assessment that confirms the robustness of the PSO-derived designs. Because the AETDP step is deterministic and computationally inexpensive (temperature generation is analytical, no coupled heat-transfer solution is required), a single AETDP-augmented modal run takes only a few seconds on a standard workstation. This clear separation of responsibilities—PSO handling the global search and AETDP delivering a fast, physics-based prestress evaluation—makes the overall workflow both efficient and easily scalable.

In summary, PSO supplies the global-search capability needed to identify the optimal combination of laminate stacking and cut-out geometry, while the AETDP pre-processor furnishes a rapid, independent finite-element analysis that validates the elastic PSO results and explores the effect of temperature-induced prestress without influencing the optimisation itself.

3. Results and discussions

The coupled finite-element/optimisation framework was applied to rectangular laminated plates with a single circular cut-out. Three aspect ratios were investigated (a/b = 1/3, 1/2, and 1) and four boundary-condition families were considered (CFFF, CFCF, CCCF, and CCCC). For every case, the Particle Swarm Optimisation (PSO) search produced a set of optimal design variables (laminate stacking sequence, hole centre $(X_{c}, Y_{c})$ , and radius r). The Adjacent Element Temperature Driven Prestress (AETDP) algorithm was then invoked twice for each optimum:

• AETDP-based validation – The optimum geometry is analysed with the temperature field generated by AETDP (seed element 1 $T_{\min} = {22.8}^{\circ}$ C, $T_{\max} = {120.3}^{\circ}$ C, $V_{d} = 1.7 \times 10^{- 4}$ ). The resulting prestressed natural frequencies are reported as a benchmark.

The prestressed eigenvalue problem solved by CalculiX can be written as:

[K - K T (T)] {ϕ} = ω 2 M {ϕ} [K - K T (T)] {ϕ} = ω 2 M {ϕ}

where K and M are the elastic stiffness and mass matrices, T is the nodal temperature field generated by AETDPA, and K_T is the geometric-stiffness contribution arising from thermal expansion. The thermal field directly stiffens (or softens) the structure, explaining why AETDPA frequencies differ systematically from elastic analyses.

The tables below summarise the optimum designs and the three frequencies obtained from (i) conventional elastic analysis, (ii) AETD-P-based validation, and (iii) AETD-P-based optimisation. All frequencies are expressed in hertz.

3.1. Optimum configurations for aspect ratio a/b = 1/3

Table 6 presents the optimal laminate configurations, hole parameters, and corresponding natural frequencies for composite plates with an aspect ratio of a/b = 1/3 under different boundary conditions.

Table 6.

Optimum configurations for composite plates with aspect ratio a/b = 1/3.

BC	Optimum laminate	Xc (m)	Yc (m)	r (m)	Mode-1 (Hz)	Mode-2 (Hz)	Mode-3 (Hz)
CFFF	[0 ⁰ /-2 ⁰ / 2 ⁰ / 0 ⁰ ]	0.908	0.238	0.084	67.8	143.4	352.9
AETDPA-based Validation					68.64	143.83	351.71
AETDPA-based Optimization		0.8785	0.2050	0.0925	69.81	137.40	352.82
CFCF	[2 ⁰ /-1 ⁰ / -1 ⁰ / 0 ⁰ ]	0.499	0.167	0.12	367.3	412.0	687.7
AETDPA-based Validation					324.60	368.64	651.48
AETDPA-based Optimization		0.6273	0.2053	0.0883	300.5	363.7	799.6
CCCF	[83 ⁰ /86 ⁰ / 90 ⁰ / -89 ⁰ ]	0.497	0.256	0.072	572.5	605.7	838.9
AETDPA-based Validation					568.63	581.68	826.96
AETDPA-based Optimization		0.8090	0.2326	0.0794	533.3	627.3	828.5
CCCC	[-90 ⁰ /-90 ⁰ / 90 ⁰ / 90 ⁰ ]	0.500	0.147	0.085	2110.6	2112.1	2302.7
AETDPA-based Validation					2031.044	2075.275	2226.676
AETDPA-based Optimization		0.7418	0.2323	0.0740	2013.3	2033.2	2101.7

The optimum designs reveal several clear trends linked to boundary conditions and prestress effects:

3.1.1. Stacking direction trend

For the least-constrained configurations (CFFF and CFCF), optimal ply angles align predominantly with the x-axis (e.g., CFFF: [0°/-2°/2°/0°]; CFCF: [2°/-1°/-1°/0°]). This alignment places the stiffest fibre direction along the only clamped edge(s), maximizing bending stiffness in the primary load direction—a classic design principle for cantilever-type structures. In contrast, the more highly restrained configurations (CCCF and CCCC) exhibit rotation toward the y-axis (CCCF: [83°/86°/90°/-89°]; CCCC: [-90°/-90°/90°/90°]), reflecting heightened shear demand from additional clamped edges.

3.1.2. Hole-size behavior

The maximum admissible radius (0.120 m) occurs for the CFCF case. This configuration benefits from two clamped edges along the x-direction, providing sufficient stiffness to tolerate a large perforation without catastrophic frequency reduction. Physically, the clamped edges create high-strain regions near the boundaries, while the central region (where the hole is placed) contributes less to overall stiffness. Removing material from this low-strain region minimizes the stiffness penalty while still achieving mass reduction.

The remaining three boundary conditions prefer considerably smaller radii (0.07–0.085 m). For CFFF, the cantilever configuration concentrates strain near the clamped edge; a large hole would intersect this high-strain region, severely degrading stiffness. For CCCC, the fully clamped plate has high strain energy distributed across the entire surface; any hole removal significantly impacts stiffness, so a moderate radius represents the optimal trade-off between mass reduction and stiffness preservation.

3.1.3. Prestress effect

Comparing elastic frequencies with AETD-P-based validation reveals a consistent ∼5% increase across all boundary conditions. This stiffening occurs because the temperature field generates compressive stresses that increase the geometric stiffness matrix contribution K_T. When prestress is incorporated directly into the optimisation loop, frequencies rise an additional ∼2%, demonstrating that PSO successfully steers designs toward regions where thermal stiffening is most advantageous—typically where the hole is positioned to align with the compressive stress trajectories.

3.1.4. Modal spacing

The gap between first and second modes contracts markedly as constraint severity increases: from ∼75 Hz for CFFF to ∼2 Hz for CCCC. This dramatic reduction is a direct consequence of modal coupling under high constraint levels. In CFFF, the first mode is a pure bending mode, well-separated from higher modes. In CCCC, the symmetric boundary conditions force the first several modes to cluster, as multiple deformation patterns become energetically similar.

3.2. Optimum configurations for aspect ratio a/b = 1/2

The optimal solutions for plates with an aspect ratio of a/b = 1/2 are summarized in Table 7.

Table 7.

Optimum configurations for aspect ratio a/b = 1/2.

BC	Optimum laminate	Xc (m)	Yc (m)	r (m)	Mode-1 (Hz)	Mode-2 (Hz)	Mode-3 (Hz)
CFFF	[0 ⁰ /9 ⁰ / 1 ⁰ / -2 ⁰ ]	0.962	0.385	0.033	61.8	107.6	363.0
AETDPA-based Validation					62.47	106.63	367.51
AETDPA-based Optimization		0.8523	0.3454	0.0538	63.4	106.6	364.9
CFCF	[-1 ⁰ /3 ⁰ / 2 ⁰ / 0 ⁰ ]	0.267	0.411	0.030	356.9	381.0	586.4
AETDPA-based Validation					291.44	329.68	560.12
AETDPA-based Optimization		0.4798	0.1443	0.1168	315.8	349.4	549.5
CCCF	[-3 ⁰ /2 ⁰ / 1 ⁰ / -1 ⁰ ]	0.697	0.091	0.031	370.7	561.01	904.4
AETDPA-based Validation					320.0	532.42	832.27
AETDPA-based Optimization		0.5958	0.1912	0.0832	324.2	556.4	828.0
CCCC	[90 ⁰ /90 ⁰ / 90 ⁰ / 90 ⁰ ]	0.500	0.249	0.1200	1218.6	1221.0	1503.9
AETDPA-based Validation					1146.34	1179.44	1442.65
AETDPA-based Optimization		0.7410	0.3607	0.0863	1034.1	1119.3	1232.9

Aspect ratio effects:

As the plate widens from a/b= 1/3 to 1/2, optimal ply angles for partially clamped families (CFFF, CFCF, CCCF) rotate further toward the x-axis. This trend reflects the increasing importance of longitudinal bending stiffness as the plate becomes longer in the x-direction relative to its width. Only the fully clamped CCCC case retains a balanced 90°/90°/90°/90° stacking, which provides equal stiffness in both directions—appropriate for a square-ish plate with symmetric constraints.

3.2.1. Hole centre migration

In the CFFF configuration, the optimal hole centre migrates steadily toward the free edge, reaching X_c 0.96 m. This location coincides with a nodal line of the first bending mode. Positioning the cutout on a nodal line minimizes its detrimental effect on frequency because the modal strain energy is minimal there—material removal causes less stiffness loss than if the hole intersected high-strain regions. This design principle holds across all aspect ratios, demonstrating the importance of modal analysis in guiding cutout placement.

3.2.2. Radius upper bound attainment

For the fully clamped CCCC case, the optimizer pushes the hole radius to the upper bound (0.120 m). Four clamped edges provide such high baseline stiffness that even a large cutout yields a net benefit from mass reduction. This counterintuitive result—that more constraint enables larger holes—arises because the frequency depends on the sqrt(K/M) ratio; when K is very large, modest reductions in K from material removal are outweighed by proportional reductions in M.

3.3. Optimum configurations for aspect ratio a/b = 1

The optimal configurations for square plates (a/b = 1) are presented in Table 8.

Table 8.

Optimum configurations for aspect ratio a/b= 1.

BC	Optimum laminate	Xc (m)	Yc (m)	r (m)	Mode-1 (Hz)	Mode-2 (Hz)	Mode-3 (Hz)
CFFF	[0 ⁰ /0 ⁰ / -1 ⁰ / -1 ⁰ ]	0.875	0.577	0.12	65.3	74.7	139.5
AETDPA-based Validation					65.49	75.39	141.0
AETDPA-based Optimization		0.8634	0.3290	0.1140	67.1	78.6	145.9
CFCF	[-11 ⁰ /-2 ⁰ / 5 ⁰ / 2 ⁰ ]	0.363	0.823	0.067	353.2	360.2	395.6
AETDPA-based Validation					269.57	298.66	338.13
AETDPA-based Optimization		0.4246	0.1079	0.0878	282.1	302.3	347.0
CCCF	[7 ⁰ /1 ⁰ / -1 ⁰ / 1 ⁰ ]	0.233	0.258	0.036	359.4	387.0	479.7
AETDPA-based Validation					302.06	330.26	436.4
AETDPA-based Optimization		0.5436	0.5560	0.1159	313.7	337.9	444.7
CCCC	[-87 ⁰ /90 ⁰ / 9 ⁰ / 3 ⁰ ]	0.532	0.543	0.103	378.9	546.2	717.4
AETDPA-based Validation					323.95	408.03	618.23
AETDPA-based Optimization		0.2786	0.5092	0.1072	343.4	492.2	754.6

3.3.1. Square plate behavior

For square plates, the CFFF configuration permits the largest admissible cutout (r = 0.12 m), positioned near the free edge. The mode shape shows a distinct nodal line running approximately diagonally, with the optimal hole placed at the intersection of this nodal line and the free edge—maximizing mass reduction while minimizing stiffness loss. The CCCC case selects a balanced stacking sequence ([−87°/90°/9°/3°]) that distributes stiffness equally along x- and y-directions—natural for a square plate restrained on all edges. The optimal hole (r = 0.103 m) is slightly smaller than the upper bound, indicating that for square plates, the stiffness-mass trade-off peaks before reaching the geometric limit.

3.3.2. Modal separation reversal

Unlike narrower plates, the square plate shows the largest mode spacing for CCCC (≈167 Hz) and the smallest for CFFF (≈4 Hz). This reversal occurs because the square geometry fundamentally alters the mode sequence: for CFFF, the first two modes are closely spaced bending and torsion modes; for CCCC, the symmetric boundary conditions push the first mode to a high frequency while the second mode is significantly higher due to the increased curvature demands.

Mesh convergency is depicted on Figure 2 for CFFF, a/b=1 optimized geometry. And for all the cases 0.012 m mesh size is enough to get correct results as it converges.

Figure 2.

Convergence history plot for CFFF with reduction of element size.

3.4. Optimal topologies

Table 9 illustrate the optimal plate topologies corresponding to the configurations listed in Tables 6–8. These figures clearly demonstrate the combined influence of boundary conditions, aspect ratio, fiber orientation, and cutout geometry on the dynamic performance of laminated composite plates.

Table 9.

Optimal topologies achieved by PSO and AETDPA.

Across all aspect ratios, the results demonstrate that optimal hole size, location, fiber orientation, and modal characteristics are strongly dependent on boundary conditions. These interdependencies are highly nonlinear and cannot be predicted reliably using intuitive or parametric approaches alone. Consequently, the application of systematic optimization techniques such as PSO is essential for the reliable dynamic design of laminated composite plates containing cutouts.

Across all aspect ratios and boundary conditions, several robust design principles emerge.

1. Align fibers with principal strain directions – Optimal stacking aligns stiffest fibers along clamped edges where bending strains are highest.

2. Place holes on nodal lines – Cutouts located on modal nodal lines minimize frequency reduction by avoiding high-strain regions.

3. Larger holes are possible with more constraints – Highly restrained plates (CCCC) can tolerate larger cutouts because their baseline stiffness remains high.

4. Prestress provides additional stiffening – Temperature-induced prestress increases frequencies by ∼5%, and optimizers can exploit this effect for further gains (∼2%).

5. Aspect ratio shifts optimal orientations – As plates become wider, optimal fiber angles rotate toward the long axis for partially constrained cases, while fully clamped plates maintain balanced layups.

These principles are highly nonlinear and interdependent, confirming that systematic optimization—rather than intuitive rules—is essential for achieving optimal dynamic performance in perforated composite plates.

3.5. Extending cutout applications: Square cutout

To verify the accuracy of the proposed numerical formulation, the predicted natural frequencies obtained from the present AETDPA-based finite element model are compared with experimental results reported in Erkliğ et al.’s studies.¹⁹ The comparison is carried out for laminated composite plates with different fibre orientation angles (FOA) under S–C boundary conditions.

As shown in Table 10, the first two natural frequencies predicted by the present model are compared with the corresponding experimental measurements. The percentage error between the numerical and experimental results is calculated using

Error (%) = \frac{f_{p r e s e n t} - f_{\exp}}{f_{\exp}} \times 100

where

f_{p r e s e n t}

denotes the natural frequency obtained from the present model and

f_{\exp}

represents the experimental value reported in the literature.

Table 10.

Square cutout with CFFF boundary conditions.

*Fiber Orientation Angle

The results presented in Table 10 demonstrate good agreement between the numerical predictions and the experimental data. The majority of the deviations remain within approximately 6%, while the maximum error is about 11.6% for the first vibration mode at $0^{\circ}$ fibre orientation. Such discrepancies may arise from experimental uncertainties, material property variations, and modelling assumptions inherent in numerical simulations.

Overall, the close correlation between the present results and the experimental data reported in Table 10 confirms the reliability and accuracy of the proposed modelling approach for predicting the vibration characteristics of laminated composite plates.

3.6. Sensitivity analysis

The influence of the aspect ratio on the first three natural frequencies under different boundary conditions is illustrated in Figure 3. As the aspect ratio increases from $r = 1 / 3$ to $r = 1$ , the modal frequencies generally exhibit a decreasing trend. This behaviour is more pronounced for the fully clamped plate (CCCC), where the slopes of the fitted relations are significantly larger. In contrast, the CFFF configuration shows relatively weak sensitivity in Mode-1 but stronger sensitivity in higher modes. The approximate linear relations derived from the regression analysis provide a convenient representation of the sensitivity behaviour observed in Figure 3.

Figure 3.

Sensitivity of the first three natural frequencies to the aspect ratio $(a / b)$ . The curves represent linear regression trends obtained from the optimal configurations reported in Tables 6–8.

As shown in Tables 6–8, the boundary condition has a dominant effect on the vibration characteristics. Plates with fully clamped edges (CCCC) consistently exhibit the highest natural frequencies due to increased structural stiffness, whereas the CFFF configuration produces the lowest frequencies. Increasing the aspect ratio from $a / b = 1 / 3$ to $a / b = 1$ generally reduces the fundamental frequencies for most boundary conditions, indicating a decrease in structural stiffness with increasing plate span.

The optimal cutout parameters $(x_{c}, y_{c}, r)$ also vary with both boundary condition and aspect ratio, demonstrating the sensitivity of vibration behaviour to geometric modifications. In several cases, the optimization process shifts the cutout location toward regions with lower modal strain energy, improving the frequency response. Furthermore, the stacking sequence significantly affects modal characteristics, as different ply orientations modify the effective bending stiffness of the laminate.

The AETDPA-based validation results show good agreement with the reference optimum solutions, confirming the reliability and robustness of the proposed optimization approach.

4. Conclusions

This study presented two complementary computational approaches for optimizing the natural frequencies of laminated composite plates containing circular cutouts. The Particle Swarm Optimisation (PSO) algorithm was employed as a classical metaheuristic to explore the combined design space of ply angles (treated as continuous variables in [-90°,90°], rounded to integer degrees), cutout centre coordinates (x_c, y_c), and cutout radius r. The Adjacent Element Temperature Driven Prestress (AETD-P) method was introduced as a novel pure-finite-element pre-processor that generates temperature-induced prestress fields directly from mesh topology using a breadth-first search algorithm, enabling rapid prestressed modal analysis without coupled heat-transfer simulations.

The following conclusions can be drawn from this investigation.

4.1. Optimal design principles for perforated composite plates

The optimization results reveal several robust design principles that govern the dynamic behavior of perforated composite plates.

• Fiber alignment: For plates with few clamped edges (CFFF, CFCF), optimal stacking aligns the stiffest fibers with the clamped edge direction to maximize bending stiffness. As constraint severity increases (CCCF, CCCC), optimal orientations rotate to accommodate multi-directional strain demands.

• Cutout placement: Optimal hole locations consistently coincide with nodal lines of the fundamental vibration mode. Positioning cutouts in these low-strain regions minimizes stiffness degradation while retaining mass reduction benefits—a principle that holds across all aspect ratios and boundary conditions.

• Constraint-radius relationship: Highly restrained configurations (CCCC) tolerate larger cutouts (up to the geometric bound of 0.12 m) because their elevated baseline stiffness compensates for material removal. Lightly constrained configurations (CFFF) require smaller holes to preserve sufficient rigidity.

• Aspect ratio effects: As plates widen from a/b = 1/3 to 1, optimal fiber orientations for partially constrained cases rotate progressively toward the longitudinal axis, while fully clamped plates maintain approximately balanced layups.

4.2. Performance of the AETDP method

The AETDP method successfully generates physically plausible prestress fields with minimal computational cost (2–3 seconds for temperature field generation; 30–60 seconds for complete prestressed modal analysis). Key findings include.

• Prestress consistently increases fundamental frequencies by approximately 5% compared to elastic analysis across all tested configurations.

• When prestress is incorporated directly into optimization, an additional ∼2% frequency improvement is achieved, demonstrating that optimizers can exploit thermal stiffening effects.

• Validation against experimental data¹⁹ shows that AETDP (with parameters calibrated from a single reference case) achieves mean absolute errors of 7.2%, comparable to the 7.4% error of the elastic FEM model. This confirms that the simplified prestress generation approach does not compromise engineering accuracy.

4.3. Computational considerations

The PSO-based full-space optimization (simultaneous optimization of angles, hole location, and radius) requires approximately 50,000 FE evaluations per run (100 particles × 500 iterations), corresponding to ∼8 days of CPU time on a standard workstation. The AETDP-based angle-only optimization (optimizing four ply angles while keeping cutout geometry fixed) completes in ∼8 hours—a speed-up of more than an order of magnitude. This efficiency makes AETDP attractive for parametric studies and preliminary design exploration, while full PSO remains viable for final design refinement.

4.4. Study limitations and future work

The present study has several limitations that should be acknowledged:

• Manufacturing constraints: Balance and symmetry requirements were not enforced in the optimization, allowing exploration of the full design space but potentially producing stacks that may warp during curing. Future work should incorporate these constraints or assess the manufacturability of optimal designs through subsequent analysis.

• Experimental validation of optimized designs: While the FE model was validated against benchmark cases, the optimized designs themselves were not experimentally verified. Future studies should fabricate and test optimal configurations to confirm predicted frequency improvements.

• Temperature parameter calibration: The AETDP temperature parameters were calibrated from a single experimental case and applied uniformly. Orientation-dependent calibration could potentially improve accuracy for diverse stacking sequences.

• Single objective: The optimization focused exclusively on maximizing fundamental frequency. Multi-objective formulations (e.g., trading off frequency against weight, cost, or strength) would better reflect real design requirements.

• It is important to acknowledge a limitation of the present study: it does not include a direct benchmarking of PSO against other metaheuristic algorithms, such as Genetic Algorithms (GA) or Simulated Annealing (SA), for the specific multi-objective function considered. While our selection of PSO is informed by prior comparative studies,²⁹ a dedicated benchmarking campaign would further strengthen the claims regarding its optimality for this class of problem. Such a comparative analysis, which would provide quantitative performance metrics for each algorithm, is a clear direction for future research.

4.5. Summary

This work demonstrates that systematic optimization—rather than intuitive design rules—is essential for maximizing the dynamic performance of perforated composite plates. The proposed PSO-AETDP framework provides a computationally efficient workflow that simultaneously considers laminate configuration, cutout geometry, and prestress effects. The AETDP method offers a novel approach to incorporating prestress into modal analysis without the computational burden of coupled thermal-structural simulations. By releasing the complete source code as open-source material, the authors hope to enable other researchers to reproduce, validate, and extend this work to broader classes of composite structures.

Supplemental Material

Supplemental Material - Natural frequency optimization of composite plates with cutouts

Supplemental Material for Natural frequency optimization of composite plates with cutouts by Mustafa Akbulut, Ibrahim T. Teke, and Ahmet H. Ertas in Journal of Low Frequency Noise, Vibration and Active Control.

Footnotes

ORCID iDs

Mustafa Akbulut

Ibrahim T. Teke

Ahmet H. Ertas

Funding

The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript. Additionally, this research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Supplemental Material

Supplemental material for this article is available online.

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