Lamination Parameters Research Articles

An efficient isogeometric buckling optimization framework for multi-patch laminated shells: Leveraging 3D solid elements, lamination parameters and penalty method

Buckling is a common failure mode of laminated shell structures during service, and isogeometric analysis methods are highly suitable for the buckling analysis due to their accurate geometric modeling capability and high-order continuity. However, the isogeometric buckling analysis for a multi-patch structure and its optimal design still face challenges in computational efficiency. To address this issue, this paper proposes an integrated framework for the buckling optimization of multi-patch laminated shells, which leverages 3D solid elements, lamination parameters, and the penalty coupling method. Solid-shell elements have advantages in shell analysis due to their simplicity, absence of rotational degrees of freedom, and adaptive layering in the thickness direction. Lamination parameters simplify the optimization of fiber orientations in laminated shells by condensing the design variable space and improving the convexity of the optimization problem. The penalty method can handle the coupling in multiple patches at a relatively low cost. Nevertheless, lamination parameters are rarely applied to the case of solid-shell elements. In addition, despite using lamination parameters, the number of design variables for multi-patch structures remains significantly large. To tackle these issues, this study derives the solid-shell formulation in lamination parameters and the sensitivity analysis formulae in the reverse mode. Experiment validation is conducted to assess the accuracy of the proposed method based on lamination parameters with solid elements, the feasibility of the multi-patch optimization model, and the computational efficiency of optimization sensitivity based on the reverse mode compared to the forward counterpart common in the literature.

Quantum computing and tensor networks for laminate design: A novel approach to stacking sequence retrieval

As with many tasks in engineering, structural design frequently involves navigating complex and computationally expensive problems. A prime example is the weight optimization of laminated composite materials, which to this day remains a formidable task, due to an exponentially large configuration space and non-linear constraints. The rapidly developing field of quantum computation may offer novel approaches for addressing these intricate problems. However, before applying any quantum algorithm to a given problem, it must be translated into a form that is compatible with the underlying operations on a quantum computer. Our work specifically targets stacking sequence retrieval with lamination parameters, which is typically the second phase in a common bi-level optimization procedure for minimizing the weight of composite structures. To adapt stacking sequence retrieval for quantum computational methods, we map the possible stacking sequences onto a quantum state space. We further derive a linear operator, the Hamiltonian, within this state space that encapsulates the loss function inherent to the stacking sequence retrieval problem. Additionally, we demonstrate the incorporation of manufacturing constraints on stacking sequences as penalty terms in the Hamiltonian. This quantum representation is suitable for a variety of classical and quantum algorithms for finding the ground state of a quantum Hamiltonian. For a practical demonstration, we performed numerical state-vector simulations of two variational quantum algorithms and additionally chose a classical tensor network algorithm, the DMRG algorithm, to numerically validate our approach. For the DMRG algorithm, we derived a matrix product operator representation of the loss function Hamiltonian and the penalty terms. Although this work primarily concentrates on quantum computation, the application of tensor network algorithms presents a novel quantum-inspired approach for stacking sequence retrieval.

Reliability based optimisation of composite plates under aeroelastic constraints via adapted surrogate modelling and genetic algorithms

Composite materials are vital in aerospace for their exceptional strength-to-weight ratio. This study delves into reliability-based optimisation of composite plates within aeroelastic constraints, employing an efficient global optimisation method and genetic algorithms. Initial analysis focuses on aeroelastic responses such as limit flutter speed, gust response, and static loads, emphasising maximum strain assessment. To tackle optimisation challenges of composite stacking sequences, a homogenisation technique with lamination parameters is applied. We then formulate a constrained optimisation problem to minimise gust response while meeting flutter and maximum strain constraints. Surrogate models based on conditioned Gaussian Processes are developed for each aeroelastic response, facilitating optimisation within the composite design space. These models, with potential for local refinement, expedite optimal solution identification. Further, we integrate reliability-based optimisation into the framework to determine a robust stacking sequence using genetic algorithms, accounting for random fibre orientation variations. This holistic approach integrates aeroelastic analysis, constrained optimisation, surrogate modelling, and reliability-based optimisation, proving effective in designing reliable, efficient composite structures for aerospace, thus enhancing performance and safety.

Conical-shaped variable stiffness composite laminates: Design and fiber path planning

This study presents an innovative design methodology for conical-shaped variable stiffness composite laminates. It employs the spectral Chebyshev method to solve dynamic equations, optimizing lamination parameters to maximize fundamental frequency. Subsequently, the fiber angles at the sampling points across the domain are retrieved. In the final step, a custom normalized cut segmentation approach is proposed to partition the domain into clusters and delineate fiber paths based on discrete fiber angles, while satisfying the manufacturing constraints such as curvature, gaps, and overlaps. The effectiveness of the proposed design methodology is demonstrated through several case studies, where maximum enhancement in fundamental frequency for the same part weight is compared to optimized constant stiffness composite laminates. The results show that an enhancement of 16% is achievable for a variable stiffness laminate disregarding manufacturing constraints for the fully clamped case study. However, the improvements reduce to 12.6% and 14% (deviating from the ideal optimal fundamental frequency by 4.5% and 3.3%), respectively, once the manufacturable constant and curvilinear fibers in each cluster are computed. This work can lead significant advances of engineering applications by enabling the design of variable stiffness laminates with complex geometries improving mechanical performance and/or reducing structural weight compared to conventional laminates.

Efficient Design Methods for Variable-Angle-Tow Composite Toroidal Pressure Vessels

Pressure vessels are widely used in various aerospace and automotive applications. For some applications with limited space, spherical and ellipsoidal shapes are less attractive than toroids. Toroidal shapes can be defined as curved, endless hollow shapes that eliminate the need for end caps when used for pressure vessels. This study presents an analytical formulation using lamination parameters for the novel design of toroidal pressure vessels. The proposed design is based on stiffness tailoring by changing the fiber tow trajectories spatially through the structure to suppress the direct stress and strain gradients in the thickness direction, leading to reduced structural weight, defined as bend-free variable-angle-tow (VAT) toroidal pressure vessels. For a toroidal pressure vessel with a fixed geometry, the VAT distribution through the structure to suppress bending is obtained. Subsequently, the bending stresses and strain levels in the designed VAT toroidal pressure vessels are assessed numerically and compared with those of their isotropic and constant-stiffness composite counterparts. Results show that the designed VAT toroidal pressure vessel generates considerably lower degrees of bending than its corresponding isotropic and constant-stiffness composite counterparts.

A hybrid path-following approach for constrained nonlinear-buckling optimization of variable stiffness composite shells with shape imperfections

Nonlinear buckling optimization of variable stiffness (VS) composite shells is computationally expensive due to frequent nonlinear analyses. Although the generalized path-following method can serve to reduce the redundant computation, it is weakened by two kinds of discontinuity. First, the design variables may not always be searched continuously in the constrained optimization. Second, the critical buckling states may not be continuous at singular points on the search path. To address this issue, a hybrid path-following approach is proposed to trace the critical buckling states in the nonlinear-buckling optimization with constraints on the realizability of the lamination parameters and the amplitude range of geometric imperfections. Firstly, the optimal search direction is determined with the sensitivity analysis of the critical buckling equations in the isogeometric continuum shell model. Then, a continuous following strategy is adopted to provide a proper initial estimate for the Newton iterations at a disconnected point. Moreover, an investigation into the singularity of the critical buckling equations is conducted to identify the discontinuous points of the critical states in a line-search. Upon detection of the singularity, a segment of path in state space is inserted to recover the continuity. Numerical experiments show that the continuity of the critical states in the optimization is well maintained, which leads to a more efficient optimization for the VS shell with shape imperfections.

A novel spectral element method with a higher-order coarse quad meshing approach to design laminated composite panels with arbitrarily shaped cutouts

Laminated panels are widely used in various industries due to their distinct advantages. The design of laminated composites requires efficient methodologies due to the vast design space. This study proposes a new spectral element modeling approach for accurate and computationally efficient analysis of laminated composites with arbitrarily shaped cutouts. The method employs a coarse quad meshing approach and a spectral Chebyshev method to determine the element matrices. The presented method enables obtaining high-fidelity models by applying h- and p-refinements following a non-dimensional approach, aiming to yield a model with the fewest degrees of freedom to achieve a desired convergence level. Benefiting from the advantages of both meshless methods (in terms of computational efficiency) and finite element methods (in terms of geometric capabilities), we performed several case studies for laminated panels with cutouts and it is shown that the presented spectral element method (SEM) enables calculating the natural frequencies and the mode shapes as accurate as FEM, yet decreases the analysis duration by 13 folds. Furthermore, the developed approach was employed with a gradient-based optimizer or genetic algorithm to demonstrate the design of (sandwich) laminated composites for obtaining optimal lamination parameters and 2D Pareto fronts.

A new framework for optimizing energy harvesting from smart composites integrated with piezoelectric patches utilizing lamination parameters

Amongst various structures, 2D panels are widely used in many industries and in each application, there are many ambient vibrations which can be converted into electrical energy. The very efficient method to utilize can be using piezoelectric patches which are stuffiest enough for various applications like energy harvesting. This study presents a novel approach for investigating energy harvesting in smart structures using composite panels integrated with piezoelectric patches. The panels are chosen to symmetry-balanced laminated composites, and modal and harmonic analysis conducted using the Rayleigh-Ritz method. To compute the kinetic and potential energy components of the piezoelectric patches at a local level, piecewise Heaviside functions are employed, and these energy components are integrated into the equations of motion together with those of the host composite plate. The results of the numerical method are validated by a commercial finite element software (FEM) COMSOL and there is a good match between FEM and this paper results. By subjecting the laminated composite with piezoelectric patches to forced vibration while varying the lamination parameter, the power output is optimized. The findings emphasize the substantial impact of the lamination parameter on power output, indicating that modifications can result in significant power output increase.

Design of manufacturable variable stiffness composite laminates using spectral Chebyshev and normalized cut segmentation methods

The in-plane fiber orientations of variable stiffness (VS) laminates can be tailored to achieve enhanced structural properties compared to conventional constant stiffness (CS) laminates. However, VS laminate manufacturing faces challenges such as wrinkles, gaps, and overlaps. To address these challenges, we present a novel three-step design methodology. First, the laminate is modeled using lamination parameters (LPs) and the spectral Chebyshev method, and the optimal LPs are determined to maximize the fundamental frequency. Then, the discrete fiber angles are retrieved using the optimal LP distribution. Lastly, a normalized-cut segmentation method is applied to divide the domain into clusters and to generate manufacturable curvilinear fiber paths. Case studies focusing on designing clusters containing both straight and curvilinear fiber paths demonstrate that the designed VS composites can significantly enhance the dynamic performance with up to 20% enhancement in the fundamental frequency compared to CS laminates, under fully clamped boundary conditions with manufacturing constraints.

Mathematical modeling and dynamic analysis of carbon fiber reinforced plastic shaft-disk-bearing system considering internal damping effect

The internal damping has significant influence on the dynamic characteristics of carbon fiber reinforced plastic drive shaft. In this paper, a dynamic model of the single-span multi-disk carbon fiber reinforced plastic shaft with elastic support is established based on layer-wise theory, damping specific capacity model, and Euler–Bernoulli beam theory. The model takes into account physical factors such as rigid body displacement caused by elastic support deformation, gyroscopic effect of concentrated mass disk, viscous internal damping of the carbon fiber reinforced plastic shaft, and lamination parameters. The model is used to analyze the natural angular frequency and critical instability speed of carbon fiber reinforced plastic shaft-disk-bearing system. The analysis results are compared with those obtained from the equivalent modulus beam theory, equivalent single layer theory, simplified homogenized beam theory, and quadrature element method. All the results demonstrate the accuracy of the proposed model, thereby validating its effectiveness. Furthermore, the model is utilized to investigate the influence of lamination parameters and mass ratio on the natural angular frequency and critical instability speed. This study provides valuable insights for the dynamics and stability design of the carbon fiber reinforced plastic shaft-disk-bearing system.

A general framework for constructing the feasibility-constraint function of lamination parameters in optimization

Lamination parameters are extensively employed as design variables in the optimization design of composite laminates. The feasibility-constraint function of the lamination parameters in optimization is essential since it is related to the accuracy of the optimal results. In the existing literature, the feasible region for lamination parameters is expressed by multiple inequalities, which is inaccurate and unsmooth at some corners and tedious in the cases of high dimensions. This paper aims at proposing a general framework for constructing the feasibility-constraint functions in optimization for any given set of lamination parameters with higher precision compared to the functions given by the inequalities. The feasibility-constraint function is generated by training an artificial neural network with the backpropagation algorithm based on the idea of the level set method. The training data mainly comes from the boundary points shaped by the hyperplanes calculated by a variational method. The proposed framework consists of four parts: obtaining the hyperplanes, generating the boundary points of the feasible region, constructing the feasibility-constraint function, and evaluating the accuracy of the generated function. Several experiments are implemented to validate the proposed method in the cases of four-, six-, and eight-dimensional lamination parameters. The 4D results show that the generated feasibility-constraint functions have the advantage in precision compared to the existing constraint functions in literature in the real application of structural optimization. Besides, in the 6D and 8D cases, the smoothness and accuracy of the generated feasible boundaries are also shown. The performance of enforcing the generated constraint in optimization is also studied in all three cases, which indicates that the generated constraint functions are applicable in the optimization design of composite laminates.

Influence of Lamination Conditions of EVA Encapsulation on Photovoltaic Module Durability

Encapsulation is a well-known impact factor on the durability of Photovoltaics (PV) modules. Currently there is a lack of understanding on the relationship between lamination process and module durability. In this paper, the effects of different lamination parameters on the encapsulant stability due to stress testing have been investigated from both on-site production quality and long-term stability viewpoints. Rather than focusing on single stability factors, this paper evaluates lamination stability using a number of indicators including EVA (ethylene-vinyl acetate copolymer) curing level, voids generation, chemical stability, optical stability, and adhesion strength. The influences of EVA curing level on the stability of other properties are also discussed. It is shown that laminates stability increases with increasing curing level to an upper limit, beyond which leading to the formation of voids, reduced transmittance stability, discoloration, and unstable interfaces. A minimum gel content is identified but an upper limit should not be surpassed. The best range of gel content for the materials tested here is 84–90%. Samples with gel content below 70% show low chemical and optical stability, weak adhesion strength, and EVA flowing. Laminates with gel content over 92% are more likely to become yellow and are less stable in adhesion.

Buckling optimization of variable-stiffness composite plates via variable stiffness optimization algorithm

A variable stiffness optimization (VSO) algorithm is presented for optimizing variable-stiffness composite (VSC) plates with linear fiber path functions. A new definition of lamination parameters characterizing the distances between stacking configurations of VSC plates is presented. Using the inherent sensitivity of bending stiffness of composite laminates, the concept of “through thickness design” is introduced. The first step is to determine a good initial point for the VSC plates using three-dimensional sampling optimization (3DSO). The fiber angle design variables are then categorized into three groups. By optimizing the three groups' design variables sequentially and iteratively, the stiffness of the VSC plates is stiffened part by part until an optimum is reached. Lastly, the obtained optimum is redesigned accounting for the curvature constraint. The finite element method (FEM) is developed for the buckling analysis of VSC plates, and both Q4 and Q8 elements are employed to verify the accuracy and convergence of the FEM. Under a variety of boundary conditions and loading cases, FEM and VSO algorithm are used to maximize the buckling load of square and rectangular symmetrical VSC plates. Optimal results are compared with those available in the literature, demonstrating the effectiveness, robustness, and efficiency of the VSO algorithm.

Advanced lamination parameter interpolation and extrapolation methods for designing manufacturable variable stiffness laminates

The design of variable stiffness laminates requires efficient methodologies due to the increased number of optimization variables associated with curvilinear fiber paths. Here, this need is addressed by the development of two novel approaches: the lamination parameter extrapolation method (LPEM) and the relaxed lamination parameter interpolation method (RLPIM). These techniques build on the previously proposed lamination parameter interpolation method (LPIM), and collectively they form a spectrum of approaches that differ in optimization capacity, conservativeness regarding fiber curvature constraints, and computational cost. The resulting governing equations are solved using the spectral Chebyshev method to further improve the efficiency of the optimization process. The case studies demonstrate the effectiveness and unique properties of the developed algorithms.

Optimization of Hybrid Composite Laminates with Distinct Ply Thicknesses Using Integer Programming

Optimization of constant-stiffness composite laminates with discrete values of fiber orientations and ply thicknesses is a nonconvex combinatorial problem. Recent studies show that this problem can be efficiently solved using the mixed integer program (MIP) by reformulating lamination parameters as functions of decision variables that represent fiber orientation and thickness choices. These formulations are limited to analyzing laminates having plies of the same thickness and material. This paper extends the previously employed lamination parameter reformulation to enable laminate designs with plies of different thicknesses and materials, thereby leading to MIP with cubic constraints. The linearization techniques are employed to convert cubic constraints of buckling load factors in terms of binary variables into sets of linear constraints that are efficiently solved with optimization algorithms for MIPs. It is shown that the use of distinct ply thicknesses in the optimization further reduces the mass of single-material laminates made with carbon-fiber-reinforced polymer when compared with previous studies using the same material. Additionally, the combination of plies made of glass-fiber-reinforced polymer and carbon-fiber-reinforced polymer with distinct thicknesses is shown to reduce the weight penalty when optimized for cost minimization in comparison to plies with uniform thickness.

Analysis of smart laminated composites integrated with piezoelectric patches using spectral element method and lamination parameters

This paper investigates the effect of stacking sequence on the power output of a smart composite panel integrated with piezoelectric patches, using lamination parameter formulation and spectral element method (SEM). The deformation of the panel is expressed using the first-order shear deformation theory. The strain energy of the host plate is formulated using lamination parameters and the governing equations are derived following Hamilton’s principle. To solve the governing equations accurately and efficiently, a spectral element method is applied where the structure is divided into regions that are continuous in terms of geometry, and element matrices of each region are calculated using the spectral Chebyshev approach. This method benefits both from the (geometry) flexibility of the finite element method and the accuracy of the meshless methods. The developed electromechanical model is used to study the effect of the number of piezo patches and their sizes. To demonstrate the accuracy and performance of the presented SEM, six case studies were investigated by comparing natural frequencies, structural/voltage frequency response functions (FRFs), and computational duration to those obtained from a finite element analysis (FEA). The maximum difference in the predicted natural frequencies between the SEM and FEA results is below 1% and the FRFs obtained using the presented solution technique excellently match the FEA results. Yet, the simulation duration is significantly reduced compared to FEA. To exploit the computational efficiency of the presented analysis approach, optimization case studies were also performed implementing a genetic algorithm to maximize the power output by optimizing the stacking sequence and patch distribution.

Buckling analysis and optimization of variable angle tow composite plates via Ritz method and variable stiffness optimization

A novel variable stiffness optimization (VSO) algorithm is proposed accounting for the inherent mechanical characteristics of variable angle tow (VAT) plates. Through sampling in the design space of lamination parameters (LPs), a good initial point is identified in stacking sequence design domain. The concept of tailoring stiffness point by point through-thickness is then introduced. By using the linear fiber path function, design variables for VAT plates are divided into three groups, and then a layerwise optimization approach (LOA) is applied to optimize the angles from the outermost to the innermost stacking position within the three groups, sequentially and iteratively. Through a point-by-point optimization of fiber paths, the stiffness of VAT plates is enhanced part by part. Ritz-type buckling solutions are presented for VAT composite plates with arbitrary boundary conditions. Using first-order shear deformation theory (FSDT), the translational and rotational degrees of freedom of the plate are divided into in-plane and out-of-plane components to solve the prebuckling and buckling problems independently. Buckling loads of VAT plates are optimized under multiple boundary conditions and loadings. Results indicate that VSO algorithm is efficient and robust, and the buckling resistance capacity of optimized VAT plates can be improved approximately twice over constant-stiffness laminates.

Vibration optimization of composite plates with multiple circular holes using two-dimensional sampling optimization method

Perforated composite plates are widely used in engineering structures, and to prevent resonance, it is important to maximize the fundamental frequency. In this paper, the Ritz method is developed to calculate the vibration frequency of laminated composite plates with multiple circular holes, and the two-dimensional sampling optimization method (2DSO) is adopted to optimize the stacking sequence for achieving maximal fundamental frequency. Accounting for a variety of boundary conditions applied to the plate contour and circular holes, Legendre polynomials are adopted as admissible functions in the Ritz method in order to predict the vibration behavior of perforated plates. Then, 2DSO uses the lamination parameter (LP) to represent the distances between different stacking sequences. By generating uniformly distributed points in the LP design space and determining their corresponding stacking sequences, several good candidate stacking sequences are identified. Subsequently, local sampling optimization is employed to find better stacking sequences around the candidate stacking sequences. Lastly, the layerwise optimization approach (LOA) is employed to refine the sampling optimum. Perforated composite plates with one, two, and four circular holes, under different boundary conditions, are optimized. Results show the accuracy of the Ritz method, and the stability and efficiency of the 2DSO algorithm.

Gluing Characteristics of Giant Bamboo Using Four Commercial Adhesives

The study aimed to determine the bonding performance of laminated giant bamboo[Dendrocalamus asper (Schult.) Backer] glued with four commercial adhesives (PVAc-D3, PUR, UF, PF) at different surface pairings (pith-pith, pith-skin, and skin-skin) and glue spread rates(100, 150, and 200 g/m2). Kiln-dried giant bamboo poles were ripsawn, planed, and cut to length to produce slats for lamination. Slats for surface roughness and wettability tests were sanded with 180-grit sandpaper on both skin and pith surfaces. Surface roughness of the skin and pith was measured using Mitutoyo SJ–210 Surftest unit, whereas wettability was determined via the sessile drop method. Giant bamboo slats were bonded using specific lamination parameters for each adhesive. Tensile shear tests at dry and wet conditions were performed on the laminates to determine bond strength. The results showed that the bamboo pith had a rougher texture than the skin but with insignificant contact angle differences. Moreover, PVAc-D3 and PUR gave the highest and lowest initial contact angles on both sides, respectively, with PUR maintaining the smallest values throughout the contact duration. Adhesive, surface pairing, and some interactions (adhesive x glue spread and adhesive x surface pairing) significantly affected the dry shear strength, whereas adhesive and adhesive x surface pairing influenced wet shear strength. PUR-bonded laminates had the highest dry shear strength, followed by PF, PVAc-D3, and UF. In terms of wet shear strength, only PVAc-D3 did not conform to the minimum glue bond strength requirement of more than 1 MPa and cohesive bamboo failure of at least 40% (PNS 2099:2015). Skin-skin and pith-pith surface pairing yielded the highest and lowest dry shear strengths, respectively. Increasing the amount of glue did not translate to a stronger bond. PUR, UF, and PF are feasible alternatives to PVAc-D3 in engineered bamboo production for various end-uses.

Isogeometric stiffness matrix formulations for buckling analysis of multi-layered shells based on lamination parameters for solid-shell elements

Multi-layered composite shells are extensively used in many industrial fields since they have superior performance in applications. As load-carrying components, buckling failure is one of the most common failure modes of multi-layered shells. Hence, the layup design of multi-layered shells based on optimal buckling factors is widely concerned, where lamination parameters are commonly used in the representation of stiffness. However, the common formulations of laminate stiffness are based on the Classical Laminate Plate Theory (CLT) or the First-order Shear Deformation Plate Theory (FSDT). This paper aims at developing stiffness formulations based on 3D continuum solid theory. The accuracy and efficiency of the stiffness formulations derived in this study are analysed and compared to those obtained by layer-by-layer integration and summation of stiffness calculations.