Continuum Approach to Shape Sensitivity Analysis in Composite Laminates

Continuum Sensitivity Analysis (CSA) is an approach for calculating analytic derivatives. A direct CSA formulation is advantageous for computing derivatives of many state variables or performance functions. An adjoint formulation of CSA is beneficial for computing derivatives with respect to many design variables, although adjoint CSA boundary conditions are often problematic. For the proposed continuum-discrete hybrid adjoint approach, the adjoint variable is introduced after discretization which simplifies boundary conditions. The sensitivity boundary conditions for the hybrid CSA are posed in terms of the continuum state variables. Thus, the hybrid adjoint formulation of CSA results in design derivatives that are as accurate as those obtained from direct CSA, in addition to making the analysis efficient for the case of large number of design variables. Two test cases, first of an axial bar and second of a cantilever beam modeled with solid elements, illustrate how the hybrid adjoint formulation inherits the benefits of the direct and adjoint CSA formulations. This is also the first application of CSA to obtain design derivatives nonintrusively using three dimensional (3-D) Spatial Gradient Reconstruction (SGR) method for 3-D solid elements.

This paper presents results for the study of equivalence between the total form continuum sensitivity equation (CSE) method and the discrete analytic method of shape design sensitivity analysis. For the discrete analytic method, the sensitivity equations are obtained by taking analytic derivatives of the discretized equilibrium equations with respect to the shape design parameters. For the CSE method, the equilibrium equations are firstly differentiated to form a set of linear continuous sensitivity equations and then discretized for solving the shape sensitivities. The sensitivity equations can be derived by taking the material derivatives (total form) or the partial derivatives (local form) of the equilibrium equations. The total form CSE method is shown for the first time to be equivalent, after finite element discretization, to the discrete analytic method for nonlinear second-order differential equations of a particular form with design dependent loads when they use the same: (1) finite element discretization, (2) numerical integration of element matrices, (3) design velocity fields that are linear with respect to the design variable and (4) shape functions for domain transformation and for design velocity field calculations. The shape sensitivity equations for three-dimensional geometric nonlinear elastic structures and linear potential flow are derived by using both total form CSE and discrete analytic method to show the equivalence of the two methods for these specific examples. The accuracy of shape sensitivity analysis is verified by potential flow around an airfoil and a joined beam with an airfoil under gust load. The results show that analytic sensitivity results are consistent with the complex step results.

Sensitivity analysis is an important component of many model activities in hydrology. Numerous studies have been conducted in calculating various sensitivities. Most of these sensitivity analysis focus on the sensitivity of state variables (e.g., hydraulic head) to parameters representing medium properties such as hydraulic conductivity or prescribed values such as constant head or flux at boundaries, while few studies address the sensitivity of the state variables to some shape parameters or design parameters that control the model domain. Instead, these shape parameters are typically assumed to be known in the model. In this study, based on the flow equation, we derive the equation (and its associated initial and boundary conditions) for sensitivity of hydraulic head to shape parameters using continuous sensitivity equation (CSE) approach. These sensitivity equations can be solved numerically in general or analytically in some simplified cases. The approach has been demonstrated through two examples and the results are compared favorably to those from analytical solutions or numerical finite difference methods with perturbed model domains, while numerical shortcomings of the finite difference method are avoided.

This paper describes a method for structural optimization considering the elastoplastic structural behaviour. The proposed method consists of three parts : elastoplastic incremental structural analysis, elastoplastic sensitivity analysis, and optimization. The structural analysis and sensitivity analysis are performed using the incremental finite element method. For structural optimization, the Sequential Unconstrained Minimization Technique (SUMT) is used.In the optimization process, reanalysis is performed using the elastoplastic sensitivities. To get rational reanalysis solutions, some limitations should be imposed on the magnitude of changes in design variables. The maximum allowable changes in design variables are automatically evaluated by applying repeated sensitivity analysis during the optimization process.After verifying the accuracy of the reanalysis solution based on repeated sensitivity analysis, a three-bar and a ten-bar plane trusses are analyzed taking the cross-sectional areas as design variables. Comparison of the results shows that the proposed method gives very good results.

In response to the current challenges of low precision and efficiency in the optimization of composite material layups, the insufficient lightweight of automotive body floors, and the high cost of carbon fiber composites, this study introduces an optimized design method for carbon fiber composite flooring layups. Based on implicit parametric technology, a hybrid PSO-BFO (Particle Swarm Optimization-Bacterial Foraging Optimization) algorithm is employed. This approach achieves an integrated optimization of materials, processes, and structures, thereby balancing and reducing costs. The SFE-CONCEPT is utilized to establish an implicit parameterization model for the body floor, which is validated through experiments and finite element simulation analysis. Concept design and modeling of the carbon fiber composite material floor laminate are performed. Continuous variable optimization is employed to determine the thickness, tile shape, and number of layers for the front, middle, and rear floors. A continuous variable discretization rounding strategy is used to obtain the discrete layer numbers for each laminate orientation of the composite material floor. The continuous fiber lamination strategy is applied to create different shared lamination regions. The PSO-BFO hybrid optimization method is proposed to optimize the lamination sequence as a multi-objective optimization, addressing the challenges of discrete lamination sequence, explosive combinations, and multiple variables in the optimization design of carbon fiber composite material floor laminates. The optimization results demonstrate improvements of 34.4% in floor quality M, 6.0% in static bending stiffness BS, and 5.3% in lightweight coefficient QLX using the proposed PSO-BFO method. PSO-BFO and PSO-GA (Particle Swarm Optimization-Genetic Algorithm) methods are more capable of obtaining global optimal solutions for complex optimization problems than single optimization algorithms. Still, the results obtained by the PSO-BFO method are more balanced.

Gradient-based optimization techniques need precise and efficient sensitivities. Numerical sensitivity methods such as finite differences are easy to implement but imprecise and computationally inefficient. In contrast, analytical sensitivity methods, such as the discrete and continuum ones are highly accurate and efficient. Continuum Sensitivity Analysis (CSA) is an analytical method used to calculate derivatives of shape or value parameters. While CSA has been successfully applied in static analysis and dynamic problems in the time domain, this work presents an extension of the approach to eigenvalue sensitivities for the first time. However, CSA revealed limitations, prompting the exploration of an alternative approach based on discrete analytical differentiation. This method is employed for the first time in shape sensitivities. The derivatives of the stiffness and mass matrices required by the method are calculated analytically, resulting in high accuracy and computational efficiency. In addition, an element agnostic approach has been developed leveraging primary analysis matrices to calculate their derivative. This characteristic, along with the nonintrusivity, makes the method employable with standard commercial software. Both approaches have been applied and validated in a wide range of scenarios, involving vibration and buckling problems.

The low velocity impact is a common phenomenon which occurs in fiber reinforced polymer composite products like LPG cylinders, fighter aircraft fuel drop tanks, aircraft wing surfaces, sports goods etc. The consequences of low velocity impact will create a considerable damage and ultimately lead to a premature failure of the structure. Hence the polymer composites for engineering applications must be provided with a better design solution. From the literature survey it is observed that, the response of composite laminates subjected to quasi-static loading, exhibits similar results as that of low velocity impact. Polymer reinforced composites are poor in damage tolerance with better strength to weight ratio than conventional materials. However composite materials can be tailored to meet the design requirements by manipulating fiber orientations and laminae stacking sequence. In the present paper, principles of classical laminate theory are considered for analysis. FEM is implemented for thorough understanding of the failure mechanism of each laminate by layer wise. Simulated quasi-static loading tests and observed the layer wise distribution of transverse strain intensity. The experimental setup is designed and fabricated as per ASTM D 6264 standards. The E-glass/epoxy composite laminate is quasi-statically loaded at its center by a steel ball indenter of diameter 8.7mm and its response is measured by the degree of opacity or translucency in terms of interlaminar and intra-laminar damage area. The stacking sequence of composite laminates are chosen as [00/600]12, [00/750]12 and [00/900]12. The damage areas obtained from numerical analysis are in good agreement with experimental results.

A model problem arising from optical design of photonic bandgap structure is investigated. That is, the optimization problem is to find the material inhomogeneity in a domain so that a particular eigenmode governed by the scalar Helmholtz equation is localized. The continuum sensitivity analysis of the objective function including the eigenmode is carried out. The derivative of the objective function with respect to the density function is obtained by the sensitivity problem and the adjoint problem in continuum systems. When the multiplicity of eigenmode happens, our strategy is to select the closest eigenmode to the current eigenmode. Four numerical examples in a square domain are studied, with different weight functions and initial density distributions. The numerical results illustrate the validity of the algorithm based on the continuum sensitivity analysis.

The weight and fuel savings offered by composite materials make them attractive not only to the military, but also to the civilian aircraft, space, and automobile industries. In these industries, bolted and riveted joints are extensively used as a primary method for structural joining. Bolted joints in composite materials have complex failure modes, and hence the demand for improving their performance exists. The main objective of this work is to improve the performance of bolted joints in composite structures by introducing nanoparticles/fibers around the expected failure zone. The literature on this issue showed shortcomings in the investigations of such materials. Most of the investigations in this field aimed to enhance the mechanical properties of epoxy materials, which cannot be used alone for high performance structural applications due to their low mechanical properties. In the present work, epoxy resin was modified with different types of nanofillers including multi walled carbon nanotubes (MWCNT). Nano-phased epoxy was used to fabricate different types of nanocomposites as well as nano-hybridized glass fiber reinforced composite laminates. Therefore, six different advanced materials were fabricated including a nanocomposite material (MWCNT/E), a quasi-isotropic nano-hybridized composite laminate (QI-GFR/MWCNT/E), a unidirectional nano-hybridized composite laminate (UD-GFR/MWCNT/E) and a control panel manufactured without nano-fillers (neat epoxy, QI-GFR/E, UD-GFR/E). The materials were characterized by tension and compression tests. The obtained properties are essential for the validation of respective finite element analysis. The results showed improvements in the tensile and compressive properties (strength and modulus) of the fabricated nanocomposites (MWCNT/E) compared with neat epoxy. The hybridized composite laminate with MWCNT showed high improvements in their mechanical properties compared to the composite laminates without nanofillers.

To enhance performance under low‐velocity impact (LVI), the ply stacking sequence of a composite laminate must be carefully selected. To guarantee that the laminate can sustain the projected impact energy, the impact load should be carefully evaluated during the design phase. To obtain an optimized stacking layup orientation, the damage behaviors of excellent specific strength carbon fiber–reinforced polymer (CFRP) composite laminates with stacking sequences of [0°]8 (unidirectional), [0°/90°]2s (cross‐ply), [0°/+45°/−45°/90°]s (quasi‐isotropic), and [0°/−30°/−60°/−90°/90°/60°/30°/0°] (antisymmetric) were numerically investigated under LVI. Throughout the investigation, 1 to 2 kg of impactors were employed at a velocity of 3.835 m/s. The damage initiations in composite laminates were assessed using both Hashin’s criteria and the Puck–Schurmann criterion. For predicting delamination between composite plies, the quadratic nominal stress delamination failure criteria were applied. For the study of composite damage behavior under LVI, the time‐dependent changes in impact‐resisting force, force versus displacement, and time‐dependent variations in the internal energy of the composite laminates were taken into consideration. The predicted results show that in comparison with the other three configurations, the composite laminate with a ply arrangement of [0°/+45°/−45°/90°]s (quasi‐isotropic) exhibited a stronger impact‐resisting force, smaller peak displacement, fewer deformations, and a smaller amount of damage with superior rebound energy.

PurposeIt is seen that little amount of work on optimization of mechanical properties taking into consideration the combined effect of design variables such as stacking angle, stacking sequence, different resins and thickness of composite laminates has been carried out. The focus of this research work is on the optimization of the design variables like stacking angle, stacking sequence, different resins and thickness of composite laminates which affect the mechanical properties of hybrid composites. For this purpose, the Taguchi technique and the method of gray relational analysis (GRA) are used to identify the optimum combination of design variables. In this case, the effect of the abovementioned design variables, particularly of the newly developed resin (NDR) on mechanical properties of hybrid composites has been investigated.Design/methodology/approachThe Taguchi method is used for design of experiments and with gray relational grade (GRG) approach, the optimization is done.FindingsFrom the experimental analysis and optimization study, it was seen that the NDR gives excellent bonding strength of fibers resulting in enhanced mechanical properties of hybrid composite laminates. With the GRA method, the initial setting (A3B2C4D2) was having GRG 0.866. It was increased by using a new optimum combination (A2B2C4D1) to 0.878. It means that there is an increment in the grade by 1.366%. Therefore, using the GRA approach of analysis, design variables have been successfully optimized to achieve enhanced mechanical properties of hybrid composite laminates.Originality/valueThis is an original research work.

Two of the most important requirements when using gradient-based optimization for fluid–structure interaction problems are efficiency and accuracy in calculating the sensitivities. As a result, analytical continuum sensitivity formulations are finding their place due to their lower cost and accuracy. However, the necessary computation of mesh sensitivities in shape optimization is becoming a bottleneck, especially when handling complex geometries. In this research, an algorithm based on continuum sensitivity analysis for structural shape design variables is developed. In the proposed method, regularized Heaviside functions are used to modify the properties of mesh cells. Therefore, mesh dependency is removed from the sensitivity equations. This method does not require moving or modifying the mesh to handle the shape changes. This method is applied to two different validation cases where continuum sensitivity equations are formulated and solved. Validation cases are selected as a structural and thermal problem to show the applicability of this method for different physics. The results demonstrate the efficiency of the proposed algorithm for design-optimization framework.

Many studies were directed toward understanding damage patterns in composite laminates and determining the damage development sequence upon high velocity impact. Damage accumulation depends on projectile velocity and on a number of other parameters, so that it is not possible to set strict limits between the different regimes. However, experiments show that, for a given set of experimental conditions where the impact speed is the only variable, there is a certain threshold velocity below which no detectable damage occurs. Above the threshold velocity, no surface damage is observed except for a small indentation at the contact point, but significant internal damage consisting of delaminating and matrix cracks is introduced. As the impact velocity increases further, surface damage due mainly to fiber breakage is introduced. For very high speeds, the target does not have time to deform, and perforation occurs, leaving a clean hole in the sample. The objective of this study is to develop a mathematical model that corresponds to the deformed geometry under high velocity impact applications for composite laminates. A total of 100 tests were conducted on composite laminates, struck by cylindrohemispherical projectiles at normal incidents with velocities up to about 100 mls. The types of materials, used this study, are AS4/3051, IM7/5250 CarbonlEpoxy and TI003 Glass/Epoxy. The strain energy was obtained by derivation of the proposed deflection function. The strain energy was plotted with respect to the deflection of the mid-plane and, then correlated through dynamic correlation factors to actual kinetic energy during the impact. The dynamic correlation factors were determined using a genetic algorithm regression analysis. Two types of materials were tested, namely plain graphite composites and hybrid composites. The growth of the delamination and also the effect of varying the stacking sequence were investigated for the different type of materials and various orientations. The mathematical model appears to provide a reasonable representation of the deformation of composite laminates during the penetration by a cylindro-hemispherical projectile. Furthermore, hybrid composites appear to provide more resistance to the impact, whereas plain composites have less resistance with respect to the higher velocities. It was concluded that, the change of the material in a hybrid composite affects the growth of the damaged area and also reduces the impact penetration resistance. Hence, IM7/E-Glass hybrid has a higher resistance to the penetration. Measurements of the energy levels of the hybrid composites indicated that they offer the highest resistance to ballistic perforation. The hybrid composites perforated at velocities between 77 mls and 83 (mls), whereas the graphite composites perforated at velocities between 48 m/s and 59 (mls). The higher perforation resistance is attributed to the reduced level of delamination generated during the impact, and also the addition of the E-Glass, which was capable of absorbing more energy during the impact. In studying the graphite composites, the best orientation in terms of the stacking sequence was found to be [(45, -45, 0, 90) 2 ] S , which indicates that this stacking sequence withstand higher velocity and hence absorbs more energy during the impact. Therefore, the quasi-isotropi corientation [(45, -45, 0, 90) 2 ] S is best for impact resistance if a laminate is not combined with E-Glass. The ballistic-limit velocity prior to perforation for the Quasi-isotropic laminate was measured as 58.9 m/s. This is a significant increase compared to the other plain graphite samples. The energy required for the complete perforation is approximately 48% higher in this stacking sequence as compared to other plain Graphite specimens. It was also found that the energy absorption capability is reduced significantly in the cross-ply laminates. The penetration resistance of the [(0,90,0,90) 2 ] S laminate and the energy required for perforation are approximately 50% less than the other plain graphite specimens.

Many studies were directed toward understanding damage patterns in composite laminates and determining the damage development sequence upon high velocity impact. Damage accumulation depends on projectile velocity and on a number of other parameters, so that it is not possible to set strict limits between the different regimes. However, experiments show that, for a given set of experimental conditions where the impact speed is the only variable, there is a certain threshold velocity below which no detectable damage occurs. Above the threshold velocity, no surface damage is observed except for a small indentation at the contact point, but significant internal damage consisting of delaminating and matrix cracks is introduced. As the impact velocity increases further, surface damage due mainly to fiber breakage is introduced. For very high speeds, the target does not have time to deform, and perforation occurs, leaving a clean hole in the sample. The objective of this study is to develop a mathematical model that corresponds to the deformed geometry under high velocity impact applications for composite laminates. A total of 100 tests were conducted on composite laminates, struck by cylindrohemispherical projectiles at normal incidents with velocities up to about 100 mls. The types of materials, used this study, are AS4/3051, IM7/5250 CarbonlEpoxy and TI003 Glass/Epoxy. The strain energy was obtained by derivation of the proposed deflection function. The strain energy was plotted with respect to the deflection of the mid-plane and, then correlated through dynamic correlation factors to actual kinetic energy during the impact. The dynamic correlation factors were determined using a genetic algorithm regression analysis. Two types of materials were tested, namely plain graphite composites and hybrid composites. The growth of the delamination and also the effect of varying the stacking sequence were investigated for the different type of materials and various orientations. The mathematical model appears to provide a reasonable representation of the deformation of composite laminates during the penetration by a cylindro-hemispherical projectile. Furthermore, hybrid composites appear to provide more resistance to the impact, whereas plain composites have less resistance with respect to the higher velocities. It was concluded that, the change of the material in a hybrid composite affects the growth of the damaged area and also reduces the impact penetration resistance. Hence, IM7/E-Glass hybrid has a higher resistance to the penetration. Measurements of the energy levels of the hybrid composites indicated that they offer the highest resistance to ballistic perforation. The hybrid composites perforated at velocities between 77 mls and 83 (mls), whereas the graphite composites perforated at velocities between 48 m/s and 59 (mls). The higher perforation resistance is attributed to the reduced level of delamination generated during the impact, and also the addition of the E-Glass, which was capable of absorbing more energy during the impact. In studying the graphite composites, the best orientation in terms of the stacking sequence was found to be [(45, -45, 0, 90) 2 ] S , which indicates that this stacking sequence withstand higher velocity and hence absorbs more energy during the impact. Therefore, the quasi-isotropi corientation [(45, -45, 0, 90) 2 ] S is best for impact resistance if a laminate is not combined with E-Glass. The ballistic-limit velocity prior to perforation for the Quasi-isotropic laminate was measured as 58.9 m/s. This is a significant increase compared to the other plain graphite samples. The energy required for the complete perforation is approximately 48% higher in this stacking sequence as compared to other plain Graphite specimens. It was also found that the energy absorption capability is reduced significantly in the cross-ply laminates. The penetration resistance of the [(0,90,0,90) 2 ] S laminate and the energy required for perforation are approximately 50% less than the other plain graphite specimens.

A computational capability is developed for the optimum design of radial drilling machine structure to satisfy static rigidity and natural frequency requirements using finite element idealization. The radial drilling machine structure is idealized with frame elements and is analyzed by using different combinations of cross sectional shapes for the radial arm and the column. From the results obtained, the best combination of cross sectional shapes is suggested for the structure. With this combination of cross sectional shapes, mathematical programming techniques are used to find the minimum weight design of the radial drilling machine structure. A sensitivity analysis is conducted about the optimum point to find the effects of changes in design variables on the structural weight and the response quantities.