This paper presents an aero load correction strategy applicable to the static aeroelastic optimization of composite wings. The optimization framework consists of a successive convex subproblem iteration procedure, in which a gradient-based optimizer consecutively solves a local approximation problem. Responses are approximated as a linear and/or reciprocal function of the laminate membrane and bending stiffness matrices. Together with the laminate thicknesses h, they constitute the design variables in the optimization process. Internally, the design space is transformed from stiffness matrices to lamination parameters, resulting in a continuous and convex optimization problem. Structural responses considered in the stiffness optimization are strength, local buckling and mass; aileron effectiveness, divergence, and twist constitute the aeroelastic responses. Steady aeroelastic loads are calculated with a doublet lattice method (DLM) embedded in the applied finite element solver, allowing for the generation of response sensitivities that incorporate the effects of displacement-dependent aeroelastic loads. To incorporate flow phenomena that cannot be reproduced with DLM, a higher order aerodynamic method is considered. The developed correction methods and their application are presented in this paper. The correction is twofold, first, aiming at a correction of DLM by means of camber and twist modifications applied directly to the doublet lattice mesh and second, by employing the capabilities of a higher order computational fluid dynamics (CFD) solver, like the DLR-based TAU code. To this end, DLM loads transferred to the structure are rectified by means of the supposedly superior CFD results. The aero load correction method is applied in the stiffness optimization of a forward swept wing. First, a trim application without structural optimization is discussed, to demonstrate the convergence behavior of the correction forces. The results of a wing skin mass minimization with balanced and unbalanced laminates are presented. In particular, the differences between optimizations with and without aero correction are highlighted. Eventually, a stacking sequence optimization based on the continuous optimization results is demonstrated.

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Composite materials are finding increasing application, for example in commercial aircraft. Traditionally fiber angles are constant in a single layer. Currently, so called variable stiffness panels with steered fibres, where the angle is changing within a layer are investigated. These panels are usually manufactured using automated fibre placement machines. Since the fibre angle is changing, and the tow paths are shifted as a whole in a single direction, gaps and/or overlaps between consecutive tows are created. This paper explores the effect of these gaps on the stiffness and buckling load of variable stiffness panels. A methodology is presented using homogenization to account for the gaps in a computational efficient way. The result shows that the stiffness results are on conservative side and are within 5% accuracy. However, the buckling results are on the unconservative side. The computational cost of the pre-processing of the proposed method is 45 times lower than the cost of the defect-ply method presented in the literature.

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A method to optimise the fibre angle distribution of variable stiffness laminates is proposed. The proposed method integrates a fibre angle retrieval step with a fibre angle optimisation procedure. A multi-level approximation approach is used in combination with the method of successive approximations. First, fibre angle retrieval is done by approximating the structural responses based on the optimal stiffness distribution found using lamination parameters. The full fibre angle optimisation is done by updating the approximations based on the current stacking sequence. Next, the actual fibre paths are optimised taking into account the actual size of a tow, and the maximum size of any gap or overlap appearing. The paths are smoothed out using CATIA, and finally spline curves are found that can be sent to a fibre placement machine for manufacturing. It is shown for a bucking optimisation with a stiffness constraint that the number of finite element analyses reduces significantly by starting the optimisation from the optimal stiffness distribution rather than from a user-specified stacking sequence. Next, it is shown that updating the approximations also leads to considerable improvements over fibre angle retrieval. Similar promising results are obtained for a stress optimisation problem.

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A way to approximate the compliance of composites for optimisation is described. A two-level approximation scheme is proposed inspired by traditional approximation concepts such as force approximations and convex linearisation. In level one, an approximation in terms of the reciprocal in-plane stiffness matrix is made. In level two, either the lamination parameters, or the nodal fibre angle distribution are used as design variables. A quadratic approximation is used to build the approximations in terms of the fibre angles. The method of conservative, convex separable approximations is used for the optimisation. Conservativeness is guaranteed by adding a convex damping function to the approximations. Two numerical examples, one optimisng the compliance of a plate clamped on the left, loaded downwards on the bottom right, another one optimising the compliance of a plate loaded with a shear force and a moment show the computational efficiency of the proposed optimisation algorithm.

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The present paper proposesa continuous-time state-space formulationofthe unsteady vortexlattice method, which is derived through a discretization of the governing advection equation for transport of vorticity in the wake. A continuous-time system isobtained byonly discretizing the advection equationinspace, while retaining the derivative with respect to time. The discretization in space is based on the discontinuous Galerkin method. The present method can be applied to any arbitrary nonuniform wake discretization and can be extended to higher-order panel methods or a nonflat wake shape. The method is extended to compressible flows by applying the Prandtl-Glauert transformation. The time-dependent terms in the small disturbance potential equation are neglected. Thus, incompressible flow solution procedures are applied with minimum modifications to unsteady compressible problems.The benefits are demonstratedby applying the modelto the gust analysisofageneral aircraft wing, varying the time step, and introducing a nonuniform wake discretization, resulting in a reduced model size for a given accuracy. The resulting continuous-time state-space model can be used for efficient loads analysis of general aircraft wings including the effects of compressibility and allows for easy integration with structural or flight dynamic models for efficient aero(servo)elastic analyses.

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The Koiter-Newton (KN) method is a combination of local multimode polynomial approximations inspired by Koiter's initial postbuckling theory and global corrections using the standard Newton method. In the original formulation, the local polynomial approximation, called a reduced-order model, is used to make significantly more accurate predictions compared to the standard linear prediction used in conjunction with Newton method. The correction to the exact equilibrium path relied exclusively on Newton-Raphson method using the full model. In this paper, we proposed a modified Newton-type KN method to trace the geometrically nonlinear response of structures. The developed predictor-corrector strategy is applied to each predicted solution of the reduced-order model. The reduced-order model can be used also in the correction phase, and the exact full nonlinear model is applied only to calculate force residuals. Remainder terms in both the displacement expansion and the reduced-order model are well considered and constantly updated during correction. The same augmented finite element model system is used for both the construction of the reduced-order model and the iterations for correction. Hence, the developed method can be seen as a particular modified Newton method with a constant iteration matrix over the single KN step. This significantly reduces the computational cost of the method. As a side product, the method has better error control, leading to more robust step size adaptation strategies. Numerical results demonstrate the effectiveness of the method in treating nonlinear buckling problems.

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This work is concerned with the development of a framework to solve shape optimization problems for transient heat conduction problems within the context of isogeometric analysis (IGA). A general objective functional is used to accommodate both shape optimization and passive control problems under transient conditions. An adjoint sensitivity analysis, which accounts for possible discontinuities in the objective functional, is performed analytically and subsequently discretized within the context of IGA. The gradient of the objective functional is used in a descent algorithm to solve optimization problems. Numerical examples are presented to validate and demonstrate the capacity to manage thermal fields under transient conditions.

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Variable stiffness composites, where fibre angles are spatially varied by steering the tows in curvilinear paths to optimise the structural response, have been a subject of intensive study. In this paper, experimental validation of the variable stiffness composite technology is carried out for a panel representing a wing lower-skin with a large access hole designed against material failure. An idealised flat panel with a large cut-out under tension or combined tension and shear is modeled using finite elements. In addition to a quasi-isotropic laminate, constant stiffness and variable stiffness laminates are designed to maximise the failure load using a multi-step optimisation framework. Three panels, one for each type of laminate, are built from thermoset prepreg material using automated fibre placement. All three panels are tested in pure tension. The failure loads, failure modes and weights of the tested panels are compared. The results indicate that the variable stiffness laminate is capable of sustaining significantly larger loads, before failure, than the constant stiffness and quasi-isotropic laminates of equal weight.

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In isogeometric shape optimization, the use of the search direction directly predicted from the discrete shape gradient makes the optimization history strongly dependent on the discretization. This discretization-dependency can affect the convergence and may lead the optimization process into a sub-optimal solution. The source of this discretization-dependency is traced to the lack of consistency with the local steepest descent search direction in the continuous formulation. In the present contribution, this inconsistency is analyzed using the shape variation equations and subsequently illustrated with a volume minimization problem. It is found that the inconsistency originates from the NURBS discretization which induces a discrete quadratic norm to represent the continuous Euclidean norm. To fix this inconsistency, three normalization approaches are proposed to obtain a discretization-independent normalized descent search direction. The discretization-independence of the proposed approaches is verified with a benchmark problem. The superiority of the proposed search direction and its suitability for numerical implementation is illustrated with examples of shape optimization for mechanical and thermal problems. Although the present work focuses on a NURBS-based discretization usually used in conjunction with isogeometric analysis, the proposed methodology may also be applied to alleviate the “mesh-dependency” in (traditional) Finite Element-based shape optimization.

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Grid-stiffened composite structures not only allow for significant structural weight reduction but also are competitive in terms of structural stability and damage tolerance compared with conventional stiffened candidates. As the development of Automated Fibre Placement (AFP) technology matures, a unitized construction of skin and stiffeners is easily manufacturable. In this paper, a curved stiffener layout is optimized to enhance the structural buckling resistance. A linear variation of stiffener angles is used, resulting in the formation of a locally rhombic lattice pattern. Due to the spatial variation of angle and spacing induced by the use of curved stiffeners, analytic solutions for the responses are not generally applicable. Thus, global and local buckling loads are calculated based on finite element models by a previously-developed global/local coupled strategy. Since the stiffeners are not explicitly modelled in the finite element calculations, a fixed mesh is used for gradient-based optimization. Both parametric design and optimization are performed in order to find the optimal curved grid pattern, whose practical performances are assessed by post-buckling analysis. A comparison between the performances of structures with curved stiffeners, with straight stiffeners, and variable-stiffness skins with curved fibres, demonstrates the potential of curved stiffener configurations in improving the structural efficiency.

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Modern composite structures offer multiple avenues of optimising performance. One avenue is to optimise a single stacking sequence over the structure leading to constant stiffness designs. Another avenue is to allow the stacking sequence to vary over the structure leading to variable stiffness laminates. This may be achieved either by dropping plies or by steering the fibres. When using ply drops to optimise the thickness distribution two different set of decisions are involved: the selection of ply drop boundaries, and the selection of the ply drop order. In this paper, the fibre angle distribution, the ply drop boundaries, and the ply drop order are simultaneously optimised. The optimisation of fibre angle distribution lends itself easily to gradient based methods. The ply drop boundary optimisation is formulated using topology optimisation techniques and is thus solvable using gradient based methods as well. The ply drop order optimisation requires discrete variables and is hence approached using an evolutionary algorithm based on stacking sequence tables. In this paper an efficient multi-step algorithm is developed to combine the optimisation of all aspects of variable stiffness laminates. The results indicate that significantly improved designs may be obtained by including the ply drop order in the optimisation.

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This article presents an optimization tool for the stacking sequence design of blended composite structures. Enforcing blending ensures the manufacturability of the optimized laminate. A novel optimization strategy is proposed combining a genetic algorithm (GA) for stacking sequence tables with a multi-point structural approximation using a modified Shepard’s interpolation in stiffness-space. A successive approximation approach is used where the set of design points used to create the structural approximations is successively enriched using the elite of the previous step. Additional improvement in the generality and efficiency of the algorithm is obtained by using load approximations thus enabling the implementation of a wide range of stress-based design criteria. A multi-panel, blended composite problem is used as an application to demonstrate the performance of the developed tool. The optimization is performed with mass as the objective to be minimized, subjected to strength and buckling constraints. The results presented show that completely blended and feasible stacking sequence designs can be obtained, having their structural performance close to the theoretical continuous optimum itself. Additionally, the multi-point Shepard’s approximation shows a considerable saving in computational costs, while the limitations of inexpensive stiffness-matching optimizations are observed.

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A buckling analysis of an isotropic circular plate with piezoceramic annular plate attached to it is presented. The annular plate is attached at the circular plate edge so that the radius of the circular plate is the inner radius of the annular plate. The piezoceramic annular plate is used as the radial in-plane load source. The radial and hoop stresses are found to be depends on the radial throughout the annular region while constants throughout the circular regions. The governing equations is solved approximately using finite difference method. The solutions to be found to be in good agreement with results from FEM analysis. A parametric study also presented.@en

Modern composite structures o er two avenues of optimising performance. One, optimising a single stacking sequence over the structure leading to constant sti ness designs. Two, varying the sti ness over the structure. This may be achieved by dropping plies, changing the thickness, or by steering the fibres, changing the fibre angles. Optimising ply drops involves two decisions: the selection of ply drop boundaries, and of the ply drop order. Previous work considered the problems of optimising the fibre angle distribution and ply drop boundaries but ply drop order was pre-specified. This paper extends the work to simultaneously optimise the fibre angle distribution, the ply drop boundaries, and order. The optimisation of fibre angle distribution lends itself to gradient-based methods. The ply drop boundary optimisation is formulated using topology optimisation techniques and is thus also solvable using gradient-based methods. The ply drop order optimisation requires discrete variables and is hence approached using an evolutionary algorithm based on stacking sequence tables. In this paper an e cient multi-step algorithm is developed combining the optimisation of all aspects of variable sti ness laminates. The results indicate that significant improvements may be obtained by including the ply drop order in the optimisation at a relatively modest computational cost. @en

Nonconventional laminates, as defined in this paper, are laminates where ply angles are not restricted to a finite set: for example 0, 45, −45 −45 and 90 deg. Removing this restriction, structural behavior (for example, postimpact behavior) can be significantly improved. Nonconventional laminates are made of straight or steered fibers leading to constant- or variable-stiffness composites. In traditional composite design, empirical guidelines are imposed, guaranteeing the robustness of the composite. This paper presents a method to take design guidelines into account during nonconventional laminate optimization. The 10% rule is interpreted as a lower bound on the degree of isotropy and formulated as a positive-semidefinite matrix constraint. Other guidelines are interpreted as bounds on ply angle or angle difference, or the number of variables is reduced by fixing variables to user-defined values. Numerical results optimizing a plate under biaxial tension for strength demonstrate the optimizer generates nonconventional laminates obeying all guidelines, performing at least as well as conventional composites. For variable-stiffness laminates, a flat plate under uniaxial compression is optimized for buckling under a stiffness constraint. Applying the 10% rule, almost half of the improvement over conventional laminates is lost compared to the unconstrained case, demonstrating the importance of including design guidelines during nonconventional laminate optimization.

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Optimisation algorithms used to automatically size structural members commonly involve stress constraints to avoid material failure. Therefore the cost of optimisation grows rapidly as the number of structural members is increased due to the corresponding increase in the number of constraints. In this work, an efficient method for large scale stress constrained structural sizing optimisation problems is proposed. A convex, separable, and scalable approximation for stress constraints which splits the approximation into a local fully stressed term and a global load distribution term is introduced. Predictor-corrector interior point method, an excellent option for large scale optimization problem, is chosen to solve the approximate subproblems. The core idea in this work is to achieve computational efficiency in the optimization procedure by avoiding the construction and the solution of the Schur complement system generated by the interior point method. Avoiding the Schur complement, and explicit sensitivity analysis, eliminates the high cost of solving stress constrained problems within the interior point optimisation. This is achieved using the preconditioned conjugate gradient method, and a new preconditioner is proposed specifically for stress constrained problems. The proposed method is applied to a number of beam sizing problems. Numerical results show that optimal complexity is achieved by the algorithm, the computational cost being linearly proportional to the number of sizing variables.

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