Mostafa M. Abdalla

Reduction Method for Finite Element Nonlinear Dynamic Analysis of Shells

DOI: 10.2514/1.J051003Theprojectionoftheequationsofmotiononasuitablemodalbasiscandrasticallyreducethenumberofdegreesoffreedom of a nonlinear dynamic !nite element analysis. A reduction method based on enriching a set of vibrationmodeswithsecond-ordermodescalculatedviaaperturbationtechniqueispresented.Thesesecond-ordermodesarereadily calculated via the solution of corresponding linear problems. The results indicate that the second-ordermodes are displacement !elds that are essential for an appropriate representation of the nonlinear behavior of thestructure. Problems exhibiting strong geometrical nonlinearities under relatively high dynamic loads aresuccessfully handled by forming the reduction basis with vibration modes and their corresponding second-ordermodes, calculated at two different static equilibrium con!gurations.

Thermomechanical Response of Variable Stiffness Composite Panels

Analysis of non-traditional Variable Stiffness (VS) laminates, obtained by steering the fiber orientation as a spatial function of location, have shown to improve buckling load carrying capacity of flat rectangular panels under axial compressive loads. In some cases the buckling load of simply supported panels doubled compared to the best conventional laminate with straight fibers. Two distinct cases of stiffness variation, one due to fiber orientation variation in the direction of the loading, and the other one perpendicular to the loading direction, were identified as possible contributors to the buckling load improvements. In the first case, the increase was attributed to the favorable distribution of the transverse in-plane stresses over the panel platform. In the second case, a higher degree of improvement was obtained due to the re-distribution of the applied in-plane loads. Experimental results, however, showed substantially higher levels of buckling load improvements compared with theoretical predictions. The additional improvement was determined to be due to residual stresses introduced during curing of the laminates. The present paper provides a simplified thermomechanical analysis of residual stress state of variable stiffness laminates. Systematic parametric analyses of both cases of fiber orientation variations show that, indeed much higher buckling loads could result from the residual stresses present in such laminates.

Optimization of a variable-stiffness skin for morphing high-lift devices

One of the possibilities for the next generation of smart high-lift devices is to use a seamless morphing structure. A passive composite variable-stiffness skin as a solution to the dilemma of designing the structure to have high enough stiffness to withstand aerodynamic loading and low stiffness to enable morphing is proposed. The variable-stiffness skin is achieved by allowing for a spatial fibre angle and skin thickness variation on a morphing high-lift system. The stiffness distribution is tailored to influence the deformation of the structure beneficially. To design a realistic stiffness distribution, it is important to take aerodynamic and actuation loads into account during the optimization. A two-dimensional aero-servo-elastic framework is created for this purpose. Skin optimization is performed using a gradient-based optimizer, where sensitivity information is found through application of the adjoint method. The implementation of the aero-servo-elastic environment is addressed and initial optimization results presented. The results indicate that a variable-stiffness skin increases the design space. Moreover, the importance of taking the change in aerodynamic loads due to morphing skin deformation into account during optimization is demonstrated.

A Generic Morphing Wing Analysis and Design Framework

A generic framework for morphing wing aeroelastic analysis and design is presented. The wing is discretised into an arbitrary number of wing segments. Two types of actuation mechanisms are identified: inter-rib mechanisms operating across a wing segment and intra-rib mechanisms acting between two adjacent wing segments. Virtually, any shape can be obtained by distributing four morphing modes over the entire morphing wing. Three are an intra-rib mechanism and one is an inter-rib mechanism. The intra-rib modes are wing shear, twist and extension, and the inter-rib mode is wing folding. The wing is modeled using a close coupling between a non-linear beam formulation and Weissinger aerodynamics. The framework is intended to aid quick preliminary design of morphing wings to trade-off contradictory requirements in a flight mission. The morphing wing can be optimized for discrete points in the flight mission, and for the entire flight mission. The framework can be used to predict aerodynamic performance, load distribution, aeroelastic deformations, and the required actuation forces and moments and corresponding actuation energy. Therefore, the performance gains of wing morphing can be weighed against the energy costs and weight penalties due to the presence of the actuators. The functionality of the framework is demonstrated by making use of a folding and sweeping wing test case.

POSTBUCKLING ANALYSIS OF VARIABLE STIFFNESS COMPOSITE PLATES USING A FINITE ELEMENT-BASED PERTURBATION METHOD

Modern fiber placement machines allow laminates with spatially varying stiffness properties to be manufactured. In earlier research, the authors optimized variable stiffness plates for maximum buckling load, demonstrating significant improvements in load-carrying capacity. In aerospace applications, panel structures are often permitted to enter the postbuckling regime during service. It is, therefore, not only important to understand their postbuckling behavior, but also to develop fast analysis methods that can subsequently be used in a design optimization framework. The aim of the present research is to study the postbuckling behavior of the optimized plates using a perturbation method that has been developed earlier within a general-purpose finite element environment. The perturbation approach is used to compute postbuckling coefficients, which are used to make a quick estimate of the postbuckling stiffness of the panel and to establish a reduced-order model. In the present work, the postbuckling analysis of variable stiffness plates is carried out using the reduced-order model, and the potential of the approach for incorporation within the optimization process is demonstrated.

Thermomechanical Design Optimization of Variable Stiffness Composite Panels for Buckling

Both numerical and experimental research have shown that variable stiffness laminates allow for significant design improvements. Variable stiffness laminates can be manufactured by exploiting the built-in steering capabilities of modern fiber placement machines (FPM). Such variable stiffness panels optimized for buckling were built and tested previously. Test results yielded higher buckling loads than those predicted numerically, which was later found to be related to the residual stresses present after curing. Hence, to benefit fully from the design freedom offered by variable stiffness panels built by FPMs it is essential to include the influence of thermal stresses in the optimization formulation. In this paper, an optimization framework, developed by the authors, is extended to include the influence of thermal loads. Numerical results confirm the importance of including thermal effects in the optimization process. Designs including thermal stresses are found to have improved buckling performance over a larger range of operating temperatures. Proper tailoring of both stiffness and thermal properties allows for ultimate buckling load improvements in the order of four to six times that of the corresponding quasi-isotropic panel.

Static/Dynamic Edge Movability Effect on Non-Linear Aerothermoelastic Behavior of Geometrically Imperfect Curved Skin Panel: Flutter and Post-Flutter Analysis

This paper addresses the problem of the aerothermoelastic modeling behavior and analyses of skin curved panels with static and dynamic edge movability effect in high supersonic flow. Flutter and post-flutter behavior will be analyzed toward determining under which conditions such panels will exhibit a benign instability, that is a stable limit cycle oscillation, or a catastrophic instability, that is an unstable LCO. The aerothermoelastic governing equations are developed from the geometrically non-linear theory of infinitely long two dimensional curved panels. Von Karman non-linear strain-displacement relation in conjunction with the Kirchhoff plate-hypothesis is adopted. A geometrically imperfect curved panel forced by a supersonic/hypersonic unsteady flow is numerically investigated using Galerkin approach. These equations are based on the third-order piston theory aerodynamic for modeling the flow-induced forces. Furthermore, the effects of thermal degradation and Kelvins model of structural damping independent of time and temperature are also considered in this model. Computational analysis and discussion of the finding along with pertinent conclusions are presented.

Cross-sectional modelling of thin-walled composite beams

A formulation for computationally efficient method of modelling the cross-sectional properties of a thin-walled, multi cell, and prismatic beam with anisotropic material properties, is presented. The formulation is straightforward to facilitate sensitivity analysis, making it suitable for optimization framework. Starting from shell kinematics, the Variational Asymptotic Method is used to formulate asymptotically correct expression of the energy functional in terms of small parameters inherent to thin-walled, and slender structures. Using the Saint Venant’s theory and the solvability conditions, the energy functional is written in terms of the cross-sectional forces and inverse of the Timoshenko stiffness matrix. Finite element method is then employed to discretize the cross-section using 1D hermitian beam elements and the cross-sectional stiffness properties are evaluated numerically. Results from the current method are then compared with VABS and NABSA, showing improved solution for the shear and coupling terms of the Timoshenko stiffness matrix.

Wind load effect in topology optimization problems

Topology optimization of two dimensional structures subject to dead and wind loading is considered. The wind loading is introduced into the formulation by using standard expressions for the drag force. A design problem formulation is constructed for minimum compliance design subject to a volume constraint and using the popular SIMP material model. The method of moving asymptotes (MMA) is used to solve the optimization problem. The MMA is modified by including line search and changing the formula for the update of asymptotes. In order to obtain black/white design, intermediate density values, which are used as design variables, are controlled by imposing an explicit constraint. Numerical examples demonstrate that the proposed formulation is successful in incorporating the effect of wind loading into the topology optimization problem.

Cellular Automata Paradigm for Topology Optimisation

Cellular Automata (CA) is an emerging paradigm for the combined analysis and design of complex systems using local update rules. Several algorithms based on of the CA paradigm have recently been demonstrated successfully for the topology optimisation of structures. In the present paper, the elements of a CA paradigm for topology optimisation are discussed. A framework for a biologically inspired CA topology optimisation is proposed and an initial effort to fit existing CA topology optimisation studies within this basic framework.