Design and optimization of variable stiffness composite structures modeled using Bézier curve

Abstract

The usage of advanced fiber-reinforced polymer (FRP) matrix composites has been dramatically increased since the first carbon fiber patented in the 1960's. Particularly, the aerospace companies' interest has been gradually grown in carbon fiber-reinforced polymer (CFRP) aircraft structures due to major performance improvements such as high strength and stiffness to weight ratios and reduced weight. The traditional design approaches and manufacturing methodologies of CFRP structures in various industries have been well established and applied for more than 50 years. They are mainly developed for straight fibers and the optimum design solutions have been achieved by the choice of constituent materials, different fiber orientation angles that are often limited to 0, ±45, and 90 degrees, laminate stacking sequence and total number of plies. However, increasing complexity of structure geometries have resulted in complex lay-ups & contours; therefore, advanced manufacturing methodologies such as Automated Fiber Placement (AFP) and Tailored Fiber Placement (TFP) are developed to improve productivity and process reliability. Following the introduction of advanced manufacturing methods CFRP structures with complex geometry, complex lay-ups & contours have been manufactured with improved productivity and process reliability. In addition to that, composite materials can be tailored more effectively to meet design requirements by changing the design approach from straight to curvilinear fibers. The composite structures designed with curvilinear fibers have spatially varying stiffness due to local fiber orientations in the ply, and accordingly they are named as variable stiffness (VS) structures. In this dissertation, the variable stiffness composite plates and circular cylindrical shells modeled using parametric Bézier curves as curvilinear fiber paths are designed and optimized. The design method with parametric Bézier curves covers a wide and complex design space from simple linear angle variation to constant curvature path to highly nonlinear angle variations. The designed VS composite structures are expressed with new lay-up definition conventions that use simple and intuitive variables such as segment/station angles and multipliers/curvatures. The optimum structural designs in the complex design space of plates and circular cylindrical shells are searched using a multi-step optimization with multi-objective such as buckling and stiffness, and a novel pre-trained multi-step/cycle surrogate-based optimization (PMSO) framework with single objective, i.e. buckling, respectively. First, VS composite plates and circular cylinders are designed with 'Direct Fiber Path Parameterization' (DFPP) that uses continuous curve functions for fiber orientation angles at each point or grid in the laminate. The cubic and quadratic Bézier curves are used as curvilinear fiber path. The fiber paths as Bézier curves are constructed with approximation and interpolation formulations. The approximation curve captures the defined angles at the start point and the end point, and the shape of the curve changes with the position of the control points intuitively. On the other end, interpolation curve follows the exact positions of control points at the expense of control of the fiber angle. Therefore, fiber angles are different from the defined sector angles. Three types of parametric curves are formulated, i.e., cubic Bézier interpolation curve and quadratic and cubic Bézier approximation curves. Cubic Bézier approximation curves are specially formulated to define constant curvature fiber paths. Considering the characteristics of Bézier curves, intuitive conventions to define lay-ups of laminated VS plates and shells are proposed. The position of course boundaries within each ply are calculated using the reference fiber path, and resulting courses are shifted along one direction to cover VS plate and cylindrical shell surfaces. The reference fiber paths are defined with design variables such as sector/station angles and multipliers/curvatures, which are used to calculate control points. Current proposal for lay-up definition allows one to move stations using multipliers within an interval, hence it is possible to find lower curvature fiber paths with the same sector angles. The minimum curvature value is a major characteristic of curvilinear fiber paths due to manufacturing constraints. Golden Section Search and Downhill Simplex methodologies are used depending on the design approach together with Bézier curve formulations. The Golden Section Search method, which is a technique for finding an extremum, (minimum or maximum) of a unimodal function, is applied to approximation curves, and Downhill Simplex method is applied to interpolation curves due to a multidimensional space with n multipliers. The curvature values are significantly minimized without changing the lay-up definitions; especially for quadratic Bézier approximation curves, the curvature distribution along characteristic length gets close to the constant curvature results. Three different geometries for VS plates (b/a ≈ 1.8) and two different geometries for VS circular cylinder Cylinder 1 (L/R ≈ 2.67) and Cylinder 2 (L/R = 2) are modeled. Considering the cylindrical coordinates, the courses laid on the cylinder are axially shifted to have circumferentially varying stiffness and strength; however, the effective width of the ply is modified to have continuous fiber paths around the circumference. To have averaged boundaries, which is called no gap condition, minimum effective course width is used as the reference shifting value. The lay-up process is completed on developed plane of the cylinder, and then translated into cylindrical coordinates. Second, finite element models of laminated VS plates and cylindrical shells are generated using Ansys Mechanical APDL codes. Four node Shell 181 quadrilateral elements with full integration are used to mesh the VS plate and the VS composite shells with Cylinder 1 geometry, and FE models of layered VS composite shells with Cylinder 2 geometry are generated using eight node Shell 281 elements with reduced integration. Both shell elements are based on the first-order shear-deformation theory (referred to as Mindlin-Reissner shell theory). These elements with six degrees of freedom at each node (translations in the nodal x, y, and z directions and rotations about the nodal x, y, and z axis) are usually used to analyze thin to moderately-thick shell structures. The mesh convergence studies of reference QI plate and VS circular shells and plates are performed, and reference element edge lengths are chosen considering accurate mapping of curvilinear fiber paths on finite element mesh, buckling results, and computational efficiency. The curvilinear fiber paths for each ply are then mapped to related element centroids by APDL functions. Next, the VS laminates and circular cylinders are optimized for maximum stiffness and/or buckling load using surrogate-based NSGA-II algorithm. The NSGA-II is an evolutionary algorithm and supports multi-objective optimizations. The design space development strategy is an important part of surrogate modeling to get optimal distribution of fewest number of points with maximum insight into the design. Thus, experimental designs are generated with Optimal Space Filling (OSF) algorithm according to specified intervals. Then, surrogate models are generated with Genetic Aggregation. The Genetic Aggregation selects the best solution from Full 2nd-Order Polynomials, Non-Parametric Regression, Kriging, and Moving Least Squares. The algorithm generates the population of all methods and then it applies single response surface or combination of response surfaces according to fitness functions. The assemblage of Genetic Aggregation surrogate model is constructed with weighted average of selected meta-models. The weight and the combination of meta-models depend on design of experiment method and the behavior of VS structures designed with the approximation and interpolation curves. Two-cycle approach is used to increase the accuracy of the surrogate models. The first cycle consists of the design space between 80° and -80°, and the second cycle searches for ±20 degree of the optimum angle calculated at the first cycle. A better lay-up for Size 1 – Case 3 compared to results in literature is found by using reduced the domain in the second cycle. The best buckling performance is found for Size 3 plate with Case 3 boundary conditions that has 103% increase in buckling load against 44% reduction in equivalent stiffness compared to reference quasi-isotropic laminate. It is clear that increase in plate size increases the buckling performance of VS plates. This is due to wider design space with relaxed curvature constraint that allows higher angle differences between edge and the middle of the plate, accordingly fiber angle at the plate edges can align closer to loading direction while the fiber angles far from edge converges to smaller angles. The quadratic Bézier approximation curve is found to be a good alternative of cubic Bézier approximation curve with constant curvature, as it has similar edge load distribution and buckling mode shapes. Additionally, the stations, which are fixed for cubic Bézier approximation curve with constant curvature, can be shifted for definition with quadratic approximation without changing the lay-up definition according to designer's need. Finally, a novel pre-trained design optimization framework is proposed to optimize buckling load of VS composite circular cylinders under pure bending with curvature and strength constraints. By using Bézier curves, designers have more effective control on the design domain to improve the buckling performance in accordance with requirements such as curvature and strength. The strength constrain is calculated by using Tsai-Wu failure criterion. The optimizations are conducted using PMSO framework that utilizes NSGA-II. The main benefit of this framework is to gather prior knowledge about the design space at the first step by conducting pre-training optimizations using laminated VS composite shells with single ply definition. This narrows down the design space significantly before conducting a full lay-up design optimization with large number of parameters at the second step. Moreover, multiple cycle approach at each step helps to reduce the complexity of the optimization together with increased surrogate model accuracy. The optimization is completed for four different laminate stack-ups that are made up of all VS plies and partial VS plies in combination with unidirectional fibers (±45°, 0° and 90°). The maximum increase in buckling load is found to be 31% for Laminate 1 and 41% for Laminate 4 compared to reference QI shells. This gives 14% and 16% higher buckling load than the literature studies, and the Laminate 4 results are achieved for two times more design variables using approximately same number of sampling points. The gain in buckling load is due to the redistribution of stresses on compression and tension side as a consequence of variable angle distribution within each ply. The fiber angles close to axial direction on the tension side increase the strength and stiffness of the structure, and angles close to circumferential axis on the compression side reduce the stiffness of buckling critical region to distribute the compressive loads onto wider region.