Stacking sequence optimization and blending design of laminated composite structures

The stacking sequence optimization problem for multi-region composite structures is studied in this work by considering both blending and design constraints. Starting from an initial stacking sequence design, unnecessary plies can be removed from this initial design and layer thicknesses of necessary plies are optimally determined. The existence of each ply is represented with discrete 0/1 variables and ply thicknesses are treated as continuous variables. A first-level approximate problem is constructed with branched multipoint approximate functions to replace the primal problem. To solve this approximate problem, genetic algorithm is firstly used to optimize discrete variables, and meanwhile, a blending design scheme is proposed to generate a blended structure. Starting from the thinnest region, this scheme shares all layers of current thinnest region with its adjacent regions. For non-shared layers in the adjacent regions, local mutation is implemented to add or delete plies to make them efficient designs. The whole process is repeated until the blending rule is satisfied. After that, a second-level approximate problem is built to optimize the continuous variables of ply thicknesses for retained layers. Those procedures are repeated until the optimal solution is obtained. Numerical applications, including a two-patch panel and a corrugated central cylinder in a satellite, are conducted to demonstrate the efficacy of the optimization strategy.

     

A bilevel integer programming method for blended composite structures

This paper proposes a new approach for the design of a composite structure. This approach is formulated as an optimization problem where the weight of the structure is minimized such that a reserve factor is higher than a threshold. The thickness of each region of the structure is optimized together with its stacking sequence and the ply drop-offs. The novelty of this approach is that, unlike in common practice, the optimization problem is not simplified and split into two steps, one for finding the thicknesses and one for the stacking sequence. The optimization problem is solved without any simplification assumption. It is formulated as a bilevel integer programming and it uses the backtracking procedure to satisfy the blending and the manufacturing rules. Some numerical experiments are performed to show the efficiency of the proposed optimization method over complex cases which cannot be solved with the existing methods.     

Laminate optimization of blended composite structures using a modified Shepard’s method and stacking sequence tables

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.     

An integrated optimization for laminate design and manufacturing of a CFRP wheel hub based on structural performance

An integrated optimization that comprehensively considers design and manufacturing factors such as the geometric appearance, laminate constitutions, laminate distribution, laminate thickness and stacking sequence, is proposed for designing a carbon fiber reinforced polymer wheel hub of a racecar. First, the driving conditions of the racecar are analyzed to determine the performance requirements. Then, under the condition that the geometric design regions are partitioned and the constitutions of fiber plies with different directions are defined, laminate design and manufacturing model is established. A multi-objective optimization is then performed to achieve a lightweight, high-stiffness laminate structure in different design regions. Next, number of plies in each region is obtained from the thickness of laminate, and then, the stacking sequence is optimized to improve the stiffness of the laminate structure. Finally, laminate transitions for different regions are investigated. The results showed that laminate design and manufacturing optimization can reduce the weight of the wheel hub and improve the performance of the wheel hub under static, dynamic and impact conditions. The proposed optimization approach provides a feasible solution for a performance-based design of composite structures.     

A repair operator for the preliminary design of a composite structure using genetic algorithms

This paper deals with the preliminary design of a composite structure where the design variables are the thicknesses and the percentages of fiber orientations in the zones of the structure. In this paper, we propose to include the design and manufacturing rules in the preliminary design. A stacking sequence generator is used to compute admissible stacking sequences with respect to these rules and which correspond to the design variables. Given that an admissible stacking sequence does not exist for every set of values of the design variables, a repair operator is proposed to cope with this problem. It aims at changing the values of the percentages of the fiber orientations in order to guarantee the existence of admissible stacking sequences with respect to the manufacturing rules. The repair operator is integrated into an optimization loop which uses a genetic algorithm to perform the preliminary design of a composite structure. Its efficiency is shown with a test case which involves a large number of fiber orientations and stacking sequences.     

Topology and thickness optimization of laminated composites including manufacturing constraints

This paper presents a gradient based optimization method for large-scale topology and thickness optimization of fiber reinforced monolithic laminated composite structures including certain manufacturing constraints to attain industrial relevance. This facilitates application of predefined fiber mats and reduces the risk of failure such as delamination and matrix cracking problems. The method concerns simultaneous determination of the optimum thickness and fiber orientation throughout a laminated structure with fixed outer geometry. The laminate thickness may vary as an integer number of plies, and possible fiber orientations are limited to a finite set. The conceptual combinatorial problem is relaxed to a continuous problem and solved on basis of interpolation schemes with penalization through the so-called Discrete Material Optimization method, explicitly including manufacturing constraints as a large number of sparse linear constraints. The methodology is demonstrated on several numerical examples.     

Artificial Bee Colony (ABC) for multi-objective design optimization of composite structures

In this paper, we present a generic method/model for multi-objective design optimization of laminated composite components, based on Vector Evaluated Artificial Bee Colony (VEABC) algorithm. VEABC is a parallel vector evaluated type, swarm intelligence multi-objective variant of the Artificial Bee Colony algorithm (ABC). In the current work a modified version of VEABC algorithm for discrete variables has been developed and implemented successfully for the multi-objective design optimization of composites. The problem is formulated with multiple objectives of minimizing weight and the total cost of the composite component to achieve a specified strength. The primary optimization variables are the number of layers, its stacking sequence (the orientation of the layers) and thickness of each layer. The classical lamination theory is utilized to determine the stresses in the component and the design is evaluated based on three failure criteria: failure mechanism based failure criteria, maximum stress failure criteria and the tsai-wu failure criteria. The optimization method is validated for a number of different loading configurations—uniaxial, biaxial and bending loads. The design optimization has been carried for both variable stacking sequences, as well fixed standard stacking schemes and a comparative study of the different design configurations evolved has been presented. Finally the performance is evaluated in comparison with other nature inspired techniques which includes Particle Swarm Optimization (PSO), Artificial Immune System (AIS) and Genetic Algorithm (GA). The performance of ABC is at par with that of PSO, AIS and GA for all the loading configurations.     

Composite stacking sequence optimization for aeroelastically tailored forward-swept wings

A method for stacking sequence optimization and aeroelastic tailoring of forward-swept composite wings is presented. It exploits bend-twist coupling to mitigate aeroelastic divergence. The method proposed here is intended for estimating potential weight savings during the preliminary aircraft design stages. A structural beam model of the composite wingbox is derived from anisotropic shell theory and the governing aeroelastic equations are presented for a spanwise discretized forward swept wing. Optimization of the system to reduce wing mass is undertaken for sweep angles of ?35° to 0° and Mach numbers from 0.7 to 0.9. A subset of lamination parameters (LPs) and the number of laminate plies in each pre-defined direction (restricted to {0°,±45°, 90°}) serve as design variables. A bi-level hybrid optimization approach is employed, making use of a genetic algorithm (GA) and a subsequent gradient-based optimizer. Constraints are implemented to match lift requirements and prevent aeroelastic divergence, excessive deformations, airfoil stalling and structural failure. A permutation GA is then used to match specific composite ply stacking sequences to the optimum design variables with a limited number of manufacturing constraints considered for demonstration purposes. The optimization results in positive bend-twist coupling and a reduced structural mass. Results are compared to an uncoupled reference wing with quasi-isotropic layups and with panel thickness alone the design variables. For a typical geometry and a forward sweep of ?25° at Mach 0.7, a wingbox mass reduction of 13 % was achieved.     

DMTO – a method for Discrete Material and Thickness Optimization of laminated composite structures

This paper presents a gradient based topology optimization method for Discrete Material and Thickness Optimization of laminated composite structures, labelled the DMTO method. The capabilities of the proposed method are demonstrated on mass minimization, subject to constraints on the structural criteria; buckling load factors, eigenfrequencies, and limited displacements. Furthermore, common design guidelines or rules, referred to as manufacturing constraints, are included explicitly in the optimization problem as series of linear inequalities. The material selection and thickness variation are optimized simultaneously through interpolation functions with penalization. Numerical results for several parameterizations of a finite element model of a generic main spar from a wind turbine blade are presented. The different parameterizations represent different levels of complexity with respect to manufacturability. The results will thus give insight into the relation between potential weight saving and design complexity. The results show that the DMTO method is capable of solving the problems robustly with only few intermediate valued design variables.     

A primal-dual backtracking optimization method for blended composite structures

A combinatorial optimization method is proposed for finding the optimal stacking sequence and the ply drop-offs of a blended composite structure. This method assumes that the thicknesses of the regions of the structure are fixed in advance. It is able to handle efficiently design and manufacturing rules which are of combinatorial type. The optimization problem is formulated as a constrained binary programming problem and it is solved by applying both a primal and a dual backtracking procedures with a local search method. Some numerical experiments are carried out to show the efficiency of the optimization method with respect to both computational time and quality criteria.     

Optimal design of sandwich and laminated composite plates based on the planning of experiments

Optimal design problems of sandwich plates with soft core and laminated composite face layers, and multilayered composite plates are investigated. The optimal design problems are solved by using the method of the planning of experiments. The optimization procedure is divided into the following stages: choice of control parameters and establishment of the domain of search, elaboration of plans of experiment for the chosen number of reference points, execution of the experiment, determination of simple mathematical models from the experimental data, design of the structure on the basis of the mathematical models discovered, and finally verification experiments at the point of the optimal solution. Vibration and damping analysis is performed by using a sandwich plate finite elements based on a broken line model. Damping properties of the core and face layers of the plate are taken into account in the optimal design. Modal loss factors are computed using the method of complex eigenvalues or the energy method. Frequencies and modal loss factors of the plate are constraints in the optimal design problem. There are also constraints on geometrical parameters and the bending stiffness of the plate. The mass of the plate is the objective function. Design parameters are the thickness of the plate layers. In the points of experiments computer simulation using FEM is carried out. Using this information, simple mathematical models for frequencies and modal loss factors for the plate are determined. These simple mathematical functions are used as constraints in the nonlinear programming problem, which is solved by random search and the penalty function method. Numerical examples of the optimal design of clamped sandwich and simply supported laminated composite plates are presented. A significant improvement of damping properties of a sandwich plate is observed in comparison with a simple plate of equal natural frequencies.     

Development of a structural optimization strategy for the design of next generation large thermoplastic wind turbine blades

This paper presents the development of a structural optimization process for the design of future large thermoplastic wind turbine blades. The optimization process proposed in this paper consists of three optimization steps. The first step is a topology optimization of a short untwisted and non tapered section of the blade, with the inner volume used as the design domain. The second step is again a topology optimization, but on the first half of a blade to study the effect of non symmetry of the structure due to blade twist and taper. Results of this optimization step are then interpreted to build a shell model of the complete blade structure to perform composite size optimization based on a minimum mass objective subjected to constraints on deflection, composite strength and structural stability. Different blade models using ribs are then optimized and compared against conventional blade structure (box spar structure without ribs and single web structure without ribs). The use of ribs in wind turbine blade structures, which is more adapted to thermoplastic composite manufacturing than for thermoset composites, leads to slightly lighter blades than conventional blade structures.     

Optimum design of composite shells subject to natural frequency constraints

A technique for the optimum (minimum weight) design of a composite shell subject to constraints on its natural frequencies is presented. The optimization problem is posed as a general mathematical programming problem in which one or more of the inequality constraints involves the shell natural frequencies, which must be evaluated numerically during the optimization. For this reason, a method for numerically evaluating the natural frequencies of composite shells is also presented. The method is based upon the finite element method of structural analysis and Rayleigh's principle. Because the element used is applicable to anisotropic shells of arbitrary shape, the method is very general. By using Rayleigh's principle, the necessity of assembling overall mass and stiffness matrices for the shell is eliminated. The optimization is performed by nondimensionalizing the mathematical programming problem and using the penalty function method of Fiacco and McCormick to transform the problem to a sequence of unconstrained minimizations having solutions which converge to the solution of the original (constrained) problem. The unconstrained minimizations are performed using the variable metric method of Fletcher and Powell. Derivatives of the nondimensional frequency constraints are evaluated numerically using difference equations. The frequency calculation method is demonstrated by calculating the fundamental frequency for the transverse vibration mode of a multilayered cylindrical shell with fixed overall geometry and variable composite geometry. Results indicate that the frequency increases with increasing fiber orientation angle, fiber volume fraction, or lamina thickness. The optimization technique is demonstrated by minimizing the weight of the shell discussed above subject to a constraint on its fundamental transverse frequency. The design variables are the fiber orientation angle, the fiber volume fraction, and the lamina thickness. Results are presented and explained in terms of the physical aspects of the problem.     

On the formulation and implementation of geometric and manufacturing constraints in node–based shape optimization

We introduce a novel method to handle geometrical and manufacturing constraints in parameter–free shape optimization. Therefore the design node coordinates are split in two sets where one set is declared as new design variables and the other set is coupled to the new design variables such that the geometrical constraint is fulfilled. Thereby no additional equations are appended to the optimization problem. In contrast the implementation of a demolding constraint is presented by formulating inequality constraints which indeed have to be attached to the optimization problem. In the context of a sensitivity–based shape optimization approach all manufacturing constraints have to be formulated in terms of the finite element node coordinates such that first order gradients with respect to the design node coordinates can be derived.     

A procedure to design optimum composite plates using implicit decision trees

This paper presents a novel methodology to minimize the weight of a composite plate, taking into account both its structural integrity and its manufacturing constraints. This optimization problem has been abstracted, and reduced to, a graph problem where finding the optimum is equivalent to finding the shortest path in the graph. This graph is an implicit ternary decision tree and path finding is accomplished with a parallelized breadth-first search procedure. As a result of this search, the procedure is able to provide both a global optimum and a Pareto front. The implementation of an implicit decision tree as a way to optimize a laminate stacking sequence is a novel idea, in spite that both issues have been tackled in numerous publications separately. Its benefits are clearly stated in this work.     

Efficient size and shape optimization of truss structures subject to stress and local buckling constraints using sequential linear programming

The advance in digital fabrication technologies and additive manufacturing allows for the fabrication of complex truss structure designs but at the same time posing challenging structural optimization problems to capitalize on this new design freedom. In response to this, an iterative approach in which Sequential Linear Programming (SLP) is used to simultaneously solve a size and shape optimization sub-problem subject to local stress and Euler buckling constraints is proposed in this work. To accomplish this, a first order Taylor expansion for the nodal movement and the buckling constraint is derived to conform to the SLP problem formulation. At each iteration a post-processing step is initiated to map a design vector to the exact buckling constraint boundary in order to facilitate the overall efficiency. The method is verified against an exact non-linear optimization problem formulation on a range of benchmark examples obtained from the literature. The results show that the proposed method produces optimized designs that are either close or identical to the solutions obtained by the non-linear problem formulation while significantly decreasing the computational time. This enables more efficient size and shape optimization of truss structures considering practical engineering constraints.     

A hierarchical genetic algorithm with age structure for multimodal optimal design of hybrid composites

A genetic algorithm aiming the optimal design of composite structures under non-linear behaviour is presented. The approach addresses the optimal material/stacking sequence in laminate construction and material distribution topology in composite structures as a multimodal optimization problem. The proposed evolutionary process is based on a sequential hierarchical relation between subpopulations evolving in separated isolation stages followed by migration. Improvements based on the species conservation paradigm are performed to avoid genetic tendencies due to elitist strategies used in the hierarchical subpopulations. The concept of species is associated with material distribution topology in composite structures, and an enlarged master population with age structure is considered concurrently with the hierarchical topology. Rules based on species concept are imposed on either isolation or migration stages to overcome the predominance of a species and to guarantee the diversity. A mutation process controlled by the stress field is implemented, improving the local genetic search. The proposed model allows multiple solutions for the optimal design problem.     

Stacked Regressions

Stacking regressions is a method for forming linear combinations of different predictors to give improved prediction accuracy. The idea is to use cross-validation data and least squares under non negativity constraints to determine the coefficients in the combination. Its effectiveness is demonstrated in stacking regression trees of different sizes and in a simulation stacking linear subset and ridge regressions. Reasons why this method works are explored. The idea of stacking originated with Wolpert (1992).     

Geological uncertainties associated with 3-D subsurface models

Subsurface models are generally built from both subjective interpretation and mathematical interpolation/extrapolation techniques. These models are therefore uncertain, but their uncertainty is rarely expressed in a geological forecast. In this paper, an evaluation method of geological uncertainties related to 3-D subsurface models is proposed and tested on a real case. This method is based on the subsurface model, which is considered the most probable prediction (best guess). The various geological interfaces are handled as Gaussian random fields to which a model of spatial variability describing possible fluctuations around the best guess is applied. Several structural constraints, such as the shape of folds and thickness of layers are accounted for in the model. At this point, the local variance can be estimated throughout the study area by application of the simple kriging technique. Finally, the variability is converted into probabilities of occurrence of the various rock masses present in the study area. The probabilities are calculated according to intersection rules governing the stratigraphic sequence of the subsurface model. They enable one to probabilistically model subsurface structures in the form of a three-dimensional (3-D) probability field.     

Automatic design of tapered member gabled frames

A program for calculating member sizes to yield a minimum with grabled frame with tapered members has been developed. The design produced satisfies in all respects the requirements of Appendix D to the 1969 AISC specification which governs the design of tapered members. The frame design is symmetrical about the vertical centerline, although the loading need not by symmetrical. The rafter can have no, one or two changes of taper within its length, at the designer's option. The matrix force method is used to do the analyses necessary in the design process. The problem is solved by finding a design which minimizes frame weight subject to constraints imposed by the design specification,such as maximum stress and maximum width/thickness ratios, to minimum or interior penalty function approach using the variable metric method of Davidon, Fletcher and Powell. A number of example penalty function design are given to show the versatility of the technique, as well as demonstrating the effect parmeters such as purlin spacing, member length, rafter slope and member depth to width ratio have on resulting designs.