Shahriar Setoodeh

Approximate Feasible Regions for Lamination Parameters

Lamination parameters represent the laminate lay-up configuration in a compact form. According to their definitions, lamination parameters are related and cannot be arbitrarily selected. The region which defines physically meaningful lamination parameters is called the feasible lamination parameters region. In order to formulate a typical structural design problem using lamination parameters, the feasible domain for the particular set of lamination parameters is required. Such a domain is known to be convex and is available in closed form only for a limited number of combinations of the lamination parameters. In the present study, we exploit the convexity of these feasible regions and outline a numerical approach using successive convex hull approximations. In the successive convex hull approximation we refine the feasible region by increasing number of layers as well as number of possible fiber orientation angles. In this process, the volume of the feasible domain is monitored to measure the convergence of the approach. The final approximation is presented in the form of linear inequality constraints (hyperplanes) that can be included in typical structural optimization problems as side constraints. The major advantage of such a methodology is that this approximation needs to be done only once and the resulting set of constraints can be stored and reused for any other structural optimization problem. First, numerical results are compared for the cases where exact definitions exist for feasible domains. Next we apply the present approach to sets of lamination parameters for which exact definitions for feasible regions do not exist and compare our results with approximate results using other methods already available in the literature.

Generating Curvilinear Fiber Paths from Lamination Parameters Distribution

Contrary to the classical stacking sequence design of composite laminates with straight fibers, each ply can be designed with curvilinear fiber paths resulting in variation of the stiffness properties over the structure. Such laminates are often denoted as “variablestiffness panels” in the literature. Instead of treating fiber orientation angles as spatial design variables, lamination parameters can be used. However, retrieving the actual stacking sequence requires additional efforts at a post processing level. In this paper, we presume that the optimal distribution of lamination parameters is already obtained for a particular design problem. Then we use curve fitting techniques to obtain continuous fiber paths that result in a distribution of the lamination parameters close to the optimal distribution in a least square sense while satisfying the manufacturing curvature constraint. The fiber orientation angle is expanded using a set of basis functions and unknown coefficients. The unknown coefficients are then computed such t hat the assumed form represents the optimal distribution of the lamination parameters in a least square sense. The key feature of such approach is that at this post-processing step, no more expensive finite element analyses are needed and the curve fitting is performed using simple polynomial and trigonometric function evaluations. The curve fitting problem is then solved using a constrained nonlinear least square solver where maximum curvature is controlled using a side constraint. Numerical results demonstrate the efficiency of the proposed formulation for minimum compliance design problems. For the cantilever plate problem investigated, the compliance of the approximate design is only than 2.5% larger than than the compliance of optimal lamination parameters design. A methodology is also proposed to estimate the thickness buildup due to the curved fiber paths.

Pipeline implementation of cellular automata for structural design on message-passing multiprocessors

The inherent structure of cellular automata is trivially parallelizable and can directly benefit from massively parallel machines in computationally intensive problems. This paper presents both block synchronous and block pipeline (with asynchronous message passing) parallel implementations of cellular automata on distributed memory (message-passing) architectures. A structural design problem is considered to study the performance of the various cellular automata implementations. The synchronous parallel implementation is a mixture of Jacobi and Gauss-Seidel style iteration, where it becomes more Jacobi like as the number of processors increases. Therefore, it exhibits divergence because of the mathematical characteristics of Jacobi iteration matrix for the structural problem as the number of processors increases. The proposed pipeline implementation preserves convergence by simulating a pure Gauss-Seidel style row-wise iteration. Numerical results for analysis and design of a cantilever plate made of composite material show that the pipeline update scheme is convergent and successfully generates optimal designs.

Design of Variable-Stiffness Composite Laminates for Maximum In-Plane Stiffness Using Lamination Parameters

Benefits of directional properties of fiber reinforced composites could be fully utilized by proper placement of the fibers in their optimal spatial orientations. This paper investigates the optimal in-plane design of variable-stiness fiber reinforced composites using lamination parameters. The classical minimum compliance design problem is formulated in the lamination parameters space and is solved using a feasible sequential quadratic programming solver at any point in the domain. Such formulation has two key features; first, it limits the number of design variables to four regardless of the actual number of layers, second, the local design problem is convex in the lamination parameters space. General laminate lay-up case and balanced symmetric laminates are studied separately. The special case of balanced symmetric laminates with equal thickness layers is studied by introducing a restricted design problem. Also a methodology for computing fiber angles for the restricted problem is presented.

DESIGN OF COMPOSITE LAYERS WITH CURVILINEAR FIBER PATHS USING CELLULAR AUTOMATA

The anisotropic advantageous properties of fiber reinforced composites may not be fully exploited unless the fibers are properly placed in their optimal spatial orientations. This paper investigates application of Cellular Automata (CA) for curvilinear fiber design of composite laminae for in-plane responses. CA use local rules to update both field and design variables in an iterative scheme till convergence. In the present study, displacement update rules are derived using a finite element model governing the equilibrium of the cell neighborhood and fiber angles are locally optimized based on a minimum strain energy criterion. A manufacturing improvement is applied on top of the local optimum orientation wherever this angle is not consistent with the cell neighborhood orientation trend. Numerical studies showed convergency of the local update rules and considerable improvements in the stiffness properties for a cantilever bending test and a square plate with a cutout.

LOCAL UPDATE SCHEMES FOR CELLULAR AUTOMATA IN STRUCTURAL DESIGN

This paper investigates original local update schemes for Cellular Automata (CA) in structural design. Local problems based on mathematical programming and optimality criteria are tested, allowing the isolation of an original local update scheme. The scheme consists of repeating analysis and optimality-based design rules locally. Several systematic experiments on various problem sizes are performed to show the efficiency and robustness of the approach. For comparison, the experiments are also run using a more traditional CA implementation.

Design of Variable Stiffness Composite Laminates For Maximum Bending Stiffness

Benefits of directional properties of fiber reinforced composites could be fully utilized by proper placement of the fibers in their optimal spatial orientations. This paper investigates an application of a Cellular Automata (CA) based strategy for design of variable stiffness composite laminates for optimal bending stiffness. CA are iterative numerical techniques that use local rules to update both field and design variables to satisfy equilibrium and optimality conditions. In the present study, displacement update rules are derived using a cell level model governing the equilibrium of the CA neighborhood. Local fiber orientation angles are treated as continuous design variables, and their spatial distribution is determined based on an optimality criterion formulation for minimum compliance design. Numerical examples for simply supported and clamped square plates are used to demonstrate the improvement in bending stiffness.