Interactive-adaptive geometrically nonlinear analysis of shell structures

This paper presents an investigation of interactive-adaptive techniques for nonlinear finite element structural analysis. In particular, effective methods leading to reliable automated, finite element solutions of nonlinear shell problems are of primary interest here. This includes automated adaptive nonlinear solution procedures based on error estimation and adaptive step length control, reliable finite elements that account for finite deformations and finite rotations, three-dimensional finite element modeling, and an easy-to-use, easy-to-learn graphical user interface with three-dimensional graphics. A computational environment, which interactively couples a comprehensive geometric modeler, an automatic three-dimensional mesh generator and an advanced nonlinear finite element analysis program with real-time computer graphics and animation tools, is presented. Three examples illustrate the merit and potential of the approaches adopted here and confirm the feasibility of developing fully automated computer aided engineering environments.     

Benchmark case studies in optimization of geometrically nonlinear structures

A structural optimization algorithm is developed for truss and beam structures undergoing large deflections against instability. The method combines the nonlinear buckling analysis using the displacement control technique, with the optimality criteria approaches. Several benchmark case studies illustrate the procedure and the results are compared with examples reported in the literature. It is shown that a design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass or overestimates the buckling load for these types of structures, so a design based on the linear buckling analysis may result in catastrophic failure. The effect of geometrical nonlinearities and element imperfections has also been studied.     

Shape preserving design of geometrically nonlinear structures using topology optimization

Structural and Multidisciplinary Optimization - Subparts of load carrying structures like airplane windows or doors must be isolated from distortions and hence structural optimization needs to take...     

A 213-line topology optimization code for geometrically nonlinear structures

Structural and Multidisciplinary Optimization - This paper presents a 213-line MATLAB code for topology optimization of geometrically nonlinear structures. It is developed based on the density...     

Discrete optimization of geometrically nonlinear truss structures under stability constraints

The paper deals with discrete optimization of elastic trusses with geometrical nonlinear behaviour and constraints on stability. The problem consists of minimizing the weight and determining the optimal member distribution so that the external load does not cause a loss of stability of the structure. Member cross-sections are selected from a catalogue of available sections. Element stresses, elment stability and global structural stability constraints are considered. A controlled enumeration method according to the increasing value of the objective function is applied. Shallow space trusses are numerically analysed.     

A displacement based optimization method for geometrically nonlinear frame structures

An extension of the displacement based optimization method to frames with geometrically nonlinear response is presented. This method, when applied to small-scale trusses with linear and nonlinear response, appeared to be efficient providing the same solutions as the classical optimization method. The efficiency of the method is due to the elimination of numerous finite element analyses that are required in using the traditional optimization approach. However, as opposed to trusses, frame problems have typically a larger number of degrees of freedom than cross sectional area design variables. This leads to difficulties in the implementation of the method compared to the truss implementation. A scheme that relaxes the nodal equilibrium equations is introduced, and the method is validated using test examples. The optimal designs obtained by using the displacement based optimization and the classical approaches are compared to validate the application to frame structures. The characteristics and limitations of the optimization in the displacement space for sizing problems, based on the current formulation, are discussed.     

Nonlinear topology optimization of layered shell structures

Topology stiffness (compliance) design of linear and geometrically nonlinear shell structures is solved using the SIMP approach together with a filtering scheme. A general anisotropic multi-layer shell model is employed to allow the formation of through-the-thickness holes or stiffening zones. The finite element analysis is performed using nine-node Mindlin-type shell elements based on the degenerated shell approach, which are capable of modeling both single and multi-layered structures exhibiting anisotropic or isotropic behavior. The optimization problem is solved using analytical compliance and constraint sensitivities together with the Method of Moving Asymptotes (MMA). Geometrically nonlinear problems are solved using iterative Newton–Raphson methods and an adjoint variable approach is used for the sensitivity analysis. Several benchmark tests are presented in order to illustrate the difference in optimal topologies between linear and geometrically nonlinear shell structures.     

Topological shape optimization of geometrically nonlinear structures using level set method

Using the level set method, a topological shape optimization method is developed for geometrically nonlinear structures in total Lagrangian formulation. The structural boundaries are implicitly represented by the level set function, obtainable from “Hamilton-Jacobi type” equation with “up-wind scheme,” embedded into a fixed initial domain. The method minimizes the compliance through the variations of implicit boundary, satisfying an allowable volume requirement. The required velocity field to solve the Hamilton-Jacobi equation is determined by the descent direction of Lagrangian derived from an optimality condition. Since the homogeneous material property and implicit boundary are utilized, the convergence difficulty is significantly relieved.     

Constrained adaptive topology optimization for vibrating shell structures

This paper describes an algorithm for structural topology optimization entitled Constrained Adaptive Topology Optimization or CATO which is applied here to produce the optimum design of shell structures under free vibration conditions. The algorithm, based on an artificial material model and an updating scheme, combines ideas from the more mathematically rigorous homogenization (h) methods and the more intuitive evolutionary (e) methods. Thus, CATO can be seen as a hybrid h/e method. The optimization problem is defined as maximizing or minimizing a chosen frequency with a constraint on the structural volume/mass by redistributing the material through the structure. The efficiency of the proposed algorithm is illustrated through several numerical examples. Received February 17, 2000     

Simultaneous shape and topology optimization of shell structures

In this research, Method of Moving Asymptotes (MMA) is utilized for simultaneous shape and topology optimization of shell structures. It is shown that this approach is well matched with the large number of topology and shape design variables. The currently practiced technology for optimization is to find the topology first and then to refine the shape of structure. In this paper, the design parameters of shape and topology are optimized simultaneously in one go. In order to model and control the shape of free form shells, the NURBS (Non Uniform Rational B-Spline) technology is used. The optimization problem is considered as the minimization of mean compliance with the total material volume as active constraint and taking the shape and topology parameters as design variables. The material model employed for topology optimization is assumed to be the Solid Isotropic Material with Penalization (SIMP). Since the MMA optimization method requires derivatives of the objective function and the volume constraint with respect to the design variables, a sensitivity analysis is performed. Also, for alleviation of the instabilities such as mesh dependency and checkerboarding the convolution noise cleaning technique is employed. Finally, few examples taken from literature are presented to demonstrate the performance of the method and to study the effect of the proposed concurrent approach on the optimal design in comparison to the sequential topology and shape optimization methods.     

Numerical methods for optimization of nonlinear shell structures

A numerical method for the optimal design of nonlinear shell structures is presented. The nonlinearity is only geometrical and the external load is assumed to be conservative. The nonlinear shell is analysed using standard nonlinear shell finite elements with the displacements and the rotation of the shell normals as independent analysis variables. Shell thicknesses and cross-sectional dimensions of beam stiffeners are used as design variables. The nonlinear optimization problem is solved using a Newton barrier method. The usefulness of the proposed method is demonstrated on shallow stiffened shell structures exhibiting significant nonlinear response.Presented at NATO ASI Optimization of Large Structural Systems, Berchtesgaden, Sept. 23 – Oct. 4, 1991     

A non-parametric free-form optimization method for shell structures

This paper presents a numerical shape optimization method for the optimum free-form design of shell structures. It is assumed that the shell is varied in the out-of-plane direction to the surface to determine the optimal free-form. A compliance minimization problem subject to a volume constraint is treated here as an example of free-form design problem of shell structures. This problem is formulated as a distributed-parameter, or non-parametric, shape optimization problem. The shape gradient function and the optimality conditions are theoretically derived using the material derivative formulae, the Lagrange multiplier method and the adjoint variable method. The negative shape gradient function is applied to the shell surface as a fictitious distributed traction force to vary the shell. Mathematically, this method is a gradient method with a Laplacian smoother in the Hilbert space. Therefore, this shape variation makes it possible both to reduce the objective functional and to maintain the mesh regularity simultaneously. With this method, the optimal smooth curvature distribution of a shell structure can be determined without shape parameterization. The calculated results show the effectiveness of the proposed method for the optimum free-form design of shell structures.     

C1 plate and shell finite elements for geometrically nonlinear analysis of multilayered structures

The aim of this work is to analyze the geometrically nonlinear mechanical behaviour of multilayered structures by a high order plate/shell finite element in order to predict displacements and stresses of such composite structures for design applications. Based on a conforming finite element method, a C¹ triangular six node finite element is developed using trigonometric functions for the transverse shear stresses. The geometric nonlinearity is based on von-Karmann assumptions and only five generalized displacements are used to ensure:

• •a cosine distribution for the transverse shear stresses with respect to the thickness co-ordinate, avoiding shear correction factors;
• •the continuity conditions between layers of the laminate for both displacements and transverse shear stresses;
• •the satisfaction of the boundary conditions at the top and bottom surfaces of the shells.

     

A multilevel genetic algorithm for optimization of geometrically nonlinear stiffened composite structures

A multilevel genetic algorithm aiming the global optimization of beam reinforced composite structures with nonlinear geometric behaviour is proposed. A unified approach based on load-displacement control for buckling and first ply failure analysis is adopted. The Newton-Raphson iterative scheme and the arc-length method are used for tracing the equilibrium path and later for updating the critical values. The proposed genetic algorithm performs several sequences of two optimization levels resulting from the decomposition of the original optimization problem. Independent genetic searches are implemented for each level where different fitness functions and sub-populations are considered. The genetic operators selection and crossover supported by an elitist strategy are used while the diversity of the sub-populations is guaranteed based on implicit mutation. A genetic material exchange between levels is performed using clones and so the offspring of matured sub-populations is guaranteed. To improve the efficiency of the multilevel genetic optimization a niche of population is induced after the first stage at both levels.     

Combined shape and reinforcement layout optimization of shell structures

This paper presents a combined shape and reinforcement layout optimization method of shell structures. The approach described in this work is applied to optimize simultaneously the geometry of the shell mid-plane as well as the layout of surface stiffeners on the shell. This formulation involves a variable ground structure, since the shape of the shell surface is modified in the course of the process. Here we shall consider a global structural design criterion, namely the compliance of the structure, following basically the classical problem of distributing a limited amount of material in the most favourable way.The solution to the problem is based on a finite element discretization of the design domain. The material within each of the elements is modelled by a second-rank layered Mindlin plate microstructure. By a simple modification, this type of microstructure can be used to find the optimum distribution of stiffeners on shell structures. The effective stiffness properties are computed analytically through a smear-out procedure. The proposed method has been implemented into a general optimization software called Odessy and satisfactorily applied to the solution of some numerical examples, which are illustrated at the end of the paper.     

On simultaneous shape and material layout optimization of shell structures

This work presents a computational method for integrated shape and topology optimization of shell structures. Most research in the last decades considered both optimization techniques separately, seeking an initial optimal topology and refining the shape of the solution later. The method implemented in this work uses a combined approach, were the shape of the shell structure and material distribution are optimized simultaneously. This formulation involves a variable ground structure for topology optimization, since the shape of the shell mid-plane is modified in the course of the process. It was considered a simple type of design problem, where the optimization goal is to minimize the compliance with respect to the variables that control the shape, material fraction and orientation, subjected to a constraint on the total volume of material. The topology design problem has been formulated introducing a second rank layered microestructure, where material properties are computed by a “smear-out” procedure. The method has been implemented into a general optimization software called ODESSY, developed at the Institute of Mechanical Engineering in Aalborg. The computational model was tested in several numerical applications to illustrate and validate the approach.     

Topology optimization for finding shell structures manufactured by deep drawing

This paper presents a new approach for optimizing shell structures considering their mid surface design including cut-outs. Therefore we introduced a manufacturing constraint to the 3D topology optimization based on the density method in order to receive an optimized structure without undercuts and with a constant wall thickness, so that these structures can be manufactured by deep drawing in one step. It is shown that introducing cut-outs while increasing the shell thickness can improve the performance of shell structures considering their stiffness at a constant mass.     

Topology optimization of stiffened plate/shell structures based on adaptive morphogenesis algorithm

This article proposes an adaptive morphogenesis algorithm to design stiffened plate/shell structures in a growth manner. The idea of this work is inspired by researches in leaf venation which indicates that the adaptive growth of leaf vein provides the relatively large structure with an effective reinforcement. This excellent performance is regarded as the contribution of two primary morphological features: branching and hierarchy. To apply the growth mechanism of leaf venation into stiffened plate/shell structures, a mathematical model describing the growth process is established. Based on this, the adaptive morphogenesis algorithm is developed to make stiffeners “grow” step by step. Besides, the “stiffness transforming operation”, a numerical treatment, is introduced to enable stiffeners to grow along arbitrary directions in the FEM model, which guarantees the design more optimized than previous methods. To obtain a further verification of the proposed method, a comparison between the proposed method and three typical methods is implemented. This comparison shows that the proposed method endows the designed object with a more excellent performance than others. Therefore, the proposed method is competent in the stiffened plate/shell structure design.