Axisymmetric shell optimization under fracture mechanics and geometric constraint

This paper describes a problem of axisymmetric shell optimization under fracture mechanics and geometric constraints. The shell is made from quasi-brittle materials, and through crack arising is admitted. It is supposed that the shell is loaded by cyclic forces. A crack propagation process related to the stress intensity factor is described by Paris fatigue law. The problem of finding the meridian shape and the thickness distribution (geometric design variables) of the shell having the smallest mass subject to constraints on the cyclic number for fatigue cracks and geometrical constraint on the shell volume is investigated. Special attention is devoted to different possibilities of problem transformation and analytical methods of their solution. Using minimax approach, optimal shapes of the shells and their thickness distributions have been found analytically.     

Reinforcement layout and sizing optimization of composite submarine sail structures

A topology optimization approach that makes use of nonlinear design variable-to-sizing relationship is presented. A finite element (FE) model is used to describe the loaded structure, but unlike the microstructure approach, the decision is whether an element in the continuum should have maximum or minimum cross-sectional dimension while its material density and moduli are held constant. This approach is applied to reinforcement layout optimization of a very large and geometrically complex Composite Advanced Sail (CAS) structure under an asymmetric wave slap loading condition. A high-complexity model in the form of multilayered shell and a low-complexity model in the form of stiffened shell are developed for the layout optimization of the CAS and solved for minimum strain energy. The effects of constraints such as buckling instability on optimal placement of internal stiffeners are also explored. Based on the results of the layout optimization, a new FE model of the CAS is developed and optimized for minimum weight. Depending upon the degree of variability in skin thickness, the results show a weight saving of up to 19% over the original model.     

Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material

The present paper studies multi-objective design of lightweight thermoelastic structure composed of homogeneous porous material. The concurrent optimization model is applied to design the topologies of light weight structures and of the material microstructure. The multi-objective optimization formulation attempts to find minimum structural compliance under only mechanical loads and minimum thermal expansion of the surfaces we are interested in under only thermo loads. The proposed optimization model is applied to a sandwich elliptically curved shell structure, an axisymmetric structure and a 3D structure. The advantage of the concurrent optimization model to single scale topology optimization model in improving the multi-objective performances of the thermoelastic structures is investigated. The influences of available material volume fraction and weighting coefficients are also discussed. Numerical examples demonstrate that the porous material is conducive to enhance the multi-objective performance of the thermoelastic structures in some cases, especially when lightweight structure is emphasized. An “optimal” material volume fraction is observed in some numerical examples.     

Optimization of mass effectiveness of axisymmetric pressure vessels

The optimization problems taken into consideration in this paper consist in finding the shape and the thickness distribution of elastic axisymmetric shells in the frame of membrane theory, loaded by fixed statical forces in such a way that the cost functional reaches a maximum, while satisfying some strength mechanics constraints. In this paper the mass effectiveness of the shell is considered as a cost function and as constraint we use the bounds on the maximum normal stress which is typical for shells made from brittle or quasi-brittle materials. The paper gives some possible formulations of optimal structural design problems. Analytical investigations and corresponding solutions are presented.     

Axisymmetric structural optimization design and void control for selective laser melting

Additive manufacturing processes, of which Selective Laser Melting (SLM) is one, provide an increased design freedom and the ability to build structures directly from CAD models. There is a growing interest in using optimization methods to design structures in place of manual designs. Three design optimization problems were addressed in this paper. The first related to axisymmetric structures and the other two addressing important design constraints when manufacturing using SLM. These solutions were developed and applied to a case study of a turbine containment ring. Firstly, many structural components such as a turbine containment ring are axisymmetric while they are subjected to a non-axisymmetric load. A solution was presented in this paper to generate optimized axisymmetric designs for a problem in which the mechanical model was not axisymmetric. The solution also worked equally well for generating a prismatic geometry with a uniform cross section, requiring no change in the procedure from axisymmetric designs to achieve this. Secondly, the SLM process experiences difficulties manufacturing structures with internal voids larger than a certain upper limit. A method was developed that allowed the designer to provide a value for this upper limit to the optimization method which would prevent the generation of internal voids larger than this value in any optimized design. The method calculated the sizes of all the voids and did not increase their size once they reached this limit. It was also aware of voids near each other, providing a minimum distance between them. Finally, in order to remove the metal powder, that fills the internal voids of structures built using SLM to reduce unnecessary weight, a method was developed to build paths to join the internal voids created during the optimization process. It allowed the analyst to nominate suitable path entrance locations from which powder could be removed, then found the shortest path connecting all voids and these locations. For axisymmetric structures it also distributed this path around the circumference to avoid generating weak points.     

Free vibration analysis and optimisation of axisymmetric plates and shells—II. Shape optimisation

The second part of this paper is concerned with the structural shape optimisation of vibrating axisymmetric shells. Natural frequencies and mode shapes are determined using curved, variable thickness, Mindlin-Reissner FEs introduced and benchmarked in the first part of the paper. The whole shape optimisation process is carried out by integrating FE analysis, cubic spline shape and thickness definitions, sensitivity analysis and mathematical programming. The semi-analytical method is used to determine the sensitivities of the objective function and constraints to changes in the design variables. Several examples are considered to illustrate and highlight various features of the optimisation, including various plates, a conical shell, a branched shell, and a church bell.     

Free vibration analysis and optimisation of axisymmetric plates and shells—I. Finite element formulation

This paper deals with the free vibration analysis of shells of revolution using the finite element method. A family of variable thickness, curved C(0) Mindlin-Reissner axisymmetric finite elements is presented which include shear deformation and rotatory inertia effects. The accuracy, convergence and efficiency of these newly developed elements are explored through a series of free vibration analyses of axisymmetric shell structures and the results are compared with those obtained by other analytical and numerical methods. The comparisons show that the method yields very good results with a relatively small number of elements and that estimates for the higher modes can be obtained without any difficulties. A companion paper will consider the structural optimisation of axisymmetric shells undergoing free vibrations.     

Design optimization of stiffened storage tank for spacecraft

Stiffened storage tank is an important structural component in spacecraft. Its structural weight is one of the key criterions in the design phase. This paper focuses on the design optimization of the structure by using finite element method, structural sensitivity analysis techniques, and sequential linear/quadratic programming aimed to reduce the structural weight. Design variables include the numbers of stiffeners, stiffeners’ section dimensions, and shell thickness distribution. Detailed finite element modeling processes are presented, which are the ways to construct the stiffener (beam orientation and offset) and shell elements and the ways to determine the analysis model and structural boundary conditions. A brief introduction to sensitivity analysis and optimization solution algorithm is also given. Main attention is paid to the studies of design optimization of the tank structure, including the selection of design cases, evaluation, and comparison of the optimal results. There are six design cases considered in the design procedures. Numerical results show that by using the above computational techniques, the structural weight is effectively reduced. In this work, MSC.Patran/Nastran is employed to construct the Finite Element Model (FEM), and JIFEX, which is developed in our group, is used to conduct the structural design optimization. JIFEX is a structural analysis and optimization software package developed by Gu and colleagues in the Dalian University of Technology Department of Engineering Mechanics. Among its many functions is the ability to analyze and optimize piezoelectric smart structures.     

Shape optimization of axisymmetric structures with adaptive finite element procedures

A robust and versatile algorithm for shape optimization with adaptive finite element procedures is developed for the design of axisymmetric structures. The algorithm is based on the use of boundary parameterization with cubic splines for describing shape changes and takes advantage of the utilities available in an advancing front type mesh generator. Six-noded triangular elements are adopted. Shape optimization examples involving solid axisymmetric structures are presented to illustrate the various features of the integrated approach.     

Application of penalty functions to a curved isoparametric axisymmetric thick shell element

A curved axisymmetric shell element with three nodes is developed. Quadratic interpolation is used and as the transverse shearing strain is included only first derivatives are required in the calculation of the strains. The element is found to yield accurate solutions for thick circular plates but a penalty factor must be used when the ratio of plate radius to thickness is of the order of 100. With appropriate values of the penalty factor, though, thin plate behaviour is reproduced with reasonable accuracy. Further, it is shown that for all practical purposes the penalty factor need only be based on the plate thickness. This is a useful conclusion in relation to shell analysis where different penalty factor values would otherwise need to be evaluated on the basis of the radius to thickness ratio. Finally the element is shown to give good results for cylindrical and spherical shells.     

Shape optimization of quasi-brittle axisymmetric shells by genetic algorithm

This paper describes the application of genetic algorithm to the shape optimization of axisymmetric shells. The primary problem of axisymmetric shell optimization under fracture mechanics constraint is formulated as the weight (volume of shell material) minimization under stress intensity constraints. It is assumed that the shells are made from quasi-brittle materials and through-thickness crack presence is admitted. Taking into account the fact of incomplete information concerning crack arising (size, location and orientation) this paper presents some numerical results based on a guaranteed approach.     

Integrated design optimization of structural size and control system of piezoelectric curved shells with respect to sound radiation

Simultaneously optimizing the thickness of the base structure and the location of piezoelectric sensors/actuators as well as control gains is investigated for minimizing the sound radiation from the vibrating curved shell integrated with sensors/actuators under harmonic excitation. The finite element formulation of the piezoelectric curved shell structure is described. The piezoelectric element is coupled into the base shell element using nodal displacement constraint equations. The active control of structural vibration-acoustic radiation is formulated using the velocity feedback algorithm. Based on both passive and active control measures, an integrated optimization model of the vibro-acoustic problem is proposed, in which the sound power is taken as the objective function. The thickness of the base shell elements and the parameters of control system, including the location of sensors/actuators and control gains, are chosen as the design variables. In order to restrict the complexity of the control system, the number of sensors/actuators is considered as a constraint. A simulated annealing algorithm is extended to handle the vibro-acoustic optimization problem with the continuous and discrete variables co-existing. Numerical examples demonstrate the effectiveness of the optimization scheme and the correctness of the computation program.     

Optimum part orientation in Rapid Prototyping using genetic algorithm

Part orientation is an important parameter in the planning of a Rapid Prototyping (RP) process as it directly governs productivity, part quality and cost of manufacturing. This paper reports the design and implementation of a system for obtaining optimum orientation of a part for RP. Developed in a modular fashion, the system comprises of functional modules for CAD model preprocessing, shelling (hollowing), part orientation and optimization. CAD part model in STL format is an input to the system. The oriented CAD model is sliced and hollowed with desired shell thickness. Genetic algorithm based strategy is used to obtain optimum orientation of the parts for RP process. The objective criteria for optimization is considered to be a weighted average of the performance measures such as build time, part quality and the material used in the hollowed model. The developed system has been tested with several case studies considering SLS process.     

An integrated approach for shape and topology optimization of shell structures

In this paper an automated approach for simultaneous shape and topology optimization of shell structures is presented. Most research in the last decades considered these optimization techniques separately, seeking an initial optimal material layout and refining the shape of the solution later. The method developed in this work combines both optimization techniques, where the shape of the shell structure and material distribution are optimized simultaneously, with the aim of finding the optimum design that maximizes the stiffness of the shell. This formulation involves a variable ground structure for topology optimization, since the shape of the shell is modified in the course of the process. The method has been implemented into a computational model and the feasibility of the approach is demonstrated using several examples.     

Minimum cost design of a welded orthogonally stiffened cylindrical shell

In this study the optimal design of a cylindrical orthogonally stiffened shell member of an offshore fixed platform truss, loaded by axial compression and external pressure, is investigated. Ring stiffeners of welded box section and stringers of halved rolled I-section are used. The design variables considered in the optimization are the shell thickness as well as the dimensions and numbers of stiffeners. The design constraints relate to the shell, panel ring and panel stringer buckling, as well as manufacturing limitations. The cost function includes the cost of material, forming of plate elements into cylindrical shape, welding and painting. In the optimization a number of relatively new mathematical optimization methods (leap-frog - LFOPC, Dynamic-Q, ETOPC, and particle swarm - PSO) are used, in order to ensure confidence that the finally computed optimum design is accurately determined, and indeed corresponds to a global minimum. The continuous optimization procedures are adapted to allow for discrete values of the design variables to be used in the final manufacturing of the truss member. A comparison of the computed optimum costs of the stiffened and un-stiffened assemblies, shows that significant cost savings can be achieved by orthogonal stiffening, since the latter allows for considerable reduction of the shell thickness, which results in large material and manufacturing cost savings.     

Comparison of shell theories in the analysis of cylindrical shells with discontinuity in the thickness

Cylindrical shells with discontinuity in the thickness and that are subjected to axisymmetric loading have been analysed. Two types of finite elements are used: the first is based on thin shell theory and the second on thick shell theory. The loadings considered are a uniform internal pressure and a circular ring load at the mid-section. The effect of these loads for various end conditions and various step-ratios in the thickness have been analysed. Numerical results are presented and compared for both the theories. It has been shown that the transverse normal stress acting along the thickness direction is not negligible compared to other stresses at places of discontinuity either in the thickness or in the loading. The weight of the shell is kept constant for various step-ratios.     

Efficient sensitivity analysis and optimization of shell structures by the ABAQUS code

In this paper, a new efficient sensitivity analysis procedure is presented for the optimization of shell structures without access to the finite element source code. It is devised as a general interface tool to extend existing finite element systems from pure structural analysis to design capability. The implementation is performed based on the ABAQUS code. Kirchhoff flat shell elements are taken into account in the study with the element thickness as design variables. To ensure the performance and the validity of the proposed procedure, satisfactory sensitivity and optimization results are illustrated for numerical examples.     

Optimal design of plastic cylindrical shells of von Mises material

Optimal design of plastic circular cylindrical shells of von Mises material is studied. The optimization problem is stated as the maximization problem of the load carrying capacity for given weight of the shell. Shells with constant and piecewise-constant thickness are considered. The maximization problem is performed under the requirement that the material volume of the stepped shell is equal to the case of the reference shell of constant thickness. The material of the shell is assumed to be an ideal rigid plastic obeying von Mises yield criterion. The considered nonlinear problems are solved by using the CASes method.     

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.