Computer Aided Optimal Design: Structural and Mechanical Systems

As a part of the broad development that has taken place in the field of structural optimization in recent years, analytical modelling for the design of continuum structures has been extended to cover a variety of new applications. Thus there are formulations available now for the optimization with respect to various modes of response or measures of performance, for most types of structural form, to be optimal relative to material distribution, shape, choice of materials, prestress, and so on. Only a modest part out of the comprehensive list of topics is to be covered in these lectures. The reader will find a good many of the major areas of application e.g., ‘design for dynamic response,’ ‘shape design’, ‘grid optimization,’ and ‘sensitivity analysis’ — to name a few, treated in separate lectures given elsewhere within the institute. (Citations to other lectures in this collection are identified by the authors name with an asterisk attached to it.) Our effort is directed more toward an exposition of methods for the interpretation of design problems into a form convenient for analysis. This is to be done mainly within the perspective of well known results from the mathematics of optimization. The material presented here is comprised for the most part of formal problem statements, listings and interpretation of necessary conditions, and the presentation of example applications.

This chapter contains notes for lectures delivered as part of the Advanced Study Institute “Computer Aided Optimal Design: Structural and Mechanical Systems” organized in Troia, Portugal, 30 June – 11 July 1986, by the Center of Mechanics and Materials of the University of Lisbon. The author gratefully acknowledges Professor Carlos Mota Soares and his collagues for a most successful arrangement of the meeting and the other lecturers for several extremely interesting contributions.

During the last decade of his immensely creative life, Professor William Prager’s research was directed at two central objectives, the derivation of a comprehensive set of static-kinematic optimality criteria and the development of an optimal layout theory. As the late Professor Prager’s closest former associate, the first author will review briefly these fields in the first part of this memorial lecture.

In this paper we consider the simultaneous optimal design of controls and structure for an active control of a flexible structure. Problem formulations related to mission control and to control of structural properties are identified and illustrated on a simple model problem, and the relationship to eigenfrequency optimization is discussed. We also discuss relevant design objectives for modal control of distributed structures or large scale structures and examples of optimal beam designs are presented.

Numerical optimization techniques provide a uniquely general and versatile tool for design automation. While these methods have been developed, to a large degree, by the operations research community, research in their application to engineering problems has been extensive as well. The first formal statement of nonlinear programming (numerical optimization) applied to structural design was offered by Schmit in 1960 [1]. Since that time, the field has evolved at an ever-increasing pace until it can now be considered to be reasonably mature.

The optimization of a structure modeled by finite elements can proceed in two diametrically opposed directions. The first direction is that of interfacing a finite element software package with an optimization package where both packages are treated primarily as black boxes. The second direction is the intimate integration of the finite element analysis and optimization processes. Many research structural optimization programs followed the second path for reasons of efficiency and convenience to the researchers who wrote these programs. However, in production codes the tendency is to follow the more modular first direction. The integrated approach is probably justified only when it reflects algorithmic integration of the analysis and optimization processes. This type of integration is presently at the research stage.

First, the historical background leading to the optimality criteria approach is discussed pointing out the role of the traditional design methods on one hand, and Prager’s work based on variational principles on the other hand as the two motivating influences. This is followed by the formal development of the method utilizing the separability properties of discretized structures or models. The importance of the single constraint case is pointed out and the associated particularly simple yet powerful optimality criteria is presented followed by extension to multiple constraints. Examples are used to illustrate the approach for displacement, stress and eigenvalue related constraints.

The optimal design problem for linear elastic structures has been the subject of abundant literature, and various behavioural constraints have been taken into account that very often concern structural deformation. On the other hand, the structural behaviour beyond the elastic limit has been considered in many papers dealing with optimal design for prescribed plastic collapse load (rigid-plastic models). Comprehensive surveys can be found, for istance, in Ref.s 1 to 5.

This study addresses the task faced by designers of structures for which “limit-states” criteria must be satisfied at one or more distinct loading levels. For example, the specified limit states may concern acceptable elastic displacements under service loads, acceptable elastic stresses under factored service loads, and adequate strength reserve under ultimate loads against buckling instability and/or plastic collapse of the structure. Ideally, while satisfying the various performance criteria, the most economical design of the structure is sought.

In the past decade the interest in using active control systems to improve structural performance has increased dramatically. The basic idea is to build structures which use an active control system as a substitute for strength, stiffness or damping. An early application of the concept was the gust-alleviation system designed to prolong the fatigue life of the aging fleet of B-52 bombers. In that application, sensors were used to anticipate gust loads on the wing, and control surface on the wing were deflected to cancel part of these gust loads. Today, these types of systems are already considered in the preliminary design stage (e.g. Ref. 1). The same idea may be used to improve the ride quality in an airplane. Other aeronautical applications such as active flutter suppression are discussed in Ref. 2.

Algorithms are presented to design a minimum weight structure and to improve the dynamic response of a closed-loop control system. Constraints are imposed either on the structural response quantities or on the complex eigenvalue distribution of the closed-loop system. Use of the algorithms is illustrated by solving different problems.

The present article deals with an extended class of design problems when besides material or dimensional variables, also structure shape, external support conditions, action through initial distortions is considered. In particular, a problem of mixed boundary conditions is discussed when imposed strength and stiffness requirements may lead to conflicting design decisions The sensitivity analysis for both linear and non-linear structures provides a uniform variational approach to such variety of problems by generating expressions of functional gradients explicitly in terms of state fields of primary and adjoint structures.

The material derivative concept of continuum mechanics and an adjoint variable method of design sensitivity analysis are used to relate variations in structural shape to measures of structural performance. A domain method of shape design sensitivity analysis is used to best utilize the basic character of the finite element method that gives accurate information not on the boundary but in the domain. Implementation of shape design sensitivity analysis using finite element computer codes is discussed. Recent numerical results are used to demonstrate accuracy that can be obtained using the method. Result of design sensitivity analysis is used to carry out design optimization of a built-up structure.

Finite element gridding is regarded as an optimal design problem which yields the optimal grid adaptively by applying the optimality criteria method. Finite element approximation error analysis is critically reviewed to determine the effect of grid distortion which is a key factor of the irregular distribution of approximation error in flux and stress. Based on this study, appropriate error measures are defined for the optimal grid design problem as well as error indicators which estimate the total amount of approximation error.

In many problems of structural mechanics there is a need to assess the effect of variation of one or several design functions or parameters /such as material properties, cross-sectional dimensions, shape or support conditions/ on the thermal and mechanical state fields within a structure. The methods of such sensitivity analysis have been explored in various fields of science and engineering. Numerous publications reveal the growing interest in the optimum design of structures subject to both mechanical and temperature constraints. When mathematical optimization techniques are used to design structures subjected to temperature, stress and/or displacement constraints, derivatives of these constraints with respect to design variables are usually required. The present paper extends the previous works [1–2] and is concerned with such class of problems for which the first and second variations of any functional that depends on displacements, strains, stresses and temperature can be explicitly expressed in terms of variations of design variables.

This paper is concerned with a new variational approach to structural design sensitivity analysis, including the consideration of shape variation. Specifically, the mutual Hu-Washizu functional is introduced for obtaining explicit sensitivity expressions for response functionals; and the Eulerian-Lagrangian kinematic description is applied to the description of design geometry variations.

Methods of design sensitivity analysis with linear response under static and dynamic loads have been developed and documented over the last fifteen years [1–5]. However, they are just beginning to be developed with nonlinear response [6–8]. Purpose of this paper is to describe a method of design sensitivity analysis of static nonlinear response using incremental finite element procedures. To accomplish this objective, nonlinear analysis of the structure must be performed which is usually quite tedious. The most effective procedure is to use load incrementation coupled with an iteration. With such a procedure, geometric as well as material nonlinearities (different material models) can be consistently and uniformly treated. However, the structure can collapse before the full load level is reached. In that case, design must be improved to have a stable structure. Thus procedures for calculation of collapse load and its sensitivity to design changes must be included in structural optimization problem with nonlinear response.

The shape optimal design of shafts and two-dimensional elastic structural components is formulated using boundary elements. The design objective is to maximize torsional rigidity of the shaft or to minimize compliance of the structure, subject to an area constrain Also a model based on minimum area and stress constraints is developed, where the real and adjoint structures are identical, but with different loading conditions. All degrees of freedom of the models are at the boundary and there is no need for calculating displacements and stresses in the domain. Formulations based on constant, linear and quadratic boundary elements are developed. A method for calculating accurately the stresses at the boundary is presented, which improves considerably the design sensitivity information. It is developed a technique for an automatic mesh refinement of boundary element models. The corresponding nonlinear programming problems are solved by Pshenichny’s linearization method. The models are applied to shape optimal design of several shafts and elastic structural components. The advantages and disadvantages of the boundary element method over the finite element technique for shape optimal design of structures are discussed with reference to applications. A literature survey of the development of the boundary element method for shape optimal design is presented.

A general method for shape design sensitivity analysis as applied to potential problems is developed with the standard direct boundary integral equation (BIE) formulation. The material derivative concept and adjoint variable method are employed to obtain an explicit expression for the variation of the performance functional in terms of the boundary shape variation. The adjoint problem defined in the present method takes a form of the indirect BIE. This adjoint problem can be solved using the same direct BIE of the original problem with a different set of boundary values, which brings about computational simplicity. The accuracy of the sensitivity formula is studied with a seepage problem. The detailed derivation of the formulas for general elliptic problems and a more elaborate numerical scheme will be described elsewhere.

Shape design sensitivity analysis of thermoelastic solids is done by using the material derivative idea and the adjoint variable method. Boundary element methods are proposed for spatial discretization of system of equations and relevant functionals.

Engineering system design used to be compartmentalized by discipline. Material specialists would design better materials, fluid mechanics specialists would design optimum shapes, structural analysts would produce optimum structural designs based on materials and loads obtained by material and fluid mechanics specialists, and so on. Occasionally, interdisciplinary effects forced cooperation between disciplines. Aeroelastic phenomena such as flutter or loss of contro1-surface effectiveness forced aerodynamic and structural analysts to cooperate in the creation of the new discipline of aeroelasticity. However, when such interdisciplinary phenomena did not force cooperation, very little existed, beyond the conceptual design level.

Methods for calculation of first and second design derivatives of performance measures for nonlinear dynamic systems are presented. Design sensitivity analysis formulations for dynamic systems are presented in two alternate forms; (1) equations of motion are written in terms of independent generalized coordinates and are reduced to first order form and (2) equations of motion of constrained systems are written in terms of a mixed system of second order differential and algebraic equations. Both first and second order design sensitivity analysis methods are developed, using a theoretically simple direct differentiation approach and a somewhat more subtle, but numerically efficient, adjoint variable method. Detailed derivations are presented and computational algorithms are discussed. Examples of first and second order design sensitivity analysis of mechanisms and machines are presented and analyzed.

In the early stages of development of new fields, it happens time and again that the imminent potential impact of the new field is exaggerated to the point of a new panacea, usually by workers at the fringe or outside the research area of the field. An attendant consequence often is increased scepticism, if not outright hostility, among the broader community of researchers who cannot separate fact from fiction. This appears to be somewhat the case currently with the emerging field of artificial intelligence (AI) and, in particular, with its applied branch of expert or knowledge-based systems. It is therefore with some caution that the present article, not addressing the mainstream research community of AI, examines the issues involved in developing knowledge-based systems for optimal design. By the nature of the present state of the art in the field, the exposition is to a large degree speculatory. There is yet very little done that would qualify as concrete results in the field. Still, the motivation for developing such systems is eminently strong and the apparent potential results are overwhelmingly desirable, so that at least some speculation is justified.

The insertion of CAD technology in optimal design is strongly related to the more general problem of integration.

This Section concentrates on two major objectives that are currently pursued at the research level, and that should soon be ready for implementation in practical Computer Aided Engineering systems. The first objective is to develop a general approach to shape optimal design of elastic structures discretized by the Finite Element Method (FEM). The key idea is to employ geometric modeling concepts typical of the Computer Aided Design (CAD) technology, in order to produce sensitivity analysis results. These sensitivity data can then be used by an optimizer to generate an improved design.The second goal is to implement an interactive redesign system that integrates optimization methods within a flexible and efficient computational tool, easy to use by design engineers. This interactive module is intended to create the missing link between FEM and CAD technologies and therefore it should constitute one of the key elements in the complex chain needed to computerize the design cycle.The approach followed can be summarized as follows. First the behavior of the structure is analyzed by using the finite element method. Subsequently a sensitivity analysis is performed to evaluate the first derivatives of the structural response quantities. These derivatives are used by an efficient optimizer, which selects an improved design. A reanalysis of the modified design is next performed after updating the finite element mesh. This iterative process is repeated until convergence to an acceptable optimum design has been achieved, which usually requires less than 10 FEM analyses.The long term objective is to create a coherent interactive system that makes the best possible use of the respective capabilities of the engineer and the computer.

This paper is intended for the potential developer and user of structural design software. A great number of recent publications have had a strong influence on the development and improvement of structural design software. New ideas have to be studied, realized and tested to develop or to use a state-of-the-art program. In this report I have attempted to make comments on some of the basic ideas. The practical knowledge and experience gained during the development of our in-house programming system, which is designated LAGRANGE, were taken into consideration. The features of about 30 internationally used program systems for structural design are listed in the appendix.

Based on: — a rapid design-oriented finite element method for analysis of large complex thin-walled steel structures such as ship hulls, — a comprehensive mathematical model for the evaluation of the capability of such structures, and — analytical expressions for the derivatives of the response and the capability with respect to the governing design variables, the paper presents a procedure for minimizing a linear combination of structural weight and production costs for a number of loading conditions. The large, highly constrained non-linear optimization problem is solved by sequential linear programming.

It has been demonstrated that optimal design is indispensable for the sizing of complex structures using composite materials. On a significant application to the RAFALE wing optimisation saved 10% on structural weight and obtained a 25% more stiffened structure in comparaison with a classical sizing. Sensitivity analysis gives a great knowledge on the structure.Multimodel optimisation, the most recent advance in our system, managing the synthesis of calculations on several static and dynamic models is described. A significant application to the lay-up of a composite wing panel is detailed (2 F.E. models with 3500 and 13000 DOF, 80 models for panel buckling, 480 design variables, 1000 static and aeroelastic constraints, 24 load cases).Then it clearly appears that the passage between the rough results of optimisation to the final shape of each lay is impossible to solve with classical drawing means. An algorithm computing the optimal shape of each lay now integrated in our C.A.D. system CATIA is described. An application on the previous wing panel is detailed.Close integration of C.A.D. systems and FEM systems, including geometry, mesh generation, analysis and optimisation is an obliged way for greater developments and applications of F.E.M.. It will enable FEM system to optimize the shape of structures. Integration of CATIA and ELFINI is described.Multimodel optimisation on complex structures needs a nearly prohibitive CPU time on classical computers. The interest of vectorial multi-processor architectures with extended core memory is showed.

Demands within the international aerospace industry for the design of primary components of ever higher quality, at minimum weight and low cost, are growing constantly while pressure to achieve the shortest time scales remains strong. As a result, the use of computers in design and for structural mechanics is of growing importance.

The general structural optimization system OPTSYS has been developed. In this system software for mathematical programming, structural, aeroelastic and aerodynamic analysis are integrated. The system design and methods used in OPTSYS are presented and applications to aircraft structures are supplied to show the capabilities of the system.

A general approach to shape optimization based on CAD formulations is proposed and implemented in the OASIS-ALADDIN system for structural optimization. Two large scale examples demonstrates the ideas.

Until recently, structural optimization has not significantly impacted the wider analysis community. Several reasons may be advanced, the most prominent in the mechanical engineering field being the requirement for shell bending elements. In addition, the capability has generally been made available as ad hoc packages loosely connected to an analysis program. SDRC’s OPTISEN offers fully integrated analysis and redesign with interactive graphics capabilities for both the specification of the design problem and the interpretation of results.

The different algorithms which are used to solve the general nonlinear programming problems, are usually assessed by means of efficiency, accuracy, reliability and robustness. The efficiency is a measure of the processing time and is considered as the most important criterion in almost all test studies. Obviously the acceptance of a method is dominated by efficiency, because of excessive execution time all other criteria are superfluous.

Optimal design of large systems in the interdisciplinary design environment needs reliable algorithms. Many algorithms have been developed and evaluated for structural optimization. In these algorithms, efficiency has been given priority over reliability and generality. To have efficiency, approximations are introduced into the algorithms. With such approximations, many algorithms loose their robustness and applicability to complex problems. In summary, approximate methods are efficient but unreliable, inaccurate and not general. Globally convergent (robust) methods are general, accurate and reliable but need more computational effort. My contention is that we should relax the efficiency “constraint” for optimum design of practical systems. I am not saying that we should use inefficient algorithms. What I am advocating is that reliability should be given more weightage over efficiency in practical design environment. We should concentrate on developing reliable algorithms that are generally applicable. For complex and interdisciplinary systems, reliability of algorithms is essential. Unreliable algorithms can be actually more expensive because they require more user interaction and time, resulting in overall inefficiency. Use of optimization in general design environment also needs reliable algorithms.

Optimal design, has, in the past, been viewed as a batch mode computational problem. Most iterative optimization methods make little or no provision for interactive control of the process by the experienced designer. A number of factors are emerging that suggest more emphasis should be placed on methods of optimal design that involve the designer, preferably an interactive mode.

AI has been “over-sold” in recent times and many exaggerated claims are made on its behalf which seems unlikely to come to fruition in the near future. Nevertheless, some important developments are taking place and the Structural Optimization community show view, therefore view AI as representing a new “box of programing tools” some of which will be beneficial in making Structural Optimization systems more user friendly.

Structural optimization is now a mature discipline, and it begins to penetrate the industrial community. Several commercially available FEM systems will soon be released with design optimization capabilities. Therefore the most important area of development in the next few years will certainly reside in creating good user interfaces aimed at facilitating the task of the designers. In particular it will be extremely important to nicely integrated FEM and CAD technologies within an optimization loop, in order to fully computerize the design cycle.