Optimization of Structural and Mechanical Systems

The following sections are included:

• PREFACE

• LIST OF CONTRIBUTORS

• CONTENTS

Basic concepts of optimization are described in this chapter. Optimization models for engineering and other applications are described and discussed. These include continuous variable and discrete variable problems. Optimality conditions for the continuous unconstrained and constrained problems are presented. Basic concepts of algorithms for continuous and discrete variable problems are described. An introduction to the topics of multiobjective and global optimization is also presented.

Numerical algorithms for real life engineering optimization must be strong and capable of solving very large problems with a small number of simulations and sensitivity analysis. In this chapter we describe some numerical techniques to solve engineering problems with the Feasible Arc Interior Point Algorithm (FAIPA) for nonlinear constrained optimization. These techniques include quasi-Newton formulations that avoid the storage of the approximation matrix. They include also numerical algorithms to solve in an efficient manner the internal linear systems of FAIPA. Numerical results with large size test problems and with a structural optimization example shows that FAIPA is strong an efficient for large size optimization.

In the past three decades, evolutionary computation has been shown to be a very powerful tool for structural engineers. Application of evolutionary computation methodologies have been spread far and wide throughout the field of structural engineering ranging from selection of shapes for relatively simple structural systems to designing active control systems to mitigate seismic response to determining the location and extent of damage within structural systems. The present chapter provides an overview of evolutionary computation including a brief history of its development and the types of algorithms that are considered to be forms of evolutionary computation. A basic discussion of the genetic algorithm and evolutionary strategy is provided within the context of application to a very simple structural engineering design problem. The chapter provides a bird's eye view and discussion of many applications of evolutionary computation in the field of structural engineering. A brief synthesis of recent applications of evolutionary computation in the field of structural engineering is provided and recommendations for future work are given.

Decision-making is critical to the success of any product or system design. Multi-objective optimization can provide effective and efficient tools for decision-making under conflicting design criteria. The concept of tradeoff is integral to multiobjective optimization; and several approaches have been developed to resolve this tradeoff – yielding the so-called Pareto optimal solutions. These approaches can be broadly classified as those that require the specification of the designer preferences, and those that generate a set of Pareto optimal solutions from which the designer can choose. These methods and their relative merits and shortcomings are the focus of this chapter. A discussion regarding implementing these methods for practical problems is presented, followed by a discussion on industrial and academic applications.

Recently, shape optimization has been implemented into several commercial finite element programs to meet industrial need to lower cost and to improve performance. This chapter provides a comparatively easy method of shape optimization to implement into a commercial finite element code by using geometric boundary method. The geometric boundary method defines design variables as CAD based curves. Surfaces and solids are consecutively created and meshes are generated within finite element analysis. Then shape optimization is performed outside of finite element program.

Taking as a starting point a design case for a compliant mechanism (a force inverter), the fundamental elements of topology optimization are described. The basis for the developments is a FEM format for this design problem and emphasis is given to the parameterization of design as a raster image and the techniques associated with solving this class of problems by computational means.

Recent developments in design sensitivity analysis of nonlinear structural systems are presented. Various aspects, such as geometric, material, and boundary nonlinearities are considered. The idea of variation in continuum mechanics is utilized in differentiating the nonlinear equations with respect to design variables. Due to the similarity between variation in design sensitivity analysis and linearization in nonlinear analysis, the same tangent stiffness is used for both sensitivity and structural analyses. It has been shown that the computational cost of sensitivity calculation is a small fraction of the structural analysis cost. Such efficiency is due to the fact that sensitivity analysis does not require convergence iteration and it uses the same tangent stiffness matrix with structural analysis. Two examples are presented to demonstrate the accuracy and efficiency of the proposed sensitivity calculation method in nonlinear problems.

Optimal controllers are presented in this chapter for control of structures with emphasis on disturbance modeling. Both time domain and frequency domain methods are presented. Advantages and disadvantages of both the methods are discussed. Techniques for incorporating the excitation characteristics and frequency response information using augmentation techniques are presented. Numerical examples illustrating the control techniques and augmentation procedures for single and multiple degrees of freedom system are presented. The robustness principles in the context of linear optimal control are also discussed briefly.

An experimentally verified approach for the optimization of systems for acoustics with passive structural modifications is given. The method is general enough to handle a variety of structural modifications and structural impedances. Following some introductory acoustics and vibrations concepts, the optimization approach is formulated. Governing equations and solution methods are given, and finally several example applications are shown.

Design optimization studies for mechanical systems have increasingly become concerned with mathematical treatment of uncertainties in system demands and capacity, boundary conditions, component interactions, and available resources. The problem of optimum design under uncertainty has been formulated as reliability-based design optimization (RBDO). Recent efforts in this context seek to integrate advances in two directions: computational reliability analysis methods and deterministic design optimization. Much current work is focused on developing computationally efficient strategies for such integration, using decoupled or single loop formulations instead of earlier nested formulations. The extension of reliability-based optimization to include robustness requirements leads to multi-objective optimization under uncertainty. Another important application concerns multidisciplinary problems, where the various reliability constraints are evaluated in different disciplinary analysis codes and there is feedback coupling between the codes. Applications of recently developed methods to automotive and aerospace design problems are discussed, and new directions for further study are outlined.

This chapter reviews design optimization approaches which account for uncertainty, life-cycle performance, and cost. State-of-the-art probabilistic methods for analyzing the stochastic response of components and structural systems are outlined and their integration into design optimization methods discussed. Formulations for including probabilistic design criteria into reliability-based optimization problems are presented. The importance of life-cycle optimization under uncertainty with multiple objectives is emphasized and optimization methods for such problems are presented. This chapter shows that accounting for uncertainty via probabilistic approaches is an important and powerful design tool in various fields of application.

Articulated linkages appear in many fields, among them, robotics, human modeling, and mechanism design. The inverse kinematics (IK) of articulated linkages forms the basic problem to solve in various scenarios. This chapter presents an optimization-based approach for IK of a specific articulated linkage, the human model. The problem is formulated as a single-objective optimization (SOO) problem with a single performance measure or as a multi-objective-optimization (MOO) problem with multiple combined performance measures. A human performance measure is a physics-based metric, such as delta potential energy, joint displacement, joint torque, or discomfort, and serves as an objective function (cost function) in an optimization formulation. The implementation of the presented approach is shown for three models: a 4-degree-of-freedom (DOF) finger model, 21-DOF torso-right hand model, and 31-DOF torso-both hands model. Preliminary validation using a motion capture system demonstrates the accuracy of the proposed method.

The concept of multidisciplinary design optimization (MDO) has been addressed to solve optimization problems with multiple disciplines. Conventional optimization generally solves the problems with a single discipline. Disciplines are coupled in MDO problems. Many MDO methods have been proposed. The methods are classified and some representative methods are introduced. The advantages and drawbacks of each method are described and discussed.

Recent developments in meshfree method and its application to shape optimization are presented. The approximation theory of the Reproducing Kernel Particle Method is first introduced. The computational issues in domain integration and imposition of boundary conditions are discussed. A stabilization of nodal integration in meshfree discretization of boundary value problems is presented. Shape optimization based on meshfree method is presented, and the treatment of essential boundary conditions as well as the dependence of the shape function on the design variation is discussed. The proposed meshfree based shape design optimization yields a significantly reduced number of design iterations due to the meshfree approximation of sensitivity information without the need of remeshing. It is shown through numerical examples that the mesh distortion difficulty exists in the finite element–based design approach for design problems with large shape changes is effectively resolved.

Sensitivity-free formulations do not require design sensitivity analysis of problem functions during optimization iterations. These formulations include some of the state variables of the problem as optimization variables in addition to the real design variables. This gives explicit dependence of the problem functions on the optimization variables. Therefore gradients of the functions with respect to the optimization variables can be calculated easily. Sensitivity-free formulations include the simultaneous analysis and design (SAND) approaches, mathematical programs with equilibrium constraints (MPEC), and partial differential equations (PDE)-constrained optimization problems. In addition to the sensitivity-free formulations, the conventional formulations for optimization of structural and mechanical systems are described. Advantages and disadvantages of the formulations are discussed. Some recent evaluations of the formulations are also described.

Kriging metamodel that is suitable for approximation of highly nonlinear functions is derived systematically and sampling techniques for kriging are summarized and compared. Then optimization of an engineering problem based on kriging metamodel is performed.

The goal of robust design is to make system performance least sensitive to uncertainties. There is, however, no dominant formulation. One approach is to minimize the standard deviation of the performance and the other a reliability-based approach where the reliability in terms of a limit function is maximized or it is imposed as a probabilistic constraint. As methods that do not require probability information, a sensitivity-based formulation using gradient index is introduced with a MEMS example comparing yield rates. A robust design in terms of relative safety index that is defined by a new concept of allowable set in the random variable space finds interesting applications to multi-body mechanism design and human motion trajectory. As a reliability-based approach, a recently developed moment method based on full factorial design of experiments is greatly improved in efficiency by using adaptive response surface constructions. An application of tolerance synthesis has illustrated the method and its practicality for industrial problems.

There is increasing evidence that computing clusters created with commodity chips are capable of out-performing traditional supercomputers. The trend of using these commodity computing systems for engineering analysis and design is rapidly gaining momentum. In this chapter we discuss the different parallel processing scenarios and the implementation in the HYI-3D design optimization software system. We examine the hardware and software issues with 32-bit and 64-bit design optimization computations. A scenario for configuring a design engineer's workbench is presented where desktop computations are combined with execution on a computing cluster so as to reduce the design cycle time. Using multi-level parallelism, not only can the function evaluation be carried out in parallel but also other steps in the design optimization algorithm can be computed in parallel – gradients, line search and direction-finding problem. Numerical examples involving sizing, shape and topology optimization show the gains obtained from coarse and fine grain parallelism for both gradient and non-gradient optimization techniques.

Basic formulations of SemiDefinite Programming (SDP) are presented, and the properties including optimality conditions and duality are summarized. It is shown that structural optimization problems considering compliance, eigenvalue of vibration, etc. can be formulated by SDP problems. Interior point methods for solving SDP are briefly introduced. Finally, it is shown that the analysis problem of a cable network can be formulated as a Second-Order Cone Programming (SOCP) problem that is a particular case of SDP.

This chapter considers the rich field of nonlinear optimal control. After providing a brief history of feedback control theory, we provide some preliminaries and some general feedback control practices for both linear and nonlinear systems. We use these to set up a review of both linear and nonlinear optimal control.

The following sections are included:

• INDEX