Erik Lund

Multiple eigenvalues in structural optimization problems

This paper discusses characteristic features and inherent difficulties pertaining to the lack of usual differentiability properties in problems of sensitivity analysis and optimum structural design with respect to multiple eigenvalues. Computational aspects are illustrated via a number of examples.Based on a mathematical perturbation technique, a general multiparameter framework is developed for computation of design sensitivities of simple as well as multiple eigenvalues of complex structures. The method is exemplified by computation of changes of simple and multiple natural transverse vibration frequencies subject to changes of different design parameters of finite element modelled, stiffener reinforced thin elastic plates.Problems of optimization are formulated as the maximization of the smallest (simple or multiple) eigenvalue subject to a global constraint of e.g. given total volume of material of the structure, and necessary optimality conditions are derived for an arbitrary degree of multiplicity of the smallest eigenvalue. The necessary optimality conditions express (i) linear dependence of a set of generalized gradient vectors of the multiple eigenvalue and the gradient vector of the constraint, and (ii) positive semi-definiteness of a matrix of the coefficients of the linear combination.It is shown in the paper that the optimality condition (i) can be directly applied for the development of an efficient, iterative numerical method for the optimization of structural eigenvalues of arbitrary multiplicity, and that the satisfaction of the necessary optimality condition (ii) can be readily checked when the method has converged. Application of the method is illustrated by simple, multiparameter examples of optimizing single and bimodal buckling loads of columns on elastic foundations.

A Method of “Exact” Numerical Differentiation for Error Elimination in Finite-Element-Based Semi-Analytical Shape Sensitivity Analyses

ABSTRACT The traditional, simple numerical differentiation of finite-element stiffness matrices by a forward difference scheme is the source of severe error problems that have been reported recently for certain problems of finite-element-based, semi-analytical shape design sensitivity analysis. In order to develop a method for elimination of such errors, without a sacrifice of the simple numerical differentiation and other main advantages of the semi-analytical method, the common mathematical structure of a broad range of finite-element stiffness matrices is studied in this paper. This study leads to the result that element stiffness matrices can generally be expressed in terms of a class of special scalar functions and a class of matrix functions of shape design variables that are defined such that the members of the classes admit “exact” numerical differentiation (exact up to round-off error) by means of very simple correction factors to upgrade standard computationally inexpensive first-order finite di...

Shape design sensitivity analysis of eigenvalues using “exact” numerical differentiation of finite element matrices

As has been shown in recent years, the approximate numerical differentiation of element stiffness matrices which is inherent in the semi-analytical method of finite element based design sensitivity analysis, may give rise to severely erroneous shape design sensitivities in static problems involving linearly elastic bending of beam, plate and shell structures.This paper demonstrates that the error problem also manifests itself in semi-analytical sensitivity analyses of eigenvalues of such structures and presents a method for complete elimination of the error problem. The method, which yields “exact” numerical sensitivities on the basis of simple first-order numerical differentiation, is computationally inexpensive and easy to implement as an integral part of the finite element analysis.The method is presented in terms of semi-analytical shape design sensitivity analysis of eigenvalues in the form of frequencies of free transverse vibrations of plates modelled by isoparametric Mindlin finite elements. Finally, the development is illustrated via two examples of occurrence of the error phenomenon when the traditional method is used and it is shown that the problem is completely eliminated by the application of the new method.

Shape Design Optimization of Steady Fluid-Structure Interaction Problems with Large Displacements

The objective of this work is to develop and implement efficient numerical procedures for gradient based shape design optimization of steady, strongly coupled fluid—structure interaction problems. Nonlinearities in the analysis arise from both the solid behaviour, the fluid behaviour, and from the deformed interface, where the deformation of the interface is large enough to significantly alter the response. The solution for state is obtained using finite element residual formulations based on a consistent weak formulation of the fluid—structure interaction problem. The resulting nonlinear equations are solved using an approximate Newton method. Design sensitivity analysis (DSA) is performed by the direct differentiation method, and the resulting sensitivity equations are solved very efficiently by an incremental iterative method. Gradient based shape optimization is illustrated for finding the shape of a body with smallest drag in a flow governed by the two-dimensional steady Navier-Stokes equations for an infinitely stiff body but also for a very flexible body in strong interaction with the surrounding fluid. The shape optimization examples illustrate the potential of the described methods to solve nonlinear multidisciplinary design problems involving strongly coupled fluid-structure interaction.

Finite Element Based Engineering Design Sensitivity Analysis and Optimization

The aim of this paper is to present basic concepts and selected finite element based methods and tools for sensitivity analysis and rational engineering design and optimization of mechanical structures and components. The main emphasis is devoted to sensitivity and optimization problems that involve shape and sizing design variables.

Concurrent Engineering Design Optimization in a CAD Environment

Concepts, methods and tools for interactive CAD-based concurrent engineering design optimization of mechanical products, systems and components which are critical in terms of cost, development time, functionality and quality, are presented. The emphasis is on formulation, development and implementation of methods and capabilities for finite element analysis, design sensitivity analysis, rational design, synthesis and optimization of mechanical systems and components, and the integration of these methods into a standard CAD modeling environment with a view to develop a concurrent engineering design optimization system. Methods for optimizing the topology of mechanical components are integrated into the system and used as a preprocessor for subsequent shape and sizing optimization. Use of the system for concurrent engineering design of mechanical components is illustrated by examples.

THE ISSUE OF GENERALITY IN DESIGN OPTIMIZATION SYSTEMS

This paper attempts to convey some of the authors experiences in constructing and using systems for design optimization with as much generality as necessary to solve practical engineering design problems. A number of examples demonstrate that real-life design often calls for facilities that were difficult to foresee when the optimization system was devised. The conclusion contains two important points: a) in todays use of design optimization systems, the definition of the problem must often be modified to suit the capabilities of the system at hand, and b) the construction of design optimization systems with the necessary generality and flexibility to function in an engineering design environment is a problem that merits independent research beyond the traditional, fundamental investigations into mechanical properties of optimum structures and solution methods. The paper points out a number of important areas for further research.

Interlaminar/interfiber failure of unidirectional glass fiber reinforced composites used for wind turbine blades

A unidirectional glass fiber/epoxy composite was characterized under multi-axial loading by testing off-axis specimens under uniaxial tension and compression at various angles relative to the fiber direction. Iosipescu shear tests were performed with both symmetric and asymmetric specimens. Tests were performed on both 1-2 and 1-3 material coordinate planes. Strain gauges and Digital Image Correlation were used to record the stress–strain responses. A new approach was used to define a ‘failure initiation strength’ by analyzing the recorded stress–strain curves. The experimentally determined failure stresses were compared with the predictions of the maximum stress, Tsai-Wu and Northwestern University failure criteria. It was found that using the approach of analyzing the stress–strain curve to define a point of material failure initiation, it was possible to obtain good correlation between the experimental data and predictions by both the Tsai-Wu and the NU failure criteria.

Eigenfrequency and Buckling Optimization of Laminated Composite Shell Structures Using Discrete Material Optimization

The design problem of maximizing the lowest eigenfrequency or the buckling load factor of laminated composite shell structures is investigated using the so-called Discrete Material Optimization (DMO) approach. The design optimization method is based on ideas from multi-phase topology optimization where the material stiffness is computed as a weighted sum of candidate materials, thus making it possible to solve discrete optimization problems using gradient based techniques and mathematical programming. The potential of the DMO method to solve the combinatorial problem of proper choice of material, stacking sequence and fiber orientation simultaneously is illustrated for two multi-layered multi-material plate examples.

Shape Optimization of Fluid-Structure Interaction Problems Using Two-Equation Turbulence Models

The objective of this work is to develop and implement efficient numerical procedures for gradient based shape optimization of steady, strongly coupled fluid–structure interaction problems involving turbulent flow. The governing equations are the Reynoldsaveraged Navier-Stokes equations combined with a large displacement formulation for the structure. The eddy viscosity is computed using either algebraic turbulence models or the two-equation k − ω model. The solution for state is obtained using finite element residual formulations based on a consistent weak formulation of the fluid–structure interaction problem. Due to the possible large displacements of the structure, the fluid mesh is updated using a modified elastic analogy. The resulting nonlinear equations are solved using an approximate Newton method, making it possible to do design sensitivity analysis (DSA) by the direct differentiation method in a very efficient way where the resulting sensitivity equations are solved by an incremental iterative method. Gradient based shape optimization is illustrated for finding the shape of both an infinitely stiff and a flexible valve. Such examples of semi-analytic DSA and gradient based shape optimization of strongly coupled fluid-structure interaction problems involving turbulent flow modeled using two-equation turbulence models have, at least to the knowledge of the authors, not been presented before in the literature and thus represent novel results.