A. J. Valido

OPTIMAL DESIGN OF NONLINEAR STRUCTURES AND MECHANICAL SYSTEMS

This paper presents an optimal structural design system based on a variational theory of design sensitivity analysis for linear and nonlinear structures and mechanical systems. So called flexible systems and structures, as well as design and control variables, are treated within a unified frame. The concept of an auxiliary system, the principle of virtual work, and a Lagrangean-Eulerian description of the deformations and design variations, are used to develop the unified viewpoint. Finite element and finite difference methods are used for spatial and time discretization of the sensitivity equations. The isoparametric concept of the finite element formulation is related to the concept of control volume. The concept of a design element is used for the design modeling of the structure and to generate the analysis model from the design model. Structural analysis and optimization codes are combined to create an optimal design capability. Optimalily criteria methods and nonlinear programming are applied as opt...

Optimal design of elastic-plastic structures with post-critical behaviour

This work deals with the optimal design of nonlinear structures where both geometric and path-dependent material nonlinearities are considered. Postcritical behaviour is allowed. Critical and postcritical constraints are considered. Constraints on local and global stability have been introduced. The classical critical load constraint against global instability is given with numerical advantage by a new method using a displacement constraint. The total Lagrangian description and a continuum variational formulation are used for the response and design sensitivity analysis. A continuation algorithm is used to implement the postcritical path. The path-dependent sensitivity problem is addressed by an incremental strategy. A direct differentiation approach is used to derive the response sensitivities with respect to both cross-section and configuration design. A finite element technique models the structure. A mathematical programming approach is used for the optimization process. Numerical examples are performed on three-dimensional truss structures.

Geometrically nonlinear composite beam structures: optimal design

This paper applies the formulation and the finite element analysis and sensitivity analysis model developed in a companion paper to the optimal design of various geometrically nonlinear composite laminate beam structures. The element design sensitivities are imbedded in the finite element code and the global sensitivities are interfacing the analysis with a nonlinear programming optimizer. Laminate thickness and lamina orientations are considered as design variables. Mass, stiffness and deflection are used as objectives or combined as multiobjective functions. The nonlinear critical load or the Tsai-Hill stress failure index, are used as constraints. Postcritical domain behavior allows the advantageous substitution of the critical load constraint by a displacement constraint.

Optimal cross-section and configuration design of elastic-plastic structures subject to dynamic cyclic loading

This paper shows an optimal design problem with continuum variational formulation, applied to nonlinear elasticplastic structures subject to dynamic loading. The total Lagrangian procedure is used to describe the response of the structure. The direct differentiation method is used to obtain the sensitivities of the structural response that are needed to solve the optimization problem. Since unloading and reloading of the structure are allowed, the structural response is path-dependent and an additional step is needed to integrate the constitutive equations. It can be shown, consequently, that design sensitivity analysis is also path-dependent. A finite element method with implicit time integration is used to discretize the state and sensitivity equations.A mathematical programming approach is used for the optimization process. Numerical applications are performed on a 3-D truss structure, where cross-sectional areas and nodal point coordinates are treated as design variables. Optimal designs have been obtained and compared by using two different strategies: a twolevel strategy where the levels are defined according to the type of design variables, cross sectional areas or node coordinates, and optimizing simultaneously with respect to both types of design variables. Comparisons have also been made between optimal designs obtained by considering or not considering the inertial term of the structural equilibrium.

Geometrically nonlinear composite beam structures: design sensitivity analysis

The purpose of this paper is to develop a finite element model for optimal design of composite laminated thin-walled beam structures, with geometrically nonlinear behavior, including post-critical behavior. A continuation paper will be presented with design optimization applications of this model. The structural deformation is described by an updated Lagrangean formulation. The structural response is determined by a displacement controlled continuation method. A two-node Hermitean beam element is used. The beams are made from an assembly of flat-layered laminated composite panels. Beam cross-section mass and stiffness property matrices are presented. Design sensitivities are imbedded into the finite element modeling and assembled in order to perform the structural design sensitivity analysis. The adjoint structure method is used. The lamina orientation and the laminate thickness are selected as the design variables. Displacement, failure index, critical load and natural frequency are considered as performance measures. The critical load constraint calculated as the limit point of the nonlinear response is also considered, but a new method is proposed, replacing it by a displacement constraint.

Design and control of nonlinear mechanical systems for minimum time

This paper presents an integrated methodology for optimal design and control of nonlinear flexible mechanical systems, including minimum time problems. This formulation is implemented in an optimum design code and it is applied to the nonlinear behavior dynamic response. Damping and stiffness characteristics plus control driven forces are considered as decision variables. A conceptual separation between time variant and time invariant design parameters is presented, this way including the design space into the control space and considering the design variables as control variables not depending on time. By using time integrals through all the derivations, design and control problems are unified. In the optimization process we can use both types of variables simultaneously or by interdependent levels. For treating minimum time problems, a unit time interval is mapped onto the original time interval, then treating equally time variant and time invariant problems. The dynamic response and its sensitivity are discretized via space-time finite elements, and may be integrated either by at-once integration or step-by-step. Adjoint system approach is used to calculate the sensitivities.

A space-time finite element model for design and control optimization of nonlinear dynamic response

A design and control sensitivity analysis and multicriteria optimization formulation is derived for flexible mechanical systems. This formulation is implemented in an optimum design code and it is applied to the nonlinear dynamic response. By extending the spatial domain to the space-time domain and treating the design variables as control variables that do not change with time, the design space is included in the control space. Thus, one can unify in one single formulation the problems of optimum design and optimal control. Structural dimensions as well as lumped damping and stiffness parameters plus control driven forces, are considered as decision variables. The dynamic response and its sensitivity with respect to the design and control variables are discretized via space-time finite elements, and are integrated at-once, as it is traditionally used for static response. The adjoint system approach is used to determine the design sensitivities. Design optimization numerical examples are performed. Nonlinear programming and optimality criteria may be used for the optimization process. A normalized weighted bound formulation is used to handle multicriteria problems.

Design variation of thin-walled composite beam cross-section properties

Purpose The purpose of this paper is to present a design sensitivity analysis continuum formulation for the cross-section properties of thin-walled laminated composite beams. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, on shear bending and shear warping, and on the individual stiffness of the laminates constituting the cross-section. Design/methodology/approach In order to determine its properties, the cross-section geometry is modeled by quadratic isoparametric finite elements. For design sensitivity calculations, the cross-section is modeled throughout design elements to which the element sensitivity equations correspond. Geometrically, the design elements may coincide with the laminates that constitute the cross-section. Findings The developed formulation is based on the concept of adjoint system, which suffers a specific adjoint warping for each of the properties depending on warping. The lamina orientation and the laminate thickness are selected as design variables. Originality/value The developed formulation can be applied in a unified way to open, closed or hybrid cross-sections.

Finite Element Analysis of Geometrically Nonlinear Thin-Walled Composite Laminated Beams

The purpose of the present work is to develop a finite element model for structural analysis of composite laminated thin-walled beam structures, with geometrically nonlinear behavior, including torsional warping deformation. The structural deformation is described by an Updated Lagrangean formulation. To define the load-deflection path, a generalized displacement control method has been implemented. We have considered a thin-walled section made from an assembly of flat layered laminated composite elements. The parts like a flange and a web in a section will be referred to as a panel. The cross-section bending-torsion properties are integrals based on the cross-section geometry, on the warping function and on the individual stiffness of the laminates that constitute the crosssection. The cross-section geometry is discretized by quadratic isoparametric finite elements in order to determine its bending-torsion properties. The structural discretization is formulated throughout three-dimensional two-node Hermitean finite beam elements, with seven degrees of freedom per node. In numerical examples a thin-walled cross-section cantilever beam is considered. The influence of the lamina orientation on the critical load is studied.

Design Sensitivity Analysis of Composite Thin-walled Profiles including Torsion and Shear Warping

The present work deals with the design sensitivity analysis of thin-walled laminated composite beam cross-section bending-torsion properties. These properties are expressed as integrals based on the cross-section geometry, on the warping functions for torsion, shear bending and shear warping, and on the individual stiffness of the laminates that constitute the cross-section. Among these properties, we may emphasize the location of the shear center, the torsion stiffness, the warping stiffness and the shear coefficients. A variational continuum formulation for the design sensitivity analysis of crosssection bending-torsion properties is presented. The formulation is based on the concept of adjoint system, which suffers a specific adjoint warping for each of the properties. The developed formulation can be applied in a unified way to open, closed or hybrid cross-sections, with simply connected or multi-connected domains. The lamina orientation and the laminate thickness are selected as the de-sign variables. For the sensitivity calculations, the cross-section may be modeled throughout design elements to which the element sensitivity equations correspond. Geometrically, the de-sign elements may coincide, with the laminates that constitute the cross-section.