J.B. Cardoso

Variational principle for shape design sensitivity analysis

In this paper, the adjoint method of design sensitivity analysis is stated as a general variational principle that is applicable to a wide range of problems. The principle gives a very simple and straightforward method of obtaining the design sensitivity expression for a functional dependent on the state fields. The sensitivity expression involves certain adjoint fields and explicit design variations of the functional and the governing equations for the primary state fields. The principle is stated in terms of an augmented functional that is defined by adding to the response functional whose sensitivity is desired, the equilibrium equation of the primary problem. Then explicit design variation of the augmented functional gives the total design variation of the response functional. Stationarity of the augmented functional with respect to the state fields defines the adjoint problem for the response functional. The principle is proved and applied to several classes of linear and nonlinear problems, such as field problems, structural problems, and dynamic response problems. It is shown that the design sensitivity expressions derived in the literature using long procedures are obtained quite routinely by use of the principle.

A design sensitivity analysis principle and its implementation into ADIANA

Abstract This paper describes a continuum principle of design sensitivity analysis of linear and nonlinear structures for shape and nonshape optimization problems. The principle has been derived previously using the virtual work equation, and the concepts of reference volume and adjoint structure. It is stated in a form that is more convenient to interpret and discretize for numerical calculations. Physical interpretation for each implicit design variation term in the principle is given relating it to the virtual work of internal or external forces, or their explicit design variations. A discretization of the principle using usual shape functions of finite element analysis is developed for implementation into computer programs, such as ADINA. Some numerical examples are solved to show practical use of the principle.

Nonlinear structural design sensitivity analysis for path dependent problems. Part 2: Analytical examples

Abstract In Part 1 of this paper a general theory of design sensitivity analysis for nonlinear structures with history dependent effects is given. The theory is also reduced to several special cases to show its generality. In the present paper, three examples are used to analytically verify the theory. Both the adjoint structure method (ASM) and the direct differentiation method (DDM) are demonstrated. The examples are (1) reaction force sensitivity for an indeterminate beam, (2) displacement sensitivity for several cases of an axial force element and (3) stress and displacement sensitivities for a visco-elastic problem. These examples provide insights for numerical implementation of the theory for more complex design problems.

Use of ADINA for design optimization of nonlinear structures

Abstract This paper describes procedures for design sensitivity analysis and optimization of nonlinear structural systems with the computer program ADINA. Formulation of the structural optimization problem, design sensitivity analysis with nonlinear response using incremental finite element procedures, and two strategies to use ADINA for design optimization are described. A database and a modem database management system are used to couple ADINA with design sensitivity analysis and optimization modules. Comparison of optimum designs with linear and nonlinear structural responses for trusses with material and geometric nonlincarities are given. More complex structures can be optimized with the developed procedures to fully exploit the capabilities of ADINA.

Design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems

This paper describes a unified variational theory for design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems for shape, nonshape, material and mechanical properties selection, as well as control problems. The concept of an adjoint system, the principle of virtual work and a Lagrangian-Eulerian formulation to describe the deformations and the design variations are used to develop a unified view point. A general formula for design sensitivity analysis is derived and interpreted for usual performance functionals. Analytical examples are utilized to demonstrate the use of the theory and give insights for application to more complex problems that must be treated numerically.

Structural shape sensitivity analysis - Relationship between material derivative and control volume approaches

The material derivative and control volume approaches for structural shape design sensitivity analysis are presented, analyzed, and compared. Starting with a continuum formulation and a general response functional needing sensitivity analysis, the two approaches are derived. It is shown that the final design sensitivity expression for one approach can be obtained from the final expression for the other. Thus, the two approaches are theoretically equivalent. Discretizations of the continuum expressions are presented, and it is shown that the two approaches can lead to different implementations for numerical calculations. A unified interpretation is developed in which the explicit design variations (partial derivatives with respect to the design variables) of the internal and external forces are the major calculations needed to implement the design sensitivity analysis. The discretized forms of the continuum sensitivity expressions are also compared with the ones obtained by starting with the discretized model ab initio. This comparison shows that these two approaches lead to similar expressions for numerical calculations. Therefore, the exact same procedures can be used for computer implementations for both of the approaches. The presented analyses give insights for implementation of the design sensitivity analysis theory with the finite element analysis programs.

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.

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.

Shape design sensitivity analysis of field problems

Abstract This paper presents a variational theory of design sensitivity analysis with respect to the selection of shape dimensions and property parameters for steady-state linear field problems. The design sensitivity expressions are derived using the concepts of reference domain and adjoint fields. The adjoint fields are interpreted as sensitivities of the functionals related to the primary fields. Two analytical examples are used to demonstrate the use of the formulation and give insight for more complex problems that must be treated numerically.

Structural shape design sensitivity analysis - A unified viewpoint

Two major approaches for structural shape design sensitivity analysis the material derivative and control volume approaches are presented, analyzed and compared. Starting with a continuum formulation and a general response functional needing sensitivity analysis, the two approaches are derived. It is shown that the final design sensitivity expression for one approach can be obtained from the final expression for the other. Thus the two approaches are theoretically the same. Discretizations of the continuum expressions are discussed and it is shown that the two approaches can lead to different implementations for numerical calculations. The control volume approach appears to be more natural for computer implementation because it is viewed as an extension of the isoparametric formulation for finite element analysis to the design sensitivity analysis problem. This view point leads to a simple explanation of the method involving explicit design variations of the internal and external forces. Using this interpretation a general implementation of the method is suggested either outside an established finite element code or inside it.