Material Derivative Method Research Articles

Design Sensitivity Analysis and Optimization of Nonlinear Structural Systems With Critical Loads

A continuum formulation for design sensitivity analysis of critical loads is developed for nonlinear structural systems that are subjected to conservative loading. Both geometric and material nonlinear effects are considered. Sizing design variables such as cross-sectional areas of beam or truss design components and thicknesses of plate or membrane design components, together with their shape design variables, are treated. A continuum approach is used to obtain design sensitivity expressions in integral form. For shape design sensitivity analysis, the material derivative concept and domain method are used to find variations of the critical load due to variations in shape of the physical domain. The total Lagrangian formulation for incremental equilibrium equation and one-point linearized eigenvalue problems are utilized. A numerical method is presented to evaluate continuum design sensitivity expressions using analysis results of established finite element codes. It is found that no adjoint system is necessary for design sensitivity analysis of the critical load. Numerical results show the proposed method for design sensitivity of critical loads is accurate for both sizing and shape design variables. A numerical procedure for optimal design of nonlinear structural systems is presented, using the proposed continuum design sensitivity analysis method. An optimal design problem with a stability constraint is solved.

An Extension of the Material Derivative Method for Configuration Design Sensitivity Analysis

Abstract Configuration design variation of built-up structures that include truss, beam, plane elastic solid, and plate design components can be characterized by changes in the domain shape and orientation of design components. The material derivative method of shape design sensitivity analysis is extended to develop a continuum configuration design sensitivity analysis method for built-up structures by accounting for effects of both shape and orientation changes in the same energy equation. Variations of energy bilinear and load linear forms, with respect to both shape and orientation design variables, are derived for each structural component. Using the adjoint variable or direct differentiation method, configuration design sensitivity results for built-up structures are obtained in terms of the design velocity fields. The relationship between design parameterizations and design velocity fields is discussed for line and surface design components. A numerical method is developed to evaluate these configu...

Optimal shapes for the control of nonlinear heat conduction

The material derivative method is developed here for the optimal shape design in the control of a nonlinear heat conduction system in steady state described by a variational inequality. It is known that this method can be used for the optimal shape design for systems described by partial differential equations; it is used here for differential inequalities by taking limits of the expressions resulting from an approximation scheme. The computations are done by the finite element method; the gradient of the criteria as a function of the coordinates nodes is computed, and the performance criterion is then minimized by the material derivative method.

Sensitivity analysis of linear elastic systems using domain parameterization and a mixed mutual energy principle

A unified approach for explicit structural design sensitivity analysis of linear elastic systems is presented. First-order sensitivity expression involving the complete set of design variable, including shape design parameters, are derived for general response functional. Both direct differentiation and adjoint variable methods are developed in a general, variational context. A domain parameterization method, related to the mixed Eulerian-Lagrangian kinematic description, is introduced for the treatment of shape variation problems. This approach is an alternative to the material derivative methods presented elsewhere in the literature. The adjoint variable derivation is based on a mixed energy principle, the mutual Hu-Washizu energy principle, which is formally stated and proved. The design sensitivity derivation based on this principle unifies and extends several of the adjoint variable methods found in the literature. The derivations presented here are extended to nonlinear systems in a subsequent paper. The sensitivity expressions are valid for use with force, displacement or mixed approximate solutions of the elasticity equations. Other formulations are generally only consistent with displacement solution methods. Numerical implementation of the design sensitivity methods is discussed and illustrative examples are included.

Shape Sensitivity Analysis of Boundary Optimal Control Problems for Parabolic Systems

This paper is concerned with the shape sensitivity analysis of a class of boundary control, constrained, optimal control problems for parabolic systems. The notion of Euler and Lagrange derivatives of a boundary optimal control in the direction of a vector field is introduced. The derivatives are obtained in the form of optimal solutions of auxiliary optimal control problems. The method of sensitivity analysis used in this paper is based on related results on the differential stability of metric projections in Hilbert space onto a convex, closed subset, combined with the material derivative method of shape sensitivity analysis. Parabolic initial-boundary value problems with Dirichlet and Neumann boundary conditions are considered in this paper.

Shape design sensitivity analysis of plates and plane elastic solids under unilateral constraints

The local dependence of static response and eigenvalues on the shape of plates and plane elastic solids is characterized. The so-called material derivative method is used. The shape sensitivity analysis includes, besides linear problems, nonlinear problems with unilateral conditions, e.g., the frictionless contact problem for an elastic body on a rigid foundation. The results on shape sensitivity analysis can be used to obtain expressions for variations of integral functionals that arise in structural optimization problems.

Numerical Considerations in Structural Component Shape Optimization

A unified design sensitivity analysis theory and a linearization method of optimization are employed for structural component shape optimization. A material derivative method for shape design sensitivity analysis, using the variational formulation of the equations of elasticity and the finite element method for numerical analysis, is used to calculate derivatives of stress and other structural response measures with respect to boundary shape. Alternate methods of boundary shape parameterization are investigated, through solution of two test problems that have been treated previously by other methods: a fillet and a torque arm. Numerical experiments with these examples and a variety of finite element models show that component shape optimization requires careful selection of boundary parameterization, finite element model, and finite element grid refinement techniques.