Computational method for optimization of structural shapes

The one- and two-dimensional inviscid Euler equations are formulated in an integro-differential form for shape design sensitivity analysis and optimization. The principal tool employed to derive the performance derivative sensitivity equations is the variational method, which is a continuous alternative to the discrete sensitivity analysis. Along with the sensitivity equations, the co-state equations and their boundary conditions are derived. Thereafter, based on the modal analysis of Von Neumann theory, the stability limits of the co-state equations are investigated. The stability characteristics of the co-state equations are compared with those of the Euler equations. Finally, using the criteria from the stability analysis of the state and co-state equations, some results of shape design optimization and sensitivity analysis for both quasi one- and two-dimensional Euler equations are presented. Copyright © 1999 John Wiley & Sons, Ltd.

A unified method for continuum shape design sensitivity analysis and optimal design of mechanical components is developed. A domain method of shape design sensitivity analysis that uses the material derivative concept of continuum mechanics is employed. For numerical implementation of shape optimal design, parameterization of the boundary shape of mechanical components is defined and illustrated using a Bezier surface. In shape design problems, nodal points of the finite element model move as the shape changes. A method of automatic regridding to account for shape change has been developed using a design velocity field in the physical domain that obeys the governing equilibrium equations of the elastic solid. For numerical implementation of the continuum shape design sensitivity analysis and automatic regridding, an established finite element analysis code is used. To demonstrate the feasibility of the method developed, shape design optimization of a main engine bearing cap is carried out as an example.

A shape design sensitivity analysis (DSA) and optimization of structural transient dynamics are proposed for the finite deformation elastoplastic materials under impact with a rigid surface. A shape variation of the structure is considered using the material derivative approach in continuum mechanics. Hyperelasticitybased multiplicatively decomposed elastoplasticity is used for the constitutive model. The implicit Newmark time integration scheme is used for the structural dynamics. The design sensitivity equation is solved at each converged time step with the same tangent stiffness matrix as response analysis without iteration. The cost of sensitivity computation is more efficient than the cost of response analysis for the implicit time integration method. The efficiency and the accuracy of the proposed method are shown through the design optimization of a vehicle bumper. *Communicated by D. Tortorelli.

A shape design sensitivity analysis and optimization procedure is proposed using a meshfree method. A CAD tool connection is established to facilitate the seamless integration of structural modeling, simulation, and design optimization. A new CAD-based method for design velocity computation is proposed, applicable to both the meshfree method and the finite element method. A shape design variable can be chosen among the geometric parameters defined in the CAD model in order to support a concurrent engineering environment. A new CAD-based method for updating the discrete model is proposed, such that the meshfree model is automatically updated in a continuous and consistent way during design optimization. The proposed method is applied to 2D and 3D linear elastic problems, as well as to a large deformation hyperelastic problem.

Shape design sensitivity analysis and optimization of three dimensional resonant systems

Shape design sensitivity analysis and optimization of elasto-plasticity with frictional contact

Using the bond-based peridynamics and the parallel computation with binary decomposition, an adjoint shape design sensitivity analysis(DSA) method is developed for the dynamic crack propagation problems. The peridynamics includes the successive branching of cracks and employs the explicit scheme of time integration. The adjoint variable method is generally not suitable for path-dependent problems but employed since the path of response analysis is readily available. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It turns out that C1-continuous volume fraction is necessary for the accurate evaluation of shape design sensitivity in peridynamic discretization.

Shape design optimization of thermoelasticity problems using isogeometric boundary element method

Error estimation for the finite element evaluation of [formula omitted] and [formula omitted] in mixed-mode linear elastic fracture mechanics

An adjoint approach for the shape design sensitivity analysis (SDSA) of plane elasticity problems with a direct boundary integral equation formulation is presented. The objective function is expressed in an integral form and augmented by incorporating the elasticity equations via adjoint functions. The variation of the augmented function is taken by performing its material derivative to obtain the sensitivity of the objective function. The adjoint functions are obtained through the solution of the adjoint problem resulting from setting the local variations of the displacement, velocity, and traction fields equal to zero. An approximate procedure for the application of concentrated adjoint load in the boundary element framework is suggested. This allows the computation of displacement, traction, and stress sensitivities at discrete nodal points. The present formulation includes the treatment of body forces in sensitivity calculations. Numerical examples demonstrate the validity of the present approach.

This paper presents a brief overview of some of the more recent advances in steady aerodynamic shape-design sensitivity analysis and optimization, based on advanced computational fluid dynamics. The focus here is on those methods particularly well- suited to the study of geometrically complex configurations and their potentially complex associated flow physics. When nonlinear state equations are considered in the optimization process, difficulties are found in the application of sensitivity analysis. Some techniques for circumventing such difficulties are currently being explored and are included here. Attention is directed to methods that utilize automatic differentiation to obtain aerodynamic sensitivity derivatives for both complex configurations and complex flow physics. Various examples of shape-design sensitivity analysis for unstructured-grid computational fluid dynamics algorithms are demonstrated for different formulations of the sensitivity equations. Finally, the use of advanced, unstructured-grid computational fluid dynamics in multidisciplinary analyses and multidisciplinary sensitivity analyses within future optimization processes is recommended and encouraged.

This paper presents a new methodology for topology optimization in which the topology of the design domain considered may change during the shape optimization process, unlike the case of classical shape or topology optimization. To achieve this, the concept of the topological gradient is introduced to compute the sensitivity of an objective function when a small hole is drilled in the domain, and this is successfully connected to the shape sensitivity. A test problem consisting of an elliptic dielectric material with a circular hole has been selected to illustrate the validity of the proposed method.

Optimal blank shape design using the initial velocity of boundary nodes

This paper deals with shape optimization of three-dimensional elastic bodies. The proposed approach is based on natural design variables and shape functions and utilizes the boundary element method. The design variables are the magnitudes of a set of fictitious loads applied on the structure. Shape optimal design problems based on minimum mean compliance, peak stress minimization and stress constraints are considered. A general method for shape design sensitivity analysis using the material derivative concept and adjoint variable method is used. The sensitivity formula for a general stress constraint imposed over the optimized boundary is derived. Optimal shapes for three-dimensional problems are presented to show numerical applications.

In this paper, a novel optimal shape design method is proposed using the finite-difference time-domain (FDTD) method and the design sensitivity analysis to obtain broad-band characteristics of microwave devices. In shape design problem, the nodes that describe the shape of geometry to be optimized are taken as design variables. The design sensitivity is evaluated using the adjoint variable equation that is obtained from a terminal-value problem. The adjoint equation can be solved by the FDTD technique with the backward time scheme. With this method, a Ka-band unilateral fin line is tested to show validity.