Journal of Optimization

Volume 2015 (2015), Article ID 345120, 14 pages

http://dx.doi.org/10.1155/2015/345120

## A Structural Optimization Framework for Multidisciplinary Design

Department of Engineering, The Pennsylvania State University, Altoona, PA 16601, USA

Received 29 September 2014; Revised 23 December 2014; Accepted 6 January 2015

Academic Editor: Qingsong Xu

Copyright © 2015 Mohammad Kurdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work describes the development of a structural optimization framework adept at accommodating diverse customer requirements. The purpose is to provide a framework accessible to the optimization research analyst. The framework integrates the method of moving asymptotes into the finite element analysis program (FEAP) by exploiting the direct interface capability in FEAP. Analytic sensitivities are incorporated to provide a robust and efficient optimization search. User macros are developed to interface the design algorithm and analytic sensitivity with the finite element analysis program. To test the optimization tool and sensitivity calculations, three sizing and one topology optimization problems are considered. In addition, flutter analysis of a heated panel is analyzed as an example of coupling to nonstructural discipline. In sizing optimization, the calculated semianalytic sensitivities match analytic and finite difference calculations. Differences between analytic designs and numerical ones are less than 2.0% and are attributed to discrete nature of finite elements. In the topology problem, quadratic elements are found robust at resolving checkerboard patterns.

#### 1. Introduction

Increasingly engineering products are required to maintain high efficiency but yet perform well under conflicting customer requirements. Examples of these include thermal loads [1], combination of aeroelastic, thermal, and buckling loads, and acoustic and impact loads. Structural optimization strives to improve performance by interrogating multiple disciplines simultaneously. However, there is continuous need to consider new disciplines within existing optimization tools.

Industry recognized potential for structural optimization since the 1980s. Now there are multiple commercial structural optimization tools that address product design. These tools have emerged primarily from the academic research community and then commercialized to focus on a particular application. Early structural optimization implementations include adding sizing and shape optimization to finite element software such as NASTRAN, ANSYS, and ABAQUS [2, page 243]. Thomas et al. [3] indicates that early version of Altair OptiStruct is based on application of homogenization method for topology optimization [4]. GENESIS [5] focused on employing advanced approximation techniques [6, 7] and sensitivity tools to provide an efficient optimization search [8]. To obtain design sensitivities in ABAQUS, Yi et al. [9] application utilized response surface technique to approximate design derivatives for structural optimization of gear damper. The author utilized python to access the ABAQUS kernel and formulate the noise criterion.

Thomas et al. [3] highlights important requirements of structural optimization in commercial tools. Of these requirements, the ease of use of the interface and narrow application focus of the tool provide a disincentive to expand the tools to wider range of applications. For example, until now topology optimization is limited to structural problems and does not consider important physics [10] critical to the structure. On the other hand, research tools frequently utilize in-house codes to model a narrow physical problem and then couple to structural optimization. Extension of the methodology into commercial tools may require significant development.

It is desired to have a design optimization tool that offers development environment intermediate between research and commercial applications. The tool should be amenable to the use of robust optimization methods, analytic sensitivities, advanced approximate methods, and inclusion of all important physics. The development of analytic sensitivities for new criteria is particularly important for effectiveness of gradient-based optimization tools. In addition and as suggested by Thomas et al. [3], structural optimization software should have a broad analysis capability by containing different types of elements and are amenable to integration of new physics and material types. The finite element analysis program (FEAP) [11], developed by Professor Taylor at UC Berkeley, is a research finite-element code available in FORTRAN. It offers wide range of elements and encompasses structural analysis and thermal capabilities in linear and nonlinear and static and transient domains. The code is designed to incorporate new elements, physics, materials, meshes, and solution procedures by development of user macros. In addition, the macros lend an interfacing capability to an optimization program.

Several works exist on the utilization of FEAP in structural optimization considering aeroelastic constraints. Moon [12] conducted an aerostructural optimization of wings under divergence constraint. Alonso et al. [13, 14] demonstrated an optimization framework by coupling high-fidelity CFD code to FEAP. In particular in [13] finite difference tools are used to compute the structural sensitivities. In the above studies, a Python-based programming interface is used to wrap the fluid code to structural analysis tool. More recently, Schafer et al. [15] carried out a derivative-free shape optimization study by coupling a finite volume flow solver (FASTEST) to FEAP using the coupling interface (MpCCI).

In this work a structural optimization and analytic sensitivity framework is developed. The framework is built from within the FEAP command structure by accessing the user interface. The advantage of this is direct access to finite element matrices necessary for formulation of analytic gradients, flexibility in the choice of the numerical optimization method, and incorporation of new physics. Analytic gradients are derived and provided to the method of moving asymptotes [16] to expedite the optimization search (this method is utilized here as an example, and the user can select other gradient-based optimization techniques depending on the application). To handle competing objectives, a Pareto approach [17] multiobjective is utilized to address the multiobjective problem. The Pareto front is computed via the -constraint method. The paper proceeds in Section 2 to provide the general form of nonlinear structural system of equations. In Section 3, the analytic sensitivity equations are summarized. In Section 4, the implementation of the structural optimization framework and sensitivity analysis is described. In Section 5, numerical studies are carried out to demonstrate application of the framework to structural problems with sizing and density design variables. Finally Section 6 demonstrates incorporation of new physics by analyzing the buckling and flutter of a heated panel.

#### 2. Structural Analysis

In static nonlinear analysis of structures, the finite element method gives a set of equilibrium equations [2, page 274]: where is a design variable, is a global response deformation, is the external applied load, is the internal load in structure due to the deformation, and is a load continuation parameter.

Newton method is used to compute the solution to system of equations. A Taylor series approximation of the residual is where is the tangent stiffness matrix due to internal and external loads as follows: The tangent stiffness matrix is calculated at the element level and then assembled for the global system. The matrix is symmetric when external loads are independent of deformations. In this case, direct solver is used to solve the system of equations for . The deformation is updated with the new solution until convergence to specified accuracy.

#### 3. Sensitivity Analysis

For performance functions with dependence on deformation, one can write the sensitivity of to design variable : In order to compute the sensitivity using finite differences, one needs to recompute the response at perturbed value of design variable . In addition to computational cost, this sensitivity is prone to inaccuracies in the finite element solution. There are two alternative options for computing the sensitivity semianalytically [2, page 274]. The direct method computes the sensitivity from equilibrium equations (1) at a converged response : Forward finite difference is used to compute sensitivity of the residual. Note that this only requires build-up of the residual at new design. The factored tangent stiffness matrix is available from final iteration in (2). Equation (5) is repeatedly solved for all design variables. The cost of computation increases for optimization problems with large number of design variables. The adjoint method avoids this repetition by premultiplying (5) with an adjoint vector and adding to (4). The adjoint vector is computed by setting the coefficient of to zero: where refers to transpose of matrix. For structural systems where the external load is independent of deformation, the tangent stiffness matrix is symmetric and this operation can be skipped. Sensitivity of response is then computed as Note that (6) is repeatedly solved for each deformation dependent performance function. In optimization problems with large number of such functions, the direct method is more appropriate.

#### 4. FEAP Implementation

The optimization framework is applied in the problem solution stage of the FEAP input file [11, page 3]. Here a direct interface (referred to as user macro [18]) to the program is developed to carry out the optimization search, sensitivity analysis, and definitions of the objective function and constraints. The macro allows access to the FEAP program pointers and defines new user pointers for use in the framework. New user commands are interweaved with existing solution commands in FEAP to achieve the algorithm shown in Figure 1, where a user command calls the user macro; see Table 1.