# Problem Formulations

Structural Model and Material Layout Description

The objective of continuum structural optimization is to find a layout of a structural material of specified properties in a defined spatial region that provides optimum structural performance. In order that the widest possible class of structural layouts can be considered, the methods in question must accommodate such generality. In this work, the spatial region that the candidate structural models can occupy is denoted Qs. To facilitate both description of the structural material layout in Qs and analysis of the performance associated with each layout considered, the domain is discretized into a relatively fine mesh of nodes and finite elements.

It is desired that at the end of the form-finding process, the structural region Qs will be decomposed into a collection of regions cumulatively denoted Oa that contain the structural material in question, and the remaining regions QB = Qs Oa that are devoid of structural material. Since solution of the form­finding problem in this way is ill-posed, an alternative relaxed approach is usually employed, wherein it is assumed that an amorphous “mixture” of structural material A and a void material B exists throughout the structural region Qs. In each region of Qs, the nature of the mixture is characterized by a local

volumetric density фА of structural material A. By permitting mixtures, the structural material A and a fictitious void material B are allowed to simultaneously occupy an infinitesimal neighborhood about each Lagrangian point x є Q s • The volumetric density of structural material A at a fixed Lagrangian

point x є Qs is denoted by фА(x) and represents the fraction of an infinitesimal region surrounding point X occupied by material A. Natural constraints upon the volumetric densities are:

0 <^A(X) < 1; 0 <^(X) < 1; ^A(X) + ^(X) = 1. (1)

Clearly, when фА(х) = 1 the point X contains solid structural material, and when фА(х) = 0 the point X is

devoid of structural material. The last physical constraint of (1) states that the material volume fractions at X are not independent and so one need only be concerned with the layout of structural material A . The design of a structure is here considered to be the spatial distribution of the structural material A in Ds.

To describe the distribution of material A throughout Ds using a finite number of design parameters, the volumetric density at each of the NUMNP nodal point forms a set of NUMNP design variables. These are then interpolated over the space of all intermediate points in the structure using the nodal shape functions:

NUMNP

9(X) = £b(X) V X efi, (2)

i=1

where bi are the nodal volumetric density values associated with the structural material; and Ni(X) are the nodal shape functions. This approach yields a C0 continuous design variable field that is not susceptible to “checkerboarding” instabilities.

Given the finite element model of the structural region Ds, the structural loads and restraints (or supports) on this region are specified as the set of design loads. For each set of design loadings, and for each realization of the design vector b = {b1,b2,K, bNUMNP}, the response performance of the structure

will be analyzed as a boundary value problem. From the computed response of the structure, the performance of the structure will be quantified, as will be the sensitivity of the performance to variations in the design variables.