For each design, a structural analysis problem is solved on the continuum domain Ds. In general terms, the structural analysis problem solved for each realization of the design vector b is the following: Find the displacement field u(X) Ds^^3 such that the variational equilibrium problem is solved:
Jо : &e dQs = Jh • £u drs + jpg ■ c)u dQs
П, Г, n.
where ct(X) is the local stress field in the structure; h is a traction vector consistent with the design loads being applied to the structure; p(X) is the local mass density of the structural material; g is the gravitational body force vector; 8u is a kinematically admissible variational displacement field; and де is the corresponding variational strain field. In the structural model, the material features linear elastic behavior such that о = C* : £ where the effective elasticity tensor is design dependent and prescribed in accordance with Eq. (3). The matrix problem associated with variational equilibrium of the discrete finite element structural model, for which u(X) = ^iNi (X) u ., is
0 = K • u – fext = fmt – fext
K LM = J BtjC*m„BMk dQs
fmt = к • u = J Бг я dQs
f= J N h drs + J N pg dQ,
In all of the above, N denote the nodal shape functions and Б denote the standard strain-displacement matrices [c. f. Bathe, 1996]. The structural stiffness matrix K is positive definite due to the characteristics of the effective elasticity tensor C*, and this guarantees a unique solution to the structural analysis problem for each realization of the design b.
Once the equilibrium solution to the problem of Eq. (5) is obtained, then the linearized geometrical stiffness matrix G can be computed based on the stress field ct in the structure:
It is worth noting that G is not necessarily positive definite but rather depends heavily upon the nature of the stress field in the structure. A purely tensile stress field clearly makes G positive definite, although for any compressive stresses, G will not be positive definite.