Isogeometric design of elastic arches for maximum fundamental frequency

Abstract

The isogeometric paradigm is aimed at unifying the geometric and analysis descriptions of engineering problems. This unification is brought about by employing the same basis functions describing the geometry to approximate the physical response. Non-uniform rational B-splines (NURBS) are commonly used for this purpose and are adopted in the present work for the design of elastic arches. Design for optimal shape and stiffness distribution is considered. Manufacturing constraints are imposed on shape and sizing variables. Shape changes are represented by altering spatial location of the control points and the associated weights. Sizing variables, that control the stiffness distribution, are defined at the control points and interpolated using the same spline basis functions. Since analysis, sizing, and shape design share the same underlying description, consistent discrete sensitivities can be easily evaluated analytically, greatly improving the performance of the optimisation process. While sizing should reflect the influence of local stress states, shape design is preferably performed at a global level. Thus, a multilevel approach is utilised, where shape design is carried out at a coarser level. Projecting the shape design sensitivities bridges the gap between the different levels. A variational formulation of essential manufacturing constraints for sizing and shape optimal design is introduced. The design framework is applied to fundamental frequency maximisation problems.

Isogeometric shape optimization in fluid mechanics

“…An important challenge in the methodology is the specification of a suitable weight˜ of the regularization. This challenge is similar in nature to the one associated with specifying a suitable minimal distance between control points (Wall et al 2008), or a maximal shape change norm (Nagy et al 2011). The specification may be partly facilitated by estimating the initial ratio between the physical objective C 0 and the regularization objective R 0 :…”

“…This is the challenge in a nut-shell: when optimizing the location of many control points (for a sufficiently unconstrained problem, and with a sufficiently tight convergence criterion), they may cluster, spuriously yielding slightly lower values of the objective function on the cost of significantly worse parametrizations and less accurate analysis, which may eventually lead to a collapse of the method. The clustering of control points is a wellknown issue in isogeometric shape optimization (Wall et al 2008;Nagy et al 2011). Related numerical problems in finite element based shape optimization, and regularization techniques to address them, are also welldescribed (Bletzinger et al 2010).…”

“…A popular choice is to put a lower bound on the distance between control points (Wall et al 2008), and although this approach does take care of the tendency of control points to cluster, it still closes the door to parts of the design space, like the sharp corners that appear at the inlet and the outlet of the pipe bend in Figure 7. Another choice is to prescribe an upper bound on a single, global measure of the shape change (Nagy et al 2011) during the optimization, thereby significantly reducing the number of constraints. Finally, very recently a regularization scheme based on a shape gradient method has been applied in an isogeometric setting (Azegami et al 2012).…”

“…Numerous studies within structural mechanics have been made, using either NURBS control points (Wall et al 2008;Cho and Ha 2009), NURBS control points and weights (Nagy et al 2010a,b;Qian 2010;Nagy et al 2011), or T-splines control points (Ha et al 2010;Seo et al 2010b) as design variables. NURBS-based isogeometric shape optimization using a boundary integral method has also been studied (Li and Qian 2011).…”

“…This also applies in an isogeometric approach. Here, the shape is given by control points, and in this setting, care has to be taken to avoid clustering and folding over of these control points during optimization, which in turn may lead to singular parametrizations (Wall et al 2008;Nagy et al 2011).…”