Optimal Sizing of Industrial Structural Mechanics Problems Using Automatic Differentiation
by
Wolfram Mühlhuber
Institute of Analysis and Computational Mathematics, University of Linz
Coauthors: Gundolf Haase (University of Linz), Ulrich Langer (University of Linz)

The industrial application of our work consists of the mass minimization of a frame in an injection moulding machine. This frame has to compensate the forces acting on the mould inside the machine and has to fulfill certain critical constraints. The deformation of that frame with constant thickness is described by the plain stress state equations for linear elasticity. If the thickness varies then we use a generalized plain stress state with constant thickness in the coarse grid elements. These direct problems are solved by an adaptive multigrid solver.

The mass minimization problem leads to a constrained minimization problem for a non-linear functional which will be solved by some standard optimization algorithm which requires the gradients with respect to design parameters.

In this optimal sizing problem we have to handle hundreds of varying thickness parameters which makes numerical differentiation non-competitive. The first approach to calculate the gradient quite fast consists in using Automatic Differentiation (AD) of our direct problem solver. This approach works fine for direct solvers for the direct problem but requires huge memory and disk capabilities to handle iterative, and especially adaptive solvers. The other approach consists in writing the total derivative of the functional and get many partial derivatives by solving only one adjoint problem by means of our adaptive multilevel solver. Unfortunately, one may have to handle and implement huge expressions for other partial derivatives so that this approach is only useful for rather simple functionals.

The best solution seems to be the combination of AD and the handcoded adjoint method. This means, that we handle derivatives with respect to the differential equations by the adjoint method and all remaining derivatives by AD. Using this hybrid technique instead of a pure AD approach, we achieved an acceleration by factor 4.

The presented numerical results show that this combination of handcoded gradient routines and AD results in a very powerful and efficient tool. The flexibility of the pure AD approach remains but more or less at the same speed as by using completely handcoded gradient routines. This is especially important in the industrial application during the design process. Furthermore the optimal sizing can also be used for finding an initial guess for the shape optimization.

http://www.sfb013.uni-linz.ac.at/~wmuehlhu/ad2000.ps.gz

Date received: January 26, 2000