Continuous design transmutation method for problems in engineering analysis

Many problems in engineering analysis and design involve searching solutions of a system of equations. In most cases, because these equations are highly coupled and non-linear and thus unlikely to yield close-form solutions, researchers seek possible solutions using numerical techniques. In this paper, we present a ‘Continuous Design Transmutation Method’, a numerical method based on the polynomial continuation theory that has been discussed in recent works appearing in engineering literature. Our model departs from similar approaches in that its start system and homotopy are based on physical design and not pure mathematical equations. To avoid singularity on real paths, we introduce a heuristic disturbance mechanism. First, we simplify target design in order to construct a design that possesses both a similar structure and known solutions. Then, we construct a homotopy between the governing equations of the simplified design and those of the target design. The solutions of the target problem emerge as one tracks the solutions of the simplified design’s governing equations as these incrementally evolve from the simplified design into the target design. Using our method, one tracks only isolated solutions of the simplified problem. All of the extraneous paths have been eliminated before the solution-tracking procedure begins. As a result, not only can one easily interpret, in physical terms, the transmutation process, but one can also monitor the design feature changes. In this paper, we demonstrate our method as applied to two sample problems: a five-point position-generation problem for a planar four-bar mechanism and a function-generation problem for an RSSR linkage