Abstract
Continuation methods are considered here in a broad sense as the collection of methods needed for the computational analysis of specified parts of the solution field of “under-determined” equations Fx = c where F: R m → R n , m >; n. is given and any suitable m− n of the variables x, are designated as parameters. Such equations arise frequently in structural mechanics. In general, the solutions are ( m− n)-dimensional manifolds in R m . Some basic existence results for the case m = n +1 are presented, and a procedure for the computational trace of the corresponding one-dimensional solution manifolds in R n +1 is discussed in detail. Then a general approach is formulated which allows, under certain assumptions, the computation of the derivative of F, and which includes, the usual incremental formulations in structural mechanics. In finite element applications it is possible to combine the continuation procedure with adaptive mesh-refinements: for a model problem it is shown that such a combined process can be surprisingly effective. The article ends with some comments about the general case m >; n + 1 and the possibility of assessing numerically the structural stability of a structure.
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