PROJECTION-ITERATIVE MODIFICATION OF THE METHOD OF LOCAL VARIATIONS FORPROBLEMS WITH A QUADRATIC FUNCTIONAL

Most variational problems of the mechanics of deformable solids are characterized by the use of a quadratic functional of the potentialenergy of deformation.An effective grid method for solving variational problems is the method of local variations (MLV) developed by F.L. Chernous’ko1andinvestigated in studies by F.L. Chernous’ko and colleagues.2–6This method has a number of advantages over other numerical methods.In the mechanics of deformable solids, it is effective for analysing significantly inhomogeneous stress states and makes it possible toexamine loading of various types and boundary conditions and the different structures of investigated systems (for a shell–plate structure– the variability of rigidity, cutouts). In computer realization, the advantage of the MLV consists in the fact that, in computations, besidesthe program, only the immediate approximation for each point of the grid, used at the given stage, is stored in the computer memory, i.e.,the memory stores a certain array of numbers that is renewed from step to step.6However, it must be pointed out that, in solving certain classes of problems (for example, variational problems, which reduce tolinear boundary-value problems), the MLV requires a longer calculation time to achieve the specified accuracy compared with finitedifference methods and Ritz-type variational methods. The search for ways to eliminate this shortcoming led to the idea of developingmore effective schemes of its realization, based on using the idea of projection-iterative methods making it possible to shorten considerablythe computer calculation time.7–12The effectiveness of projection-iterative schemes of realization of finite difference and finite elementmethods for solving a wide class of problems of elasticity and plasticity theory has been shown.13–19The development of a projection-iterative modification of a further numerical method for solving variational problems – the MLV – is of undoubted interest. Note that issuesof reducing the computing time for the MLV have been addressed previously.