On an effective algorithm for minimizing generalized Trefftz functionals of linear elasticity theory

Abstract

The problem of minimizing the generalized Trefftz functionals of three-dimensional elasticity theory results in a minimax problem for the Lagrangian. An algorithm is proposed for searching for the saddle point in coordinate functions not subjected to any constraints in the domain and on the boundary (this is the efficiency of the algorithm). The convergence of the approximate solution is investigated. The Trefftz variational method /1/ is convenient for solving boundary value problems of mathematical physics in that the dimensionality of the problem being solved is reduced because of its reduction to the solution of equations defined on the domain boundary. At the same time, when constructing the solution using the Ritz process, say, the coordinate functions should be selected so that they satisfy the differential equation of the boundary value problem in the domain, which is a serious constraint. An approach is proposed below that uses Lagrange multipliers to reduce this constraint when minimizing the generalized Trefftz functionals of the fundamental boundary value problems of linear elasticity theory. The results obtained can also be used to minimize the classical Trefftz functionals of the boundary value problems of mathematical physics /1/. Generalized Trefftz functionals were constructed in /2, 3/ for the fundamental problems of linear elasticity theory with continuous and discontinuous elasticity coefficients. The functionals are minimized in solutions (ordinary or generalized) for the linear equilibrium equation for an elastic medium in displacements. Assuming the existence of a coordinate system of functions satisfying the equilibrium equation (in the generalized sense) in /4/, the Ritz process was investigated for solving problems to minimize the generalized Trefftz functionals in an example of the second boundary value problem of three-dimensional elasticity theory. The practical construction of the above-mentioned coordinate system is a fairly complex problem. At the same time, the differential equation of the boundary value problem in whose solutions the minimum of the functionals is sought, can be considered as a linear constraint in the problem of minimizing the Trefftz functionals. Then such a minimization problem with linear constraints can be reduced to the minimax problem of a certain Lagrangian (by using reciprocity theory).