[2011] The Global Nonlinear Galerkin Method for the Solution of von Ka…
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Tech Science Press, CMC, vol.23, no.2, pp.155-186, 2011
The Global Nonlinear Galerkin Method for the Solution of von Karman Nonlinear Plate Equations: An Optimal & Faster Iterative Method for the Direct Solution of Nonlinear Algebraic Equations F(x) = 0, using dx = l[aF + (1 - a)BT F]
Author(s): H.H. Dai, J.K. Paik and S.N. Atluri
Abstract:
The application of the Galerkin method, using global trial functions which satisfy the boundary conditions, to nonlinear partial differential equations such as those in the von Karman nonlinear plate theory, is well-known. Such an approach using trial function expansions involving multiple basis functions, leads to a highly coupled system of nonlinear algebraic equations (NAEs). The derivation of such a system of NAEs and their direct solutions have hitherto been considered to be formidable tasks. Thus, research in the last 40 years has been focused mainly on the use of local trial functions and the Galerkin method, applied to the piecewise linear system of partial differential equations in the updated or total Lagrangean reference frames. This leads to the so-called tangent-stiffness finite element method. The piecewise linear tangent-stiffness finite element equations are usually solved by an iterative Newton-Raphson method, which involves the inversion of the tangent-stiffness matrix during each iteration. However, the advent of symbolic computation has made it now much easier to directly derive the coupled system of NAEs using the global Galerkin method. Also, methods to directly solve the NAEs, without inverting the tangent-stiffness matrix during each iteration, and which are faster and better than the Newton method are slowly emerging. In a previous paper [Dai, Paik and Atluri (2011a)], we have presented an exponentially convergent scalar homotopy algorithm to directly solve a large set of NAEs arising out of the application of the global Galerkin method to von Karman plate equations. While the results were highly encouraging, the computation time increases with the increase in the number of NAEs-the number of coupled NAEs solved by Dai, Paik and Atluri (2011a) was of the order of 40. In this paper we present a much improved method of solving a larger system of NAEs, much faster. If F(x) = 0 [Fi (xj ) = 0] is the system of NAEs governing the modal amplitudes xj [j = 1, ...N], for large N, we recast the NAEs into a system of nonlinear ODEs: Fx = l[aF + ( 1 - a )BT F], where l and a are scalars, and Bij = �Fi / �xj . We derive a purely iterative algorithm from this, with optimum value for l and a being determined by keeping x on a newly defined invariant manifold [Liu and Atluri (2011b)]. Several numerical examples of nonlinear von Karman plates, including the post-buckling behavior of plates with initial imperfections are presented to show that the present algorithms for directly solving the NAEs are several orders of magnitude faster than those in Dai, Paik and Atluri (2011a). This makes the resurgence of simple global Galerkin methods, as alternatives to the finite element method, to directly solve nonlinear structural mechanics problems without piecewise linear formulations, entirely feasible.
The Global Nonlinear Galerkin Method for the Solution of von Karman Nonlinear Plate Equations: An Optimal & Faster Iterative Method for the Direct Solution of Nonlinear Algebraic Equations F(x) = 0, using dx = l[aF + (1 - a)BT F]
Author(s): H.H. Dai, J.K. Paik and S.N. Atluri
Abstract:
The application of the Galerkin method, using global trial functions which satisfy the boundary conditions, to nonlinear partial differential equations such as those in the von Karman nonlinear plate theory, is well-known. Such an approach using trial function expansions involving multiple basis functions, leads to a highly coupled system of nonlinear algebraic equations (NAEs). The derivation of such a system of NAEs and their direct solutions have hitherto been considered to be formidable tasks. Thus, research in the last 40 years has been focused mainly on the use of local trial functions and the Galerkin method, applied to the piecewise linear system of partial differential equations in the updated or total Lagrangean reference frames. This leads to the so-called tangent-stiffness finite element method. The piecewise linear tangent-stiffness finite element equations are usually solved by an iterative Newton-Raphson method, which involves the inversion of the tangent-stiffness matrix during each iteration. However, the advent of symbolic computation has made it now much easier to directly derive the coupled system of NAEs using the global Galerkin method. Also, methods to directly solve the NAEs, without inverting the tangent-stiffness matrix during each iteration, and which are faster and better than the Newton method are slowly emerging. In a previous paper [Dai, Paik and Atluri (2011a)], we have presented an exponentially convergent scalar homotopy algorithm to directly solve a large set of NAEs arising out of the application of the global Galerkin method to von Karman plate equations. While the results were highly encouraging, the computation time increases with the increase in the number of NAEs-the number of coupled NAEs solved by Dai, Paik and Atluri (2011a) was of the order of 40. In this paper we present a much improved method of solving a larger system of NAEs, much faster. If F(x) = 0 [Fi (xj ) = 0] is the system of NAEs governing the modal amplitudes xj [j = 1, ...N], for large N, we recast the NAEs into a system of nonlinear ODEs: Fx = l[aF + ( 1 - a )BT F], where l and a are scalars, and Bij = �Fi / �xj . We derive a purely iterative algorithm from this, with optimum value for l and a being determined by keeping x on a newly defined invariant manifold [Liu and Atluri (2011b)]. Several numerical examples of nonlinear von Karman plates, including the post-buckling behavior of plates with initial imperfections are presented to show that the present algorithms for directly solving the NAEs are several orders of magnitude faster than those in Dai, Paik and Atluri (2011a). This makes the resurgence of simple global Galerkin methods, as alternatives to the finite element method, to directly solve nonlinear structural mechanics problems without piecewise linear formulations, entirely feasible.
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