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PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping
Reducing a Two-Point Nonlinear Boundary Value Problem of a Deformable System to a Sequence of Linear Problems
National Building Research Institute, Technion-Israel Institute of Technology, Israel
A.G. Razdolsky, "Reducing a Two-Point Nonlinear Boundary Value Problem of a Deformable System to a Sequence of Linear Problems", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 134, 2007. doi:10.4203/ccp.86.134
Keywords: nonlinear, boundary value problem, increment, parameter, deformable, thin flexure rod.
An equilibrium state of a deformable system described by a two-point nonlinear boundary value problem can be treated as a result of a deformation process of the system when a load parameter is raised from zero to the given final magnitude. This deformation process can be considered as a sequence of a large number of small increments of the system displacements caused by the small increases of the load parameter. A determination of the displacement increments is reduced to the solution of a linear boundary value problem for the functions which characterize a rate of the displacement increments when the load parameter sustains a small increment. This linear problem is obtained by a linearization of the differential equations and boundary conditions which result from the difference quotient expression for the source nonlinear equilibrium equations and boundary conditions. The final values of the system displacements result from summing over all the increments of the system displacements. This approach of solving an equilibrium problem of a deformable system is taken as the straight point for a development of the method of solving a two-point nonlinear boundary value problem that depends on some numerical parameter, action parameter.
Solving such a nonlinear problem is reduced to solving a sequence of linear boundary value problems for partial derivatives of the unknown functions with respect to the action parameter, function increment rates. The unknown functions and the function increment rates are defined by their values at the specified points of the definition domain of functions. The linear boundary value problem with respect to the function increment rates can be solved in the small vicinity of any given value of the action parameter if the solution of the source nonlinear problem is known at the given value of the action parameter. This problem is derived by a linearization of the difference quotient expressions for the differential equations and boundary conditions of the source nonlinear problem which correspond to the two near values of the action parameter. It is assumed that the initial values of the unknown functions are known when the action parameter takes the initial value. The solution of the linear problem with respect to the function increment rates enables an approximation of the solution of the source nonlinear problem in the mentioned small vicinity of the given value of the action parameter. The increment of each required function is computed as a product of the action parameter increment by the function increment rate of the corresponding function. The linear boundary value problems are solved by the superposition method of particular solutions of the Cauchy problem which satisfy the boundary conditions at the initial point. Solving the differential equations is carried out by numerical methods. The final solution of the nonlinear boundary value problem is determined by moving with small steps from the initial value of the action parameter to its final value and repeating the described procedure.
A use of the present method is demonstrated by solving the two-point boundary value problem which describes the equilibrium states of thin flexible rod under the action of the uniform transverse load. A computer program is developed for solving this problem. The obtained relationships between the deformation parameters of the stressed rod and its bending rigidity can be used in practice for the design of cables and flexible steel ties
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