engleski

Convergence and accuracy of numerical models using dynamic analogies

This paper describes a procedure for proving the convergence and for the estimation of the numerical solution error of boundary problems based on the transformation of the discrete boundary problem into an equivalent discrete dynamic problem. The convergence and accuracy of numerical solutions depend on the convergence of dynamic eigenproblem spectrum. In this paper we are dealing with the equilibrium boundary problem over domain and the appropriate boundary conditions. The standard eigenproblem can be associated with the boundary problem. The solutions of the standard eigenproblem are pairs of eigenvalues and orthonormalized eigenfunctions spanning the separable Hilbert space. The generalized eigenproblem is obtained by analogy with the standard eigenproblem if the mass functions can be treated as a mass distributed over the whole domain. The numerical model of the boundary problem, has an appertained standard eigenproblem and a dynamic eigenproblem where solutions are eigenvalues and eigenvectors spanning a separable Hilbert spaces. If a mass matrix is diagonal and especially if Mn=(1/n)I, then matrix Mn represents discretization measure of domain W. When nŽĽ, the discrete mass is transformed to a continuous function derivable over parts of the domain. If eigenvalues of the dynamic eigenproblem converge when nŽĽ, the numerical solution exists and converges towards the exact solution. Once we have proved that a numerical solution is convergent on the given class of boundary problems, the next step is to find out how the discretization and the load type affect the accuracy of the results. The procedure for the estimation of the error consists of two phases. The relative error in any component of each eigenvector has to be computed firstly. It is independent of the applied load and, for the observed discretization, it can be evaluated in relation to the analytical solution or to the solution obtained by an other discretization. Afterwards, the error of each eigenvector, based on the influence of each eigenvector on the numerical solutions, is computed. The total error is obtained by summing the errors of all eigenvectors. Several examples of boundary problems used to prove the liaison in this paper are taken from the linear and nonlinear structural analysis. The applied numerical method is the finite element method, although any numerical method can be used. The developed procedure for proving the convergence and for the estimation of numerical solution error of boundary problems is relatively simple, easy to perform and guarantees the upper bounds of the error for determined discretization.

convergence; accuracy; error analysis; numerical models; dynamic analogies; eigenproblem

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