Computable Estimates of the Modeling Error Related to Kirchhoff-love Plate Model

The Kirchhoff-Love plate model is a widely used in the analysis of thin elastic plates. It is well known that Kirchhoff-Love solutions can be viewed as certain limits of displacements and stresses for elastic plates where the thickness tends to zero. In this paper, we consider the problem from a different point of view and derive computable upper bounds of the difference between the exact three-dimensional solution and a solution computed by using the Kirchhoff-Love hypotheses. This estimate is valid for any value of the thickness parameter. In combination with a posteriori error estimates for approximation errors, this estimate allows the direct measurement of both, approximation and modeling errors, encompassed in a numerical solution of the Kirchhoff-Love model. We prove that the upper bound possess necessary asymptotic properties and, therefore, does not deteriorate as the thickness tends to zero. COMPUTABLE ESTIMATES OF THE MODELING ERROR RELATED TO KIRCHHOFF-LOVE PLATE MODEL SERGEY REPIN AND STEFAN SAUTER A!"#$%&#. The Kirchhoff-Love plate model is a widely used in the analysis of thin elastic plates. It is well known that Kirchhoff-Love solutions can be viewed as certain limits of displacements and stresses for elastic plates where the thickness tends to zero. In this paper, we consider the problem from a different point of view and derive computable upper bounds of the difference between the exact three-dimensional solution and a solution computed by using the Kirchhoff-Love hypotheses. This estimate is valid for any value of the thickness parameter. In combination with a posteriori error estimates for approximation errors, this estimate allows the direct measurement of both, approximation and modeling errors, encompassed in a numerical solution of the Kirchhoff-Love model. We prove that the upper bound possess necessary asymptotic properties and, therefore, does not deteriorate as the thickness tends to zero. Mathematics Subject Classification 2000: 35J35, 65N15, 74K20 1. I!"#$%&'"($! In many practically important cases, an approximation of the solution u(d) of a d-dimensional problem (we call it Problem P(d) and assume that ud belongs to a Banach space V ) is found by solving some simplified problem P(d−k) (where k is a positive integer number). A solution of this problem we denote by u(d−k). An approximation of u(d−k) is usually obtained by projecting P(d−k) onto a finite dimensional space and solving the corresponding discrete problem P τ , where τ is a small parameter related to the respective mesh Tτ . Thus, instead of u(d) we compute u (d−k) τ (see Fig. 1). The general purpose of u (d−k) τ is to present a reliable information on u(d). It should be outlined that the functions u(d), u(d−k) and u (d−k) τ belong to different spaces, so that to compare these functions we need a dimension reconstruction operator R : V (d−k) → V that forms d-dimensional images of (d−k)-dimensional solutions. One can construct such an operator by different methods, but obviously it must satisfy two conditions: computational simplicity and boundedness. Additionally, we assume that R satisfies the