PREDICTION OF ERROR IN FINITE ELEMENT RESULTS A. Umar,t H. Abbas,f A. Qadeerf and D. K. Sehgals tuniversity Polytechnic, Aligarh Muslim University, Aligarh 202 002, India fDepartment of Civil Engineering, Aligarh Muslim University, Aligarh 202 002, India §Department of Applied Mechanics, Indian Institute of Technology, New Delhi 110 016, India zyxwvutsr (Received 13 January 1995) Abstract-The precision in the results of analysis using the finite element (FE) technique requires that a structure to be analyzed shall be discretized using a finer mesh. The number of elements into which the structure shall be discretized so as to obtain the FE results within an acceptable tolerance is usually decided by experience or a guess. Moreover, the error in the results of FE analysis is an unknown, therefore, it is difficult to say whether the results are within an acceptable tolerance. It is due to these reasons that an extrapolation technique is presented in the present paper to predict the results more precisely by carrying out the analysis for two mesh configurations which have a different number of elements in each case, but possess a similar discretization pattern. This is done by taking into consideration the variation of parameters within the body of the structure. The error in the results of analysis of plane strain and plate bending problems has been found to be inversely proportional to the number of elements. Two other algorithms for the modeling of error have also been tested. Several numerical examples under the two categories of problems considered in the study are presented to demonstrate the utility of the proposed extrapolation technique. Copyright 0 1996 Elsevier Science Ltd. zyxwvutsrqponmlkjihgfedcbaZYX 1. INTRODUCTION The precision in the results of analysis using the finite element method (FEM) requires that the structure to be analyzed shall be discretized using a finer mesh in addition to the correct representation of the structure [ 1,2]. However, finer mesh discretization requires a large computer memory as well as execution time. The variation of FE results with the increase in the number of elements is shown in Fig. 1. The whole range of number of elements has been divided into three zones. When the number of elements is small (n < n,) then the FE results are not reliable; this has been named as the zone of improper representation (IR). The increase in the number of elements beyond ni leads to the gradual convergence of test results as shown in Fig. l(a), provided the discretization is done by taking into consideration the expected variation of parameters within the body of the structure. Otherwise the convergence is achieved only when very large number of elements are taken, as shown in Fig. l(b). This zone has been known as the zone of proper representation (PR) which is the zone II. The increase in the number of elements beyond n, (Fig. 1) leads to the accumulation of round off error consequently FE results become deviated from the correct results instead of getting converged. It is obvious that the increase in the number of elements gives rise to the increase in the requirement of computer memory and execution time, shown in Fig. I(c). If the minimum number of the elements n required for the acceptable error suits the machine as well as the user (i.e. n < nr and n <n,) then the FE results will be reliable. However, when the number of elements n required for the acceptable error does not suit the machine or the user (i.e. n > nT or n > n,) then the FE results become unreliable. Moreover, the minimum number of elements required for the acceptable error is an unknown. Therefore, it is always advisable to carry out the analysis for two or three mesh configurations having different number of elements in each case, but having similar discretiza- tion pattern. This takes into consideration the vari- ation of parameters within the structure, with better results being predicted, or the error in the analysis may be checked using the algorithm presented in the paper. To demonstrate the versatility of the approach, some numerical examples have been presented. 2. EXTRAPOLATION TECHNIQUE Let R, be the correct results and R be the result of a coarse mesh analysis obtained by using n number of elements. If e is the error in the result, then R,= R fe. (1) Three different algorithms tested in the present study for modeling of the error e are; e cc: (Algorithm I),