1 Reply Latest reply on Aug 21, 2009 4:14 PM by Jose Eduardo Fernandes de Camargo

    ENERGY NORM ERROR?

      what is ENERGY NORM ERROR and when we have to use it, is there any limit for it thanks.
        • Re: ENERGY NORM ERROR?
          Jose Eduardo Fernandes de Camargo

          ENERGY NORM ERROR, is a norm that helps to find out how precisely is your analysis:

           

          Below you'd be able to find out how SolidWorks Simulation deals with it:

          Explanation of the  ERR stress error estimation in COSMOS Products

           

          Discussion

           

          Introduction

          The ERR option is currently available  for the TRIANG, TETRA4, TETRA4R, TETRA10, SHELL3 and SHELL4 elements only. The  estimator is based on energy error norm and allows good evaluation of local  errors.

           

          Actually, the stress error estimation is based on the principle of the  continuity of stress. The resulting stress distribution of a finite element  analysis is generally discontinuous. The nodal stresses of each element are  averaged to smooth the discontinuity in the element stresses. The stress error  in each element is defined as the difference between the element stress and the  average of the nodal stresses corrected using the form functions. This error is  used to calculate the energy norm error for each element. COSMOS Products allow  the user to plot a contour of the percentage of the elemental energy norm error  relative to the average elemental energy.

           

          Zienkiewicz's method - simple error estimation

           

          In finite element analysis of elastic problems, linear or higher order  polynomial shape functions are used to model the behavior of discretized  structures. As a result, a discontinuous stress field is generated across  inter-element boundaries. A linear finite element approximation û of the displacement field u and the corresponding stress field  are illustrated in Fig. 1, for a one-dimensional  problem.

           

           

          To obtain a better stress field, a nodal averaging process is used to obtain  the averaged nodal stresses and the displacement shape functions N are used to interpolate the new stress  field

          It is obvious that is a better approximation than . Therefore, a pointwise stress error   can be estimated by

          The error estimate of the ith element  can be evaluated in the energy norm as

          Where D is the constitutive matrix  and is the vector of the stress component error. The global  error estimate can be computed by summing over the entire domain, using

          where NE is the total number of  elements.

           

          The elemental percentage error if the ERR stress error plotted by COSMOS  Products, noted , is

          where is twice the strain energy obtained from the finite element analysis, i.e.

          where is the vector of strains.

          In COSMOSWorks and COSMOSDesignSTAR, the ERR value distribution can be  obtained as a stress plot with result type set to Element values and component set to ERR: Energy Norm Error (in COSMOSWorks 2004 and  older, it was called ERR: Stress Error).  In COSMOSM, you can use the Results, Plot, Stress menu or ACTSTR command. Note  that ERR is unitless and represents a percentage.

          Note:

          • The plotted value is only an indirect representation of the error  in stress, as it is actually an estimation of the error in energy norm in each  element. Nevertheless, it can be considered to represent the relative  distribution of stress errors in homogeneous meshes.

          References

          • A Simple Error Estimator and Adaptive  Procedure for Practical Engineering Analysis by O.C. Zienkiewicz and J.  Z. Zhu in International Journal for Numerical Methods in Engineering, vol. 24,  337-357 (1987)

          • An error analysis and mesh adaptation method  for shape design of structural components, by K.-H. Chang and . K. Choi  (1991) in Computers 1 Structures Vol. 44. No. 6. pp. 1275-1289, 1992 Pergamon  Press Ltd.