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:

Article ID: x174 - Explanation of the ERR stress error estimation

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.

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

DiscussionIntroduction

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.