Assessing the Numerical Error in the Finite Element Solution


Assessing the Numerical Error in the Finite Element Solution

The most important step in any finite element analysis is clearly choosing the appropriate mathematical model to solve. This model must contain all the important ingredients regarding the geometry, loading, boundary conditions, and material data. The mathematical model must be reliable and effective (see K. J. Bathe, Chapter 1 of Finite Element Procedures, 1996) and is solved using a finite element program with a mesh defined by the analyst.

The finite element solution of the mathematical model will contain some numerical error which the analyst clearly wants to be sufficiently small. This will of course be the case if a very fine mesh is used with appropriate finite elements (as available in ADINA). But, in practice, when the mesh used is not very fine, the assessment of the numerical error between the "exact" solution and the finite element solution can be important. Since the exact solution is unknown, lower and upper bounds, proven to bracket closely the exact solution, and obtainable with an acceptable computational effort, would be very valuable. Unfortunately, these bounds are not available for general analysis, not yet even in academic research. Hence for general analyses, the currently available error assessments are based on estimates without these bounds.

Since this error estimation is an important issue, and presents an excellent research challenge, a very large research effort by engineers and mathematicians has focused on the development of error estimators. The current state of error estimation has been surveyed, for example, in ref. [1], where also the error estimators available in ADINA are mentioned. These are based on the work published already in 1985 (see refs. [2] and [3]) and are still believed to be the most effective available in general practical analyses.

The figures below show results obtained in the study of a cantilever structure with the left end fixed, in linear, nonlinear and FSI solutions. Of particular interest is the stress solution around the elliptical hole in the middle of the cantilever. In the linear analysis, the structure is simply subjected to the uniform pressure p. In the nonlinear analysis, this pressure is increased to 6p and causes plasticity and large deformations. Finally, in the fully coupled FSI solution, the steady-state force effects of a Navier-Stokes fluid flow around the structure (of magnitude about 2p) are considered in addition to the pressure of 6p.

In each case, a coarse mesh ADINA solution, the ADINA error estimation for the longitudinal (bending) stress using the coarse mesh and the "exact" error are shown. The exact errors have been obtained by comparing the coarse mesh solutions with extremely fine mesh solutions (that in practice would of course not be computed). The ADINA error estimation is seen in these cases to be conservative and not far from the exact error.

While we have shown the error estimation in a demonstrative 2D problem, the same procedures are of course available in ADINA for the stress and thermal analyses of 3D solids and shells, including contact conditions and FSI, and hence can be used in many practical analyses. However, a word of caution is necessary: As with all existing practical error estimation techniques, there is no proof that the error estimate is always accurate and a conservative prediction. Hence the error procedures are primarily useful to estimate whether the mesh is fine enough and if a refinement is necessary where such refinement should be concentrated.



Figure 1  Original coarse mesh of cantilever with boundary conditions





Figure 2(a)  Cantilever structure — linear analysis,
estimated error in region of interest shown



Figure 2(b)  Cantilever structure — linear analysis,
exact error in region of interest shown



Figure 3(a)  Cantilever structure — nonlinear analysis,
estimated error in region of interest shown



Figure 3(b)  Cantilever structure — nonlinear analysis,
exact error in region of interest shown



Figure 4(a)  Cantilever structure — FSI analysis,
estimated error in region of interest shown



Figure 4(b) Cantilever structure — FSI analysis,
exact error in region of interest shown


References

  1. T. Grätsch and K. J. Bathe, “A posteriori Error Estimation Techniques in Practical Finite Element Analysis”, Computers & Structures, Vol. 83, pp. 235-265, 2005.

  2. T. Sussman and K. J. Bathe, “Studies of Finite Element Procedures — On Mesh Selection”, Computers & Structures, Vol. 21, pp. 257-264, 1985.

  3. T. Sussman and K. J. Bathe, “Studies of Finite Element Procedures — Stress Band Plots and the Evaluation of Finite Element Meshes”, Engineering Computations, Vol. 3, pp. 178-191, 1986.

Keywords:
Error measure, error estimator, error indicator, meshing, mesh refinement, reliability