Visualization of errors of finite element solutions P. Beckers, H.G. Zhong, Ph. Andry Aerospace Department, University of Liege, B-4000 Liege, Belgium Abstract The aim of this paper is to show how to use graphical tools and new output of finite element software in order to improve results interpretation. This methodology is necessary because previous experiences have shown that all the aspects of the analysis have to be used if we want to get a correct decision based on finite element simulation. 1 Introduction The question that has to be addressed is how to exploit the benefits of a numerical simulation by finite element method? The most important tool is an efficient visualization of the results. However the difficulty is to extract the right information. To achieve this goal, the results can be classified into three main categories: direct results of the discretized field; secondary results like stress, in general they need some smoothing technique; and finally third level output resulting from error estimation. The first two groups are usually provided by standard finite element packages. The third one involves mainly some improvement of the stress solution followed by an error estimation. The main objective of this paper is to show how the utilization of visualization techniques in the tree levels of output interpretation will help the user to be more confident in the computation and to get better insight of the analyzed problem.
250 Visualization and Intelligent Design in Engineering 2. Presentation of a typical application 2.5 2 1.5 1 0.5 0 Figure 1: Axonometric view of exact Von Mises stress, mode I In order to illustrate the proposed methodology, we will consider as in Szabo & BabuSka [1], an edge cracked panel shown in figure 3. Along external sides, the surface traction corresponding to stresses of mode I (figure 1) and/or mode II fields, e.g. Barthelemy [2], are imposed. Plane strain conditions and Poisson's ratio of 0.3 are assumed. p Relative energy eiror - uniform degree 1 - uniform degree 2 - adaptative degree 1 - adaptative degree 2 - adaptative degree 1 (mesh 2) - adaptative degree 2 (mesh 2) - p convergence (mesh 2) Degrees of freedom Figure 2: Energy convergence curves, modes I & II
Visualization and Intelligent Design in Engineering 251 According to Szabo & BabuSka [1] the exact strain energy is known to be = 10.5412281, where a is half the size of the panel, t the thickness and E the Young's modulus. Specialized finite element formulations can give a very precise and low cost computation of the stress intensity factors. But the purpose of this study is to show how a full automatic procedure of mesh adaptation is able to converge to the exact solution despite the presence of a strong singularity that has to be captured by the procedure. This example is also very well designed to prove how a proper use of visualization techniques can give a clear idea of the characteristics of the numerical model. 3. Methodology of the analysis In order to show the behaviour of several adaptive techniques, convergence studies are performed either on mode I either on modes I & II loading cases. Uniform refinements are compared to adaptive ones based on a stress estimator very similar to the S. P. R. (Superconvergent Patch Figure 3: Mode I, exact section along A... E Recovery) technique of Zienkiewicz & Zhu [3]. The starting mesh is a very coarse uniform one composed of four elements or a predefined mesh initially designed by Szabo & BabuSka [1], (figure 4) for p- refinement method and very well suited for the present problem. In the analysis, emphasis will be put on the graphical techniques able to show the behaviour of the solution. Another important point is to comment what type of variable has to be selected for graphical displays. Figure 4: Initial mesh 2, ref. [1]
252 Visualization and Intelligent Design in Engineering Geometric scale Figure 5: Mode I & II deformation, adaptive mesh 4. Primary variables output In the first level of output, the opportunity is given to analyze the general behaviour of the solution and to have a general insight of its characteristics. The most general and compact information about a finite element simulation is the global energy. Although in standard applications, it is not possible to perform several analyses, here for an academic example, the presentation of Figure 6: typical adaptive mesh convergence curves allows to understand the main trends when using different element models, or different types of meshes, figure 2. As noted earlier by many authors, uniformly refined meshes converge according to the singularity factor of the solution (here 0.5). This convergence rate is independent of the degree of the approximation. Adaptive meshes are converging with a velocity related to the degree of the elements, but when a very good initial mesh is chosen, this convergence is still faster because the mesh is close to the optimal one everywhere in the domain, see mesh 2, with intermediate adapted mesh in figure 6. It seems also that in this application, the solution of h-refinement is
Visualization and Intelligent Design in Engineering 253 as good as the results obtained in the p-method used by Szabo & BabuSka [1], at least for degree 2 approximation. This remark is confirming the excellent behaviour of h-p methods and the high quality of second degree elements in linear elastic problems. 5. Second level output Figure 7: Mode I, F.E. cut, along A...E, A Analyzing and displaying the stress output is the main difficulty in the finite element post-processing. The well known intrinsic difficulty for representing derivatives needs an enhanced method able to improve their quality. Techniques such as stress averaging used for a long time and now S.P.R. methods give an opportunity to obtain the same quality of results as for primary displacements, if they are Figure 8: Modes I & II, section, uniform and used with care. adaptive meshes, von Mises stress Moreover they also constitute a very good basis to perform error estimation. Very often contouring methods are used to show the map of stress variations, figure 9. However they don't provide a very accurate basis to measure the quality or to identify the defaults of the solution. Sectioning methods are more able to give a precise visualization of stress variation, see figure 3, for the exact solution andfigures7 & 8, for different cutting lines. Figure 7 shows that the finite element solution is giving erratic results at point A, this is reflecting the presence of the singularity. 6. Third level output Since the development of error estimation techniques it is possible to show everywhere in the domain the level of error what is equivalent to show the quality of the approximate solution. Applying some technique for smoothing the stressfield, e.g. Beckers, Zhong & Maunder [4], this error estimation
254 Visualization and Intelligent Design in Engineering Figure 9: von Mises stress, adaptive mesh, mode I & II is obtained through the computation of the error energy. Local density of error is displayed in figure 11 using a logarithmic scale. This output gives a good idea of the quality of the results but something more has to be done to know how to modify the mesh for obtaining a desired level of accuracy: this is the study of local convergence rates. 7. Computation of the required mesh density This is the last step that has to be performed for extracting the full information of the present model. Wherever important gap is found in the inter element stress discontinuities or when important stress gradients are localized, an automatic procedure calculate the convergence behaviour of the solution, e.g. Beckers & Zhong [5], this analysis has to tell the user what level of mesh refinement should be necessary. A global mesh density map is obtained that describes the theoretical optimal mesh necessary to reach a given level of accuracy. This step has to be followed by a remeshing or by the refinement of the actual mesh and the next step of the iterative procedure can start again. When the solution is assumed to be sufficient, the user is able to understand all the consequences of the analysis and to make all the decisions concerning the problem. A zoom of the domain, figure 10, help to observe details and to control the distortion of the elements. It shows a
Visualization and Intelligent Design in Engineering 255 Geometric scale.001» Figure 10: Shear stress, zoom of the bottom of the crack detail of the crack, with shear stress contours superposed to the mesh. Note that the size variation of the elements is higher than 1000. If all the steps of the procedure are analyzed in details and if all the checks are performed concerning the range of validity of the solution, the user can be very confident of the knowledge acquired about the numerical simulation. This is a necessary condition that has to be fulfilled before transmitting the conclusions of the study to another department. 8. Conclusion In the design of finite element models of complex structures all the available tools of graphics, error analysis and results displays have to be used to help the user to make the right decision. If part of these results is not exploited there is a big danger to miss some important characteristic of the solution and to misunderstand the behaviour of the structure. This is a good argument to convince all the analysts of exploring the full range of tools and criticizing all the components of the solution. In many modern finite element packages, the now available tools are sufficient, in the authors opinion, to drive very efficiently and safely most linear applications.
256 Visualization and Intelligent Design in Engineering Estimated i: Exact Figure 11: Exact and estimated error densities, uniform mesh, mode I References 1. Szabo B.A. & BabuSka I. Finite element analysis, John Wiley & Sons, New York, 1991. 2. Barthelemy B. Notions pratiques de Mtcanique de la rupture, editions EyroUes, Paris, 1980. 3. Zienkiewicz O.C. & Zhu J.Z., The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique; Part 2: Error estimates and adaptivity, Int. J. Num. Meth. Engng., 1992, 33, 1331-1382. 4. Beckers P., Zhong H.G. & Maunder E.A.W. Numerical comparison of several a posteriori error estimators for 2D stress analysis, Euro. J. Finite Elements, 1993, 2-2, 155-178. 5. Beckers P. & Zhong H.G. Mesh adaptation for 2D stress analysis (ed. B.H.V. Topping & M. Papadrakakis), pp. 2.47 to 2.60, Proceedings of the Second International Conference on Computational Structures Technology, Athens, Greece, 1994, Civil-comp Press, 1994.