The proceedings of the conferences have been published on CD since the initiative taken by Prof. G. M. Carlomagno at the Sorrento meeting allowing authors to include in their papers color, video clips and animations.The aim of this work is the study and the implementation of appropriate visualization techniques for high-order discontinuous finite element data in two and three-dimensions. In particular, we are dealing with field discontinuity and deformed cells. Such data are produced for example by chemical simulations, by fluid dynamics simulations, or, in general, anywhere high accuracy on boundary domain description is required

Gobbetti, Enrico

Leone, Andrea O.

Marzano, Paola

Pedinotti, Sergio

Scateni, Riccardo

P-arch

8th International Symposium on Flow Visualisation (1998)DISCONTINUOUS FINITE ELEMENTVISUALIZATIONA. O. Leone P. Marzano E. Gobbetti R. Scateni S. PedinottiKeywords: visualization, high-order finite elements, discontinuous fields, mesh refinementAbstractThe aim of this work is the study and the im-plementation of appropriate visualization tech-niques for high-order discontinuous finite ele-ment data in two and three-dimensions. In par-ticular, we are dealing with field discontinuityand deformed cells. Such data are producedfor example by chemical simulations, by fluiddynamics simulations, or, in general, anywherehigh accuracy on boundary domain descriptionis required.1 MotivationsFinite element methods are well suited to repre-sent physical processes in simulation programs.Several applications in the field of fluid dynam-ics need to represent discontinuous fields definedon high-order finite elements cells[1]. Such dataare produced for example as an output fromchemical simulation processes.The combination of discontinuous finite el-ement simulation methods, topologically un-structured grids, and high-order polynomial ap-proximations of fields and geometries allowsprograms to handle a variety of situations inwhich standard approaches usually fail[2]. Inparticular, structured grids suffer from heavylimitations when dealing with complicated ge-Received January 1, 1998. Accepted March 10, 1998.Contact author: A. O. Leone1, P. Marzano, E. Gobbetti, R.Scateni, S. Pedinotti21CRS4 - Centro di Ricerca, Sviluppo e Studi Superi-ori in Sardegna, Via N. Sauro 10, 09123 Cagliari, Italy.(http://www.crs4.it/)2ENEL - Centro Ricerca Termica, Pisa, Italyometry or when trying to adapt to local featuresof the solution. Furthermore, discontinuous fi-nite element methods ensure time accuracy ofthe solution to be obtained by using high orderpolynomial approximations within each element(cell), thus being particularly well suited to un-structured grids. Finally, when high accuracyin the computational domain boundary descrip-tion is required by a finite element simulationprogram, it is often necessary to deal with ge-ometrically deformed cells, as the higher-ordershape of these cells may give a better approxi-mation of the boundaries than a linear one[3].Common visualization approaches do not of-fer the possibility to handle discontinuities andthis research area seems to be completely un-investigated when discontinuities are combinedwith unstructured topology.The aim of our research is to supply gen-eral purpose visualization tools and techniquesfor representing meshes made up of topolog-ically different cells (a combination of trian-gles, quadrilaterals, and other generic convexpolygons in the bidimensional case; a com-bination of tetrahedra together with hexahe-dra, and generic convex polyhedra in the three-dimensional case) on which discontinuous scalarand vector fields are defined. Also, high-orderpolynomial parametric description of the geom-etry must be represented graphically along withthe field values defined over it.1–1A. O. Leone, P. Marzano, E. Gobbetti, R. Scateni, S. Pedinotti2 Research Activity2.1 High order polynomial approxima-tionStandard visualization tools and techniques areoptimized for the case when each variable de-fined over a cell has a linear behavior. In par-ticular, current graphics hardware only imple-ments linear interpolation of colors (Gouraudshading). In order to obtain interactive perfor-mance, one of the main problems that has tobe faced is to linearly approximate polynomialfunctions for rendering purposes.In our approach, such approximation hasbeen performed using the Longest Side Mid-point Insertion mesh refinement algorithm[4].At each step of the iterative method, a cellto be refined, with respect to a defined errorestimation and a given threshold, is split alongits longest side in two cell. The same splittingis recursively applied to any cell adjacent to thesame side (see fig. 1).Fig. 1. The Longest Side Midpoint Insertionalgorithm. The top cell starts the recursionThis method has been proven to producea conforming and good quality mesh[5] and toconverge in a finite number of computationalstep to a linear approximation of the functiondefined on the original mesh, given a tolerancevalue, both in 2D and 3D.The use of a grid refinement approach makesit possible to use, on the refined set of cells,standard visualization methods for linear func-tions. As the visualization algorithms have todeal only with linear functions, it is possible toobtain interactive performance.Figure 2 shows the isolines representationof a field defined over a quadratic triangularelement where the refinement alghoritm hasbeen applied.Fig. 2. A quadratic field represented by isolines2.2 Discontinuity RepresentationAn effective representation of discontinuities isa major problem to be solved for discontinuousfields. Discontinuity is a fundamental informa-tion in the simulation process, as discrepancy infunction values at cell interfaces provides infor-mation on solution convergence.Fig. 3. Visualization of discontinuitiesTo visualize discontinuity we decided to han-dle each cell as a separate entity by represent-ing function values at a vertex independently oneach cell sharing that vertex (see fig. 3). Thisway to describe the actual solution has thenbeen combined with the representation of theaverage solution for each vertex obtained by thevalues of the function in the cells incident on it.The possibility to describe both the actual and1–2Discontinuous Finite Element Visualizationthe average solution showed to be a good wayfor emphasizing discontinuities. Both solutionsare represented by classical visualization tech-niques (i.e., by isocontours and color mapping),as described in the following section.Another approach for discontinuity repre-sentation consists in mapping the variance ofthe discontinuous solution on the edges of theunderlying grid. In this way discontinuities aremade evident within the mesh and their visu-alization can be combined with isocontours orcolor mapping of either the actual or the aver-age solution. Figure 4 shows one of these pos-sible combinations applied to the same datasetdepicted in figure 3.Fig. 4. Visualization of derived average andvariance fields2.3 Deformed GeometryWhile visualizing this kind of data, we had theproblem to represent deformed cells with com-puter graphics primitive that are linear. Oursolution is to produce a good approximation ofsuch cells with a set of linear cells. We ap-plied the Longest Side Midpoint Insertion al-gorithm even in this case by considering thephysical coordinates of the mapped cell controlpoints as fields defined over the cell. This way,the same finite element interpolation formulascan be used to compute the geometric mappingof the referrer cell into the physical space (seefig. 5).Figure 6 shows how quadratic boundaries ofa bidimensional dataset are well represented byreferrer space physical spaceFig. 5. Deformation of a P2 cell. All sides aregiven by 3 points coordinates in the physical spaceusing a small amount of additional linear cellsobtained by the refinement process.Fig. 6. Visualization of quadratic boundaries.Thick lines represent sides of added elements2.4 Error estimationOne of the critical points of the method pre-sented resides in estimating the error introducedby considering an element as if it was linear.Such error drives the refinement of the mesh andit is used to stop the splitting process when theaccuracy reaches a value below a given thresh-old.Because the refinement is performed bothfor geometric that for field representation, weused two different error measures.For each cell, the geometric error is com-puted as the maximum distance of the controlpoints in the physical space from the corre-sponding point of the linearly mapped cell (seefig. 7).1–3A. O. Leone, P. Marzano, E. Gobbetti, R. Scateni, S. PedinottiFig. 7. Error estimation: distance C-C’ is thegeometric error for the deformed cellThe global geometric error for a mesh isdefined by the maximum error computed oneach cell. If this is greater than the giventhreshold, then the cell with the biggest error issplitted according to the refinement alghorithm.The field error, on the other hand, is com-puted as the maximum absolute difference be-tween the field values on the control points andthe corresponding linearly interpolated values.In the case of the second order element showedin figure 8, the error is computed by the formulaerr = max |F2 −F1+F32||F4 − F3+F52 ||F6 − F5+F12 | (1)where Fi is the field value on the i-th controlpoint.Here again, the global field error for a meshis defined by the maximum error computed oneach cell. If this is greater than the giventhreshold, then the cell with the biggest error issplitted according to the refinement alghorithm.3 ImplementationThe first practical result of our research was theimplementation of a visualization system for therepresentation of discontinuous fields defined onhigh order finite element grids. The system,called MIGAVIS, combines different techniquesfor representing discontinuities on unstructuredFig. 8. Error estimation: field error is givenby formula (1)topologies within a simple parametric user in-terface.MIGAVIS has been developed by augment-ing the object-oriented Visualization Toolkit bySchroeder, Martin and Lorensen[6] with newclasses for handling high-order finite elementsand discontinuity information. The applicationhas been completely developed in C++ and thedevelopment platform was an IRIX 5.3 SiliconGraphics workstation. A parametric user inter-face implemented in OSF/Motif allows an easyinteraction with data. The system offers thepossibility of visualizing:• the wire-frame grid on which the solutionis defined;• the actual discontinuous solution;• the average solution.Both the actual and the average solutioncan be described by either a scalar represen-tation (through isocontours or color mapping)or a vector representation (through glyphs orhedgehogs). The user can interactively selectthe scalar field (i.e., a specific solution corre-sponding to a variable in the simulation process)for scalar visualization and/or the scalar fieldsfor vectorial visualization. The vector visualiza-tion is obtained combining different scalar solu-tions.1–4Discontinuous Finite Element VisualizationThe variance of the solution can be mappedon grid edges whenever desired, thus empha-sizing discrepant values at cell interfaces just incorrespondence of their location within the grid.All the visualization methods can be combinedto produce a single image.The visualization system is integrated withan interface that gives the possibility to inter-actively modify the parameters of the visualiza-tion process:• parameters for mesh refinement process;• colormaps for variance and for all solutionand grid representations;• value ranges for variance and for all solu-tion representations;• number of contours;• glyphs and hedgehogs parameters (height,scale, etc. );• parameters for linear and non-linear col-ormaps;• scale factor for Cartesian axes;• offset factor for visualized objects withrespect to cartesian axes.4 ResultsThe MIGAVIS visualization system is currentlybeing used at ENEL Research Department inPisa (Polo Termico) as a suitable post-processorfor MIGALE, a discontinuous finite elementssolver for compressible and viscous flows. Themain application of MIGALE is the study ofturbomachinery flows.The main reason to use MIGAVIS as vi-sualizer for a specific CFD solver is given bythe particularity of the discontinuous finite ele-ments method. As a matter of fact, the visual-ization of solution discontinuities is very usefulin order to highlight computational lack eitherin convergence or in precision. The visualiza-tion of discontinuities is especially important in3D simulation, that represents the most chal-lenging task.As an example, we can consider two 2Dcases recently studied. The first one is theflow calculation for a first stage steam turbinenozzle. In this case the computation was aimedat evaluating the behaviour of a new profiledesigned to reduce the blades erosion due tooxide particles. Figures 9 and 10 show theMach isolines for the old and for the new slantedprofile.Fig. 9. 2811 vertex and 5304 cellsFig. 10. 3430 vertex and 6486 cellsThe second case is an optical probe for den-sity measurements in the wet steam region ofturbines. The probe consists of a steel cylin-der having a lenghtwise cavity, the laser beamemitted from one side of the cavity is collectedby a detector placed on the opposite side. Fig-ures 11 and 12 shows the fully developed com-putations of different precisions. Mach contours1–5A. O. Leone, P. Marzano, E. Gobbetti, R. Scateni, S. Pedinottifor isoparametric linear elements have evidentdiscontinuities in the wake and near the solidboundary. Quadratic elements coupled withcubic boundaries representation eliminate suchdiscontinuities.Fig. 11. 2534 vertex and 4919 linear elementsFig. 12. 2534 vertex and 4919 quadraticelementsFigure 13 shows the same dataset refined upto a 50% added cells to obtain a more accuraterepresentation of the field under analysis.References[1] Bassi F, Mariotti G, Pedinotti S, and Rebay S.Recent developments on the solution of the fullnavier-stokes equations using a discontinuousfinite element method. Proc IX Int.l Conf.on Finite Elements in Fluids, New Trends andApplications, Venice, Italy, pp 529–539, Oct15–21 1995.Fig. 13. The same dataset of fig 12 refined upto 7337 cells[2] Bassi F, Rebay S, Mariotti G, and PedinottiS. A high order accurate discontinuous finiteelement method for inviscid and viscous turbo-machinery flows. Proc 2nd European Confer-ence on Turbomachinery - Fluid Dynamics andThermodynamics, Antwerp, Belgium, Mar 5–71997.[3] Kardenstuncer K and Norrie D. Finite ElementHandbook. McGraw-Hill, 1995.[4] Rivara M and Inostroza P. A discussion onmixed (longest-side midpoint insertion) delau-nay techniques for the triangulation refine-ment problem. Proc 4th International Mesh-ing Roundtable, Sandia Nat.l Labs, pp 335–346,October 1995.[5] Rivara M and Levin C. A 3-d refinement algo-rithm suitable for adaptive and multi-grid tech-niques. Communications in Applied NumericalMethod, Vol. 8, pp 281–290, 1992.[6] Schroeder W, Martin K, and Lorensen B. TheVisualization Toolkit - An Object-Oriented Ap-proach to 3D Graphics. Prentice-Hall, 1996.1–6

Discontinuous finite element visualization

http://www.p-arch.it/bitstream/handle/11050/684/fv98.pdf?sequence=1&isAllowed=y

Discontinuous finite element visualization

Abstract

Similar works

Full text

Available Versions

P-arch