FEA_MNE772 Finite Element Application for Three-Dimensional Stress - PowerPoint PPT Presentation

FEA_MNE772 Finite Element Application for Three-Dimensional Stress Analysis An Approach to Scientific Visualization Techniques of Numerical Results Ivan Assing da Silva Klaus de Geus Sergio Scheer Topics in Scientific Visualization – MNE772 PPGMNE – UFPR 2017

Summary • Overview • Workflow • Numerical Results • ViSC Techniques • Scalar Color Mapping • Scalar Isosurface • Scalar Cutting Plane • Vector Glyphs • Tensor Ellipsoid Glyphs • Tensor Superquadric Glyphs • Tensor Hyperstreamlines • Example Models and Solver Times • Aspects of Implementation • Bibliography

Overview ViSC timeline 3D views solver techniques controls selection of scalar model attributes visualization configurations execution log (1) All images and graphics in this paper were produced by the author.

Workflow POST PROCESSING ViSC TECHNIQUES • Scalar Color Mapping INPUT MODELS • Scalar Isosurface Solid3D • Scalar Cutting Plane • CDB Ansys Mesh File • Vector Glyphs • FSXL XML Native File • Tensor Ellipsoid Glyphs Truss3D (1) • Tensor Superquadric Glyphs • DXF Drawing Exchange Format • Tensor Hyperstreamlines • FTXL XML Native File PROCESSING NUMERICAL RESULTS PRE PROCESSING • Stiffness Matrix • Displacement Vector Field • File Loading • Loading Vector • Stress Tensor • Model Validation • Boundary conditions setup • von Mises Stress • Model Rendering • Solver Linear System • Principal Stresses (1) The application solves Truss 3D models, but the visualization techniques were not implemented for this model type in this time.

Numerical Results • Displacement Field (vector) 𝒗 = (𝑣 𝑦 , 𝑣 𝑧 , 𝑣 𝑨 ) • Stress Tensor (tensor) σ 𝑦𝑦 σ 𝑦𝑧 σ 𝑦𝑨 σ 𝑦𝑧 σ 𝑧𝑧 σ 𝑧𝑨 𝝉 = σ 𝑦𝑨 σ 𝑧𝑨 σ 𝑨𝑨 • von Mises Stress (scalar) 2 + σ 𝑧𝑨 2 + σ 𝑦𝑨 2 ) (σ 𝑦𝑦 − σ 𝑧𝑧 ) 2 +(σ 𝑧𝑧 − σ 𝑨𝑨 ) 2 +(σ 𝑨𝑨 − σ 𝑦𝑦 ) 2 +6(σ 𝑦𝑧 𝜏 𝑤 = 2 • Absolute Displacement (scalar) 𝑣 = 𝒗

Numerical Results • Principal Stress (tensor) Eigenanalysis for stress tensor transformation (diagonalization of tensor). Eigenvalues are the values of principal stresses and the eigenvectors are vectors of the normal basis of principal stresses space. σ 𝑦𝑦 σ 𝑦𝑧 σ 𝑦𝑨 σ 1 0 0 σ 𝑦𝑧 σ 𝑧𝑧 σ 𝑧𝑨 𝑉 𝑈 0 σ 2 0 = 𝑉 σ 𝑦𝑨 σ 𝑧𝑨 σ 𝑨𝑨 0 0 σ 3 SCALARS DESCRIPTION 𝑣 𝑦 - displacement on x axis 𝜏 𝑧𝑧 - normal stress on y axis 𝜏 𝑤 - von Mises stress 𝑣 𝑧 - displacement on y axis 𝜏 1 - major principal stress 𝜏 𝑨𝑨 - normal stress on z axis 𝑣 𝑧 - displacement on z axis 𝜏 𝑦𝑧 - shear stress on xy plane 𝜏 2 - medium principal stress 𝑣 - absolute displacement 𝜏 𝑧𝑨 - shear stress on yz plane 𝜏 3 - minor principal stress 𝜏 𝑦𝑦 - normal stress on x axis 𝜏 𝑦𝑨 - shear stress on xz plane

ViSC Techniques – brief description • Scalar Color Mapping Associates a color with the scalar value. • Scalar Isosurface Shows the surfaces where the scalar has equal values. Along the surface the scalar values are constants. • Scalar Cutting Plane Shows the scalars values by color mapping along the defined intersection plane. • Vector Glyphs Associates a object (glyph) at every vector’s point, oriented and scaled according vector information. • Tensor Ellipsoid Glyphs Associates a ellipsoid at every tensor’s point. The ellipsoid axes are oriented according the principal stress eigenvectors a nd scaled according the corresponding principal stress values. • Tensor Superquadric Glyphs Associates a superquadric form at every tensor’s point. The superquadric glyph is oriented and scaled similarly to ellipsoid glyph, and the superquadric’s shape is define according the principal stresses isotropy. • Tensor Hyperstreamlines Defines a streamline following a principal stress direction and assumes an ellipse cross section form along the streamline. The ellipse cross section is scaled according the corresponding principal stress values.

Scalar Color Mapping

Scalar Isosurface

Scalar Cutting Plane

Vector Glyphs

Tensor Ellipsoid Glyphs

Tensor Superquadric Glyphs

Tensor Hyperstreamlines

Example Models and Solver Time (1)(2) 10.000 50 9.000 45 8.000 40 7.000 35 6.000 30 Number of Nodes Solver Time (s) 5.000 25 4.000 20 3.000 15 2.000 10 1.000 5 0 0 Models Number of Nodes Solver Time (s) (1) Solver time with Iterative Solver on GPU processing. (2) Reference hardware: CPU Intel Corei7 1743 MHz + GPU Nvidia GeForce 460M with 192 Cuda cores.

Model: cube_crack.fsxl [4126; 19829; 5,641]  [Number of Nodes; Number of Elements; Solver Time (s)]

Model: beam.fsxl [1653; 4788; 4,382]

Model: gear_v3.fsxl [5177; 18268; 12,525]

Model: cube_v2.fsxl [6618; 34025; 12,706]

Model: cube_v3.fsxl [1147; 5281; 0,897]

Model: cube_v4.fsxl [4812; 24279; 8,664]

Model: connection_v2.fsxl [1815; 6681; 2,744]

Model: spherical tank_v2.fsxl [4141; 13002; 7,975]

Model: plate.fsxl [5667; 26395; 11,604]

Model: cylinder_v3.fsxl [7080; 35780; 16,109]

Aspects of Implementation • Core programming language: C++ • Compilers: GCC 6.4 on Linux and Visual Studio 2015 on MS Windows • Main Libraries: • VTK 8.0.0 [https://www.vtk.org/] • Qt 5.9.1 [https://www.qt.io/] • Atlas Blas 3.10.3 [http://math-atlas.sourceforge.net/] • Lapack 3.7.0 [http://www.netlib.org/lapack/] • Magma 2.2.0 [http://icl.cs.utk.edu/magma/] • Source code available at https://github.com/IvanAssing/FEA_MNE772 • Compiled program for MS Windows available at https://drive.google.com/open?id=1c1ZvDc2-h56PWIPmBy8pd_pomQQrthwt • Bug reports, comments or suggestions: please send email to ivanassing@gmail.com

Bibliography • Kindlmann, G., “ Superquadric Tensor Glyphs ” . IEEE TCVG Symposium on Visualization, 2004. • Kratz, A., Meyer, B., Hotz, I., “ A Visual Approach to Analysis of Stress Tensor Fields ” . Zuse Institute Berlin, 2010. • Lai, M., Rubin, D., Krempl, E., “ Introduction to Continuum Mechanics ” . 4th. ed. Elsevier Inc. 2010. • Laidlaw, D.H., Vilanova, A., et.al. “ New Developments in the Visualization and Processing of Tensor Fields ” . Springer, 2012. • Munzner, T., “ Visualization Analysis & Design ” . CRC Press, 2014. • Schroeder, W., Martin, K. Lorensen, B., “ The Visualization Toolkit: an object-oriented approach to 3D graphics ” . 4th. ed. Kitware, Inc, 2006. • Schroeder, W., et.al., “ The VTK User’s Guide ” . 11th. ed. Kitware, Inc. 2010 • Telea, A., “ Data Visualization: Principles and Practice ” . 2nd. ed. CRC Press, 2015. • Zienkiewicz, O.C., Taylor, R.L., “ The Finite Element Method, Volume 1: The Basis ”, 5th. ed., Butterworth Heinemann, 2000.