Application to complex solids of adaptive isogeometric analysis using T-splines

http://www.dca.iusiani.ulpgc.es/proyecto2012-2014 Application to Complex Solids of Adaptive Isogeometric Analysis using T-splines MINECO y FEDER Project: CGL2011-29396-C03-00 PEMEX & CONACYT-SENER Project, Fondo Sectorial, contract: 163723 R. Montenegro(1)* , J.M. Cascón(2) , E. Rodríguez(1) , J.M. Escobar (1) M. Brovka(1) , J.I. López(1) , J. Ramírez(1) (1) University Institute SIANI, University of Las Palmas de Gran Canaria, Spain (2) Department of Economics and History of Economics, University of Salamanca, Spain 10th WCCM , 8–13 July 2012, São Paulo, Brazil Motivation: Solid Modeling with Trivariate T-splines ● INPUT: Surface Triangulation ● OUTPUT: Trivariate T-spline ● 3-D T-Mesh of the Meccano The Meccano Method for Isogeometric Solid Modeling Motivation: Solid Modeling with Trivariate T-splines ● INPUT: Solid Surface ● OUTPUT: Trivariate T-spline The Meccano Method for Isogeometric Solid Modeling ● 3-D T-Mesh of the Meccano 16th IMR (2007) Motivation: Simultaneous Mesh Generation and Volume Parameterization 18th IMR (2009) ● INPUT: Surface Triangulation ● OUTPUT: Tetrahedral Mesh ● Meccano Tetrahedral Mesh The Meccano Method for 3-D Mesh Generation Algorithm Steps: Surface information as input data; explicit in this case Isogeometric Modeling of a Genus-one Solid Algorithm Steps: The meccano approximation Isogeometric Modeling of a Genus-one Solid Algorithm Steps: Coarse tetrahedral mesh (tet-subdivision of the polycube) Isogeometric Modeling of a Genus-one Solid Algorithm Steps: Local refined tetrahedral mesh Isogeometric Modeling of a Genus-one Solid  Initial cube and its subdivision after three consecutive tetrahedron bisection Local Refinement: Kossaczky’s Algorithm (JCAM 1994) Refinement of a cube http://www.alberta-fem.de/, ALBERTA code From a the i-th solid surface triangulation patch to the i-th meccano face Surface Parameterization of M.S. Floater (CAGD 1997) http://www.sintef.no/math software, GoTools from SINTEF ICT Free node Local optimization New position for the free node v(x,y,z) v(x,y,z) Objective: Improve the quality of the local mesh N(v) by minimising an objective function Local mesh N(v) Tetrahedral Mesh Optimization SUS Code: Freely-available in http://www.dca.iusiani.ulpgc.es/proyecto2008-2011 Simultaneous Untangling and Smoothing (CMAME 2003) Application to the Armadillo: A surface triangulation as input datum Meccano Method for a Complex Genus-Zero Solid http://graphics.stanford.edu/data/3Dscanrep/, Stanford Computer Graphics Laboratory T Tt Physical Element T Target Element Tt Optimization (to get less distortion in the parameterization) T-mesh and anchor tα Bivariate Cubic T-spline Basis Function support of the T-spline 1 ξ 2 ξ × = ( ) ( ) ( ) 221121, ξξξξ ααα NNB = ( ) 11 ξ α N ( ) 22 ξ α N Knots associated to anchor tα : { } 1 6 1 5 1 4 1 2 1 1 1,,,, ξξξξξ α =Ξ { } 2 6 2 5 2 4 2 3 2 2 2 ,,,, ξξξξξ α =Ξ Isogeometric Modeling and Analysis Example of T-mesh and T-splines in 2-D Cube tetrahedral mesh Octree division of the cube Cube T-mesh Each cube of the octree does not contain any node of the Kossaczky mesh in its inner Obtained by using the meccano method with Kossacsky refinement Automatic Adaptation of Inner and Boundary Discretizations Construction of the T-mesh for the Bunny Interpolation points (the anchors) are mapped to the solid by using the volumetric parameterization that was obtained by the meccano method Mapping maintaining baricentric coordinates Mapping of the interpolation points Construction of the T-mesh for the Bunny ( ) ( ) ∑ ∈ = A R α αα ξξξξξξ 321321 ,,,, PS ( ) ( ) ( ) ∑ ∈ = A B B R β β α α ξξξ ξξξ ξξξ 321 321 321 ,, ,, ,, ( ) ( ) ( ) ( ) 332211321 ,, ξξξξξξ αααα NNNB = With: Blending functions Trivariate basis splines volumetric parameterization ( ) ( ) AR A ∈∀= ∑ ∈ β α βααβ tPtS Control points are calculated by solving the sparse linear system α P ( ) β tS β t The Spline Interpolation Physical space location Parametric space location Calculation of Control Points by Fulfilling the Interpolation Conditions T-mesh of a bone T-mesh T-spline T-mesh and T-spline of the Bone Automatic Adaptation of Inner and Boundary Discretizations Cross-sections of the Bone T-spline Automatic Adaptation of Inner and Boundary Discretizations Adaptive Isogeometric Refinement Igea: T-spline of Numerical Solution 2nd local refinement 6021 cells, 9807 DOF Adaptive Isogeometric Refinement Igea: T-spline of Numerical Solution 5th local refinement 6756 cells, 10838 DOF Adaptive Isogeometric Refinement Igea: Numerical solution in a cross section of the parametric space Initial T-mesh 5692 cells, 9304 DOF 5th local refinement 6756 cells, 10838 DOF 2nd local refinement 6021 cells, 9807 DOF Adaptive Isogeometric Refinement uh: Initial T-mesh (green) uh: 5th local refinement (blue) u: exact solution (red) Igea: Exact and numerical solution in a cross section of the parametric space Adaptive Isogeometric Refinement Igea: Rate of convergence Error estimator (slope: -30.7) Exact error in L2 norm (slope: -28.2) Exact error in H1 seminorm (slope: -17.2) Parametric space Physical space Scaled Jacobian Final Comments and Future Works Valid and Invalid Configurations in IGA Parametric space Physical space Scaled Jacobian Valid and Invalid Configurations in IGA Remarks: • In 2-D: Problems could appear in the corners • In 3-D: Problems could appear in the corners and edges Final Comments and Future Works Parametric space (Meccano T-mesh) Physical space (Tangled T-mesh) The Meccano Method for T-mesh: T-spline Optimization Physical space (Optimized T-mesh) Final Comments and Future Works Automatic Construction of the Meccano Final Comments and Future Works Automatic Construction of the Meccano Final Comments and Future Works