Polygonal Finite Element Methods N. Sukumar University of - PowerPoint PPT Presentation

University of California, Davis Polygonal Finite Element Methods N. Sukumar University of California, Davis Workshop on Discretization Methods for Polygonal and Polyhedral Meshes Milano, September 18, 2012

Collaborators and Acknowledgements • Collaborators  Alireza Tabarraei (UNC, Charlotte)  Seyed Mousavi (University of Texas, Austin)  Kai Hormann (University of Lugano) • Research support of the NSF is acknowledged

Outline  Motivation: Why Polygons in Computations?  Strong and Weak/Variational Forms of Boundary-Value Problems  Conforming Polygonal Finite Elements  Maximum-Entropy Approximation Schemes  Summary and Outlook

Motivation: Voronoi Tesellations in Mechanics Polycrystalline Fiber-matrix Osteonal bone alloy composite (Martin and Burr, 1989) (Courtesy of (Bolander and Kumar, LLNL) S, PRB, 2004)

Motivation: Flexibility in Meshing & Fracture Modeling Convex Mesh Nonconvex Mesh

Motivation: Transition Elements, Quadtree Meshes A B Quadtree Transition elements A B Zoom

Galerkin Finite Element Method (FEM) 2 FEM: Function-based method to solve 3 x partial differential equations steady-state heat conduction, DT diffusion, or electrostatics 1 Strong Form: Variational Form:

Galerkin FEM (Cont’d) Variational Form must vanish on the boundary Finite-dimensional approximations for trial function and admissible variations

Galerkin FEM (Cont’d) Discrete Weak Form and Linear System of Equations

Biharmonic Equation Strong Form Variational (Weak) Form

Elastostatic BVP: Strong Form BCs

Elastostatic BVP: Weak Form/PVW Kinematic relation Constitutive relation Approximation for trial function and admissible variations basis function

Elastostatic BVP: Discrete Weak Form , Material moduli matrix

Finite Element versus Polygonal Approximations Data Approximation Finite Element Polygonal Element Quadrilateral e e e Triangle `shape’ function

Three-Node FE versus Polygonal FE (Cont’d) FEM (3-node) Polygonal

Three-Node FE versus Polygonal FE (Cont’d) Assembly FEM Polygonal

Barycentric Coordinates on Polygons • Wachspress basis functions (Wachspress, 1975; Meyer et al., 2002; Malsch and Dasgupta, 2004) • Mean value coordinates (Floater, 2003; Floater x and Hormann, 2006) x • Laplace and maximum-entropy basis functions (S, 2004; S and Tabarraei, 2004)

Properties of Barycentric Coordinates • Non-negative • Partition of unity • Linear reproducing conditions

Wachspress Basis Functions: Reference Elements Canonical Elements

Isoparametric Transformation (S and Tabarraei, IJNME, 2004)

Nonconvex Polygons (Floater, CAGD, 2003; Hormann and Floater, ACM TOG, 2006) Mean Value Coordinates (Tabarraei and S, CMAME, 2008)

Issues in the Numerical Implementation Mesh Generation and Numerical Integration  Mesh generation for polygonal (Sieger et al., 2010; Ebeida et al., ACM TOG, 2011; Talischi et al., 2012) and polyhedral meshes (Ebeida and Mitchell, 2011)  Numerical integration of bivariate polynomials and generalized barycentric coordinates on polygons (Mousavi and S, 2010; 2011)

Patch Test Quadtree mesh Mesh a Mesh b Mesh c Linear essential (Dirichlet) BCs are imposed on Error in the norm = Error in the energy norm =

Poisson Problem: Localized Potential Potential (Tabarraei and S, CMAME, 2007)

Poisson Problem: Mesh Refinements Mesh a Mesh b Mesh c Mesh d Mesh e Mesh f

Principle of Maximum Entropy (Shannon, Bell. Sys. Tech. J., 1948; Jaynes, Phy. Rev., 1957)  discrete set of events  possibility of each event  uncertainty of each event  Shannon entropy  average uncertainty  concave functional a  unique maximum  Jaynes’s principle of maximum entropy  maximizing s.t. , gives the least-biased probability distribution

Entropy to Generalized Barycentric Coordinates WPC  convex polygon with vertices MVC  for any , maximize HC subject to MEC  maximum entropy basis functions (S, IJNME, 2004)

Max-Ent Basis Functions: Unit Square 4(0,1) 3(1,1) x 2 = ( 0 , 1 ) � 1(0,0) 2(1,0) which simplifies to

Max-Ent Basis Functions: Unit Square (Cont’d) Since , we obtain and therefore which are the same as bilinear finite element shape functions

Maximum-Entropy Meshfree Basis Functions  scattered nodes in with coordinates  for any , maximize subject to (Arroyo & Ortiz, IJNME, 2006; S & Wright, IJNME, 2007) pos-def mass matrix convex basis convex hull property functions no Runge phenomenon

Boundary Behavior (Arroyo and Ortiz, IJNME, 2006) Interior basis functions vanish on the boundary  Lagrange multipliers blow-up on the boundary! Derivatives are computed by taking appropriate limits (Greco and S, preprint)

Second-Order Max-Ent Basis Functions Using the second-order reproducing constraints results in the constraints being an unfeasible set if  Ortiz (Caltech) and Arroyo (UPC, Barcelona) : relaxed the quadratic constraint (gap function) to realize second-order completeness a.e.; Gonzalez et al. (Zaragoza) adopted de Boor’s algorithm for higher-order max-ent  Non-negative restriction on the basis functions is relaxed and a modified entropy functional is used to construct higher-order signed max-ent basis functions (S and Wright, IJNME, 2007; Bompadre et al., CMAME, 2012)

• Second-Order Max-Ent Basis Functions (Cont’d) Relaxation of second-order constraints (Cyron et al., IJNME, 2009) (Rosolen et al., in review, 2012) convex hull (Courtesy of Adrian Rosolen, UPC/MIT)

Second-Order Max-Ent: Nonuniform Grids (Courtesy of Adrian Rosolen, UPC/MIT)

Non-Negative Max-Ent Coordinates (Hormann and S, Comp. Graph. Forum, 2008) Prior is based on edge weight functions

Quadratic Max-Ent Coordinates on Polygons  Use notion of a prior in the modified entropy measure (signed basis functions) introduced by Bompadre et al., CMAME, 2012  Adopt the linear constraints for quadratic precision proposed by Rand et al., arXiv, 2011  Use nodal priors (Hormann and S, CGF, 2008) based on edge weights in the max-ent variational formulation  Construction applies to convex and nonconvex planar polygons. On each boundary facet, one-dimensional Bernstein bases (Farouki, CAGD, 2012) are obtained

Quadratic Reproducing Conditions Pairwise products of generalized barycentric coordinates: (Rand et al., arXiv, 2011)

Quadratic Max-Ent: Formulation  planar polygon with vertices  for any subject to 6 linearly independent equality constraints: PU, linear reproducing conditions and

Constraint Equations Define: Constraints:

Quadratic Max-Ent: Formulation and Solution  Karush-Kuhn-Tucker (KKT) first-order optimality conditions  Lagrangian dual function  Solved using Newton’s method with line search (3 to 7 iterations needed for convergence to 1e-15)

Gradient of the Shape Functions (Hessian)

Polygonal Elements Square Pentagon Saw-tooth L-Shaped

Newton Iterations for Shape Function Computations

Quadratic Precision Shape Functions: Square Gaussian prior uniform prior Gaussian prior edge prior

Quadratic Precision Shape Functions: Pentagon edge prior

Quadratic Precision Shape Functions: Nonconvex edge prior

Quadratic Precision Shape Functions: L-Shaped edge prior

Derivatives of Shape Functions square L-shaped

Approximation over an L-Shaped Polygon Approximation error for an arbitrary bivariate polynomial

Polygonal and Polyhedral Meshes Self-similar trapezoids (Arnold et al., MC, 2002)

Patch Test: 3-point 12-point 2 x 2 Gauss 4 x 4 Gauss 6 x 6 Gauss

Efficient Integration in Meshfree: Elasticity  Use of corrected, smoothed, or assumed shape function derivatives in meshfree methods have been introduced: Krongauz and Belytschko (1997); Chen et al. (2001); Belytschko et al. (2008-2010); Duan et al. (2012)  Following Duan et al. (2012)

Efficient Integration (Cont’d)  For the higher-order (quadratic) patch test, the stress tensor is an affine function: linear combination of  Obtain corrected shape function derivative using 3-point Gauss rule within each sub-triangle of the polygon

Summary  Introduced generalized barycentric coordinates and the discrete equations for standard and polygonal FE  Discussed construction of linearly precise shape functions on polygonal meshes and implementation of polygonal finite elements  Used relative entropy to construct quadratically precise shape functions on planar polygons  Interesting links with the virtual element method (Brezzi and collaborators in Pavia and Milan)