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

 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

Outline

Motivation: Voronoi Tesellations in Mechanics

Polycrystalline alloy

(Courtesy of Kumar, LLNL) (Martin and Burr, 1989) (Bolander and S, PRB, 2004)

Fiber-matrix composite Osteonal bone

Motivation: Flexibility in Meshing & Fracture Modeling

Convex Mesh Nonconvex Mesh

Motivation: Transition Elements, Quadtree Meshes

Quadtree Zoom Transition elements

A A B B

Galerkin Finite Element Method (FEM)

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

Galerkin FEM (Cont’d)

Variational Form Finite-dimensional approximations for trial function and admissible variations must vanish on the boundary

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

, Material moduli matrix

Elastostatic BVP: Discrete Weak Form

`shape’ function Data Approximation

Finite Element versus Polygonal Approximations

Triangle Quadrilateral

Finite Element Polygonal Element

e e e

FEM (3-node) Polygonal

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

Assembly

FEM Polygonal

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

• Wachspress basis functions (Wachspress, 1975;

Meyer et al., 2002; Malsch and Dasgupta, 2004)

• Mean value coordinates

(Floater, 2003; Floater and Hormann, 2006)

• Laplace and maximum-entropy basis functions

x (S, 2004; S and Tabarraei, 2004)

Barycentric Coordinates on Polygons

x

• Non-negative
• Partition of unity
• Linear reproducing conditions

Properties of Barycentric Coordinates

Wachspress Basis Functions: Reference Elements

Canonical Elements

Isoparametric Transformation

(S and Tabarraei, IJNME, 2004)

Nonconvex Polygons

Mean Value Coordinates

(Floater, CAGD, 2003; Hormann and Floater, ACM TOG, 2006)

(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)

Mesh a Mesh b Mesh c

Patch Test

Quadtree mesh

Error in the norm = Error in the energy norm =

Linear essential (Dirichlet) BCs are imposed on

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

 discrete set of events  possibility of each event  uncertainty of each event  Shannon entropy

• average uncertainty
• concave functional
• unique maximum

 Jaynes’s principle of maximum entropy

• maximizing s.t. ,

gives the least-biased probability distribution

(Shannon, Bell. Sys. Tech. J., 1948; Jaynes, Phy. Rev., 1957)

a

Entropy to Generalized Barycentric Coordinates

 convex polygon with vertices  maximum entropy basis functions (S, IJNME, 2004)

WPC HC MVC MEC

subject to  for any , maximize

Max-Ent Basis Functions: Unit Square

1(0,0) 3(1,1) 2(1,0) 4(0,1)

2

) 1 , ( =

• x

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

(Arroyo & Ortiz, IJNME, 2006; S & Wright, IJNME, 2007)

convex basis

pos-def mass matrix convex hull property no Runge phenomenon

functions subject to

Boundary Behavior

Interior basis functions vanish on the boundary

(Arroyo and Ortiz, IJNME, 2006)

 Lagrange multipliers blow-up on the boundary! Derivatives are computed by taking appropriate limits (Greco and S, preprint)

Second-Order Max-Ent Basis Functions

 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) results in the constraints being an unfeasible set if Using the second-order reproducing constraints

Second-Order Max-Ent Basis Functions (Cont’d)

• Relaxation of second-order constraints

(Cyron et al., IJNME, 2009) (Rosolen et al., in review, 2012)

(Courtesy of Adrian Rosolen, UPC/MIT)

convex hull

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

• n 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

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

Constraint Equations

Constraints: Define:

Quadratic Max-Ent: Formulation and Solution

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

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 edge prior Gaussian 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

• f 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)