Transfinite Interpolation Based Analysis
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1 Transfinite Interpolation Based Analysis Nathan Collier 1 V.M. Calo 1 Javier Principe 2 1 King Abdullah University of Science and Technology 2 International Center for Numerical Methods in Engineering 7 June 2010 Collier (KAUST) Transfinite Element Analysis 7 June / 26
2 Motivation - Collier (KAUST) Transfinite Element Analysis 7 June / 26
3 Overview 1 Transfinite Element Analysis Basis Function Construction Sample Basis Functions 2 Application 1: Terrain Modeling Automatic Refinement Insert Features 3 Application 2: Finite Elements with Simplified Meshing Current Challenges 4 Conclusions Collier (KAUST) Transfinite Element Analysis 7 June / 26
4 Interpolate Features 1 u1 d1 u d2 u2 2 1 f1(u1) F (u) f2(u2) 2 d3 3 u3 f3(u3) 3 i 1 d 1 F (u) = f 1 (u 1 ) 1 d d f 2 (u 2 ) 1 d 3 d d f 3 (u 3 ) 1 d 3 d d d 3 n f W i n f = f i (u i ) nf = f i (u i )Ŵ i (d 1, d 2, d 3 ) j W j i 1 d 2 1 d 3 Collier (KAUST) Transfinite Element Analysis 7 June / 26
5 Basis Function Construction n f F (u) = f i (u i )Ŵi(d) Each footprint can then be parameterized by its own one-dimensional basis, B i,k n b f i (u) = B i,k (u)c i,k which substituted into the surface approximation becomes n f n b F (u) = B i,k (u)c i,k Ŵ i (d) i k n f n b i k ( ) = B i,k (u)ŵ i (d) C i,k = N j C j i k }{{} j N i,k Collier (KAUST) Transfinite Element Analysis 7 June / 26
6 Sample Basis Functions Linear Footprints, Convex Polygons Collier (KAUST) Transfinite Element Analysis 7 June / 26
7 Sample Basis Functions Gradients N N x N y Node 0 Node 1 Node 2 Collier (KAUST) Transfinite Element Analysis 7 June / 26
8 Derivative Control: Ribbons Not only can we construct functions which interpolate features, but we can also control the derivative as the feature is crossed. 1 u1 d1 u d2 u2 2 1 f1(u1) F (u) f2(u2) 2 d3 3 u3 f3(u3) 3 Collier (KAUST) Transfinite Element Analysis 7 June / 26
9 Bi-cubic: 28 vs 16 Collier (KAUST) Transfinite Element Analysis 7 June / 26
10 Non-standard Basis Functions Collier (KAUST) Transfinite Element Analysis 7 June / 26
11 Overview 1 Transfinite Element Analysis Basis Function Construction Sample Basis Functions 2 Application 1: Terrain Modeling Automatic Refinement Insert Features 3 Application 2: Finite Elements with Simplified Meshing Current Challenges 4 Conclusions Collier (KAUST) Transfinite Element Analysis 7 June / 26
12 Projection onto Function Space Currently using L 2 projections. Find U A such that argmin U A Z H(U A ) 2 where Z is a function to approximate, and H is the interpolation which is controlled by parameters U A. Collier (KAUST) Transfinite Element Analysis 7 June / 26
13 Footprint Refinement Each footprint can be refined locally due to the surface interpolating each footprint. Collier (KAUST) Transfinite Element Analysis 7 June / 26
14 Footprint Refinement Tensor Product (Old way) Transfinite (New way) Collier (KAUST) Transfinite Element Analysis 7 June / 26
15 Footprint Refinement Other degrees of freedom can then be obtained via a surface L 2 projection G 1 Cubic G 2 Quintic Collier (KAUST) Transfinite Element Analysis 7 June / 26
16 Footprint Insertion Combine image processing with projective interpolation Collier (KAUST) Transfinite Element Analysis 7 June / 26
17 Footprint Insertion Combine image processing with projective interpolation Collier (KAUST) Transfinite Element Analysis 7 June / 26
18 Footprint Insertion C 0 Continuity C 1 Continuity Collier (KAUST) Transfinite Element Analysis 7 June / 26
19 Footprint Insertion Collier (KAUST) Transfinite Element Analysis 7 June / 26
20 Overview 1 Transfinite Element Analysis Basis Function Construction Sample Basis Functions 2 Application 1: Terrain Modeling Automatic Refinement Insert Features 3 Application 2: Finite Elements with Simplified Meshing Current Challenges 4 Conclusions Collier (KAUST) Transfinite Element Analysis 7 June / 26
21 Motivation - Collier (KAUST) Transfinite Element Analysis 7 June / 26
22 Laplace Equation Find θ(x) such that with boundary conditions θ = 0 x Ω θ x=0 = 0, n θ x=1 = 1, n θ y=0,1 = 0 Collier (KAUST) Transfinite Element Analysis 7 June / 26
23 Integration Check the relative change in the stiffness matrix diagonal terms as refining the integration mesh. Gaussian Quadrature 4x4 rule used with uniform refinements. Collier (KAUST) Transfinite Element Analysis 7 June / 26
24 Discontinuity B 5 (x, y) d dx B 5(x, y) d dy B 5(x, y) Collier (KAUST) Transfinite Element Analysis 7 June / 26
25 Discontinuity - Removed by Assembly B 5 (x, y) d dx B 5(x, y) d dy B 5(x, y) Pre-Assembly Post-Assembly Collier (KAUST) Transfinite Element Analysis 7 June / 26
26 Conclusion Transfinite Interpolation is a powerful, flexible way to define a basis Grants the freedom to place degrees of freedom at will without propagating them to unintentional regions of the domain Ties in directly to CAD programs whose geometries are watertight and requires no meshing in the traditional sense Current work is focused in accurate integration techniques Collier (KAUST) Transfinite Element Analysis 7 June / 26
Motivation Patch tests Numerical examples Conclusions
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