Algebraic Flux Correction Schemes for High-Order B-Spline Based Finite Element Approximations Numerical Analysis group Matthias Möller m.moller@tudelft.nl 7th DIAM onderwijs- en onderzoeksdag, 5th November 2014
Outline 1 Motivation Finite elements in a nutshell 2 Introduction to B-splines Knot insertion (h-refinement) Order elevation (p-refinement) Geometric mapping Definition of ansatz spaces Properties of B-splines 3 Constrained data projection Numerical examples 4 Constrained transport Numerical examples M. Möller (NA group) AFC for B-Spline based FEM 2 / 34
Finite elements in a nutshell Strong problem: find u C k (Ω) such that Lu = f in Ω + bc s Weighted residual formulation: find u V such that w[lu f] dx = 0 w W Ω Boundary conditions: V (trial) and W (test spaces) contain essential bc s natural bc s are incorporated via integration by parts M. Möller (NA group) AFC for B-Spline based FEM 3 / 34
Finite elements in a nutshell, cont d Galerkin finite elements: choose finite-dimensional spaces V h := {ϕ j } V and W h := {φ i } W and find u h = j u jϕ j V h such that φ i [Lu h f] dx = 0 i = 1,..., dim(w h ) Ω h neglecting complications due to bc s this yields [ ] φ i Lϕ j dx u j = φ i f dx i = 1,..., dim(w h ) Ω h Ω h j M. Möller (NA group) AFC for B-Spline based FEM 4 / 34
Problem Poor approximation of discontinuities/steep gradients if standard Galerkin methods are used without proper stabilization L 2 -projection convective transport M. Möller (NA group) AFC for B-Spline based FEM 5 / 34
Problem Poor approximation of discontinuities/steep gradients if standard Galerkin methods are used with proper stabilization L 2 -projection convective transport M. Möller (NA group) AFC for B-Spline based FEM 5 / 34
Algebraic Flux Correction Methodology based on algebraic design criteria to derive robust and accurate high-resolution finite element schemes for Constrained data projection [5] Convection-dominated transport processes [3, 5, 6, 7, 10, 11] Anisotropic diffusion processes [3, 9] Processes with maximum-packing limit [4]... Many success stories published... ;-) M. Möller (NA group) AFC for B-Spline based FEM 6 / 34
Algebraic Flux Correction Methodology based on algebraic design criteria to derive robust and accurate high-resolution finite element schemes for Constrained data projection [5] Convection-dominated transport processes [3, 5, 6, 7, 10, 11] Anisotropic diffusion processes [3, 9] Processes with maximum-packing limit [4]... Many success stories published... ;-) mostly for P 1 and Q 1 finite elements ;-( M. Möller (NA group) AFC for B-Spline based FEM 6 / 34
Algebraic Flux Correction, cont d Extension of AFC to P1 / Q 1 elements [12]: P 2 elements [2]: CR-FE satisfy the necessary prerequisites [...] fail completely [...] yielding overdiffusive approximate solutions. RT-FE provides an accurate resolution [...] if the integral mean value based variant is adopted. In summary, algebraic flux correction for quadratic finite elements seems to be feasible but gives rise to many challenging open problems. Objective: to extend AFC to high-order B-spline basis functions M. Möller (NA group) AFC for B-Spline based FEM 7 / 34
B-splines in a nutshell Define knot vector Ξ = (ξ 1, ξ 2,..., ξ n+p+1 ) as a sequence of non-decreasing coordinates in the parameter space Ω 0 = [0, 1]: ξ i R is the i th knot with index i = 1, 2,..., n + p + 1 p is the polynomial order of the B-splines n is the number of B-spline functions Cox-de Boor recursion formula for N i,p : Ω 0 R p = 0 : N i,0 = { 1 if ξi ξ < ξ i+1 0 otherwise define 0 0 = 0 p > 0 : N i,p = ξ ξ i N i,p 1 + ξ i+p+1 ξ N i+1,p 1 ξ i+p ξ i ξ i+p+1 ξ i+1 M. Möller (NA group) AFC for B-Spline based FEM 8 / 34
Knot insertion (h-refinement) Ξ = (0, 0, 1, 1), p = 1 Ξ = (0, 0, 0.5, 1, 1), p = 1 1 1 0.8 0.8 0.6 0.6 0.4 0.2 0 0 0.5 1 one element N 0,1 N 1,1 0.4 0.2 0 0 0.5 1 N 0,1 N 1,1 N 2,1 two elements, C 0 -continuity In general we have C p m i -continuity across element boundaries, where m i is the multiplicity of the value of ξ i in the knot vector. M. Möller (NA group) AFC for B-Spline based FEM 9 / 34
Knot insertion, cont d Ξ = (0, 0, 1, 1), p = 1 Ξ = (0, 0, 0.5, 0.5, 1, 1), p = 1 1 1 0.8 0.8 0.6 0.6 0.4 0.2 0 0 0.5 1 one element N 0,1 N 1,1 0.4 0.2 0 0 0.5 1 N 0,1 N 1,1 N 2,1 N 3,1 two elements, discontinuity In general we have C p m i -continuity across element boundaries, where m i is the multiplicity of the value of ξ i in the knot vector. M. Möller (NA group) AFC for B-Spline based FEM 10 / 34
Order elevation 1 Ξ = (0, 0, 1, 1), p = 1 1 Ξ = (0, 0, 0, 1, 1, 1), p = 2 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0 0 0.5 1 one element N 0,1 N 1,1 0.2 0 0 0.5 1 one element N 0,2 N 1,2 N 2,2 In contrast to standard Lagrange finite element basis functions, the B-spline functions never become negative over their support. M. Möller (NA group) AFC for B-Spline based FEM 11 / 34
Nonuniform continuity at element boundaries 1 0.8 0.6 0.4 0.2 4 th -order B-spline functions 0 0 0.2 0.4 0.6 0.8 1 Ξ = (0, 0, 0, 0, 0, 1/3, 2/3, 2/3, 2/3, 1, 1, 1, 1, 1) }{{}}{{}}{{}}{{} discontinuity C 3 C 1 discontinuity N 0,4 N 1,4 N 2,4 N 3,4 N 4,4 N 5,4 N 6,4 N 7,4 N 8,4 M. Möller (NA group) AFC for B-Spline based FEM 12 / 34
Nonuniform continuity at element boundaries First derivatives of 4 th -order B-spline functions 10 5 0 5 10 0 0.2 0.4 0.6 0.8 1 Ξ = (0, 0, 0, 0, 0, 1/3, 2/3, 2/3, 2/3, 1, 1, 1, 1, 1) }{{}}{{}}{{}}{{} discontinuity C 3 C 1 discontinuity N 0,4 N 1,4 N 2,4 N 3,4 N 4,4 N 5,4 N 6,4 N 7,4 N 8,4 M. Möller (NA group) AFC for B-Spline based FEM 13 / 34
Geometric mapping Define mapping between parameter space Ω 0 = [0, 1] and the computational domain Ω using control points p i R d, d = 1, 2, 3 G : Ω 0 Ω, G = n N i,p p i i=1 Examples: G : [0, 1] [a, b] R G : [0, 1] curve in R 2 or R 3 (linear mapping) (work in progress) M. Möller (NA group) AFC for B-Spline based FEM 14 / 34
Ansatz spaces Construct ansatz space from B-spline basis functions V h (Ω 0, p, Ξ, G) = span{ϕ i (x) = N i,p G 1 (x)} Approximate the solution the standard way u(x) u h (x) = n ϕ i (x)u i, i=1 x Ω Approximate fluxes by Fletcher s group formulation, e.g., v(x)u(x) (vu) h (x) = n ϕ i (x)(v i u i ), i=1 x Ω M. Möller (NA group) AFC for B-Spline based FEM 15 / 34
Properties of B-splines Derivative of p th order B-spline is a B-spline of order p 1 N i,p = p p N i,p 1 N i+1,p 1 ξ i+p ξ i ξ i+p+1 ξ i+1 B-splines form a partition of unity. That is, for all ξ [a, b] n N i,p = 1 i=1 n N i,p = 0 i=1 AFC-1: Edge-wise flux decomposition b n c ij = ϕ i ϕ j dx, c ij = 0 a j=1 n j=1 c ij u j = j i c ij (u j u i ) M. Möller (NA group) AFC for B-Spline based FEM 16 / 34
Properties of B-splines, cont d B-splines of order p have compact support supp N i,p = [ξ i, ξ i+p+1 ), i = 1,..., n B-splines are strictly positive over the interior of their support N i,p > 0 for ξ (ξ i, ξ i+p+1 ), i = 1,..., n AFC-2: positive consistent and lumped mass matrices b n m ij = ϕ i ϕ j dx > 0 m i = m ij > 0 a j=1 M. Möller (NA group) AFC for B-Spline based FEM 17 / 34
Constrained data projection Find u L 2 ([a, b]) s.t. b a w(u f)dx = 0 w L 2 ([a, b]) Consistent L 2 -projection n b m ij u H j = ϕ i f dx j=1 a Lumped L 2 -projection m i u L i = b a ϕ i f dx Constrained L 2 -projection [5] u i = u L i + 1 α ij fij H, m i 0 α ij = α ji 1, fji H = fij H j i M. Möller (NA group) AFC for B-Spline based FEM 18 / 34
Symmetric flux limiting algorithm [5] 1 Prelimited raw antidiffusive fluxes { fij H m ij (u H i u H j = ), if (uh i u H j )(ul i u L j ) > 0 0, otherwise (!) 2 Bounds and antidiffusive increments Q ± i = max min j i (0, u L i u L j ), P ± i = max min j i (0, f ij ) 3 Nodal and edge-wise limiting coefficients R ± i = m iq ± i P ± i, α ij = { min(r + i, R j ), if f ij > 0 min(r + j, R i ), if f ij < 0 M. Möller (NA group) AFC for B-Spline based FEM 19 / 34
Test case: Semi-ellipse of McDonald p = 1, n = 32 1 L 1 -/L 2 -errors 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 u H u 1 = 0.0653 u L u 1 = 0.0684 u u 1 = 0.0606 u H u 2 = 0.1774 u L u 2 = 0.1686 u u 2 = 0.1677 M. Möller (NA group) AFC for B-Spline based FEM 20 / 34
Test case: semi-ellipse of McDonald p = 2, n = 32 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 L 1 -/L 2 -errors u H u 1 = 0.0351 u L u 1 = 0.0568 u u 1 = 0.0352 u H u 2 = 0.0748 u L u 2 = 0.1121 u u 2 = 0.0892 M. Möller (NA group) AFC for B-Spline based FEM 21 / 34
Test case: semi-ellipse of McDonald p = 3, n = 32 1 L 1 -/L 2 -errors 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 u H u 1 = 0.0595 u L u 1 = 0.0788 u u 1 = 0.0615 u H u 2 = 0.1213 u L u 2 = 0.1435 u u 2 = 0.1317 M. Möller (NA group) AFC for B-Spline based FEM 22 / 34
Test case: semi-ellipse of McDonald p = 4, n = 32 1 L 1 -/L 2 -errors 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 u H u 1 = 0.0480 u L u 1 = 0.0896 u u 1 = 0.0608 u H u 2 = 0.1014 u L u 2 = 0.1505 u u 2 = 0.1252 M. Möller (NA group) AFC for B-Spline based FEM 23 / 34
Thought experiment: What are ideal knots? 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 M. Möller (NA group) AFC for B-Spline based FEM 24 / 34
Thought experiment: What are ideal knots? 0.8 0.6 0.4 L 1 -/L 2 -errors u H u 1 = 0.0037 u u 1 = 0.0220 0.2 0 0 0.2 0.4 0.6 0.8 1 u H u 2 = 0.0091 u u 2 = 0.0615 Deduce ideal knots from nodal correction factors or residual Ξ = (0, 0, 0, 0, 0.1, 0.2, 0.2, 0.2, 0.2,0.3, 0.4, 0.4, 0.4, 0.4,0.5, 0.6, 0.6, 0.6, 0.6,0.7, 0.7, 0.7,0.8, 0.8, 0.8, 0.8,0.9, 1, 1, 1, 1) Use smoothness indicator [8] to avoid peak clipping. M. Möller (NA group) AFC for B-Spline based FEM 24 / 34
Constrained transport b Find u HD 1 ([a, b]) s.t. w(vu) x + dw x u x dx = 0 j=1 j=1 a w H 1 0 ([a, b]) Galerkin scheme Discrete upwind scheme n (k ij + s ij )u H j = 0 n (k ij + d ij + s ij )u L j = 0 b k ij = v j ϕ i ϕ jdx, a s ij = d b a ϕ iϕ jdx, d ij = max(k ij, 0, k ji ) High-resolution TVD-type scheme [11] n ij + d ij + s ij )u j=1(k j + α ij f ij = 0, j i f ji = f ij M. Möller (NA group) AFC for B-Spline based FEM 25 / 34
Test case: steady convection-diffusion v d = 100 1.5 1 0.5 p = 1, n = 16 1.2 1 0.8 0.6 0.4 0.2 p = 2, n = 16 1 0.8 0.6 0.4 0.2 p = 3, n = 16 0 0 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1 u H u 1 = 0.0325 u L u 1 = 0.0056 u u 1 = 0.0028 u H u 1 = 0.0415 u L u 1 = 0.0061 u u 1 = 0.0031 u H u 1 = 0.0365 u L u 1 = 0.0059 u u 1 = 0.0029 M. Möller (NA group) AFC for B-Spline based FEM 26 / 34
Summary Algebraic flux correction concept has been generalized to higher-order approximations based on B-spline bases Original lowest-order approximation is naturally included Nodal correction factors/residual provide information to locally reduce inter-element continuity by increasing knot multiplicity Peak clipping at smooth extrema is prevented by locally deactivating the flux limiter using the smoothness indicator [8] M. Möller (NA group) AFC for B-Spline based FEM 27 / 34
Current and future research Analysis for general geometric mappings G : Ω 0 Ω Extension to multi-dimensions by tensor-product construction G(ξ, η, ζ) = n m l N i,p N j,q (η)n k,r (ζ)p ijk i=1 j=1 k=1 Embed local B-spline based AFC-scheme into global outer Galerkin method with unstructured quad/hexa macro mesh Exploit potential of fully structured data per macro element A. Jaeschke (COSSE-MSc), S-R. Janssen (DD: AM-LR) M. Möller (NA group) AFC for B-Spline based FEM 28 / 34
Outlook: Isogeometric Analysis [1] Poor approximation of curved boundaries (with low-order FEs) Ω Ω h IgA approach adopts the same (hierarchical) B-spline, NURBS, etc. basis functions for the approximate solution u h u and for exactly representing the geometry Ω h = Ω Our activities in IgA: bi-weekly PhD-seminar (participants from LR) PhD-candidate in CSC-15 and/or EU-project (?) M. Möller (NA group) AFC for B-Spline based FEM 29 / 34
References I T.J.R. Hughes, J.A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194:4135 4195, 2005. D. Kuzmin. On the design of algebraic flux correction schemes for quadratic finite elements. Journal of Computational and Applied Mathematics, 218(1):79 87, 2008. Special Issue: Finite Element Methods in Engineering and Science (FEMTEC 2006) Special Issue: Finite Element Methods in Engineering and Science (FEMTEC 2006). M. Möller (NA group) AFC for B-Spline based FEM 30 / 34
References II D. Kuzmin. Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes. Journal of Computational and Applied Mathematics, 236(9):2317 2337, 2012. D. Kuzmin and Y. Gorb. A flux-corrected transport algorithm for handling the close-packing limit in dense suspensions. Journal of Computational and Applied Mathematics, 236(18):4944 4951, 2012. {FEMTEC} 2011: 3rd International Conference on Computational Methods in Engineering and Science, May 913, 2011. M. Möller (NA group) AFC for B-Spline based FEM 31 / 34
References III D. Kuzmin, M. Möller, J.N. Shadid, and M. Shashkov. Failsafe flux limiting and constrained data projections for equations of gas dynamics. Journal of Computational Physics, 229(23):8766 8779, 2010. D. Kuzmin, M. Möller, and S. Turek. Multidimensional FEM-FCT schemes for arbitrary time stepping. International Journal for Numerical Methods in Fluids, 42(3):265 295, 2003. D. Kuzmin, M. Möller, and S. Turek. High-resolution FEMFCT schemes for multidimensional conservation laws. Computer Methods in Applied Mechanics and Engineering, 193(4547):4915 4946, 2004. M. Möller (NA group) AFC for B-Spline based FEM 32 / 34
References IV D. Kuzmin and F. Schieweck. A parameter-free smoothness indicator for high-resolution finite element schemes. Central European Journal of Mathematics, 11(8):1478 1488, 2013. D. Kuzmin, M.J. Shashkov, and D. Svyatskiy. A constrained finite element method satisfying the discrete maximum principle for anisotropic diffusion problems. Journal of Computational Physics, 228(9):3448 3463, 2009. D. Kuzmin and S. Turek. Flux correction tools for finite elements. Journal of Computational Physics, 175(2):525 558, 2002. M. Möller (NA group) AFC for B-Spline based FEM 33 / 34
References V D. Kuzmin and S. Turek. High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. Journal of Computational Physics, 198(1):131 158, 2004. M. Möller. Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems. Computing, 95(5):425 448, 2013. M. Möller (NA group) AFC for B-Spline based FEM 34 / 34