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Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Roy Stogner Computational Fluid Dynamics Lab Institute for Computational Engineering and Sciences University of Texas at Austin August 12, 2008

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Outline 1 Goals Macroelement Spaces 2 Adaptive Mesh Refinement / Coarsening 3 4 Software Implementation Solver Details 5 Divergence-free Flow 6 7 Thin Film Flow Cahn-Hilliard Phase Decomposition 8 Contributions 9

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Goals Dissertation Goals Goals Parallel adaptive solution of fourth order problems Adaptive mesh refinement of C 1 macroelements Error estimation on conforming formulations of fourth order problems New weak formulations of thin film flow problems Numerical experiments of thin film flow phenomena

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Macroelement Spaces Application Classes Fourth Order Terms Streamfunction Viscosity Thin Film Surface Tension Material Interface Diffusion

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Macroelement Spaces C 1 Finite Element Spaces In Galerkin approximations of fourth order problems we find integrated products of second derivatives of trial and test functions. Conforming finite element approximations require at least H 2 conforming functions. We can use C 1 continuous (and W 2 , ∞ bounded, W 2 , p conforming) finite elements: Macroelement Types Powell-Sabin 6-split triangle Powell-Sabin-Heindl 12-split triangle Clough-Tocher 3-split triangle

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Macroelement Spaces Macroelements Constraining C 1 continuity on arbitrary meshes requires quintics, a higher degree than desired for many approximations. Instead, we subdivide each macroelement:

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Macroelement Spaces Constructing Macroelements Macroelement Basis Precalculation Index the “raw” degrees of freedom “Write” all constraints as symbolic matrix rows Put constraint matrix in row reduced form Add boundary DoF equations (checking each for linear independence) If necessary, make matrix square with interior DOF equations (again checking for linear independence) Invert matrix, multiply by ˆ e i to get basis coefficients corresponding to the DOF on row i

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Macroelement Spaces Clough-Tocher 3-split

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Macroelement Spaces Interpolation, H 2 Approximation Convergence Powell-Sabin and Clough-Tocher macroelements can exactly reproduce quadratics and cubics ( k ≡ 2 , 3 ) , respectively. Standard interpolation, H 2 approximation rules apply. For w ∈ H n (Ω) , n ≤ k + 1 , N � P h ( f ) ≡ σ i ( f ) φ i i = 1 Ch n − m | w | H n (Ω) || w − P h w || H m (Ω) ≤ Ch n − 2 | u | H n (Ω) || u − u h || H 2 (Ω) ≤

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Macroelement Spaces L 2 Approximation Convergence Clough-Tocher elements obtain an additional power of h over Powell-Sabin elements in H 2 norm. The difference is greater in L 2 . With η ≡ min ( 2 ( k + 1 − m ) , k + 1 − r , n − r ) , for Galerkin approximation u h to an elliptic problem on H m (Ω) , || u − u h || H r (Ω) ≤ Ch η || u || H n (Ω) For k = 3 , or k = 2 , r ≥ 1 , this is familiar. For fourth order problems ( m = 2 ), quadratic elements ( k = 2 ), in the L 2 norm ( r = 0 ), η = 2 ( k + 1 − m ) = 2 .

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Adaptive Mesh Refinement / Coarsening h Adaptivity Macroelement splitting, adaptive refinement subdivision must match along element sides 12-split, 3-split triangles, 4-split tetrahedra are compatible 6-split triangles, 12-split tetrahedra are not

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Adaptive Mesh Refinement / Coarsening h Adaptivity Maintaining function space continuity requires constraining some degrees of freedom on fine elements in terms of degrees of freedom on coarse neighbor elements. u F u C = � � u F i φ F u C j φ C = i j i j A ki u i = B kj u j A − 1 = u i ki B kj u j Integrated values (and fluxes, for C 1 continuity) give element-independent matrices: ( φ F i , φ F ≡ k ) A ki ( φ C j , φ F ≡ k ) B kj

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Adaptive Mesh Refinement / Coarsening Error Indicators Integration by parts gives an upper error bound on subelements S for the biharmonic problem: � �� �� � f − ∆ 2 u h � �� S h 2 || e || H 2 (Ω) ≤ S + C Ω � S � 1 + 1 n ∆ u h ]] || ∂ S h 3 / 2 2 || [[∆ u h ]] || ∂ S h 1 / 2 2 || [[ ∂ � S S The most significant term gives a simple indicator on elements K for more general fourth order problems: � η K ≡ h K || [[∆ u h ]] || ∂ K

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Software Implementation libMesh C++ Finite Element Library Initial developers: Benjamin Kirk, John Peterson Contributions from Michael Anderson, Bill Barth, Daniel Dreyer, Derek Gaston, David Knezevic, Hendrik van der Heijden, Steffen Petersen, Florian Prill, others Key Features Mixed element geometries in unstructured grids Adaptive mesh h-refinement with hanging nodes Parallel system assembly and solution Integration w/ PETSc, LASPack, METIS, ParMETIS Export/import to common data formats 200 downloads/month, 100 current users, 20 papers

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Software Implementation libMesh Usage FEMSystem +element_solution libMesh library provides +element_residual +element_jacobian +FE_base elements, linear +*_time_derivative(request_jacobian) +*_constraint(request_jacobian) +*_postprocess() algebra, common tools Application code NavierStokesSystem CahnHilliardSystem implements physical equations, control loops LaplaceYoungSystem SurfactantSystem

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Software Implementation Finite Element Classes FEBase +phi: vector<vector<Real> > Finite Element object +dphi: vector<vector<RealGradient> > +d2phi: vector<vector<RealTensor> > computes data for each +JxW: vector<Real> +quadrature_rule: QRule geometric Elem object +reinit(Elem) +reinit(Elem,side,) Application code is element independent Lagrange Hierarchic Hermite Monomial CloughTocher

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Software Implementation New Contributions to libMesh Current chief software architect: Roy Stogner Key Features Added Macroelement construction and quadrature C 1 macroelement, Hermite classes Hessian calculations Parallel adaptivity for general element types Parallel unstructured meshing Projection, interpolation for general elements New nonlinear solver, timestepping frameworks Additional error estimators, adaptivity strategies

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Solver Details Adaptive Time Stepping 2 10 Trapezoidal H 2 Based Timesteps L 2 Based Timesteps integration to 0 10 avoid −2 10 extrapolation Timestep Length −4 10 failures Truncation error −6 10 compares 2 δ t to −8 10 δ t in relative H r −10 10 −10 −8 −6 −4 −2 0 2 4 10 10 10 10 10 10 10 10 norm Time

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Solver Details Adaptive Time Stepping Adaptive Time Step Lengths for Imposed Bias Magnitude Study Time step length 100 may be limited by 1 random topological Time Step Length 0.01 events at any 0.0001 time Mean step 1e-06 lengths grow 1e-08 0.0001 0.001 0.01 0.1 1 10 100 smoothly. Time

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Solver Details Newton-Krylov Nonlinear Solver Adaptively reduced linear residual reduction tolerance Inner GMRES iteration, Block Jacobi/ILU preconditioning Reliability Improvements Brent’s Method line search to find residual reduction Feedback to adaptive time stepping or continuation solvers

Parallel Adaptive C 1 Macro-Elements for Nonlinear Thin Film and Non-Newtonian Flow Problems Divergence-free Flow Divergence-free Elements Curls of C 1 basis functions become div-free C 0 spanning functions Constraining kernel of ∇× is simple in 2D Pressure term disappears from Navier-Stokes equations: � � � ∇ P · � v d Ω = P � v · � n dS − P ∇ · � v d Ω = 0 Ω ∂ Ω Ω