Enhanced Discretization and Space-Time Refinement in Moving-Boundary Flow Simulation Marek Behr Chair for Computational Analysis of Technical Systems RICAM Special Semester November 8, 2016
Outline • Why moving boundaries? • melt flow processes • solid mechanics • Which frame of reference? • Lagrangian, Eulerian, mixed • How to follow the interface? • tracking and capturing • Enhanced surface discretization • NEFEM, IGA S. Elgeti and H. Sauerland Deforming Fluid Domains Within • Space-time refinement the Finite Element Method: Five Mesh-Based Tracking Methods in • simplex space-time grids Comparison Archives of Computational Methods in Engineering 23 (2016) 323–361 arXiv:1501.05878 2
Why Moving Boundaries? • Fluid: • filling • swelling • solidification SFB 1120 TP B5 XC Production TP A3 • Solid: • shrinking • deformation SFB 1120 TP B2 3
Which Frame of Reference? • Lagrangian: • frame attached to moving material • no inflow or outflow of material • Eulerian: • frame fixed in space • material flows relative to the frame • Lagrangian-Eulerian: • neither Lagrangian nor Eulerian • Lagrangian and Eulerian as special cases 4
How to Follow the Interface? Interface tracking Interface capturing interface = grid boundary cut interface cell empty cell deforming ALE grid fixed Eulerian grid 5
Interface Tracking • Advantages: • low discretization errors at the interface • easy inclusion of interface effects • Disadvantages: • deformation limited if no remeshing • projection errors if remeshing • Arbitrary Lagrangian-Eulerian (ALE), space-time finite elements 6
Interface Capturing • Advantages: • easy handling of large deformations • allows topology changes • Disadvantages: • high discretization errors at the interface • load imbalance • Particle methods, volume-of-fluid, level-set, phase field 7
Interface Capturing • Level Set • Level set function φ ( x , t ) • fluid A domain φ < 0 • fluid B domain φ > 0 • interface φ = 0 • Due to Osher and Sethian (1988) • Advected with fluid velocity: ∂φ ∂ t + v · r φ = 0 • Determines material properties at integration point: ( ρ A , µ A , φ ( x , t ) < 0 ρ , µ ( φ ) = φ ( x , t ) > 0 ρ B , µ B , 8
Interface Capturing • Level Set Reinitialization • Initial value is signed distance function • Signed-distance property degrades: • How to reinitialize? • search for closest interface node • narrow band search • iterative solution of Eikonal equation • Mass conservation is not guaranteed 9
Interface Capturing • Level Set Sharpening • XFEM-based integration: level set linearization quadrature • Adaptive refinement: 10
Interface-Tracking Problem • Three governing equations • kinematic condition • interior mesh update • incompressible Navier-Stokes • Solved within nonlinear iteration loop begin time step loop begin nonlinear iteration loop solve elevation equation solve deformation equation solve flow equation end loop end loop 11
Governing Equations: Flow • Incompressible Navier-Stokes equations: • Constitutive equation: • Boundary conditions: • For plastics flow: • Navier-Stokes → Stokes • Newtonian fluid → shear-thinning or viscoelastic (Giesekus, Oldroyd-B) 12
Galerkin Least-Squares Formulation: Flow • Find such that : • Notation: 13
Governing Equations: Deformation • Displacements governed by linear elasticity: • Boundary conditions: • Mesh update: 14
Galerkin Formulation: Deformation • Find such that : • Coefficients and adjusted to stiffen small cells: • Alternate approaches: ▶ geometric (Masud, 1997 & 2007), ▶ stress-based (Oñate et al., 2000), ▶ ... 15
Governing Equations: Elevation • Kinematic condition at the free surface: • One condition but two or three unknowns per node • Whether explicitly stated or not, equivalent to: where is the generalized elevation along direction e • Often applied point-wise; OK in tanks, but not in channels or jets: 16
Galerkin Least-Squares Formulation: Elevation • Find such that : • Stabilization and DC parameters, using residual : 17
Example: Olmsted Dam • Spillway of Olmsted dam on Ohio River • Experiments in a scale model at ERDC: • Periodic spillway section 301.5ft • Obstacles dissipate flow energy and reduce erosion: 239ft 182ft 280ft 79ft 32ft 18
Example: Olmsted Dam • Computed using 418K tetrahedra and 1000 time steps: 19
Example: Olmsted Dam • Mesh adapts to free surface movement: 20
Example: Olmsted Dam • Quasi-steady state reached without remeshing • Stream-wise component of velocity shown • Hydraulic jump position accurately predicted 21
Example: Olmsted Dam • Comparison of space-time (left) and level-set with XFEM (right) t = 3 . 5 s t = 6 . 1 s t = 3 . 5 s t = 6 . 1 s t = 8 . 5 s t = 11 . 6 s t = 8 . 5 s t = 11 . 6 s t = 27 . 2 s t = 36 . 1 s t = 27 . 2 s t = 36 . 1 s Sauerland et al., Computers and Fluids , 87 (2013) 41–49 22
NURBS-Enhanced Finite Elements (Huerta et al.) θ • Most elements are standard finite elements • Elements on the NURBS boundary represent the exact geometry Stavrev, Knechtges, Elgeti and Huerta, International Journal for Numerical Methods in Fluids , 81 (2016) 426–450
NEFEM in Die Swell • Transient Stokes equations; parabolic inflow; no slip on the walls • FEM: linear space-time elements • NEFEM: linear space-time approximation; geometry with cubic NURBS Swell Factor NEFEM Swell Factor FEM Mesh I: 768 DOF 1.1262 1.1374 Mesh II: 4074 DOF 1.1257 1.1271 Mesh III: 6006 DOF 1.1220 1.1220
Isogeometric Analysis (Hughes et al.) • Analogous to isoparametric concept • IGA uses NURBS to represent both the geometry and the unknown: n u h = X R i,p ( θ ) u i i =1 Hughes et al., Isogeometric analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement, Computer Methods in Applied Mechanics and Engineering , 194 , (2005) 4135–4195
Spline-Based Coupling for Fluid-Structure-Interaction Standard FE IGA + NEFEM IGA + Standard FE
Storage Tanks • plant with industrial storage tanks • stability under earthquakes (seismic loading) is an important design requirement diamond-shaped pressure bumps elephant footing buckling
Storage Tanks Requirements Methods • free-surface flow • interface tracking • deformable domain with • NURBS-enhanced arbitrary wall shapes finite elements • arbitrary boundary description through • deformable structure NURBS • Isogeometric Analysis for elastodynamics
3D Cylindrical Tank • Liquid-filled storage tank subjected to seismic excitation thickness Navier - slip BC diameter height excitation • Linear FEM for fluid, IGA shell for solid
3D Cylindrical Tank • Fluid Flow Field • Fluid velocity (left) and mesh deformation (right) in rigid tank: • Notice mesh nodes sliding across the NURBS edge
3D Cylindrical Tank • Structural Solution • Structural deformation without excitation maximal and minimal deformation equilibrium vertical displacement • Excitation started after equilibrium reached
3D Cylindrical Tank • FSI Solution surface grid point height in rigid (red) and flexible (blue) tank
Mesh Generation for Space-Time • Hughes and Hulbert (1988), Maubach (1991): • 3D case presents obvious difficulties … 33
Prisms versus Simplices • Standard prismatic extensions of 3D elements: 3d6n 4d8n • Simplex elements required for fully-unstructured meshes: 3d4n 4d5n 34
Prisms to Simplices • Start with prismatic 4D mesh • Add temporal refinement nodes along the spines: in2 in4 jn2 jn3 in3 kn2 kn2 in2 jn2 in1 in1 jn1 jn1 kn1 kn1 • Perturb temporal coordinates: • Delaunay triangulation using, e.g., qhull package 35
Final Steps • Sliver elimination: d2 d1 • Temporally refined 3D space-time mesh for 2D cylinder: 36
Test Case: Gaussian Hill in 2D and 3D • Initial profile: advected according to: with and • Spatial meshes in 2D and 3D: z y x 0.01 y 0.00 x 0.00 0.01 0.99 1.00 • Numerical solutions at compared to exact solution 37
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