A Mesh-less method for solutions of the Stefan Problem Vaughan Voller If A man knows he is going to be hanged tomorrow it concentrates the mind wonderfully— Dr Samuel Johnson 1709-1784 Preliminary Results

Essence of a Numerical Solution Cover Domain with Field of NODES with locations xi 2. Unknowns are nodal values of dependent variable 3. By appropriate APPROXIMATIONS of Governing Equation Obtain a SET Of Discrete Algebraic equations That relate the nodal value at P to Values at the neighbors Process is facilitated by DATA Structure of node points P

The most Straight Forward Is a STRUCTURED GRID OF NODES Approximation Process is facilitated by-- DATA Structure of node points P Location of any node point given a Row, Column index The most Straight Forward Is a STRUCTURED GRID OF NODES If Grid is “square” immediate application of Taylor Series Row i Col. j

A more flexible approach is use Nodes to Define Approximation Process is facilitated by-- DATA Structure of node points A more flexible approach is use Nodes to Define An UNSTRUCTURED MESH OF ELEMENTS n 3 In Each Element obtain a Continuous Approximation Use this approximation with governing equation e.g. in CVFEM--use 2 1

Approximation Process is facilitated by-- DATA Structure of node points The most flexible approach is to have no mesh ---CLOUDS of Near Neighbor Nodes Very limited restriction on placement of nodes But may have to be inventive In arriving at sound discrete equations

Easy to adapt Simplistic Summary Increasing Flexibility GRID---Structure MESH---FEM “CLOUD”---SPH More Complex approximation Less Efficient solution Ax=b — Easy to Fit Geometry Could be Difficult to adapt “Poor” approximation Even Less Efficient solution Ax=b — Very Easy to Fit Geometry Easy to adapt Easy Approximation Efficient solution Ax=b — Difficult to Fit Geometry

Corrective Smooth Particle Method CSPM -related to SPH Chen et al IJNME 1999 P Taylor Series about node P Multiply by Weighting Factor associated with node P

Properties of W Symmetric about point P Finite region of support Differentiable P 2hp 1.5 > a > 0.5 Char. Length multiple of nearest neig. distance r=R/h

similar manipulations for first derivatives Corrective Smooth Particle Method CSPM -related to SPH Chen et al IJNME 1999 P Taylor Series about node P Multiply by Weighting Factor associated with node P Integrate over support similar manipulations for first derivatives

Integrate numerically Corrective Smooth Particle Method CSPM -related to SPH Chen et al IJNME 1999 Integrate numerically Using particles as integration points P 2hp 1.5 > a > 0.5 If Rnb is the radial distance to the neighbors of P r=R/h Weak point --- Critical Feature

Application to The Stefan Problem--- General problem of Interest Solid T < Tm b.) time t > 0 Liquid T > Tm liquid-solid interface T = Tm T=Ta >Tm n At time t>=0 apply a fixed temperature T=Ta >Tm to a patch of the boundary so as to cause a melt region that grows with time Initial state Insulated region containing solid at Temperature T; < Tm (melt temp) a.) time t = 0 Solid Ti < Tm Objective track the movement of the melt front Gmelt

Governing Equations Assume constant density and specific heat c — but jump in conductivity K. with time scale and space scales Dimensionless temperature And dimensionless grouping , DH-latent heat, St-Stefan number, L dim. Lat. heat liquid-solid interface T = 0 n Two-Domain Stefan Model Diffusive Interface—Single Domain T=0 g=0 g=1 Phase change occurs smoothly across A “narrow” temperature range

Particular Versions Two Dimension one-dimension s(t) K=0.25 T= 0.5 Melting of unit cylinder, initially at phase Change temperature. Has analytical solution Solve in Cartesian Check with fine grid FD solution Using radial symmetry

CSPM Solution Use CSPM approximation of derivative twice Backward Euler (explicit) in time P 2hp Data Structure Global Number nodes 1-----n Identify “cloud” of neighbors associated with each node---- make a list of nodes (global numbers) that fall within a radial distance 2hP Can be manipulated into general form

Dt = 0.002 One-D solution Front Movement K = 0.25 T=1 T=-0.5 o o o CSPM Analytical h = 0.033-0.05 Dt = 0.002 Temperature Profile at time t =2.8 Temperature Histories At ref points Plateau at phase change temp. A feature in all fixed grid solutions

Ti = 0 T=1 For the 2-D problem Need to consider a means of placing out points Two Methods

Discretization: Two steps Put points along boundary of domain— with equal arc spacing Make a structured mesh with spacing SPH Nodes are a List i=1 to nbound Then Lay boundary Mesh Over Structure Mesh Add structured points to SPH node List if they are a distance INSIDE boundary

Black Dot: structured mesh point excluded from SPH list Blue Circle: Boundary Polygon Red Circle: structured mesh point Included in SPH list IDEA stolen from Immersed Boundary Methods of Fotis Sotiropoulos Gives a reasonably Well spaced grid Easy to identify “node Clouds” h = Dr

Results Radial Movement of Front with Time Fine grid radial symmetry solution Melt pattern at An intermediate times

Also works when points are “Jostled”

“Patchy” Unstructured Mesh Delaunay Good Results But sensitive to choice of h P

So far very basic calculations But they show promise— Need to look at Adaptivity Lagrangian

Moving Boundaries in Sediment Transport Two Sedimentary Moving Boundary Problems of Interest Shoreline Fans Toes

Badwater Deathvalley Examples of Sediment Fans Moving Boundary 1km How does sediment- basement interface evolve

Melting vs. Shoreline movement An Ocean Basin

Enthalpy Sol. A 2-D Front -Limit of Cliff face Shorefront But Account of Subsidence and relative ocean level Enthalpy Sol. land surface shoreline ocean h(x,y,t) x = s(t) G(x,y,t) Solve on fixed grid in plan view bed-rock y Track Boundary by calculating in each cell x y

A 2-D problem Sediment input into an ocean with an evolving trench driven By hinged subsidence

With Trench

Need to account for Interaction with channels which can avulse

Easy to adapt Increasing Flexibility GRID---Structure MESH---FEM “CLOUD”---SPH Easy Approximation Efficient solution Ax=b — Difficult to Fit Geometry More Complex approximation Less Efficient solution Ax=b — Easy to Fit Geometry Could be Difficult to adapt “Poor” approximation Even Less Efficient solution Ax=b — Very Easy to Fit Geometry Easy to adapt