Moving Fluid Elements

Transcription

1 Moving Fluid Elements Mike Baines ICFD25, September 28 Matthew Hubbard, Peter Jimack, UoR students

2 Some history 983 ICFD formed emphasising algorithmic development introduced a series of workshops (3 a year = 75 so far!) established a triennial Conference series on Numerical Methods for Fluid Dynamics, which continues to flourish stimulated a training program for young researchers, whose success is exemplified here introduced an early form of knowledge transfer, through industrial contacts and overseas visitors

3 Moving mesh methods This talk is about velocity-based moving mesh finite element methods for the numerical solution of time-dependent partial differential equations and systems; with applications to moving boundary problems, blow-up problems and the tracking of singularities, among others.

4 Outline Moving Finite Elements Moving Fluid Elements Scale-invariance and similarity Examples and Discussion

5 FIXED finite elements, for u t = Lu Residual: U t LU (U is finite element approximation) Expansion: U t = j U j φ j (x) (φ i are basis functions) Least squares Residual minimisation is equivalent to min U i b a {U t LU} 2 dx b a φ i (x){u t LU}dx = Galerkin

6 Moving finite elements for u u x ẋ = Lu Residual: U U x Ẋ LU Expansions: U = j U j φ j, Ẋ = j Ẋ j φ j Least squares Residual minimisation (Miller) min U i,ẋ i b a { U U x Ẋ LU} 2 dω, equivalent to { b a φ i { U U x Ẋ LU}dx = Double- b a ( U x)φ i { U U x Ẋ LU}dx = Galerkin

7 Remarks on Moving Finite Elements (MFE) Implemented using piecewise linear approximation of U and X. General purpose, multidimensional, capable of high resolution. Contains inherent singularities when nodes collinear. Too reliant on parameters needed to control the regularisation. For certain first order equations (of the form u t + H(x, u x ) = ) MFE generates a finite dimensional Hamiltonian structure for the nodal positions and element slopes (Baines 994).

8 Non-conservative and conservative forms. u = u t + u x ẋ u u x ẋ Lu = Chain rule Non-conservative form... d dt d dt udx = u t dx + [uẋ] b(t) udx [uẋ] b(t) = Ludx Leibnitz rule Conservative form

9 Non-conservative and conservative weak forms φ i { U U x Ẋ LU}dx = Non-conservative form... d dt φ i Udx φ i (UẊ ) x dx φ i LUdx = Conservative form - taking the test functions φ i to move with Ẋ, i.e. t φ i + Ẋ x φ i =

10 Moving (Mesh) Fluid (Finite) Elements (MMFE) We start from the conservative weak form, viz. d dt W i Udx φ i (UẊ ) x dx = φ i LUdx... A second equation is needed to move the nodes. Set φ i Udx = constant in time φ i (UẊ ) x dx = φ i LUdx (LAGRANGIAN Conservation) (EULERIAN Conservation)

11 Lagrangian conservation Evolution of a constant local mass

12 MMFE Algorithm At each time step:. Use Eulerian conservation to obtain the velocity Ẋ, φ i (UẊ ) x dx = φ i LUdx 2. Advance X in time 3. Use Lagrangian conservation for the solution U, φ i Udx = its initial value

13 Remarks on MMFE Implemented using piecewise linear approximation for U, X. Generalises to multidimensions. No collinearity singularity (as in MFE). Extends to conservation of monitor function integrals. General purpose, with the added bonus of asymptotically approximating self-similar solutions.

14 Scale-invariance A PDE is scale-invariant if there exists a scaling parameter λ and scaling indices β, γ such that the mapping leaves the equation unaltered. t λt, x λ β x, u λ γ u To achieve this property all variables must be interdependent. Numerically this demands solution-adaptive methods which move the spatial mesh in time. Fixed mesh methods cannot do this. Both MFE and MMFE are scale-invariant moving-mesh finite element methods. As we shall see MFE mimics self-similarity.

15 Self-similarity A self-similar solution is a relation between the (scale-invariant) similarity variables x/t β, u/t γ, of the form u ( x ) t γ = f t β (exhibiting a scaling symmetry), where the function f satisfies an ODE (or at least a reduced form of the PDE). Self-similar solutions play an important role in identifying geometric properties of PDEs, such as intermediate asymptotics and comparison theorems.

16 u u Porous Medium Equation For example, the nonlinear porous medium equation (PME) u t = (u n u x ) x (n ), with u = at the boundaries, has a compact self-similar source solution for u of the form: Porous Medium Equation: n= Porous Medium Equation: n= n =.7 n > x x

17 Propagation of L 2 projections of self-similar solutions MMFE is based on the Lagrangian conservation principle φ i Udx is time-invariant From this it can be deduced that under spatial similarity the L 2 projection U of a self-similar solution into the space spanned by the φ i is carried in time.

18 Radially-symmetric porous medium equation The radially-symmetric porous medium equation (PME) u t = r (run u r ) r (n ) with u = on a (moving) boundary has an exact self-similar source solution of the form u ss = A ( r ) 2 { t β t β where β = 2/(2 + 2n) and A is a constant. } /n

19 Results Initial and final profiles for PME solutions for n = 3. Porous Medium Equation: n=3 Porous Medium Equation: n=

20 Accuracy - against self-similar solutions Porous Medium Equation: n = Porous Medium Equation: n = LOG(error) LOG(error) Recovered solution ALE solution Recovered mesh 3 Recovered solution ALE solution Recovered mesh 5.5 ALE mesh slope = 2 ALE mesh slope = LOG(dx) LOG(dx) L 2 errors in one dimension for the radially symmetric self-similar to the PME problem for n = and 3 (with equispaced initial data).

21 Numerical invariance of the similarity variables Porous Medium Equation: n=3.5 Porous Medium Equation: n= r/t β u/t γ t t

22 A numerical comparison property PME: m = ; t =..2 PME: m = ; t = u.5 u x.5.5 x A random perturbation applied to the initial solution and its evolution compared with two self-similar solutions, for n =.

23 A comparison property of the region of support PME: m = 4; t =..75 PME: m = 4; t = y y x x A perturbation of the initial region of support and its evolution compared with that of two self-similar solutions, for n =.

24 Waiting time Three profiles showing the waiting time experienced by the boundary (for certain initial data) before it moves. The middle profile shows the moment at which the boundary moves. Porous Medium Equation: n=2 Porous Medium Equation: n=2 Porous Medium Equation: n= (Waiting times are due to the passage of time that a shock in the velocity takes to travel to the boundary.)

25 u Moving profiles for a heat equation with a sink term The Crank-Gupta equation u t = u xx with u = u x = at the moving boundary and u = + e (t ) at x = (taken from an exact solution) Oxygen Diffusion/Absorption t=. t=.2 t=.4 t=.6 t=.8 t=.97 Exact x

26 Two phase flow with a moving internal boundary A solidification problem governed by the radially symmetric heat equation with a Stefan condition at the (radially symmetric) moving boundary Two Phase Stefan Problem Two Phase Stefan Problem

27 y y Euler equations Results for a converging and diverging cylindrical shock x x

28 Comments and extensions The method targets the mesh motion, which is governed by an elliptic equation. The solution appears as an algebraic by-product. No regularisation is needed. The method extends to systems of equations (in conjunction with a solver). The underlying conservation principle can be cast in terms of a general monitor function, allowing physically or geometrically based mesh velocities to be generated.

29 Applications The method gives efficient numerical solutions to various nonlinear time-dependent PDE problems with implicit moving boundaries or singularities, for example: Moving-boundary problems, e.g. glaciers, biological growth models Blow-up problems (reaction-diffusion, NLS) Two-phase problems with an internal boundary The Euler equations of gasdynamics (in conjunction with a solver) The shallow water equations (in conjunction with a solver)

30 Summary Based on weak forms of Lagrangian and Eulerian conservation laws for the dependent variable, the MMFE method is a general scale-invariant velocity-based moving-mesh finite-element method which has wide application to time-dependent PDE problems. The algorithm generates an approximation to the self-similar velocity, approximately carrying the L 2 projection of self-similar solutions in time. The method has been validated on a number of moving-boundary problems with exact solutions, including nonlinear diffusion, diffusion-reaction and Stefan problems, and can be used for systems of equations in conjunction woth a solver.

31 Acknowledgements and References Chris Budd and co-workers at Bath: Weizhang Huang, Bob Russell and co-workers: References: M.J.Baines, M.E.Hubbard and P.K.Jimack, A moving mesh finite-element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries, Appl. Numer. Math., 54:45-469, 25. M.J.Baines, M.E.Hubbard, P.K.Jimack and A.C.Jones, Scale-invariant moving finite-elements for nonlinear partial differential equations in two dimensions, Appl. Numer. Math., 56:23-252, 26.