Automatic hp -adaptivity for inductively heated incompressible flow - - PowerPoint PPT Presentation

Automatic hp-adaptivity for inductively heated incompressible flow of liquid metal

Lenka Dubcov´ a, Ivo Doleˇ zel, Pavel ˇ Sol´ ın S¨ udostdeutsche Kolloquium zur Numerischen Mathematik Dresden May 1, 2010

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 1 / 24

Inductively heated flow

couples 3 physics:

◮ electromagnetic field ◮ flow fields ◮ temperature

examples:

◮ heating of a molten metal by an inductor ◮ electromagnetic pumping ◮ electromagnetic stirring of molten metals

usually hard to solve numerically

◮ strongly coupled nonlinear nonstationary process ◮ all physics exhibit different behaviour ◮ several monocodes must be combined Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 2 / 24

Outline

1

Mathematical model

2

Numerical solution of the problem

3

Automatic adaptivity

4

Coupling strategies

5

Numerical results

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 3 / 24

Model problem

electrically conductive liquid (molten sodium) in a basalt pipe time-variable magnetic field generated by a harmonic current liquid is heated up by Joule losses liquid is accelerated by corresponding Lorentz forces axisymmetric arrangement

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 4 / 24

Mathematical model - magnetic field

curl(curl A) + iωγµA − γµ u × curl A = µJext, µ – magnetic permeability γ – electric conductivity ω – angular frequency u – flow velocity vector A – phasor of magnetic vector potential Jext – phasor

• f

the external harmonic current density in the inductor In the axisymmetric arrangement −∂2Aϕ ∂z2 − ∂2Aϕ ∂r 2 − ∂ ∂r Aϕ r

• + iωµγAϕ

+γ µur(Aϕ r + ∂Aϕ ∂r ) + γ µuz ∂Aϕ ∂z = µJext,ϕ. with boundary condition Aϕ = 0 on artificial boundary ∂Ω sufficiently far-away.

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 5 / 24

Mathematical model - magnetic field

heating is caused by the Joule losses pJ = |J|2 γ = | − iωγ A + γ u × curl A|2 γ liquid is accelerated by local (volumetric) Lorentz forces (fr, fz) f = J × B = J × curl A

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 6 / 24

Mathematical model - temperature

div(λ∇T) − ρc ∂T ∂t + u · ∇T

• = pJ

λ – thermal conductivity ρ – specific mass c – specific heat u – flow velocity vector T – temperature pJ – Joule losses In axisymmetric arrangement λ ∂2T ∂r 2 + 1 r ∂T ∂r + ∂2T ∂z2

• − ρ C

∂T ∂t + ur ∂T ∂r + uz ∂T ∂z

• = pJ

temperature field is calculated only in the liquid and pipe BC: along the pipe surface – convection, temperature of liquid at the inlet is known, axis of symmetry – zero Neumann condition

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 7 / 24

Mathematical model - flow field

The incompressible flow obeys the Navier-Stokes equations: ρ ∂ur ∂t + ur ∂ur ∂r + uz ∂ur ∂z

• + ν

∂2ur ∂r 2 + 1 r ∂ur ∂r − ur r 2 + ∂2ur ∂z2

• + ∂p

∂r = fr ρ ∂uz ∂t + ur ∂uz ∂r + uz ∂uz ∂z

• + ν

∂2uz ∂r 2 + 1 r ∂uz ∂r + ∂2uz ∂z2

• + ∂p

∂z = fz ∂ur ∂r + ur r + ∂uz ∂z = 0 ur – radial velocity uz – axial velocity p – pressure ν – dynamic viscosity ρ – specific mass f – Lorentz forces gravitational force acting on particles is neglected BC: on internal wall velocity vanishes, ur vanishes along the z-axis, velocity profile at the inlet is known, on the outlet “do nothing” condition is satisfied: (p − pref) n + 1 2(u · n)− u = ν ∂u ∂n

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 8 / 24

Numerical solution of the problem

Rothe’s method for time discretization Higher-order finite element method (hp-FEM) for spatial discretization

◮ different sizes (h) of elements and different pol. degrees (p) on elements ◮ hp-FEM is capable of exponential convergence

Each field is solved in a geometrically different domain Accuracy of the solution is controlled via hp-adaptive algorithm Individual meshes for all physics – no data-transfers between meshes Nonlinearities are resolved using picard’s iterations (future plans: Newton’s method)

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 9 / 24

Computational domains, initial meshes

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 10 / 24

Automatic Adaptivity - hanging nodes

Motivation arbitrary-level hanging nodes technique by recursive algorithm ⇒ no mesh regularization required basis functions satisfy conformity requirements (continuity) ⇒ no changes in discrete formulation ⇒ no additional constraints ⇒ positive-definiteness of stiffness matrix preserved

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 11 / 24

Automatic Adaptivity in hp-FEM

Adaptivity on meshes with arbitrary-level hanging nodes is local. Elements adjacent to refined element are not affected. Forced refinements avoided. ⇒ Error is reduced optimally Standard error estimaters are not sufficient. Many options how to refine an element. Information about shape of the error needed. ⇒ Reference solution

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 12 / 24

Automatic Adaptivity - Algorithm

1

calculate the solution on the current mesh

2

calculate reference solution on refined mesh

3

calculate the error on each element and total error

◮ if the total error is less than prescribed tolerance → stop 4

sort all elements by their error and mark certain number for refinement

5

for each element find the best refinement from many options:

◮ For each candidate (functions spaces Sk): 1

project the reference solution onto space Sk = Span(ϕj) X

i

ai (ϕi, ϕj) = (uref , ϕj), ∀ϕj ∈ Sk

2

compute error between projected solution and reference solution

◮ Find the best candidate – large decrease in error

– reasonable increase in DOFs

6

recalculate the solution

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 13 / 24

Magnetic potential - Convergence

0.1 1 10 100 2000 4000 6000 8000 10000 12000 relative H1 error [%] DOFs hp-FEM h-FEM, p = 2 h-FEM, p = 1

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 14 / 24

Magnetic potential - comparison of meshes

hp-mesh: 1932 Degrees of freedom, H1 error 0.38% h-mesh (p = 2): 11262 Degrees of freedom, H1 error 0.72%

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 15 / 24

Coupling - Multi-Mesh Assembling

All fields strongly coupled

◮ monolithic discretization ◮ fields exhibit different behaviour – different meshes ◮ data exchanges between physical fields

Multi-mesh assembling technique

◮ very coarse master mesh shared by all components (some elements

missing)

◮ meshes can be refined individually – individual adaptive processes starting

from a coarse mesh

◮ union mesh containing all refinements from all components is never

physically constructed

◮ multi-mesh allows also dynamic meshes for nonstationary problems

(meshes travelling with solution features)

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 16 / 24

Individual behavior for all physics

Mesh for magnetic potential Mesh for flow field

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 17 / 24

Assembling over Multiple Meshes

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 18 / 24

Numerical results

Magnetic potential A

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 19 / 24

Numerical results

Magnetic induction B

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 20 / 24

Numerical results

Lorentz forces Joule looses

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 21 / 24

Numerical results

Velocity field Temperature

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 22 / 24

Conclusions

Monolithic discretization of coupled magnetohydrodynamic problem performed Individual features of each physics resolved using unequal meshes Meshes dynamically change with the solution Meshes adapted automatically, accuracy controlled Numerical solution obtained by our adaptive software Hermes2D (hp-adaptivity, arbitrary-level hanging nodes, multi-mesh technology, dynamical meshes) For download, tutorial, and examples visit http://hpfem.org/

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 23 / 24

Thank you for your attention.

Lenka Dubcov´ a (MFF UK) hp-adaptivity for inductively heated flow Dresden, May 1, 2010 24 / 24