Automatic hp -adaptivity for inductively heated incompressible flow of liquid metal zel, Pavel ˇ Lenka Dubcov´ a, Ivo Doleˇ 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 Mathematical model 1 Numerical solution of the problem 2 Automatic adaptivity 3 Coupling strategies 4 Numerical results 5 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 ) + i ωγµ A − γµ u × curl A = µ J ext , µ – magnetic permeability γ – electric conductivity ω – angular frequency u – flow velocity vector A – phasor of magnetic vector J ext – phasor of the external potential harmonic current density in the inductor In the axisymmetric arrangement − ∂ 2 A ϕ ∂ z 2 − ∂ 2 A ϕ ∂ r 2 − ∂ � A ϕ � + i ωµγ A ϕ ∂ r r + γ µ u r ( A ϕ + ∂ A ϕ ∂ A ϕ ∂ r ) + γ µ u z = µ J ext ,ϕ . r ∂ z with boundary condition A ϕ = 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 p J = | J | 2 = | − i ωγ A + γ u × curl A | 2 γ γ liquid is accelerated by local (volumetric) Lorentz forces ( f r , f z ) 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 � ∂ T � div ( λ ∇ T ) − ρ c ∂ t + u · ∇ T = p J – thermal conductivity – specific mass λ ρ c – specific heat u – flow velocity vector T – temperature p J – Joule losses In axisymmetric arrangement � ∂ 2 T ∂ r + ∂ 2 T ∂ r 2 + 1 ∂ T � � ∂ T ∂ T ∂ T � λ − ρ C ∂ t + u r ∂ r + u z = p J ∂ z 2 r ∂ z 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: � ∂ 2 u r r 2 + ∂ 2 u r � ∂ u r ∂ u r ∂ u r � ∂ r 2 + 1 ∂ u r ∂ r − u r � + ∂ p ρ ∂ t + u r ∂ r + u z + ν ∂ r = f r ∂ z 2 ∂ z r � ∂ 2 u z ∂ r + ∂ 2 u z � ∂ u z ∂ u z ∂ u z � ∂ r 2 + 1 ∂ u z � + ∂ p ρ ∂ t + u r ∂ r + u z + ν ∂ z = f z ∂ z 2 ∂ z r ∂ u r ∂ r + u r r + ∂ u z ∂ z = 0 u r – radial velocity u z – 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, u r vanishes along the z -axis, velocity profile at the inlet is known, on the outlet “do nothing” condition is satisfied: ( p − p ref ) 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 calculate the solution on the current mesh 1 calculate reference solution on refined mesh 2 calculate the error on each element and total error 3 ◮ if the total error is less than prescribed tolerance → stop sort all elements by their error and mark certain number for refinement 4 for each element find the best refinement from many options: 5 ◮ For each candidate (functions spaces S k ): project the reference solution onto space S k = Span ( ϕ j ) 1 X a i ( ϕ i , ϕ j ) = ( u ref , ϕ j ) , ∀ ϕ j ∈ S k i compute error between projected solution and reference solution 2 ◮ Find the best candidate – large decrease in error – reasonable increase in DOFs recalculate the solution 6 Lenka Dubcov´ a (MFF UK) hp -adaptivity for inductively heated flow Dresden, May 1, 2010 13 / 24
Magnetic potential - Convergence 100 hp-FEM h-FEM, p = 2 h-FEM, p = 1 relative H 1 error [%] 10 1 0.1 0 2000 4000 6000 8000 10000 12000 DOFs Lenka Dubcov´ a (MFF UK) hp -adaptivity for inductively heated flow Dresden, May 1, 2010 14 / 24
Magnetic potential - comparison of meshes 1932 Degrees of freedom, H 1 error 0.38% hp -mesh: 11262 Degrees of freedom, H 1 error 0.72% h -mesh ( p = 2): 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
Recommend
More recommend
