ParaPhase: Spacetime parallel adaptive simulation of phase-field - - PowerPoint PPT Presentation
ParaPhase: Spacetime parallel adaptive simulation of phase-field models on HPC architectures Oliver Sander 29. 11. 2016 ParaPhase Spacetime parallel adaptive simulation of phase-field models on HPC architectures Heike Emmerich
◮ Modelling technique for problem with moving interfaces ◮ Sharp interfaces are smeared out over a finite width ǫ
◮ Demixing of alloys ◮ Solidification dynamics ◮ Viscous fingering ◮ Fracture formation [Keip, Uni Stuttgart] ◮ Liquid-phase epitaxy [Emmerich, Uni Bayreuth]
◮ Very localized features ◮ High grid resolution necessary ◮ Key phenomena may emerge only for large domains and simulation times
◮ Nonlinear and nonsmooth equations ◮ Explicit methods: very short time steps ◮ Implicit methods: Newton-methods work badly, if they work at all
◮ Adaptive Finite-Element methods for phase-field demixing problems
◮ Energy functional, e.g.,
◮ Gradient flow
◮ We use implicit time discretization, e.g.,
◮ Sequence of non-quadratic minimization problems
c
k (c)
◮ Non-smooth parts, but block-separable
m
◮ Frequently convex, or at least close to convex
◮ Generalizes standard multigrid to nonsmooth convex minimization
◮ Provable global convergence for strictly convex problems ◮ No regularization parameters ◮ Convergence rates independent of the mesh resolution
◮ MPI-parallel implementation
◮ Implement TNNMG nonsmooth multigrid for a model of brittle fracture
◮ Model developed and analyzed by Christian Miehe, Stuttgart ◮ Previously: Operator splitting ◮ Extend the convergence proof to certain biconvex functionals
◮ Unknowns: displacement u : Ω → Rd, fracture phase field d : Ω → [0, 1] ◮ Elastic bulk energy density ψ(u) = λ 2 (tr ∇symu)2 + µ tr(∇symu)2 ◮ Regularized crack surface density γ(d) = 1 2l(d2 + l2∇d2) ◮ Total energy
◮ Time evolution of u and d are determined by minimization principle
˙ u∈W ˙
u
˙ d∈W ˙
d
notch
◮ State-of-the-art solution scheme (Operator split):
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
Operator split TNNMG 40 × 40 60 × 60 80 × 80 TNNMG Operator split grid resolution [# elements]
◮ TNNMG and operator split perform at the same speed for small problems ◮ With increasing grid resolution, the operator split method needs more and
◮ Iteration numbers for the nonsmooth multigrid method remain bounded
◮ Relevant engineering problems demand a fine grid to properly resolve
◮ Uniform grids too expensive −
◮ Previous work: adaptive phase field simulations for demixing [Gr¨
◮ Nonsmooth multigrid in an MPI-parallel situation for nonlinear/nonsmooth
◮ Dynamic load balancing
◮ Dynamic load-balancing will not scale to large processor numbers ◮ Therefore: parallelize in time!
◮ Parallel-in-time method ◮ Compute fine and coarse defect problems in parallel ◮ Related to space–time multigrid ◮ Expected to integrate nicely with nonsmooth multigrid method TNNMG
coarse sweep fine sweep coarse comm. fine comm.
predictor
Distributed and Unified Numerics Environment
◮ Separate libraries for
◮ Grids ◮ Shape functions ◮ Linear algebra ◮ etc.
dune-common dune-geometry dune-grid dune-localfunctions dune-istl dune-grid-glue dune-foamgrid dune-fem dune-pdelab dune-mc ... ...
Application 1 Application 2
◮ A great common platform for joint development!
◮ Refinement/coarsening ◮ Different refinement strategies
◮ Distributed grids ◮ MPI communication ◮ Dynamic load balancing
◮ Work in progress
◮ C++ implementation of the parallel full approximation scheme in space
◮ Time parallel algorithm for solving ODEs and PDEs ◮ Contains basic implementations of the spectral deferred correction (SDC)
◮ Transparent development through Github:
◮ MPI-parallel version of TNNMG ◮ Dynamic load-balancing
◮ Discontinuous-Galerkin discretizations ◮ Increase arithmetic density ◮ Towards GPU programming
◮ Combine PFASST and FE and multigrid ◮ Apply to simple phase-field equations
◮ Test the TNNMG method for the brittle-fracture model ◮ Combine with grid adaptivity ◮ Extend to ductile materials