Trellis: A Framework for Adaptive Numerical Analysis Based on Multiparadigm Programming in C++ Jean-Francois Remacle, Ottmar Klaas and Mark Shephard Scientific Computation Research Center Rensselaer Polytechnic Institute

Scope of the presentation Aim of Trellis: find y(x,t)  Y(  ) such that Trellis modular design –A parallel adaptive mesh library, takes care of  –A discretization library, takes care of Y(  ) –A core library, takes care of f –A solver library for algebraic systems

Linearization We usually need a linearization of The aim of Trellis is to provide M, C, K and f Trellis interacts with external solvers like PetSC or DASPK

Parallel Algorithm Oriented Mesh Data-structure Aim of AOMD: providing services to mesh users –Basic services, iterators to various ranges of entities, iterators on adjacencies, input-output... –Geometry based analysis, relation mesh to model is maintained –Support of dynamic mesh adjacencies –Parallel services: message passing and load balancing capabilities Open source:

Parallel Algorithm Oriented Mesh Data-structure AOMD extensions –Conforming (anisotropic) and non-conforming adaptive capabilities, available in parallel –Calculus toolkit, integration, curvilinear elements and their mappings (Bezier, Lagrange) –Computational Geometry toolkit (Octree, ADT) –Interface to solid modelers (e.g. Parasolid), vertex snapping –TSTT interface

Example of AOMD capabilities Parallel Adaptive D.G. Solver Load Balancing High order

The Discretization Library Representing components y i of a tensor field y With –A functional basis: –Coefficients (DOF’s):

Degrees of Freedom Aim: flexibility –parallel, h-p adaptive –multiple fields –multi-methods, multi-physics Representation –constant part, DofKey –variable part DofData –The idea of a general DOF representation is far more important than the implementation

Degrees of Freedom Manager Design –Contains all degrees of freedom –Container: std::map or std::hash_map if available e.g. at –Singleton pattern i.e. one only instance in the program –Parallel capabilities

Function Spaces Provide C and N of Hierarchy of classes Available: –Hierarchical, p<15 –Lagrange, p<10 –L 2 -Orthogonal, p<15 –Crouzeix-Raviart –Enriched X-fem basis, to come...

Examples of Function Spaces

Linear operators Aim: take tensor components and build a tensorial representation –A field with 3 component may be a covariant vector, a vector or 3 scalars (Euler 1-D e.g.) We call with and we have the expansion

Examples of Operators

Scalar product, dual pairing Consider –Operators F i acting on y i –Contraction :: between operator results produces a scalar Particular case: bilinear density –Linearisation of the general case –Representation: dim( L 1 )  dim( L 2 ) matrix (not tensor!)

Some other densities Linear Form –Representation: column vector, dim( L ) Trilinear Form –Automatic linearization

Contributors Matrix Contributor Representation

Implementation Generic: Template parameters: operators, material law –Efficient (inlining) and very general –An operator that computes must exist –That type safety helps developer not to make mistakes

Algebraic and ODE Solvers Interfaces –to serial linear system solvers: Sparskit, IML,… –to parallel solvers: PetSC, SuperLU –to ODE solvers: PesSC, DASPK Internal Trellis solvers –Newton, BFGS –classical ODE solvers: CN, RK...

Navier-Stokes in 4 lines of code Constraints: fix components to a value

Channel flow, Re=625

Natural convection (time dependant)

Heated from below Natural convection –Ra = 10 5 –Semi-implicit

Magneto-hydrodynamics Tilt instability –Dipole of current (  b) oppositely directed (repelling forces) in a constant b ( confining field) –dipole starts turning in order to align the external magnetic field (minimize magnetic energy) –repelling effect is able to expel vortices –Instability: kinetic energy grows like exp(  t) with  = O(1.4)

Magneto-hydrodynamics Characterization of ker (div) –From “inside”, with potentials –From “outside” with Lagrange multipliers (pressure and electric potential). SUPG stabilization (modified upwind operators b’ and ’ )

Results for a Tilt instability –Magnetic potential a with b =  ( ae z ), p=1 and p=3 (v and b)

Results for the Tilt instability Magnetic Flux Density and Velocity

Results for the Tilt instability Kinetic energy vs. time

Current Current density j e z =  b Oscillations observed –SUPG Stabilization for higher order (p=3) may not be sufficient

Conclusions Multiparadigm design in C++ –Higher level objects, Object Oriented –Kernel, Generic Trellis –Operator based, linear and non-linear –Complex physics easy to implement Future –Parallel (in progress) and adaptive (in progress)