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: www.scorec.rpi.edu/AOMD
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 yi 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 www.stlport.org
– 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– L2-Orthogonal, p<15– Crouzeix-Raviart– Enriched X-fem basis,
to come...
Examples of Function Spaces
Examples of Function Spaces
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 Fi acting on yi
– Contraction :: between operator results produces a scalar
• Particular case: bilinear density
– Linearisation of the general case
– Representation: dim(L1)dim(L2) 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 = 105
– 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 = (aez ) , 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
0.00E+00
1.00E-01
2.00E-01
3.00E-01
4.00E-01
5.00E-01
6.00E-01
2.00E-03 2.90E+00 4.94E+00 8.00E+00 1.41E+01
Current
• Current density j ez= 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)