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)