Nektar++: A Progress Report Spencer Sherwin, Chris Cantwell, Dave - - PowerPoint PPT Presentation

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Nov 14, 2023 224 likes •588 views

Nektar++: A Progress Report Spencer Sherwin, Chris Cantwell, Dave Moxey, Mike Kirby Department of Aeronautics, Imperial College London SCI Institute, University of Utah Outline What are we doing? Optimizing our implementations

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Nektar++: A Progress Report

Spencer Sherwin, Chris Cantwell, Dave Moxey, Mike Kirby

Department of Aeronautics, Imperial College London SCI Institute, University of Utah

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Outline

  • What are we doing?
  • Optimizing our implementations
  • Variable p
  • Cloud computing
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Nektar++: An h to p finite element framework

h-type geometric flexibility

p-type exponential accuracy

Provide an unified interface to an open environment which blends high- and low-order finite element methods.

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Nektar++: www.nektar.info

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Nektar++: www.nektar.info

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Nektar++: www.nektar.info

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Nektar++: www.nektar.info

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Nektar++: www.nektar.info

  • 

  

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SLIDE 9

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

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



   

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   



 



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  • 

  

  • Homogeneous Expansions
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

  • 



  • Direct Stability Analysis

Complex Geometry LNS & DNS

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Hybrid Numbering & Mixed Parallelisation

Global numbering Local numbering Hybrid numbering



Spectral/hp plane parallelisation



Fourier parallelisation



Mixed parallelisation

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Outline

  • What are we doing?
  • Optimizing our implementations
  • From h to p efficiently
  • Variable p
  • Cloud Computing
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Evaluation Strategies for iterative solvers

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P

t tLocMat

Local Sum-factorisation Local Matrix Global Matrix 5 10 15 1 2 3 4 5 6 7

T TLocMat

=

=

2 x

  • mass matrix operator: ˆ

gi =

  • j

(Φi, Φj)Ω ˆ fj ∀i

Computational results

Vos, Sherwin, Kirby, JCP, 2010

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  • ⇥2u(x) λu(x) = f(x)

Local Sum-factorisation Local Matrix Global Matrix Optimal 100 200 300 400 500 600 RunTime P + 1 2 4 6 8 10 12 14 16 10 100 1000 10000 100000 5 10 15 20 25

  • minimal run-time

Error vs computational cost?

¡9 ¡Elements, ¡P=6 ¡ ¡

10−1

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P + 1 2 4 6 8 10 12 14 16 10 100 1000 10000 100000 5 10 15 20 25

  • ¡ ¡ ¡ ¡ ¡ ¡ ¡minimal ¡run-­‑time ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡-­‑ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡path ¡of ¡optimal ¡discretisations

P 1/h Sum-factorisation Local Matrix Global Matrix Optimal 1 5 10 15 5 10 15 20 25

Error vs computational cost?

10−1

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P 1/h 1 5 10 15 101 102 103 104 105 2 4 6 8 10 12 14 16 18 20

Computational results: Non smooth solution

  • ¡minimal ¡run-­‑time ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡-­‑ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡corner ¡problem ¡(error ¡= ¡10-­‑4)

⇥2u(x) λu(x) = f(x) Helmholtz ¡equation

irregular ¡solution

  • u(r, θ) = r

2 3 sin(2

3θ + π 3 )

xi = ✓ i nel ◆3

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P 1/h 1 5 10 15 101 102 103 104 105 2 4 6 8 10 12 14 16 18 20

Computational results

  • ¡minimal ¡run-­‑time ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡-­‑ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡path ¡of ¡optimal ¡discretisations

P 1/h Dof Sum-factorisation Local Matrix Global Matrix Optimal 1 5 10 15 2 4 6 8 10 12 14 16 18 20

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Outline

  • What are we doing?
  • Optimizing our implementations
  • Variable p
  • Cloud Computing
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Local Matrix: Hardware diversity

n=4 n=8 n=6 n=10 n=18 n=27 n=343

P=6

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6 5 4 3 2 1

0.5 1

  • 1
  • 0.5

0.5 1 x

  • 1
  • 0.5

0.5 1 y

Solution Variable P Variable P Error Fixed P error

Variable matrix size due to variable p

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How many parameters?

P1 P2 P3 P4 P5,P6 6 parameters P1 P2 P3 P4,P5 5 parameters

Hexahedral: 12 edges, 8 faces, 1 interior = 31 parameters Tetrahedral: 6 edges, 4 faces, 1 interior = 17 parameters

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Tensor product design

φpq(ξ1,ξ2) = hp(ξ1) hq(ξ2)

hp(ξ1)

ξ1 ξ2

hq(ξ2)

p q

p q

φpq(ξ1,ξ2) = ψp(ξ1) ψq(ξ2)

ψp(ξ1)

ξ1 ξ2

ψq(ξ2)

p q

p q

a a

a a
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Zero packing

P1 P2 P3 P4 P5,P6 6 parameters

φpq(ξ1,ξ2) = ψp(ξ1) ψq(ξ2)

ψp(ξ1)

ξ1 ξ2

ψq(ξ2)

p q p q a a a a

P1=3 P3=1 P2=2 P4=2 P5=3 P6=2 max(P1,P3,P5)=3 max(P2,P4,P5)=2 2 parameters

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Helmholtz example

P=4 P=8 P=12 P=17

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LibHPC

  • J. Cohen, P. Burovskiy, J. Darlington
  • Target software on multi-core, distributed &

hetrogeneous platforms

  • Run on Infrastructure-as-a-service (IaaS) clouds
  • Why?
  • Intermittent running makes access to HPC difficult
  • Scale resource beyond local capacity
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Heterogeneous Computing Infrastructure, e.g. Infrastructure as a Service Cloud Computing Platforms Disk Image Repository

Nektar++ Disk Image Nektar++ libhpc Job Manager Nektar++ Disk Image Developer Builds image Published to repository

Libhpc Deployer User

User Job Specification

Nektar++ Virtual Machine Disk image deployed to execution platform Deployer requests startup of cloud resources Nektar++ Virtual Machine Nektar++ Virtual Machine Nektar++ Virtual Machine Nektar++ Virtual Machine Communication between VM instances for parallel computations

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) (1) 0.5 1 1.5 2 2.5 3 3.5 1 2 4 8 16 32 Runtime (s x103) Cores Non-virtual Virtual Libhpc

5 10 15 20 25 30 35 5 10 15 20 25 30 35 Speedup Cores Ideal Non-virtual Virtual Libhpc

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Summary

  • Presented implementation optimisations to blend

high and low order polynomial order

  • Mixed implementation of basic operators
  • Mixed Fourier discretisations.
  • Does cloud computing offer possibilities?