Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Convergence and error analysis of an automatically differentiated finite volume based heat conduction code

Convergence and error analysis of an automatically differentiated finite volume based heat... This paper aims to investigate the convergence and error properties of a finite volume-based heat conduction code that uses automatic differentiation to evaluate derivatives of solutions outputs with respect to arbitrary solution input(s). A problem involving conduction in a plane wall with convection at its surfaces is used as a test problem, as it has an analytical solution, and the error can be evaluated directly.Design/methodology/approachThe finite volume method is used to discretize the transient heat diffusion equation with constant thermophysical properties. The discretized problem is then linearized, which results in two linear systems; one for the primary solution field and one for the secondary field, representing the derivative of the primary field with respect to the selected input(s). Derivatives required in the formation of the secondary linear system are obtained by automatic differentiation using an operator overloading and templating approach in C++.FindingsThe temporal and spatial discretization error for the derivative solution follows the same order of accuracy as the primary solution. Second-order accuracy of the spatial and temporal discretization schemes is confirmed for both primary and secondary problems using both orthogonal and non-orthogonal grids. However, it has been found that for non-orthogonal cases, there is a limit to the error reduction, which is concluded to be a result of errors in the Gauss-based gradient reconstruction method.Originality/valueThe convergence and error properties of derivative solutions obtained by forward mode automatic differentiation of finite volume-based codes have not been previously investigated. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Numerical Methods for Heat & Fluid Flow Emerald Publishing

Convergence and error analysis of an automatically differentiated finite volume based heat conduction code

Download PDF
18 pages

Loading next page...
 
/lp/emerald-publishing/convergence-and-error-analysis-of-an-automatically-differentiated-boeZJOuTX6
Publisher
Emerald Publishing
Copyright
© Emerald Publishing Limited
ISSN
0961-5539
DOI
10.1108/hff-09-2018-0489
Publisher site
See Article on Publisher Site

Abstract

This paper aims to investigate the convergence and error properties of a finite volume-based heat conduction code that uses automatic differentiation to evaluate derivatives of solutions outputs with respect to arbitrary solution input(s). A problem involving conduction in a plane wall with convection at its surfaces is used as a test problem, as it has an analytical solution, and the error can be evaluated directly.Design/methodology/approachThe finite volume method is used to discretize the transient heat diffusion equation with constant thermophysical properties. The discretized problem is then linearized, which results in two linear systems; one for the primary solution field and one for the secondary field, representing the derivative of the primary field with respect to the selected input(s). Derivatives required in the formation of the secondary linear system are obtained by automatic differentiation using an operator overloading and templating approach in C++.FindingsThe temporal and spatial discretization error for the derivative solution follows the same order of accuracy as the primary solution. Second-order accuracy of the spatial and temporal discretization schemes is confirmed for both primary and secondary problems using both orthogonal and non-orthogonal grids. However, it has been found that for non-orthogonal cases, there is a limit to the error reduction, which is concluded to be a result of errors in the Gauss-based gradient reconstruction method.Originality/valueThe convergence and error properties of derivative solutions obtained by forward mode automatic differentiation of finite volume-based codes have not been previously investigated.

Journal

International Journal of Numerical Methods for Heat & Fluid Flow – Emerald Publishing

Published: Aug 30, 2019

Keywords: Convergence; Finite volume method; Automatic differentiation; Conduction; Error

References

  • Computing and visualization in science automatic differentiation in C++ using expression templates and application to a flow control problem
    Aubert, P.; Césaré, N.D.; Pironneau, O.
  • Efficient management of parallelism in object oriented numerical software libraries
    Arge, E.; Bruaset, A.; Langtangen, H.
  • PETSc users manual
    Balay, S.; Zhang, H.
  • PETSc web page
    Balay, S.; Zhang, H.
  • An accurate gradient and hessian reconstruction method for cell-centered finite volume discretizations on general unstructured grids
    Betchen, L.J.; Straatman, A.G.
  • Adifor 2.0: automatic differentiation of fortran 77 programs
    Bischof, C.; Khademi, P.; Mauer, A.; Carle, A.
  • Automatic differentiation of the general-purpose computational fluid dynamics package FLUENT
    Bischof, C.H.; Bücker, H.M.; Rasch, A.; Slusanschi, E.; Lang, B.
  • A general sensitivity equation formulation for turbulent heat transfer
    Colin, E.; Étienne, S.; Pelletier, D.; Borggaard, J.
  • Automatic differentiation of a finite-volume-based transient heat conduction code for sensitivity analysis
    DeGroot, C.T.
  • WEdiff: Western engineering automatic differentiation library
    DeGroot, C.T.
  • A finite-volume model for fluid flow and nonequilibrium heat transfer in conjugate fluid-porous domains using general unstructured grids
    DeGroot, C.T.; Straatman, A.G.
  • An improved continuous sensitivity equation method for optimal shape design in mixed convection
    Duvigneau, R.; Pelletier, D.; Borggaard, J.
  • Recipes for adjoint code construction
    Giering, R.; Kaminski, T.
  • The tapenade automatic differentiation tool: principles, model, and specification
    Hascoet, L.; Pascual, V.
  • Fast reverse-mode automatic differentiation using expression templates in C++
    Hogan, R.J.
  • OpenFOAM: a C ++ library for complex physics simulations
    Jasak, H.; Jemcov, A.; Tukovic, Z.
  • Arithmetic parameterization of general purpose CFD code
    Jemcov, A.
  • Algorithmic differentiation of general purpose CFD code: implementation and verification
    Jemcov, A.; Mathur, S.; Neittaanmäki, P.; Rossi, T.; Majava, K.; Pironneau, O.; Lasiecka, I. (Eds.)
  • Discretization and parallel performance of an unstructured finite volume Navier-Stokes solver
    Karimian, S.A.M.; Straatman, A.G.
  • Algorithmic differentiation of numerical methods
    Naumann, U.; Lotz, J.; Leppkes, K.; Towara, M.
  • Numerical simulation and sensitivity analysis of effective parameters on natural convection and entropy generation in a wavy surface cavity filled with a nanofluid using RSM. Numer
    Shirvan, K.M.; Mirzakhanlari, S.; Chamkha, A.J.; Mamourian, M.
  • A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods
    Syrakos, A.; Varchanis, S.; Dimakopoulos, Y.; Goulas, A.; Tsamopoulos, J.
  • A tensorial approach to computational continuum mechanics using object-oriented techniques
    Weller, H.; Tabor, G.; Jasak, H.; Fureby, C.
  • Numerical simulation and optimization of turbulent flows through perforated circular pin fin heat sinks
    Wu, C.-H.; Tang, H.-W.; Yang, Y.-T.
  • An integrated space-time finite-volume method for moving-boundary problems
    Zwart, P.J.; Raithby, G.D.; Raw, M.J.