skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

This content will become publicly available on August 19, 2021

Title: Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures

Abstract

Problems with topological uncertainties appear in many fields ranging from nano-device engineering to the design of bridges. In many of such problems, a part of the domains boundaries is subjected to random perturbations making inefficient conventional schemes that rely on discretization of the whole domain. Here, we study elliptic PDEs in domains with boundaries comprised of a deterministic part and random apertures, and apply the method of modified potentials with Green’s kernels defined on the deterministic part of the domain. This approach allows to reduce the dimension of the original differential problem by reformulating it as a boundary integral equation posed on the random apertures only. The multilevel Monte Carlo method is then applied to this modified integral equation and its optimal ϵ-2 asymptotical complexity is shown. Finally, we provide the qualitative analysis of the proposed technique and support it with numerical results.

Authors:
ORCiD logo [1];  [2]
  1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
  2. Middle Tennessee State Univ., Murfreesboro, TN (United States)
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF)
OSTI Identifier:
1665995
Grant/Contract Number:  
AC05-00OR22725; DMS1620280
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Scientific Computing
Additional Journal Information:
Journal Volume: 84; Journal Issue: 3; Journal ID: ISSN 0885-7474
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
Green's function; Green's potential; boundary integral equations; random boundaries; multilevel Monte Carlo

Citation Formats

Reshniak, Viktor, and Melnikov, Yuri. Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures. United States: N. p., 2020. Web. doi:10.1007/s10915-020-01296-9.
Reshniak, Viktor, & Melnikov, Yuri. Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures. United States. doi:10.1007/s10915-020-01296-9.
Reshniak, Viktor, and Melnikov, Yuri. Wed . "Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures". United States. doi:10.1007/s10915-020-01296-9.
@article{osti_1665995,
title = {Method of Green’s Potentials for Elliptic PDEs in Domains with Random Apertures},
author = {Reshniak, Viktor and Melnikov, Yuri},
abstractNote = {Problems with topological uncertainties appear in many fields ranging from nano-device engineering to the design of bridges. In many of such problems, a part of the domains boundaries is subjected to random perturbations making inefficient conventional schemes that rely on discretization of the whole domain. Here, we study elliptic PDEs in domains with boundaries comprised of a deterministic part and random apertures, and apply the method of modified potentials with Green’s kernels defined on the deterministic part of the domain. This approach allows to reduce the dimension of the original differential problem by reformulating it as a boundary integral equation posed on the random apertures only. The multilevel Monte Carlo method is then applied to this modified integral equation and its optimal ϵ-2 asymptotical complexity is shown. Finally, we provide the qualitative analysis of the proposed technique and support it with numerical results.},
doi = {10.1007/s10915-020-01296-9},
journal = {Journal of Scientific Computing},
number = 3,
volume = 84,
place = {United States},
year = {2020},
month = {8}
}

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on August 19, 2021
Publisher's Version of Record

Save / Share:

Works referenced in this record:

Stochastic BEM with spectral approach in elastostatic and elastodynamic problems with geometrical uncertainty
journal, May 2005


Computing Green'S Functions for flow in Heterogeneous Composite Media
journal, January 2013


On integral equations of the first kind with logarithmic kernels
journal, December 1988


X-SFEM, a computational technique based on X-FEM to deal with random shapes
journal, January 2007

  • Nouy, Anthony; Schoefs, Franck; Moës, Nicolas
  • European Journal of Computational Mechanics, Vol. 16, Issue 2
  • DOI: 10.3166/remn.16.277-293

Stochastic analysis of transport in tubes with rough walls
journal, September 2006


Stochastic smoothed profile method for modeling random roughness in flow problems
journal, August 2013

  • Zayernouri, Mohsen; Park, Sang-Woo; Tartakovsky, Daniel M.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 263
  • DOI: 10.1016/j.cma.2013.05.007

An extended stochastic finite element method for solving stochastic partial differential equations on random domains
journal, October 2008

  • Nouy, A.; Clément, A.; Schoefs, F.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 51-52
  • DOI: 10.1016/j.cma.2008.06.010

The numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs
journal, January 1991


Stochastic finite elements of discretely parameterized random systems on domains with boundary uncertainty: STOCHASTIC FINITE ELEMENTS ON UNCERTAIN DOMAIN
journal, July 2014

  • Kundu, A.; Adhikari, S.; Friswell, M. I.
  • International Journal for Numerical Methods in Engineering, Vol. 100, Issue 3
  • DOI: 10.1002/nme.4733

Sparse second moment analysis for elliptic problems in stochastic domains
journal, April 2008

  • Harbrecht, Helmut; Schneider, Reinhold; Schwab, Christoph
  • Numerische Mathematik, Vol. 109, Issue 3
  • DOI: 10.1007/s00211-008-0147-9

Stochastic projection schemes for deterministic linear elliptic partial differential equations on random domains
journal, August 2010

  • Mohan, P. Surya; Nair, Prasanth B.; Keane, Andy J.
  • International Journal for Numerical Methods in Engineering, Vol. 85, Issue 7
  • DOI: 10.1002/nme.3004

Homogenization of random heterogeneous media with inclusions of arbitrary shape modeled by XFEM
journal, July 2014

  • Savvas, Dimitris; Stefanou, George; Papadrakakis, Manolis
  • Computational Mechanics, Vol. 54, Issue 5
  • DOI: 10.1007/s00466-014-1053-x

Qualocation
journal, December 2000


An Unconventional Quadrature Method for Logarithmic-Kernel Integral Equations Equations on Closed Curves
journal, March 1992

  • Sloan, Ian H.; Burn, B. J.
  • Journal of Integral Equations and Applications, Vol. 4, Issue 1
  • DOI: 10.1216/jiea/1181075670

Iterated Galerkin versus Iterated Collocation for Integral Equations of the Second Kind
journal, January 1985

  • Graham, Ivan G.; Joe, Stephen; Sloan, Lan H.
  • IMA Journal of Numerical Analysis, Vol. 5, Issue 3
  • DOI: 10.1093/imanum/5.3.355

Some applications of the Greens' function method in mechanics
journal, January 1977


A semi-analytical approach to Green׳s functions for heat equation in regions of irregular shape
journal, September 2014


Impact of endothelium roughness on blood flow
journal, May 2012

  • Park, Sang Woo; Intaglietta, Marcos; Tartakovsky, Daniel M.
  • Journal of Theoretical Biology, Vol. 300
  • DOI: 10.1016/j.jtbi.2012.01.017

Boundary integral formulation for 2D and 3D thermal problems exibiting a linearly varying stochastic conductivity
journal, April 1996

  • Manolis, G. D.; Shaw, R. P.
  • Computational Mechanics, Vol. 17, Issue 6
  • DOI: 10.1007/BF00363984

Multilevel Monte Carlo Path Simulation
journal, June 2008


On Using a Modified Nyström Method to Solve the 2-D Potential Problem
journal, June 1993


Extended stochastic FEM for diffusion problems with uncertain material interfaces
journal, September 2012


Numerical Methods for Differential Equations in Random Domains
journal, January 2006

  • Xiu, Dongbin; Tartakovsky, Daniel M.
  • SIAM Journal on Scientific Computing, Vol. 28, Issue 3
  • DOI: 10.1137/040613160

eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces
journal, August 2010

  • Nouy, A.; Clément, A.
  • International Journal for Numerical Methods in Engineering, Vol. 83, Issue 10
  • DOI: 10.1002/nme.2865

A stochastic Lagrangian approach for geometrical uncertainties in electrostatics
journal, September 2007


Analytic regularity and collocation approximation for elliptic PDEs with random domain deformations
journal, March 2016

  • Castrillón-Candás, Julio E.; Nobile, Fabio; Tempone, Raúl F.
  • Computers & Mathematics with Applications, Vol. 71, Issue 6
  • DOI: 10.1016/j.camwa.2016.01.005

A Posteriori Error Estimation for a cut cell Finite Volume Method with Uncertain Interface Location
journal, January 2015


A fictitious domain approach to the numerical solution of PDEs in stochastic domains
journal, May 2007


Some numerical results using the modified Nyström method to solve the 2-D potential problem
journal, January 1994