Modeling the flash-heat experiment on porous domains

Authors:
H. T. Banks, D. Cioranescu, A. K. Criner and W. P. Winfree

Journal:
Quart. Appl. Math. **70** (2012), 53-67

MSC (2000):
Primary 35B27, 76R50, 78M40

Published electronically:
September 16, 2011

MathSciNet review:
2920615

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Abstract | References | Similar Articles | Additional Information

Abstract: We discuss a mathematical model for the flash-heat experiment in homogeneous isotropic media. We then use this model to investigate the use of homogenization techniques in approximating models for interrogation via flash-heating in porous materials. We represent porous materials as both randomly perforated domains and periodically perforated domains.

**1.**H.T. Banks, B. Boudreaux, A. K. Criner, K. Foster, C. Uttal, T. Vogel, A.K. Criner and W.P. Winfree, Thermal based damage detection in porous materials, Tech. Rep. CRSC-TR08-11, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, September, 2008;*Inverse Probl. Sci. Engr.,***18**(2009), 835-851.**2.**H.T. Banks, D. Cioranescu, A.K. Criner and W.P. Winfree, Modeling the flash-heat experiment on porous domains, Tech. Rep. CRSC-TR10-06, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, May, 2010.**3.**H. T. Banks, N. L. Gibson and W. P. Winfree, Void detection in complex geometries, Tech. Rep. CRSC-TR08-09, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, May, 2008.**4.**H. T. Banks, Michele L. Joyner, Buzz Wincheski, and William P. Winfree,*Nondestructive evaluation using a reduced-order computational methodology*, Inverse Problems**16**(2000), no. 4, 929–945. MR**1776475**, 10.1088/0266-5611/16/4/304**5.**H. T. Banks and Fumio Kojima,*Boundary shape identification problems in two-dimensional domains related to thermal testing of materials*, Quart. Appl. Math.**47**(1989), no. 2, 273–293. MR**998101****6.**H. T. Banks and F. Kojima,*Identification of material damage in two-dimensional domains using the SQUID-based nondestructive evaluation system*, Inverse Problems**18**(2002), no. 6, 1831–1855. Special section on electromagnetic and ultrasonic nondestructive evaluation. MR**1955921**, 10.1088/0266-5611/18/6/324**7.**H. T. Banks, Fumio Kojima, and W. P. Winfree,*Boundary estimation problems arising in thermal tomography*, Inverse Problems**6**(1990), no. 6, 897–921. MR**1082231****8.**D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in domains with holes, to appear.**9.**Doina Cioranescu and Jeannine Saint Jean Paulin,*Homogenization of reticulated structures*, Applied Mathematical Sciences, vol. 136, Springer-Verlag, New York, 1999. MR**1676922****10.**Doina Cioranescu and Patrizia Donato,*An introduction to homogenization*, Oxford Lecture Series in Mathematics and its Applications, vol. 17, The Clarendon Press, Oxford University Press, New York, 1999. MR**1765047****11.**Patrizia Donato and Aïssam Nabil,*Homogenization and correctors for the heat equation in perforated domains*, Ricerche Mat.**50**(2001), no. 1, 115–144. MR**1941824****12.**The Mathworks, Inc.,*Partial Differential Equation Toolbox 1: User's Guide*, The Mathworks, Inc., Natick, MA, 2008.**13.**W. J. Parker, R. J. Jenkins, C. P. Butler and G. L. Abbott, Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity,*J. Appl. Phys.***32**(9):1679-1684, 1961.**14.**Pavel Šolín,*Partial differential equations and the finite element method*, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. MR**2180081****15.**Wenping Wang, Jiaye Wang, and Myung-Soo Kim,*An algebraic condition for the separation of two ellipsoids*, Comput. Aided Geom. Design**18**(2001), no. 6, 531–539. MR**1843064**, 10.1016/S0167-8396(01)00049-8

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Additional Information

**H. T. Banks**

Affiliation:
Department of Mathematics, Center for Research in Scientific Computation, North Carolina State University, Raleigh, North Carolina 27695-8212

**D. Cioranescu**

Affiliation:
Laboratoire J. L. Lions, Université Pierre et Marie Curie, 175 rue du Chevaleret, 75013 Paris, France

**A. K. Criner**

Affiliation:
Department of Mathematics, Center for Research in Scientific Computation, North Carolina State University, Raleigh, North Carolina 27695-8212

**W. P. Winfree**

Affiliation:
Nondestructive Evaluation Science Branch, NASA Langley Research Center, MS 231, Hampton, Virginia 23681

DOI:
https://doi.org/10.1090/S0033-569X-2011-01230-8

Keywords:
Modeling porous media,
thermal diffusion,
homogenization

Received by editor(s):
May 16, 2010

Published electronically:
September 16, 2011

Article copyright:
© Copyright 2011
Brown University