### Abstract

From an analytical standpoint, a majority of calculations use known boundary conditions (temperature or flux) and the so-called direct route to determine internal temperatures, strains, and/or stresses. For such problems where the thermal boundary condition is known a priori, the analytical procedure and solutions are tractable for the linear case where the thermophysical properties are independent of temperature. On the other hand, the inverse route where the boundary conditions must be determined from remotely determined temperature and/or flux data is much more difficult mathematically, as well as inherently sensitive to data errors (i.e., ill-posed). When solutions are available, they are often restricted to a harsh, albeit unrealistic step change in temperature or flux and/or are only valid for relatively short time frames before temperature changes occur at the far boundary. While the two approaches may seem to be at odds with each other, a generalized direct solution based on polynomial temperature or strain-histories can also be used to determine unknown boundary conditions via least-squares determination of coefficients. Once the inverse problem (and unknown boundary condition) is solved via these coefficients, the resulting polynomial can then be used with the generalized direct solution to determine the thermal-and stress-states as a function of time and position. When used for both thick slabs and tubes, excellent agreement was seen for various test cases. In fact, the derived solutions appear to be well suited for many thermal scenarios, provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties that do not vary with temperature. Since temperature dependent properties can certainly be an issue that affects accuracy in these types of calculations, some recent analytical procedures for both direct and inverse solutions are also discussed.

Original language | English (US) |
---|---|

Article number | 041202 |

Journal | Journal of Pressure Vessel Technology, Transactions of the ASME |

Volume | 134 |

Issue number | 4 |

DOIs | |

State | Published - Jul 17 2012 |

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### All Science Journal Classification (ASJC) codes

- Safety, Risk, Reliability and Quality
- Mechanics of Materials
- Mechanical Engineering

### Cite this

*Journal of Pressure Vessel Technology, Transactions of the ASME*,

*134*(4), [041202]. https://doi.org/10.1115/1.4006353

}

*Journal of Pressure Vessel Technology, Transactions of the ASME*, vol. 134, no. 4, 041202. https://doi.org/10.1115/1.4006353

**Direct and inverse solutions for thermal-and stress-transients and the analytical determination of boundary conditions using remote temperature or strain data.** / Segall, A. E.; Drapaca, C.; Engels, D.; Zhu, T.; Yang, H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Direct and inverse solutions for thermal-and stress-transients and the analytical determination of boundary conditions using remote temperature or strain data

AU - Segall, A. E.

AU - Drapaca, C.

AU - Engels, D.

AU - Zhu, T.

AU - Yang, H.

PY - 2012/7/17

Y1 - 2012/7/17

N2 - From an analytical standpoint, a majority of calculations use known boundary conditions (temperature or flux) and the so-called direct route to determine internal temperatures, strains, and/or stresses. For such problems where the thermal boundary condition is known a priori, the analytical procedure and solutions are tractable for the linear case where the thermophysical properties are independent of temperature. On the other hand, the inverse route where the boundary conditions must be determined from remotely determined temperature and/or flux data is much more difficult mathematically, as well as inherently sensitive to data errors (i.e., ill-posed). When solutions are available, they are often restricted to a harsh, albeit unrealistic step change in temperature or flux and/or are only valid for relatively short time frames before temperature changes occur at the far boundary. While the two approaches may seem to be at odds with each other, a generalized direct solution based on polynomial temperature or strain-histories can also be used to determine unknown boundary conditions via least-squares determination of coefficients. Once the inverse problem (and unknown boundary condition) is solved via these coefficients, the resulting polynomial can then be used with the generalized direct solution to determine the thermal-and stress-states as a function of time and position. When used for both thick slabs and tubes, excellent agreement was seen for various test cases. In fact, the derived solutions appear to be well suited for many thermal scenarios, provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties that do not vary with temperature. Since temperature dependent properties can certainly be an issue that affects accuracy in these types of calculations, some recent analytical procedures for both direct and inverse solutions are also discussed.

AB - From an analytical standpoint, a majority of calculations use known boundary conditions (temperature or flux) and the so-called direct route to determine internal temperatures, strains, and/or stresses. For such problems where the thermal boundary condition is known a priori, the analytical procedure and solutions are tractable for the linear case where the thermophysical properties are independent of temperature. On the other hand, the inverse route where the boundary conditions must be determined from remotely determined temperature and/or flux data is much more difficult mathematically, as well as inherently sensitive to data errors (i.e., ill-posed). When solutions are available, they are often restricted to a harsh, albeit unrealistic step change in temperature or flux and/or are only valid for relatively short time frames before temperature changes occur at the far boundary. While the two approaches may seem to be at odds with each other, a generalized direct solution based on polynomial temperature or strain-histories can also be used to determine unknown boundary conditions via least-squares determination of coefficients. Once the inverse problem (and unknown boundary condition) is solved via these coefficients, the resulting polynomial can then be used with the generalized direct solution to determine the thermal-and stress-states as a function of time and position. When used for both thick slabs and tubes, excellent agreement was seen for various test cases. In fact, the derived solutions appear to be well suited for many thermal scenarios, provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties that do not vary with temperature. Since temperature dependent properties can certainly be an issue that affects accuracy in these types of calculations, some recent analytical procedures for both direct and inverse solutions are also discussed.

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U2 - 10.1115/1.4006353

DO - 10.1115/1.4006353

M3 - Article

AN - SCOPUS:84863759571

VL - 134

JO - Journal of Pressure Vessel Technology, Transactions of the ASME

JF - Journal of Pressure Vessel Technology, Transactions of the ASME

SN - 0094-9930

IS - 4

M1 - 041202

ER -