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Inverse Problems in Science and Engineering Volume 18, 2010 - Issue 1: Proceedings of the 4th International Conference on Inverse Problems: Modelling and Simulation, May 26th–30th, 2008, held in Fethiye, Turkey

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Original Articles

Different finite element approaches for inverse heat conduction problems

Krzysztof GrysaFaculty of Management and Computer Modelling, Kielce University of Technology, al.1000-lecia P.P. 7, 25-314, Kielce, Poland

Renata LeśniewskaFaculty of Management and Computer Modelling, Kielce University of Technology, al.1000-lecia P.P. 7, 25-314, Kielce, Poland

Pages 3-17 | Received 15 Oct 2008, Accepted 15 Apr 2009, Published online: 08 Oct 2009

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https://doi.org/10.1080/17415970903233556

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Figure 1. Time-space elements in the case of temperature continuous in the nodes.

Figure 2. Time-space elements in the case of temperature discontinuous in the nodes.

Figure 3. The boundary conditions (3–5), the ITR placement and the finite elements.

Table 1. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for FEM with the condition of continuity of temperature in the common nodes.

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Table 2. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for FEM with no continuity of temperature in the common nodes.

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Table 3. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for nodeless FEM.

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Table 4. Errors of the identified temperature at the boundary x = 1 for δ_b ∈ (0, 0.99).

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Table 5. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for successive time steps k for nodeless FEMT.

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Table 6. Errors of the identified temperature at the boundary x = 1 for δ_b = 0.1 for different energetic minimizing terms and successive time steps k for nodeless FEMT.

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Figure 4. The maximum norm (15) of the temperature field error for 12 and for 15 T-functions approximation in the nodeless FEMT with heat flux regularization and exact ITRs.

Table 7. δL₂ norm of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 and of errors of the identified temperature at the boundary x = 1 for noisy ITRs.

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Figure 5. Norm (16) of the temperature field error for noisy and smoothed ITRs.

Table 8. δL₂ norm of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 and of errors of the identified temperature at the boundary x = 1 for smoothed ITRs.

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Different finite element approaches for inverse heat conduction problems

Table 1. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for FEM with the condition of continuity of temperature in the common nodes.

Table 2. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for FEM with no continuity of temperature in the common nodes.

Table 3. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for nodeless FEM.

Table 4. Errors of the identified temperature at the boundary x = 1 for δ_b ∈ (0, 0.99).

Table 5. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for successive time steps k for nodeless FEMT.

Table 6. Errors of the identified temperature at the boundary x = 1 for δ_b = 0.1 for different energetic minimizing terms and successive time steps k for nodeless FEMT.

Table 7. δL₂ norm of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 and of errors of the identified temperature at the boundary x = 1 for noisy ITRs.

Table 8. δL₂ norm of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 and of errors of the identified temperature at the boundary x = 1 for smoothed ITRs.

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Different finite element approaches for inverse heat conduction problems

Figures & data

Table 1. Norms of errors of the approximated temperature field as a function of the distance δb from the boundary x = 1 for different energetic minimizing terms and for FEM with the condition of continuity of temperature in the common nodes.

Table 2. Norms of errors of the approximated temperature field as a function of the distance δb from the boundary x = 1 for different energetic minimizing terms and for FEM with no continuity of temperature in the common nodes.

Table 3. Norms of errors of the approximated temperature field as a function of the distance δb from the boundary x = 1 for different energetic minimizing terms and for nodeless FEM.

Table 4. Errors of the identified temperature at the boundary x = 1 for δb ∈ (0, 0.99).

Table 5. Norms of errors of the approximated temperature field as a function of the distance δb from the boundary x = 1 for different energetic minimizing terms and for successive time steps k for nodeless FEMT.

Table 6. Errors of the identified temperature at the boundary x = 1 for δb = 0.1 for different energetic minimizing terms and successive time steps k for nodeless FEMT.

Table 7. δL2 norm of errors of the approximated temperature field as a function of the distance δb from the boundary x = 1 and of errors of the identified temperature at the boundary x = 1 for noisy ITRs.

Table 8. δL2 norm of errors of the approximated temperature field as a function of the distance δb from the boundary x = 1 and of errors of the identified temperature at the boundary x = 1 for smoothed ITRs.

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Table 1. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for FEM with the condition of continuity of temperature in the common nodes.

Table 2. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for FEM with no continuity of temperature in the common nodes.

Table 3. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for nodeless FEM.

Table 4. Errors of the identified temperature at the boundary x = 1 for δ_b ∈ (0, 0.99).

Table 5. Norms of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 for different energetic minimizing terms and for successive time steps k for nodeless FEMT.

Table 6. Errors of the identified temperature at the boundary x = 1 for δ_b = 0.1 for different energetic minimizing terms and successive time steps k for nodeless FEMT.

Table 7. δL₂ norm of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 and of errors of the identified temperature at the boundary x = 1 for noisy ITRs.

Table 8. δL₂ norm of errors of the approximated temperature field as a function of the distance δ_b from the boundary x = 1 and of errors of the identified temperature at the boundary x = 1 for smoothed ITRs.