A new method for identifying temperature-dependent thermal conductivity in transient heat conduction problems based on element differential method

Identification of force loadings on the inner circumference of a finite‐length elastic cylinder

“…TA B L E 1 Case-study parameters used in formula (24). Consider a direct axisymmetric elasticity problem for a finite-length hollow cylinder  imposing the boundary conditions ( 5), (6), and (8), where…”

“…F I G U R E 3 Full-field distributions of the radial (first row) and axial (second row) displacements in a finite-length hollow cylinder 𝑟 in = 0.4, 𝑏 = 2 computed from the direct problem under the loadings ( 5), ( 6), (8), and (24) for the cases 𝜉 = 1, 2, 3 shown in the first, second, and third column, respectively.…”

“…Having solved the system (23), we can reconstruct the coefficients in (12). Then, within the context of formulae ( 14) and ( 17), we can calculate the stresses and displacements in the cylinder and ultimately, manage to restore the unknown normal and tangential loading on the inner circumference of the cylinder 𝜎 in (𝑧) and 𝜏 in (𝑧) which are indicated in (8).…”

“…Such information about the "side" or "secondary" parameters along with the constitutive equations of the mathematical model for the considered coupled processes allows for formulating inverse problems on the identification of unknown loadings acting over the "forbidden" area [6]. Quite often such problems appear to be ill-posed in the Hadamard sense which requires advanced approaches towards their formulation and solution [7][8][9][10].…”

“…𝑙 , 𝑢 𝑟 (1, 𝑧 𝑙 )), 𝑙 = 0, 1, … , 𝐿} , ve out (𝑧) = linappr 𝑧∈[−𝑏,𝑏]{(𝑧 𝑙 , 𝑢 𝑧 (1, 𝑧 𝑙 )), 𝑙 = 0, 1, … , 𝐿} and used as the input data for the inverse problem. In order to simulate the small errors in the input data for the inverse problem concerned with the possible imperfection of the displacements measurement on the outer circumference of the cylinder, we consider the following input data 𝑢 e out (𝑧) = ūe out (𝑧) + 0.01𝛿 0 𝜀 0 , 𝑣 e out (𝑧) = ve out (𝑧) + 0.01𝛿 0 𝜀 0(25) F I G U R E 2 Full-field distributions of the dimensionless radial (first row) and tangential (second row) stresses in a finite-length hollow cylinder 𝑟 in = 0.4, 𝑏 = 2computed from the direct problem under the loadings (5), (6),(8), and (24) for the cases 𝜉 = 1, 2, 3 shown in the first, second, and third column, respectively.…”