This study applies a hybrid technique of the Laplace transform and finite-difference methods in conjunction with the least-squares method and experimental temperature data inside the test material to predict the unknown surface temperature, heat flux and absorptivity for various surface conditions in the laser surface heating process. In this study, the functional form of the surface temperature is unknown a priori and is assumed to be a function of time before performing the inverse calculation. In addition, the whole time domain is divided into several analysis sub-time intervals and then these unknown estimates on each analysis interval can be predicted. In order to show the accuracy of the present inverse method, comparisons are made among the present estimates, direct results and previous results, showing that the present estimates agree with the direct results for the simulated problem. However, the present estimates of the surface absorptivity deviate slightly from previous estimated results under the assumption of constant thermal properties. The effect of the surface conditions on the surface absorptivity and temperature is not negligible.

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SUMMARYA hybrid scheme of the Laplace transform, finite difference and least-squares methods in conjunction with a sequential-in-time concept, cubic spline and temperature measurements is applied to predict the heat transfer coefficient distribution on a boundary surface in two-dimensional transient inverse heat conduction problems. In this study, the functional form of the heat transfer coefficient is unknown a priori. The whole spatial domain of the unknown heat transfer coefficient is divided into several analysis sub-intervals. Later, a series of connected cubic polynomial function in space and a linear function in time can be applied to estimate the unknown surface conditions. Due to the application of the Laplace transform, the unknown heat transfer coefficient can be estimated from a specific time. In order to evidence the accuracy of the present inverse scheme, comparisons among the present estimates, previous results and exact solution are made. The results show that the present inverse scheme not only can reduce the number of the measurement locations but also can increase the accuracy of the estimated results. Good estimation on the heat transfer coefficient can be obtained from the knowledge of the transient temperature recordings even in the case with measurement errors.

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When a Hamiltonian can be split into integrable and nonintegrable parts, the former part is solved analytically, and the latter one is integrated numerically by means of implicit symplectic integrators such as the first-order semi-implicit Euler method or the second-order implicit midpoint rule. These analytical and numerical solutions are used to construct a second-order mixed symplectic integrator with the semi-implicit Euler method and one with the implicit midpoint rule. A theoretical analysis shows that the Euler mixed integrator is inferior to the midpoint one in the sense of numerical stability. Numerical simulations of the circularly-restricted three-body problem also support this fact. It is further shown through numerical integrations of the post-Newtonian Hamiltonian of spinning compact binaries that the qualities of the Euler mixed integrator and the midpoint mixed method do depend on the type of orbits. Especially for chaotic orbits, the Euler mixed integrator often becomes unstable. In addition, the Euler mixed integrator has an advantage over the midpoint mixed method in computational efficiency, and is almost equivalent to the latter in the numerical accuracy if the two mixed integrators are stable. In spite of this, the midpoint mixed integrator is worth recommending for the study of the dynamics of post-Newtonian Hamiltonians of spinning compact binaries.