Global space–time multiquadric method for inverse heat conduction problem

Abstract

SUMMARYIn this paper, a radial basis collocation method (RBCM) based on the global space-time multiquadric (MQ) is proposed to solve the inverse heat conduction problem (IHCP). The global MQ is simply constructed by incorporating time dimension into the MQ function as a new variable in radial coordinate. The method approximates the IHCP as an over-determined linear system with the use of two sets of collocation points: one is satisfied with the governing equation and another is for the given conditions. The least-square technique is introduced to find the solution of the over-determined linear system. The present work investigates two types of the ill-posed heat conduction problems: the IHCP to recover the surface temperature and heat flux history on a source point from the measurement data at interior locations, and the backward heat conduction problem (BHCP) to retrieve the initial temperature distribution from the known temperature distribution at a given time. Numerical results of four benchmark examples show that the proposed method can provide accurate and stable numerical solutions for one-dimensional and two-dimensional IHCP problems. The sensitivity of the method with respect to the measured data, location of measurement, time step, shape parameter and scaling factor is also investigated.

Estimation of river pollution source using the space-time radial basis collocation method

“…3 and 4. These figures illustrate that the errors E C and E S vary in a cone-shaped distribution with respect to c and w, which is similar to the results reported by Li and Mao [28], and that the global optimal value ranges are as shown in the core of the cone. Interestingly, comparison of the optimal region of E C errors with that of the E S errors (Fig.…”

“…To solve the time-dependent inverse model, a new Euclidian distance, i.e., (||x−x k || 2 +w 2 ||t−t k || 2 ) 1/2 , where w is a scale parameter, was proposed by normalizing the time dimension to be a pseudo-spatial dimension in [28,29]. Taking a two-dimensional problem as an example, the new Euclidean distance is given by:…”

“…As mentioned in the Introduction, although Li and Mao [28,29]investigated the convergence of the global space-time MQ solution with respect to the shape parameter c and the scaling factor w, it is still necessary to seek the best performance of the RBCM based on selection of the optimal model parameters, especially when the method is used in practical applications.…”

“…Cheng and Cabral [13] and Mao and Li [31] employed the inverse multiquadric (MQ) as a RBF to solve ill-posed inverse boundary problems for the Laplace equation. As for the timedependent inverse problems, the global space-time RBCM was then constructed for inverse and backward heat conduction problems [28] and for estimations of the spatial plume distribution and source release history in groundwater [29]. Owing to the complexity of the source identification problem, sensitivity analysis associated with the number, location and error level of observations and model parameters was also performed by Li and Mao [28,29] The advantages of the space-time RBCM mainly lie in its efficiency due to the truly meshfree and non-iterative property, super-convergence and insensitivity to the direction of time.…”

“…For problems of interpolating scattered data, Foley [18] improved the RBF shape parameter based on the root mean square deviation between the MQ and inverse MQ models at a set of test points, and Rippa [34] proposed instead a cross-validation technique for computing the RBF shape parameter by minimizing the cost function based on the residual errors of only the observation data; for solving PDE systems, Roque and Ferreira [35] built the cost function based only on the residual errors of the governing equations, but Tsaia et al [41] applied the golden search algorithm to choose a good shape parameter based on the residual errors of both the governing equation and Dirichlet boundary conditions. Li and Mao [28,29] investigated convergence of the global space-time MQ solution with respect to the shape parameter and the scaling factor. The results showed that a reasonable (although not necessarily the best) numerical solution exists within a wide range of both parameters, with a V-shaped or U-shaped error profile with respect to the shape parameter and scaling parameter.…”