Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method

Abstract

The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Computational cost may dramatically increase, especially when dealing with strong-from equations where high-order derivative operators are required as compared to weak-form approaches for obtaining results with promising levels of accuracy. Under the framework of gradient approximation, the derivatives of reproducing kernel shape functions can be constructed synchronically, thereby alleviating the complexity in computation. In view of this, the present work first introduces the weighted high-order gradient reproducing kernel collocation method in the inverse analysis. The convergence of the method is examined through the weights imposed on the boundary conditions. Then, several configurations of multiply connected domains are provided to numerically investigate the stability and efficiency of the method. To reach the desired accuracy in detecting the outer boundary for two special cases, special treatments including allocation of points and use of ghost points are adopted as the solution strategy. From four benchmark examples, the efficacy of the method in detecting the unknown boundary is demonstrated.

Direct Collocation with Reproducing Kernel Approximation for Two-Phase Coupling System in a Porous Enclosure

“…For the problem in consideration, the same location and number of source points and collocation points are utilized herein, i.e., N s = N c . Unlike the weighted collocation method with least-squares formulation in the previous studies [11][12][13]19], there are typically more collocation points than source points in the collocation system, and boundary conditions are imposed by weights to balance the errors in the domain and on the boundary. With the RK approximation in the direct collocation method, a sparse system of Jacobian matrix is reached, thereby making the Newton-Raphson collocation method efficient.…”

“…As the resulting system is often ill-conditioned with a large condition number, numerical instability might be raised; for more information, see [8][9][10]. In contrast, the reproducing kernel (RK) shape function is a local function; although it maintains an algebraic convergence rate, the resulting system is more stable to yield promising results [11][12][13]. Other types of approximation adopted in the collocation method, such as global expansion in Bernstein polynomials, exponential functions, and Taylor matrix, can be referred to the studies [14][15][16] for solving nonlinear ODE systems.…”