DSpace 集合:

MQ-RBF with an optimal value of shape parameter determined by a size-independence technique

2026-01-01T00:00:00Z

標題: MQ-RBF with an optimal value of shape parameter determined by a size-independence technique 作者: Liu, Chein-Shan; Tsai, Chia-Cheng 摘要: This paper proposes a new method to find the optimal value of shape parameter in the multiquadric radial basis function (MQ-RBF), which is formulated by a size-independence technique (SIT). The size n for a high-dense MQ-RBF is reduced to m < n, with n simultaneously denoting the number of radial bases, the number of unknown coefficients and the dimension of linear system. Then a lower m-dimensional linear system is solved several times to determine the optimal value of shape parameter, using the golden section search algorithm to minimize the maximal error to fit the given data. Inserting the optimal value obtained from a lower m dimensional linear system into a high-dense MQ-RBF, one can construct a highly accurate interpolant. To solve elliptic type equations we adapt the MQ-RBF by SIT. For searching a good shape parameter in the high-dense MQ-RBF the SIT can improve the accuracy and save CPU time.

Projective Solutions Methods Automatically Satisfying the Stokes, Oseen and Brinkman Equations

2026-01-01T00:00:00Z

標題: Projective Solutions Methods Automatically Satisfying the Stokes, Oseen and Brinkman Equations 作者: Liu, Chein-Shan; Hsu, Tai-Wen; Tsai, Chia-Cheng 摘要: The new projective solutions methods (PSMs) for solving the Stokes, Oseen, and Brinkman flow problems are presented in this paper. They automatically satisfy the governing equations and are therefore Trefftz-type methods. Utilizing the third-order formulation and three-dimensional analytic functions, we derive a meshless Trefftz-type method to solve three-dimensional Stokes flow problems. The Oseen and Brinkman equations are transformed into four coupled third-order/first-order partial differential equations. The projective-type particular solution (PTPS) is obtained via a projective function in terms of the projective variable; the third-order ordinary differential equations (ODEs) with constant coefficients are derived to determine the projective functions. The Trefftz-type PSM is extremely accurate, because the governing equations (including the incompressibility condition) are implemented automatically. For the Brinkman equations, the general solutions of velocity and pressure are presented by using the Helmholtz function and a harmonic function, whose corresponding Trefftz-type numerical method is developed. Upon comparison with the method of fundamental solutions (MFS), the new methods exhibit some advantages, including lower condition numbers, faster convergence, and better accuracy. We also apply the Trefftz-type PSM to solve the exterior problem of the Stokes equations, where the velocity tends to zero at infinity.

Data Interpolation by Using Two-Parameter Boundary Shape Functions

2026-01-01T00:00:00Z

標題: Data Interpolation by Using Two-Parameter Boundary Shape Functions 作者: Liu, Chein-Shan; Chang, Chih-Wen 摘要: This paper addresses the interpolation of noisy data, a problem in which high-order polynomial methods frequently exhibit ill-conditioning and the Runge phenomenon. To mitigate these issues, a hybrid mixed-type method (MTM) is first developed by augmenting classical collocation equations with Galerkin-derived moment equations and optimizing a characteristic length parameter within the polynomial representation. Subsequently, the study develops a boundary shape function interpolation method (BSFIM) for two-dimensional interpolation over rectangular and arbitrary planar domains. In the BSFIM, a family of two-parameter boundary shape functions is systematically derived such that each basis function automatically and exactly satisfies the measured boundary data. This boundary-conforming and term-wise separable basis function design yields a low-dimensional, and well-conditioned approximation space that allows the extraction of highly accurate interpolants from substantially fewer interior data points than those required in traditional polynomial or radial basis function (RBF) methods. Extensive numerical examples, including problems on irregular domains, discontinuous scattered data, and Stokes-flow-based pressure profile construction, demonstrate that the BSFIM can achieve interpolation errors below the noise level and exhibit robust performance under conditions involving high measurement noise.

Integral and Numerical Formulations for Seeking the Period of Non-Conservative Nonlinear Oscillator With/Without the First Integral

2025-01-01T00:00:00Z

標題: Integral and Numerical Formulations for Seeking the Period of Non-Conservative Nonlinear Oscillator With/Without the First Integral 作者: Liu, Chein-Shan; Tsai, Chia-Cheng; Chang, Chih-Wen 摘要: For a non-conservative nonlinear oscillator (NCNO) having a periodic solution, the existence of the first integral is a certain symmetry of the nonlinear dynamical system, which signifies the balance of kinetic energy and potential energy. A first-order nonlinear ordinary differential equation (ODE) is used to derive the first integral, which, equipped with a right-end boundary condition, can determine an implicit potential function for computing the period by an exact integral formula. However, the integrand is singular, which renders a less accurate value of the period. A generalized integral conservation law endowed with a weight function is constructed, which is proved to be equivalent to the exact integral formula. Minimizing the error to satisfy the periodicity conditions, the optimal initial value of the weight function is determined. Two non-iterative methods are developed by integrating three first-order ODEs or two first-order ODEs to compute the period. Very accurate value of the period can be observed upon testing five examples. For the NCNO without having the first integral, the integral-type period formula is derived. Four examples belong to the Li & eacute;nard equation, involving the van der Pol equation, are evaluated by the proposed iterative method to determine the oscillatory amplitude and period. For the case with one or more limit cycles, the amplitude and period can be estimated very accurately. For the NCNO of a broad type with or without having the first integral, the present paper features a solid theoretical foundation and contributes integral-type formulations for the determination of the oscillatory period. The development of new numerical algorithms and extensive validation across a diverse set of examples is given.