Hsin-Yun Hu's research while affiliated with Tunghai University and other places

... Galerkin meshfree methods [9] are a unique class of numerical methods based on a purely point-based discretization. They offer advantages in classes of problems where mesh-based finite elements encounter difficulty, such as those involving extreme-deformation, multi-body evolving contact, fragmentation, among others; they also offer other attractive features like arbitrary smoothness or roughness uncoupled with the order of accuracy, ease of discretization, ease of adaptivity, and intrinsic enrichment [3,7,9,24]. However, their implementation is less trivial than the finite proximation has been introduced which possesses the weak Kronecker delta property, and can thus strictly satisfy the requirements on the test function (and for simple boundary conditions, the trial function) in the weak formulation [20]. ...

... Trefftz methods originate from Trefftz (1926) and have since been developed for a wide range of problems, for an overview see Zienkiewicz (1997), Hiptmair et al. (2016), Qin (2005), Li et al. (2008), Kita & Kamiya (1995). The central principle of Trefftz methods is the construction of a discrete basis of solutions to the differential operator under consideration, making the space of Trefftz functions problem dependent. ...

... Popular linear projection techniques include the proper orthogonal decomposition (POD) [9], the reduced basis method [10], and the balanced truncation method [11], while autoencoders [12,13] are often applied for nonlinear projection [14,15,16]. The linear-subspace ROM (LS-ROM) has been successfully applied to various problems, such as nonlinear heat conduction [17], Lagrangian hydrodynamics [18,19,20], nonlinear diffusion equations [17,21], Burgers equations [20,22,23,24], convection-diffusion equations [25,26], Navier-Stokes equations [27,28], Boltzmann transport problems [29,30], fracture mechanics [31,32], molecular dynamics [33,34], fatigue analysis under cycling-induced plastic deformations [35], topology optimization [36,37], structural design optimization [38,39], etc. Despite successes of the classical LS-ROM in many applications, it is limited to the assumption that intrinsic solution space falls into a low-dimensional subspace, which means the solution space has a small Kolmogorov n-width. ...

... Transfer function approach was used to formulate a solution of lateral vibration of Euler-Bernoulli axially moving beam over elastic constraints [45]. A radial-based collocation technique, allowing the approximation of higher-order derivatives, was used to solve the governing equation of axially moving beam compromised of second-order differential in time and 4th order differential in space [46][47][48]. In an attempt to study the free vibration in pipes conveying fluids, differential transformation method (DTM) was introduced by Ni et al. [49]. ...

... However, the computational cost of computing this gradient is non-trivial (cf. [30]). Therefore a so-called implicit gradient (which originated from the synchronized derivative [35]) has been developed [16,43] to save computational cost. ...

... Analyzing deformations in incompressible solid mechanics has become increasingly important in recent years due to its wide applications in industrial and research fields, and it is currently the subject of an active research, see [1][2][3][4][5][6][7][8] and therein references. There are numerous methodologies devised to approximate the incompressible linear elasticity equations, including stabilized finite element methods (FEM) [9][10][11][12][13], discontinuous Galerkin methods [14], methods based on the least square approach [15], finite volume methods [1], collocation approaches [16], isogeometric approaches [17][18][19], and boundary element methods [20]. ...

... Gradient smoothing technique can not only reduce the complexity of derivative calculation but also improve the smoothness of shape function derivative. This is beneficial to the calculation of high-order derivatives of meshfree functions such as RKPM [50][51][52]. Liu et al. [53,54] introduced the smoothing technique from the mathematical point of view when setting the finite element model, and formed a series of smooth finite element methods (SFEMs). Node-based SFEM (NSFEM) [55], Edge-based SFEM (ESFEM) [56], Face-based SFEM (FSFEM) [57], etc. [58] are developed from SFEM. ...

... 4 This is the useful FD discretization of the elliptic PDE as using the second-order FD scheme over the nonrectangular domain, and until recently, various case studies and accuracy verifications have been performed. [5][6][7][8][9] Over the past several decades, numerical calculation methods based on the Cartesian coordinate system with complex domains have been rapidly developed. Especially in the field of hydrodynamics, many immersed boundary methods that use the FDM have been proposed (e.g., Refs. ...

... Several studies have addressed the one order of superconvergence, especially for finite element [40,23,2] and finite volume type methods [8,3,38]. Regarding finite difference methods on polygonal domains, Ferreira and Grigorieff have shown in [13] one level of superconvergence at the second order of accuracy for general elliptic operators while Li et al. in [26,24,25] specifically studied the Shortley-Weller scheme. In [27], Matsunaga and Yamamoto proved that the Shortley-Weller scheme provides a third order accuracy near the interface, which can be seen as an essential aspect of the superconvergence property. ...

... In this article, we consider the problem (2.1) with f a specific function. In fact, more general cases can be dealt with similarly (see [7]) under the assumption for ∈ (1/2, 2) ...