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

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## Publications (22)

In this work we aim to provide a fundamental theory of the reproducing kernel particle method for solving elliptic eigenvalue problems. We concentrate on the convergence analysis of eigenvalues and eigenfunctions, as well as the optimal estimations which are shown to be related to the reproducing degree, support size, and overlapping number in the...

Model order reduction (MOR) techniques for enriched reproducing kernel meshfree methods are proposed for analysis of Poisson problems with mild and strong singularities. The employment of an integrated singular basis function method (ISBFM), in conjunction with the selection of harmonic near-tip asymptotic basis functions, leads to a Galerkin formu...

A weighted strong form collocation framework with mixed radial basis
approximations for the pressure and displacement fields is proposed for
incompressible and nearly incompressible linear elasticity. It is shown
that with the proper choice of independent source points and collocation
points for the radial basis approximations in the pressure and
d...

The earlier work in the development of direct strong form collocation methods, such as the reproducing kernel collocation method (RKCM), addressed the domain integration issue in the Galerkin type meshfree method, such as the reproducing kernel particle method, but with increased computational complexity because of taking higher order derivatives o...

Solving partial differential equations using strong form collocation with nonlocal approximation functions such as orthogonal polynomials and radial basis functions offers an exponential convergence, but with the cost of a dense and ill-conditioned linear system. In this work, the local approximation functions based on reproducing kernel approximat...

Solving partial differential equations (PDE) with strong form collocation and nonlocal approximation functions such as orthogonal polynomials, trigonometric functions, and radial basis functions exhibits exponential convergence rates; however, it yields a full matrix and suffers from ill conditioning. In this work, we discuss a reproducing kernel c...

Meshfree methods have been developed based on Galerkin type weak formulation and strong formulation with collocation. Galerkin type formulation in conjunction with the compactly supported approxima-tion functions and polynomial reproducibility yields algebraic convergence, while strong form collocation method with nonlocal approximation such as rad...

Strong form collocation with radial basis approximation is introduced for the numerical solution of transient dynamics. Von
Neumann stability analysis of this radial basis collocation method is performed to obtain the stability conditions for second
order wave equation with central difference temporal discretization. The shape parameter of the radi...

The direct approximation of strong form using radial basis functions (RBFs), commonly called the radial basis collocation method (RBCM), has been recognized as an effective means for solving boundary value problems. Nevertheless, the non-compactness of the RBFs precludes its application to problems with local features, such as fracture problems, am...

We introduce a radial basis collocation method to solve axially moving beam problems which involve order differentiation in time and order differentiation in space. The discrete equation is constructed based on the strong form of the governing equation. The employment of multiquadrics radial basis function allows approximation of higher order deriv...

Strong form collocation in conjunction with radial basis approximation functions offer implementation simplicity and exponential convergence in solving partial differential equations. However, the smoothness and nonlocality of radial basis functions pose considerable difficulties in solving problems with local features and heterogeneity. In this wo...

In this work, we discuss a reproducing kernel collocation method (RKCM) for solving order PDE based on strong formulation, where the reproducing kernel shape functions with compact support are used as approximation functions. The method based on strong form collocation avoids the domain integration, and leads to well-conditioned discrete system of...

This paper presents a superconvergence analysis for the Shortley-Weller finite difference approximation of Poisson’s equation with unbounded derivatives on a polygonal domain [cf. Z.-C. Li, T. Yamamoto and Q. Fang, J. Comput. Appl. Math. 151, No. 2, 307–333 (2003; Zbl 1030.65108); Z.-C. Li, H.-Y. Hu, Q. Fang, and T. Yamamoto, Numer. Funct. Anal. Op...

This paper presents a superconvergence analysis for the Shortley–Weller finite difference approximation of second-order self-adjoint elliptic equations with unbounded derivatives on a polygonal domain with the mixed type of boundary conditions. In this analysis, we first formulate the method as a special finite element/volume method. We then analyz...

In this paper, we provide an analysis on the collocation methods (CM), which uses a large scale of admissible functions such as orthogonal polynomials, trigonometric functions, radial basis functions and particular solutions, etc. The admissible functions can be chosen to be piecewise, i.e., different functions are used in different subdomains. The...

In this paper, we provide a framework of combinations of collocation method (CM) with the finite-element method (FEM). The key idea is to link the Galerkin method to the least squares method which is then approximated by integration approximation, and led to the CM. The new important uniformly V0h-elliptic inequality is proved. Interestingly, the i...

This paper introduces radial basis functions (RBF) into the collocation methods and the combined methods for elliptic boundary value problems. First, the Ritz-Galerkin method (RGM) is chosen using the RBF, and the integration approximation leads to the collocation method of RBF for Poisson's equation. Next, the combinations of RBF with finite-eleme...

In this short article, we recalculate the numerical example in Křížek and Neittaanmäki (1987) for the Poisson solution u= x σ(1− x)sinπ y in the unit square S as . By the finite difference method, an error analysis for such a problem is given from our previous study by where h is the meshspacing of the uniform square grids used, and C 1 and C 2 are...

In this paper, the harmonic functions of Laplace's equations are derived explicitly for the Dirichlet and the Neumann boundary conditions on the boundary of a sector. Those harmonic functions are more explicit than those of Volkov [Volkov EA, Block method for solving the Laplace equation and for constructing conformal mappings. Boca Raton: CRC Pres...

The purpose of this paper is to extend the boundary approximation method proposed by Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], i.e. the collocation Trefftz method called in this paper, for biharmonic equations with singularities. First, this paper derives the Green formulas for biharmonic equations on bounded domains with a non-smooth boundar...

This is a continued analysis on superconvergence of solution derivatives for the Shortley–Weller approximation in Li (Li, Z. C., Yamamoto, T., Fang, Q. ([2003]): Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson's equation, Part I. Smoothness problems. J. Comp. and Appl. Math. 152(2):307–333), whic...

## Citations

... 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. ...

Reference: 5.0018915

... 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) ...