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Yonghoon Kwon - One of the best experts on this subject based on the ideXlab platform.

  • Reproducing polynomial particle methods for boundary integral equations
    Computational Mechanics, 2011
    Co-Authors: Hae-soo Oh, Christopher Davis, Yonghoon Kwon
    Abstract:

    Since meshless methods have been introduced to alleviate the difficulties arising in conventional finite element method, many papers on applications of meshless methods to boundary element method have been published. However, most of these papers use moving least squares approximation Functions that have difficulties in prescribing essential boundary conditions. Recently, in order to strengthen the effectiveness of meshless methods, Oh et al. developed meshfree reproducing polynomial particle (RPP) Shape Functions, patchwise RPP and reproducing singularity particle (RSP) Shape Functions with use of flat-top partition of unity. All of these approximation Functions satisfy the Kronecker delta property. In this paper, we report that meshfree RPP Shape Functions, patchwise RPP Shape Functions, and patchwise RSP Shape Functions effectively handle boundary integral equations with (or without) domain singularities.

Hae-soo Oh - One of the best experts on this subject based on the ideXlab platform.

  • Reproducing polynomial particle methods for boundary integral equations
    Computational Mechanics, 2011
    Co-Authors: Hae-soo Oh, Christopher Davis, Yonghoon Kwon
    Abstract:

    Since meshless methods have been introduced to alleviate the difficulties arising in conventional finite element method, many papers on applications of meshless methods to boundary element method have been published. However, most of these papers use moving least squares approximation Functions that have difficulties in prescribing essential boundary conditions. Recently, in order to strengthen the effectiveness of meshless methods, Oh et al. developed meshfree reproducing polynomial particle (RPP) Shape Functions, patchwise RPP and reproducing singularity particle (RSP) Shape Functions with use of flat-top partition of unity. All of these approximation Functions satisfy the Kronecker delta property. In this paper, we report that meshfree RPP Shape Functions, patchwise RPP Shape Functions, and patchwise RSP Shape Functions effectively handle boundary integral equations with (or without) domain singularities.

  • The closed form reproducing polynomial particle Shape Functions for meshfree particle methods
    Computer Methods in Applied Mechanics and Engineering, 2007
    Co-Authors: Hae-soo Oh, Jaewoo Jeong
    Abstract:

    Abstract It has been known that reproducing kernel particle (RKP) Shape Functions with Kronecker delta property are not available in simple forms. Thus, in this paper, we construct highly regular piecewise polynomial reproducing polynomial particle (RPP) Shape Functions that satisfy the Kronecker delta property. Like RKP Shape Functions, the RPP Shape Functions reproduce the complete polynomials of degree k for an integer k ⩾ 0 . Moreover, the particles associated with these RPP Shape Functions are either uniformly distributed in ( - ∞ , ∞ ) or non-uniformly distributed in [ 0 , ∞ ) (or in a compact set [ a ,  b ]).

Ingrid Daubechies - One of the best experts on this subject based on the ideXlab platform.

  • recursive diffeomorphism based regression for Shape Functions
    Siam Journal on Mathematical Analysis, 2018
    Co-Authors: Jieren Xu, Haizhao Yang, Ingrid Daubechies
    Abstract:

    This paper proposes a recursive diffeomorphism-based regression method for the one-dimensional generalized mode decomposition problem that aims at extracting generalized modes $\alpha_k(t)s_k(2\pi N_k\phi_k(t))$ from their superposition $\sum_{k=1}^K \alpha_k(t)s_k(2\pi N_k\phi_k(t))$. We assume that the instantaneous information, e.g., $\alpha_k(t)$ and $N_k\phi_k(t)$, is determined by, e.g., a one-dimensional synchrosqueezed transform or some other methods. Our main contribution is to propose a novel approach based on diffeomorphisms and nonparametric regression to estimate wave Shape Functions $s_k(t)$. This leads to a framework for the generalized mode decomposition problem under a weak well-separation condition. Numerical examples of synthetic and real data are provided to demonstrate the successful application of our approach.

  • recursive diffeomorphism based regression for Shape Functions
    arXiv: Numerical Analysis, 2016
    Co-Authors: Jieren Xu, Haizhao Yang, Ingrid Daubechies
    Abstract:

    This paper proposes a recursive diffeomorphism based regression method for one-dimensional generalized mode decomposition problem that aims at extracting generalized modes $\alpha_k(t)s_k(2\pi N_k\phi_k(t))$ from their superposition $\sum_{k=1}^K \alpha_k(t)s_k(2\pi N_k\phi_k(t))$. First, a one-dimensional synchrosqueezed transform is applied to estimate instantaneous information, e.g., $\alpha_k(t)$ and $N_k\phi_k(t)$. Second, a novel approach based on diffeomorphisms and nonparametric regression is proposed to estimate wave Shape Functions $s_k(t)$. These two methods lead to a framework for the generalized mode decomposition problem under a weak well-separation condition. Numerical examples of synthetic and real data are provided to demonstrate the fruitful applications of these methods.

Hamid Ahmadian - One of the best experts on this subject based on the ideXlab platform.

  • An inverse approach in obtaining Shape Functions for a superconvergent thin plate element
    Inverse Problems in Science and Engineering, 2015
    Co-Authors: Shirko Faroughi, Hamid Ahmadian
    Abstract:

    This article presents an inverse approach to provide Shape Functions associate with a superconvergent thin plate element formulation. In the proposed approach, candidates for Shape Functions of an element are selected using a series of Functions such as, trigonometric series, simple and hierarchical polynomials or a combination of them. Next, one imposes all the physical, geometrical, compatibility and completeness constraints associated with the concerned element on these function. In the final stage, the unknown parameters of the Shape Functions are determined by minimizing the discrimination errors in the element formulation. The proposed method is employed to determine the Shape Functions associate with the superconvergent plate element formulation. The accuracy of the obtained formulation is examined against previously developed plate models using several numerical examples. These comparisons indicate the developed model provides more accurate results both in local and global coordinates system.

  • Shape Functions of superconvergent finite element models
    Thin-walled Structures, 2011
    Co-Authors: Hamid Ahmadian, Shirko Farughi
    Abstract:

    In structural dynamics superconvergent element models are obtained by eigen-value convergence analysis, or minimizing the discretization errors leading to maximum convergence rates in their eigen-solutions. The element formulations developed by these inverse strategies are obtained in local coordinates. As no Shape Functions are employed in their development transforming them to global coordinates is a challenge and prevents their use in practical finite element models. To remove this obstacle a new method is proposed to obtain Shape Functions for superconvergent element models attained directly from the eigen-value convergence analysis or discretization error analysis. The method employs series of trigonometric Functions to obtain Shape Functions corresponding to the superconvergent element formulations. Using the proposed strategy, the Shape Functions for superconvergent rod, beam and transverse vibration membrane are obtained. It is shown transformation of the superconvergent element formulation to the global coordinates using the obtained Shape Functions does not affect the eigen-value convergence rates.

  • Shape Functions associated with super-convergent mass matrix
    Inverse Problems in Science and Engineering, 2011
    Co-Authors: Shirko Faroughi, Hamid Ahmadian, A. Ghareghani
    Abstract:

    This article presents a novel approach in obtaining Shape Functions associated with super convergent element formulations with and continuity. In structural dynamics super-convergent element models are obtained by eigen-value convergence analysis, or minimizing the discretization errors leading to maximum convergence rates in their eigen-solutions. The element formulations developed by these inverse strategies are obtained without the need to define any Shape Functions. This article proposes a methodology to define the inverse model associated Shape Functions. The method employs trigonometric Functions to define Shape Functions corresponding to the super convergent element formulations. Using the proposed strategy, the Shape Functions of super-convergent rod and beam elements are obtained. The presented rod and beam Shape Functions corresponding to the inverse formulations are just examples on how the presented methodology can be implemented.

  • Shape Functions associated with inverse element formulations
    Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science, 2010
    Co-Authors: Shirko Faroughi, Hamid Ahmadian
    Abstract:

    Super-convergent element formulations in local co-ordinates are obtained using inverse strategies. In the inverse approach discretization errors of the element formulation are minimized leading to super-convergent solutions. In the development of the inverse element model, no Shape Functions are introduced and therefore the task of element transformation from local to global co-ordinates system remains a challenge. In this paper, a procedure is proposed to produce Shape Functions associated with the inverse element formulations via hierarchical polynomials. A membrane element formulation is

Oleg Dmitrochenko - One of the best experts on this subject based on the ideXlab platform.

  • A triangular plate element 2343 using second-order absolute-nodal-coordinate slopes: numerical computation of Shape Functions
    Nonlinear Dynamics, 2013
    Co-Authors: Alexander Olshevskiy, Oleg Dmitrochenko
    Abstract:

    In this paper, the process by which geometrical and structural matrices of plate finite elements employing absolute nodal coordinate formulation (ANCF) are constructed is studied. The kinematic and topological properties of an arbitrary plate finite element are described using universal digital code dncm that provides systematic enumeration of finite elements. This code is formed using the element’s dimension d , the number of nodes it possesses  n , the number of scalar coordinates per node  c , and a multiplier describing the process of transforming a conventional finite element to an ANCF element  m . The detailed generation of a new type of triangular plate finite element 2343 using numerical computation of Shape Functions is also discussed in the paper. The new triangular element employs position vectors and slope vectors up to second-order mixed-derivative slope vector. A detailed derivation of the equations of motion of the element is also provided and examples of its numerical simulation and validation presented.