The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Yu Huang - One of the best experts on this subject based on the ideXlab platform.
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retracted a hybrid finite element meshfree method based on Partition of Unity for transient wave propagation problems in homogeneous and inhomogeneous media
Applied Mathematical Modelling, 2020Co-Authors: Yingbin Chai, Cong Cheng, Yu HuangAbstract:Abstract The quality of the classical finite element solutions for wave propagation problems are always plagued by the numerical dispersion error and this issue is even more severe in the relatively large wave number range. The present paper focused on employing a hybrid Finite element-Meshfree method based on Partition of Unity concept to solve transient wave propagation problems in homogeneous and inhomogeneous media. In this novel method, the classical parametric shape functions of the linear triangular element are used to construct the Partition of Unity and the radial-polynomial basis functions from the radial point interpolation meshfree method are used for the local approximation. As a consequence, both the individual properties of the finite element and meshfree method are strengthened in this triangular element. Through the obtained numerical results, it is confirmed that the ability of the present triangular element to suppress the numerical dispersion error surpasses the original linear triangular element and this new triangular element has great potential for solving practical wave propagation problems in engineering application.
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A hybrid Finite element-Meshfree method based on Partition of Unity for transient wave propagation problems in homogeneous and inhomogeneous media
Applied Mathematical Modelling, 2020Co-Authors: Yingbin Chai, Cong Cheng, Yu HuangAbstract:Abstract The quality of the classical finite element solutions for wave propagation problems are always plagued by the numerical dispersion error and this issue is even more severe in the relatively large wave number range. The present paper focused on employing a hybrid Finite element-Meshfree method based on Partition of Unity concept to solve transient wave propagation problems in homogeneous and inhomogeneous media. In this novel method, the classical parametric shape functions of the linear triangular element are used to construct the Partition of Unity and the radial-polynomial basis functions from the radial point interpolation meshfree method are used for the local approximation. As a consequence, both the individual properties of the finite element and meshfree method are strengthened in this triangular element. Through the obtained numerical results, it is confirmed that the ability of the present triangular element to suppress the numerical dispersion error surpasses the original linear triangular element and this new triangular element has great potential for solving practical wave propagation problems in engineering application.
Wing Kam Liu - One of the best experts on this subject based on the ideXlab platform.
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reproducing kernel hierarchical Partition of Unity part i formulation and theory
International Journal for Numerical Methods in Engineering, 1999Co-Authors: Wing Kam LiuAbstract:This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical Partition of Unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical Partition, a class of basic wavelet functions are introduced. Based upon the built-in consistency conditions, the differential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with different combinations of the hierarchical kernels, a synchronized convergence effect may be observed. Being different from the conventional Legendre function based p-type hierarchical basis, the new hierarchical basis is an intrinsic pseudo-spectral basis, which can remain as a Partition of Unity in a local region, because the discrete wavelet kernels form a ‘Partition of nullity’. These newly developed kernels can be used as the multi-scale basis to solve partial differential equations in numerical computation as a p-type refinement. Copyright © 1999 John Wiley & Sons, Ltd.
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reproducing kernel hierarchical Partition of Unity part ii applications
International Journal for Numerical Methods in Engineering, 1999Co-Authors: Wing Kam LiuAbstract:In this part of the work, the meshless hierarchical Partition of Unity proposed in [1], referred here as Part I, is used as a multiple scale basis in numerical computations to solve practical problems. The applications discussed in the present work fall into two categories: (1) a wavelet adaptivity refinement procedure; and (2) a so-called wavelet Petrov–Galerkin procedure. In the applications of wavelet adaptivity, the hierarchical reproducing kernels are used as a multiple scale basis to compute the numerical solutions of the Helmholtz equation, a model equation of wave propagation problems, and to simulate shear band formation in an elasto-viscoplastic material, a problem dictated by the presence of the high gradient deformation. In both numerical experiments, numerical solutions with high resolution are obtained by inserting the wavelet-like basis into the primary interpolation function basis, a process that may be viewed as a spectral p-type refinement. By using the interpolant that has synchronized convergence property as a weighting function, a wavelet Petrov–Galerkin procedure is proposed to stabilize computations of some pathological problems in numerical computations, such as advection–diffusion problems and Stokes' flow problem; it offers an alternative procedure in stablized methods and also provides some insight, or new interpretation of the method. Detailed analysis has been carried out on the stability and convergence of the wavelet Petrov–Galerkin method. Copyright © 1999 John Wiley & Sons, Ltd.
Yingbin Chai - One of the best experts on this subject based on the ideXlab platform.
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retracted a hybrid finite element meshfree method based on Partition of Unity for transient wave propagation problems in homogeneous and inhomogeneous media
Applied Mathematical Modelling, 2020Co-Authors: Yingbin Chai, Cong Cheng, Yu HuangAbstract:Abstract The quality of the classical finite element solutions for wave propagation problems are always plagued by the numerical dispersion error and this issue is even more severe in the relatively large wave number range. The present paper focused on employing a hybrid Finite element-Meshfree method based on Partition of Unity concept to solve transient wave propagation problems in homogeneous and inhomogeneous media. In this novel method, the classical parametric shape functions of the linear triangular element are used to construct the Partition of Unity and the radial-polynomial basis functions from the radial point interpolation meshfree method are used for the local approximation. As a consequence, both the individual properties of the finite element and meshfree method are strengthened in this triangular element. Through the obtained numerical results, it is confirmed that the ability of the present triangular element to suppress the numerical dispersion error surpasses the original linear triangular element and this new triangular element has great potential for solving practical wave propagation problems in engineering application.
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A hybrid Finite element-Meshfree method based on Partition of Unity for transient wave propagation problems in homogeneous and inhomogeneous media
Applied Mathematical Modelling, 2020Co-Authors: Yingbin Chai, Cong Cheng, Yu HuangAbstract:Abstract The quality of the classical finite element solutions for wave propagation problems are always plagued by the numerical dispersion error and this issue is even more severe in the relatively large wave number range. The present paper focused on employing a hybrid Finite element-Meshfree method based on Partition of Unity concept to solve transient wave propagation problems in homogeneous and inhomogeneous media. In this novel method, the classical parametric shape functions of the linear triangular element are used to construct the Partition of Unity and the radial-polynomial basis functions from the radial point interpolation meshfree method are used for the local approximation. As a consequence, both the individual properties of the finite element and meshfree method are strengthened in this triangular element. Through the obtained numerical results, it is confirmed that the ability of the present triangular element to suppress the numerical dispersion error surpasses the original linear triangular element and this new triangular element has great potential for solving practical wave propagation problems in engineering application.
Cong Cheng - One of the best experts on this subject based on the ideXlab platform.
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retracted a hybrid finite element meshfree method based on Partition of Unity for transient wave propagation problems in homogeneous and inhomogeneous media
Applied Mathematical Modelling, 2020Co-Authors: Yingbin Chai, Cong Cheng, Yu HuangAbstract:Abstract The quality of the classical finite element solutions for wave propagation problems are always plagued by the numerical dispersion error and this issue is even more severe in the relatively large wave number range. The present paper focused on employing a hybrid Finite element-Meshfree method based on Partition of Unity concept to solve transient wave propagation problems in homogeneous and inhomogeneous media. In this novel method, the classical parametric shape functions of the linear triangular element are used to construct the Partition of Unity and the radial-polynomial basis functions from the radial point interpolation meshfree method are used for the local approximation. As a consequence, both the individual properties of the finite element and meshfree method are strengthened in this triangular element. Through the obtained numerical results, it is confirmed that the ability of the present triangular element to suppress the numerical dispersion error surpasses the original linear triangular element and this new triangular element has great potential for solving practical wave propagation problems in engineering application.
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A hybrid Finite element-Meshfree method based on Partition of Unity for transient wave propagation problems in homogeneous and inhomogeneous media
Applied Mathematical Modelling, 2020Co-Authors: Yingbin Chai, Cong Cheng, Yu HuangAbstract:Abstract The quality of the classical finite element solutions for wave propagation problems are always plagued by the numerical dispersion error and this issue is even more severe in the relatively large wave number range. The present paper focused on employing a hybrid Finite element-Meshfree method based on Partition of Unity concept to solve transient wave propagation problems in homogeneous and inhomogeneous media. In this novel method, the classical parametric shape functions of the linear triangular element are used to construct the Partition of Unity and the radial-polynomial basis functions from the radial point interpolation meshfree method are used for the local approximation. As a consequence, both the individual properties of the finite element and meshfree method are strengthened in this triangular element. Through the obtained numerical results, it is confirmed that the ability of the present triangular element to suppress the numerical dispersion error surpasses the original linear triangular element and this new triangular element has great potential for solving practical wave propagation problems in engineering application.
Marc Alexander Schweitzer - One of the best experts on this subject based on the ideXlab platform.
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Embedding Enriched Partition of Unity Approximations in Finite Element Simulations
Meshfree Methods for Partial Differential Equations VIII, 2017Co-Authors: Marc Alexander Schweitzer, Albert ZiegenhagelAbstract:In this paper we present a general approach to embed arbitrary approximation spaces into classical finite element simulations in a non-intrusive fashion. To this end, we employ a global Partition of Unity method to splice the two independent approximation spaces together. The main goal of this research is to enable the timely evaluation of novel discretization approaches and meshfree techniques in an industrial context by embedding them into large scale finite element simulations. We present some numerical results showing the generality and effectiveness of our approach.
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Variational Mass Lumping in the Partition of Unity Method
SIAM Journal on Scientific Computing, 2013Co-Authors: Marc Alexander SchweitzerAbstract:This paper is concerned with the construction of a variational mass lumping scheme for the Partition of Unity methods. The presented approach is applicable to arbitrary local approximation spaces and any nonnegative Partition of Unity. We give improved error bounds for the Partition of Unity method using a nonnegative Partition of Unity and show that our lumped mass matrix is conservative at least for any $f \in V^{\rm PU}$ such that $f|_{\Omega\cap\omega_i} \in V_i(\omega_i)$ for all patches $\omega_i$. We present numerical results using smooth, higher order, discontinuous, and singular local approximation spaces confirming our theoretical results.
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stable enrichment and local preconditioning in the particle Partition of Unity method
Numerische Mathematik, 2011Co-Authors: Marc Alexander SchweitzerAbstract:This paper is concerned with the stability and approximation properties of enriched meshfree and generalized finite element methods. In particular we focus on the particle-Partition of Unity method (PPUM) yet the presented results hold for any Partition of Unity based enrichment scheme. The goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h-version independent of the regularity of the solution. Hence, we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme. To this end we enrich our PPUM function space in an enrichment zone hierarchically near the singularities of the solution. This initial enrichment however can lead to a severe ill-conditioning and can compromise the stability of the discretization. To overcome the ill-conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable and well-conditioned basis independent of the employed initial enrichment. The construction of this preconditioner is of linear complexity with respect to the number of discretization points. We obtain optimal error bounds for an enriched PPUM discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we observe a kind of super-convergence. The results of our numerical experiments clearly show that our enriched PPUM with local preconditioning recovers the optimal convergence rate of O(h p ) of the uniform h-version globally. For the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of O(h p+δ ) with $${\delta\geq\frac{1}{2}}$$ for p = 1. The convergence rate of our multilevel solver is essentially the same for a purely polynomial approximation and an enriched approximation.
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an adaptive hp version of the multilevel particle Partition of Unity method
Computer Methods in Applied Mechanics and Engineering, 2009Co-Authors: Marc Alexander SchweitzerAbstract:Abstract This paper is concerned with the hp-adaptive multilevel solution of second order elliptic partial differential equations using the meshfree particle–Partition of Unity method. The proposed refinement scheme automatically constructs new discretization points (or particles), the meshfree analogue of an adaptive h-refinement, and local approximation spaces with better local resolution, a p-refinement. The refinement process is steered with the help of an a-posteriori subdomain-type error indicator. We present results of numerical experiments in two and three space dimensions which demonstrated the overall efficiency of the proposed scheme.
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a parallel multilevel Partition of Unity method for elliptic partial differential equations
2003Co-Authors: Marc Alexander SchweitzerAbstract:1 Introduction.- 2 Partition of Unity Method.- 2.1 Construction of a Partition of Unity Space.- 2.2 Properties.- 2.3 Basic Convergence Theory.- 3 Treatment of Elliptic Equations.- 3.1 Galerkin Discretization.- 3.2 Boundary Conditions.- 3.3 Numerical Results.- 4 Multilevel Solution of the Resulting Linear System.- 4.1 Multilevel Iterative Solvers.- 4.2 Multilevel Partition of Unity Method.- 4.3 Numerical Results.- 5 Tree Partition of Unity Method.- 5.1 Single Level Cover Construction.- 5.2 Construction of a Sequence of PUM Spaces.- 5.3 Numerical Results.- 6 Parallelization and Implementational Details.- 6.1 Parallel Data Structures.- 6.2 Parallel Tree Partition of Unity Method.- 6.3 Numerical Results.- 7 Concluding Remarks.- Treatment of other Types of Equations.- A.1 Parabolic Equations.- A.2 Hyperbolic Equations.- Transformation of Keys.- Color Plates.- References.
