Reproducing kernel element method Part II: Globally conforming Im/Cn hierarchies

2004 ◽  
Vol 193 (12-14) ◽  
pp. 953-987 ◽  
Author(s):  
Shaofan Li ◽  
Hongsheng Lu ◽  
Weimin Han ◽  
Wing Kam Liu ◽  
Daniel C. Simkins
Keyword(s):  
Reproducing Kernel ◽  
Element Method ◽  
Reproducing Kernel Element

The quasi-uniformity condition for reproducing kernel element method meshes

2009 ◽  
Vol 44 (3) ◽  
pp. 333-342 ◽  
Author(s):  
Nathan Collier ◽  
Daniel Craig Simkins
Keyword(s):  
Reproducing Kernel ◽  
Uniformity Condition ◽  
Element Method ◽  
Reproducing Kernel Element

scholarly journals Reproducing Kernel Element Method for Galerkin Solution of Elastostatic Problems

2012 ◽  
Vol 8 (16) ◽  
pp. 71-96
Author(s):  
Mario J Juha
Keyword(s):  
Stiffness Matrix ◽  
Reproducing Kernel ◽  
Dimensional Space ◽  
Shape Functions ◽  
Computational Implementation ◽  
Galerkin Solution ◽  
Order Smoothness ◽  
High Order Smoothness ◽  
Element Method ◽  
Reproducing Kernel Element

The Reproducing Kernel Element Method (RKEM) is a relatively new technique developed to have two distinguished features: arbitrary high order smoothness and arbitrary interpolation order of the shape functions. This paper provides a tutorial on the derivation and computational implementation of RKEM for Galerkin discretizations of linear elastostatic problems for one and two dimensional space. A key characteristic of RKEM is that it do not require mid-side nodes in the elements to increase the interpolatory power of its shape functions, and contrary to meshless methods, the same mesh used to construct the shape functions is used for integration of the stiffness matrix. Furthermore, some issues about the quadrature used for integration arediscussed in this paper. Its hopes that this may attracts the attention of mathematicians.

Reproducing kernel element method Part III: Generalized enrichment and applications

2004 ◽  
Vol 193 (12-14) ◽  
pp. 989-1011 ◽  
Author(s):  
Hongsheng Lu ◽  
Shaofan Li ◽  
Daniel C. Simkins ◽  
Wing Kam Liu ◽  
Jian Cao
Keyword(s):  
Reproducing Kernel ◽  
Element Method ◽  
Reproducing Kernel Element

Reproducing kernel element method. Part I: Theoretical formulation

2004 ◽  
Vol 193 (12-14) ◽  
pp. 933-951 ◽  
Author(s):  
Wing Kam Liu ◽  
Weimin Han ◽  
Hongsheng Lu ◽  
Shaofan Li ◽  
Jian Cao
Keyword(s):  
Reproducing Kernel ◽  
Theoretical Formulation ◽  
Element Method ◽  
Reproducing Kernel Element

Reproducing kernel element method. Part IV: Globally compatible triangular hierarchy

2004 ◽  
Vol 193 (12-14) ◽  
pp. 1013-1034 ◽  
Author(s):  
Daniel C. Simkins ◽  
Shaofan Li ◽  
Hongsheng Lu ◽  
Wing Kam Liu
Keyword(s):  
Reproducing Kernel ◽  
Element Method ◽  
Reproducing Kernel Element

Treatment of discontinuity in the reproducing kernel element method

2005 ◽  
Vol 63 (2) ◽  
pp. 241-255 ◽  
Author(s):  
Hongsheng Lu ◽  
Do Wan Kim ◽  
Wing Kam Liu
Keyword(s):  
Reproducing Kernel ◽  
Element Method ◽  
Reproducing Kernel Element

scholarly journals Enrichment of the Finite Element Method With the Reproducing Kernel Particle Method

1997 ◽  
Vol 64 (4) ◽  
pp. 861-870 ◽  
Author(s):  
Wing Kam Liu ◽  
R. A. Uras ◽  
Y. Chen
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Reproducing Kernel ◽  
Particle Method ◽  
Computational Procedure ◽  
Reproducing Kernel Particle Method ◽  
The Finite Element Method ◽  
Window Functions ◽  
Reproducing Kernel Particle ◽  
Element Method

The reproducing kernel particle method (RKPM) has attractive properties in handling high gradients, concentrated forces, and large deformations where other widely implemented methodologies fail. In the present work, a multiple field computational procedure is devised to enrich the finite element method with RKPM, and RKPM with analytical functions. The formulation includes an interaction term that accounts for any overlap between the fields, and increases the accuracy of the computational solutions in a coarse mesh or particle grid. By replacing finite element method shape Junctions at selected nodes with higher-order RKPM window functions, RKPM p-enrichment is obtained. Similarly, by adding RKPM window functions into a finite element method mesh, RKPM hp-enrichment is achieved analogous to adaptive refinement. The fundamental concepts of the multiresolution analysis are used to devise an adaptivity procedure.

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