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( )Tj
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(Civil Engineering Dimension, Vol. 17, No. 3, December 2015 \(Special Edi\
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( DOI: 10.9744/CED 17.3.152-157 ISSN 1410-9530 print / ISSN 1979-570X onl\
ine Generalization of FEM)Tj
T*
( Using Node-Based Shape Functions Kanok-Nukulchai, W.1*, Wong, F.T.2, an\
d Sommanawat, W.3)Tj
T*
( Abstract: In standard FEM, the stiffness of an element is exclusively i\
nfluenced by nodes associated with)Tj
T*
( the element via its element-based shape functions. In this paper, the a\
uthors present a method that can be)Tj
T*
( viewed as a generalization of FEM for which the influence of a node is \
not limited by a hat function around)Tj
T*
( the node. Shape functions over an element can be interpolated over a pr\
edefined set of nodes around the)Tj
T*
( element. These node-based shape functions employ Kriging Interpolations\
commonly found in geostatistical)Tj
T*
( technique. In this study, a set of influencing nodes are covered by sur\
rounding layers of elements defined)Tj
T*
( as its domain of influence \(DOI\). Thus, the element stiffness is infl\
uenced by not only the element nodes,)Tj
T*
( but also satellite nodes outside the element. In a special case with ze\
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T*
( specialized to the conventional FEM.)Tj
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( Node )Tj
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(FEM)Tj
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( or )Tj
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(K-FEM.)Tj
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(2D elastostatic, Reissner-Mindlin\222s plate and shell problems.)Tj
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( with )Tj
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( Kringing)Tj
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( shape )Tj
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(functions can be used to interpolate the mesh geometry.)Tj
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( very )Tj
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(useful)Tj
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( for representing the )Tj
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(curved)Tj
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(shells. The)Tj
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d for Analyses of Reissner-)Tj
T*
(Mindlin Plates, Emerging Trends: Key- note Lectures and Symposia, Proc. \
10th East Asia-Pacific)Tj
T*
( Conference Structure Engineering Construction \(EASEC-10\), Bangkok, Th\
ailand, August 3-5, 2006, Asian)Tj
T*
( Institute of Technology, 2006, pp. 509-514. 13. Dai, K.Y., Liu, G.R., L\
im, K.M., and Gu, Y.T., Comparison)Tj
T*
( between the Radial Point Inter- polation and the Kriging Interpolation \
Used in Meshfree Methods,)Tj
T*
( Computational Mechanics, 32, 2003, pp. 60-70. 14. Wong, F.T. and Kanok-\
Nukulchai, W., On the)Tj
T*
( Convergence of the Kriging-based Finite Element Method, Proc. 3rd Asia-\
Pacific Congress on)Tj
T*
( Computational Mechanics \(APCOM\22207\) in conjunction with 11th Intern\
ational Conference Enhancement)Tj
T*
( and Promotion of Computational Methods in Engineering and Science \(EPM\
ESC XI\), Kyoto, Japan,)Tj
T*
( December 3-6, 2007, Asian- Pacific Association for Computational Mecha-\
nics, Paper No. MS38-2-3,)Tj
T*
( 2007. 15. Wong, F.T. and Kanok-Nukulchai, W., On the Convergence of the\
Kriging-based Finite Ele- ment)Tj
T*
( Method, International Journal of Com- putational Methods, 2008. 16. Mas\
ood, Z. and Kanok-Nukulchai, W.,)Tj
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( An Adaptive Mesh Generation for Kriging Element- free Galerkin Method B\
ased on Delaunay Triangulation,)Tj
T*
( Emerging Trends: Keynote Lec- tures and Symposia, Proc. 10th East Asia-\
Pacific Conf. Struct. Eng.)Tj
T*
( Constr. \(EASEC-10\), Bang- kok, Thailand, August 3-5, 2006, Asian Inst\
itute of Technology, 2006, pp. 499-)Tj
T*
(508. 17. Sommanawat, W. and Kanok-Nukulchai, W., The Enrichment of Mater\
ial Discontinuity in Moving)Tj
T*
( Kriging Methods, Emerging Trends: Keynote Lectures and Symposia, Proc. \
10th East Asia-Pacific)Tj
T*
( Conference Structure Engineering Construction \(EASEC-10\), Bangkok, Th\
ailand, August 3-5, 2006, Asian)Tj
T*
( Institute of Technology, 2006, pp. 525-530. 18. Wicaksana, C. and Kanok\
-Nukulchai, W., Dynamic Analysis)Tj
T*
( of Timoshenko Beam and Mindlin Plate by Kriging-Based Finite Element Me\
thods, Emerging Trends:)Tj
T*
( Keynote Lectures and Symposia, Proc. 10th East Asia-Pacific Conf. Struc\
t. Eng. Constr. \(EASEC-10\),)Tj
T*
( Bangkok, Thailand, August 3-5, 2006, Asian Institute of Technology, 200\
6, pp. 515-524. 19. Wong, F.T. and)Tj
T*
( Kanok-Nukulchai, W., A Kriging-based Finite Element Method for Ana- lys\
es of Shell Structures,)Tj
T*
( Proceedings of the 8th World Congress on Computational Mechanics \(WCCM\
8\) and the 5th European)Tj
T*
( Congress on Computational Methods in Applied Sciences and Engineering \(\
ECCOMAS 2008\), Venice,)Tj
T*
( Italy, June 30-July 5, 2008, International Association for Computationa\
l Mechanics, Paper No. 1247, 2008.)Tj
T*
( Kanok-Nukulchai, W. et al. / Generalization of FEM Using Node-Based Sha\
pe Functions / CED, Vol. 17,)Tj
T*
( No. 3, December 2015, pp. 152\226157 Kanok-Nukulchai, W. et al. / Gener\
alization of FEM Using Node-Based)Tj
T*
( Shape Functions / CED, Vol. 17, No. 3, December 2015, pp. 152\226157 Ka\
nok-Nukulchai, W. et al. /)Tj
T*
( Generalization of FEM Using Node-Based Shape Functions / CED, Vol. 17, \
No. 3, December 2015, pp.)Tj
T*
( 152\226157 152 153 154 155 156 157)Tj
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( modified for this new concept. Figure 7 shows the flow diagram of a typ\
ical FEM code extended for K-FEM.)Tj
0 -1.5 TD
( After the modification, the standard FEM becomes in fact a subclass of \
K-FEM. With this convenience, K-)Tj
T*
(FEM)Tj
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( Figure 6. Convergence of the Cantilever Cylindrical Shell: Isoparametri\
c Triangular r K-FEM Shell Element)Tj
T*
( \(with shell surface generated by the same Kriging shape func- tions\) \
Versus Flat Triangular K-FEM Shell)Tj
T*
( Element \(with shell surface interpolated exclusively from its own 3 no\
des\). Figure 7. Flow Chart of a Typical)Tj
T*
( FEM Code Extended to Include K-FEM Kanok-Nukulchai, W. et al. / General\
ization of FEM Using Node-)Tj
T*
(Based Shape Functions / CED, Vol. 17, No. 3, December 2015, pp. 152\2261\
57 Conclusions The basic)Tj
T*
( concept and the advantages of K-FEM have been described. The present me\
thod is as simple as the)Tj
T*
( conventional FEM in terms of its implementation; yet it retains much of\
the advan- tages of mesh-free)Tj
T*
( methods. K.Y. Dai et al. [13] pointed out that the method using standar\
d Galerkin weak form with KI is)Tj
T*
( noncom- forming and so is K-FEM. This means the elemental piecewise KI \
is not fully compatible across)Tj
T*
( the inter- element boundaries. Its effect on the convergence was studie\
d in the context of 2D elastostatic)Tj
T*
( pro- blems [14, 15]. It was found that K-FEM with appropriate choice of\
correlation function passes the)Tj
T*
( weak patch test and therefore the convergence can be guaranteed. One po\
ssible drawback of K-FEM is its)Tj
T*
( excessive demand of the computational time, as Kriging shape functions \
are constructed element by)Tj
T*
( element during the computation. Moreover, a larger DOI means a longer t\
ime for stiffness formation and for)Tj
T*
( solving a system with larger average bandwidth. However under the curre\
nt trend, the cost of running a)Tj
T*
( FEM project is heavily weighted on the engineer\222s time for preparing\
meshes, rather than on the)Tj
T*
( computational time. Several investigations have been carried out succes\
s- fully on different applications of)Tj
T*
( K-FEM. Aside from plane elasticity problems [9, 16, 17], so far Kriging\
- based finite elements have been)Tj
T*
( developed for degenerated solid beams, plates and shells [12, 18, 19]. \
The results confirmed that K-FEM is)Tj
T*
( indeed a viable alternative to the conventional FEM and has great poten\
tial in engineering applications.)Tj
T*
( Future research may be directed at \(1\) applications of K- FEM to nonl\
inear problems and \(2\) improvement)Tj
T*
( of its computational efficiency References 1. Belytschko, T., Krongauz,\
Y., Organ, D., Fleming, M., and)Tj
T*
( Krysl, P., Meshless Methods: An Overview and Recent Developments, Compu\
ter Methods in Applied)Tj
T*
( Mechanics and Engineering, 139, 1996, pp. 3-47. 2. Fries, T.P. and Matt\
hies, H.G., Classification and)Tj
T*
( Overview of Meshfree Methods, Institute of Scientific Computing, Techni\
cal University Braunschweig,)Tj
T*
( Brunswick, Germany, 2004. 3. Gu, Y.T., Meshfree Methods and Their Compa\
risons, International Journal of)Tj
T*
( Compu- tational Methods, 2, 2005. pp. 477-515. 4. Belytschko, T., Lu, Y\
.Y., and Gu, L., Element- free)Tj
T*
( Galerkin Methods, International Journal for Numerical Methods in Engine\
ering, 37, 1994, pp. 229-256. 5.)Tj
T*
( Nayroles, B., Touzot, G., and Villon, P., Gene- ralizing the Finite Ele\
ment Method: Diffuse Approximation)Tj
T*
( and Diffuse Elements, Computa- tional Mechanics, 10, 1992, pp. 307-318.\
6. Gu, L., Moving Kriging)Tj
T*
( Interpolation and Ele- ment-free Galerkin Method, International Jour- n\
al for Numerical Methods in)Tj
T*
( Engineering, 56, 2003, pp. 1-11. 7. Tongsuk, P. and Kanok-Nukulchai, W.\
, Further Investigation of Element-)Tj
T*
(Free Galerkin Method using Moving Kriging Interpolation, Internatio- nal\
Journal of Computational Methods,)Tj
T*
( 1, 2004, pp. 345-365. 8. Sayakoummane, V. and Kanok-Nukulchai, W., A Me\
shless Analysis of Shells)Tj
T*
( Based on Moving Kriging Interpolation, International Journal of Computa\
tional Methods, 4, 2007, pp. 543-)Tj
T*
(565. 9. Plengkhom, K. and Kanok-Nukulchai, W., An Enhancement of Finite \
Element Methods with Moving)Tj
T*
( Kriging Shape Functions, International Journal of Computational Methods\
, 2, 2005, pp. 451-475. 10. Olea,)Tj
EMC
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(1)Tj
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/Article <>BDC
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/Article <>BDC
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10 36 592 730 re
W n
BT
/TT1 1 Tf
10.5 0 0 10.5 53.5 756 Tm
( \(P2\) and cubic \(P3\) polynomial basis functions. For P1, it is possi\
ble to use 1, 2, or 3 element layers for the)Tj
ET
Q
BT
/TT1 1 Tf
10.5 0 0 10.5 53.5 740.25 Tm
( DOI. However, at least 2 layers must be used for P2 and at least 3 laye\
rs for P3, following the general rule)Tj
0 -1.5 TD
( that the number of nodes covered in the DOI must not be fewer than the \
number of terms in the polynomial)Tj
T*
( basis. Results of the end deflection, normalized by the exact solution,\
for all cases are presented in Table 1)Tj
T*
( together with the corresponding computational times Normal Stress Norma\
l Stress Shear Stress Figure 2.)Tj
T*
( Stress Contours of Cantilever Plane-stress Beam by K-FEM with Cubic Bas\
is Function and Three Element)Tj
T*
( Layers of DOI. Table 1. Results Obtained from K-FEM with Different Opti\
ons for the Plane-stress Model of a)Tj
T*
( Cantilever Beam h-refine- ment p-refine- ment l-refine- Normalized Time\
* ment solution \(sec\) 1 \(FEM\) 0.928)Tj
T*
( 1.22 P1-Basis 2 layers 0.979 6.86 6x10 3 layers 0.986 23.17 P2-Basis 2 \
layers 0.999 7.02 3 layers 0.998)Tj
T*
( 23.41 P3-Basis 3 layers 1.000 23.69 1 \(FEM\) 0.981 4.81 P1-Basis 2 lay\
ers 0.994 30.06 12x20 3 layers)Tj
T*
( 0.997 115.00 P2-Basis 2 layers 1.000 30.20 3 layers 1.000 116.45 P3-Bas\
is 3 layers 1.000 111.78 24x40 1)Tj
T*
( \(FEM\) 0.995 19.34 P1-Basis 2 layers 0.998 134.92 3 layers 0.999 527.4\
4 P2-Basis P3-Basis 2 layers 3)Tj
T*
( layers 3 layers 1.000 1.000 1.000 136.55 527.64 531.36 * Note: Executio\
n on Laptop PC with Core2)Tj
T*
( DuoT5200 processor, 1.6 GHz Accuracy performance and computational time\
s over the matrix of the h-)Tj
T*
(refinement and the l-refinement, all using linear basis function, are pr\
esented in Figure 3. For relatively)Tj
T*
( crude meshes, the accuracy can be enhanced by adopting a larger DOI wit\
h more layers of elements.)Tj
T*
( Almost the same accuracy can be achieved by h-refinement from 6x10 to 2\
4x40 mesh sizes, or by l-)Tj
T*
(refinement from 1 to 3 element layers. The latter requires about 20% mor\
e computational time. However for)Tj
T*
( the case of h-refinement, we do not consider engineer\222s time for the\
remesh. A more detailed comparison of)Tj
T*
( beam displacement profile between h-refinement and l-refinement is illu\
strated in Figure 4. Figure 3.)Tj
T*
( Matrices of Solution Accuracy and Computa- tional Times for h-refinemen\
t and l-refinement, all using Linear)Tj
T*
( Basis Function Accuracy performance and computational times over the ma\
trix for h-refinement and p-)Tj
T*
(refinement, all using DOI of three element layers, are presented in Figu\
re 5. From the figure, higher)Tj
T*
( accuracy can be achieved for a fixed mesh by simply adopting a higher o\
rder basis function without)Tj
T*
( significantly increasing the computing time. Figure 4. Cantilever Beam \
Modeled by Tetrahedral Solid)Tj
T*
( Elements: Comparison of h-refinement vs l-refinement. Figure 5. Matrice\
s of Solution Accuracy and)Tj
T*
( Computatio- nal Times for h-refinement and p-refinement, all using Thre\
e Element Layers DOI. Geometry of)Tj
T*
( Curved Domain can be Repre- sented More Accurately by KI Isoparame- tri\
c Mapping The same set of)Tj
T*
( Kriging shape functions for field variable)Tj
/CS1 cs 1 0 0 scn
/TT0 1 Tf
11.25 0 0 11.25 99.25 273.75 Tm
(can be used to interpolate the)Tj
/CS0 cs 0.333 scn
/TT1 1 Tf
10.5 0 0 10.5 258.6506 273.75 Tm
( geometric field. )Tj
/CS1 cs 1 0 0 scn
/TT0 1 Tf
11.25 0 0 11.25 335.6831 273.75 Tm
(This is)Tj
/CS0 cs 0.333 scn
/TT1 1 Tf
10.5 0 0 10.5 371.3173 273.75 Tm
( very )Tj
/CS1 cs 1 0 0 scn
/TT0 1 Tf
11.25 0 0 11.25 396.9879 273.75 Tm
(useful)Tj
/CS0 cs 0.333 scn
/TT1 1 Tf
10.5 0 0 10.5 430.1172 273.75 Tm
( for)Tj
-31.511 -1.929 Td
( )Tj
/CS1 cs 1 0 0 scn
/TT0 1 Tf
11.25 0 0 11.25 102.1672 253.5 Tm
(curved)Tj
/CS0 cs 0 scn
/TT1 1 Tf
10.5 0 0 10.5 53.5 212.25 Tm
( shell problems. To demonstrate this advantage, a cantilever quarter cyl\
inder shell under pure bending is)Tj
T*
( modeled by triangular elements as shown in Figure 6. K-FEM is used to s\
olve the shell problem with quartic)Tj
T*
( basis functions and a DOI of 4 element layers. This shell problem will \
be tested for two different situations,)Tj
T*
( one with and the other without isoparametric mapping. In the first case\
, the geometry of individual shell)Tj
T*
( elements shall be interpolated by Kriging shape functions. In the latte\
r case, the geometry of individual shell)Tj
T*
( element is basically a flat facet. The results clearly confirm the adva\
ntage of the Kriging interpolated shell)Tj
T*
( geometry. Implementation of K-FEM can be Easily Incor- porated into Exi\
sting FEM Codes. As K-FEM)Tj
T*
( inherits)Tj
/CS1 cs 1 0 0 scn
/TT0 1 Tf
11.25 0 0 11.25 99.25 60.75 Tm
(the computational procedure of FEM, existing)Tj
/CS0 cs 0.333 scn
/TT1 1 Tf
10.5 0 0 10.5 343.6464 60.75 Tm
( general-purpose )Tj
/CS1 cs 1 0 0 scn
/TT0 1 Tf
11.25 0 0 11.25 425.9493 60.75 Tm
(FE)Tj
/CS0 cs 0.333 scn
/TT1 1 Tf
10.5 0 0 10.5 99.25 40.5 Tm
( programs )Tj
ET
q
10 36 592 730 re
W n
BT
/CS1 cs 1 0 0 scn
/TT0 1 Tf
11.25 0 0 11.25 149.4326 40.5 Tm
(can be easily)Tj
ET
EMC
Q
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q
10 36 592 730 re
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BT
0 scn
/TT0 1 Tf
10.5 0 0 10.5 53.5 756 Tm
(FEM is then iden- tical to the conventional FEM. Kriging Interpolation N\
amed after Danie G. Krige, a South)Tj
ET
Q
BT
0 scn
/TT0 1 Tf
10.5 0 0 10.5 53.5 740.25 Tm
( African mining engineer, Kriging is a well-known geostatistical techniq\
ue for spatial data interpolation in)Tj
0 -1.5 TD
( geology and mining. Using this interpolation, unknown at any point can \
be interpolated from known values)Tj
T*
( at scattered points in its specified neighborhood. The basic concepts o\
f the KI in the context of K-FEM are)Tj
T*
( presented in the following. A detail explanation and derivation of Krig\
ing can be found in the geos- tatistics)Tj
T*
( literatures \(e.g. [10, 11]\). Consider a two-dimensional domain modele\
d by a mesh of triangular elements)Tj
T*
( \(Figure 1\). Suppose there is a single field variable over the domain,\
u\(x\). For each element, the KI is)Tj
T*
( constructed over a set of nodes in a sub-domain ?E ? ? encompassing a p\
redetermined number of layers of)Tj
T*
( elements. The KI over sub-domain ?E can be expressed in the usual FE fo\
rm, i.e., uh \(x\) ? N\(x\) d , where)Tj
T*
( N\(x\) is the 1? n matrix of Kriging shape functions and d is the n ?1 \
matrix of field values at the nodes. In)Tj
T*
( contrast to the FEM, here n is not necessarily only the number of nodes\
associated with the element, but)Tj
T*
( also includes all its satellite nodes. InKrigingformulation,thefieldvar\
iableu\(x\),which is a deterministic)Tj
T*
( function, is viewed as the realization of a random function U\(x\). The\
shape function matrix can be)Tj
T*
( expressed as N\(x\)?pT\(x\)A?rT\(x\)B,wherepT\(x\) isthe1?m vectorofm-t\
erms-polynomialbasisandrT\(x\) isthe 1?)Tj
T*
(n vectorofcovarianceassociatedwithrespective randomfunctionUatnodesi=1,\205\
,n,andUatthe point under)Tj
T*
( consideration, x. Matrices Am?nand B n?n are defined as A ? \(PTR?1P\)?\
1PTR?1 and B ? R ?1 \(I ? PA\) , in)Tj
T*
( which P is the n ? m matrix of polynomial values at the nodes in the DO\
I, R is the n ? n matrix of covariance)Tj
T*
( between U\(x\) at a pair of nodes, and I is the n ? n identity matrix. \
From the above formulation, constructing)Tj
T*
( Kriging shape functions requires a polynomial basis function and a corr\
elation function. For the basis)Tj
T*
( function, Kanok-Nukulchai, W. et al. / Generalization of FEM Using Node\
-Based Shape Functions / CED,)Tj
T*
( Vol. 17, No. 3, December 2015, pp. 152\226157 besides complete polynomi\
al bases, it is also possible to use)Tj
T*
( incomplete polynomial bases such as bi-linear, bi-quadratic and bi-cubi\
c bases. A widely used correlation)Tj
T*
( function in the area of computational mechanics is the Gaussian correla\
tion function [6-9]. This function)Tj
T*
( contains an important parameter affecting the quality of KI, known as t\
he correlation parameter )Tj
/C2_0 1 Tf
42.412 0 Td
<0219>Tj
/TT0 1 Tf
(. In order)Tj
-42.412 -1.5 Td
( to obtain reasonable results in K-FEM, K. Plengkhom and W. Kanok-Nukulc\
hai [9] suggested a criterion for)Tj
T*
( choosing a stable range of )Tj
/C2_0 1 Tf
12.286 0 Td
<0219>Tj
/TT0 1 Tf
(. Recently the authors introduced a new correlation function [12] in the\
form of)Tj
-12.286 -1.5 Td
( a quartic spline \(QS\) correlation function. Our studies indicate a su\
perior performance of QS to the)Tj
T*
( Gaussian correlation function, as the resulting Kriging shape functions\
are less sensitive to the change of )Tj
/C2_0 1 Tf
46.916 0 Td
<0219>Tj
/TT0 1 Tf
(.)Tj
-46.916 -1.5 Td
( In Figure 1, to illustrate the concept of element- layered DOI, suppose\
that the element of interest in a)Tj
T*
( square domain is Element 1, the choices of DOI, comprising one up to fo\
ur element layers, are shown in the)Tj
T*
( Figure 1. It is noted that the DOI does not have to be convex. If one u\
ses quadratic basis func- tion \(m=6\))Tj
T*
( and choose to use three-layered DOI to construct KI over Element 1, the\
DOI will encom- pass 30 \(n=30\))Tj
T*
( nodes. The plot of Kriging shape function associated with node I, based\
on QS correlation function, is)Tj
T*
( shown in the right-hand side of Figure 1. Key Advantages of K-FEM The S\
tress Field can be Obtained with)Tj
T*
( Remar- kable Accuracy and Global Smoothness Using the same mesh size, K\
-FEM yields a stress field)Tj
T*
( with higher accuracy and better smoothness than that of the standard FE\
M. This is because one can freely)Tj
T*
( adopt a higher-order basis function and a larger DOI for any fixed mesh\
. To show this, a cantilever plane-)Tj
T*
(stress beam, Figure 2, under end parabolic shear is modeled with a crude\
mesh of 6x10 triangular elements.)Tj
T*
( In the same figure, the quality of stress output obtained by K-FEM usin\
g cubic basis and three-layered DOI)Tj
T*
( is demonstrated by the stress contours generated directly from nodal va\
lues with no post-processing)Tj
T*
( manipulation. Like FEM, there is no guarantee for stress field to be pe\
rfectly continuous across the inter-)Tj
T*
(element boun- daries; however, the degree of discontinuity is found to b\
e rather insignificant. Solution)Tj
T*
( Refinements can be Achieved with no Re-meshing In K-FEM, quality improv\
ement of solutions can be)Tj
T*
( achieved by: \(a\) increasing the order of the basis function or p-refi\
nement, or \(b\) enlarging the element-)Tj
T*
(layered DOI or l-refinement. For illustra- tion, the cantilever plane-st\
ress beam is modeled with 3 mesh)Tj
T*
( sizes, i.e., with 6x10, 12x20 and 24x40 triangular elements. Each mesh \
is tested with linear \(P1\), quadratic)Tj
ET
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