This paper describes a numerical technique for solving engineering analysis problems that combine radial basis functions and collocation technique with meshfree method with distance fields, also known as solution structure method. The proposed hybrid technique enables exact treatment of all prescribed boundary conditions at every point on the geometric boundary and can be efficiently implemented for both structured and unstructured grids of basis functions. Ability to use unstructured grids empowers the meshfree method with distance fields with higher level of geometric flexibility. By providing exact treatment of the boundary conditions, the new approach makes it possible to exclude boundary conditions from the collocation equations. This reduces the size of the algebraic system, which results in faster solutions. At the same time, the boundary collocation points can be used to enforce the governing equation of the problem, which enhances the solution’s accuracy. Application of the proposed method to solution of heat transfer problems is illustrated on a number of benchmark problems. Modeling results are compared with those obtained by the traditional collocation technique and meshfree method with distance fields.

References

References
1.
Belytschko
,
T.
,
Krongauz
,
Y.
,
Organ
,
D.
,
Fleming
,
M.
, and
Krysl
,
P.
, 1996, “
Meshless Methods: An Overview and Recent Developments
,”
Comput. Methods Appl. Mech. Eng.
,
139
(
1–4
), pp.
3
47
.
2.
Babuska
,
I.
,
Banerjee
,
U.
, and
Osborn
,
J. E.
, 2003, “
Survey of Meshless and Generalized Finite Element Methods: A Unified Approach
,”
Acta Numerica
,
12
, pp.
1
125
.
3.
Günther
,
F. C.
, and
Liu
,
W. K.
, 1998, “
Implementation of Boundary Conditions for Meshless Methods
,”
Comput. Methods Appl. Mech. Eng.
,
163
, pp.
205
230
.
4.
Freytag
,
M.
,
Shapiro
,
V.
, and
Tsukanov
,
I.
, 2006, “
Field Modeling With Sampled Distances
,”
Comput.-Aided Des.
,
38
(
2
), pp.
87
100
.
5.
Rvachev
,
V. L.
, 1982,
Theory of R-Functions and Some Applications
,
Naukova Dumka
, Kiev, Ukraine, in
Russian
.
6.
Rvachev
,
V. L.
,
Sheiko
,
T. I.
,
Shapiro
,
V.
, and
Tsukanov
,
I.
, 2000, “
On Completeness of RFM Solution Structures
,”
Comput. Mech.
,
25
, pp.
305
317
.
7.
Tsukanov
,
I.
,
Shapiro
,
V.
, and
Zhang
,
S.
, 2003, “
A Meshfree Method for Incompressible Fluid Dynamics Problems
,”
Int. J. Numer. Methods Eng.
,
58
(
1
), pp.
127
158
.
8.
Kantorovich
,
L. V.
, and
Krylov
,
V. I.
, 1958,
Approximate Methods of Higher Analysis
,
Interscience
,
New York
.
9.
Tsukanov
,
I.
, and
Shapiro
,
V.
, 2005, “
Meshfree Modeling and Analysis of Physical Fields in Heterogeneous Media
,”
Adv. Comput. Math.
,
23
(
1–2
), pp.
95
124
.
10.
Rvachev
,
V. L.
,
Sheiko
,
T. I.
,
Shapiro
,
V.
, and
Tsukanov
,
I.
, 2001, “
Transfinite Interpolation Over Implicitly Defined Sets
,”
Comput. Aided Geom. Des.
,
18
, pp.
195
220
.
11.
Shapiro
,
V.
, and
Tsukanov
,
I.
, 1999, “
Meshfree Simulation of Deforming Domains
,”
Comput.-Aided Des.
,
31
(
7
), pp.
459
471
.
12.
Shapiro
,
V.
, and
Tsukanov
,
I.
, 2002, “
The Architecture of SAGE—A Meshfree System Based on RFM
,”
Eng. Comput.
,
18
(
4
), pp.
295
311
.
13.
Belytschko
,
T.
,
Parimi
,
C.
,
Moës
,
N.
,
Sukumar
,
N.
, and
Usui
,
S.
, 2003, “
Structured Extended Finite Element Methods for Solids Defined by Implicit Surfaces
,”
Int. J. Numer. Methods Eng.
,
56
(
4
), pp.
609
635
.
14.
Höllig
,
K.
, 2003,
Finite Element Methods With B-Splines
(Frontiers in Applied Mathematics),
Society Industrial and Applied Mathematics
,
Philadelphia, PA
.
15.
Kansa
,
E. J.
, (1999), “
Motivation for Using Radial Basis Functions to Solve PDEs
,” http://www.cityu.edu.hk/rbfpde/files/overview-pdf.pdfhttp://www.cityu.edu.hk/rbfpde/files/overview-pdf.pdf
16.
Schaback
,
R.
, 1995, “
Error Estimates and Condition Numbers for Radial Basis Function Interpolation
,”
Adv. Comput. Math.
, (
3
), pp.
251
264
.
17.
Buhmann
,
M. D.
, 2001, “
A New Class of Radial Basis Functions With Compact Support
,”
Math. Comput.
,
70
(
233
), pp.
307
318
.
18.
Buhmann
,
M. D.
, 2009,
Radial Basis Functions: Theory and Implementations
(Cambridge Monographs on Applied and Computational Mathematics),
Cambridge University Press
,
Cambridge
.
19.
Ling
,
L.
, and
Hon
,
Y. C.
, 2005, “
Improved Numerical Solver for Kansa’s Method Based on Affine Space Decomposition
,”
Eng. Anal. Boundary Elem.
,
29
, pp.
1077
1085
.
20.
Larsson
,
E.
, and
Fornberg
,
B.
, 2003, “
A Numerical Study of Some Radial Basis Function Based Solution Methods for Elliptic PDEs
,”
Comput. Math. Appl.
,
46
(
5–6
), pp.
891
902
.
21.
Atluri
,
S. N.
,
Kim
,
H.-G.
, and
Cho
,
J. Y.
, 1999, “
A Critical Assessment of the Truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) Methods
,”
Comput. Mech.
,
24
, pp.
348
372
.
22.
Atluri
,
S. N.
, and
Zhu
,
T.
, 1998, “
A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics
,”
Comput. Mech.
,
22
(
2
), pp.
117
127
.
23.
Jin
,
X.
,
Li
,
G.
, and
Aluru
,
N. R.
, 2005, “
New Approximations and Collocation Schemes in the Finite Cloud Method
,”
Comput. Struct.
,
83
(
17–18
), pp.
1366
1385
.
24.
Hon
,
Y. C.
, and
Schaback
,
R.
, 2001, “
On Unsymmetric Collocation by Radial Basis Functions
,”
Appl. Math. Comput.
,
119
, pp.
177
186
.
25.
Li
,
G.
,
Paulino
,
G. H.
, and
Aluru
,
N. R.
, 2003, “
Coupling of the Meshfree Finite Cloud Method With the Boundary Element Method: A Collocation Approach
,”
Comput. Methods Appl. Mech. Eng.
,
192
(
20–21
), pp.
2355
2375
.
26.
Shapiro
,
V.
, and
Tsukanov
,
I.
, 2002,
From Geometric Modeling to Shape Modeling
,
Kluwer Acedemic Publishers
,
Norwell, MA
, pp.
127
136
.
27.
Bloomenthal
,
J.
, 1997,
Introduction to Implicit Surfaces
,
Morgan Kaufmann
,
San Francisco, CA
.
28.
Shapiro
,
V.
, and
Tsukanov
,
I.
, 1999, “
Implicit Functions With Guaranteed Differential Properties
,”
Fifth ACM Symposium on Solid Modeling and Applications
,
Ann Arbor
,
MI
, pp.
258
269
.
29.
Shapiro
,
V.
, and
Tsukanov
,
I.
, 1999, “
Implicit Functions With Guaranteed Differential Properties
,”
SMA’99: Proceedings of the Fifth ACM Symposium on Solid Modeling and Applications
,
ACM
,
New York
, pp.
258
269
.
30.
Shapiro
,
V.
, 2007, “
Semi-Analytic Geometry With R-Functions
,”
Acta Numerica
,
16
, pp.
239
303
.
You do not currently have access to this content.