This paper describes the pioneering application of the meshfree solution structure method (SSM) to computer simulation of natural vibrations of 3D mechanical parts and structures. Using several carefully chosen examples, we investigate the accuracy and convergence of the computed natural frequencies. The salient feature of our approach is exact treatment of the prescribed boundary conditions that are enforced using approximate distance functions that vanish on the boundaries of a geometric model. Ability to use spatial meshes that do not necessarily conform to the shape of the geometric model makes it possible to eliminate or substantially simplify the finite element meshing. This defines unprecedented geometric flexibility of the SSM as well as the complete automation of the solution procedure.

References

References
1.
Archer, J. S.
,
1968
, “NASA Space Vehicle Design Criteria (Structures): Natural Vibration Modal Analysis,” NASA Langley Research Center, Langley, VA, Report No. NASA SP-8012.
2.
Liu
,
G.
, and
Liu
,
M.
,
2003
,
Smoothed Particle Hydrodynamics
,
World Scientific
, Singapore.
3.
Randles
,
P. W.
, and
Libersky
,
L. D.
,
1996
, “
Smoothed Particle Hydrodynamics: Some Recent Improvements and Applications
,”
Comput. Methods Appl. Mech. Eng.
,
139
(1–4), pp.
375
408
.10.1016/S0045-7825(96)01090-0
4.
Nayroles
,
B.
,
Touzot
,
G.
, and
Villon
,
P.
,
1992
, “
Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements
,”
Computat. Mech.
,
10
(5), pp.
301
318
.10.1007/BF00364252
5.
Liu
,
W. K.
,
Jun
,
S.
, and
Zhang
,
Y. F.
,
1995
, “
Reproducing Kernel Particle Methods
,”
Int. J. Numer. Methods Fluids
,
20
(8–9), pp.
1081
–1106.10.1002/fld.1650200824
6.
Chen
,
J.-S.
,
Pan
,
C.
,
Roque
,
C.
, and
Wang
,
H.-P.
,
1998
, “
A Lagrangian Reproducing Kernel Particle Method for Metal Forming Analysis
,”
Comput. Mech.
,
22
(3), pp.
289
307
.10.1007/s004660050361
7.
Duarte
,
C. A.
, and
Oden
,
J. T.
,
1996
, “
H-p Clouds—An h-p Meshless Method
,”
Numer. Methods Partial Differ. Equ.
,
12
(6), pp.
673
705
.10.1002/(SICI)1098-2426(199611)12:6<673::AID-NUM3>3.0.CO;2-P
8.
Atluri
,
S.
, and
Zhu
,
T.
,
1998
, “
A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics
,”
Comput. Mech.
,
22
(
2
), pp.
117
127
.10.1007/s004660050346
9.
Melenk
,
J. M.
, and
Babuska
,
I.
,
1996
, “
The Partition of Unity Finite Element Method: Basic Theory and Applications
,”
Comput. Methods Appl. Mech. Eng.
,
139
(1–4), pp.
289
314
.10.1016/S0045-7825(96)01087-0
10.
Gunter
,
F. C.
, and
Liu
,
W. K.
,
1998
, “
Implementation of Boundary Conditions for Meshless Methods
,”
Comput. Methods Appl. Mech. Eng.
,
163
(1–4), pp.
205
230
.10.1016/S0045-7825(98)00014-0
11.
Zhu
,
P.
,
Zhang
,
L.
, and
Liew
,
K.
,
2014
, “
Geometrically Nonlinear Thermomechanical Analysis of Moderately Thick Functionally Graded Plates Using a Local Petrov–Galerkin Approach With Moving Kriging Interpolation
,”
Compos. Struct.
,
107
, pp.
298
314
.10.1016/j.compstruct.2013.08.001
12.
Zhang
,
L.
,
Zhu
,
P.
, and
Liew
,
K.
,
2014
, “
Thermal Buckling of Functionally Graded Plates Using a Local Kriging Meshless Method
,”
Compos. Struct.
,
108
, pp.
472
492
.10.1016/j.compstruct.2013.09.043
13.
Zhang
,
L.
,
Lei
,
Z.
,
Liew
,
K.
, and
Yu
,
J.
,
2014
, “
Static and Dynamic of Carbon Nanotube Reinforced Functionally Graded Cylindrical Panels
,”
Compos. Struct.
,
111
, pp.
205
212
.10.1016/j.compstruct.2013.12.035
14.
Rvachev
,
V. L.
,
Sheiko
,
T. I.
,
Shapiro
,
V.
, and
Tsukanov
,
I.
,
2000
, “
On Completeness of RFM Solution Structures
,”
Comput. Mech.
,
25
(2–3), pp.
305
317
.10.1007/s004660050479
15.
Rvachev
,
V. L.
,
Sheiko
,
T. I.
,
Shapiro
,
V.
, and
Tsukanov
,
I.
,
2001
, “
Transfinite Interpolation Over Implicity Defined Sets
,”
Comput. Aided Geom. Des.
,
18
(3), pp.
195
220
.10.1016/S0167-8396(01)00015-2
16.
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
.10.1002/nme.760
17.
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
.10.1007/s10444-004-1835-3
18.
Tsukanov
,
I.
, and
Posireddy
,
S. R.
,
2011
, “
Hybrid Method of Engineering Analysis: Combining Meshfree Method With Distance Fields and Collocation Technique
,”
ASME J. Comput. Inf. Sci. Eng.
,
11
(
3
), p.
031001
.10.1115/1.3572035
19.
Freytag
,
M.
,
Shapiro
,
V.
, and
Tsukanov
,
I.
,
2011
, “
Finite Element Analysis in Situ
,”
Finite Elem. Anal. Des.
,
47
(9), pp.
957
972
.10.1016/j.finel.2011.03.001
20.
Krysl
,
P.
, 2006,
A Pragmatic Introduction to the Finite Element Method for Thermal And Stress Analysis: With the Matlab Toolkit SOFEA
, World Scientific Publishing, Singapore.
21.
Zienkiewicz
,
O.
, and
Taylor
,
R.
,
2005
,
The Finite Element Method
,
Butterworth-Heinemann
, Amsterdam.
22.
Tsukanov
,
I.
, and
Shapiro
,
V.
,
2002
, “
The Architecture of SAGE—A Meshfree System Based on RFM
,”
Eng. Comput.
,
18
(
4
), pp.
295
311
.10.1007/s003660200027
23.
Luft
,
B.
,
Shapiro
,
V.
, and
Tsukanov
,
I.
,
2008
, “
Geometrically Adaptive Numerical Integration
,”
ACM Symposium on Solid and Physical Modeling
(
SPM '08
), Stony Brook, NY, June 2–4, pp.
147
157
.10.1145/1364901.1364923
24.
Tsukanov
,
I.
, and
Hall
,
M.
,
2003
, “
Data Structure and Algorithms for Fast Automatic Differentiation
,”
Int. J. Numer. Methods Eng.
,
56
(
13
), pp.
1949
1972
.10.1002/nme.647
25.
Komzsik
,
L.
, 2003, “
Software, Environments and Tools
,”
The Lanczos Method: Evolution and Application
, Society for Industrial and Applied Mathematics, Philadelphia, PA.
26.
Hughes
,
T.
, 2012,
The Finite Element Method: Linear Static and Dynamic Finite Element Analysis
, Courier Dover Publications, Mineola, NY.
27.
Bloomenthal
,
J.
,
1997
,
Introduction to Implicit Surfaces
,
Morgan Kaufmann Publishers
, San Francisco, CA.
28.
Rvachev
,
V. L.
,
1982
,
Theory of R-Functions and Some Applications
,
Naukova Dumka
, Kiev, Ukraine (in Russian).
29.
Shapiro
,
V.
, and
Tsukanov
,
I.
,
1999
, “
Implicit Functions With Guaranteed Differential Properties
,”
Fifth ACM Symposium on Solid Modeling and Applications
(
SMA '99
), Ann Arbor, MI, June 8–11, pp.
258
269
.10.1145/304012.304038
30.
Shapiro
,
V.
,
2007
, “
Semi-Analytic Geometry With R-Functions
,”
Acta Numer.
,
16
, pp.
239
303
.10.1017/S096249290631001X
31.
Pasko
,
A.
, and
Adzhiev
,
V.
,
2004
, “
Function-Based Shape Modeling: Mathematical Framework and Specialized Language
,”
Automated Deduction in Geometry
(Lecture Notes in Artificial Intelligence, Vol. 2930),
F.
Winkler
, ed.,
Springer-Verlag
, Berlin, pp.
132
160
.10.1007/978-3-540-24616-9_9
32.
Fryazinov
,
O.
,
Vilbrandt
,
T.
, and
Pasko
,
A.
,
2013
, “
Multi-Scale Space-Variant FRep Cellular Structures
,”
Comput. Aided Des.
,
45
(
1
), pp.
26
34
.10.1016/j.cad.2011.09.007
33.
Ricci
,
A.
,
1973
, “
A Constructive Geometry for Computer Graphics
,”
Comput. J.
,
16
(
2
), pp.
157
160
.10.1093/comjnl/16.2.157
34.
Freytag
,
M.
,
Shapiro
,
V.
, and
Tsukanov
,
I.
,
2007
, “
Scan and Solve: Acquiring the Physics of Artifacts
,”
ASME
Paper No. DETC2007-35701.10.1115/DETC2007-35701
35.
Rvachev
,
V. L.
,
1963
, “
Analytical Description of Some Geometric Objects
,”
Dokl Akad. Nauk Ukr. SSR
,
153
(
4
), pp.
765
768
.
36.
Freytag
,
M.
,
Shapiro
,
V.
, and
Tsukanov
,
I.
,
2006
, “
Field Modeling With Sampled Distances
,”
Comput. Aided Des.
,
38
(
2
), pp.
87
100
.10.1016/j.cad.2005.06.004
37.
Tsukanov
,
I.
, and
Hall
,
M.
,
2000
, “
Fast Forward Automatic Differentiation Library (FFADLib): A User Manual
,” Spatial Automation Laboratory, University of Wisconsin-Madison, Madison, WI, Technical Report No. 2000-4, http://sal-cnc.me.wisc.edu
38.
Gasparini
,
R.
,
Kosta
,
T.
, and
Tsukanov
,
I.
,
2013
, “
Engineering Analysis in Imprecise Geometric Models
,”
Finite Elem. Anal. Des.
,
66
, pp.
96
109
.10.1016/j.finel.2012.10.011
39.
Barrett
,
R.
,
Berry
,
M.
,
Chan
,
T. F.
,
Demmel
,
J.
,
Donato
,
J.
,
Dongarra
,
J.
,
Eijkhout
,
V.
,
Pozo
,
R.
,
Romine
,
C.
, and
der Vorst
,
H. V.
,
1994
,
Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods
,
SIAM
,
Philadelphia, PA
.
You do not currently have access to this content.