A method for finite element analysis using a regular or structured grid is described that eliminates the need for generating conforming mesh for the geometry. The geometry of the domain is represented using implicit equations, which can be generated from traditional solid models. Solution structures are constructed using implicit equations such that the essential boundary conditions are satisfied exactly. This approach is used to solve boundary value problems arising in thermal and structural analysis. Convergence analysis is performed for several numerical examples and the results are compared with analytical and finite element analysis solutions to show that the method gives solutions that are similar to the finite element method in quality but is often less computationally expensive. Furthermore, by eliminating the need for mesh generation, better integration can be achieved between solid modeling and analysis stages of the design process.

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.
0045-7825,
139
, pp.
3
47
.
2.
Nayroles
,
B.
,
Touzat
,
G.
, and
Villon
,
P.
, 1992, “
Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements
,”
Comput. Mech.
0178-7675,
10
, pp.
307
318
.
3.
Belytschko
,
T.
,
Lu
,
Y. Y.
, and
Gu
,
L.
, 1994, “
Element Free Galerkin Methods
,”
Int. J. Numer. Methods Eng.
0029-5981,
37
, pp.
229
256
.
4.
Atluri
,
S. N.
, and
Zhu
,
T. L.
, 2000, “
The Meshless Local Petrov-Galerkin (MLPG) Approach for Solving Problems in Elasto-Statics
,”
Arch. Technol.
1361-326X,
25
, pp.
169
179
.
5.
Liu
,
G. R.
, and
Gu
,
Y. T.
, 2001, “
A Point Interpolation Method for Two-Dimensional Solids
,”
Int. J. Numer. Methods Eng.
0029-5981,
50
, pp.
937
951
.
6.
Melenk
,
J. M.
, and
Babuska
,
I.
, 1996, “
The Partition of Unity Finite Element Method: Basic Theory and Applications
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
139
, pp.
289
314
.
7.
De
,
S.
, and
Bathe
,
K. J.
, 2000, “
Method of Finite Spheres
,”
Comput. Mech.
0178-7675,
25
, pp.
329
345
.
8.
De
,
S.
, and
Bathe
,
K. J.
, 2001, “
The Method of Finite Spheres with Improved Numerical Integration
,”
Comput. Struct.
0045-7949,
79
, pp.
2183
2196
.
9.
Sukumar
,
N.
,
Moran
,
B.
, and
Belytschko
,
T.
, 1998, “
The Natural Element Method in Solid Mechanics
,”
Int. J. Numer. Methods Eng.
0029-5981,
43
, pp.
839
887
.
10.
Dolbow
,
J.
, and
Belytschko
,
T.
, 1999, “
Numerical Integration of the Galerkin Weak Form in Meshfree Methods
,”
Comput. Mech.
0178-7675,
23
, pp.
219
230
.
11.
Gonzalez
,
D.
,
Cueto
,
E.
,
Martinez
,
M. A.
, and
Doblare
,
M.
, 2004, “
Numerical integration in Natural Neighbour Galerkin methods
,”
Int. J. Numer. Methods Eng.
0029-5981,
60
, pp.
2077
2104
.
12.
Laguardia
,
J. J.
,
Cueto
,
E.
, and
Doblare
,
M.
, 2005, “
A Natural Neighbour Galerkin Method with Quadtree Structure
,”
Int. J. Numer. Methods Eng.
0029-5981,
63
, pp.
789
812
.
13.
Krongauz
,
Y.
, and
Belytschko
,
T.
, 1996, “
Enforcement of Essential Boundary Conditions in Meshless Approximations Using Finite Elements
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
131
, pp.
133
145
.
14.
Kumar
,
A. V.
, and
Lee
,
J. H.
, 2006, “
Step Function Representation of Solid Models and Application to Mesh Free Engineering Analysis
,”
ASME J. Mech. Des.
1050-0472,
128
(
1
), pp.
46
56
.
15.
Kantorovich
,
L. W.
, and
Krylov
,
W. I.
, 1956, “
Näherungsmethoden der Höheren Analysis
,”
VEB Deutscher Verlag der Wissenschaften
,
Berlin
.
16.
Shapiro
,
V.
, and
Tsukanov
,
I.
, 1999, “
Meshfree Simulation of Deforming Domains
,”
Comput.-Aided Des.
0010-4485,
31
, pp.
459
471
.
17.
Rvachev
,
V. L.
, and
Shieko
,
T. I.
, 1995, “
R-functions in Boundary Value Problems in Mechanics
,”
Appl. Mech. Rev.
0003-6900,
48
, pp.
151
188
.
18.
Belytschko
,
T.
,
Parimi
,
C.
,
Moes
,
N.
,
Sukumar
,
N.
, and
Usui
,
S.
, 2003, “
Structured Extended Finite Element Methods for Solids Defined by Implicit Surfaces
,”
Int. J. Numer. Methods Eng.
0029-5981,
56
, pp.
609
635
.
19.
Clark
,
B. W.
, and
Anderson
,
D. C.
, 2002, “
Finite Element Analysis in 3D Using Penalty Boundary Method
,”
Proceedings of Design Engineering Technical Conferences
,
Montreal, Canada
.
20.
Osher
,
S.
, and
Fedkiw
,
R.
, 2002,
Level Set Methods and Dynamic Implicit Surfaces
,
Springer
,
New York
.
21.
Mortenson
,
M. E.
, 1997,
Geometric Modeling
,
Wiley
,
New York
.
22.
Rvachev
,
V. L.
,
Sheiko
,
T. I.
,
Shapiro
,
V.
, and
Tsukanov
,
I.
, 2001, “
Transfinite Interpolation Over Implicitly Defined Sets
,”
Comput. Aided Geom. Des.
0167-8396,
18
, pp.
195
220
.
23.
Timoshenko
,
S. P.
, and
Goodier
,
J. N.
, 1969,
Theory of Elasticity
,
3rd ed.
,
McGraw-Hill
,
New York
.
24.
Gu
,
Y. T.
, and
Liu
,
G. R.
, 2001, “
A Local Point Interpretation Method for Static and Dynamic Analysis of Thin Beams
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
190
, pp.
515
552
.
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