Biomedical Engineering Reference
In-Depth Information
penalty factors are positive, then the positive definite property of discrete system
of equations is preserved. Additionally, as it is perceptible, the Galerkin procedure
produces symmetric and banded stiffness matrices K and K
a
.
Nevertheless, the penalty method presents some disadvantages. It is not pos-
sible to enforce exactly the essential boundary conditions with the penalty method.
The accuracy of the solution depends on the magnitude of the penalty factors and
the obtained results are less accurate than the results obtained with the Lagrange
multipliers method. Experience shows that higher penalty factors lead to more
accurate results, however larger penalty factors could lead to ill-conditioned
stiffness matrix. In addition, the appropriate magnitude of the penalty factors vary
with the analysed problem, therefore it is not possible to define a universally
acceptable penalty factor.
References
1. Liu GR (2002) A point assembly method for stress analysis for two-dimensional solids. Int J
Solid Struct 39:261-276
2. Liu GR (2002) Mesh free methods-moving beyond the finite element method. CRC Press,
Boca Raton
3. Belytschko T, Lu YY, Gu L (1994) Element-free galerkin method. Int J Numer Meth Eng
37:229-256
4. Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for
Galerkin mesh-free methods. Int J Numer Methods Eng 50(2):435-466
5. Sze KY, Chen JS, Sheng N, Liu XH (2004) Stabilized conforming nodal integration:
exactness and variational. Finite Elem Anal Des 41(2):147-171
6. Elmer W, Chen JS, Puso M, Taciroglu E (2012) A stable, meshfree, nodal integration method
for nearly incompressible solids. Finite Elem Anal Des 51:81-85
7. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Meth
Fluids 20(6):1081-1106
8. Atluri SN, Zhu T (1998) A new meshless local Petrov-Galerkin (MLPG) approach in
computational mechanics. Comput Mech 22(2):117-127
9. Wang JG, Liu GR (2002) A point interpolation meshless method based on radial basis
functions. Int J Numer Meth Eng 54:1623-1648
10. Nguyen VP, Rabczuk T, Bordas S, Duflot M (2008) Meshless methods: a review and
computer implementation aspects. Math Comput Simul 79(3):763-813
11. Liu GR, Gu YT (2001) A point interpolation method for two-dimensional solids. Int J Numer
Meth Eng 50:937-951
12. Dinis LMJS, Jorge RMN, Belinha J (2007) Analysis of 3D solids using the natural neighbour
radial point interpolation method. Comput Methods Appl Mech Eng 196(13-16):2009-2028
13. Sibson R (1981) A brief description of natural neighbor interpolation. In: Barnett V (ed)
Interpreting multivariate data. Wiley, Chichester, pp 21-36
14. Boots BN (1986) Voronoï (Thiessen) polygons. Geo Books, Norwich
15. Preparata FP, Shamos MI (1985) Computational geometry—an introduction. Springer, New
York
16. Okabe A, Boots BN, Sugihara K, Chiu SN (2000) Spatial tessellations: concepts and
applications of Voronoï diagrams, 2nd edn. Wiley, Chichester
17. Lawson CL (1977) Software for C1 surface interpolation. In: Rice JR (ed) Mathematical
software III, 3rd edn. Academic Press, New York
