From Wikipedia, the free encyclopedia Weakened weak form
Weakened weak form
Weakened weak form (or W2 form) [1] is used in the for-
mulation of general numerical methods based on mesh-
Features of W2 formulations
free methods and/or finite element method settings. Th- 1) The W2 formulation offers possibilities for formulate
ese numerical methods are applicable to solid mechanics various (uniformly) "soft" models that works well with
as well as fluid dynamics problems. triangular meshes. Because triangular mesh can be gen-
erated automatically, it becomes much easier in re-mesh-
ing and hence automation in modeling and simulation.
Description This is very important for our long term goal of develop-
For simplicity we choose elasticity problems (2nd order ment of fully automated computational methods.
PDE) for our discussion[2]. Our discussion is also most 2) In addition, W2 models can be made soft enough
convenient in reference to the well-known Weak form (in uniform fashion) to produce upper bound solutions
and strong form. In a strong formulation for an approx- (for force-driving problems). Together with stiff models
imate solution, we need to assume displacement func- (such as the fully-compatible FEM models), one can con-
tions that are 2nd order differentiable. In a weak for- veniently bound the solution from both sides. This allows
mulation, we create linear and bilinear forms and then easy error estimation for generally complicated prob-
search for a particular function (an approximate solu- lems, as long as a triangular mesh can be generated. This
tion) that satisfy the weak statement. The bilinear form is important for producing so-called certified solutions.
uses gradient of the functions that has only 1st order dif- 3) W2 models can be built free from volumetric lock-
ferentiation. Therefore, the requirement on the continu- ing, and possibly free from other types of locking phe-
ity of assumed displacement functions is weaker than in nomena.
the strong formulation. In a discrete form (such as the Fi- 4) W2 models provide the freedom to assume sep-
nite element method, or FEM), a sufficient requirement arately the displacement gradient of the displacement
for a assumed displacement function is piecewise contin- functions, offering opportunities for ultra-accurate and
uous over the entire problems domain. This allows us to super-convergent models. It may be possible to construct
construct the function using elements (but making sure linear models with energy convergence rate of 2.
it is continuous a long all element interfaces), leading to 5) W2 models are often found less sensitive to mesh
the powerful FEM. distortion.
Now, in a weakened weak (W2) formulation, we fur- 6) W2 models are found effective for low order meth-
ther reduce the requirement. We form a bilinear form us- ods.
ing only the assumed function (not even the gradient).
This is done by using the so-called generalized gradient
smoothing technique [3], with which one can approxi-
Existing W2 models
mate the gradient of displacement functions for certain Typical W2 models are the Smoothed Point Interpolation
class of discontinuous functions, as long as they are in Methods (or S-PIM) [6]. The S-PIM can be node-based
a proper G space [4]. Since we do not have to actually (known as NS-PIM or LC-PIM) [7], edge-based (ES-PIM)
perform even the 1st differentiation to the assumed dis- [8], and cell-based (CS-PIM) [9]. The NS-PIM was devel-
placement functions, the requirement on the consistence oped using the so-called SCNI technique [10]. It was then
of the functions are further reduced, and hence the discovered that NS-PIM is capable of producing upper
weakened weak or W2 formulation. bound solution and volumetric locking free [11]. The ES-
PIM is found superior in accuracy, and CS-PIM behaves in
between the NS-PIM and ES-PIM. Moreover, W2 formula-
History tions allow the use of polynomial and radial basis func-
The development of systematic theory of the weakened tions in the creation of shape functions (it accommodates
weak form started from the works on meshfree methods the discontinuous displacement functions, as long as it is
[5]. It is relatively new, but had very rapid development in G1 space), which opens further rooms for future de-
in the past few years. velopments. The S-FEM is largely the linear version of S-
PIM, but with most of the properties of the S-PIM and
much simpler. It has also variations of NS-FEM, ES-FEM
1
From Wikipedia, the free encyclopedia Weakened weak form
and CS-FEM. The major property of S-PIM can be found [6] Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC
also in S-FEM[12]. The S-FEM models are: Press. 978-1-4200-8209-9
• Node-based Smoothed FEM (NS-FEM) [13] [7] Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li
• Edge-based Smoothed FEM (NS-FEM) [14] GY and Han X, A linearly conforming point
• Face-based Smoothed FEM (NS-FEM) [15] interpolation method (LC-PIM) for 2D solid
• Cell-based Smoothed FEM (NS-FEM) [16][17][18] mechanics problems, International Journal of
• Edge/node-based Smoothed FEM (NS/ES-FEM) [19] Computational Methods, 2(4): 645-665, 2005.
• Alpha FEM method (Alpha FEM) [20][21] [8] G.R. Liu, G.R. Zhang. Edge-based Smoothed Point
Interpolation Methods. International Journal of
Applications [9]
Computational Methods, 5(4): 621-646, 2008
G.R. Liu, G.R. Zhang. A normed G space and
Some of the applications of W2 models are: weakened weak (W2) formulation of a cell-based
1) Mechanics for solids, structures and piezoelectrics Smoothed Point Interpolation Method.
[22] Liu GR, Nguyen-Xuan H, Nguyen-Thoi T, A the- International Journal of Computational Methods,
oretical study on the smoothed FEM (S-FEM) models: 6(1): 147-179, 2009
Properties, accuracy and convergence rates, [10] Chen, J. S., Wu, C. T., Yoon, S. and You, Y. (2001). A
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS stabilized conforming nodal integration for
IN ENGINEERING Vol. 84 Issue: 10, 1222-1256, 2010 ; Galerkin mesh-free methods. Int. J. Numer. Meth.
2) Fracture mechanics and crack propagation[23] [24] Eng. 50: 435–466.
[25]; [11] G. R. Liu and G. Y. Zhang. Upper bound solution to
3) Heat transfer[26][27]; elasticity problems: A unique property of the
4) Structural acoustics[28][29] [30]; linearly conforming point interpolation method
5) Nonlinear and contact problems[31]; (LC-PIM). International Journal for Numerical
6) Adaptive Analysis [32] [33]; Methods in Engineering, 74: 1128-1161, 2008.
7) Phase change problem [34]; [12] Zhang ZQ, Liu GR, Upper and lower bounds for
8) Limited analysis [35]. natural frequencies: A property of the smoothed
finite element methods, INTERNATIONAL JOURNAL
See also FOR NUMERICAL METHODS IN ENGINEERING Vol.
84 Issue: 2, 149-178, 2010
• G space [13] Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Lam KY
• Meshfree methods (2009) A node-based smoothed finite element
• Smoothed finite element method method (NS-FEM) for upper bound solutions to
• Smoothed point interpolation method solid mechanics problems. Computers and
• Finite element method Structures; 87: 14-26.
[14] Liu GR, Nguyen-Thoi T, Lam KY (2009) An edge-
References based smoothed finite element method (ES-FEM)
for static, free and forced vibration analyses in
[1] G.R. Liu. A G space theory and a weakened weak solids. Journal of Sound and Vibration; 320:
(W2) form for a unified formulation of compatible 1100-1130.
and incompatible methods: Part I theory and Part II [15] Nguyen-Thoi T, Liu GR, Lam KY, GY Zhang (2009) A
applications to solid mechanics problems. Face-based Smoothed Finite Element Method (FS-
International Journal for Numerical Methods in FEM) for 3D linear and nonlinear solid mechanics
Engineering, 81: 1093-1126, 2010 problems using 4-node tetrahedral elements.
[2] Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC International Journal for Numerical Methods in
Press. 978-1-4200-8209-9 Engineering; 78: 324-353
[3] Liu GR, A GENERALIZED GRADIENT SMOOTHING [16] Liu GR, Dai KY, Nguyen-Thoi T (2007) A smoothed
TECHNIQUE AND THE SMOOTHED BILINEAR FORM finite element method for mechanics problems.
FOR GALERKIN FORMULATION OF A WIDE CLASS Computational Mechanics; 39: 859-877
OF COMPUTATIONAL METHODS, INTERNATIONAL [17] Dai KY, Liu GR (2007) Free and forced vibration
JOURNAL OF COMPUTATIONAL METHODS Vol.5 analysis using the smoothed finite element method
Issue: 2, 199-236, 2008 (SFEM). Journal of Sound and Vibration; 301:
[4] Liu GR, ON G SPACE THEORY, INTERNATIONAL 803-820.
JOURNAL OF COMPUTATIONAL METHODS, Vol. 6 [18] Dai KY, Liu GR, Nguyen-Thoi T (2007) An n-sided
Issue: 2,257-289, 2009 polygonal smoothed finite element method
[5] Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC (nSFEM) for solid mechanics. Finite Elements in
Press. 978-1-4200-8209-9 Analysis and Design; 43: 847－860.
2
From Wikipedia, the free encyclopedia Weakened weak form
[19] Li Y, Liu GR, Zhang GY, An adaptive NS/ES-FEM [29] He ZC, Liu GR, Zhong ZH, et al. A coupled ES-FEM/
approach for 2D contact problems using triangular BEM method for fluid-structure interaction
elements, FINITE ELEMENTS IN ANALYSIS AND problems, ENGINEERING ANALYSIS WITH
DESIGN Vol.47 Issue: 3, 256-275, 2011 BOUNDARY ELEMENTS Vol. 35 Issue: 1, 140-147,
[20] Liu GR, Nguyen-Thoi T, Lam KY (2009) A novel FEM 2011
by scaling the gradient of strains with factor [30] Zhang ZQ, Liu GR, Upper and lower bounds for
(FEM). Computational Mechanics; 43: 369-391 natural frequencies: A property of the smoothed
[21] Liu GR, Nguyen-Xuan H, Nguyen-Thoi T, Xu X finite element methods, INTERNATIONAL JOURNAL
(2009) A novel weak form and a superconvergent FOR NUMERICAL METHODS IN ENGINEERING
alpha finite element method (SFEM) for Vol.84 Issue: 2,149-178, 2010
mechanics problems using triangular meshes. [31] Zhang ZQ, Liu GR, An edge-based smoothed finite
Journal of Computational Physics; 228: 4055-4087 element method (ES-FEM) using 3-node triangular
[22] Cui XY, Liu GR, Li GY, et al. A thin plate formulation elements for 3D non-linear analysis of spatial
without rotation DOFs based on the radial point membrane structures, INTERNATIONAL JOURNAL
interpolation method and triangular cells, FOR NUMERICAL METHODS IN ENGINEERING, Vol.
INTERNATIONAL JOURNAL FOR NUMERICAL 86 Issue: 2 135-154, 2011
METHODS IN ENGINEERING Vol.85 Issue: 8 , [32] Nguyen-Thoi T, Liu GR, Nguyen-Xuan H, et al.
958-986, 2011 Adaptive analysis using the node-based smoothed
[23] Liu GR, Nourbakhshnia N, Zhang YW, A novel finite element method (NS-FEM), INTERNATIONAL
singular ES-FEM method for simulating singular JOURNAL FOR NUMERICAL METHODS IN
stress fields near the crack tips for linear fracture BIOMEDICAL ENGINEERING Vol. 27 Issue: 2,
problems, ENGINEERING FRACTURE MECHANICS 198-218, 2011
Vol.78 Issue: 6 Pages: 863-876, 2011 [33] Li Y, Liu GR, Zhang GY, An adaptive NS/ES-FEM
[24] Liu GR, Chen L, Nguyen-Thoi T, et al. A novel approach for 2D contact problems using triangular
singular node-based smoothed finite element elements, FINITE ELEMENTS IN ANALYSIS AND
method (NS-FEM) for upper bound solutions of DESIGN Vol.47 Issue: 3, 256-275, 2011
fracture problems, INTERNATIONAL JOURNAL FOR [34] Li E, Liu GR, Tan V, et al. An efficient algorithm for
NUMERICAL METHODS IN ENGINEERING Vol.83 phase change problem in tumor treatment using
Issue: 11, 1466-1497, 2010 alpha FEM, INTERNATIONAL JOURNAL OF
[25] Liu GR, Nourbakhshnia N, Chen L, et al. A NOVEL THERMAL SCIENCES Vol.49 Issue: 10, 1954-1967,
GENERAL FORMULATION FOR SINGULAR STRESS 2010
FIELD USING THE ES-FEM METHOD FOR THE [35] Tran TN, Liu GR, Nguyen-Xuan H, et al. An edge-
ANALYSIS OF MIXED-MODE CRACKS, based smoothed finite element method for primal-
INTERNATIONAL JOURNAL OF COMPUTATIONAL dual shakedown analysis of structures,
METHODS Vol. 7 Issue: 1, 191-214, 2010 INTERNATIONAL JOURNAL FOR NUMERICAL
[26] Zhang ZB, Wu SC, Liu GR, et al. Nonlinear Transient METHODS IN ENGINEERING Vol.82 Issue: 7,
Heat Transfer Problems using the Meshfree ES- 917-938, 2010
PIM, INTERNATIONAL JOURNAL OF NONLINEAR
SCIENCES AND NUMERICAL SIMULATION Vol.11
Issue: 12, 1077-1091, 2010
External links
[27] Wu SC, Liu GR, Cui XY, et al. An edge-based • [1]
smoothed point interpolation method (ES-PIM) for
heat transfer analysis of rapid manufacturing
system, INTERNATIONAL JOURNAL OF HEAT AND
MASS TRANSFER Vol.53 Issue: 9-10, 1938-1950, 2010
[28] He ZC, Cheng AG, Zhang GY, et al. Dispersion error
reduction for acoustic problems using the edge-
based smoothed finite element method (ES-FEM),
INTERNATIONAL JOURNAL FOR NUMERICAL
METHODS IN ENGINEERING Vol. 86 Issue: 11 Pages:
1322-1338, 2011
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From Wikipedia, the free encyclopedia Weakened weak form
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