# Weakened weak form

Weakened weak form (or W2 form)[1] is used in the formulation of general numerical methods based on meshfree methods and/or finite element method settings. These numerical methods are applicable to solid mechanics as well as fluid dynamics problems.

## Description

For simplicity we choose elasticity problems (2nd order PDE) for our discussion.[2] Our discussion is also most convenient in reference to the well-known weak and strong form. In a strong formulation for an approximate solution, we need to assume displacement functions that are 2nd order differentiable. In a weak formulation, we create linear and bilinear forms and then search for a particular function (an approximate solution) that satisfy the weak statement. The bilinear form uses gradient of the functions that has only 1st order differentiation. Therefore, the requirement on the continuity of assumed displacement functions is weaker than in the strong formulation. In a discrete form (such as the Finite element method, or FEM), a sufficient requirement for an assumed displacement function is piecewise continuous over the entire problems domain. This allows us to construct the function using elements (but making sure it is continuous a long all element interfaces), leading to the powerful FEM.

Now, in a weakened weak (W2) formulation, we further reduce the requirement. We form a bilinear form using 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 approximate the gradient of displacement functions for certain class of discontinuous functions, as long as they are in a proper G space.[4] Since we do not have to actually perform even the 1st differentiation to the assumed displacement functions, the requirement on the consistence of the functions are further reduced, and hence the weakened weak or W2 formulation.

## History

The development of systematic theory of the weakened weak form started from the works on meshfree methods.[2] It is relatively new, but had very rapid development in the past few years.[when?]

## Features of W2 formulations

1. The W2 formulation offers possibilities for formulate various (uniformly) "soft" models that works well with triangular meshes. Because triangular mesh can be generated automatically, it becomes much easier in re-meshing and hence automation in modeling and simulation. This is very important for our long-term goal of development of fully automated computational methods.
2. In addition, W2 models can be made soft enough (in uniform fashion) to produce upper bound solutions (for force-driving problems). Together with stiff models (such as the fully compatible FEM models), one can conveniently bound the solution from both sides. This allows easy error estimation for generally complicated problems, as long as a triangular mesh can be generated. This is important for producing so-called certified solutions.
3. W2 models can be built free from volumetric locking, and possibly free from other types of locking phenomena.
4. W2 models provide the freedom to assume separately the displacement gradient of the displacement functions, offering opportunities for ultra-accurate and super-convergent models. It may be possible to construct linear models with energy convergence rate of 2.
5. W2 models are often found less sensitive to mesh distortion.
6. W2 models are found effective for low order methods

## Existing W2 models

Typical W2 models are the smoothed point interpolation methods (or S-PIM).[5] The S-PIM can be node-based (known as NS-PIM or LC-PIM),[6] edge-based (ES-PIM),[7] and cell-based (CS-PIM).[8] The NS-PIM was developed using the so-called SCNI technique.[9] It was then discovered that NS-PIM is capable of producing upper bound solution and volumetric locking free.[10] The ES-PIM is found superior in accuracy, and CS-PIM behaves in between the NS-PIM and ES-PIM. Moreover, W2 formulations allow the use of polynomial and radial basis functions in the creation of shape functions (it accommodates the discontinuous displacement functions, as long as it is in G1 space), which opens further rooms for future developments. 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 and CS-FEM. The major property of S-PIM can be found also in S-FEM.[11] The S-FEM models are:

## Applications

Some of the applications of W2 models are:

1. Mechanics for solids, structures and piezoelectrics;[22][23]
2. Fracture mechanics and crack propagation;[24][25][26][27]
3. Heat transfer;[28][29]
4. Structural acoustics;[30][31][32]
5. Nonlinear and contact problems;[33][34]
6. Stochastic analysis;[35]
8. Phase change problem;[37]
9. Crystal plasticity modeling.[38]
10. Limited analysis.[39]

## References

1. ^ G.R. Liu. "A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory and Part II applications to solid mechanics problems". International Journal for Numerical Methods in Engineering, 81: 1093–1126, 2010
2. ^ a b Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9
3. ^ Liu GR, "A Generalized Gradient Smoothing Technique and the Smoothed Bilinear Form for Galerkin Formulation of a Wide Class of Computational Methods", International Journal of Computational Methods Vol.5 Issue: 2, 199–236, 2008
4. ^ Liu GR, "On G Space Theory", International Journal of Computational Methods, Vol. 6 Issue: 2, 257–289, 2009
5. ^ Liu, G.R. 2nd edn: 2009 Mesh Free Methods, CRC Press. 978-1-4200-8209-9
6. ^ Liu GR, Zhang GY, Dai KY, Wang YY, Zhong ZH, Li GY and Han X, "A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems", International Journal of Computational Methods, 2(4): 645–665, 2005.
7. ^ G.R. Liu, G.R. Zhang. "Edge-based Smoothed Point Interpolation Methods". International Journal of Computational Methods, 5(4): 621–646, 2008
8. ^ G.R. Liu, G.R. Zhang. "A normed G space and weakened weak (W2) formulation of a cell-based Smoothed Point Interpolation Method". International Journal of Computational Methods, 6(1): 147–179, 2009
9. ^ Chen, J. S., Wu, C. T., Yoon, S. and You, Y. (2001). "A stabilized conforming nodal integration for Galerkin mesh-free methods". International Journal for Numerical Methods in Engineering. 50: 435–466.
10. ^ G. R. Liu and G. Y. Zhang. Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM). International Journal for Numerical Methods in Engineering, 74: 1128–1161, 2008.
11. ^ Zhang ZQ, Liu GR, "Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods", International Journal for Numerical Methods in Engineering Vol. 84 Issue: 2, 149–178, 2010
12. ^ Liu GR, Nguyen-Thoi T, Nguyen-Xuan H, Lam KY (2009) "A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems". Computers and Structures; 87: 14–26.
13. ^ Liu GR, Nguyen-Thoi T, Lam KY (2009) "An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses in solids". Journal of Sound and Vibration; 320: 1100–1130.
14. ^ Nguyen-Thoi T, Liu GR, Lam KY, GY Zhang (2009) "A Face-based Smoothed Finite Element Method (FS-FEM) for 3D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements". International Journal for Numerical Methods in Engineering; 78: 324–353
15. ^ Liu GR, Dai KY, Nguyen-Thoi T (2007) "A smoothed finite element method for mechanics problems". Computational Mechanics; 39: 859–877
16. ^ Dai KY, Liu GR (2007) "Free and forced vibration analysis using the smoothed finite element method (SFEM)". Journal of Sound and Vibration; 301: 803–820.
17. ^ Dai KY, Liu GR, Nguyen-Thoi T (2007) "An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics". Finite Elements in Analysis and Design; 43: 847－860.
18. ^ a b Li Y, Liu GR, Zhang GY, "An adaptive NS/ES-FEM approach for 2D contact problems using triangular elements", Finite Elements in Analysis and Design Vol.47 Issue: 3, 256–275, 2011
19. ^ Liu GR, Nguyen-Thoi T, Lam KY (2009) "A novel FEM by scaling the gradient of strains with factor α (αFEM)". Computational Mechanics; 43: 369–391
20. ^ Liu GR, Nguyen-Xuan H, Nguyen-Thoi T, Xu X (2009) "A novel weak form and a superconvergent alpha finite element method (SαFEM) for mechanics problems using triangular meshes". Journal of Computational Physics; 228: 4055–4087
21. ^ Zeng W, Liu GR, Li D, Dong XW (2016) A smoothing technique based beta finite element method (βFEM) for crystal plasticity modeling. Computers and Structures; 162: 48-67
22. ^ Cui XY, Liu GR, Li GY, et al. A thin plate formulation without rotation DOFs based on the radial point interpolation method and triangular cells, International Journal for Numerical Methods in Engineering Vol.85 Issue: 8 , 958–986, 2011
23. ^ Liu GR, Nguyen-Xuan H, Nguyen-Thoi T, A theoretical study on the smoothed FEM (S-FEM) models: Properties, accuracy and convergence rates, International Journal for Numerical Methods in Engineering Vol. 84 Issue: 10, 1222–1256, 2010
24. ^ Liu GR, Nourbakhshnia N, Zhang YW, A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems, Engineering Fracture Mechanics Vol.78 Issue: 6 Pages: 863–876, 2011
25. ^ Liu GR, Chen L, Nguyen-Thoi T, et al. A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems, International Journal for Numerical Methods in Engineering Vol.83 Issue: 11, 1466–1497, 2010
26. ^ Liu GR, Nourbakhshnia N, Chen L, et al. "A Novel General Formulation for Singular Stress Field Using the Es-Fem Method for the Analysis of Mixed-Mode Cracks", International Journal of Computational Methods Vol. 7 Issue: 1, 191–214, 2010
27. ^ Zeng W, Liu GR, Jiang C, Dong XW, Chen HD, Bao Y, Jiang Y. "An effective fracture analysis method based on the virtual crack closure-integral technique implemented in CS-FEM", Applied Mathematical Modelling Vol. 40, Issue: 5-6, 3783-3800, 2016
28. ^ Zhang ZB, Wu SC, Liu GR, et al. "Nonlinear Transient Heat Transfer Problems using the Meshfree ES-PIM", International Journal of Nonlinear Sciences and Numerical Simulation Vol.11 Issue: 12, 1077–1091, 2010
29. ^ Wu SC, Liu GR, Cui XY, et al. "An edge-based 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
30. ^ 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
31. ^ He ZC, Liu GR, Zhong ZH, et al. "A coupled ES-FEM/BEM method for fluid-structure interaction problems", Engineering Analysis With Boundary Elements Vol. 35 Issue: 1, 140–147, 2011
32. ^ Zhang ZQ, Liu GR, "Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods", International Journal for Numerical Methods in Engineering Vol.84 Issue: 2, 149–178, 2010
33. ^ Zhang ZQ, Liu GR, "An edge-based smoothed finite element method (ES-FEM) using 3-node triangular elements for 3D non-linear analysis of spatial membrane structures", International Journal for Numerical Methods in Engineering, Vol. 86 Issue: 2 135–154, 2011
34. ^ Jiang C, Liu GR, Han X, Zhang ZQ, Zeng W, A smoothed finite element method for analysis of anisotropic large deformation of passive rabbit ventricles in diastole, International Journal for Numerical Methods in Biomedical Engineering, Vol. 31 Issue: 1,1-25, 2015
35. ^ Liu GR, Zeng W, Nguyen-Xuan H. Generalized stochastic cell-based smoothed finite element method (GS_CS-FEM) for solid mechanics, Finite Elements in Analysis and Design Vol.63, 51-61, 2013
36. ^ Nguyen-Thoi T, Liu GR, Nguyen-Xuan H, et al. "Adaptive analysis using the node-based smoothed finite element method (NS-FEM)", International Journal for Numerical Methods in Biomedical Engineering Vol. 27 Issue: 2, 198–218, 2011
37. ^ Li E, Liu GR, Tan V, et al. "An efficient algorithm for phase change problem in tumor treatment using alpha FEM", International Journal of Thermal Sciences Vol.49 Issue: 10, 1954–1967, 2010
38. ^ Zeng W, Larsen JM, Liu GR. Smoothing technique based crystal plasticity finite element modeling of crystalline materials, International Journal of Plasticity Vol.65, 250-268, 2015
39. ^ Tran TN, Liu GR, Nguyen-Xuan H, et al. "An edge-based smoothed finite element method for primal-dual shakedown analysis of structures", International Journal for Numerical Methods in Engineering Vol.82 Issue: 7, 917–938, 2010