The smoothed finite element method (S-FEM): A framework for the design of numerical models for desired solutions

doi:10.1007/s11709-019-0519-5

Front. Struct. Civ. Eng.

2019, Vol. 13

Issue (2) : 456-477 https://doi.org/10.1007/s11709-019-0519-5

REVIEW

The smoothed finite element method (S-FEM): A framework for the design of numerical models for desired solutions

Gui-Rong Liu(

)

Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati 45219, USA

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Abstract

The smoothed finite element method (S-FEM) was originated by G R Liu by combining some meshfree techniques with the well-established standard finite element method (FEM). It has a family of models carefully designed with innovative types of smoothing domains. These models are found having a number of important and theoretically profound properties. This article first provides a concise and easy-to-follow presentation of key formulations used in the S-FEM. A number of important properties and unique features of S-FEM models are discussed in detail, including 1) theoretically proven softening effects; 2) upper-bound solutions; 3) accurate solutions and higher convergence rates; 4) insensitivity to mesh distortion; 5) Jacobian-free; 6) volumetric-locking-free; and most importantly 7) working well with triangular and tetrahedral meshes that can be automatically generated. The S-FEM is thus ideal for automation in computations and adaptive analyses, and hence has profound impact on AI-assisted modeling and simulation. Most importantly, one can now purposely design an S-FEM model to obtain solutions with special properties as wish, meaning that S-FEM offers a framework for design numerical models with desired properties. This novel concept of numerical model on-demand may drastically change the landscape of modeling and simulation. Future directions of research are also provided.

Keywords computational method finite element method smoothed finite element method strain smoothing technique smoothing domain weakened weak form solid mechanics softening effect upper bound solution

Corresponding Author(s): Gui-Rong Liu

Online First Date: 29 January 2019 Issue Date: 12 March 2019

Cite this article:

Gui-Rong Liu. The smoothed finite element method (S-FEM): A framework for the design of numerical models for desired solutions[J]. Front. Struct. Civ. Eng., 2019, 13(2): 456-477.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-019-0519-5
https://academic.hep.com.cn/fsce/EN/Y2019/V13/I2/456

Fig.1 Smoothing domains used in a CS-FEM model. A quadrilateral element may be divided into smoothing cells (SCs) by connecting the mid-segment-points of opposite segments of smoothing domains [³⁶]. (a)

n S C = 1

; (b)

n S C = 2

; (c)

n S C = 3

; (d)

n S C = 4

; (e)

n S C = 8

; (f)

n S C = 16

(from [⁹⁸]). In applications, four SCs for each element are often used. Use of one SC can be more efficient and sometimes can produce upper bound solutions, but may have the so-called “hourglass” instability.

Fig.2 Edge-based smoothing domains on a Tr3-mesh. Shaded areas are typical smoothing domains. The smoothing domain

Ω l s

is for edge

l

on the problem domain boundary, and is a triangle

A O C

for a boundary edge. Smoothing domain

Ω k s

is for interior edge

k

that is inside the problem domain, and it is a four-sided convex polygon

D P F Q

. (from [¹⁶⁷])

Fig.3 Node-based smoothing domains for an NS-FEM model using Tr3-mesh. The smoothing domain

Ω k s

for node

k

is a polygon with

2 n k e

sides (where

n k e

is the number of elements surrounding node

k

). (from [¹⁶⁷])

Fig.4 Edge-based smoothing domain on a Te4-mesh for ES-FEM-Te4 model. Only the part of the smoothing domain

Ω k, j s

for edge

k

is shown. It is located inside element

j

, and is a double tetrahedron ACPOQ. If the are other elements connected to edge k, similar partial smoothing domains need to be constructed. (from [¹⁶⁷])

Fig.5 Node-based smoothing domain on a part of Te4-mesh. Only part of the smoothing domain

Ω k, j s

is shown. It is for node

k

in the element

j

, and it is a polyhedron

A E L G M F K O

. If the are other elements connected to node k, similar partial smoothing domains need to be constructed. (from [¹⁶⁷])

Fig.6 Schematic illustration of face-based smoothing domains based on tetrahedral elements: a face-based smoothing domain

Ω k s

created from two adjacent tetrahedral elements based upon their interface

k

. (from [¹⁶⁷])

Type*	method for creation and number of SD’s ( $N s$ )	S-FEM models	dimension of problem; properties
Cell-based SD (CSD)	SD’s or smoothing cells (SC’s) are divided from and located within the elements ( $N s = ∑ i = 1 N e n s c i$ , $n s c i = 1, 2, 3, 4, ...$ )	CS-FEM nCS-FEM	1D, 2D, 3D softer; high accuracy; insensitive to mesh distortion
Edge-based SD (ESD)	SD’s are created based on edges by connecting portions of the surrounding elements sharing the associated edge ( $N s = N e d g e$ )	ES-FEM	2D, 3D softer; very high accuracy, less insensitive to mesh distortion
Node-based SD (NSD)	SD’s are created based on nodes by connecting portions of the surrounding elements sharing the associated node ( $N s = N n o d e$ )	NS-FEM	1D, 2D, 3D soft; upper bound, very insensitive to mesh distortion, volumetric locking free
Face-based SD (FSD)	SD’s are created based on faces by connecting portions of the surrounding elements sharing the associated face ( $N s = N f a c e$ )	FS-FEM	3D softer; very high accuracy, less insensitive to mesh distortion

Tab.1 Existing types of smoothing domains (SD’s) used in S-FEM models

Fig.7 Types of smoothing domains created on a 3D mechanical component (engine connection bar) discretized with 4-noded tetrahedral elements. (a) Face-based smoothing domains (on the surface the FS smoothing domains cannot be seen, and hence it appears like the element mesh); (b) edge-based smoothing domains; (c) node-based smoothing domains, and (d) an example of a normal stress s_xx solution using the ES-FEM-Te4 model [¹⁶³]

Fig.8 Types of smoothing domains created on a 3D mechanical component (socket) discretized with 4-noded tetrahedral elements. (a) Face-based smoothing domains (on the surface the FS smoothing domains cannot be seen, and hence it appears like the element mesh); (b) edge-based smoothing domains; (c) node-based smoothing domains, and (d) an example of a solution of displacement in the z-direction using the FS-FEM-Te4 model [¹⁶³]

dimension of the problem	minimum number of smoothing domains
1D	$N s min ?$ = n_t
2D	$N s min ?$ = 2n_t /3
3D	$N s min ?$ = 3n_t /6= n_t /2

Tab.2 Minimum number of smoothing domains

N s min ?

for solid mechanics problems with

n t

(unconstrained) total nodes [⁵] [⁸]

Fig.9 Positions of Gauss points at mid-segment-points on segments of smoothing domains; a) Four quadrilateral smoothing domains in a Q4 element; b) Six triangular smoothing domains in a 6-sided polygonal element (from [⁸])

point	N₁	N₂	N₃	N₄	description
1	1.0	0	0	0	field node
2	0	1.0	0	0	field node
3	0	0	1.0	0	field node
4	0	0	0	1.0	field node
5	1/2	1/2	0	0	side midpoint
6	0	1/2	1/2	0	side midpoint
7	0	0	1/2	1/2	side midpoint
8	1/2	0	0	1/2	side midpoint
9	1/4	1/4	1/4	1/4	intersection of two bi-medians
g1	3/4	1/4	0	0	Gauss point (mid-segment point of $Γ k, p s$ )
g2	3/8	3/8	1/8	1/8	Gauss point (mid-segment point of $Γ k, p s$ )
g3	3/8	1/8	1/8	3/8	Gauss point (mid-segment point of $Γ k, p s$ )
g4	3/4	0	0	1/4	Gauss point (mid-segment point of $Γ k, p s$ )
g5	1/4	3/4	0	0	Gauss point (mid-segment point of $Γ k, p s$ )
g6	0	3/4	1/4	0	Gauss point (mid-segment point of $Γ k, p s$ )
g7	1/8	3/8	3/8	1/8	Gauss point (mid-segment point of $Γ k, p s$ )
g8	0	1/4	3/4	0	Gauss point (mid-segment point of $Γ k, p s$ )
g9	0	0	3/4	1/4	Gauss point (mid-segment point of $Γ k, p s$ )
g10	1/8	1/8	3/8	3/8	Gauss point (mid-segment point of $Γ k, p s$ )
g11	0	0	1/4	3/4	Gauss point (mid-segment point of $Γ k, p s$ )
g12	1/4	0	0	3/4	Gauss point (mid-segment point of $Γ k, p s$ )

Tab.3 Values of 4 nodal shape functions at different points within a Q4 element [⁸] (shown in Fig. 9(a)

point	N_1’	N_2’	N_3’	N_4’	N_5’	N_6’	description
1’	1.0	0	0	0	0	0	field node
2’	0	1.0	0	0	0	0	field node
3’	0	0	1.0	0	0	0	field node
4’	0	0	0	1.0	0	0	field node
5’	0	0	0	0	1.0	0	field node
6’	0	0	0	0	0	1.0	field node
O	1/6	1/6	1/6	1/6	1/6	1/6	centroid point
g1’	7/12	1/12	1/12	1/12	1/12	1/12	Gauss point (mid-segment point of $Γ k, p s$ )
g2’	1/2	1/2	0	0	0	0	Gauss point (mid-segment point of $Γ k, p s$ )
g3’	1/12	7/12	1/12	1/12	1/12	1/12	Gauss point (mid-segment point of $Γ k, p s$ )
g4’	0	1/2	1/2	0	0	0	Gauss point (mid-segment point of $Γ k, p s$ )
g5’	1/12	1/12	7/12	1/12	1/12	1/12	Gauss point (mid-segment point of $Γ k, p s$ )
g6’	0	0	1/2	1/2	0	0	Gauss point (mid-segment point of $Γ k, p s$ )
g7’	1/12	1/12	1/12	7/12	1/12	1/12	Gauss point (mid-segment point of $Γ k, p s$ )
g8’	0	0	0	1/2	1/2	0	Gauss point (mid-segment point of $Γ k, p s$ )
g9’	1/12	1/12	1/12	1/12	7/12	1/12	Gauss point (mid-segment point of $Γ k, p s$ )
g10’	0	0	0	0	1/2	1/2	Gauss point (mid-segment point of $Γ k, p s$ )
g11’	1/12	1/12	1/12	1/12	1/12	7/12	Gauss point (mid-segment point of $Γ k, p s$ )
g12’	1/2	0	0	0	0	1/2	Gauss point (mid-segment point of $Γ k, p s$ )

Tab.4 Values of six nodal shape functions at different points within a 6-sided polygonal element [⁸] (shown in Fig. 9(b))

Fig.10 A 2D cantilever beam loaded by a downward parabolically distributed shear stress at the right end

Fig.11 Convergence of numerical solution in the strain energy for the 2D cantilever problem (from [¹⁶³]). A set of uniformly distributed 3-noded triangular elements are used to discretize the problem domain, and the density of the mesh is controlled by the degrees of freedom (DoFs). It is found that the FEM solution is a lower bound, the NS-FEM gives an upper bound, and the ES-FEM gives ultra-accurate solution. The all use exactly the same element mesh, but different types of smoothing domain. The findings are predicted by S-FEM theory. This demonstrates that we can now design numerical models with different properties by simply using different types of smoothing domains

Fig.12 A 3D cantilever cubic solid fixed on its left face, and it is subjected to a uniformly distributed pressure loading on the top surface

Fig.13 Convergence of numerical solution in strain energy for the 3D cantilever cubic solid (from [¹⁶³]). A set of uniformly distributed 4-noded tetrahedral elements are used to discretize the problem domain, and the density of the mesh is controlled by the degrees of freedom (DoFs). It is found again that the FEM solution is a lower bound, the NS-FEM gives an upper bound, and the ES-FEM gives ultra-accurate solution. The all use exactly the same element mesh, but different types of smoothing domain.

Fig.14 In the S-FEM framework, we have now two knobs: one on the left tuns the stiffening effects by properly assuming the displacement field for the model, and another on the right tuns the softening effects by strain smoothing operations

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