Novel unified finite element schemes for computational solid mechanics based on Bézier elements

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Novel unified finite element schemes for computational
solid mechanics based on B´ezier elements
Chennakesava Kadapa
Swansea Academy of Advanced Computing
Email: c.kadapa@swansea.ac.uk
UKACM 2019 Conference, London, 10-12 April, 2019.
Chennakesava Kadapa (SA2C) Unified FE schemes for Computational Mechanics UKACM 2019 Conference 1 / 38

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Introduction
Introduction
Why do we need new finite element techniques for solid mechanics?
Lack of
Accurate, robust and computationally efficient
Explicit schemes for elastodynamics and wave propagation
Incompressible material models
Polymers
Biological soft tissues
Soils
With solid-solid contact
Adaptive refinement
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Introduction
Explicit schemes - introduction
Governing equations in infinitesimal (small) strain regime
ρ
∂2
u
∂t2
− · σ = f in Ω (1a)
u = g on ΓD (1b)
σ · n = t on ΓN (1c)
u(x, 0) = u0 in Ω (1d)
v(x, 0) = v0 in Ω (1e)
Finite element discretisation with u = Nu u
M a + Fint
= Fext
(2)
M =
Ω
ρ NT
u Nu dΩ, Fint
=
Ω
BT
σ dΩ
Fext
=
Ω
NT
u f dΩ +
ΓN
NT
u t dΓ
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Introduction
Explicit schemes - introduction (cont’d)
Chung and Lee scheme [1]
M an+1 = Fext
n − Fint
n (3a)
un+1 = un + ∆t vn + ∆t2 1
2
− β an + β an+1 (3b)
vn+1 = vn + ∆t [(1 − γ) an + γ an+1] (3c)
∆t = CFL
h
c
(4)
Mass lumping for M
1
3
1
3
1
3
1
4
1
4
1
4
1
4
Advantages
No need for matrix solvers
Computationally appealing for dynamic
problems with short-term response
Blast and impact loading
Wave propagation
Dynamic fracture
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Introduction
Explicit schemes - fundamental issues
1
3
1
3
1
3
(a) Row-Sum
1
4
1
4
1
4
1
4
(b) Row-Sum
0 0
0
1
3
1
3
1
3
(c) Row-Sum
3/57 3/57
3/57
16/57
16/5716/57
(d) Proportional
Figure: Lagrange elements: mass contribution for each node using mass lumping
Issues (for compressible linear elastic materials (ν < 0.35))
Linear triangle/tetrahedron - stiff behaviour, especially in bending
Linear quad/hex - difficulty in mesh generation for complex 3D geometries
Quadratic tria/tetra - not recommended for contact problems in dynamics
ANSYS explicit - does not support any higher-order elements
Abaqus explicit - C3D10M but is very expensive
Cubic and higher-order - very expensive for any practical applications.
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Introduction
Explicit schemes - additional issues due to incompressibility
At this point we are practically left with linear triangular/tetrahedral elements
only for which
Pure displacement formulation results in
Volumetric and shear locking
Spurious oscillations in pressure field
Reduced integration
Not applicable
Selective reduced integration
Not applicable
B-bar formulation
Not applicable
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Literature
Literature
1.) Fractional-step-based projection schemes by Zienkiewicz and co. [2]
2.) Averaged nodal pressure approach by Bonet and Burton [3]
3.) Stabilised nodally integrated elements by Puso and Solberg [4]
4.) F-bar patch for triangular/tetrahedral elements by de Souza Neto et al. [5]
5.) F-bar aided edge-based smoothed method by Onishi et al. [6]
6.) D-VMS mixed formulations by Scovazzi et al [7, 8, 9]
7.) Mixed displacement-stress & displacement-strain by Cervera et al. [10, 11]
8.) First-order conservation laws by Bonet and Gil group [12, 13]
Disadvantages
First-order accuracy for stresses
Significant number of additional variables for second-order accurate stresses
Ad-hoc stabilisation parameters that control accuracy and stability
Unsuitability of dynamic variables to elastostatic problems (occupy major
share of problems in solid mechanics)
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B´ezier elements
Alternatives and a solution
Sticking with the Lagrange elements does not offer efficient solutions.
Isogeometric analysis (IGA) - B-Splines, NURBS, T-Splines etc.
Explicit dynamics - Anitescu et al [14], Evans et al [15]
× Major portion of research on IGA is limited to tensor-product meshes.
× No preprocessors (mesh generators) for IGA.
× Pose difficulties in applying Dirichlet BCs.
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B´ezier elements
Alternatives and a solution
Sticking with the Lagrange elements does not offer efficient solutions.
Isogeometric analysis (IGA) - B-Splines, NURBS, T-Splines etc.
Explicit dynamics - Anitescu et al [14], Evans et al [15]
× Major portion of research on IGA is limited to tensor-product meshes.
× No preprocessors (mesh generators) for IGA.
× Pose difficulties in applying Dirichlet BCs.
But
Relax requirements on isogeometry.
For practical applications, quadratic elements are sufficient enough.
For quadratic non-isogeometric B´ezier elements, existing mesh generators can
be leveraged by exploiting the properties of B´ezier curve.
Dirichlet BCs can be applied using elimination approach.
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B´ezier elements
Quadratic B´ezier elements - properties
P1(X1) P2(X2)
P3(X3)
P4
X4
P5
X5
P6
X6
(a) Curved edges
P1(X1) P2(X2)
P3(X3)
P4(X4)
P5(X5)P6(X6)
(b) Straight edges
Figure: Quadratic B´ezier triangle.
- Control point. - Node.
0
1
0.2
0.4
1
0.6
0.8
2
0.8
0.5 0.6
1
1
0.4
0.2
0 0
0
1
0.2
0.4
0.6
0.8
1
2
0.5
1
0.8
1
0.6
0.4
0.2
0 0
0
1
0.2
0.4
1
0.6
0.8
2
0.8
0.5 0.6
1
1
0.4
0.2
0 0
0
1
0.2
0.4
1
0.6
0.8
2
0.8
0.5 0.6
1
1
0.4
0.2
0 0
0
1
0.2
0.4
1
0.6
0.8
2
0.8
0.5 0.6
1
1
0.4
0.2
0 0
0
1
0.2
0.4
1
0.6
0.8
2
0.8
0.5 0.6
1
1
0.4
0.2
0 0
Figure: Shape functions
Advantages
Non-negative basis functions
Mass lumping - ideal for explicit schemes
Uniform nodal loads due to applied pressure
Smooth curve/surface - good for contacts
1
6
1
6
1
6
1
6
1
6
1
6
Figure: Row-sum lumping
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B´ezier elements
Quadratic B´ezier elements - mesh generation
P1(X1) P3(X3)
P2
X2
(a) Curved edge
P1(X1) P3(X3)P2(X2)
(b) Straight edge
Figure: Quadratic B´ezier
curve
A point at parametric coordinate, ξ(0 ≤ ξ ≤ 1)
X(ξ) = (1 − ξ)2
P1 + 2 ξ (1 − ξ) P2 + ξ2
P3
X(ξ = 0) = P1 = X1
X(ξ = 1) = P3 = X3
For any other ξ = ˆξ corresponding to node X2,
P2 =
1
2 ˆξ (1 − ˆξ)
X2 − (1 − ˆξ)2
X1 − ˆξ2
X3 (5)
When X2 is exactly in the middle, then ξ = ˆξ = 0.5.
P2 = 2 [X2 − 0.25 X1 − 0.25 X3] (6)
Chennakesava Kadapa (SA2C) Unified FE schemes for Computational Mechanics UKACM 2019 Conference 10 / 38

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B´ezier elements
Quadratic B´ezier elements - mesh generation
P1(X1) P2(X2)
P3(X3)
P4
X4
P5
X5
P6
X6
Figure: Quadratic B´ezier triangle
P1 = X1
P2 = X2
P3 = X3
P4 = 2 [X4 − 0.25 X1 − 0.25 X2]
P5 = 2 [X5 − 0.25 X2 − 0.25 X3]
P6 = 2 [X6 − 0.25 X3 − 0.25 X1]
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B´ezier elements
Quadratic B´ezier elements - Dirichlet BCs
Mesh generation
P1 = X1
P2 = X2
P3 = X3
P4 = 2 [X4 − 0.25 X1 − 0.25 X2]
P5 = 2 [X5 − 0.25 X2 − 0.25 X3]
P6 = 2 [X6 − 0.25 X3 − 0.25 X1]
Dirichlet BCs
uB
1 = uL
1
uB
2 = uL
2
uB
3 = uL
3
uB
4 = 2 uL
4 − 0.25 uL
1 − 0.25 uL
2
uB
5 = 2 uL
5 − 0.25 uL
2 − 0.25 uL
3
uB
6 = 2 uL
6 − 0.25 uL
3 − 0.25 uL
1
0.4 0.8 1.2 1.6
-log(h)
10−8
10−6
10−4
10−2
errornorm
1.0
3.0
1.0
2.0
L2 error
H1 error
0.4 0.8 1.2 1.6
-log(h)
10−8
10−6
10−4
10−2
errornorm
1.0
3.0
1.0
2.0
L2 error
H1 error
Figure: Poisson equation with solution u(r, θ) = 2
3
(r − 1
r
) sin θ
Chennakesava Kadapa (SA2C) Unified FE schemes for Computational Mechanics UKACM 2019 Conference 12 / 38

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B-bar formulation
B-bar formulation - behaviour in bending (Kadapa [16])
0 10 20 30 40 50 60
Number of elements per side
0
2
4
6
8
10
Y-displacementofthetip(mm)
Ref
TRI3
TRIB6
TRIB6B
(a) Convergence
-14.00
-7.00
0.00
7.00
-20.00
12.00
pressure
(b) Pressure - TRIB6
-14.00
-7.00
0.00
7.00
-20.00
12.00
pressure
(c) Pressure - TRIB6B
0 5 10 15 20
Number of elements along length
0
10
20
30
40
50
60
70
Y-displacementofthetip
TET4
TETB10
TETB10B
(a) Convergence
-3.25 0.00 3.25-7.00 6.00
pressure
(b) TETB10
-3.25 0.00 3.25-7.00 6.00
pressure
(c) TETB10B
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B-bar formulation
B-bar formulation - elastodynamics - complex geometries
(a) Mesh M1 (b) Mesh M2
0.000 0.005 0.010 0.015 0.020
Time (s)
−20
−15
−10
−5
0
5
10
15
20
Y-displacementofpointA(mm)
TETB10-M1
TETB10-M2
TETB10B-M1
TETB10B-M2
(c) Time Vs Displacement
0.0e+00
-2.0e+06
2.0e+06
sigma_xx
(a) TETB10
0.0e+00
-2.0e+06
2.0e+06
sigma_xx
(b) TETB10B
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F-bar formulation
F-bar formulation
0 10 20 30 40 50 60 70
Number of elements per side
0
2
4
6
8
10
Y-displacementofpointA
Reference
TRIB6
Q2/Q2-SD
TRIB6F
(a) Convergence
-13.6
-7.2
-0.8
5.6
-20.0
12.0
pressure
(b) TRIB6
-13.6
-7.2
-0.8
5.6
-20.0
12.0
pressure
(c) TRIB6F
-1.0e+05
3.0e+05
7.0e+05
1.1e+06
-5.0e+05
1.5e+06
pressure
(a) Implicit
-1.0e+05
3.0e+05
7.0e+05
1.1e+06
-5.0e+05
1.5e+06
pressure
(b) Explicit
Issues
No reduction in number of load steps
Requires excessive numerical damping for
high-frequency modes
TVD-RK2 method - not computationally
appealing due to two-stage process
Not applicable for truly incompressible
models
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Mixed formulation
Mixed displacement-pressure formulation (Kadapa [17])
Modified Cauchy stress:
σ = σdev + m p (7)
Nearly incompressible materials:
p = κ mT
ε − Small strains (8)
p =
∂U
∂J
− Finite strains (9)
Static and implicit elastodynamics:
Kuu Kup
Kpu Kpp
∆u
∆p = −
Ru
Rp
(10)
Explicit elastodynamics:
Muu an+1 = Fext
n − Fint,mixed
n (11)
un+1 = un + ∆t vn + ∆t2 1
2
− β an + β an+1 (12)
vn+1 = vn + ∆t [(1 − γ) an + γ an+1] (13)
Mpp pn+1 =
Ω
NT
p κ mT
εn+1 dΩ − Small strains (14)
Mpp pn+1 =
Ω
NT
p
∂U
∂J un+1
dΩ − Finite strains (15)
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Mixed formulation
Displacement-pressure combinations - Inf-Sup stability and Accuracy
BT2/BT0 — Quadratic B´ezier triangle/tetrahedron for displacement
and element-wise constant value for pressure (25%)
BT2/BT1 — Quadratic B´ezier triangle/tetrahedron for displacement
and linear B´ezier triangle/tetrahedron for pressure (5%)
0.1 0.4 0.7 1.0 1.3
-log(h)
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
log(βh)
BT2/BT0 (2D)
BT2/BT0 (3D)
BT2/BT1 (2D)
BT2/BT1 (3D)
(a) Inf-Sup constants
0.0 0.3 0.6 0.9 1.2 1.5 1.8
-log(h)
10−10
10−8
10−6
10−4
10−2
100
102
errornorm
1
4.2
1
2.1
1
2
1
1
||eu||L2 (BT2/BT0)
||eu||L2 (BT2/BT1)
||eσ||L2 (BT2/BT0)
||eσ||L2 (BT2/BT1)
(b) 2D problem
0.0 0.3 0.6 0.9 1.2 1.5
-log(h)
10−10
10−8
10−6
10−4
10−2
100
102
errornorm
1
4.2
1
2.1
1
1.8
1 0.5
||eu||L2 (BT2/BT0)
||eu||L2 (BT2/BT1)
||eσ||L2 (BT2/BT0)
||eσ||L2 (BT2/BT1)
(c) 3D problem
Figure: Stability and accuracy characteristics.
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Fully-Explicit scheme - examples
Mixed formulation - examples - elastostatics
0 5 10 15 20
Number of elements per side
0
20
40
60
80
100
120
%Compression
p/p0 =20
p/p0 =40
p/p0 =60
p/p0 =80
Reference
BT2
BT2/BT0
BT2/BT1
Q2/Q2-SD
(a) Convergence
-186.0
-124.0
-62.0
0.0
-250.0
60.0
pressure
(b) BT2
-186.0
-124.0
-62.0
0.0
-250.0
60.0
pressure
(c) BT2/BT0
-186.0
-124.0
-62.0
0.0
-250.0
60.0
pressure
(d) BT2/BT1
Figure: Compression of block: Neo-Hookean model, ν = 0.4999.
0 25 50 75 100 125 150 175 200
Z-displacement of point A (x-1)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Loadfactor
Reference
M1-BT2
M1-BT2/BT0
M1-BT2/BT1
M2-BT2
M2-BT2/BT0
M2-BT2/BT1
(a) Convergence
-3.00e+05
-1.00e+05
1.00e+05
3.00e+05
-5.00e+05
5.00e+05
sigma_zz
(b) BT2
-3.00e+05
-1.00e+05
1.00e+05
3.00e+05
-5.00e+05
5.00e+05
sigma_zz
(c) BT2/BT0
-3.00e+05
-1.00e+05
1.00e+05
3.00e+05
-5.00e+05
5.00e+05
sigma_zz
(d) BT2/BT1
Figure: Compression of block: Neo-Hookean model, ν = 0.3.
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Fully-Explicit scheme - examples
Mixed formulation - examples - elastodynamics
-1.0e+05
3.0e+05
7.0e+05
1.1e+06
-5.0e+05
1.5e+06
pressure
(i) (ii) (iii) (iv)
Figure: Neo-Hookean model, ν = 0.499.
Implicit or Explicit.
0 10 20 30 40 50 60 70 80
Time (μs)
3
4
5
6
7
8
Radius(mm)
Reference
BT2-Implicit
BT2-Explicit
BT2/BT0-Implicit
BT2/BT0-Explicit
BT2/BT1-Implicit
BT2/BT1-Explicit
-140.0
-70.0
0.0
70.0
-200.0
150.0
pressure
Figure: Elastoplastic. Implicit or Explicit.
0.0 0.2 0.4 0.6 0.8 1.0
Time (ms)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Radialdisplacement(in)
Experiment
Belytschko-Tsay (fine)
NURBS-Shell-p4
BT2-Explicit
BT2/BT0-Explicit
BT2/BT1-Explicit
(a) Comparison
-1.00e+04
0.00e+00
1.00e+04
2.00e+04
-2.00e+04
3.00e+04
pressure
(b) BT2
-1.00e+04
0.00e+00
1.00e+04
2.00e+04
-2.00e+04
3.00e+04
pressure
(c) BT2/BT1
Figure: Elastoplastic. Implicit or Explicit.
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Semi-implicit scheme
Semi-implicit scheme for mixed formulation
Weak form:
Muu an+1 +
Ω
BT
m pn+1 dΩ = Fext
n −
Ω
BT
σdev(un) dΩ (16)
Ω
NT
p mT
εn+1 −
pn+1
κ
dΩ = 0 (17)
Discretised system:
Kuu Kup
Kpu Kpp
∆u
∆p = −
Ru
Rp
(18)
where Kuu =
αm
β∆t2
Muu; Kpp = −
Ω
1
κ
NT
p Np dΩ
Solution: ∆p = S−1
−Rp + Kpu K−1
uu Ru (19)
∆u = K−1
uu [−Ru − Kup ∆p] (20)
Schur complement, S = Kpp − Kpu K−1
uu Kup (21)
Advantages:
Using BT2/BT1 element, size of S is only about 5% of that of global matrix.
Critical time step is limited only by shear wave speed
Straightforward to add contacts - Lagrange multipliers or penalty or Nitsche
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Semi-implicit scheme
Semi-implicit scheme - Fully-Explicit Vs Semi-Implicit
(a) M1 (b) M2
0 5 10 15 20 25 30 35 40
Time (ms)
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
Y-displacement(cm)
Explicit
Semi-implicit
(c) M1, ν = 0.45
0 5 10 15 20 25 30 35 40
Time (ms)
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
Y-displacement(cm)
Explicit
Semi-implicit
(d) M1, ν = 0.499
Figure: Twisting column: Neo-Hookean model.
ν = 0.3 ν = 0.45 ν = 0.48 ν = 0.499 ν = 0.49999
Mesh M1
Fully-explicit (FE) 10.8 17.6 27.4 119.2 1171.7
Semi-implicit (SI) 9.9 9.4 9.6 9.1 9.1
Ratio (FE/SI) 1.1 1.9 2.9 13.1 128.8
Mesh M2
Fully-explicit (FE) 161.7 286.8 429.8 1857.4 17838.1
Semi-implicit (SI) 199.1 188.3 185.9 183.0 183.0
Ratio (FE/SI) 0.8 1.5 2.3 10.1 97.5
Table: Twisting column: time taken in seconds for each simulation to reach 10 ms.
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Semi-implicit scheme
Semi-implicit scheme - Fully-Implicit Vs Semi-Implicit
0 5 10 15 20 25 30 35 40
Time (ms)
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
Y-displacement(cm)
Implicit (CM)
Implicit (LM)
Semi-implicit
(a) M1
0 5 10 15 20 25 30 35 40
Time (ms)
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
Y-displacement(cm)
Implicit (CM)
Implicit (LM)
Semi-implicit
(b) M2
1.29e+04
3.57e+04
5.86e+04
8.14e+04
1.04e+05
1.27e+05
-1.00e+04
1.50e+05
pressure
(c) FI
1.29e+04
3.57e+04
5.86e+04
8.14e+04
1.04e+05
1.27e+05
-1.00e+04
1.50e+05
pressure
(d) SI
Figure: Twisting column: Neo-Hookean model, ν = 0.5.
Mesh M1 Mesh M2
Fully-implicit scheme (FI) 319 12884
Semi-implicit scheme (SI) 10 218
Ratio (FI/SI) 32 60
Table: Twisting column: time taken in seconds for each simulation to reach 10 ms.
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Semi-implicit scheme
Semi-implicit scheme - complex geometry and wave propagation
-0.2 0.0 0.2-0.5 0.5
pressure
0.0 1.2 2.4 3.6-1.0 5.0
pressure
Figure: Stent model: Ogden model with ν = 0.5.
0.1 0.2 0.3 0.40.0 0.5
Displacement
-3.0 0.0 3.0-7.9 7.3
sigma_xy
Figure: Wave propagation: shear wave in linear elastic medium, ν = 0.5.
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Summary
Summary
Novel unified finite element formulations using B´ezier elements
Introduced B-bar and F-bar formulations for BT2 element
Introduced BT2/BT0 and BT2/BT1 elements
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Summary
Acknowledgements
Acknowledges the support of the Supercomputing Wales project, which is
part-funded by the European Regional Development Fund (ERDF) via the Welsh
Government.
Thank you
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References
References I
J. Chung and J. M. Lee.
A new family of explicit time integration methods for linear and non-linear structural
dynamics.
International Journal for Numerical Methods in Engineering, 37:3961–3976, 1994.
O. C. Zienkiewicz, J. Rojek, R. L. Taylor, and M. Pastor.
Triangles and Tetrahedra in explicit dynamic codes for solids.
International Journal for Numerical Methods in Engineering, 43:565–583, 1998.
J. Bonet and A. J. Burton.
A simple average nodal pressure tetrahedral element for incompressible and nearly
incompressible dynamic explicit applications.
Communications in Numerical Methods in Engineering, 14:437–449, 1998.
M. A. Puso and J. Solberg.
A stabilized nodally integrated tetrahedral.
International Journal of Numerical Methods in Engineering, 67:841–867, 2006.
E. A. de Souza Neto, F. M. Andrade Pires, and D. R. J. Owen.
F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible
solids. Part I: formulation and benchmarking.
International Journal of Numerical Methods in Engineering, 62:353–383, 2005.
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References
References II
Y. Onishi, R. Iida, and K. Amaya.
F-bar aided edge-based smoothed finite element method using tetrahedral elements for
finite deformation analysis of nearly incompressible solids.
International Journal for Numerical Methods in Engineering, 109:1582–1606, 2017.
G. Scovazzi, B. Carnes, X. Zeng, and S. Rossi.
A simple, stable, and accurate linear tetrahedral finite element for transient, nearly, and
fully incompressible solid dynamics: a dynamic variational multiscale approach.
International Journal for Numerical Methods in Engineering, 106:799–839, 2016.
S. Rossi, N. Abboud, and G. Scovazzi.
Implicit finite incompressible elastodynamics with linear finite elements: A stabilized
method in rate form.
Computer Methods in Applied Mechanics and Engineering, 311:208–249, 2016.
G. Scovazzi, T. Song, and X. Zeng.
A velocity/stress mixed stabilized nodal finite element for elastodynamics: Analysis and
computations with strongly and weakly enforced boundary conditions.
Computer Methods in Applied Mechanics and Engineering, 325:532–576, 2017.
M. Cervera, M. Chiumenti, and R. Codina.
Mixed stabilized finite element methods in nonlinear solid mechanics. Part I: formulation.
Computer Methods in Applied Mechanics and Engineering, 199:2559–2570, 2010.
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References
References III
M. Cervera, M. Chiumenti, and R. Codina.
Mixed stabilized finite element methods in nonlinear solid mechanics. Part II: strain
localization.
Computer Methods in Applied Mechanics and Engineering, 199:2571–2589, 2010.
A. J. Gil, C. H. Lee, J. Bonet, and M. Aguirre.
A stabilised Petrov-Galerkin formulation for linear tetrahedral elements in compressible,
nearly incompressible and truly incompressible fast dynamics.
Computer Methods in Applied Mechanics and Engineering, 276:659–690, 2014.
J. Bonet, A. J. Gil, C. H. Lee, M. Aguirre, and R. Ortigosa.
A first order hyperbolic framework for large strain computational solid dynamics. Part I:
total Lagrangian isothermal elasticity.
Computer Methods in Applied Mechanics and Engineering, 283:689–732, 2015.
C. Anitescu, C. Nguyen, T. Rabczuk, and X. Zhuang.
Isogeometric analysis for explicit elastodynamics using a dual-basis diagonal mass
formulation.
Computer Methods in Applied Mechanics and Engineering, 346:574–591, 2019.
J. A. Evans, R. R. Hiemstra, T. J. R. Hughes, and A. Reali.
Explicit higher-order accurate isogeometric collocation methods for structural dynamics.
Computer Methods in Applied Mechanics and Engineering, 338:208–240, 2018.
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References
References IV
C. Kadapa.
Novel quadratic B´ezier triangular and tetrahedral elements using existing mesh generators:
Applications to linear nearly incompressible elastostatics and implicit and explicit
elastodynamics.
International Journal for Numerical Methods in Engineering, 117:543–573, 2019.
C. Kadapa.
Novel quadratic B´ezier triangular and tetrahedral elements using existing mesh generators:
Extension to nearly incompressible implicit and explicit elastodynamics in finite strains.
International Journal for Numerical Methods in Engineering, 2019.
T. J. R. Hughes.
The Finite Element Method: Linear Static and Dynamic Finite Element Analysis.
Dover Publications, 2000.
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Computational Methods for Plasticity, Theory and Applications.
John Wiley and Sons, United Kingdom, 2008.
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References
Appendix
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Appendix
B-bar formulation - basics
Idea: Hughes [18]
¯σ = D ¯ε, ¯ε = Idev ε + ¯εvol, ¯εvol =
1
V e
Ωe
εvol dΩ (22)
Fint,Bbar
=
Ω
¯BT
¯σ dΩ (23)
¯Ba =










( ¯B1 + 2B1)/3 ( ¯B2 − B2)/3 ( ¯B3 − B3)/3
( ¯B1 − B1)/3 ( ¯B2 + 2B2)/3 ( ¯B3 − B3)/3
( ¯B1 − B1)/3 ( ¯B2 − B2)/3 ( ¯B3 + 2B3)/3
B2 B1 0
0 B3 B2
B3 0 B1










(24)
B1 =
∂Na
∂x
; B2 =
∂Na
∂y
; B3 =
∂Na
∂z
(25)
¯B1 =
1
V e
Ωe
∂Na
∂x
dΩ; ¯B2 =
1
V e
Ωe
∂Na
∂y
dΩ; ¯B3 =
1
V e
Ωe
∂Na
∂z
dΩ. (26)
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Appendix
B-bar formulation - Implicit and explicit schemes
Implicit scheme:
M an+αm + K un+αf = Fext
n+αf
(27)
where, K =
Ω
¯BT
D¯B dΩ. (28)
Explicit scheme:
Muu an+1 = Fext
n − Fint,Bbar
n
un+1 = un + ∆t vn + ∆t2 1
2
− β an + β an+1
vn+1 = vn + ∆t [(1 − γ) an + γ an+1]
Fint,Bbar
n =
Ω
¯BT
Ω ¯σn dΩ
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Appendix
B-bar formulation - Thick cylinder under internal pressure
−2.0 −1.5 −1.0 −0.5
-log(h)
10−8
10−6
10−4
10−2
100
102
L2
errornormindisplacement
1.0
2.0
1.0
3.0
TRI3
TRIB6
TRIB6B
(a) ν = 0.3
−2.0 −1.5 −1.0 −0.5
-log(h)
10−8
10−6
10−4
10−2
100
102
L2
errornormindisplacement
1.0
2.0
1.0
3.0
TRI3
TRIB6
TRIB6B
(b) ν = 0.48
−2.0 −1.5 −1.0 −0.5
-log(h)
10−8
10−6
10−4
10−2
100
102
L2
errornormindisplacement
1.0
2.2
1.0
3.0
TRI3
TRIB6
TRIB6B
(c) ν = 0.49999
−2.0 −1.5 −1.0 −0.5
-log(h)
10−6
10−4
10−2
100
102
L2
errornorminstress
1.0
1.0
1.0
2.0
TRI3
TRIB6
TRIB6B
(d) ν = 0.3
−2.0 −1.5 −1.0 −0.5
-log(h)
10−6
10−4
10−2
100
102
L2
errornorminstress
1.0
1.0
1.0
2.0
TRI3
TRIB6
TRIB6B
(e) ν = 0.48
−2.0 −1.5 −1.0 −0.5
-log(h)
10−6
10−4
10−2
100
102
L2
errornorminstress
1.0
2.0
TRI3
TRIB6
TRIB6B
(f) ν = 0.49999
Figure: Error norms in displacement and stress
-0.06
0
0.06
0.12
-0.10
0.17
sigma_xx
(a) Displacement
formulation
-0.06
0
0.06
0.12
-0.10
0.17
sigma_xx
(b) B-bar formulation
Figure: σxx stress
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Appendix
B-bar formulation - Thick sphere under internal pressure
−2.25 −2.00 −1.50 −1.00 −0.75
-log(h)
10−6
10−4
10−2
100
102
L2
errornormindisplacement
1.0
2.0
1.0
3.0
TET4
TETB10
TETB10B
(a) ν = 0.3
−2.25 −2.00 −1.50 −1.00 −0.75
-log(h)
10−6
10−4
10−2
100
102
L2
errornormindisplacement
1.0
1.8
1.0
3.0
TET4
TETB10
TETB10B
(b) ν = 0.48
−2.25 −2.00 −1.50 −1.00 −0.75
-log(h)
10−6
10−4
10−2
100
102
L2
errornormindisplacement
1.0
3.0
TET4
TETB10
TETB10B
(c) ν = 0.49999
−2.25 −2.00 −1.50 −1.00 −0.75
-log(h)
10−2
100
102
104
L2
errornorminstress
1.0
1.0
1.0
2.0
TET4
TETB10
TETB10B
(d) ν = 0.3
−2.25 −2.00 −1.50 −1.00 −0.75
-log(h)
10−2
100
102
104
L2
errornorminstress
1.0
0.75
1.0
2.0
TET4
TETB10
TETB10B
(e) ν = 0.48
−2.25 −2.00 −1.50 −1.00 −0.75
-log(h)
10−2
100
102
104
L2
errornorminstress 1.0
0.75
1.0
2.0
TET4
TETB10
TETB10B
(f) ν = 0.49999
Figure: Error norms in displacement and stress
-0.080
-0.040
0.000
0.040
-0.098
0.071
sigma_xx
(a) Displacement
formulation
-0.080
-0.040
0.000
0.040
-0.098
0.071
sigma_xx
(b) B-bar formulation
Figure: σxx stress
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Appendix
B-bar formulation - behaviour in bending
0 10 20 30 40 50 60
Number of elements per side
0
2
4
6
8
10
Y-displacementofthetip(mm)
Ref
TRI3
TRIB6
TRIB6B
(a) Convergence
-14.00
-7.00
0.00
7.00
-20.00
12.00
pressure
(b) Pressure - TRIB6
-14.00
-7.00
0.00
7.00
-20.00
12.00
pressure
(c) Pressure - TRIB6B
0 5 10 15 20
Number of elements along length
0
10
20
30
40
50
60
70
Y-displacementofthetip
TET4
TETB10
TETB10B
(a) Convergence
-3.25 0.00 3.25-7.00 6.00
pressure
(b) TETB10
-3.25 0.00 3.25-7.00 6.00
pressure
(c) TETB10B
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Appendix
B-bar formulation - elastodynamics - complex geometries
(a) Mesh M1 (b) Mesh M2
0.000 0.005 0.010 0.015 0.020
Time (s)
−20
−15
−10
−5
0
5
10
15
20
Y-displacementofpointA(mm)
TETB10-M1
TETB10-M2
TETB10B-M1
TETB10B-M2
(c) Time Vs Displacement
0.0e+00
-2.0e+06
2.0e+06
sigma_xx
(a) TETB10
0.0e+00
-2.0e+06
2.0e+06
sigma_xx
(b) TETB10B
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Appendix
F-bar formulation - Implicit and explicit schemes
Idea: de Souza Neto et al [19]
¯σ = σ(¯F), ¯F =
J0
J
1
dim
F, J = detF, J0 = J|centroid (29)
Implicit scheme:
αm
β ∆t2
M + αf (KM + KG + Kq) ∆u = −R (30)
KM =
ω
BT
D B dω, KG =
ω
GT
Σ G dω, Kq =
ω
GT
q (G0 − G) dω (31)
Explicit scheme:
Muu an+1 = Fext
n − Fint,Fbar
n
un+1 = un + ∆t vn + ∆t2 1
2
− β an + β an+1
vn+1 = vn + ∆t [(1 − γ) an + γ an+1]
Fint,Fbar
=
ω
BT
ω ¯σ dω
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Appendix
F-bar formulation - results and issues
0 10 20 30 40 50 60 70
Number of elements per side
0
2
4
6
8
10
Y-displacementofpointA
Reference
TRIB6
Q2/Q2-SD
TRIB6F
(a) Convergence
-13.6
-7.2
-0.8
5.6
-20.0
12.0
pressure
(b) TRIB6
-13.6
-7.2
-0.8
5.6
-20.0
12.0
pressure
(c) TRIB6F
-1.0e+05
3.0e+05
7.0e+05
1.1e+06
-5.0e+05
1.5e+06
pressure
(a) Implicit
-1.0e+05
3.0e+05
7.0e+05
1.1e+06
-5.0e+05
1.5e+06
pressure
(b) Explicit
Issues
No reduction in number of load steps
Requires excessive numerical damping for
high-frequency modes
TVD-RK2 method - not computationally
appealing due to two-stage process
Not applicable for truly incompressible
models
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