The document presents coupling approaches for combining classical linear elasticity models with non-local peridynamic models for applications in computational mechanics. It describes three coupling methods - matching displacements (MDCM), matching stresses (MSCM), and variable horizon (VHCM). Numerical examples are presented to compare the accuracy of the three methods on manufactured solutions using cubic and quartic polynomials, demonstrating that all methods converge with refinement but VHCM typically has the lowest error.

1. On the coupling of classical and non-local models for
applications in computational mechanics
Patrick Diehlb and Serge Prudhommea
aDépartement de Mathématiques et de Génie Industriel
Polytechnique Montréal
bCenter for Computation and Technology
Louisiana State University
Society of Engineering Science Annual Technical Meeting
October 16–18, 2022
P. Diehl (CCT/Physics @ LSU) Coupling Approaches October 16–18, 2022 1 / 28

2. Content
Model problem in 1D: linear elasticity and peridynamic models
Coupling formulations for classical linear elasticity models
Coupling methods for linear elasticity and peridynamic models
Discrete formulation
Numerical examples
Non-equidistant meshes
Damage
Concluding remarks
Presentation partly based on:
P. Diehl and S. Prudhomme,
Coupling approaches for classical linear elasticity and bond-based peridynamic models,
Journal of Peridynamics and Nonlocal Modeling, 2021.
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3. Model Problem
Classical linear elasticity model in 1D (with cross-sectional area A = 1):
−Eu′′
(x) = fb(x), ∀x ∈ Ω = (0, ℓ),
u(x) = 0, at x = 0
Eu′
(x) = g, at x = ℓ
Coupling with peridynamic model:
Nonlocal model in Ωδ = (a, b) ⊂ Ω where δ = horizon.
0 a − δ a x − δ x x + δ b b + δ ℓ
Ω
Ωδ
Hδ(x)
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4. Peridynamics
Linearized microelastic bond-based model [Silling, 2000]:
−
Z
Hδ(x)
κ
ξ ⊗ ξ
∥ξ∥3
(u(y) − u(x))dy = fb(x)
In 1D:
−
Z x+δ
x−δ
κ
u(y) − u(x)
|y − x|
dy = fb(x)
Taylor expansion:
Z x+δ
x−δ
κ
u(y) − u(x)
|y − x|
dy =
κδ2
2
u′′
(x) +
δ2
24
u′′′′
(x) + . . .
,
Approximation:
−
κδ2
2
u′′
(x) +
δ2
24
u′′′′
(x) + . . .
= fb(x), ∀x ∈ Ωδ
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5. Compatibility of the two models
By taking the limit δ → 0, one then recovers the local model
pointwise whenever κ is chosen as:
E =
κδ2
2
i.e. κ =
2E
δ2
The two models are fully compatible if
u(k)
(x) = 0, ∀x ∈ Ωδ, ∀k ≥ 4
The peridynamic model provides an approximation of the LE model
with a degree of precision equal to d = 3 w.r.t. the horizon δ.
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6. Compatibility of the two models
Stress at a point ([Silling, 2000], [Ongaro et al., 2021]):
σ±
(u)(x) =
Z x
x−δ
Z z±δ
x
κ
u(y) − u(z)
|y − z|
dydz
= Eu′
(x) +
Eδ2
24
u′′′
(x) + O(δ3
)
In order to obtain approximations with a degree of precision of three,
one needs
σ±
(u)(x) =
Z x
x−δ
Z z±δ
x
κ
u(y) − u(z)
|y − z|
dydz −
κδ4
48
u′′′
(x)
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7. Coupling Approaches
[1] Bauman, Ben Dhia, Elkhodja, Oden, and Prudhomme,
On the application of the Arlequin method to the coupling of particle and
continuum models,
Computational Mechanics, Vol. 42, 511–530 (2008).
[2] Seleson, Beneddine, and Prudhomme,
A force-based coupling scheme for peridynamics and classical elasticity,
Computational Materials Science, Vol. 66, 34–49 (2013).
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8. Coupling of linear elasticity models
0 a − ε a ℓ
b b + ε
Ω
Ω1
Γa Ω2 Γb
Ω1
In Ωi, i = 1, 2:
−Eiu′′
i (x) = fb(x), ∀x ∈ Ωi
Boundary conditions:
u1(x) = 0, at x = 0
E1u′
1(x) = g, at x = ℓ
Interface conditions:
Continuity of displacement: u1(x) − u2(x) = 0, at x = a, b
Continuity of stress: E1u′
1(x) − E2u′
2(x) = 0, at x = a, b
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9. Modified formulation if E1 = E2 := E
0 a − ε a ℓ
b b + ε
Ω
Ω1
Γa Ω2 Γb
Ω1
−Eu′′
1(x) = fb(x), ∀x ∈ Ω1
−Eu′′
2(x) = fb(x), ∀x ∈ (a − ε, b + ε)
u1(x) = 0, at x = 0
Eu′
1(x) = g, at x = ℓ
u1(x) − u2(x) = 0, at x = a, b
u1(x − ε) − u2(x − ε) = 0, at x = a
u1(x + ε) − u2(x + ε) = 0, at x = b
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10. Coupling methods for local and peridynamic models
0 a − δ a ℓ
b b + δ
Ω
Ωe
Γa Ωδ Γb
Ωe
We consider three different approaches:
MDCM = Coupling method with matching displacements
[Zaccariotto and Galvanetto, et al.], [Kilic and Madenci, 2018], [Sun
and Fish, 2019], [D’Elia and Bochev, 2021], etc.
MSCM = Coupling method with matching stresses
[Silling, Sandia Report, 2020]
VHCM = Coupling method with variable horizon
[S. Silling et al., 2015], [Nikpayam and Kouchakzadeh, 2019]
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11. MDCM formulation
0 a − δ a ℓ
b b + δ
Ω
Ωe
Γa Ωδ Γb
Ωe
−Eu′′
(x) = fb(x), ∀x ∈ Ωe
−
Z x+δ
x−δ
κ
u(y) − u(x)
|y − x|
dy = fb(x), ∀x ∈ Ωδ
u(x) = 0, at x = 0
Eu′
(x) = g, at x = ℓ
u(x) − u(x) = 0, ∀x ∈ Γa ∪ Γb
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12. MSCM formulation
0 a − δ a ℓ
b b + δ
Ω
Ωe
Γa Ωδ Γb
Ωe
−Eu′′
(x) = fb(x), ∀x ∈ Ωe
−
Z x+δ
x−δ
κ
u(y) − u(x)
|y − x|
dy = fb(x), ∀x ∈ Ωδ
u(x) = 0, at x = 0
Eu′
(x) = g, at x = ℓ
u(x) − u(x) = 0, at x = a, b
σ+
(u)(x) − Eu′
(x) = 0, ∀x ∈ Γa
σ−
(u)(x) − Eu′
(x) = 0, ∀x ∈ Γb
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13. VHCM formulation
δv
δ
0 a a + δ b − δ b x
Variable horizon function:
δv(x) =



x − a, a x ≤ a + δ
δ, a + δ x ≤ b − δ
b − x, b − δ x b
κ̄(x)δ2
v(x) = κδ2
, ∀x ∈ Ωδ
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14. VHCM formulation
−Eu′′
(x) = fb(x), ∀x ∈ Ωe
−
Z x+δv(x)
x−δv(x)
κ̄(x)
u(y) − u(x)
|y − x|
dy = fb(x), ∀x ∈ Ωδ
u(x) = 0, at x = 0
Eu′
(x) = g, at x = ℓ
u(x) − u(x) = 0, at x = a, b
σ+
(u)(x) − Eu′
(x) = 0, at x = a
σ−
(u)(x) − Eu′
(x) = 0, at x = b
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15. Discretization
x0 x1 xn1 xn1+nδ xn
Ω
Ω1
Γa Ωδ Γb
Ω2
For a given δ, we choose h such that δ/h = m is a positive integer and
such that the grid is uniform in each subinterval:
h = (b − a)/nδ = a/n1 = (ℓ − b)/n2
Classical linear elasticity model: Finite differences method with
2nd-order central difference stencil.
Peridynamic model: Collocation approach with 2nd-order trapezoidal
integration rule.
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16. Numerical experiments
Configurations:
Ω = [0, 3], a = 1, b = 2,
Manufactured problems:
Mixed BC and pure homogeneous Dirichlet BC.
Cubic polynomial solutions:
Mixed BC: u(x) = x3
Dirichlet BC: u(x) =
2
3
√
3
x(3 − 2x)(3 − x)
Quartic polynomial solutions:
Mixed BC: u(x) = x4
Dirichlet BC: u(x) =
16
81
x2
(3 − x)2
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17. Cubic manufactured solutions
Horizon δ = 1/8 and m = 2:
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
0
5
10
15
20
25
Displacement
Example with cubic solution with =1/8 and m=2
Exact solution
MDCM
MSCM
VHCM
Mixed BC
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00
Displacement
Example with cubic solution with =1/8 and m=2
Exact solution
MDCM
MSCM
VHCM
Homogeneous Dirichlet BC
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18. Quartic manufactured solutions
Error ∆max Relative error Er
δ m MDCM MSCM VHCM MDCM MSCM VHCM
1/8 2 0.0189985 0.0266890 0.0225350 0.18940 0.13873 0.15379
4 0.0234427 0.0252280 0.0194695 0.00022 0.07640 0.00316
8 0.0245057 0.0250035 0.0195917 0.04558 0.06682 0.00309
vN,max 0.0234375 0.0234375 0.0195312
1/16 2 0.0045721 0.0055334 0.0050141 0.21970 0.05564 0.06646
4 0.0056769 0.0059001 0.0051803 0.03114 0.00695 0.03553
8 0.0059471 0.0060094 0.0053329 0.01498 0.02560 0.00712
vN,max 0.0058594 0.0058594 0.0053711
1/32 2 0.0011208 0.0012410 0.0011761 0.23485 0.15282 0.16222
4 0.0013963 0.0014242 0.0013342 0.04682 0.02778 0.04960
8 0.0014644 0.0014721 0.0013876 0.00032 0.00499 0.01156
vN,max 0.0014648 0.0014648 0.0014038
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19. Quartic manufactured solutions
Error ∆max Relative error Er
δ m MDCM MSCM VHCM MDCM MSCM VHCM
1/8 2 0.0015401 0.0020472 0.0017665 0.20162 0.06127 0.06434
4 0.0019049 0.0020226 0.0016350 0.01249 0.04850 0.01493
8 0.0019929 0.0020258 0.0016605 0.03313 0.05017 0.00047
vD,max 0.0019290 0.0019290 0.0016598
1/16 2 0.0003734 0.0004367 0.0004021 0.22578 0.09444 0.10521
4 0.0004642 0.0004789 0.0004310 0.03746 0.00697 0.04090
8 0.0004865 0.0004906 0.0004455 0.00880 0.01731 0.00856
vD,max 0.0004823 0.0004823 0.0004493
1/32 2 0.0000919 0.0000998 0.0000955 0.23789 0.17224 0.18027
4 0.0001145 0.0001164 0.0001104 0.04997 0.03473 0.05218
8 0.0001202 0.0001207 0.0001151 0.00340 0.00085 0.01231
vD,max 0.0001206 0.0001206 0.0001165
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20. Challenges with damage and non-uniform meshes
In the vicinity of damage and a uniform mesh
The three methods recover the solution to the classical linear
elasticity model for polynomial solutions of degree up to three.
MSCM behaves generally better than MDCM.
VHCM and MSCM have similar behaviors, but VHCM avoids
introducing an overlap region.
Non-uniform meshes and damage
MDCM and MSCM require interpolation in the overlap region.
VHM requires interpolation as well, if the interfaces doe not overlap
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21. All with a non-uniform mesh
hPD = 1/2hFD
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
0.000000
0.000050
0.000100
0.000150
0.000200
0.000250
0.000300
0.000350
Error
in
displacement
w.r.t
FDM
Example with quartic solution for MDCM with m = 2
=1/8
=1/16
=1/32
=1/64
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
-0.000001
0.000000
0.000001
0.000002
0.000003
0.000004
0.000005
0.000006
0.000007
Error
in
displacement
w.r.t
FDM
Example with quartic solution for VHCM with m = 2
=1/8
=1/16
=1/32
=1/64
The error for all three methods for mixed boundary conditions.nMDCM
and MSCM using cubic interpolation and have a degree of precision two.
For the VHCM no interpolation is needed and has a degree of precision
three. Therefore, smaller errors are obtained.
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22. Condition numbers
1/8 1/16 1/32 1/64
105
106
107
108
109
Condition
number
Mixed boundary conditions
Quadratic interpolation
MDCM
MSCM
VHCM
1/8 1/16 1/32 1/64
105
106
107
108
Condition
number
Homogeneous Dirichlet boundary conditions
Quadratic interpolation
MDCM
MSCM
VHCM
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23. MDCM with a non-uniform mesh
hPD = 1/5hFD
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
-0.000400
-0.000350
-0.000300
-0.000250
-0.000200
-0.000150
-0.000100
-0.000050
0.000000
Error
in
displacement
w.r.t
FDM
Example with quartic solution for MDCM with m = 2
n=5
n=6
n=7
n=8
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
0.0035
0.0030
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
Error
in
displacement
w.r.t
FDM
Example with quartic solution for MDCM with m = 2
n=5
n=6
n=7
n=8
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24. MDCM with a non-uniform mesh and non-alinged
inferfaces
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
0.000000
0.001000
0.002000
0.003000
0.004000
0.005000
0.006000
Error
in
displacement
w.r.t
exact
solution
Example with quartic solution for MDCM with m = 2
m=4
m=5
m=6
m=7
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
0.000000
0.002000
0.004000
0.006000
0.008000
0.010000
0.012000
0.014000
0.016000
Error
in
displacement
w.r.t
exact
solution
Example with quartic solution for MDCM with m = 2
n=5
n=6
n=7
n=8
Error w.r.t exact solution for MDCM with mixed boundary conditions
using cubic interpolation. Here, the nodal spacing of the nonlocal region is
five time less the one in the local region. In addition, the nodes at the
interfaces x = 1 and x = 2 are not aligned. This results in one additional
interpolation in the coupling region.
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25. MDCM with damage I
1.3 1.4 1.5 1.6 1.7
0.90
0.92
0.94
0.96
0.98
1.00
x
Figure: Cubic spline interpolation for the variation of E(x) and c = 0.9. The
black dots represent the data used for the spline interpolation.
This function needs to be smooth to obtain convergence
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26. MDCM with damage II
Load / δ Linear Quadratic Cubic Quartic
c=0.9
1
⁄8 0.0004744 0.0008049 0.0008551 0.0004092
1
⁄16 0.0001200 0.0002108 0.0002271 0.0001320
1
⁄32 0.0000304 0.0000542 0.0000587 0.0000367
1
⁄64 0.0000076 0.0000137 0.0000149 0.0000096
c=0.1
1
⁄8 0.0580609 0.0618445 0.0493398 0.0346054
1
⁄16 0.0258549 0.0618445 0.0210322 0.0145342
1
⁄32 0.0078533 0.0618445 0.0063260 0.0043566
1
⁄64 0.0078533 0.0618445 0.0063260 0.0043566
Table: Maximal error w.r.t to FD for mixed boundary conditions with various
damage profiles for MDCM using cubic interpolation.
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27. Condition numbers
1/8 1/16 1/32 1/64
106
107
108
Condition
number
Mixed boundary conditions
Damage
0.1
0.25
0.75
0.9
1/8 1/16 1/32 1/64
105
106
107
108
Condition
number
Homogeneous boundary conditions
Damage
0.1
0.25
0.75
0.9
Condition number of the stiffness matrix for MDCM with variation of E
with respect to the horizon δ for the case with mixed boundary conditions
and homogeneous Dirichlet boundary conditions.
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28. Concluding Remarks
Non-uniform meshes show convergence in the error for all three
methods
VHCM performs better due to no interpolation
Non-aliened interfaces show still convergence for MDCM
Even with damage convergence is obtained for MDCM Future work
will extend the coupling approaches to other discretization methods
and to 2D/3D problems.
Investigate the damage via bond-breaking in 2D/3d
No mathematical proofs for the existence and uniqueness of the solution
for the couple system is available including damage.
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