A tale of two approaches for coupling nonlocal and local models

A tale of two approaches for coupling nonlocal and
local models
Patrick Diehl
Joint work with Serge Prudhomme, Rob Lipton, Matthias Birner and Alex
Schweitzer
Center for Computation and Technology
Department of Physics and Astronomy
Louisiana State University
Continuum Mechanics Seminar (CMS)
November 10, 2022
P. Diehl (CCT/Physics @ LSU) Coupling Approaches November 10, 2022 1 / 57

Motivation
Crack in a steel beam on the Interstate
40 bridge, near Memphis, Tenn.
Fatigue cracking of the skin panel on
Aloha Airlines Flight 243
Accurate crack simulations are important for the public safety
Peridynamic models can predict the crack path very precise, but are
computationally expensive
Coupling local and nonlocal models allows use to save computations
while using peridynamic only in the region where crack and fracture
arises

Outline
Part I
A Fracture Multiscale Model for Peridynamic enrichment within the
Partition of Unity Method
Reference:
Birner, Matthias, et al. ”A Fracture Multiscale Model for Peridynamic enrichment within the Partition of Unity Method.” arXiv
preprint arXiv:2108.02336 (2021). Accepted in Advances in Engineering Software.
Part II
On the coupling of classical and non-local models for applications in
computational mechanics
Reference:
Diehl, Patrick, and Serge Prudhomme. ”Coupling approaches for classical linear elasticity and bond-based peridynamic
models.” Journal of Peridynamics and Nonlocal Modeling (2022): 1-31.
Results with adaptive meshes and damage in preparation for Computational Mechanics.

PART I

Outline
1 Brief Introduction: Partition of Unity Methods
2 Brief Introduction: Peridynamics
3 Algorithm
4 Validation of the proposed approach
5 Numerical results
6 Conclusion

Partion of Unity Method (PUM)
Given a computational domain Ω, we assume to have a partition of unity
{ϕi} with ϕi ≥ 0 and
X
i
ϕi x

= 1 ∀x ∈ Ω
that covers the domain. We call the support of a PU function ϕi a patch
ωi := supp(ϕi).
To construct a higher order basis, each PU function ϕi is multiplied with a
local approximation space
Vi := Pi ⊕ Ei = span ψs
i , ηt
i ,
of dimension ni, where Pi are spaces of polynomials of degree dPi .

Partion of Unity Method (PUM)
To construct a higher order basis, each PU function ϕi is multiplied with a
local approximation space
Vi := Pi ⊕ Ei = span ψs
i , ηt
i ,
of dimension ni, where Pi are spaces of polynomials of degree dPi .
In the PUM the global approximation space then reads as
VPU
:=
X
i
ϕiVi =
X
i
ϕiPi + ϕiEi.

PUM equation of motion
Let us first introduce our general model problem, the equation of (linear
elastic) motion. On the global domain Ω we consider
%(x)ü(t, x) = −∇ · σ + b in [0, T] × Ω0,
where b are the volume forces acting on the body, e.g. gravity, σ is the
Cauchy stress tensor and % the mass density. The latter is computed from
the linear strain tensor ε via Hooke’s law
σ = C : ε = 2µε(u) + λtr (ε(u)) I,
with C denoting Hooke’s tensor, µ, λ the Lame parameters, tr(·) the trace
and I the identity. The linear strain tensor is computed from the
displacement field u by
ε u

(t, x) =
1
2

∇u(t, x) + ∇u(t, x)
T

.

Discrertization in space
The weak formulation of (7) is given by: For fixed t ∈ [0, T] find
u ∈ VPU(Ω) such that
Z
Ω
%üv dx = −
Z
Ω
σ u

: ε v

dx +
Z
ΓN
t̄v ds +
Z
Ω
bv dx
for all test functions v ∈ VPU that vanish on ΓD.

Bond-based PD equation of motion
%(x)ü(x)(x, t) =
Z
Bδ(x)
f(t, x0
− x, u(x0
, t) − u(x, t)) dx0
+ b(x, t)
rc
−rc
r+
−r+
r
g0(r)
Figure 1: Plot of the derivative g0
(r) of the potential function g(r) used in the
cohesive force. The force goes smoothly to zero at ±r+
.
References
Lipton, Robert. ”Dynamic brittle fracture as a small horizon limit of peridynamics.” Journal of Elasticity 117.1 (2014): 21-50.

Enriching PUM with PD using a global-local approach
global problem
solve
identify
region
boundary
conditions
PD
local problem
local
solve
extract
crack path
PUM
crack
enrichments

Two-dimensional bar (pure linear elasticity)
−F F
1
0.1
x
y
Figure 2: Sketch of the two-dimensional bar which is used to study the influence of
bond softening on the displacement. Initially, a force of F = ±9 × 105
N is applied
on the left-hand side and right-hand side of the bar. The force increases linearly
to 2F, 4F, and 8F which results in a bond damages of 7.3% up to 106.1%.
Compatibility of the models
Both models only match while the PD model stays in the linear
regime.

Simulation setup
Table 1: Simulation parameters for the discretization in time and space for the
two-dimensional bar problem.
Force F =9 × 105N Time steps tn=50 000
Node spacing hPD =0.0005m Time step size ts =2 × 10−8s
and hPUM =0.007 812 5m
Horizon δ = 4hPD =0.002m Final time T=0.001s

Simulation results
Table 2: The maximal displacement magnitude Umax obtained by the quasi-static
PUM simulation and by the explicit PD simulation for the two-dimensional bar,
see Figure 2. The traction condition’s load is increased up to twelve times to
showcase the influence of softened bond to the displacement field. As long as the
PD model stays in the linear regime, both methods result in a similar
displacement and once the softening starts the results diverges as expected.
Load [N] Umax [m] Damage [%]
PUM PD
9 × 105 1.235 × 10−4 1.203 × 10−4 7.3
4×9 × 105 4.941 × 10−4 4.890 × 10−4 29.8
8×9 × 105 9.883 × 10−4 1.020 × 10−3 62.7
12×9 × 105 1.482 × 10−3 1.675 × 10−3 106.1

Stationary Mode I crack
0.1
0.1
0.01
0.02
0.02
x
y
−F F
u = 0
Figure 3: Sketch of the square plate (0.1m × 0.1m) with an initial crack of length
0.02m. For the local PUM model a constant traction boundary condition of
±1 × 103
N is applied to the left-hand side and right-hand side of the initial crack.
A Dirichlet boundary condition is applied on the top of the plate and the
displacement is fixed in both directions.

Simulation details
stationary crack problem.
Force F =1 × 103N Time steps tn=50 000
Node spacing hPD =0.0005m Time step size ts =2 × 10−8s
and hPUM =0.000 781 25m
Horizon δ = 4hPD =0.002m Final time T=0.001s

Simulation results I
Figure 4: Left: displacement magnitude (Umax) of the global solution. Right:
Umax in the local region obtained by the PUM.

Simulation results II
δcoarse = 0.00123 and hcoarse
PD = δcoarse/4 δfine = 0.00062 and hfine
PD = δfine/8
(a) (b)

Simulation results III
δcoarse = 0.00123 and hcoarse
PD = δcoarse/4 δfine = 0.00062 and hfine
PD = δfine/8
(c) (d)

Inclined crack
0.1
0.1
0.02
Θ = 72.5°
x
y
Figure 5: Sketch of the square plate (0.1m × 0.1m) with an initial inclined crack
of length 0.02m. For the local PUM model a constant traction boundary
condition of ±1 × 103
N was applied to the top and bottom of the square plate.

Simulation details
inclined crack problem.
Force Fcoarse =4.25 × 106N Final time T=0.001s
Force Ffine =9 × 106N Time steps tn=50 000
Node spacing: Time step size ts =2 × 10−8s
hPUM =0.000 781 25m Horizon:
hcoarse
PD =0.0005m δcoarse = 4hcoarse
PD
hfine
PD =0.000 125m δfine = 8hfine
PD

Extracted crack path
0 2 · 10−2 4 · 10−2 6 · 10−2 8 · 10−2 0.1
0
2 · 10−2
4 · 10−2
6 · 10−2
8 · 10−2
0.1
(a) δcoarse = 0.002 and hcoarse
PD = δcoarse/4
0 2 · 10−2 4 · 10−2 6 · 10−2 8 · 10−2 0.1
0
2 · 10−2
4 · 10−2
6 · 10−2
8 · 10−2
0.1
(b) δfine = 0.001 and hfine
PD = δfine/8
Figure 6: Extracted crack tip positions obtained by the PD simulation. The black
line is the initial inclined crack. The red and blue lines are the crack branches and
each dot is the crack position extracted between the explicit time step 47 500 to
50 000 with a resolution of 500 explicit time steps in between.

Simulation results I
(a) PUM (b) PD

Simulation results II
(c) Damage (d) Point-wise difference

Differences in displacement field
Sharp crack vs crack zone:
Using the extracted crack path within the PUM yields in a sharp
crack, however in the PD model the crack is instead a narrow zone of
failed bonds of finite width equal to twice the peridynamic horizon δ.
In addition, at the crack tip there is a process zone on the order of
the horizon size δ where the bonds are in the process of softening to
failure. Because of this, PUM exhibits a strain singularity where PD
does not. This difference between the strain fields surrounding the
cohesive zone in PD and the strain field around the sharp crack in
PUM contributes to the difference in displacement around the crack
seen for these two models.

Differences in displacement field
Boundary conditions:
In the PUM simulation, a traction condition is applied. However, in
the PD simulation the traction condition is applied within a layer of
horizon size. It has shown that if the horizon goes to zero, the same
traction condition as in the local PUM model is applied. In the future,
the application of the non-local traction condition can be improved.
Local interactions vs nonlocal interactions
The both models only agree in the linear regime of the potential in
Figure 1. Once the inflection point rC is approached and bonds start
to soften, the two models do not agree anymore. Hence, the
displacement field around the crack looks different due to the
non-local effects in the PD model. Note that in the area where bonds
soften in the PD model, we use still linear elasticity in the PUM
model.

Conclusion and Outlook
Conclusion
The primary focus of this work has been to introduce a constructive
multiscale enrichment approach.
We only use the crack geometry and use standard enrichment
functions.
Outlook
Automatic identification of the non-local domain
Suitable discretization around the crack
Suitable time stepping between two models
Preprint
Birner, Matthias, et al. ”A Fracture Multiscale Model for Peridynamic enrichment within the Partition of Unity Method: Part I.”
arXiv preprint arXiv:2108.02336 (2021).

PART II

Outline
1 Model Problem
2 Coupling of LE models
3 Coupling Methods
4 Discretization
5 Numerical experiments
6 Conclusion

Model Problem
Classical linear elasticity model in 1D (with cross-sectional area A = 1):
−Eu00
(x) = fb(x), ∀x ∈ Ω = (0, `),
u(x) = 0, at x = 0
Eu0
(x) = g, at x = `
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 `
Ω
Ωδ
Hδ(x)

Peridynamics
Linearized microelastic bond-based model [Silling, 2000]:
−
Z
Hδ(x)
κ
ξ ⊗ ξ
kξk3
(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

u00
(x) +
δ2
24
u0000
(x) + . . .

,
Approximation:
−
κδ2
2

u00
(x) +
δ2
24
u0000
(x) + . . .

= fb(x), ∀x ∈ Ωδ

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.

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
= Eu0
(x) +
Eδ2
24
u000
(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
u000
(x)

Coupling of linear elasticity models
0 a − ε a `
b b + ε
Ω
Ω1
Γa Ω2 Γb
Ω1
In Ωi, i = 1, 2:
−Eiu00
i (x) = fb(x), ∀x ∈ Ωi
Boundary conditions:
u1(x) = 0, at x = 0
E1u0
1(x) = g, at x = `
Interface conditions:
Continuity of displacement: u1(x) − u2(x) = 0, at x = a, b
Continuity of stress: E1u0
1(x) − E2u0
2(x) = 0, at x = a, b

Modified formulation if E1 = E2 := E
0 a − ε a `
b b + ε
Ω
Ω1
Γa Ω2 Γb
Ω1
−Eu00
1(x) = fb(x), ∀x ∈ Ω1
−Eu00
2(x) = fb(x), ∀x ∈ (a − ε, b + ε)
u1(x) = 0, at x = 0
Eu0
1(x) = g, at x = `
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

Coupling methods for local and peridynamic models
0 a − δ a `
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]

MDCM formulation
0 a − δ a `
b b + δ
Ω
Ωe
Γa Ωδ Γb
Ωe
−Eu00
(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
Eu0
(x) = g, at x = `
u(x) − u(x) = 0, ∀x ∈ Γa ∪ Γb

MSCM formulation
0 a − δ a `
b b + δ
Ω
Ωe
Γa Ωδ Γb
Ωe
−Eu00
(x) = fb(x), ∀x ∈ Ωe
−
Z x+δ
x−δ
κ
u(y) − u(x)
|y − x|
u(x) = 0, at x = 0
Eu0
(x) = g, at x = `
u(x) − u(x) = 0, at x = a, b
σ+
(u)(x) − Eu0
(x) = 0, ∀x ∈ Γa
σ−
(u)(x) − Eu0
(x) = 0, ∀x ∈ Γb

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

VHCM formulation
−Eu00
(x) = fb(x), ∀x ∈ Ωe
−
Z x+δv(x)
x−δv(x)
κ̄(x)
u(y) − u(x)
|y − x|
u(x) = 0, at x = 0
Eu0
(x) = g, at x = `
u(x) − u(x) = 0, at x = a, b
σ+
(u)(x) − Eu0
(x) = 0, at x = a
σ−
(u)(x) − Eu0
(x) = 0, at x = b

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 = (` − 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.

Discretization: MDCM stiffness matrix
In grey: coupling constraint equations

Discretization: MSCM stiffness matrix

Discretization: VHCM stiffness matrix

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

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

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

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

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

Interpolation for MDCM
x0 x1 x2x2,3x3
Ω1
x4 x5 x6 x7 x8 x9
Γa Ωδ
Figure 7: The left coupling region Γa for the matching displacement coupling
approach (MDCM), where interpolation is needed to match the displacement of
the PD node at x5 with the interpolated node x2,3.
Note a cubic interpolation is required to recover solutions up to a degree
of three:

1
16
u0 −
5
16
u1 +
15
16
u2 +
5
16
+ u3

− u2,3 = 0.

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.

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

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
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
n=5
n=6
n=7
n=8

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
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
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.

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 8: 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

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 5: Maximal error w.r.t to FD for mixed boundary conditions with various
damage profiles for MDCM using cubic interpolation.

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.

Concluding Remarks
Non-uniform meshes show convergence in the error for all three
methods
VHCM performs better due to no interpolation
Non-aligned 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

Summary
Common challenges
Identifying the PD region withing the local region - Use ML?
Both models are compatible without damage
Discretizaiton around the crack
Specific challenges
Multiscale approach
Suitable time stepping between two models
Same scale approach
Which method is the best?
Interpolation in 2D and 3D might be challenging
No mathematical proofs for the existence and uniqueness of the solution or
convergence rates including damage are available. I hope y’all could help
here?

A tale of two approaches for coupling nonlocal and local models

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