On the treatment of boundary conditions for bond-based peridynamic models

On the treatment of boundary conditions for
bond-based peridynamic models
Patrick Diehl
Joint work with Serge Prudhome
Center of Computation & Technology (Louisiana State University)
patrickdiehl@lsu.edu
October 17, 2020
Patrick Diehl (LSU/CCT) Treatment of BC in bond-based PD October 17, 2020 1 / 40

Motivation
Treatment of local boundary conditions in non-local models is
challenging
However, many approaches are present in literature
Extended Domain approach
Variable Horizon approach
We study these approaches from the mathematical aspects, e.g.
convergence rates
Reference
Prudhomme S, Diehl P. On the treatment of boundary conditions for
bond-based peridynamic models. Computer Methods in Applied
Mechanics and Engineering. 2020;372:113391.

Overview
1 Model problem
Continuum local model
Peridynamic model
2 Extended Domain Method
3 Variable horizon method
4 Numerical results
δ-convergence numerical study
Effect of correction factor in EDM
m-convergence numerical study
Manufactured solution with steep gradient
5 Conclusion & Outlook

Model problem

Continuum local model
Let Ω = (0, 1). The continuum local problem consists in finding the
displacement u in the bar such that:
− EAu00
= fb, ∀x ∈ Ω, (1)
u = 0, at x = 0, (2)
EAu0
= g, at x = 1, (3)
where E and A are the constant modulus of elasticity and cross-sectional
area of the bar, respectively, g ∈ R is the traction force applied at end
point x = 1, and the scalar function fb = fb(x) is the external body force
density (per unit length).

Peridynamic model
0 δ x − δ x x + δ 1 − δ 1
Ω
Ωδ
Hδ(x)
Bδ
0 Bδ
1
Figure: Definition of the computational domains for the peridynamic model.
1D linearized microelastic model
Z
Hδ(x)∩Ω
κ
ξ ⊗ ξ
kξk3
(u(y) − u(x))dy + fb(x) = 0, (4)

Relation of material properties
Supposing that the displacement u(x) is sufficiently smooth, one can
write, for all y ∈ Ωδ:
u(y) − u(x) = u0
(x)(y − x) +
1
2
u00
(x)(y − x)2
+
1
6
u000
(x)(y − x)3
+ O(|y − x|4
),
(5)
so that the integral in (4) reduces to:
Z x+δ
x−δ
κ
u(y) − u(x)
|y − x|
dy =
κδ2
2
(u00
(x) + O(δ2
)). (6)
Substituting the integral in (4) with the above result yields:
−
κδ2
2
(u00
+ O(δ2
)) = fb, ∀x ∈ Ωδ
. (7)

Relation of material properties
By taking the limit when δ → 0, one then recovers the differential
equation (1) pointwise whenever κ is chosen as:
κδ2
2
= EA, that is κ =
2EA
δ2
, (8)
References
P. Seleson, Q. Du, M. L. Parks, On the consistency between nearest-neighbor
peridynamic discretizations and discretized classical elasticity models, Computer
Methods in Applied Mechanics and Engineering 311 (2016) 698–722

Discretization (Collocation approach)
We describe here the discretization of Problem (10)-(14) using a
collocation approach. For a given δ,
we introduce a uniform grid spacing h chosen such that, as it is
customarily done in the literature,
δ is a multiple of h, i.e. δ/h = m, with m a positive integer.
For the sake of simplicity, but without loss of generality, we shall choose
here m = 2. In this case, choosing h = 1/n, with n a positive integer, the
discretization of domain b
Ωδ is determined by the grid points:
xi = ih, i = 0, 1 . . . , n, (9)

Extended Domain Method

Extended Domain Method
−δ 0 x − δ x x + δ 1 1 + δ
b
Ωδ
Ω
Hδ(x)
Dδ
0 Dδ
1
Figure: Definition of domains for the extended domain approach.
Note many references for this approach are available and presented in the
paper. This paper focus on the mathematical aspects.

Remarks on the EMD
(a) the problem is solved on a larger domain than the domain occupied
by the bar, which increases the number of degrees of freedom for a
given discretization parameter h. This is in particular true when δ/h
is large and in higher dimensions;
(b) odd extension of the displacement field outside of Ω may become
difficult to carry on in two and three dimensions for domains with
complex geometries, especially when boundaries involve corners and
edges;
(c) conceptually, there may still be an ambiguity in the application of the
Neumann boundary condition, which is derived from the local model,
while the behavior in the vicinity of the boundary is governed by the
non-local model.

Problem definition
The problem consists then in finding u ∈ b
Ωδ such that:
−
Z x+δ
x−δ
κ
u(y) − u(x)
|y − x|
dy = fb(x), ∀x ∈ Ω, (10)
u(x) − 2u(0) + u(−x) = 0, ∀x ∈ Dδ
0, (11)
u(x) − 2u(1) + u(2 − x) = 0, ∀x ∈ Dδ
1, (12)
u = 0, at x = 0, (13)
κδ2
2
u0
= g, at x = 1, (14)

Discretized system



















1 0 −2 0 1 0 0 0
0 1 −2 1 0 0 0 0
0 0 1 0 0 0 0 0 · · ·
0 −α −4α 10α −4α −α 0 0
0 0 −α −4α 10α −4α −α 0
.
.
.
.
.
.
0 −α −4α 10α −4α −α 0 0
0 0 −α −4α 10α −4α −α 0
0 0 0 2αh −8αh 0 8αh −2αh
0 0 0 0 1 −2 1 0
0 0 0 1 0 −2 0 1






































u−2
u−1
u0
u1
u2
.
.
.
un−2
un−1
un
un+1
un+2



















=



















0
0
0
fb(x1)
fb(x2)
.
.
.
fb(xn−2)
fb(xn−1)
g
0
0



















Figure: System of linear equations associated with the discretization of the
extended domain method where α =
κδ2
16h2
.

Discretized reduced system











1 0 0 0 0 0 · · ·
−6α 11α −4α −α 0 0
−α −4α 10α −4α −α 0
.
.
.
..
.
0 −α −4α 10α −4α −α
0 0 −α −4α 11α −6α
0 0 0 4αh −16αh 12αh






















u0
u1
u2
.
.
.
un−2
un−1
un











=











0
fb(x1)
fb(x2)
.
.
.
fb(xn−2)
fb(xn−1)
g











Figure: Reduced system of linear equations associated with the discretization of
the extended domain method where α =
κδ2
16h2
.

Variable horizon method

Variable horizon method
δv
0 δ 1 − δ 1 x
Figure: Example of a variable horizon function δv (x). The circles centered at
points x ∈ Bδ
0 = (0, δ) are introduced to represent the associated domain Hδ(x).
References
S. Silling, D. Littlewood, P. Seleson, Variable horizon in a peri-dynamic medium,
Journal of Mechanics of Materials and Struc-tures 10 (5) (2015) 591–612

Choice of the variable horizon δv (x)
Requirements, ∀x ∈ Ω:
0 ≤ δv (x) ≤ δ,
x − δv (x) ≥ 0,
x + δv (x) ≤ 1.
(15)
Simple variable horizon
δv (x) =



x, 0 < x ≤ δ,
δ, δ < x ≤ 1 − δ,
1 − x, 1 − δ < x < 1.
(16)
Horizon-dependent material properties
κ̄(x)δ2
v (x) = κδ2
, ∀x ∈ Ω, (17)
We note that this scaling of the material parameter is different from
F. Bobaru, M. Yabg, L. F. Alves, S. A. Silling, E. Askari, J. Xu,Convergence,
adaptive refinement, and scaling in 1D peridy-namics, International Journal for
Numerical Methods in Engi-neering 77 (2009) 852–877

Discretized system (VHM)











1 0 0 0 0 0 · · ·
−8α 16α −8α 0 0 0
−α −4α 10α −4α −α 0
.
.
.
..
.
0 −α −4α 10α −4α −α
0 0 0 −8α 16α −8α
0 0 0 4αh −16αh 12αh






















u0
u1
u2
.
.
.
un−2
un−1
un











=











0
fb(x1)
fb(x2)
.
.
.
fb(xn−2)
fb(xn−1)
g











Figure: System of linear equations associated with the discretization of the
variable horizon method where α =
κδ2
16h2
.

Numerical results

Solutions
Quadratic solution
fb(x) = 1 and u(x) = x(4−x)
2
Cubic solution
fb(x) = x and u(x) = x(3−x)(3+x)
6
Quadratic solution
fb(x) = x2 and u(x) = x(16−x2)
12
Steep gradient1
u(x) =
x− e−(1−x)/−e−1/
1−e−1/
Maximum relative error
En = maxi=1,...,n

Note that u is the exact solution and u is the numerical obtained solution.
1
The corresponding force was calculates using the computer algebra program Maxima as shown in the paper

Quadratic solution
Maximum relative error En
n δ LLEM EDM VHM
4 0.5 0 0.05098 0
8 0.25 0 0.02443 0
16 0.125 0 0.01202 0
32 0.0625 0 0.00596 0
Table: Maximum relative error En for the quadratic solution obtained with the
local linear elasticity model (LLEM), the extended domain method (EDM), and
the variable horizon method (VHM).

Quadratic solution (EDM)
0.0 0.2 0.4 0.6 0.8 1.0
x
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0.00
Error
in
displacement
Example with Quadratic Solution using EDM
δ=0.5
δ=0.25
δ=0.125
δ=0.0625
Figure: Error in displacement for the quadratic solution using the extended
domain method (EDM) for various values of horizon δ.

Cubic solution
n δ LLEM EDM VHM
4 0.5 0.01481 0.02774 0.01481
8 0.25 0.00379 0.01764 0.00379
16 0.125 0.00096 0.00983 0.00096
32 0.0625 0.00024 0.00517 0.00024
Table: Maximum relative error En for the cubic solution obtained with the local
linear elasticity model (LLEM), the extended domain method (EDM), and the
variable horizon method (VHM).

Cubic solution (EDM)
0.0 0.2 0.4 0.6 0.8 1.0
x
−0.030
−0.025
−0.020
−0.015
−0.010
−0.005
0.000
Error
in
displacement
Example with Cubic Solution using EDM
δ=0.5
δ=0.25
δ=0.125
δ=0.0625
Figure: Error in displacement for the cubic solution using the extended domain
method (EDM) for various values of horizon δ.

Quartic solution
n δ LLEM EDM VHM
4 0.5 0.03226 0.00663 0.03617
8 0.25 0.00891 0.01267 0.01085
16 0.125 0.00233 0.00889 0.00294
32 0.0625 0.00060 0.00511 0.00076
Table: Maximum relative error En for the quartic solution obtained with the local
linear elasticity model (LLEM), the extended domain method (EDM), and the
variable horizon method (VHM).

Quartic solution (EDM)
0.0 0.2 0.4 0.6 0.8 1.0
x
−0.014
−0.012
−0.010
−0.008
−0.006
−0.004
−0.002
0.000
Error
in
displacement
Example with Quartic Solution using EDM
δ=0.5
δ=0.25
δ=0.125
δ=0.0625
Figure: Error in displacement for the quartic solution using the variable horizon
method (VHM) for various values of horizon δ.

Quartic solution (VHM)
0.0 0.2 0.4 0.6 0.8 1.0
x
0.00
0.01
0.02
0.03
0.04
Error
in
displacement
Example with Quartic Solution using VHM
δ=0.5
δ=0.25
δ=0.125
δ=0.0625
method (VHM) for various values of horizon δ.

Correction factor I (EDM)
0.0 0.2 0.4 0.6 0.8 1.0
x
1.00
1.05
1.10
1.15
1.20
1.25
κ(x)
/
κ
x1
x2
xn−1
xn−2
Correction Factor for EDM I
κ(x)/κ, x ∈ (0, δ)
κ(x)/κ, x ∈ (1 − δ, 1)
Figure: Plot of the correction factor k(x)/κ provided by (18) on the left-hand side
and by (19) on the right-hand side for δ = 0.125.
For x ∈ (0, δ)
κ̄(x) = κ

4
x
δ
−

3 − 2 ln
x
δ
x2
δ2

−1
(18)
For x ∈ (1 − δ, 1)
κ̄(x) = κ

4
1 − x
δ
−

3 − 2 ln
1 − x
δ
(1 − x)2
δ2

−1
(19)

Solution for correction factor I
0.0 0.2 0.4 0.6 0.8 1.0
x
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
Error
in
displacement
Example with Quartic Solution using EDM I
δ = 0.125
δ = 0.0625
δ = 0.03125
δ = 0.015625
Figure: Error in displacement using EDM I with correction factor k(x)/κ given
by (18) and (19) for various values of the horizon δ.

Solution for correction factor II
0.0 0.2 0.4 0.6 0.8 1.0
x
−0.0016
−0.0014
−0.0012
−0.0010
−0.0008
−0.0006
−0.0004
−0.0002
0.0000
Error
in
displacement
Example with Quartic Solution using EDM II
δ = 0.125
δ = 0.0625
δ = 0.03125
δ = 0.015625
Figure: Error in displacement using EDM II with correction factor
k(x1)/κ = k(xn−1)/κ = 8/7 for various values of the horizon δ.

Accuracy Study
0.0 0.2 0.4 0.6 0.8 1.0
x
0.000
0.001
0.002
0.003
0.004
0.005
0.006
Absolute
error
in
displacement
Accuracy Study with Quartic Solution for δ = 0.0625
EDM
EDM I
EDM II
Figure: Comparison of the absolute error in displacement for EDM without
correction, EDM I, and EDM II for δ = 0.0625.

m-convergence numerical study
m h Maximum relative error En
2 0.125 0.008 072 916 666 666 785
4 0.0625 0.001 751 730 081 305 069 6
8 0.03125 0.000 409 015 407 927 338 05
Table: Maximum relative error En for the quartic solution obtained with the
variable horizon method (VHM) for δ = 1/4 and h = δ/m, m = 2, 4, and 8.
Peridynamic load term
fb(x) = x2 + δ2
v (x)
12

Convergence study
0.0 0.2 0.4 0.6 0.8 1.0
x
0.000
0.002
0.004
0.006
0.008
0.010
Error
in
displacement
Convergence Study with Quartic Solution using VHM
m = 2
m = 4
m = 8
method (VHM) for δ = 1/4 and various values of m, with h = δ/m.

On the treatment of boundary conditions for bond-based peridynamic models

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