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Numerical solution of singularODE-BVPs arising in bio-mechanics
Ivonne Sgura
Math Department - University of Salento, Lecce, Italy
joint work with
Giuseppe Saccomandi (Perugia) & Michel Destrade (Dublin)
Hairer60th Conference on Scientific Computing, 18 June 2009 , Geneve
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.1/30
Motivation
We analyze an ODE boundary value problemdescribing a deformation of fiber reinforced materials,which application is relevant in the biomechanics of softtissues
The interesting feature of the problem is that, for somechoices of the material parameters, multiple and nonsmooth solutions can exist.
Standard numerical approach cannot provide goodresults in the present situations
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.2/30
Outline
Biomechanical problems: introduction
Nonlinear elasticity framework for fiber reinforcedmaterials
BVPs of interest for the rectilinear shear deformation:singular and multiple solutions
Numerical issues and proposed approximation methods
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.3/30
BiomechanicsMany soft tissues (connective, epithelial, muscle andnervous) consist in an isotropic ground-matrix (elastin) inwhich are embedded collagen filaments.
Collagen tearing and defects may result in disease andalteration of the biomechanical behavior of the tissue(arterial wall or skin structure)
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.4/30
Anisotropic nonlinear elasticityAim: to link collagen fibers mechanics with macroscopicmechanical behaviors of the tissue.
The presence of oriented collagen fiber bundles callsfor the consideration of anisotropy in the mathematicalmodelling of the mechanics of tissues.
IDEA: extension of the mathematical models ofnonlinear elasticity from rubber to soft tissues.
We study a composite incompressible slab of thickness Lmade of an isotropic matrix reinforced with two families ofparallel extensible fibers. In this case material anisotropy iscalled orthotropy.
Destrade, Saccomandi, Sgura - Int. J. of Engin. Sc. in press
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.5/30
Rectilinear shear deformationX = (X1, X2, X3) material particle position in theundeformed configurationx = (x1, x2, x3) position in the deformed configuration,F = ∂x
∂Xdeformation gradient,
B = FTF left Cauchy Green strain tensor.
The shear deformation is given by
x1 = X1 + L f(X3/L), x2 = X2, x3 = X3,
If η ≡ X3/L, f = f(η), 0 ≤ η ≤ 1 is the unknowndeformation.
K = f ′ = df/dη is called amount of shear.K = const. ⇒ simple shear otherwise rectilinearinhomogeneous shear.
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.6/30
Rectilinear shear deformationTransverse section of a reinforced slab (Φ = π/3)
0
0.2
0.4
0.6
0.8
1
–1 –0.5 0.5 1 1.50
0.2
0.4
0.6
1
–1 –0.5 0.5 1 1.5
M
ΦX1 1
x
n mN
Fiber directions in reference and deformed configuration are~M = cosΦ~E1 + sinΦ~E3, ~N = − cosΦ~E1 + sinΦ~E3
~m = (cosΦ + K sinΦ)~e1 + sinΦ~e3,~n = (− cosΦ + K sinΦ)~e1 + sinΦ~e3
We consider the shearing along the bisectrix of the angleΨ = π − 2Φ between the two families.
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.7/30
The Standard Reinforcing ModelTo describe the hyperelastic stress response of softcollagenous tissues, is postulated the existence of aHelmholtz free-energy function (strain energy)W = Wiso(I1, I2) + Wani(I4, I6), Ii strain invariants of ~B
Simplest model for orthotropy [Holzapfel, Ogden 2000]:
W = µ(I1 − 3)/2 + F(I4) + G(I6),
F(I4) = µE1(I4 − 1)2/4 and G(I6) = µE2(I6 − 1)2/4
I1 = 3 + K2 modeling the elastin matrix (iso)I4 ≡ ~m · m, I6 ≡ ~n · n modeling fibers
I4 squared stretch in the fiber direction: I4 > 1 ⇔ fibersin extension along ~m, I4 ≤ 1 in compression.
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.8/30
Governing equationThe equilibrium equations are [Ogden, 1984]: div~σ = ~0
~σ = −p~I + µ~B + 2dF
dI4~m ⊗ ~m + 2
dG
dI6~n ⊗ ~n,
is the Cauchy stress tensor, p Lagrange multiplier
Let α pressure gradient, the BVP in normal form is
d2fdη2 = α
1+γ sin2 Φ[2 cos2 Φ+6β cosΦ sinΦf ′+3 sin2 Φf ′2],
(i) Mixed : f(0) = 0, f ′(1) = K1,
(ii) Dirichlet: f(0) = 0, f(1) = 0
K1 shear stress assigned on the upper face of the slab
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.9/30
BVP for Orthogonal fibersγ = E1 + E2 collagen/elastin strength ratio. γ = 0 ⇔isotropic material.
β = (E1 − E2)/(E1 + E2) measure of orthotropy.β = ±1, (~m ⇈ ~n) transverse isotropy ( wrt ~m) ;β = 0 (E1 = E2) mechanically equivalent fibers.
If Φ = π/4 and K = f ′ the corresponding BVP is:
d2f
dη2=
α
P(K),
with P(K) := 1 + γ4
[2 + 6βK + 3K2
]∈ P2 K = f ′.
If the determinant ∆(β, γ) ≥ 0, f ′′ can develop singularities....
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.10/30
Smooth solutions∆ = (3β2 − 2)γ − 4 < 0 ⇔existence and uniqueness of a smooth solution areguaranteed by general theorems
0.8 0.85 0.9 0.95 10
2
4
6
8
10
12
14
16
18
20
β
γ
case (A) ∆(β,γ)=0
0 0.2 0.4 0.6 0.8 1
−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
η
f(η)
β=0.5, γ=3 : ∆(β,γ)< 0,
case (B)∆(β,γ) >0
∆(β,γ)< 0
α=1
α=5
α=10
Any BVP numerical solver could easily track thesesolutions.
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.11/30
Loss of regularity and multiple solutionsBy integrating
∫P (K)dK = α
∫dη, we have
g(K) :=(1 +
γ
2
)K +
3βγ
4K2 +
γ
4K3 = α(η − η0)
η0 ∈ [0, 1] integration const.
g(K) has maximum and minimum K1,2 = −β ∓√
∆3γ .
g1,2 ≡ g(K1,2) identify the points η1,2 = g1,2/α + η0.
case (A) : ∆ = 0 ⇔ K1 = K2 = −β inflection point!if η1 = η2 := ηs ∈ [0,1] ⇒ f * C2, f ∈ C1
case (B) : ∆ > 0 ⇔ K1 6= K2
if η1 or η2 ∈ [0, 1] , ∀ηs ∈ [η1, η2] ∩ [0,1], f ′′(ηs) blows upand there are two or three possible values of K(ηs)!!!⇒ f * C1, f ∈ C.
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.12/30
Case (B)∆ > 0: Multiple solutionsK(η) can jump in infinitely many ηs ∈ [η1, η2] ∩ [0, 1], s.t.[K]ηs
:= K(η+s ) − K(η−s ) > 0 : which jump?
By the strain energy W = W (I1, I4, I6) = W (K) we have
g(K) =1
µ
dW (K)
dK:= WK(K) ∈ P3
and for all K, η = η(K) = g(K)/α + η0.
Hence, the energy is not a convex function and for thisthe uniqueness of the BVP solution is not guaranteedfor all parameter values.
This suggests to choose special singularity points ηs
and then special solution jumps [K]ηsaccording to
energetic motivations.Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.13/30
Energetic guidelines(a) Maxwell rule convention : jump at ηs giving equal areassuch that: K−
s = min{z ∈ R|g(z) − α(ηs − η0) = 0} := K−
M
K+s = max{z ∈ R|g(z) − α(ηs − η0) = 0} := K+
M
(b) Maximum delay convention : jump at ηs = g2/α + η0 s. t.K−
s = K2 := K−
D K+s = max{z |g(z)−α(η2−η0) = 0} := K+
D
−3 −2 −1 0 1−50
0
50
100
K
Free−energy plot: W(K)− τ K
−60 −40 −20 0 20 40−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1Mixed bc., β=1
g(K)=WK(K)
K
g1
g2
τ=g2: Maximum Delay sol
τ=g1
Maxwell solution
KM−
KM+
KD−=K
2
KD+
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.14/30
Regularity Map
Mixed bc’s: the singularity location ηs can be a priori knownDirichlet bc’s: the singularity location ηs is unknown
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.15/30
Numerical literature on singular BVPsxNy′ = f(x, y) 0 < x < 1, y ∈ C1(0, 1]N = 1: singularity of 1st kind (polar coord),N = 2 sing. of 2nd kind (change of var t = a/x :[a,+∞) −→ (0, 1]
Spline and collocation methods: Agarwal-Kadalbajoo;Shampine, Auzinger-Koch-Weinmuller (2002-2005);Ford-Pennline (2009), ...Review of numerical techniques for many kind ofsingular BVPs Kumar-Singh (Adv. in Eng. Soft. 2009)
In any case:i) the singularity point is at either one or both of the bounda rypoints
ii) its location is a priori known
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.16/30
Mixed BC’s: the singular BVPSince g(K) = α(η − η0), by K(1) = K1: η0 = 1 − g(K1)/α
A singularity point is obtained by ηs = g(K−
s )/α + η0
CASE (A) (∆ = 0): the unique singularity point isηA = g(−β)/α + η0, where [K]ηA = 0 andKA := K−
s = K1,2 = −β.
CASE (B) (∆ > 0): ηs and K−
s are known by the energeticchoice (Maxwell or Maximum delay sol.)
We solve the singular BVP (f ′′(ηs) → ∞) as a multipointproblem on I0 ∪ I1 =[0, ηs] ∪ [ηs, 1], with the conditions:
f(0) = 0, K(ηs) = K−
s , [f ]ηs= 0, K(1) = K1.
where [f ]ηs:= f(η−s ) − f(η+
s ) = 0 continuity conditionNumerical solution of singular ODE-BVPs arising in bio-mechanics – p.17/30
A finite difference approachThe blow up of f ′′ implies that also finite differenceschemes must be carefully used.We apply the Extended Central Difference Formulas in order to:
discretize directly the derivatives in the 2nd order BVP
discretize BC’s with the same (high) order than internalpoints
The nonlinear BVP is solved via quasi-linearization ⇒sequence of linear BVPs:
f ′′ = α/P(f ′) ⇔
{y0 giveny′′
k+1 + ck y′
k+1 = gk, k = 0,1, . . .
ck = 32
αγ(β+y′
k)P 2(y′
k) , gk =α[1+ γ
4(2+12βy′
k+9(y′
k)2)]
P 2(y′
k)
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.18/30
High order schemes: D2ECDFWe consider a stencil of k + 1 points, k ≥ 2. For s = 0, . . . ,k,
there exist k + 1 possible schemes M(ν)s , ν = 1,2, such that
each formula requires s initial values and k − s final values.
M(2)s : y′′(ξi) ≈ y′′
i := 1h2
k−s∑
j=−s
α(s)j+s
yi+j, s = 0, . . . ,k
M(1)s : y′(ξi) ≈ y′
i := 1h
k−s∑
j=−s
β(s)j+s
yi+j, s = 0, . . . ,k.
In [Amodio-Sgura ’04] have been obtained the coefficients of
each M(ν)s , ν = 1, 2, in order to have for all s the same maxi-
mum possible order p = k.
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.19/30
Example: k = 4
C(2) =
1 1 1
−2 −2 2
1 1 1
C(1) =
−3/2 1/2 1/2
2 0 −2
−1/2 −1/2 3/2
C(2) =
3512
11
12− 1
12− 1
12
1112
− 263
−5
3
4
3
1
3− 14
3192
1
2−5
2
1
2
192
− 143
1
3
4
3−5
3− 26
31112
− 1
12− 1
12
11
12
3512
C(1) =
− 2512
−1
4
1
12− 1
12
14
4 −5
6−2
3
1
2− 4
3
−33
20 −3
23
43
−1
2
2
3
5
6−4
− 14
1
12− 1
12
1
4
2512
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.20/30
Matrix formulationOn I0 = [0, ηs] and I1 = [ηs, 1] we use the same constantstepsize s.t. h = ηs/n0 and n1 = (1 − ηs)/h.Let be n ∈ {n0, n1}, y0 ∈ {0, η−s }, yf ∈ {η+
s , 1}. On each Ij ,j = 0, 1, we have the unknowns:
Y = [y1,y2, . . . ,yn]T, Y = [y0, YT, yf ]T.
On each Ij , our FD schemes in matrix form are
Y′′(x) ≈ d2Y :=1
h2AY, Y′(x) ≈ d1Y :=
1
hBY,
A = [aI, A, aF], B = [bI, B, bF] ∈ R(n+2)×(n+2)
A, B main and additional methods of same order;
aI,aF, bI,bF row vectors for initial and final BC’s.Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.21/30
D2: A = [aI; A; aF] =
aI,0 aI,1 . . . aI,k 0 . . .
α(1)0 α
(1)1 . . . α
(1)k
......
...
α(k/2−1)0 α
(k/2−1)1 . . . α
(k/2−1)k
α(k/2)0 α
(k/2)1 . . . α
(k/2)k
α(k/2)0
. . .
. . .. . .
α(k/2)0 . . . α
(k/2)k−1 α
(k/2)k
α(k/2+1)0 . . . α
(k/2+1)1 α
(k/2+1)k
......
α(k−1)0 . . . α
(n)k−1 α
(k−1)k
0 aF,0 . . . aF,k−1 aF,k
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.22/30
Global discretization∀k the global discretization of each linear BVP has globalorder p = k and produces the linear system MkZ = qk
If m = n0 + n1 + 4:Z = [Y0, Y1]T = [y0,Y0,y
−
s ,y+s ,Y1,yn1+1]. ∈ Rm,.
qk = Gk ∈ Rm,qk,1 = 0, qk,n0+1 = K−
s , qk,n0+2 = 0, qk,m = K1
Mk =1
h2
(A0 0
0 A1
)+
1
hDck
(B0 0
0h B1
)∈ Rm×m
Dirichlet in η = 0: a(0)I = 0, b
(0)I = [h,0, . . . ,0]
Neumann in η−s : a(0)F = 0; b
(0)F = [0, . . . ,0, β
(k)0 , β
(k)1 , . . . , β
(k)k ]
Continuity [f ]ηs = 0: a(0)F = 0, b
(1)I,1 = −h, 0 oth.
Neumann in η = 1: a(1)I = 0; b
(1)F = b
(0)FNumerical solution of singular ODE-BVPs arising in bio-mechanics – p.23/30
Mixed BC’s: simulations Case (A)η0 = 1 − γ((f ′(1) + β)3 − β3)/(4α)
f(η) = 3γ16α
{[β3 + 4α(η − η0)/γ
]4/3−[β3 − 4α/γη0
]4/3}− βη
K(η) =[β3 + 4α(η − η0)/γ
]1/3− β
γ = 4, α = 10,K1 = 0.5, β ≤ 1.0
0 0.2 0.4 0.6 0.8 1−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
η
f(η
)
0 0.2 0.4 0.6 0.8 1−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
η
K(η
)
D2ECDF4: γ=4/(3β2−2), α=10
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
140
η
f’(η
)
β=0.875
β=0.95
β=1
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.24/30
Mixed bc: absolute errorsβ = 1.0, γ = 4, α = 10,K1 = 0.5: ηs = 0.6625h = 6.625e-3, n0 = 100, m = 152
p = 21 p = 2 p = 4
‖f − f∗‖∞ 0.0746 0.0662 0.0389
‖K − K∗‖∞ 5.7521 0.4120 0.2179
0 0.2 0.4 0.6 0.8 1
10−4
10−3
10−2
η
err(f)
0 0.2 0.4 0.6 0.8 1
10−5
10−4
10−3
10−2
10−1
100
η
err(K)
p=4p=2p=2bc
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.25/30
Mixed BC’s: bvp4c and shootingThe Matlab code bvp4c , multipoint collocation methods
0 0.5 1−1200
−1000
−800
−600
−400
−200
0
η
f(η)
0 0.5 1−2000
−1500
−1000
−500
0
500
K(η)
η
BVP4c − multipoint
0 0.5 10
0.5
1
1.5
2
2.5
3
3.5
4x 10
7
f’’(η)
η
Shooting : Solve two IVPs on I0 ∪ I1 s.t. :for IVP0: f(0) = 0,K(0) = K0, K0 root of g(x) + α η0 = 0
for IVP1: f(ηs) = fs ≈ f(ηs) from IV P0, K(η+s ) = K+
s = −β.
ode45 : Failure at ηs = 6.625e-1. Unable to meet integration tol without
reducing the step size below the smallest value allowed (1.776357e-015) at time t.
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.26/30
Mixed bc: numerical solutions case (B)Maxwell solution :ηs = ηM and K−
s = K−
M
Maximum delay solution : ηs = ηD = g2/α + η0 and K−
s = K2.bvp4c : β = 1, γ = 10, error singular Jacobian
p = 21 p = 2 p = 4
|K+s − K+
M | 1.5957e-2 3.9086e-3 7.2377e-4
|K+s − K+
D | 1.6239e-2 7.1699e-3 4.606e-3
0 0.2 0.4 0.6 0.8 1−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
η
f(η)
γ=10, α=10, f ’(1) =0.5
0 0.2 0.4 0.6 0.8 1−2.5
−2
−1.5
−1
−0.5
0
0.5
η
f ’ (η
)
Max delay solMaxwell sol
β=1
β=0.95β=0.95
β=1
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.27/30
Dirichlet BC’s: f(0) = 0, f(1) = 0The singularity location ηs is unknown!Look for η∗0 ≈ η0 ∈ [0, 1] as zero of F (z) := f(z; η = 1) = 0.F (z) has no explicit form and ∀zi, Fi ≈ F (zi) numerically.If F0 · F1 < 0 then the bisection method on [0, 1] converges.k = 0:- ηk
0 = 12, Fk ≈ F(ηk
0), Kk0 root of g(x) + α ηk
0 = 0.
- Obtain the jump K∗ = K−
sk by energy consideration then:
ηks = g(K∗)
α + ηk0 ⇒ ⇒ Ik0 = [0, ηk
s ], Ik1 = [ηks ,1]
For k = 0,1, . . . until |Fk| ≤ tol solve the IVPs
IVPk0 on Ik0 s.t. f(0) = 0, f ′(0) = Kk
0,
IVPk1 on Ik1 s.t. f(ηk
s ) = fks ,K(ηks ) = K−
s , fks ≡ fks numapprox given by IV P k
0 in ηks .
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.28/30
Dirichlet BC’s: numerical solutions ∆ > 0γ = 10.0, α = 10.0, with tol=1e-6β = 1.0 (transverse isotropy), ηM
0 = 0.2423, ηD0 = 0.21725;
β = 0.95 (orthotropy), we find ηM0 = 0.13818, ηD
0 = 0.05014.The singularity location ηs = ηM,D
s are obtained numerically.
0 0.2 0.4 0.6 0.8 1−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
η
f(η)
γ=10, α=10, f(0)=f(1)=0
0 0.2 0.4 0.6 0.8 1−2.5
−2
−1.5
−1
−0.5
0
0.5
1
η
K(η)
Maximum delay solMaxwell sol
β=0.95
β=1β=1
β=0.95
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.29/30
Work in progressMixed BC’s : explain the convergence behavior ofD2ECDF near the singularity; to apply variable stepsize
Dirichlet BC’s : Combine the bisection method withD2ECDF solving a sequence of multipoint BVPs(change the stop criterion)
....
Numerical solution of singular ODE-BVPs arising in bio-mechanics – p.30/30