Numerical solution of singular ODE-BVPs arising in bio ... · ODE-BVPs arising in bio-mechanics Ivonne Sgura Math Department - University of Salento, Lecce, Italy joint work with

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


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


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)


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


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.


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.


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.


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


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....


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.


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.


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+


Regularity Map

Mixed bc’s: the singularity location ηs can be a priori knownDirichlet bc’s: the singularity location ηs is unknown


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


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)


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.


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


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


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


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


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.


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


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 .


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


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 ... · ODE-BVPs arising in bio-mechanics Ivonne Sgura Math Department - University of Salento, Lecce, Italy joint work with

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