A semi-analytical method to solve the equilibrium ... · MotivationAnalytical solutionBoundary conditions on the pitNumerical resultsConclusions 1 Motivation and basic equations 2

(m2op)III

Third International Workshop on Multiphysics,Multiscale and Optimization Problems

A semi-analytical method to solve the equilibriumequations of axisymmetric elasticity in a half-space

with a hemispherical pit

Eduardo Godoy1 Mario Durán2

1Ingenieros Matemáticos Consultores Asociados S.A. (INGMAT), Santiago, Chile.

2Facultad de Ingenieŕıa, Pontificia Universidad Católica de Chile, Santiago, Chile.

22 May 2014

Motivation Analytical solution Boundary conditions on the pit Numerical results Conclusions

Overview

1 Motivation and basic equations

2 Analytical solution in series form

3 Boundary conditions on the hemispherical pit

4 Numerical results and validation

5 Conclusions and perspectives for future work

E. Godoy INGMAT

A semi-analytical method to solve the equilibrium equations of axisymmetric elasticity in a half-space with a hemispherical pit

E. Godoy INGMAT

Main motivation: Stresses in surface excavations

Need for determining gravity stresses in the surrounding rock mass

Estimate potential slope failure or landslide occurring

E. Godoy INGMAT

Main motivation: Stresses in surface excavations

Need for determining gravity stresses in the surrounding rock mass

Estimate potential slope failure or landslide occurring

E. Godoy INGMAT

Difficulties associated with this problem

When trying to solve this problem using mathematical andnumerical approaches, the following difficulties are encountered

1 Complex geometry of the excavation and its surrounding area

2 Complex internal structure of the rock mass (inhomogeneous,anisotropic and with discontinuities)

3 The surrounding area is in practice unbounded (to be storedin a computer it needs to be truncated)

Most commercial software to solve this type of problems (usuallybased on FEM) are able to deal with difficulties 1 and 2, but theyfail in resolving adequately difficulty 3 (problem of unboundedness)

E. Godoy INGMAT

Approach considered in this work

Difficulties associated with complex geometry and internalstructure are simplified, focusing the attention on treatingappropriately the problem of unboundedness

We thus assume the following hypotheses

1 Excavation is an hemispherical pit surrounded by a half-space

2 Half-space occupied by a homogenous isotropic elastic solid

3 Gravity force acting downwards everywhere in the half-space

A simplified mathematical model of our problem is built

A semi-analytical solution method is considered, which takesadvantage of the axisymmetry of the domain

E. Godoy INGMAT

Axisymmetric domain

Axisymmetric spherical coordinates (r, φ)

Unit vectors r̂, φ̂ and k̂

Pit radius h, gravity acceleration ~g

Domain and boundaries:

Ω = {(r, φ) : h < r

Axisymmetric domain

φ

rAxisymmetric spherical coordinates (r, φ)

Ω = {(r, φ) : h < r

Axisymmetric domain

φ

r

r̂

φ̂

k̂

Ω = {(r, φ) : h < r

Axisymmetric domain

h

φ

r

~g

r̂

φ̂

k̂

Ω = {(r, φ) : h < r

Axisymmetric domain

Γh

Γ∞

h

Ω

Γs

φ

r

~g

r̂

φ̂

k̂

Ω = {(r, φ) : h < r

Basic equations

Displacement field u, stress tensor σ(u)

Isotropic Hooke’s law σ(u) = λ(∇ · u)I + µ(∇u+∇uT )with λ ≥ 0 and µ > 0 the Lamé constants of the elastic solid

Poisson’s ratio ν =λ

2(λ+ µ)

Equation of elastostatic equilibrium ∇ · σ(u) = −ρgk̂with ρ the density of the elastic solid

Traction-free boundary conditions (Neumann-type) σ(u)n̂ = 0

with n̂ the outward unit normal vector

E. Godoy INGMAT

Basic equations

2(λ+ µ)

E. Godoy INGMAT

Basic equations

2(λ+ µ)

E. Godoy INGMAT

Basic equations

2(λ+ µ)

E. Godoy INGMAT

Basic equations

2(λ+ µ)

E. Godoy INGMAT

Elasticity in axisymmetric spherical coordinates

A generic displacement field u is given by components as

u(r, φ) = ur(r, φ) r̂ + uφ(r, φ) φ̂

The stress tensor σ has the following four components:

σr(u) = (λ+ 2µ)∂ur∂r

+λ

r

[2ur +

∂uφ∂φ

+ cotφuφ

]σφ(u) = λ

∂ur∂r

+1

r

[2(λ+ µ)ur + (λ+ 2µ)

∂uφ∂φ

+ λ cotφuφ

]σθ(u) = λ

∂ur∂r

+1

r

[2(λ+ µ)ur + λ

∂uφ∂φ

+ (λ+ 2µ) cotφuφ

]σrφ(u) = µ

[1

r

∂ur∂φ

+∂uφ∂r− uφ

r

]E. Godoy INGMAT

Elasticity in axisymmetric spherical coordinates

A generic displacement field u is given by components as

u(r, φ) = ur(r, φ) r̂ + uφ(r, φ) φ̂

The stress tensor σ has the following four components:

σr(u) = (λ+ 2µ)∂ur∂r

+λ

r

[2ur +

∂uφ∂φ

+ cotφuφ

]σφ(u) = λ

∂ur∂r

+1

r

[2(λ+ µ)ur + (λ+ 2µ)

∂uφ∂φ

+ λ cotφuφ

]σθ(u) = λ

∂ur∂r

+1

r

[2(λ+ µ)ur + λ

∂uφ∂φ

+ (λ+ 2µ) cotφuφ

]σrφ(u) = µ

[1

r

∂ur∂φ

+∂uφ∂r− uφ

r

]E. Godoy INGMAT

Boundary-value problem (BVP) in u

In absence of pit the half-space undergoes lithostatic displacement

ug(x, y, z) = −ρgz2

2(λ+ 2µ)k̂

In presence of pit the displacement field u satisfies the BVP

∇ · σ(u) = −ρgk̂ in Ωσ(u)k̂ = 0 on Γ∞

σ(u)r̂ = 0 on Γh

σ(u)φ̂ · r̂ = u · φ̂ = 0 on Γs|u− ug| = O

(1r

)as r →∞

E. Godoy INGMAT

Boundary-value problem (BVP) in u

In absence of pit the half-space undergoes lithostatic displacement

ug(x, y, z) = −ρgz2

2(λ+ 2µ)k̂

In presence of pit the displacement field u satisfies the BVP

∇ · σ(u) = −ρgk̂ in Ωσ(u)k̂ = 0 on Γ∞

σ(u)r̂ = 0 on Γh

σ(u)φ̂ · r̂ = u · φ̂ = 0 on Γs|u− ug| = O

(1r

)as r →∞

E. Godoy INGMAT

Boundary-value problem (BVP) in v

We use the decomposition u = ug + v, with v satisfying the BVP

∇ · σ(v) = 0 in Ωσ(v)k̂ = 0 on Γ∞

σ(v)r̂ = f on Γh

σ(v)φ̂ · r̂ = v · φ̂ = 0 on Γs|v| = O

(1r

)as r →∞

with f(φ) =ρgh cosφ

1− ν((ν + (1− 2ν) cos2φ

)r̂ − (1− 2ν) cosφ sinφ φ̂

)In what follows, this BVP in v is solved

E. Godoy INGMAT

∇ · σ(v) = 0 in Ωσ(v)k̂ = 0 on Γ∞

σ(v)r̂ = f on Γh

σ(v)φ̂ · r̂ = v · φ̂ = 0 on Γs|v| = O

(1r

)as r →∞

1− ν((ν + (1− 2ν) cos2φ

E. Godoy INGMAT

∇ · σ(v) = 0 in Ωσ(v)k̂ = 0 on Γ∞

σ(v)r̂ = f on Γh

σ(v)φ̂ · r̂ = v · φ̂ = 0 on Γs|v| = O

(1r

)as r →∞

1− ν((ν + (1− 2ν) cos2φ

E. Godoy INGMAT

Displacement in terms of Boussinesq potentials

The displacement v is sought under the form

2µv = ∇(Φ + zΨ)− 4(1− ν)Ψ k̂

where Φ and Ψ are scalar functions (Boussinesq potentials)

This v satisfies the equation of elastostatic equilibrium providedthat Φ and Ψ are harmonic functions in Ω

As v has to fulfil the decaying condition |v| = O(

1r

)as r →∞,

Boussinesq potentials Ψ and Φ must satisfy ∆Ψ = 0 in ΩΨ = O(1r

)as r →∞

∆Φ = 0 in Ω|∇Φ| = O(1r

)as r →∞

E. Godoy INGMAT

2µv = ∇(Φ + zΨ)− 4(1− ν)Ψ k̂

1r

)as r →∞,

)as r →∞

∆Φ = 0 in Ω|∇Φ| = O(1r

)as r →∞

E. Godoy INGMAT

2µv = ∇(Φ + zΨ)− 4(1− ν)Ψ k̂

1r

)as r →∞,

)as r →∞

∆Φ = 0 in Ω|∇Φ| = O(1r

)as r →∞

E. Godoy INGMAT

Expressions for the Boussinesq potentials

Laplace equation in spherical coordinates (r, φ) for F = Ψ,Φ

1

r2∂

∂r

(r2∂F (r, φ)

∂r

)+

1

r2 sinφ

∂

∂φ

(sinφ

∂F (r, φ)

∂φ

)= 0

Standard separation of variables in Ω yields Fn(r, φ) =Pn(cosφ)rn+1

where Pn(·) is the Legendre polynomial of order n ≥ 0Solutions Ψ:Decaying condition is Ψ = O

(1r

)as r →∞

Ψn(r, φ) =Pn(cosφ)

rn+1n ≥ 0

E. Godoy INGMAT

1

r2∂

∂r

(r2∂F (r, φ)

∂r

)+

1

r2 sinφ

∂

∂φ

(sinφ

∂F (r, φ)

∂φ

)= 0

(1r

)as r →∞

rn+1n ≥ 0

E. Godoy INGMAT

1

r2∂

∂r

(r2∂F (r, φ)

∂r

)+

1

r2 sinφ

∂

∂φ

(sinφ

∂F (r, φ)

∂φ

)= 0

(1r

)as r →∞

rn+1n ≥ 0

E. Godoy INGMAT

1

r2∂

∂r

(r2∂F (r, φ)

∂r

)+

1

r2 sinφ

∂

∂φ

(sinφ

∂F (r, φ)

∂φ

)= 0

where Pn(·) is the Legendre polynomial of order n ≥ 0Solutions Φ:Decaying condition is |∇Φ| = O

(1r

)as r →∞

Φn(r, φ) =

ln(r − r cosφ) n = −1Pn(cosφ)

rn+1n ≥ 0

E. Godoy INGMAT

Obtaining the series

Solutions in series form are built by substituting the following twocombinations of Boussinesq potentials

1 Φ = Φn Ψ = 0 n ≥ −12 Ψ = (2n+ 1)Ψn Φ = −(n− 4 + 4ν)Φn−1 n ≥ 0

in the expression

2µv = ∇(Φ + zΨ)− 4(1− ν)Ψ k̂

These displacements are denoted respectively by v(1)n and v

(2)n

The associated stress tensors, denoted by σ(1)n and σ

(2)n

respectively, are obtained explicitly with the aid of the Hooke’s lawin spherical coordinates

E. Godoy INGMAT

in the expression

2µv = ∇(Φ + zΨ)− 4(1− ν)Ψ k̂

(2)n

E. Godoy INGMAT

in the expression

2µv = ∇(Φ + zΨ)− 4(1− ν)Ψ k̂

(2)n

E. Godoy INGMAT

in the expression

2µv = ∇(Φ + zΨ)− 4(1− ν)Ψ k̂

(2)n

E. Godoy INGMAT

in the expression

2µv = ∇(Φ + zΨ)− 4(1− ν)Ψ k̂

(2)n

E. Godoy INGMAT

in the expression

2µv = ∇(Φ + zΨ)− 4(1− ν)Ψ k̂

(2)n

E. Godoy INGMAT

The basic displacements and stresses can be reexpressed as

v(1)n (r, φ) =w

(1)n (φ)

rn+2σ(1)n (r, φ) =

τ(1)n (φ)

rn+3n ≥ −1

v(2)n (r, φ) =w

(2)n (φ)

rn+1σ(2)n (r, φ) =

τ(2)n (φ)

rn+2n ≥ 0

where for j = 1, 2, w(j)n , τ

(j)n are functions depending on φ

The general solution (v,σ) is then expressed as

v(r, φ) =

∞∑n=−1

1

rn+2

(a(1)n w

(1)n (φ) + a

(2)n+1w

(2)n+1(φ)

)σ(r, φ) =

∞∑n=−1

1

rn+3

(a(1)n τ

(1)n (φ) + a

(2)n+1τ

(2)n+1(φ)

)where for n ≥ −1, a(1)n and a(2)n are generic real coefficients

E. Godoy INGMAT

The basic displacements and stresses can be reexpressed as

v(1)n (r, φ) =w

(1)n (φ)

rn+2σ(1)n (r, φ) =

τ(1)n (φ)

rn+3n ≥ −1

v(2)n (r, φ) =w

(2)n (φ)

rn+1σ(2)n (r, φ) =

τ(2)n (φ)

rn+2n ≥ 0

where for j = 1, 2, w(j)n , τ

(j)n are functions depending on φ

The general solution (v,σ) is then expressed as

v(r, φ) =

∞∑n=−1

1

rn+2

(a(1)n w

(1)n (φ) + a

(2)n+1w

(2)n+1(φ)

)σ(r, φ) =

∞∑n=−1

1

rn+3

(a(1)n τ

(1)n (φ) + a

(2)n+1τ

(2)n+1(φ)

)where for n ≥ −1, a(1)n and a(2)n are generic real coefficients

E. Godoy INGMAT

Traction-free boundary conditions on Γ∞

Let us recall the boundary conditions imposed on the plane surface

σ(u)k̂ = 0 on Γ∞

For z = 0, k̂ = −φ̂ ⇒ σ(v)k̂ = −σrφ(v)r̂ − σφ(v)φ̂Imposing σφ(v) = σrφ(v) = 0 to the general solution yields the

following relations for the coefficients a(1)n and a

(2)n

a(1)−1 = (3− 2ν)a

(2)0

(2n+ 1)a(1)2n = α2n a

(2)2n+1 n ≥ 0

(2n+ 2)a(1)2n+1 = α2n+2 a

(2)2n+2 n ≥ 0

with α2n = (2n+ 1)2 − 2(1− ν)

E. Godoy INGMAT

σ(u)k̂ = 0 on Γ∞

(2)n

a(1)−1 = (3− 2ν)a

(2)0

(2n+ 1)a(1)2n = α2n a

(2)2n+1 n ≥ 0

(2n+ 2)a(1)2n+1 = α2n+2 a

(2)2n+2 n ≥ 0

with α2n = (2n+ 1)2 − 2(1− ν)

E. Godoy INGMAT

σ(u)k̂ = 0 on Γ∞

(2)n

a(1)−1 = (3− 2ν)a

(2)0

(2n+ 1)a(1)2n = α2n a

(2)2n+1 n ≥ 0

(2n+ 2)a(1)2n+1 = α2n+2 a

(2)2n+2 n ≥ 0

with α2n = (2n+ 1)2 − 2(1− ν)

E. Godoy INGMAT

Analytical solution expressed in series form

Rearranging appropriately the general solution, it is reexpressed as

v(r, φ) =

∞∑n=0

An

(hr

)2n+2w(A)n (φ) +

∞∑n=−1

Bn

(hr

)2n+3w(B)n (φ)

σ(r, φ) =1

h

[ ∞∑n=0

An

(hr

)2n+3τ (A)n (φ) +

∞∑n=−1

Bn

(hr

)2n+4τ (B)n (φ)

]

where functions w(A)n , w

(B)n , τ

(A)n and τ

(B)n are defined in terms of

w(1)n , w

(2)n , τ

(1)n and τ

(2)n , and An, Bn are generic real coefficients

It is easily verified that this general solution also satisfies theaxisymmetric boundary conditions on Γs: vφ = σrφ(v) = 0

E. Godoy INGMAT

v(r, φ) =

∞∑n=0

An

(hr

)2n+2w(A)n (φ) +

∞∑n=−1

Bn

(hr

)2n+3w(B)n (φ)

σ(r, φ) =1

h

[ ∞∑n=0

An

(hr

)2n+3τ (A)n (φ) +

∞∑n=−1

Bn

(hr

)2n+4τ (B)n (φ)

]

(B)n , τ

(A)n and τ

w(1)n , w

(2)n , τ

(1)n and τ

E. Godoy INGMAT

v(r, φ) =

∞∑n=0

An

(hr

)2n+2w(A)n (φ) +

∞∑n=−1

Bn

(hr

)2n+3w(B)n (φ)

σ(r, φ) =1

h

[ ∞∑n=0

An

(hr

)2n+3τ (A)n (φ) +

∞∑n=−1

Bn

(hr

)2n+4τ (B)n (φ)

]

(B)n , τ

(A)n and τ

w(1)n , w

(2)n , τ

(1)n and τ

E. Godoy INGMAT

Truncation of the series

Up to now we have obtained a solution that is fully analytical

It satisfies the equation of elastostatic equilibrium, boundaryconditions on Γ∞ and Γs, and a decaying condition at infinity

The boundary condition on Γh cannot be imposed analytically, soit is enforced numerically, giving rise to a semi-analytical solution

First of all the infinite series are truncated at a finite order N :

v(r, φ) =

N∑n=0

An

(hr

)2n+2w(A)n (φ) +

N∑n=−1

Bn

(hr

)2n+3w(B)n (φ)

σ(r, φ) =1

h

[ N∑n=0

An

(hr

)2n+3τ (A)n (φ) +

N∑n=−1

Bn

(hr

)2n+4τ (B)n (φ)

]

E. Godoy INGMAT

v(r, φ) =

N∑n=0

An

(hr

)2n+2w(A)n (φ) +

N∑n=−1

Bn

(hr

)2n+3w(B)n (φ)

σ(r, φ) =1

h

[ N∑n=0

An

(hr

)2n+3τ (A)n (φ) +

N∑n=−1

Bn

(hr

)2n+4τ (B)n (φ)

]

E. Godoy INGMAT

v(r, φ) =

N∑n=0

An

(hr

)2n+2w(A)n (φ) +

N∑n=−1

Bn

(hr

)2n+3w(B)n (φ)

σ(r, φ) =1

h

[ N∑n=0

An

(hr

)2n+3τ (A)n (φ) +

N∑n=−1

Bn

(hr

)2n+4τ (B)n (φ)

]

E. Godoy INGMAT

v(r, φ) =

N∑n=0

An

(hr

)2n+2w(A)n (φ) +

N∑n=−1

Bn

(hr

)2n+3w(B)n (φ)

σ(r, φ) =1

h

[ N∑n=0

An

(hr

)2n+3τ (A)n (φ) +

N∑n=−1

Bn

(hr

)2n+4τ (B)n (φ)

]

E. Godoy INGMAT

Quadratic functional of energy

Let us recall the boundary condition on the hemispherical surface

σ(v)r̂ = f on Γh

The numerical enforcement of this boundary condition yieldsapproximate values of a finite number of coefficients An and Bn

For this we define the following quadratic functional of energy

J(v) = − 12h2

∫Γh

σr̂ · v ds+ 1h2

∫Γh

f · v ds

The numerical minimisation of this functional leads to a linearsystem of equations for the coefficients An and Bn, whoseparticular structure allows us to solve it in a very efficient form

E. Godoy INGMAT

σ(v)r̂ = f on Γh

J(v) = − 12h2

∫Γh

σr̂ · v ds+ 1h2

∫Γh

f · v ds

E. Godoy INGMAT

σ(v)r̂ = f on Γh

J(v) = − 12h2

∫Γh

σr̂ · v ds+ 1h2

∫Γh

f · v ds

E. Godoy INGMAT

σ(v)r̂ = f on Γh

J(v) = − 12h2

∫Γh

σr̂ · v ds+ 1h2

∫Γh

f · v ds

E. Godoy INGMAT

Matrix form of the quadratic functional

Substituting in J the expressions for v and σ as truncated seriesevaluated at r = h, and rearranging appropriately yields

J(v) =1

2

N∑n=0

N∑k=0

Q(AA)nk AnAk +

1

2

N∑n=0

N∑k=−1

Q(AB)nk AnBk

+1

2

N∑n=−1

N∑k=0

Q(BA)nk BnAk +

1

2

N∑n=−1

N∑k=−1

Q(BB)nk BnBk

−N∑n=0

c(A)n An −N∑

n=−1c(B)n Bn

for matrices Q(AA), Q(AB), Q(BA), Q(BB) and vectors c(A), c(B)

E. Godoy INGMAT

Substituting in J the expressions for v and σ as truncated seriesevaluated at r = h, and rearranging appropriately yields

J(v) =1

2

N∑n=0

N∑k=0

Q(AA)nk AnAk +

1

2

N∑n=0

N∑k=−1

Q(AB)nk AnBk

+1

2

N∑n=−1

N∑k=0

Q(BA)nk BnAk +

1

2

N∑n=−1

N∑k=−1

Q(BB)nk BnBk

−N∑n=0

c(A)n An −N∑

n=−1c(B)n Bn

for matrices Q(AA), Q(AB), Q(BA), Q(BB) and vectors c(A), c(B)

E. Godoy INGMAT

For i, j = A,B the entries of matrices Q(ij) and vectors c(i) aredefined as

Q(ij)nk = −

∫ ππ/2

([w(i)n ]r(φ)[τ

(j)k ]r(φ) + [w

(i)n ]φ(φ)[τ

(j)k ]rφ(φ)

)sinφdφ

c(i)n = −h∫ ππ/2

([w(i)n ]r(φ)fr(φ) + [w

(i)n ]φ(φ)fφ(φ)

)sinφdφ

All these entries are calculated explicitly with the aid of integralformulae for Legendre polynomials and their derivatives, e.g.∫ π

π/2

P2n(cosφ)P2k+1(cosφ) sinφ dφ = −(2k + 1)P2n(0)P2k(0)

(2k + 1− 2n)(2k + 2 + 2n)

E. Godoy INGMAT

For i, j = A,B the entries of matrices Q(ij) and vectors c(i) aredefined as

Q(ij)nk = −

∫ ππ/2

([w(i)n ]r(φ)[τ

(j)k ]r(φ) + [w

(i)n ]φ(φ)[τ

(j)k ]rφ(φ)

)sinφdφ

c(i)n = −h∫ ππ/2

([w(i)n ]r(φ)fr(φ) + [w

(i)n ]φ(φ)fφ(φ)

)sinφdφ

All these entries are calculated explicitly with the aid of integralformulae for Legendre polynomials and their derivatives, e.g.∫ π

π/2

P2n(cosφ)P2k+1(cosφ) sinφ dφ = −(2k + 1)P2n(0)P2k(0)

(2k + 1− 2n)(2k + 2 + 2n)

E. Godoy INGMAT

Structure of matrices

N+1 N+2 N+2

Q(AA) Q(AB) Q(BB)

N+1 N+2

Q(AA) and Q(BB) are symmetric and positive definite matrices

Q(BA) = [Q(AB)]T

E. Godoy INGMAT

N+1 N+2 N+2

Q(AA) Q(AB) Q(BB)

N+1 N+2

Q(BA) = [Q(AB)]T

E. Godoy INGMAT

N+1 N+2 N+2

Q(AA) Q(AB) Q(BB)

N+1 N+2

Q(BA) = [Q(AB)]T

E. Godoy INGMAT

Minimisation of the functional and linear system

Defining the vectors x(A) = (A0 A1 . . . AN )T ∈ RN+1 and

x(B) = (B−1 B0 . . . BN )T ∈ RN+2

and the matrices and vectors by blocks

Q =

[Q(AA) Q(AB)

[Q(AB)]T Q(BB)

]x =

[x(A)

x(B)

]c =

[c(A)

c(B)

]the quadratic functional is reexpressed as J(x) = 12x

TQx− xTcAs the matrix Q is symmetric and positive definite, J has a globalminimum, which is reached when ∇J(x) = 0This is equivalent to Qx = c, which corresponds to a linear systemof equations for the coefficients An and Bn, stored in the vector x

E. Godoy INGMAT

x(B) = (B−1 B0 . . . BN )T ∈ RN+2

Q =

[Q(AA) Q(AB)

[Q(AB)]T Q(BB)

]x =

[x(A)

x(B)

]c =

[c(A)

c(B)

E. Godoy INGMAT

x(B) = (B−1 B0 . . . BN )T ∈ RN+2

Q =

[Q(AA) Q(AB)

[Q(AB)]T Q(BB)

]x =

[x(A)

x(B)

]c =

[c(A)

c(B)

E. Godoy INGMAT

x(B) = (B−1 B0 . . . BN )T ∈ RN+2

Q =

[Q(AA) Q(AB)

[Q(AB)]T Q(BB)

]x =

[x(A)

x(B)

]c =

[c(A)

c(B)

E. Godoy INGMAT

x(B) = (B−1 B0 . . . BN )T ∈ RN+2

Q =

[Q(AA) Q(AB)

[Q(AB)]T Q(BB)

]x =

[x(A)

x(B)

]c =

[c(A)

c(B)

E. Godoy INGMAT

Solution of the linear system

To solve [Q(AA) Q(AB)

[Q(AB)]T Q(BB)

] [x(A)

x(B)

]=

[c(A)

c(B)

]we use the Schur-Banachiewicz blockwise inversion formula:

x(A) =([Q(AA)]−1 + [Q(AA)]−1Q(AB)[Q̃(BB)]−1[Q(AB)]T [Q(AA)]−1

)c(A)

− [Q(AA)]−1Q(AB)[Q̃(BB)]−1c(B)

x(B) = −[Q̃(BB)]−1[Q(AB)]T [Q(AA)]−1c(A) + [Q̃(BB)]−1c(B)

whereQ̃(BB) = Q(BB) − [Q(AB)]T [Q(AA)]−1Q(AB)

is the Schur complement of Q(BB) in Q

E. Godoy INGMAT

[Q(AB)]T Q(BB)

] [x(A)

x(B)

]=

[c(A)

c(B)

)c(A)

− [Q(AA)]−1Q(AB)[Q̃(BB)]−1c(B)

E. Godoy INGMAT

[Q(AB)]T Q(BB)

] [x(A)

x(B)

]=

[c(A)

c(B)

)c(A)

− [Q(AA)]−1Q(AB)[Q̃(BB)]−1c(B)

E. Godoy INGMAT

To evaluate the Schur-Banachiewicz blockwise inversion formula itis required to invert two matrices:

The symmetric tridiagonal matrix Q(AA): It is efficientlyinverted using the Thomas algorithm for tridiagonal systems

The positive definite symmetric full matrix Q̃(BB): It isefficiently inverted with the aid of its Cholesky factorisation

The evaluation of the formula yields approximate values ofcoefficients A0, A1, . . . , AN and B−1, B0, B1, . . . , BN

Substituting these coefficients in the expressions of v and σ astruncated series, we obtain explicitly the semi-analytical solution

E. Godoy INGMAT

Evaluation of the solution

The semi-analytical solution was numerically evaluated using thefollowing parameter values:

Parameter h g ρ E ν

Value 600 m 9.81 m/s2 2725 Kg/m3 70.2 GPa 0.3

The Lamé’s constants are obtained through the formulae

λ =νE

(1 + ν)(1− 2ν) , µ =E

2(1 + ν)

A truncation order of N = 50 was considered

Recall that the physical displacement is obtained as u = ug + v

E. Godoy INGMAT

Value 600 m 9.81 m/s2 2725 Kg/m3 70.2 GPa 0.3

λ =νE

(1 + ν)(1− 2ν) , µ =E

2(1 + ν)

E. Godoy INGMAT

Value 600 m 9.81 m/s2 2725 Kg/m3 70.2 GPa 0.3

λ =νE

(1 + ν)(1− 2ν) , µ =E

2(1 + ν)

E. Godoy INGMAT

Value 600 m 9.81 m/s2 2725 Kg/m3 70.2 GPa 0.3

λ =νE

(1 + ν)(1− 2ν) , µ =E

2(1 + ν)

E. Godoy INGMAT

2D Plots of the solution

E. Godoy INGMAT

Validation of the solution

To validate the semi-analytical procedure of solution, the sameaxisymmetric boundary-value problem was numerically solved usingthe commercial FEM software COMSOL Multiphysics

Dirichlet boundary conditions on theright and bottom boundaries

Different values of the square length Lwere considered

It is expected that as L increases, thenumerical solution approaches thesemi-analytical solution

E. Godoy INGMAT

h

L

Ωtr

E. Godoy INGMAT

h

L

Ωtr

E. Godoy INGMAT

h

L

Ωtr

E. Godoy INGMAT

h

L

Ωtr

E. Godoy INGMAT

Plots of the semi-analytical and numerical solution

Displacements on Γh obtained by the semi-analytical procedureand numerically by FEM for different values of L

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2−0.05

0

0.05

0.1

0.15

0.2

Displacement component ur

Angle (rad)

Dis

plac

emen

t (m

)

Semi−analyticalNumerical L=1000 mNumerical L=2000 mNumerical L=5000 mNumerical L=10000 m

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20

0.02

0.04

0.06

0.08

0.1

0.12

Displacement component uφ

Angle (rad)

Dis

plac

emen

t (m

)

Semi−analyticalNumerical L=1000 mNumerical L=2000 mNumerical L=5000 mNumerical L=10000 m

E. Godoy INGMAT

Conclusions

A semi-analytical procedure was developed to solve theproblem of an elastic half-space with a hemispherical pit

The procedure imposes numerically boundary conditions onthe hemispherical pit to a previously obtained fully analyticalsolution in series form

This numerical imposition is performed by minimising aquadratic functional of energy, leading to a symmetric andpositive definite linear system for the coefficients of the series,whose structure allows to solve it efficiently

The obtained semi-analytical solution was evaluated,numerical results were provided and a validation procedurewas carried out

E. Godoy INGMAT

Conclusions

E. Godoy INGMAT

Conclusions

E. Godoy INGMAT

Conclusions

E. Godoy INGMAT

Conclusions

E. Godoy INGMAT

Perspectives for future work

Calculation of an axisymmetric DtN (Dirichlet-to-Neumann)operator for the elastic half-space

Coupling of this DtN operator with FEM 2D to treat pits withgeometries other than hemispherical but still axisymmetric

Calculation of a non-axisymmetric DtN operator for theelastic half-space

Coupling of this DtN operator with FEM 3D to treat pits withnon-axisymmetric geometry

E. Godoy INGMAT

Thanks for your attention!

Any question?

E. Godoy INGMAT

Motivation and basic equationsAnalytical solution in series formBoundary conditions on the hemispherical pitNumerical results and validationConclusions and perspectives for future work

Documents

A semi-analytical method to solve the equilibrium ... · MotivationAnalytical solutionBoundary conditions on the pitNumerical resultsConclusions 1 Motivation and basic equations 2