-
(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
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Axisymmetric domain
φ
rAxisymmetric spherical coordinates (r, φ)
Unit vectors r̂, φ̂ and k̂
Pit radius h, gravity acceleration ~g
Domain and boundaries:
Ω = {(r, φ) : h < r
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Axisymmetric domain
φ
r
r̂
φ̂
k̂
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
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Axisymmetric domain
h
φ
r
~g
r̂
φ̂
k̂
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
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Axisymmetric domain
Γh
Γ∞
h
Ω
Γs
φ
r
~g
r̂
φ̂
k̂
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
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Obtaining the series
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Obtaining the series
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Matrix form of the quadratic functional
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Matrix form of the quadratic functional
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Solution of the linear system
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Solution of the linear system
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Solution of the linear system
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Solution of the linear system
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Solution of the linear system
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
2D Plots of the solution
E. Godoy INGMAT
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
h
L
L
Ωtr
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
h
L
L
Ωtr
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
h
L
L
Ωtr
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
h
L
L
Ωtr
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
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
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
-
Motivation Analytical solution Boundary conditions on the pit
Numerical results Conclusions
Thanks for your attention!
Any question?
E. Godoy INGMAT
A semi-analytical method to solve the equilibrium equations of
axisymmetric elasticity in a half-space with a hemispherical
pit
Motivation and basic equationsAnalytical solution in series
formBoundary conditions on the hemispherical pitNumerical results
and validationConclusions and perspectives for future work