Scaled Boundary Methods: an introduction Charles Augarde Mechanics Research Group, School of Engineering & Computing Sciences, Durham University
4 April 2011
1st ACME School, Heriot-Watt University.
Overview ........
1
Background
2
History
3
Basic formulation for solid mechanics
4
Examples of use
5
Meshless and hybrid versions
6
Other aspects & summary
7
References
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
2 / 39
Background Location w.r.t. other methods
Features The scaled boundary method is an order reduction method for solid mechanics problems, with links to both finite elements and boundary elements. In its statics form it is particularly useful for modelling infinite domains, for geomechanics problems for instance. It is a novel semi-analytical method for continuum analysis requiring no fundamental solution (as in BEM for example).
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
3 / 39
Background Location w.r.t. other methods
Features The scaled boundary method is an order reduction method for solid mechanics problems, with links to both finite elements and boundary elements. In its statics form it is particularly useful for modelling infinite domains, for geomechanics problems for instance. It is a novel semi-analytical method for continuum analysis requiring no fundamental solution (as in BEM for example).
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
3 / 39
Background Location w.r.t. other methods
Features The scaled boundary method is an order reduction method for solid mechanics problems, with links to both finite elements and boundary elements. In its statics form it is particularly useful for modelling infinite domains, for geomechanics problems for instance. It is a novel semi-analytical method for continuum analysis requiring no fundamental solution (as in BEM for example).
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
3 / 39
Background History
The SB method was originally devised by John Wolf in Zurich in the 1990s, further developed by Wolf in collaboration with Chongmin Song (UNSW) and more recently elucidated by Andrew Deeks (Durham) working with Wolf. Wolf developed the method firstly for dynamic problems in the frequency domain but the method was complex and received relatively little attention. Deeks and Wolf [1] recast the method for solid mechanics using virtual work, which made it much more accessible. This is what I will cover in today’s lecture
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
4 / 39
Background History
The SB method was originally devised by John Wolf in Zurich in the 1990s, further developed by Wolf in collaboration with Chongmin Song (UNSW) and more recently elucidated by Andrew Deeks (Durham) working with Wolf. Wolf developed the method firstly for dynamic problems in the frequency domain but the method was complex and received relatively little attention. Deeks and Wolf [1] recast the method for solid mechanics using virtual work, which made it much more accessible. This is what I will cover in today’s lecture
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
4 / 39
Background History
The SB method was originally devised by John Wolf in Zurich in the 1990s, further developed by Wolf in collaboration with Chongmin Song (UNSW) and more recently elucidated by Andrew Deeks (Durham) working with Wolf. Wolf developed the method firstly for dynamic problems in the frequency domain but the method was complex and received relatively little attention. Deeks and Wolf [1] recast the method for solid mechanics using virtual work, which made it much more accessible. This is what I will cover in today’s lecture
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
4 / 39
Background History
The SB method was originally devised by John Wolf in Zurich in the 1990s, further developed by Wolf in collaboration with Chongmin Song (UNSW) and more recently elucidated by Andrew Deeks (Durham) working with Wolf. Wolf developed the method firstly for dynamic problems in the frequency domain but the method was complex and received relatively little attention. Deeks and Wolf [1] recast the method for solid mechanics using virtual work, which made it much more accessible. This is what I will cover in today’s lecture
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
4 / 39
Formulation The scaled boundary coordinate system
Consider a 2D bounded domain. The scaled boundary and Cartesian coordinate systems are related by the scaling equations x = x0 + ξxs (s) y = y0 + ξys (s)
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
5 / 39
Formulation Solid mechanics
An approximate solution for the displacement is sought {uh (ξ, s)} =
n X
[Ni (s)] uhi (ξ) = [N (s)] {uh (ξ)} .
i=1
[N (s)] are FE shape functions around the boundary s. {uh (ξ)} is unknown but contains values of an analytical function in ξ. ∂ ∂x
Strains (Cartesian) {εh (x, y)} = [L] {uh (x, y)} where [L] = 0
∂ ∂y
0 ∂ ∂y ∂ ∂x
Strains (scaled boundary) {εh (ξ, s)} = [L∗ ] {uh (ξ, s)} where [L∗ ] = b1 (s) ∂ + 1ξ b2 (s) ∂ ∂ξ ∂s where b1 (s) and b2 (s) are functions of [L] and a Jacobian linking the Cartesian and SB frames. Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
6 / 39
Formulation Solid mechanics
An approximate solution for the displacement is sought {uh (ξ, s)} =
n X
[Ni (s)] uhi (ξ) = [N (s)] {uh (ξ)} .
i=1
[N (s)] are FE shape functions around the boundary s. {uh (ξ)} is unknown but contains values of an analytical function in ξ. ∂ ∂x
Strains (Cartesian) {εh (x, y)} = [L] {uh (x, y)} where [L] = 0
∂ ∂y
0 ∂ ∂y ∂ ∂x
Strains (scaled boundary) {εh (ξ, s)} = [L∗ ] {uh (ξ, s)} where [L∗ ] = b1 (s) ∂ + 1ξ b2 (s) ∂ ∂ξ ∂s where b1 (s) and b2 (s) are functions of [L] and a Jacobian linking the Cartesian and SB frames. Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
6 / 39
Formulation Solid mechanics
An approximate solution for the displacement is sought {uh (ξ, s)} =
n X
[Ni (s)] uhi (ξ) = [N (s)] {uh (ξ)} .
i=1
[N (s)] are FE shape functions around the boundary s. {uh (ξ)} is unknown but contains values of an analytical function in ξ. ∂ ∂x
Strains (Cartesian) {εh (x, y)} = [L] {uh (x, y)} where [L] = 0
∂ ∂y
0 ∂ ∂y ∂ ∂x
Strains (scaled boundary) {εh (ξ, s)} = [L∗ ] {uh (ξ, s)} where [L∗ ] = b1 (s) ∂ + 1ξ b2 (s) ∂ ∂ξ ∂s where b1 (s) and b2 (s) are functions of [L] and a Jacobian linking the Cartesian and SB frames. Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
6 / 39
Formulation Solid mechanics
Strains {εh (ξ, s)} = [L∗ ] {uh (ξ, s)} = [L∗ ] [N (s)] {uh (ξ)} Substituting for the transformed [L∗ ] we get {εh (ξ, s)} = B 1 (s) {uh (ξ)},ξ + 1 B 2 (s) {uh (ξ)} ξ where 1 1 B (s) = b (s) [N (s)] and B 2 (s) = b2 (s) [N (s)],s Stresses {σh (ξ, s)} = [D] B 1 (s) {uh (ξ)},ξ + 1 [D] B 2 (s) {uh (ξ)} ξ Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
7 / 39
Formulation Solid mechanics
Strains {εh (ξ, s)} = [L∗ ] {uh (ξ, s)} = [L∗ ] [N (s)] {uh (ξ)} Substituting for the transformed [L∗ ] we get {εh (ξ, s)} = B 1 (s) {uh (ξ)},ξ + 1 B 2 (s) {uh (ξ)} ξ where 1 1 B (s) = b (s) [N (s)] and B 2 (s) = b2 (s) [N (s)],s Stresses {σh (ξ, s)} = [D] B 1 (s) {uh (ξ)},ξ + 1 [D] B 2 (s) {uh (ξ)} ξ Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
7 / 39
Formulation Solid mechanics
Strains {εh (ξ, s)} = [L∗ ] {uh (ξ, s)} = [L∗ ] [N (s)] {uh (ξ)} Substituting for the transformed [L∗ ] we get {εh (ξ, s)} = B 1 (s) {uh (ξ)},ξ + 1 B 2 (s) {uh (ξ)} ξ where 1 1 B (s) = b (s) [N (s)] and B 2 (s) = b2 (s) [N (s)],s Stresses {σh (ξ, s)} = [D] B 1 (s) {uh (ξ)},ξ + 1 [D] B 2 (s) {uh (ξ)} ξ Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
7 / 39
Formulation Virtual work approach
We use virtual work to introduce the equilibrium requirement. Ignoring body forces for now we have
boundary tractions
Virtual displacement field {δu(ξ, s)} = [N (s)] {δu(ξ)} Virtual strainfield {δε(ξ, s)} = B 1 (s) {δu(ξ)},ξ + 1 B 2 (s) {δu(ξ)} ξ
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
8 / 39
Formulation Virtual work approach
We use virtual work to introduce the equilibrium requirement. Ignoring body forces for now we have
boundary tractions
Virtual displacement field {δu(ξ, s)} = [N (s)] {δu(ξ)} Virtual strainfield {δε(ξ, s)} = B 1 (s) {δu(ξ)},ξ + 1 B 2 (s) {δu(ξ)} ξ
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
8 / 39
Formulation Virtual work approach
We use virtual work to introduce the equilibrium requirement. Ignoring body forces for now we have
boundary tractions
Virtual displacement field {δu(ξ, s)} = [N (s)] {δu(ξ)} Virtual strainfield {δε(ξ, s)} = B 1 (s) {δu(ξ)},ξ + 1 B 2 (s) {δu(ξ)} ξ
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
8 / 39
Formulation The internal virtual work term
R
{δε(ξ, s)}T {σh (ξ, s)} dV
V
Z
T
Z
{δε(ξ, s)} {σh (ξ, s)} dV = V
1 B (s) {δu(ξ)} ,ξ
V
T 1 2 B (s) {δu(ξ)} + ξ × [D] B 1 (s) {uh (ξ)} ,ξ 2 1 + [D] B (s) {uh (ξ)} dV ξ
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
9 / 39
Formulation The internal virtual work term
R
{δε(ξ, s)}T {σh (ξ, s)} dV
V
|J| is the determinant of the Jacobian linking the Cartesian 1 and scaled boundary coordinate systems, and appears in b (s) , b2 (s) The integrations above are carried out using Green’s theorem which produces line integrals on the boundary ξ = 1. Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
10 / 39
Formulation The internal virtual work term
R
{δε(ξ, s)}T {σh (ξ, s)} dV
V
By setting the following
and using {u} to mean {u(ξ = 1)} the internal virtual work term can be expressed as n o T {δu} T E 0 {uh } ,ξ + E 1 {uh } Z 1 0 T − {δu(ξ)} E ξ {uh (ξ)} ,ξξ 0 h 1 i 2 1 0 1 T + E + E − E {uh (ξ)} ,ξ − E {uh (ξ)} dξ ξ Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
11 / 39
Formulation The internal virtual work term
R
{δε(ξ, s)}T {σh (ξ, s)} dV
V
By setting the following
and using {u} to mean {u(ξ = 1)} the internal virtual work term can be expressed as n o T {δu} T E 0 {uh } ,ξ + E 1 {uh } Z 1 0 T − {δu(ξ)} E ξ {uh (ξ)} ,ξξ 0 h 1 i 2 1 0 1 T + E + E − E {uh (ξ)} ,ξ − E {uh (ξ)} dξ ξ Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
11 / 39
Formulation The external virtual work term
R
{δu(s)}T {t(s)} ds
S
The external virtual work term is much easier Z T = {δu} [N (s)]T {t(s)} ds = {δu}T {P } S
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
12 / 39
Formulation The external virtual work term
R
{δu(s)}T {t(s)} ds
S
The external virtual work term is much easier Z T = {δu} [N (s)]T {t(s)} ds = {δu}T {P } S
so the complete virtual work expression becomes n o T {δu} T E 0 {uh } ,ξ + E 1 {uh } Z 1 0 T T − {δu} {P } − {δu(ξ)} E ξ {uh (ξ)} ,ξξ 0
h 1 i 2 1 0 1 T {uh (ξ)} dξ = 0 + E + E {uh (ξ)} ,ξ − E − E ξ
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
12 / 39
Formulation The external virtual work term
R
{δu(s)}T {t(s)} ds
S
The external virtual work term is much easier Z T = {δu} [N (s)]T {t(s)} ds = {δu}T {P } S
and there are clearly two conditions to be met
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
12 / 39
Formulation Conditions arising from virtual work expressions
For the virtual work expression to be satisfied both of these conditions must be met: T {P } = E 0 {uh },ξ + E 1 {uh }
...
(A)
The second of these equations is known as the scaled boundary finite element equation in displacement and can be arrived at via other methods due to Wolf and Song [2]. However this derivation is nice because it is so closely related to finite elements.
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
13 / 39
Formulation Solutions: condition (B)
Eqn (B) is a homogeneous set of Euler-Cauchy differential equations and by inspection the solution must look like this {uh (ξ)} = c1 ξ −λ1 {φ1 } + c2 ξ −λ2 {φ2 } . . . Each term represents a contribution to the deformation comprising a scaling factor λi , a mode shape {φi } and a coefficient ci representing the contribution of each mode. The displacements for each mode take this form {u(ξ)} = ξ −λ {φ} which substituted into (B) gives this quadratic eigenproblem (ξ = 1) h h i i T λ2 E 0 − λ E 1 − E 1 − E 2 {φ} = {0} Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
14 / 39
Formulation Solutions: condition (B)
Eqn (B) is a homogeneous set of Euler-Cauchy differential equations and by inspection the solution must look like this {uh (ξ)} = c1 ξ −λ1 {φ1 } + c2 ξ −λ2 {φ2 } . . . Each term represents a contribution to the deformation comprising a scaling factor λi , a mode shape {φi } and a coefficient ci representing the contribution of each mode. The displacements for each mode take this form {u(ξ)} = ξ −λ {φ} which substituted into (B) gives this quadratic eigenproblem (ξ = 1) h h i i T λ2 E 0 − λ E 1 − E 1 − E 2 {φ} = {0} Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
14 / 39
Formulation Solutions: condition (B)
Eqn (B) is a homogeneous set of Euler-Cauchy differential equations and by inspection the solution must look like this {uh (ξ)} = c1 ξ −λ1 {φ1 } + c2 ξ −λ2 {φ2 } . . . Each term represents a contribution to the deformation comprising a scaling factor λi , a mode shape {φi } and a coefficient ci representing the contribution of each mode. The displacements for each mode take this form {u(ξ)} = ξ −λ {φ} which substituted into (B) gives this quadratic eigenproblem (ξ = 1) h h i i T λ2 E 0 − λ E 1 − E 1 − E 2 {φ} = {0} Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
14 / 39
Formulation Condition (A)
The nodal forces {q} required at the boundary ξ = 1 to equilibrate each displacement mode can be found by using condition (A) taking the expression for one modal displacement {u(ξ)} = ξ −λ {φ} . . . substituting this into (A) and setting ξ = 1 giving h i T {q} = E 1 − λ E 0 {φ} Now rearranging i −1 h 1 T {φ} − {q} . . . (C) λ {φ} = E 0 E and the quadratic eigenproblem which was . . . h substituting h into T 1 i 2 i 2 0 1 λ E −λ E − E − E {φ} = {0} Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
15 / 39
Formulation Condition (A)
The nodal forces {q} required at the boundary ξ = 1 to equilibrate each displacement mode can be found by using condition (A) taking the expression for one modal displacement {u(ξ)} = ξ −λ {φ} . . . substituting this into (A) and setting ξ = 1 giving h i T {q} = E 1 − λ E 0 {φ} Now rearranging i −1 h 1 T {φ} − {q} . . . (C) λ {φ} = E 0 E and the quadratic eigenproblem which was . . . h substituting h into T 1 i 2 i 2 0 1 λ E −λ E − E − E {φ} = {0} Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
15 / 39
Formulation Virtual work approach
gives i h 1 T {φ} − {q} − E 2 {φ} λ {q} = E 1 E 0 E Now we combine this equation with (C) into a linear eigenproblem. " 0 −1 1 T −1 # E E − E0 φ φ 1 0 −1 1 T 2 1 0 −1 =λ q q − E E E E − E E However you should have noticed we have doubled the number of degrees of freedom in the problem . . .
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
16 / 39
Formulation Virtual work approach
gives i h 1 T {φ} − {q} − E 2 {φ} λ {q} = E 1 E 0 E Now we combine this equation with (C) into a linear eigenproblem. " 0 −1 1 T −1 # E E − E0 φ φ 1 0 −1 1 T 2 1 0 −1 =λ q q − E E E E − E E However you should have noticed we have doubled the number of degrees of freedom in the problem . . .
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
16 / 39
Formulation Towards a stiffness matrix
Solving the linear eigenproblem gives 2n modes and 2n forces at the boundary. However only the n modes with non-positive real values for λ are relevant (as these give finite displacements at the scaling centre. Look at {u(ξ)} = ξ −λ {φ}).
How do we obtain a stiffness matrix for the domain? Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
17 / 39
Formulation Towards a stiffness matrix
How do we obtain a stiffness matrix for the domain?
Subset of modal displacements in columns placed in a matrix [Φ1 ]. Subset of modal forces in columns placed in a matrix [Q1 ].
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
18 / 39
Formulation Towards a stiffness matrix
Returning to the solution {uh (ξ)} = c1 ξ −λ1 {φ1 } + c2 ξ −λ2 {φ2 } . . . we can see that if we are on the boundary ξ = 1 then {c} = [Φ1 ]−1 {uh } and the equivalent nodal forces on that boundary are {P } = [Q1 ] {c} = [Q1 ] [Φ1 ]−1 {uh } so that the stiffness matrix [K] = [Q1 ] [Φ1 ]−1 . Displacements {uh } on the boundary are calculated, hence {c} and then displacements throughout the domain can be found. Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
19 / 39
Formulation Towards a stiffness matrix
Returning to the solution {uh (ξ)} = c1 ξ −λ1 {φ1 } + c2 ξ −λ2 {φ2 } . . . we can see that if we are on the boundary ξ = 1 then {c} = [Φ1 ]−1 {uh } and the equivalent nodal forces on that boundary are {P } = [Q1 ] {c} = [Q1 ] [Φ1 ]−1 {uh } so that the stiffness matrix [K] = [Q1 ] [Φ1 ]−1 . Displacements {uh } on the boundary are calculated, hence {c} and then displacements throughout the domain can be found. Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
19 / 39
Solution The full procedure 1
2
3
Form the matrix in the eigenproblem. φ Solve the eigenproblem for and λi . q Discard modes with Re(λi ) > 0.
4
Form [Φ1 ] and [Q1 ] from what is left.
5
Stiffness matrix [K] = [Q1 ] [Φ1 ]−1 .
6
Form known entries in the nodal force vector {P } .
7
Set up reduced linear system from [K] {uh } = {P } and solve for displacements on the scaled boundary (ξ = 1) {uh }.
8 9
Then find {c} = [Φ1 ]−1 {uh }. Recover the displacement field throughout the domain using n P {uh (ξ, s)} = [N (s)] ci ξ −λi {φi }. i=1 Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
20 / 39
Solution The full procedure 1
2
3
Form the matrix in the eigenproblem. φ Solve the eigenproblem for and λi . q Discard modes with Re(λi ) > 0.
4
Form [Φ1 ] and [Q1 ] from what is left.
5
Stiffness matrix [K] = [Q1 ] [Φ1 ]−1 .
6
Form known entries in the nodal force vector {P } .
7
Set up reduced linear system from [K] {uh } = {P } and solve for displacements on the scaled boundary (ξ = 1) {uh }.
8 9
Then find {c} = [Φ1 ]−1 {uh }. Recover the displacement field throughout the domain using n P {uh (ξ, s)} = [N (s)] ci ξ −λi {φi }. i=1 Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
20 / 39
Solution The full procedure 1
2
3
Form the matrix in the eigenproblem. φ Solve the eigenproblem for and λi . q Discard modes with Re(λi ) > 0.
4
Form [Φ1 ] and [Q1 ] from what is left.
5
Stiffness matrix [K] = [Q1 ] [Φ1 ]−1 .
6
Form known entries in the nodal force vector {P } .
7
Set up reduced linear system from [K] {uh } = {P } and solve for displacements on the scaled boundary (ξ = 1) {uh }.
8 9
Then find {c} = [Φ1 ]−1 {uh }. Recover the displacement field throughout the domain using n P {uh (ξ, s)} = [N (s)] ci ξ −λi {φi }. i=1 Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
20 / 39
Solution The full procedure 1
2
3
Form the matrix in the eigenproblem. φ Solve the eigenproblem for and λi . q Discard modes with Re(λi ) > 0.
4
Form [Φ1 ] and [Q1 ] from what is left.
5
Stiffness matrix [K] = [Q1 ] [Φ1 ]−1 .
6
Form known entries in the nodal force vector {P } .
7
Set up reduced linear system from [K] {uh } = {P } and solve for displacements on the scaled boundary (ξ = 1) {uh }.
8 9
Then find {c} = [Φ1 ]−1 {uh }. Recover the displacement field throughout the domain using n P {uh (ξ, s)} = [N (s)] ci ξ −λi {φi }. i=1 Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
20 / 39
Solution The full procedure 1
2
3
Form the matrix in the eigenproblem. φ Solve the eigenproblem for and λi . q Discard modes with Re(λi ) > 0.
4
Form [Φ1 ] and [Q1 ] from what is left.
5
Stiffness matrix [K] = [Q1 ] [Φ1 ]−1 .
6
Form known entries in the nodal force vector {P } .
7
Set up reduced linear system from [K] {uh } = {P } and solve for displacements on the scaled boundary (ξ = 1) {uh }.
8 9
Then find {c} = [Φ1 ]−1 {uh }. Recover the displacement field throughout the domain using n P {uh (ξ, s)} = [N (s)] ci ξ −λi {φi }. i=1 Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
20 / 39
Solution The full procedure 1
2
3
Form the matrix in the eigenproblem. φ Solve the eigenproblem for and λi . q Discard modes with Re(λi ) > 0.
4
Form [Φ1 ] and [Q1 ] from what is left.
5
Stiffness matrix [K] = [Q1 ] [Φ1 ]−1 .
6
Form known entries in the nodal force vector {P } .
7
Set up reduced linear system from [K] {uh } = {P } and solve for displacements on the scaled boundary (ξ = 1) {uh }.
8 9
Then find {c} = [Φ1 ]−1 {uh }. Recover the displacement field throughout the domain using n P {uh (ξ, s)} = [N (s)] ci ξ −λi {φi }. i=1 Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
20 / 39
Solution The full procedure 1
2
3
Form the matrix in the eigenproblem. φ Solve the eigenproblem for and λi . q Discard modes with Re(λi ) > 0.
4
Form [Φ1 ] and [Q1 ] from what is left.
5
Stiffness matrix [K] = [Q1 ] [Φ1 ]−1 .
6
Form known entries in the nodal force vector {P } .
7
Set up reduced linear system from [K] {uh } = {P } and solve for displacements on the scaled boundary (ξ = 1) {uh }.
8 9
Then find {c} = [Φ1 ]−1 {uh }. Recover the displacement field throughout the domain using n P {uh (ξ, s)} = [N (s)] ci ξ −λi {φi }. i=1 Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
20 / 39
Solution The full procedure 1
2
3
Form the matrix in the eigenproblem. φ Solve the eigenproblem for and λi . q Discard modes with Re(λi ) > 0.
4
Form [Φ1 ] and [Q1 ] from what is left.
5
Stiffness matrix [K] = [Q1 ] [Φ1 ]−1 .
6
Form known entries in the nodal force vector {P } .
7
Set up reduced linear system from [K] {uh } = {P } and solve for displacements on the scaled boundary (ξ = 1) {uh }.
8 9
Then find {c} = [Φ1 ]−1 {uh }. Recover the displacement field throughout the domain using n P {uh (ξ, s)} = [N (s)] ci ξ −λi {φi }. i=1 Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
20 / 39
Solution The full procedure 1
2
3
Form the matrix in the eigenproblem. φ Solve the eigenproblem for and λi . q Discard modes with Re(λi ) > 0.
4
Form [Φ1 ] and [Q1 ] from what is left.
5
Stiffness matrix [K] = [Q1 ] [Φ1 ]−1 .
6
Form known entries in the nodal force vector {P } .
7
Set up reduced linear system from [K] {uh } = {P } and solve for displacements on the scaled boundary (ξ = 1) {uh }.
8 9
Then find {c} = [Φ1 ]−1 {uh }. Recover the displacement field throughout the domain using n P {uh (ξ, s)} = [N (s)] ci ξ −λi {φi }. i=1 Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
20 / 39
Formulation Treatment of other loads
Body loads and tractions on side faces can be dealt with straightforwardly. The scaled boundary finite element equation in displacement gets some additional terms h i T E 0 ξ 2 {uh (ξ)},ξξ + E 0 + E 1 − E 1 ξ {uh (ξ)},ξ − 2 E {uh (ξ)} + ξ 2 {Fb (ξ)} + ξ {Ft (ξ)} = {0}
Solutions to this non-homogeneous set of equations are found by a linear combination of solutions to the original homogeneous equations and particular solutions for the added terms. These can be thought of as additional modes required to satisfy equilibrium.
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
21 / 39
Formulation Treatment of other loads
Body loads and tractions on side faces can be dealt with straightforwardly. The scaled boundary finite element equation in displacement gets some additional terms h i T E 0 ξ 2 {uh (ξ)},ξξ + E 0 + E 1 − E 1 ξ {uh (ξ)},ξ − 2 E {uh (ξ)} + ξ 2 {Fb (ξ)} + ξ {Ft (ξ)} = {0}
Solutions to this non-homogeneous set of equations are found by a linear combination of solutions to the original homogeneous equations and particular solutions for the added terms. These can be thought of as additional modes required to satisfy equilibrium.
Charles Augarde (DU)
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Formulation Options on boundaries
The development above has used a bounded domain with an internal scaling centre, however it is fine to have the scaling centre on the boundary. In that event there are side faces along which there is no need for discretisation.
(From [1]) Charles Augarde (DU)
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Formulation Unbounded domains
It is also easy to model unbounded domains, again with the scaling centre inside or on the boundary. Since the radial direction solution is analytical the infinite domain is modelled much more accurately than if the solution was discretised in ξ.
(From [1])
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Formulation Unbounded domains
It is also easy to model unbounded domains, again with the scaling centre inside or on the boundary. Since the radial direction solution is analytical the infinite domain is modelled much more accurately than if the solution was discretised in ξ.
(From [1])
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
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Formulation Substructuring
The scaling centre has to “see” all of the domain. If this is not possible, then the domain can be substructured.
(From [1]) Charles Augarde (DU)
Scaled Boundary Methods: an introduction
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A modal view This is a square bounded domain with a single quadratic element along each boundary (8 nodes, 2D, n = 16). If the number of nodes is increased, so does the number of modes thus increasing the displacement solution space.
Charles Augarde (DU)
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he surface of the half-plane is modelled without discretization usin Examples aints are placed on the side-faces, computation of the stiffness of the Footings ves both on of half thespaces logarithmic modes discussed in Section 5. This mode ressFlexible along footing. lines radiating from theFEcentre of the to be comput If comparing with solutions you footing need to remember ith that the exact solution. Three meshes of increasing fineness arewhile used: a c the FE solution cannot model the infinite domain accurately the SB does it very well
y
p
x a Bounded sub-domain
a
Charles Augarde (DU)
a
Scaled Boundary Methods: an introduction
Scaling centre
Unbounded sub-domain 4 April 2011
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Examples Efficiency
While calculations for the SB method take longer due to the need to solve the eigenproblem, the accuracy per ndof is much better than the FEM for a given problem (because of the semi-analytical solution along ξ) as shown here in results for the flexible footing problem. Footing edge
SB
Charles Augarde (DU)
FE
Scaled Boundary Methods: an introduction
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Examples Use of scaling centres at stress discontinuities
Footing problem again, taking advantage of symmetry with scaling centre placed edge where stress discontinuity occurs Figure at 7. Displacements computed with the fine mesh of the first model for Example 1. y p x a
a
Scaling centre (unbounded sub-domain) Bounded sub-domain
a
Scaling centre (bounded sub-domain)
Unbounded sub-domain
a
Figure 8. Second model for the flexible strip footing of Example 1, taking advantage of the symmetry.
From [4] Copyright # 2002 John Wiley & Sons, Ltd. Charles Augarde (DU)
Numer. Anal. Meth. Geomech. 2002; 2011 26:1031–1057 Scaled Boundary Methods:Int. an J. introduction 4 April 28 / 39
Examples Circular hole in infinite plate
e 8. Scaled boundary element contributions to the error energy norm for the bounded mo
Side-face 1.0
B
Discretised boundary
1.0 Scaling centre
A
Side-face
1.0
Figure 9. Unbounded model representing innite plate in uni-axial stress eld.
From [3]
he exactCharles solution used to provide boundary conditions) in order 4for the nite Augarde (DU) Scaled Boundary Methods: an introduction April 2011 29 / 39ele
Examples Use of scaling centres at stress singularities
Positioning the scaling centre at a crack tip in a LEFM problem provides an accurate means of dealing with the stress singularity there. 2026
S.R. Chidgzey, A.J. Deeks / Engineering Fracture Mechanics 72 (2005) 2019–2036
(7)
h
(8)
(5 )
c Side faces
h
(6)
Scaling centre
(1)
(4) (2)
(3)
b (a)
(b)
Fig. 3. A single edge cracked panel: (a) geometry and (b) scaled boundary finite element model.
From [5] For clarity only the end nodes of each element are illustrated. The dimensions of the panel are h = b/2 = c = 1 and a PoissonÕs ratio of 0.25 is used. Boundary conditions are not applied at this stage, as only the deformation modes corresponding to the eigenfunctions need be identified. Two further meshes obtained by successive congruent subdivision of the original mesh and designated ÔmediumÕ and ÔfineÕ are also used. Charles (DU) Scaled Methods: an introduction 4 April / 39 Table 1Augarde indicates the convergence of Boundary the computed eigenvalues to the expected values. The2011 negative30 sign
Hybrid methods Meshless SBM
One is not restricted to using finite elements in the s direction. A meshless version of the SB method using the MLPG approach has also been developed [6] which leads to greater accuracy. What changes in the derivation? This is a PG approach so that test and shape functions are not identical (as in the Galerkin approach) In terms of virtual work this means {uh (ξ, s)} = N 1 (s) {uh (ξ)} Virtual displacement 2 field {δu(ξ, s)} = N (s) {δu(ξ)}
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
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Hybrid methods Meshless SBM
One is not restricted to using finite elements in the s direction. A meshless version of the SB method using the MLPG approach has also been developed [6] which leads to greater accuracy. What changes in the derivation? This is a PG approach so that test and shape functions are not identical (as in the Galerkin approach) In terms of virtual work this means {uh (ξ, s)} = N 1 (s) {uh (ξ)} Virtual displacement 2 field {δu(ξ, s)} = N (s) {δu(ξ)}
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
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Hybrid methods Meshless SBM
Recall the differential operator for the SB system [L∗ ] = b1 (s) ∂ + 1ξ b2 (s) ∂ ∂ξ ∂s 1 2 where b (s) and b (s) are functions of [L] and a Jacobian linking the Cartesian and SB frames. Now that we have different shape functions for the approximate and virtual displacements we have 11 1 1 B (s) = b (s) N (s) and B 21 (s) = b2 (s) N 1 (s) ,s but 12also 1 2 B (s) = b (s) N (s) and B 22 (s) = b2 (s) N 2 (s) ,s
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Scaled Boundary Methods: an introduction
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Hybrid methods Meshless SBM
Recall the differential operator for the SB system [L∗ ] = b1 (s) ∂ + 1ξ b2 (s) ∂ ∂ξ ∂s 1 2 where b (s) and b (s) are functions of [L] and a Jacobian linking the Cartesian and SB frames. Now that we have different shape functions for the approximate and virtual displacements we have 11 1 1 B (s) = b (s) N (s) and B 21 (s) = b2 (s) N 1 (s) ,s but 12also 1 2 B (s) = b (s) N (s) and B 22 (s) = b2 (s) N 2 (s) ,s
Charles Augarde (DU)
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Hybrid methods Meshless SBM
Running these through the same procedure as we have seen for the Galerkin FE Z SB method we get a new set of these 11 12 T E = B (s) [D] B 11 (s) |J|ds ZS 12 12 T E = B (s) [D] B 21 (s) |J|ds ZS 21 22 T E = B (s) [D] B 11 (s) |J|ds ZS 22 22 T E = B (s) [D] B 21 (s) |J|ds S
and the final conditions to be satisfied, which lead to the eigenproblem, become T {P } = E 11 {uh },ξ + E 12 {uh }
E 11 ξ 2 {u} ,ξξ + E 11 + E 12 − E 21 ξ {u(ξ)} ,ξ − E 22 {u(ξ)} = {0} Charles Augarde (DU)
Scaled Boundary Methods: an introduction
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Hybrid methods Meshless SBM
Running these through the same procedure as we have seen for the Galerkin FE Z SB method we get a new set of these 11 12 T E = B (s) [D] B 11 (s) |J|ds ZS 12 12 T E = B (s) [D] B 21 (s) |J|ds ZS 21 22 T E = B (s) [D] B 11 (s) |J|ds ZS 22 22 T E = B (s) [D] B 21 (s) |J|ds S
and the final conditions to be satisfied, which lead to the eigenproblem, become T {P } = E 11 {uh },ξ + E 12 {uh }
E 11 ξ 2 {u} ,ξξ + E 11 + E 12 − E 21 ξ {u(ξ)} ,ξ − E 22 {u(ξ)} = {0} Charles Augarde (DU)
Scaled Boundary Methods: an introduction
4 April 2011
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Hybrid methods Coupled meshless SBM
SB methods are restricted to linear materials so what about modelling non-linear materials? This can be done by coupling a SB method to a finite element or a standard meshless method. A good example is a for geomechanics geotechnical problem where we do not wish to impose boundaries on the model that do not exist in practice [7,8]. p a Meshless method
+
Scaled Boundary method
Ground surface
Tunnel
∂
A coupled meshless method for geomechanics Footing problems
Scaling centre
MF domain
∂
Meshfree zone Infinite scaled boundary zone
2B y Footing traction MLPG zone
x
Unbounded SB domain
SBM nodes SBM domain
Deeks and Augarde (2007) [CMAME] Charles Augarde (DU)
Scaled Boundary Methods: an introduction
41
4 April 2011
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Hybrid methods Coupled meshless SBM
SB methods are restricted to linear materials so what about modelling non-linear materials? This can be done by coupling a SB method to a finite element or a standard meshless method. A good example is a for geomechanics geotechnical problem where we do not wish to impose boundaries on the model that do not exist in practice [7,8]. p a Meshless method
+
Scaled Boundary method
Ground surface
Tunnel
∂
A coupled meshless method for geomechanics Footing problems
Scaling centre
MF domain
∂
Meshfree zone Infinite scaled boundary zone
2B y Footing traction MLPG zone
x
Unbounded SB domain
SBM nodes SBM domain
Deeks and Augarde (2007) [CMAME] Charles Augarde (DU)
Scaled Boundary Methods: an introduction
41
4 April 2011
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Hybrid methods Coupling to BEM ARTICLE IN PRESS
SB methods can also be coupled to Boundary Element Methods [9] (the topic of the next lecture in the School). This is a model of a centre crack problem from [10]. The SB method is used to capture the stress singularity at the crack tip while the BEM is used elsewhere in the domain due to its efficiency.
G.E. Bird et al. / Engineering Analysis with Boundary Elements ] (]]]]) ]]]–]]]
BEM ΩB
ΓI
h
h
J2
Interface
SBFEM ΩS
crack tip
a
AL b
AR ˆ y) ˆ (x,
J1 b
Fig. 8. (a) The finite plate with through crack and (b) the BE–SBFEM model.
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
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Hybrid methods Coupling to BEM ARTICLE IN PRESS
SB methods can also be coupled to Boundary Element Methods [9] (the topic of the next lecture in the School). This is a model of a centre crack problem from [10]. The SB method is used to capture the stress singularity at the crack tip while the BEM is used elsewhere in the domain due to its efficiency.
G.E. Bird et al. / Engineering Analysis with Boundary Elements ] (]]]]) ]]]–]]]
BEM ΩB
ΓI
h
h
J2
Interface
SBFEM ΩS
crack tip
a
AL b
AR ˆ y) ˆ (x,
J1 b
Fig. 8. (a) The finite plate with through crack and (b) the BE–SBFEM model.
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
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Other aspects Not covered in this talk
I have not shown the method in 3D or for other PDEs, or for dynamics. SB alone can only model linear materials, but coupled to other methods non-linear problems can be tackled. Robust error estimation procedures have been developed based on Zienkiewicz-Zhu type estimators . . . . . . as have h− and p−adaptive procedures.
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Summary +++++++
And to summarize The scaled boundary FE method (in 2D) uses 1D finite elements around a boundary and combines this with an analytical solution in a radial direction. It is often of higher accuracy for a given level of computational resource and can model singularities and infinite domains very well. It requires no fundamental solution.
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Summary +++++++
And to summarize The scaled boundary FE method (in 2D) uses 1D finite elements around a boundary and combines this with an analytical solution in a radial direction. It is often of higher accuracy for a given level of computational resource and can model singularities and infinite domains very well. It requires no fundamental solution.
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
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Summary +++++++
And to summarize The scaled boundary FE method (in 2D) uses 1D finite elements around a boundary and combines this with an analytical solution in a radial direction. It is often of higher accuracy for a given level of computational resource and can model singularities and infinite domains very well. It requires no fundamental solution.
Charles Augarde (DU)
Scaled Boundary Methods: an introduction
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References .......... [1] Deeks, A.J. and Wolf, J.P. 2002. A virtual work derivation of the scaled boundary finite element method, CM 28:489-504. [2] Wolf, J.P. and Song, Ch. 1996. Finite element modelling of unbounded media, John Wiley and Sons, Chichester . [3] Deeks, A.J. and Wolf, J.P. 2002. Stress recovery and error estimation for the scaled boundary finite-element method, IJNME, 54:557-583. [4] Deeks, A.J. and Wolf, J.P. 2002. Semi-analytical elastostatic analysis of unbounded two-dimensional domains, IJNAMG, 26:1031-1057. [5] Chidgzey and Deeks, A.J. 2005. Determination of coefficients of crack tip asymptotic fields using the scaled boundary finite element method, EFM, 13:2019-2036. [6] Deeks, A.J. and Augarde, C.E. 2005. A meshless local Petrov-Galerkin scaled boundary method. CM, 36:159-170.
Charles Augarde (DU)
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References .......... [7] Deeks, A.J., Augarde, C.E. 2007. A hybrid meshless Petrov-Galerkin method for unbounded domains, CMAME, 196, 843-852. [8] Heaney, C.E., Augarde, C.E., Deeks, A.J. 2010. Modelling elasto-plasticity using the hybrid MLPG method. CMES, 56(2)153-178. [9] Chidgzey, S.R., Trevelyan, J., Deeks, A.J. 2008. Coupling of the boundary element method and the scaled boundary finite element method for computations in fracture mechanics, CAS, 86:1198-1203. [10] Bird, G.E., Trevelyan, J., Augarde, C.E. 2010. A coupled BEM Scaled boundary FEM formulation for accurate computations in linear elastic fracture mechanics. EBoundE, 34(6):599-610.
www.dur.ac.uk/charles.augarde
[email protected]
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