Meshless methods for the Reissner-Mindlin plate problem based on - - PowerPoint PPT Presentation

Meshless methods for the Reissner-Mindlin plate problem based on mixed variational forms

• J. S. Hale

31st October 2012 Universidad de Chile

• J. S. Hale

Meshless mixed methods for plates 1

Overview

▶ Meshless numerical methods

▶ Similarities with nite element methods ▶ Differences with nite element methods

▶ Reissner-Mindlin plate problem

▶ Physics of the problem ▶ Scaling ▶ The Kirchhoff limit

▶ Shear-locking

▶ Numerical demonstration in 1D ▶ Why does it happen? ▶ What are the potential solutions?

▶ Mixed variational form

▶ Projection Operator ▶ Stability ▶ Results

• J. S. Hale

Meshless mixed methods for plates 2

Meshless numerical methods

.

What’s the difference with Finite Element Methods?

. . Well, of course, there is no mesh. But really, at least from a mathematical perspective, there is very little difference between a mesh-based and a mesh-less numerical method. .

Theorem (Partition of Unity, Babuška and Melenk 1993)

. .

i i

1

(1)

• J. S. Hale

Meshless mixed methods for plates 3

Meshless numerical methods

.

What’s the difference with Finite Element Methods?

. . Well, of course, there is no mesh. But really, at least from a mathematical perspective, there is very little difference between a mesh-based and a mesh-less numerical method. .

Theorem (Partition of Unity, Babuška and Melenk 1993)

. .

∑

i

ϕi = 1

(1)

• J. S. Hale

Meshless mixed methods for plates 3

Meshless numerical methods

So, meshless methods just use different techniques to construct the Partition of Unity, or basis, to approximate the functions within the domain.

Figure : Domain

• J. S. Hale

Meshless mixed methods for plates 4

Meshless numerical methods

So, meshless methods just use different techniques to construct the Partition of Unity, or basis, to approximate the functions within the domain.

Figure : Seed with nodes

• J. S. Hale

Meshless mixed methods for plates 4

Meshless numerical methods

So, meshless methods just use different techniques to construct the Partition of Unity, or basis, to approximate the functions within the domain.

Figure : Mesh

• J. S. Hale

Meshless mixed methods for plates 4

Meshless numerical methods

So, meshless methods just use different techniques to construct the Partition of Unity, or basis, to approximate the functions within the domain. Reference Mesh

F

Figure : Construct basis

• J. S. Hale

Meshless mixed methods for plates 4

Meshless numerical methods

So, meshless methods just use different techniques to construct the Partition of Unity, or basis, to approximate the functions within the domain.

Figure : Support dened by mesh

• J. S. Hale

Meshless mixed methods for plates 4

Meshless numerical methods

So, meshless methods just use different techniques to construct the Partition of Unity, or basis, to approximate the functions within the domain.

Figure : Meshless; support no longer dened by mesh

• J. S. Hale

Meshless mixed methods for plates 4

Meshless numerical methods

So, meshless methods just use different techniques to construct the Partition of Unity, or basis, to approximate the functions within the domain.

Figure : Give a node a support area

• J. S. Hale

Meshless mixed methods for plates 4

Meshless numerical methods

So, meshless methods just use different techniques to construct the Partition of Unity, or basis, to approximate the functions within the domain.

Figure : Give every node a support area

• J. S. Hale

Meshless mixed methods for plates 4

Construct a meshless PU

Typically by minimisation or convex optimisation process. .

Moving Least-Squares (Shepard, Lancaster and Salkauskas)

. . Quadratic Weighted Least-Squares Minimisation

min

a

1 2

N i 1

wi pTa ui 2

.

Maximum-Entropy (Sukumar, M. Ortiz and Arroyo)

. . Entropy functional maximisation

min

N i 1 i ln i

wi

• J. S. Hale

Meshless mixed methods for plates 5

Construct a meshless PU

Typically by minimisation or convex optimisation process. .

Moving Least-Squares (Shepard, Lancaster and Salkauskas)

. . Quadratic Weighted Least-Squares Minimisation

min

a

1 2

N

∑

i=1

wi[pTa − ui]2

.

Maximum-Entropy (Sukumar, M. Ortiz and Arroyo)

. . Entropy functional maximisation

min

N i 1 i ln i

wi

• J. S. Hale

Meshless mixed methods for plates 5

Construct a meshless PU

Typically by minimisation or convex optimisation process. .

Moving Least-Squares (Shepard, Lancaster and Salkauskas)

. . Quadratic Weighted Least-Squares Minimisation

min

a

1 2

N

∑

i=1

wi[pTa − ui]2

.

Maximum-Entropy (Sukumar, M. Ortiz and Arroyo)

. . Entropy functional maximisation

min

φ N

∑

i=1

ϕi ln (ϕi wi )

• J. S. Hale

Meshless mixed methods for plates 5

FE

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 φ Figure : Finite Element (P1) basis functions on unit interval

• J. S. Hale

Meshless mixed methods for plates 6

MLS

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 φ Figure : Moving Least-Squares basis functions on unit interval

• J. S. Hale

Meshless mixed methods for plates 7

MaxEnt

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 φ Figure : Maximum-Entropy basis functions on unit interval

• J. S. Hale

Meshless mixed methods for plates 8

Summary

.

Question

. . What are the key differences between a mesh-based and a mesh-less PU? Property Mesh-based Mesh-less Space local element + map global Connectivity mesh support Support local, lower-bandwidth local, higher-bandwidth Integration polynomial rational Continuity

C0 easy, C1 hard

up to C Character interpolant approximant Kronecker delta yes sometimes

• J. S. Hale

Meshless mixed methods for plates 9

Summary

.

Question

. . What are the key differences between a mesh-based and a mesh-less PU? Property Mesh-based Mesh-less Space local element + map global Connectivity mesh support Support local, lower-bandwidth local, higher-bandwidth Integration polynomial rational Continuity

C0 easy, C1 hard

up to C∞ Character interpolant approximant Kronecker delta yes sometimes

• J. S. Hale

Meshless mixed methods for plates 9

Solving the problem

Step 1: Begin with the weak (variational) form of your problem .

Poisson Problem (Weak Form)

. . Find u ∈ V such that for all v ∈ V where V ≡ H1

0(Ω):

∫

Ω

∇u · ∇v dx = ∫

Ω

fv dx

Step 2: Construct a suitable Partition of Unity Vh

V

.

Poisson Problem (Discrete Form)

. . Find uh

Vh such that for all v Vh: uh v dx fv dx

• J. S. Hale

Meshless mixed methods for plates 10

Solving the problem

Step 1: Begin with the weak (variational) form of your problem .

Poisson Problem (Weak Form)

. . Find u ∈ V such that for all v ∈ V where V ≡ H1

0(Ω):

∫

Ω

∇u · ∇v dx = ∫

Ω

fv dx

Step 2: Construct a suitable Partition of Unity Vh ⊂ V .

Poisson Problem (Discrete Form)

. . Find uh ∈ Vh such that for all v ∈ Vh:

∫

Ω

∇uh · ∇v dx = ∫

Ω

fv dx

• J. S. Hale

Meshless mixed methods for plates 10

Solving the problem

Problem 1: Typically for a meshless PU we do not have Vh ⊂ V ≡ H1

0(Ω) so

we cannot enforce Dirichlet (essential) boundary conditions in a straightforward manner as with nite elements

0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 φ

• J. S. Hale

Meshless mixed methods for plates 11

Solving the problem

Step 3a: Modify the weak form of the problem to enforce boundary conditions .

Poisson Problem (Discrete Form + Constraint)

. . Find uh

h

Vh Wh such that for all v Vh Wh: uh v dx

hv ds

fv dx uh ds

Problem 2: More unknowns, non positive-denite matrix, possible stability problems

• J. S. Hale

Meshless mixed methods for plates 12

Solving the problem

Step 3a: Modify the weak form of the problem to enforce boundary conditions .

Poisson Problem (Discrete Form + Constraint)

. . Find (uh, λh) ∈ Vh × Wh such that for all (v, γ) ∈ Vh × Wh:

∫

Ω

∇uh · ∇v dx + ∫

Γ

λhv ds = ∫

Ω

fv dx ∫

Γ

uhγ ds = 0

Problem 2: More unknowns, non positive-denite matrix, possible stability problems

• J. S. Hale

Meshless mixed methods for plates 12

Solving the problem

Step 3b: Use Maximum-Entropy basis functions Vh ⊂ V ≡ H1

0(Ω) 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 φ

• J. S. Hale

Meshless mixed methods for plates 13

Solving the problem

Step 4: Substitute in trial and test basis functions .

Poisson Problem (Linear System Form)

. .

Au b

where

Aij

i j dx

bi

i f dx

• J. S. Hale

Meshless mixed methods for plates 14

Solving the problem

Step 4: Substitute in trial and test basis functions .

Poisson Problem (Linear System Form)

. .

Au = b

where

Aij = ∫

Ω

∇ϕi · ∇ϕj dx bi = ∫

Ω

ϕi f dx

• J. S. Hale

Meshless mixed methods for plates 14

Solving the problem

• J. S. Hale

Meshless mixed methods for plates 15

The Reissner-Mindlin Problem

• J. S. Hale

Meshless mixed methods for plates 16

The Reissner-Mindlin Problem

.

Displacement Weak Form

. . Find (z3, θ) ∈ (V3 × R) such that for all (y3, η) ∈ (V3 × R):

∫

Ω0

Lϵ(θ) : ϵ(η) dΩ + λ¯ t−2 ∫

Ω0

(∇z3 − θ) · (∇y3 − η) dΩ = ∫

Ω0

gy3 dΩ

(2)

• r:

ab(θ; η) + λ¯ t−2as(θ, z3; η, y3) = f(y3)

(3) .

Locking Problem

. . Whilst this problem is always stable, it is poorly behaved in the thin-plate limit t

• J. S. Hale

Meshless mixed methods for plates 17

The Reissner-Mindlin Problem

.

Displacement Weak Form

. . Find (z3, θ) ∈ (V3 × R) such that for all (y3, η) ∈ (V3 × R):

∫

Ω0

Lϵ(θ) : ϵ(η) dΩ + λ¯ t−2 ∫

Ω0

(∇z3 − θ) · (∇y3 − η) dΩ = ∫

Ω0

gy3 dΩ

(2)

• r:

ab(θ; η) + λ¯ t−2as(θ, z3; η, y3) = f(y3)

(3) .

Locking Problem

. . Whilst this problem is always stable, it is poorly behaved in the thin-plate limit¯

t → 0

• J. S. Hale

Meshless mixed methods for plates 17

Cantilever Beam Problem

Figure : Cantilever Beam with Point Load

• J. S. Hale

Meshless mixed methods for plates 18

Analytical Solution

Scaling with:

ϵ = 1 L √ EI Gbtκ, ˜ p = L2/EI

.

Kirchhoff Theory (t

0, thin)

. .

z3 x1 L PL3 3EI pL 3

.

Timoshenko Theory (t

0, thin through moderately thick)

. .

z3 x1 L PL3 3EI 1 3EI Gbt L2 pL 3 1 3 2

• J. S. Hale

Meshless mixed methods for plates 19

Analytical Solution

Scaling with:

ϵ = 1 L √ EI Gbtκ, ˜ p = L2/EI

.

Kirchhoff Theory (t = 0, thin)

. .

z3(x1 = L) = PL3 3EI = ˜ pL 3

.

Timoshenko Theory (t

0, thin through moderately thick)

. .

z3 x1 L PL3 3EI 1 3EI Gbt L2 pL 3 1 3 2

• J. S. Hale

Meshless mixed methods for plates 19

Analytical Solution

Scaling with:

ϵ = 1 L √ EI Gbtκ, ˜ p = L2/EI

.

Kirchhoff Theory (t = 0, thin)

. .

z3(x1 = L) = PL3 3EI = ˜ pL 3

.

Timoshenko Theory (t ≥ 0, thin through moderately thick)

. .

z3(x1 = L) = PL3 3EI ( 1 + 3EI GbtκL2 ) = ˜ pL 3 ( 1 + 3ϵ2)

• J. S. Hale

Meshless mixed methods for plates 19

Analytical Solution

10−4 10−3 10−2 10−1 100 ǫ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 z3(x1 = L)

Kirchhoff Timoshenko

• J. S. Hale

Meshless mixed methods for plates 20

Thick ϵ = 1

0.0 0.2 0.4 0.6 0.8 1.0 x1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 z3

Exact MaxEnt N = 10

• J. S. Hale

Meshless mixed methods for plates 21

Thin ϵ = 0.01

0.0 0.2 0.4 0.6 0.8 1.0 x1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 z3

Exact MaxEnt N = 10

• J. S. Hale

Meshless mixed methods for plates 22

Very thin ϵ = 0.001

0.0 0.2 0.4 0.6 0.8 1.0 x1 0.0 0.2 0.4 0.6 0.8 1.0 z3

Exact MaxEnt N = 10

• J. S. Hale

Meshless mixed methods for plates 23

Summary

10−4 10−3 10−2 10−1 100 ǫ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 z3(x1 = L)

Kirchhoff Timoshenko P1 FE MaxEnt

• J. S. Hale

Meshless mixed methods for plates 24

Table : The effect of h-renement on the error z3h(L)/z3(L) at the tip of the cantilever beam using P1 nite elements

N

dofs

ϵ = 1 ϵ = 0.1 ϵ = 0.01 ϵ = 0.001 ϵ = 0.0001 1

4 0.92308 0.10714 0.00120 0.00001 0.00000

10

22 0.99917 0.92308 0.10714 0.00120 0.00001

100

202 0.99999 0.99917 0.92308 0.10714 0.00120

1000

2002 1.00000 0.99999 0.99917 0.92308 0.10714

10000

20002 1.00000 1.00000 0.99999 0.99942 0.92292

.

Conclusion

. . To obtain uniform convergence with respect to using standard methods we must use a huge number of elements

• J. S. Hale

Meshless mixed methods for plates 25

Table : The effect of h-renement on the error z3h(L)/z3(L) at the tip of the cantilever beam using P1 nite elements

N

dofs

ϵ = 1 ϵ = 0.1 ϵ = 0.01 ϵ = 0.001 ϵ = 0.0001 1

4 0.92308 0.10714 0.00120 0.00001 0.00000

10

22 0.99917 0.92308 0.10714 0.00120 0.00001

100

202 0.99999 0.99917 0.92308 0.10714 0.00120

1000

2002 1.00000 0.99999 0.99917 0.92308 0.10714

10000

20002 1.00000 1.00000 0.99999 0.99942 0.92292

.

Conclusion

. . To obtain uniform convergence with respect to ϵ using standard methods we must use a huge number of elements

• J. S. Hale

Meshless mixed methods for plates 25

Shear Locking

.

The Problem

. . Inability of the basis functions to represent the limiting Kirchhoff mode

∇z3 − η = 0

(4) .

A solution?

. . Move to a mixed weak form

• J. S. Hale

Meshless mixed methods for plates 26

Shear Locking

.

The Problem

. . Inability of the basis functions to represent the limiting Kirchhoff mode

∇z3 − η = 0

(4) .

A solution?

. . Move to a mixed weak form

• J. S. Hale

Meshless mixed methods for plates 26

Mixed Weak Form

Treat the shear stresses as an independent variational quantity:

γ = λ¯ t−2(∇z3 − θ) ∈ S

(5) .

Mixed Weak Form

. .

Find (z3, θ, γ) ∈ (V3 × R × S) such that for all (y3, η, ψ) ∈ (V3 × R × S):

ab(θ; η) + (γ; ∇y3 − η)L2 = f(y3)

(6a)

(∇z3 − θ; ψ)L2 − ¯ t2 λ (γ; ψ)L2 = 0

(6b)

.

Stability Problem

. . Whilst this problem is well-posed in the thin-plate limit, ensuring stability is no longer straightforward

• J. S. Hale

Meshless mixed methods for plates 27

Mixed Weak Form

Treat the shear stresses as an independent variational quantity:

γ = λ¯ t−2(∇z3 − θ) ∈ S

(5) .

Mixed Weak Form

. .

Find (z3, θ, γ) ∈ (V3 × R × S) such that for all (y3, η, ψ) ∈ (V3 × R × S):

ab(θ; η) + (γ; ∇y3 − η)L2 = f(y3)

(6a)

(∇z3 − θ; ψ)L2 − ¯ t2 λ (γ; ψ)L2 = 0

(6b)

.

Stability Problem

. . Whilst this problem is well-posed in the thin-plate limit, ensuring stability is no longer straightforward

• J. S. Hale

Meshless mixed methods for plates 27

Stabilised Mixed Weak Form

.

Displacement Formulation

. . Locking as¯

t → 0

.

Mixed Formulation

. . Not necessarily stable .

Solution

. . Combine the displacement and mixed formulation to retain the advantageous properties of both

• J. S. Hale

Meshless mixed methods for plates 28

Stabilised Mixed Weak Form

.

Displacement Formulation

. . Locking as¯

t → 0

.

Mixed Formulation

. . Not necessarily stable .

Solution

. . Combine the displacement and mixed formulation to retain the advantageous properties of both

• J. S. Hale

Meshless mixed methods for plates 28

Stabilised Mixed Weak Form

Split the discrete shear term with a parameter 0 < α < ¯

t−2 that is

independent of the plate thickness:

as = αadisplacement + (¯ t−2 − α)amixed

(7)

• J. S. Hale

Meshless mixed methods for plates 29

Stabilised Mixed Weak Form

.

Mixed Weak Form

. .

Find (z3, θ, γ) ∈ (V3 × R × S) such that for all (y3, η, ψ) ∈ (V3 × R × S):

ab(θ; η) + (γ; ∇y3 − η)L2 = f(y3)

(8a)

(∇z3 − θ; ψ)L2 − ¯ t2 λ (γ; ψ)L2 = 0

(8b)

.

Stabilised Mixed Weak Form (Brezzi and Arnold 1993, Boffi and Lovadina 1997)

. .

ab(θ; η) + λαas(θ, z3; η, y3) + (γ, ∇y3 − η)L2 = f(y3)

(9a)

(∇z3 − θ, ψ)L2 − ¯ t2 λ(1 − α¯ t2)(γ; ψ)L2 = 0

(9b)

• J. S. Hale

Meshless mixed methods for plates 30

Stabilised Mixed Weak Form

.

Mixed Weak Form

. .

Find (z3, θ, γ) ∈ (V3 × R × S) such that for all (y3, η, ψ) ∈ (V3 × R × S):

ab(θ; η) + (γ; ∇y3 − η)L2 = f(y3)

(8a)

(∇z3 − θ; ψ)L2 − ¯ t2 λ (γ; ψ)L2 = 0

(8b)

.

Stabilised Mixed Weak Form (Brezzi and Arnold 1993, Boffi and Lovadina 1997)

. .

ab(θ; η) + λαas(θ, z3; η, y3) + (γ, ∇y3 − η)L2 = f(y3)

(9a)

(∇z3 − θ, ψ)L2 − ¯ t2 λ(1 − α¯ t2)(γ; ψ)L2 = 0

(9b)

• J. S. Hale

Meshless mixed methods for plates 30

Eliminating the Stress Unknowns

▶ Find a (cheap) way of eliminating the extra unknowns associated with

the shear-stress variables

γh = λ(1 − α¯ t2) ¯ t2 Πh(∇z3h − θh, ψh)

(10)

Figure : The Projection Πh represents a softening of the energy associated with the shear term

• J. S. Hale

Meshless mixed methods for plates 31

Eliminating the Stress Unknowns

▶ We use a version of a technique proposed by A. Ortiz, Puso and

Sukumar for the Incompressible-Elasticity/Stokes’ ow problem which they call the “Volume-Averaged Nodal Pressure” technique.

▶ A more general name might be the “Local Patch Projection” technique.

• J. S. Hale

Meshless mixed methods for plates 32

Eliminating the Stress Unknowns

• J. S. Hale

Meshless mixed methods for plates 33

Eliminating the Stress Unknowns

For one component of shear (for simplicity):

(z3,x −θ1, ψ13)L2 − ¯ t2 λ(1 − α¯ t2)(γ13; ψ13)L2 = 0

(11) Substitute in meshfree and FE basis, perform row-sum (mass-lumping) and rearrange to give nodal shear unknown for a node a. Integration is performed over local domain Ωa:

γ13a =

N

∑

i=1

∫

Ωa Na {−ϕi

ϕi,x} dΩ ∫

Ωa Na dΩ

{ϕi z3i }

(12)

• J. S. Hale

Meshless mixed methods for plates 34

Choosing α

The dimensionally consistent choice for α is length−2. In the FE literature typically this paramemeter has been chosen as either h−1 or h−2 where h is the local mesh size. .

Meshless methods

. . A sensible place to start would be

2 where

is the local support size.

• J. S. Hale

Meshless mixed methods for plates 35

Choosing α

The dimensionally consistent choice for α is length−2. In the FE literature typically this paramemeter has been chosen as either h−1 or h−2 where h is the local mesh size. .

Meshless methods

. . A sensible place to start would be ρ−2 where ρ is the local support size.

• J. S. Hale

Meshless mixed methods for plates 35

2.4 2.6 2.8 3.0 3.2 3.4 3.6 log10(dim U) −2 −1 1 2 3 4 log10(α) α ∼ 1/ρ2 Convergence surface for eL2(z3) −4.0 −3.5 −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0

• J. S. Hale

Meshless mixed methods for plates 36

2.4 2.6 2.8 3.0 3.2 3.4 3.6 log10(dim U) −2 −1 1 2 3 4 log10(α) α ∼ 1/ρ2 Convergence surface for eH1(z3) −3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0

• J. S. Hale

Meshless mixed methods for plates 37

Results - Convergence

102 103 104

dim U

10−5 10−4 10−3 10−2 10−1

e

Convergence of deflection in norms with α ∼ O(h−2)

eL2 (z3) eH1 (z3)

• J. S. Hale

Meshless mixed methods for plates 38

Results - α independence

10−4 10−3 10−2 10−1

thickness ¯ t

10−3 10−2 10−1

eH1(z3)

α independence with fixed discretisation h = 1/8, α = 32.0

eH1 (z3)

• J. S. Hale

Meshless mixed methods for plates 39

Results - Surface Plots

Figure : Displacement z3h of SSSS plate on 12 × 12 node eld + ‘bubbles’ ,

t = 10−4, α = 120

• J. S. Hale

Meshless mixed methods for plates 40

Results - Surface Plots

Figure : Rotation component θ1 of SSSS plate on 12 × 12 node eld + ‘bubbles’ ,

t = 10−4, α = 120

• J. S. Hale

Meshless mixed methods for plates 41

Summary

A method:

▶ using (but not limited to) Maximum-Entropy basis functions for the

Reissner-Mindlin plate problem that is free of shear-locking based on a stabilised mixed weak form where secondary stress are eliminated from the system of equations a priori using “Local Patch Projection” technique

• J. S. Hale

Meshless mixed methods for plates 42

Summary

A method:

▶ using (but not limited to) Maximum-Entropy basis functions for the

Reissner-Mindlin plate problem that is free of shear-locking

▶ based on a stabilised mixed weak form

where secondary stress are eliminated from the system of equations a priori using “Local Patch Projection” technique

• J. S. Hale

Meshless mixed methods for plates 42

Summary

A method:

▶ using (but not limited to) Maximum-Entropy basis functions for the

Reissner-Mindlin plate problem that is free of shear-locking

▶ based on a stabilised mixed weak form ▶ where secondary stress are eliminated from the system of equations a

priori using “Local Patch Projection” technique

• J. S. Hale

Meshless mixed methods for plates 42

Summary

A method:

▶ using (but not limited to) Maximum-Entropy basis functions for the

Reissner-Mindlin plate problem that is free of shear-locking

▶ based on a stabilised mixed weak form ▶ where secondary stress are eliminated from the system of equations a

priori using “Local Patch Projection” technique

• J. S. Hale

Meshless mixed methods for plates 42

Acknowledgements:

▶ Dr. Alejandro A. Ortiz Bernardin ▶ FONDECYT Grant Grant #11110389 ▶ EPSRC Doctoral Training Award via the Department of Aeronautics,

Imperial College London

▶ Dr. Pedro M. Baiz

• J. S. Hale

Meshless mixed methods for plates 43