Rapid testing of stabilised finite element formulations for the Reissner-Mindlin plate problem using the FEniCS project

Rapid testing of stabilised finite element

formulations for the Reissner-Mindlin plate

problem using the FEniCS project

J. S. Hale*, P. M. Baiz

18th June 2012

Outline

I Main research goal

I Plate theories

I Numerical demonstration of locking using FEniCS

I Mixed formulation

I Stabilisation

I A few results

I Summary

Outline

I Main research goal

I Plate theories

I Numerical demonstration of locking using FEniCS

I Mixed formulation

I Stabilisation

I A few results

I Summary

Outline

I Main research goal

I Plate theories

I Numerical demonstration of locking using FEniCS

I Mixed formulation

I Stabilisation

I A few results

I Summary

Outline

I Main research goal

I Plate theories

I Numerical demonstration of locking using FEniCS

I Mixed formulation

I Stabilisation

I A few results

I Summary

Outline

I Main research goal

I Plate theories

I Numerical demonstration of locking using FEniCS

I Mixed formulation

I Stabilisation

I A few results

I Summary

Outline

I Main research goal

I Plate theories

I Numerical demonstration of locking using FEniCS

I Mixed formulation

I Stabilisation

I A few results

I Summary

Outline

I Main research goal

I Plate theories

I Numerical demonstration of locking using FEniCS

I Mixed formulation

I Stabilisation

I A few results

I Summary

Our Research

A meshless method for the Reissner-Mindlin plate problem that:

I is based on a sound variational principle

I is free from shear-locking

I avoids problems of previous approaches

I can be extended to the more complicated shell problem

“A locking-free meshfree method for the simulation of shear-deformable plates based on a mixed variational formulation”, Accepted in Computer Methods in Applied Mechanics and Engineering

Our Research

A meshless method for the Reissner-Mindlin plate problem that:

I is based on a sound variational principle

I is free from shear-locking

I avoids problems of previous approaches

I can be extended to the more complicated shell problem

“A locking-free meshfree method for the simulation of shear-deformable plates based on a mixed variational formulation”, Accepted in Computer Methods in Applied Mechanics and Engineering

Our Research

A meshless method for the Reissner-Mindlin plate problem that:

I is based on a sound variational principle

I is free from shear-locking

I avoids problems of previous approaches

I can be extended to the more complicated shell problem

“A locking-free meshfree method for the simulation of shear-deformable plates based on a mixed variational formulation”, Accepted in Computer Methods in Applied Mechanics and Engineering

Our Research

A meshless method for the Reissner-Mindlin plate problem that:

I is based on a sound variational principle

I is free from shear-locking

I avoids problems of previous approaches

I can be extended to the more complicated shell problem

“A locking-free meshfree method for the simulation of shear-deformable plates based on a mixed variational formulation”, Accepted in Computer Methods in Applied Mechanics and Engineering

Our Research

A meshless method for the Reissner-Mindlin plate problem that:

I is based on a sound variational principle

I is free from shear-locking

I avoids problems of previous approaches

I can be extended to the more complicated shell problem

“A locking-free meshfree method for the simulation of shear-deformable plates based on a mixed variational formulation”, Accepted in Computer Methods in Applied Mechanics and Engineering

Our Research

A meshless method for the Reissner-Mindlin plate problem that:

I is based on a sound variational principle

I is free from shear-locking

I avoids problems of previous approaches

I can be extended to the more complicated shell problem

“A locking-free meshfree method for the simulation of shear-deformable plates based on a mixed variational formulation”, Accepted in Computer Methods in Applied Mechanics and Engineering

Finite Elements

Of course, there are many successful approaches to the Reissner-Mindlin problem in the Finite Element literature.

How can FE approaches inform the design of a new meshless method?

Unifying themes with FEniCS

I Mixed variational form (robust, general)*

I Stabilisation (bubbles, parameters)*

I Reduction and projection operators to eliminate extra unknowns (‘tricks’ become rigorous)**

*Easy with FEniCS, **Doable, but could be easier

Finite Elements

Of course, there are many successful approaches to the Reissner-Mindlin problem in the Finite Element literature.

How can FE approaches inform the design of a new meshless method?

Unifying themes with FEniCS

I Mixed variational form (robust, general)*

I Stabilisation (bubbles, parameters)*

I Reduction and projection operators to eliminate extra unknowns (‘tricks’ become rigorous)**

*Easy with FEniCS, **Doable, but could be easier

Finite Elements

Of course, there are many successful approaches to the Reissner-Mindlin problem in the Finite Element literature.

How can FE approaches inform the design of a new meshless method?

Unifying themes with FEniCS

I Mixed variational form (robust, general)*

I Stabilisation (bubbles, parameters)*

I Reduction and projection operators to eliminate extra unknowns (‘tricks’ become rigorous)**

*Easy with FEniCS, **Doable, but could be easier

Finite Elements

Of course, there are many successful approaches to the Reissner-Mindlin problem in the Finite Element literature.

How can FE approaches inform the design of a new meshless method?

Unifying themes with FEniCS

I Mixed variational form (robust, general)*

I Stabilisation (bubbles, parameters)*

I Reduction and projection operators to eliminate extra unknowns (‘tricks’ become rigorous)**

*Easy with FEniCS, **Doable, but could be easier

Finite Elements

Of course, there are many successful approaches to the Reissner-Mindlin problem in the Finite Element literature.

How can FE approaches inform the design of a new meshless method?

Unifying themes with FEniCS

I Mixed variational form (robust, general)*

I Stabilisation (bubbles, parameters)*

I Reduction and projection operators to eliminate extra unknowns (‘tricks’ become rigorous)**

*Easy with FEniCS, **Doable, but could be easier

The Reissner-Mindlin Plate Problem

The Kirchhoff Limit

Reissner-Mindlin Kirchhoff

γ = λ¯t ⁻² (∇z ₃ − θ) = 0

The Kirchhoff Limit

Reissner-Mindlin Kirchhoff

γ = λ¯t ⁻² (∇z ₃ − θ) = 0

Reissner-Mindlin Equations

Discrete Displacement Weak Form

Find (z3h, θ_h) ∈ (V3h× R_h) such that for all (y3, η) ∈ (V_3h× R_h):

R

Ω0L(θh) : (η) dΩ + λ¯t⁻²R

Ω0(∇z₃− θh) · (∇y₃− η) dΩ =R

Ω0gy₃dΩ or:

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

Reissner-Mindlin Equations

Discrete Displacement Weak Form

Find (z3h, θ_h) ∈ (V3h× R_h) such that for all (y3, η) ∈ (V_3h× R_h):

R

Ω0L(θh) : (η) dΩ R

Ω0L(θh) : (η) dΩ + λ¯t⁻²R

Ω0(∇z₃− θh) · (∇y₃− η) dΩ =R

Ω0gy₃dΩ or:

a_b(θh; η)

a_b(θh; η) + λ¯t⁻²as(θh, z₃; η, y3) = f (y3) R

Ω0L(θh) : (η) dΩ

a_b(θh; η)

Reissner-Mindlin Equations

Discrete Displacement Weak Form

Find (z3h, θ_h) ∈ (V3h× R_h) such that for all (y3, η) ∈ (V_3h× R_h):

R

Ω0L(θh) : (η) dΩ + λ¯t⁻²R

Ω0(∇z₃− θh) · (∇y₃− η) R

Ω0(∇z₃− θh) · (∇y₃− η) dΩ =R

Ω0gy₃dΩ or:

a_b(θh; η) + λ¯t⁻²aass(θ(θhh, z, z₃₃; η, y; η, y33)) = f (y3) R

Ω0(∇z₃− θh) · (∇y₃− η)

as(θh, z₃; η, y3)

Reissner-Mindlin Equations

Discrete Displacement Weak Form

Find (z3h, θ_h) ∈ (V3h× R_h) such that for all (y3, η) ∈ (V_3h× R_h):

R

Ω0L(θh) : (η) dΩ + λ¯t⁻²R

Ω0(∇z₃− θh) · (∇y₃− η) dΩ =R

Ω0gy₃dΩ R

Ω0gy₃dΩ or:

a_b(θh; η) + λ¯t⁻²as(θh, z₃; η, y3) = f (yf(y33)) R

Ω0gy₃dΩ

f(y3)

Reissner-Mindlin Equations

Discrete Displacement Weak Form

Find (z3h, θ_h) ∈ (V3h× R_h) such that for all (y3, η) ∈ (V_3h× R_h):

R

Ω0L(θh) : (η) dΩ + λ¯t⁻²R

Ω0(∇z₃− θh) · (∇y₃− η) dΩ =R

Ω0gy₃dΩ or:

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

Find (z3h, θh) ∈ (V3h× R_h) such that for all (y3, η) ∈ (V3h× R_h):

...

d e g r e e = 1

V_3 = F u n c t i o n S p a c e ( mesh , " L a g r a n g e ", d e g r e e ) R = V e c t o r F u n c t i o n S p a c e ( mesh , " L a g r a n g e ",

degree , dim=2 )

U = M i x e d F u n c t i o n S p a c e ([V_3 , R]) z_3 , t h e t a = T r i a l F u n c t i o n s ( U ) y_3 , eta = T e s t F u n c t i o n s ( U ) ...

Find (z3h, θh) ∈ (V3h× R_h) such that for all (y3, η) ∈ (V3h× R_h):

...

d e g r e e = 1

V_3 = F u n c t i o n S p a c e ( mesh , " L a g r a n g e ", d e g r e e ) R = V e c t o r F u n c t i o n S p a c e ( mesh , " L a g r a n g e ",

degree , dim=2 )

U = M i x e d F u n c t i o n S p a c e ([V_3 , R]) z_3 , t h e t a = T r i a l F u n c t i o n s ( U ) y_3 , eta = T e s t F u n c t i o n s ( U ) ...

_ij(u) :=¹/2(∇u + (∇u)^T) L(ij) := D((1 − ν) + ν tr I ) ab(θh, η) :=

Z

Ω0

L(θh) : (η) dΩ

...

e = l a m b d a t h e t a: 0 . 5*( g r a d ( t h e t a ) + g r a d ( t h e t a ) . T )

L = l a m b d a e: D*(( 1 - nu )*e + nu*tr ( e )*I d e n t i t y ( 2 ) )

a_b = i n n e r ( L ( e ( t h e t a ) ) , e ( eta ) )*dx ...

_ij(u) :=¹/2(∇u + (∇u)^T) L(ij) := D((1 − ν) + ν tr I ) ab(θh, η) :=

Z

Ω0

L(θh) : (η) dΩ

...

e = l a m b d a t h e t a: 0 . 5*( g r a d ( t h e t a ) + g r a d ( t h e t a ) . T )

L = l a m b d a e: D*(( 1 - nu )*e + nu*tr ( e )*I d e n t i t y ( 2 ) )

a_b = i n n e r ( L ( e ( t h e t a ) ) , e ( eta ) )*dx ...

a_s(θh, z₃; η, y3) = Z

Ω0

(∇z3− θ_h) · (∇y3− η) dΩ

...

a_s = i n n e r ( g r a d ( z_3 ) - theta , g r a d ( y_3 ) - eta )*dx

...

a_s(θh, z₃; η, y3) = Z

Ω0

(∇z3− θ_h) · (∇y3− η) dΩ

...

a_s = i n n e r ( g r a d ( z_3 ) - theta , g r a d ( y_3 ) - eta )*dx

...

a_b(θh; η) + λ¯t⁻²a_s(θh, z_3h; η, y3) = f (y3)

...

a = a_b + l m b d a*t* * -2*a_s f = f o r c e*y_3*dx

u_h = s o l v e( a = = f , bcs= [bc1 , bc2]) z_3h , t h e t a _ h = u_h . s p l i t ()

Done!

a_b(θh; η) + λ¯t⁻²a_s(θh, z_3h; η, y3) = f (y3)

...

a = a_b + l m b d a*t* * -2*a_s f = f o r c e*y_3*dx

u_h = s o l v e( a = = f , bcs= [bc1 , bc2]) z_3h , t h e t a _ h = u_h . s p l i t ()

Done!

10² 10³ dofs

10⁻¹ 10⁰

eL2

t = 0.1 t = 0.01 t = 0.001

Figure: Locking; Fix discretisation, decrease ¯t

Locking

Locking

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

V_b= {(y3, η) ∈ (V3× R) | ∇y3− η= 0}

V_b∩ V_h= {0}

Locking

Locking

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

V_b= {(y3, η) ∈ (V3× R) | ∇y3− η= 0}

V_b∩ V_h= {0}

Locking

Locking

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

V_b= {(y3, η) ∈ (V3× R) | ∇y3− η= 0}

V_b∩ V_h= {0}{{0}0}

for d e g r e e in r a n g e( 0 , 6 ):

V_3 = F u n c t i o n S p a c e ( mesh , " L a g r a n g e ", d e g r e e )

R = V e c t o r F u n c t i o n S p a c e ( mesh , " L a g r a n g e ", degree , dim=2 )

...

10² 10³ 10⁴ dofs

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

eH1(θ)

p = 1 p = 2 p = 3 p = 4 p = 5

Figure: Fix ¯t, increase polynomial order p

10⁰ 10¹ 10² 1/h

10⁻¹ 10⁰

eL2

t = 0.01

Figure: Fix ¯t, refine mesh by decreasing h

Error estimate

ku − u_hk

ku − u_hk ≤ C(Ω0, κ, E , ν)hp

¯t |u|

Conclusion

We can never fully eliminate locking with these approaches. It would be better to remove the dependence on ¯t entirely.

ku − u_hk

Error estimate

ku − u_hk ≤ CC(Ω(Ω00, κ, E , ν), κ, E , ν)hp

¯t |u|

Conclusion

We can never fully eliminate locking with these approaches. It would be better to remove the dependence on ¯t entirely.

C(Ω0, κ, E , ν)

Error estimate

ku − u_hk ≤ C(Ω0, κ, E , ν)hp

¯t¯t |u|

Conclusion

We can never fully eliminate locking with these approaches. It would be better to remove the dependence on ¯t entirely.

¯t

Error estimate

ku − u_hk ≤ C(Ω0, κ, E , ν)hhp

¯t |u|

Conclusion

We can never fully eliminate locking with these approaches. It would be better to remove the dependence on ¯t entirely.

h

Error estimate

ku − u_hk ≤ C(Ω0, κ, E , ν)hpp

¯t |u|

Conclusion

We can never fully eliminate locking with these approaches. It would be better to remove the dependence on ¯t entirely.

p

Error estimate

ku − u_hk ≤ C(Ω0, κ, E , ν)hp

¯t |u|

Conclusion

We can never fully eliminate locking with these approaches. It would be better to remove the dependence on ¯t entirely.

Mixed Form

Treat the shear stress as an independent variational quantity:

γh= λ¯t⁻²(∇z3h− θ_h) ∈ Sh

Discrete Mixed Weak Form

Find (z3h, θh, γh) ∈ (V3h, Rh, Sh) such that for all (y3h, η, ψ) ∈ (V_3h, R_h, S_h):

ab(θh; η) + (γh; ∇y3− η)_L² = f (y3) (∇z3h− θ_h; ψ)_L²−¯t²

λ(γh; ψ)_L² = 0

Mixed Form

Treat the shear stress as an independent variational quantity:

γh= λ¯t⁻²(∇z3h− θ_h) ∈ Sh

Discrete Mixed Weak Form

Find (z3h, θh, γh) ∈ (V3h, Rh, Sh) such that for all (y3h, η, ψ) ∈ (V_3h, R_h, S_h):

ab(θh; η) + (γh; ∇y3− η)_L² = f (y3) (∇z3h− θ_h; ψ)_L²−¯t¯t²²

λ(γh; ψ)_L² = 0

¯t²

J. S. Hale

Stability

Theorem (Brezzi 1974)

The classical saddle point problem (¯t = 0) is stable, if and only if, the following conditions hold:

1. (Z-Ellipticity of a)(Z-Ellipticity of a) There exists a constantα ≥ 0 such that:

a(v, v ) ≥ αkv k²_X ∀v ∈ Z where Z is the kernel of the bilinear form b:

Z := {v ∈ X | b(v , q) = 0 ∀q ∈ M}

2. (inf-sup condition on b) The bilinear form b satisfies an inf-sup condition:

inf

q∈M

sup

v∈X

b(v, q)

kv k_Xkqk_M =β > 0 ...

(Z-Ellipticity of a)

Stability

Theorem (Brezzi 1974)

The classical saddle point problem (¯t = 0) is stable, if and only if, the following conditions hold:

1. (Z-Ellipticity of a) There exists a constantα ≥ 0 such that:

a(v, v ) ≥ αkv k²_X ∀v ∈ Z where Z is the kernel of the bilinear form b:

Z := {v ∈ X | b(v , q) = 0 ∀q ∈ M}

2. (inf-sup condition on b)(inf-sup condition on b) The bilinear form b satisfies an inf-sup condition:

inf

q∈M

sup

v∈X

b(v, q)

kv k_Xkqk_M =β > 0 ...

(inf-sup condition on b)

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

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

a_s = αadisplacement+ (¯t⁻²− α)a^mixed

Displacement

Mixed

Stabilised Mixed Weak Form

Discrete Mixed Weak Form

a_b(θh; η) + (γh; ∇y3− η)_L² = f (y3) (∇z3h− θ_h; ψ)_L²−¯t²

λ(γh; ψ)_L² = 0

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

a_b(θ; η) + λαas(θ, z3; η, y3) + (γ, ∇y3− η)_L2 = f (y3) (∇z3− θ, ψ)_L²−_λ(1^¯t2

−α¯t²)(γ; ψ)L² = 0

Stabilised Mixed Weak Form

Discrete Mixed Weak Form

a_b(θh; η) + (γh; ∇y3− η)_L² = f (y3) (∇z3h− θ_h; ψ)_L²−¯t²

λ(γh; ψ)_L² = 0

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

a_b(θ; η) + λαaλαas_s(θ, z(θ, z33; η, y; η, y33)) + (γ, ∇y3− η)_L2 = f (y3) (∇z3− θ, ψ)_L²−_λ(1^¯t2

−α¯t²)

¯t²

λ(1−α¯t²)(γ; ψ)L² = 0

¯t²

λαa_s(θ, z3; η, y3)

¯t² λ(1−α¯t²)

J. S. Hale

Rapid Testing with FEniCS - FEniCS@Imperial Day 22

Stability revisited

Theorem (Brezzi 1974)

The classical saddle point problem (¯t = 0) is stable, if and only if, the following conditions hold:

1. (Z-Ellipticity of a)(Z-Ellipticity of a) There exists a constantα ≥ 0 such that:

a(v, v ) ≥ αkv k²_X ∀v ∈ Z where Z is the kernel of the bilinear form b:

Z := {v ∈ X | b(v , q) = 0 ∀q ∈ M}

2. (inf-sup condition on b) The bilinear form b satisfies an inf-sup condition:

inf

q∈M

sup

v∈X

b(v, q)

kv k_Xkqk_M =β > 0 ...

(Z-Ellipticity of a)

Stability revisited

Theorem (Brezzi 1974)

The classical saddle point problem (¯t = 0) is stable, if and only if, the following conditions hold:

1. (Z-Ellipticity of a) There exists a constantα ≥ 0 such that:

a(v, v ) ≥ αkv k²_X ∀v ∈ Z where Z is the kernel of the bilinear form b:

Z := {v ∈ X | b(v , q) = 0 ∀q ∈ M}

2. (inf-sup condition on b)(inf-sup condition on b) The bilinear form b satisfies an inf-sup condition:

inf

q∈M

sup

v∈X

b(v, q)

kv k_Xkqk_M =β > 0 ...

(inf-sup condition on b)

...

S = V e c t o r F u n c t i o n S p a c e ( mesh , " DG ", 0 , dim=2 ) ...

g a m m a = T r i a l F u n c t i o n ( S ) psi = T e s t F u n c t i o n ( S ) ...

a = a_b + a l p h a*l m b d a*a_s + i n n e r ( gamma , g r a d ( y_3 ) - eta ) + i n n e r ( g r a d ( z_3 ) - theta , psi ) - t* *2/( l m b d a*( 1 . 0 - a l p h a*t* *2 ) )*i n n e r ( gamma , psi ) ...

MINI2 MINI1 TRIA0220 TRIA1B20

10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ α

10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹ 10²

eH1(z3h)

Convergence in z3h for varying α with t = 10⁻³, h = 1/8

TRIA1B20 TRIA0220 MINI2 MINI1

Figure: Various elements, Fix h, vary α.

10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³ 10⁴ 10⁵ α

10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹ 10²

e

O(h⁻¹K) O(h⁻²K)

eL²(θ) eH¹(θ) eL²(z3) eH¹(z3)

Figure: TRIA0220 element. Fix h, vary α.

h = C e l l S i z e ( m e s h )

a l p h a = h* *(-2 . 0 )*c o n s t a n t

10⁰ 10¹ 10² 1/h

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

eH1(z3h)

TRIA0220. Convergence of z3h in H¹ norm Varying α recipes and t = 10⁻³

α = 1.0 α = ₅₍₁^h−1_−ν)

α = _5(1−ν)^h−2 α = 100 1/h 1/h² 1/h³

Figure: Trying different α recipes; convergence can be improved (Lovadina)

Conclusions

I The FEniCS project allows for rapid testing of different Finite Element strategies.

I ∼400 lines of code; Displacement, Mixed, Projections, Errors, Command Line Interface, Output Results etc.

I The stabilisation parameter α should be chosen based on some local discretisation dependent length measure.

I Convergence rates can even be improved by a ‘good’ choice of α

I Based on these results, we have designed a novel meshfree method based on a stabilised weak form with a Local Patch Projection technique to eliminate the shear-stress unknowns a priori

Conclusions

I The FEniCS project allows for rapid testing of different Finite Element strategies.

I ∼400 lines of code; Displacement, Mixed, Projections, Errors, Command Line Interface, Output Results etc.

I The stabilisation parameter α should be chosen based on some local discretisation dependent length measure.

I Convergence rates can even be improved by a ‘good’ choice of α

I Based on these results, we have designed a novel meshfree method based on a stabilised weak form with a Local Patch Projection technique to eliminate the shear-stress unknowns a priori

Conclusions

I The FEniCS project allows for rapid testing of different Finite Element strategies.

I ∼400 lines of code; Displacement, Mixed, Projections, Errors, Command Line Interface, Output Results etc.

I The stabilisation parameter α should be chosen based on some local discretisation dependent length measure.

I Convergence rates can even be improved by a ‘good’ choice of α

I Based on these results, we have designed a novel meshfree method based on a stabilised weak form with a Local Patch Projection technique to eliminate the shear-stress unknowns a priori

Conclusions

I The FEniCS project allows for rapid testing of different Finite Element strategies.

I ∼400 lines of code; Displacement, Mixed, Projections, Errors, Command Line Interface, Output Results etc.

I The stabilisation parameter α should be chosen based on some local discretisation dependent length measure.

I Convergence rates can even be improved by a ‘good’ choice of α

I Based on these results, we have designed a novel meshfree method based on a stabilised weak form with a Local Patch Projection technique to eliminate the shear-stress unknowns a priori

Conclusions

I The FEniCS project allows for rapid testing of different Finite Element strategies.

I ∼400 lines of code; Displacement, Mixed, Projections, Errors, Command Line Interface, Output Results etc.

I The stabilisation parameter α should be chosen based on some local discretisation dependent length measure.

I Convergence rates can even be improved by a ‘good’ choice of α

I Based on these results, we have designed a novel meshfree method based on a stabilised weak form with a Local Patch Projection technique to eliminate the shear-stress unknowns a priori

Thanks for listening.