A Maximum-Entropy Meshfree Method for the Reissner-Mindlin Plate Problem based on a Stabilised Mixed Weak Form

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A Maximum-Entropy Meshfree Method for the Reissner-Mindlin Plate Problem based on a

Stabilised Mixed Weak Form

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

27th March 2012

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Introduction

Aim of the research project

Develop a meshless method for the simulation of Reissner-Mindlin plates that is free of shear locking.

Figure: 6th free vibration mode of SSSS plate

J.S. Hale

Mixed MaxEnt Method for Plates - ACME 2012

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Existing Approaches (FE and Meshless)

I Reduced Integration (Many authors)

I Assumed Natural Strains (ANS) (eg. MITC elements, Bathe)

I Enhanced Assumed Strains (EAS) (Hughes, Simo etc.)

I Discrete Shear Gap Method (DSG) (Bletzinger, Bischoff, Ramm)

I Smoothed Conforming Nodal Integration (SCNI) (Wang and Chen)

I Matching Fields Method (Donning and Liu)

Many of these methods have been designed using (or shown to be equivalent to) a mixed weak form.

With respect to the more complex Naghdi shell problem, stabilised mixed weak forms have been shown to be particularly useful (Bathe, Arnold, Lovadina etc.).

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Existing Approaches (FE and Meshless)

J.S. Hale

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Existing Approaches (FE and Meshless)

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Existing Approaches (FE and Meshless)

J.S. Hale

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Existing Approaches (FE and Meshless)

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Existing Approaches (FE and Meshless)

J.S. Hale

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Existing Approaches (FE and Meshless)

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Key Features of the Method

Weak Form

Design is based upon on aStabilised Mixed Weak Form, like many successful approaches in the Finite Element literature.

J.S. Hale

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Key Features of the Method

Basis Functions

Uses (but is not limited to!) Maximum-Entropy Basis Functions which have a weak Kronecker-delta property. On convex node sets boundary conditions can be imposeddirectly.

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Key Features of the Method

Localised Projection Operator

Shear Stresses areeliminatedon the ‘patch’ level using a localised projection operator which leaves a final system of equations in the displacement unknowns only.

J.S. Hale

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The Reissner-Mindlin Problem

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The Reissner-Mindlin Problem

Displacement Weak Form

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

Z

Ω0

L(θ) :(η)dΩ +λ¯t⁻² Z

Ω0

(∇z₃−θ)·(∇y₃−η)dΩ

= Z

Ω0

gy3dΩ

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or:

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

Locking Problem

Whilst this problem is always stable, it is poorly behaved in the thin-plate limit ¯t→0

J.S. Hale

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The Reissner-Mindlin Problem

Displacement Weak Form

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

Z

Ω0

L(θ) :(η)dΩ +λ¯t⁻² Z

Ω0

(∇z₃−θ)·(∇y₃−η)dΩ

= Z

Ω0

gy3dΩ

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or:

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

Whilst this problem is always stable, it is poorly behaved in the thin-plate limit ¯t→0

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Shear Locking

10² 10³

number of degrees of freedom

10⁻⁷ 10⁻⁶

L2 errorinz3

t= 0.002 t= 0.02 t= 0.2

J.S. Hale

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Mixed Weak Form

Treat the shear stresses as anindependentvariational quantity:

γ =λ¯t⁻²(∇z₃−θ)∈ S (3)

Mixed Weak Form

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

a_b(θ;η) + (γ;∇y₃−η)L² =f(y₃) (4a)

(∇z3−θ;ψ)L²−¯t²

λ(γ;ψ)L² = 0 (4b)

Stability Problem

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

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Mixed Weak Form

Treat the shear stresses as anindependentvariational quantity:

γ =λ¯t⁻²(∇z₃−θ)∈ S (3)

Mixed Weak Form

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

a_b(θ;η) + (γ;∇y₃−η)L² =f(y₃) (4a)

(∇z3−θ;ψ)L²−¯t²

λ(γ;ψ)L² = 0 (4b)

Stability Problem

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

J.S. Hale

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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

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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

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Stabilised Mixed Weak Form

Split the discrete shear term with a parameter 0< α <¯t⁻² that is independent of the plate thickness:

a_s =αadisplacement+ (¯t⁻²−α)a^mixed (5)

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Stabilised Mixed Weak Form

Mixed Weak Form

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

a_b(θ;η) + (γ;∇y3−η)_L2 =f(y₃) (6a)

(∇z3−θ;ψ)L²−¯t²

λ(γ;ψ)L² = 0 (6b)

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) (7a) (∇z₃−θ, ψ)_L²− ¯t²

λ(1−α¯t²)(γ;ψ)_L² = 0 (7b)

J.S. Hale

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Stabilised Mixed Weak Form

Mixed Weak Form

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

a_b(θ;η) + (γ;∇y3−η)_L2 =f(y₃) (6a)

(∇z3−θ;ψ)L²−¯t²

λ(γ;ψ)L² = 0 (6b)

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) (7a) (∇z₃−θ, ψ)_L²− ¯t²

λ(1−α¯t²)(γ;ψ)_L² = 0 (7b)

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α independence

10⁻⁴ 10⁻³ 10⁻² 10⁻¹

thickness ¯t 10⁻³

10⁻² 10⁻¹

eH1(z3)

αindependence with fixed discretisationh= 1/8,α= 32.0

eH1 (z3 )

J.S. Hale

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Eliminating the Stress Unknowns

I Find a (cheap) way of eliminating the extra unknowns associated with the shear-stress variables

γ_h= λ(1−α¯t²)

¯t² Π_h(∇z_3h−θ_h, ψ_h) (8) Figure: The Projection Πh represents a softening of the energy associated with the shear term

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Eliminating the Stress Unknowns

I We use a version of a technique proposed by Ortiz, Puso and Sukumar for the Incompressible-Elasticity/Stokes’ flow problem which they call the “Volume-Averaged Nodal Pressure” technique.

I A more general name might be the “Local Patch Projection”

technique.

J.S. Hale

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Eliminating the Stress Unknowns

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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 (9) Substitute in meshfree and FE basis, perform row-sum

(mass-lumping) and rearrange to give nodal shear unknown for a nodea. Integration is performed over local domain Ωa:

γ_13a =

N

X

i=1

R

ΩaN_a{−φ_i φ_i_,x} dΩ R

ΩaN_a dΩ

φi

z_3i

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J.S. Hale

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Results - α sensitivity

10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³ 10⁴ α

10⁻³ 10⁻² 10⁻¹ 10⁰

eL2(θx)

Effect of stabilisationαon convergence, ¯t= 0.001

dim(U) = 363 dim(U) = 1371

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Results - α sensitivity

10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³ 10⁴ α

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

eL2(z3)

Effect of stabilisationαon convergence, ¯t= 0.001

dim(U) = 363 dim(U) = 1371

J.S. Hale

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Results - Convergence

10² 10³ 10⁴

dimU 10⁻⁵

10⁻⁴ 10⁻³ 10⁻² 10⁻¹

e

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

eL2 (z3 ) eH1 (z3 )

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Results - Surface Plots

Figure: Displacement z3hof SSSS plate on 12×12 node field +

‘bubbles’,t= 10⁻⁴,α= 120

J.S. Hale

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Results - Surface Plots

Figure: Rotation componentθ1 of SSSS plate on 12×12 node field +

‘bubbles’,t= 10⁻⁴,α= 120

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Summary

A method:

I using (but not limited to) Maximum-Entropy basis functions for the Reissner-Mindlin plate problem that is free of

shear-locking

I based on a stabilised mixed weak form

I where secondary stress are eliminated from the system of equations a prioriusing “Local Patch Projection” technique Possible future work:

I Investigate the choice of the parameterα. Robust choices exist in the FE literature.

I Examine results using other meshless basis functions (MLS, RPIM)

I Extension to Naghdi Shell model

I Investigate locking-free PUM enriched methods

J.S. Hale

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Summary

A method:

shear-locking

I where secondary stress are eliminated from the system of equations a prioriusing “Local Patch Projection” technique Possible future work:

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Summary

A method:

shear-locking

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

Possible future work:

J.S. Hale

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Summary

A method:

shear-locking

I where secondary stress are eliminated from the system of equationsa priori using “Local Patch Projection” technique Possible future work:

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Summary

A method:

shear-locking

J.S. Hale

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Summary

A method:

shear-locking

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Summary

A method:

shear-locking

J.S. Hale

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Thanks for listening.

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LBB Stability Conditions

Theorem (LBB Stability)

The discretised mixed problem is uniquely solvable if there exists two positive constantsα_h andβ_h such that:

a_b(η_h;η_h)≥α_hkη_hk²_R

h ∀η_h∈ K_h (11a)

ψinfh∈S_h sup

(ηh,y3h)∈(R_h×V_3h)

((∇y_3h−ηh), ψh)_L2

(kηhk_R_h+ky_3hk_V_3h)kψhk_S0 h

≥βh (11b)

J.S. Hale

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LBB Stability Conditions

The Problem

I To satisfy the second condition 11b make displacement spaces R_h× V_3h ‘rich’ with respect to the shear spaceS_h

I IfR_h× V_3h is too ‘rich’ then the first condition 11a may fail as K_h grows.

I Balancing these two competing requirements makes the design of a stable formulation difficult.