A meshless method for the Reissner-Mindlin plate equations based on - - PowerPoint PPT Presentation

A meshless method for the Reissner-Mindlin plate equations based on a stabilized mixed weak form using maximum-entropy basis functions

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

11th September 2012

J.S. Hale Mixed MaxEnt Method for Plates - ECCOMAS 2012 1

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

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Existing Approaches (FE and Meshless)

◮ Reduced Integration (Many authors) ◮ Assumed Natural Strains (ANS) (eg. MITC elements, Bathe) ◮ Enhanced Assumed Strains (EAS) (Hughes, Simo etc.) ◮ Discrete Shear Gap Method (DSG) (Bletzinger, Bischoff, Ramm) ◮ Smoothed Conforming Nodal Integration (SCNI) (Wang and Chen) ◮ Matching Fields Method (Donning and Liu) ◮ Direct Application of Mixed Methods (Hale and Baiz)

The Connection

Many of these methods are based on, or have been shown to be equivalent to, mixed variational methods.

J.S. Hale Mixed MaxEnt Method for Plates - ECCOMAS 2012 3

Key Features of the Method

Weak Form

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

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

J.S. Hale Mixed MaxEnt Method for Plates - ECCOMAS 2012 4

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 imposed directly.

J.S. Hale Mixed MaxEnt Method for Plates - ECCOMAS 2012 5

Key Features of the Method

Localised Projection Operator

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

J.S. Hale Mixed MaxEnt Method for Plates - ECCOMAS 2012 6

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):

• Ω0

Lǫ(θ) : ǫ(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 Mixed MaxEnt Method for Plates - ECCOMAS 2012 8

Shear Locking

102 103

number of degrees of freedom

10−7 10−6

L2 error in z3

t = 0.002 t = 0.02 t = 0.2

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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

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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

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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

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)

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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): 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)

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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 Mixed MaxEnt Method for Plates - ECCOMAS 2012 15

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

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Eliminating the Stress Unknowns

◮ 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.

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

technique.

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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 (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)

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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

• r 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.

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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

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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

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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 Mixed MaxEnt Method for Plates - ECCOMAS 2012 23

Results - Surface Plots

Figure : Displacement z3h of SSSS plate on 12 × 12 node field + ‘bubbles’, t = 10−4, α = 120

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Results - Surface Plots

Figure : Rotation component θ1 of SSSS plate on 12 × 12 node field + ‘bubbles’, t = 10−4, α = 120

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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 Possible future work:

◮ Extension to Naghdi Shell model ◮ Investigate locking-free PUM enriched methods

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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 constants αh and βh such that: ab(ηh; ηh) ≥ αhηh2

Rh

∀ηh ∈ Kh (13a) inf

ψh∈Sh

sup

(ηh,y3h)∈(Rh×V3h)

((∇y3h − ηh), ψh)L2 (ηhRh + y3hV3h)ψhS′

h

≥ βh (13b)

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LBB Stability Conditions

The Problem

◮ To satisfy the second condition 13b make displacement spaces

Rh × V3h ‘rich’ with respect to the shear space Sh

◮ If Rh × V3h is too ‘rich’ then the first condition 13a may fail

as Kh grows.

◮ Balancing these two competing requirements makes the

design of a stable formulation difficult.

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