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 1 Mixed MaxEnt Method for Plates - ECCOMAS 2012
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 2 Mixed MaxEnt Method for Plates - ECCOMAS 2012
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 3 Mixed MaxEnt Method for Plates - ECCOMAS 2012
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) a b ( θ ; η ) + λα a s ( θ, z 3 ; η, y 3 ) + ( γ, ∇ y 3 − η ) L 2 = f ( y 3 ) (1a) t 2 ¯ ( ∇ z 3 − θ, ψ ) L 2 − t 2 )( γ ; ψ ) L 2 = 0 (1b) λ (1 − α ¯ J.S. Hale 4 Mixed MaxEnt Method for Plates - ECCOMAS 2012
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 5 Mixed MaxEnt Method for Plates - ECCOMAS 2012
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 6 Mixed MaxEnt Method for Plates - ECCOMAS 2012
The Reissner-Mindlin Problem J.S. Hale 7 Mixed MaxEnt Method for Plates - ECCOMAS 2012
The Reissner-Mindlin Problem Displacement Weak Form Find ( z 3 , θ ) ∈ ( V 3 × R ) such that for all ( y 3 , η ) ∈ ( V 3 × R ): � � L ǫ ( θ ) : ǫ ( η ) d Ω + λ ¯ t − 2 ( ∇ z 3 − θ ) · ( ∇ y 3 − η ) d Ω Ω 0 Ω 0 (2) � = gy 3 d Ω Ω 0 or: t − 2 a s ( θ, z 3 ; η, y 3 ) = f ( y 3 ) a b ( θ ; η ) + λ ¯ (3) Locking Problem Whilst this problem is always stable, it is poorly behaved in the thin-plate limit ¯ t → 0 J.S. Hale 8 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Shear Locking 10 − 6 L 2 error in z 3 10 − 7 t = 0 . 002 t = 0 . 02 t = 0 . 2 10 2 10 3 number of degrees of freedom J.S. Hale 9 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Shear Locking The Problem Inability of the basis functions to represent the limiting Kirchhoff mode ∇ z 3 − η = 0 (4) A solution? Move to a mixed weak form J.S. Hale 10 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Mixed Weak Form Treat the shear stresses as an independent variational quantity: t − 2 ( ∇ z 3 − θ ) ∈ S γ = λ ¯ (5) Mixed Weak Form Find ( z 3 , θ, γ ) ∈ ( V 3 × R × S ) such that for all ( y 3 , η, ψ ) ∈ ( V 3 × R × S ): a b ( θ ; η ) + ( γ ; ∇ y 3 − η ) L 2 = f ( y 3 ) (6a) t 2 ¯ ( ∇ z 3 − θ ; ψ ) L 2 − λ ( γ ; ψ ) L 2 = 0 (6b) Stability Problem Whilst this problem is well-posed in the thin-plate limit, ensuring stability is no longer straightforward J.S. Hale 11 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Stabilised Mixed Weak Form Displacement Formulation Mixed Formulation Locking as ¯ Not necessarily stable t → 0 Solution Combine the displacement and mixed formulation to retain the advantageous properties of both J.S. Hale 12 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Stabilised Mixed Weak Form t − 2 that is Split the discrete shear term with a parameter 0 < α < ¯ independent of the plate thickness : a s = α a displacement + (¯ t − 2 − α ) a mixed (7) J.S. Hale 13 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Stabilised Mixed Weak Form Mixed Weak Form Find ( z 3 , θ, γ ) ∈ ( V 3 × R × S ) such that for all ( y 3 , η, ψ ) ∈ ( V 3 × R × S ): a b ( θ ; η ) + ( γ ; ∇ y 3 − η ) L 2 = f ( y 3 ) (8a) ¯ t 2 ( ∇ z 3 − θ ; ψ ) L 2 − λ ( γ ; ψ ) L 2 = 0 (8b) Stabilised Mixed Weak Form (Brezzi and Arnold 1993, Boffi and Lovadina 1997) a b ( θ ; η ) + λα a s ( θ, z 3 ; η, y 3 ) + ( γ, ∇ y 3 − η ) L 2 = f ( y 3 ) (9a) t 2 ¯ ( ∇ z 3 − θ, ψ ) L 2 − t 2 )( γ ; ψ ) L 2 = 0 (9b) λ (1 − α ¯ J.S. Hale 14 Mixed MaxEnt Method for Plates - ECCOMAS 2012
α independence α independence with fixed discretisation h = 1 / 8, α = 32 . 0 10 − 1 eH 1 ( z 3) e H 1 ( z 3 ) 10 − 2 10 − 3 10 − 4 10 − 3 10 − 2 10 − 1 thickness ¯ t J.S. Hale 15 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Eliminating the Stress Unknowns ◮ Find a (cheap) way of eliminating the extra unknowns associated with the shear-stress variables γ h = λ (1 − α ¯ t 2 ) Π h ( ∇ z 3 h − θ h , ψ h ) (10) t 2 ¯ Figure : The Projection Π h represents a softening of the energy associated with the shear term J.S. Hale 16 Mixed MaxEnt Method for Plates - ECCOMAS 2012
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. J.S. Hale 17 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Eliminating the Stress Unknowns J.S. Hale 18 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Eliminating the Stress Unknowns For one component of shear (for simplicity): t 2 ¯ ( z 3 , x − θ 1 , ψ 13 ) L 2 − t 2 )( γ 13 ; ψ 13 ) L 2 = 0 (11) λ (1 − α ¯ 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 Ω a : N � Ω a N a {− φ i φ i , x } d Ω � φ i � � γ 13 a = (12) � z 3 i Ω a N a d Ω i =1 J.S. Hale 19 Mixed MaxEnt Method for Plates - ECCOMAS 2012
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 20 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Convergence surface for e L 2 ( z 3 ) 4 0 . 0 − 0 . 5 3 − 1 . 0 2 − 1 . 5 α ∼ 1 /ρ 2 log 10 ( α ) − 2 . 0 1 − 2 . 5 0 − 3 . 0 − 1 − 3 . 5 − 2 − 4 . 0 2 . 4 2 . 6 2 . 8 3 . 0 3 . 2 3 . 4 3 . 6 log 10 (dim U ) J.S. Hale 21 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Convergence surface for e H 1 ( z 3 ) 4 0 . 0 − 0 . 5 3 − 1 . 0 2 α ∼ 1 /ρ 2 log 10 ( α ) − 1 . 5 1 − 2 . 0 0 − 1 − 2 . 5 − 2 − 3 . 0 2 . 4 2 . 6 2 . 8 3 . 0 3 . 2 3 . 4 3 . 6 log 10 (dim U ) J.S. Hale 22 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Results - Convergence Convergence of deflection in norms with α ∼ O ( h − 2 ) 10 − 1 eL 2 ( z 3) eH 1 ( z 3) 10 − 2 10 − 3 e 10 − 4 10 − 5 10 2 10 3 10 4 dim U J.S. Hale 23 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Results - Surface Plots Figure : Displacement z 3 h of SSSS plate on 12 × 12 node field + ‘bubbles’, t = 10 − 4 , α = 120 J.S. Hale 24 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Results - Surface Plots Figure : Rotation component θ 1 of SSSS plate on 12 × 12 node field + ‘bubbles’, t = 10 − 4 , α = 120 J.S. Hale 25 Mixed MaxEnt Method for Plates - ECCOMAS 2012
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 J.S. Hale 26 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Thanks for listening. J.S. Hale 27 Mixed MaxEnt Method for Plates - ECCOMAS 2012
LBB Stability Conditions Theorem (LBB Stability) The discretised mixed problem is uniquely solvable if there exists two positive constants α h and β h such that: a b ( η h ; η h ) ≥ α h � η h � 2 ∀ η h ∈ K h (13a) R h (( ∇ y 3 h − η h ) , ψ h ) L 2 inf sup ≥ β h (13b) ( � η h � R h + � y 3 h � V 3 h ) � ψ h � S ′ ψ h ∈S h ( η h , y 3 h ) ∈ ( R h ×V 3 h ) h J.S. Hale 28 Mixed MaxEnt Method for Plates - ECCOMAS 2012
LBB Stability Conditions The Problem ◮ To satisfy the second condition 13b make displacement spaces R h × V 3 h ‘rich’ with respect to the shear space S h ◮ If R h × V 3 h is too ‘rich’ then the first condition 13a may fail as K h grows. ◮ Balancing these two competing requirements makes the design of a stable formulation difficult . J.S. Hale 29 Mixed MaxEnt Method for Plates - ECCOMAS 2012
Recommend
More recommend
