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 J. S. Hale Meshless mixed methods for plates 2 ▶ 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
Meshless numerical methods . Meshless mixed methods for plates J. S. Hale (1) 1 i i . Theorem (Partition of Unity, Babuška and Melenk 1993) . . mesh-less numerical method. perspective, there is very little difference between a mesh-based and a Well, of course, there is no mesh. But really, at least from a mathematical . . What’s the difference with Finite Element Methods? 3
Meshless numerical methods . Meshless mixed methods for plates J. S. Hale (1) i . . Theorem (Partition of Unity, Babuška and Melenk 1993) . mesh-less numerical method. perspective, there is very little difference between a mesh-based and a Well, of course, there is no mesh. But really, at least from a mathematical . . What’s the difference with Finite Element Methods? 3 ∑ ϕ i = 1
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 de�ned 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 de�ned 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
u i 2 w i p T a i ln Maximum-Entropy (Sukumar, M. Ortiz and Arroyo) Meshless mixed methods for plates J. S. Hale w i i 1 i N min Entropy functional maximisation . . Construct a meshless PU . Typically by minimisation or convex optimisation process. 1 i N 2 1 a min Quadratic Weighted Least-Squares Minimisation . . Moving Least-Squares (Shepard, Lancaster and Salkauskas) . 5
i ln Construct a meshless PU Maximum-Entropy (Sukumar, M. Ortiz and Arroyo) Meshless mixed methods for plates J. S. Hale w i i 1 i N min Entropy functional maximisation . . . Typically by minimisation or convex optimisation process. N 2 1 a min Quadratic Weighted Least-Squares Minimisation . . Moving Least-Squares (Shepard, Lancaster and Salkauskas) . 5 ∑ w i [ p T a − u i ] 2 i = 1
Construct a meshless PU . Meshless mixed methods for plates J. S. Hale w i N min Entropy functional maximisation . . Typically by minimisation or convex optimisation process. Maximum-Entropy (Sukumar, M. Ortiz and Arroyo) 5 min . Moving Least-Squares (Shepard, Lancaster and Salkauskas) . Quadratic Weighted Least-Squares Minimisation . a 1 2 N ∑ w i [ p T a − u i ] 2 i = 1 ( ϕ i ) ∑ ϕ i ln φ i = 1
FE Figure : Finite Element ( P 1 ) basis functions on unit interval Meshless mixed methods for plates J. S. Hale 6 1 . 0 0 . 8 0 . 6 φ 0 . 4 0 . 2 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 x
MLS Figure : Moving Least-Squares basis functions on unit interval Meshless mixed methods for plates J. S. Hale 7 1 . 0 0 . 8 0 . 6 φ 0 . 4 0 . 2 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 x
MaxEnt Figure : Maximum-Entropy basis functions on unit interval Meshless mixed methods for plates J. S. Hale 8 1 . 0 0 . 8 0 . 6 φ 0 . 4 0 . 2 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 x
C 0 easy, C 1 hard Summary local, higher-bandwidth Meshless mixed methods for plates J. S. Hale sometimes yes Kronecker delta approximant interpolant Character up to C Continuity rational polynomial Integration local, lower-bandwidth . Support support mesh Connectivity global local element + map Space Mesh-less Mesh-based Property What are the key differences between a mesh-based and a mesh-less PU? . . Question 9
Summary . Meshless mixed methods for plates J. S. Hale sometimes yes Kronecker delta approximant interpolant Character Continuity rational polynomial Integration local, higher-bandwidth local, lower-bandwidth Support Mesh-less Question . . What are the key differences between a mesh-based and a mesh-less PU? Property Mesh-based Space support local element + map global Connectivity mesh 9 up to C ∞ C 0 easy, C 1 hard
V h such that for all v Solving the problem V Meshless mixed methods for plates J. S. Hale fv dx v dx u h V h : Find u h . . Poisson Problem (Discrete Form) . Step 2: Construct a suitable Partition of Unity V h Step 1: Begin with the weak (variational) form of your problem fv dx . . Poisson Problem (Weak Form) . 10 Find u ∈ V such that for all v ∈ V where V ≡ H 1 0 (Ω) : ∫ ∫ ∇ u · ∇ v dx = Ω Ω
Solving the problem fv dx Meshless mixed methods for plates J. S. Hale fv dx . . Poisson Problem (Discrete Form) . Step 1: Begin with the weak (variational) form of your problem 10 . . . Poisson Problem (Weak Form) Find u ∈ V such that for all v ∈ V where V ≡ H 1 0 (Ω) : ∫ ∫ ∇ u · ∇ v dx = Ω Ω Step 2: Construct a suitable Partition of Unity V h ⊂ V Find u h ∈ V h such that for all v ∈ V h : ∫ ∫ ∇ u h · ∇ v dx = Ω Ω
Solving the problem J. S. Hale Meshless mixed methods for plates 11 straightforward manner as with �nite elements we cannot enforce Dirichlet (essential) boundary conditions in a Problem 1: Typically for a meshless PU we do not have V h ⊂ V ≡ H 1 0 (Ω) so 1 . 0 0 . 8 0 . 6 φ 0 . 4 0 . 2 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 x
W h such that for all v Solving the problem v dx Meshless mixed methods for plates J. S. Hale problems Problem 2: More unknowns, non positive-de�nite matrix, possible stability 0 ds u h fv dx h v ds u h Step 3a: Modify the weak form of the problem to enforce boundary conditions W h : V h V h h Find u h . . Poisson Problem (Discrete Form + Constraint) . 12
Solving the problem Step 3a: Modify the weak form of the problem to enforce boundary conditions Meshless mixed methods for plates J. S. Hale problems Problem 2: More unknowns, non positive-de�nite matrix, possible stability fv dx 12 . . Poisson Problem (Discrete Form + Constraint) . Find ( u h , λ h ) ∈ V h × W h such that for all ( v , γ ) ∈ V h × W h : ∫ ∫ ∫ ∇ u h · ∇ v dx + λ h v ds = Ω Γ Ω ∫ u h γ ds = 0 Γ
Solving the problem J. S. Hale Meshless mixed methods for plates 13 Step 3b: Use Maximum-Entropy basis functions V h ⊂ V ≡ H 1 0 (Ω) 1 . 0 0 . 8 0 . 6 φ 0 . 4 0 . 2 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 x
j dx i f dx Solving the problem A ij Meshless mixed methods for plates J. S. Hale b i i where Step 4: Substitute in trial and test basis functions b Au . . Poisson Problem (Linear System Form) . 14
Solving the problem where Meshless mixed methods for plates J. S. Hale Step 4: Substitute in trial and test basis functions 14 . . Poisson Problem (Linear System Form) . Au = b ∫ A ij = ∇ ϕ i · ∇ ϕ j dx Ω ∫ b i = ϕ i f dx Ω
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
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