Applications in Solid Mechanics Fifth deal.II Users and Developers Workshop Texas A&M University August 4, 2015 A McBride BD Reddy S Bartle, A de Villiers, B Grieshaber, M Hamed, J-P Pelteret, T Povall, N Richardson Centre for Research in Computational and Applied Mechanics, University of Cape Town, South Africa
DEAL . II AND C ERECAM Centre for Research in Computational and Applied Mechanics: ๏ multidisciplinary research centre with 12 members from various faculties and departments ๏ postgraduate eduction in mechanics is a key objective ๏ 6 courses in FE, continuum mechanics, C++, deal.II (computational FEA) ๏ MSc and PhD supervision ๏ unique facility in (South) Africa deal.II and C ERECAM ๏ decision made in 2008 to use deal.II as base for code development ๏ SIAM news article on deal.II winning the Wilkinson Prize in 2007 ๏ lacked the critical mass to develop own in-house code fmuid mechanics. Areas of activity in solid ๏ even if we had the mass, we would be small number of isolated individuals reinventing the wheel in a country far away… ๏ 5 MSc projects and 8 PhD projects used deal.II in C ERECAM to date
O VERVIEW OF P RESENTATION ๏ Overview of several of the research projects using deal.II 1. IP methods for problems in elasticity 2. Thin-shells with applications in biomechanics 3. Patient-specific FSI for vascular access in haemodialysis patients 4. Surface elasticity 5. Single crystal plasticity 6. Homogenisation of material layers 7. Friction-stir welding ๏ What would I like to see (or be encouraged to implement) in deal.II
IP M ETHODS FOR P ROBLEMS IN E LASTICITY OR W HY IT ’ S GOOD NOT TO HAVE TETRAHEDRAL ELEMENTS IN DEAL . II ๏ Locking: ๏ The poor behaviour of low-order conforming finite element approximations for problems of near-incompressible or incompressible elasticity � div σ ( u ) = f ๏ Some remedies: σ ( u ) := 2 µ ε ( u ) + λ tr ε ( u ) 1 = 2 µ ε ( u ) + λ ( r · u ) 1 , ๏ selective reduced integration (SRI) E ν E λ = µ = (1 + ν ) (1 � 2 ν ) , 2 (1 + ν ) As ν ! 1 o λ ! 1 . 2 ๏ mixed DG methods for meshes of low-order quadrilateral or hexahedral cells ๏ primal DG methods for triangular elements…convergence analysis similarly restrictive ๏ Context: ๏ we assumed we could simply use a primal DG formulation in deal.II, circumvent locking and move on to the actual problem… Contrary to initial expectations, numerical experiments with the IP methods show that bilinear quadrilateral elements, while performing very well for highly com- pressible materials, perform poorly for the nearly incompressible case, unlike their ❖ Grieshaber (2013) PhD triangular counterparts. Figures 5.2 to 5.4 show sets of results that will be discussed ๏ our remedy: SRI on edge terms for IP methods ❖ G RIESHABER , M C B, R EDDY ( ACCEPTED ), SINUM
DG F ORMULATION ∈ a UI h ( u h , v ) = l UI h ( v ) Z X a UI h ( u , v ) = σ ( u ) : ε ( v ) dx Ω e Ω e ∈ T h ngp Z X X X + θ 2 µ b b u c c : { { ε ( v ) } } ds + θ λ [ b b u c c : { { r · v1 } } ] | p i w i E E ∈ Γ iD E ∈ Γ iD i =1 ngp Z u = g on Γ D , X X X � 2 µ { { ε ( u ) } } : b b v c c ds � λ [ { { r · u1 } } : b b v c c ] | p i w i σ ( u ) n = h on Γ N . E E ∈ Γ iD E ∈ Γ iD i =1 ngp 1 Z 1 X X X + k µ µ b b u c c : b b v c c ds + k λ λ [[[ u ]][[ v ]]] | p i w i , h E h E E E ∈ Γ iD E ∈ Γ iD i =1 Z Z X X l UI h ( v ) = f · v dx + θ 2 µ ( g ⌦ n ) : ε ( v ) ds standard formulation: Ω e E Ω e ∈ T h E ∈ Γ D to ngp = 2) ngp NIPG: X X + θ λ [( g ⌦ n ) : ( r · v 1 )] | p i w i θ = 1 E ∈ Γ D i =1 SIPG: θ = − 1 1 Z X + k µ µ ( g ⌦ n ) : ( v ⌦ n ) ds IIPG: h E θ = 0 E E ∈ Γ D ngp 1 Z X X X + k λ λ [( g · n ) ( v · n )] | p i w i + h · v ds. h E E E ∈ Γ D i =1 E ∈ Γ N ❖ G RIESHABER , M C B, R EDDY ( ACCEPTED ), SINUM
C OOK ’ S M EMBRANE P ROBLEM locking-free response for triangular elements (a) NIPG (b) IIPG (c) SIPG ). ν = 0 . 49995. locking for quadrilateral element , E = 250. (a) NIPG (b) IIPG (c) SIPG
C OOK ’ S M EMBRANE P ROBLEM A locking-free response for quadrilateral elements using modified formulation (a) Modified SIPG (b) Modified IIPG neps SG SG Q2 NIPG — UI IIPG — UI SIPG — UI Skewed mesh 4 1.3855 5.99901 2.07077 4.67802 2.08011 4.6961 2.21427 4.72803 8 1.95749 7.06822 2.12496 6.13406 2.08011 4.6961 2.21427 4.72803 16 2.01279 7.49608 2.24957 7.14275 2.15449 7.14171 2.28465 7.14403 32 2.19539 7.64827 2.633127 7.54839 2.38855 7.54625 2.50211 7.54509 Unstructured mesh 32 3.054 7.66333 5.44538 7.61069 4.42923 7.60888 4.5467 7.60599 vertical deflection of point A
3 D C UBE S UBJECT TO B ODY F ORCE locking-free response for locking for quadrilateral element quadrilateral elements using modified formulation convergence deteriorates upon refinement (a) Exact solution (b) Modified NIPG (a) Exact solution (b) NIPG (c) IIPG (d) SIPG (c) Modified IIPG (d) Modified SIPG ❖ G RIESHABER , M C B, R EDDY ( IN REVIEW )
A PPLICATIONS OF T HIN S HELL T HEORY IN B IOMECHANICS OR H OW TO TRICK DEAL .II TO HANDLE LOWER - DIMENSIONAL MANIFOLDS PRIOR TO CODIMENSION ONE AND MANIFOLDS ๏ Treat the 3D body as a 2D surface with a director at each point ๏ Formulate problem in terms of quantities averaged through the thickness (see S IMO & F OX ) ๏ SRI through the thickness to prevent membrane locking ๏ account for transverse isotropy and incompressibility artificial heart valves aortic clamping ❖ Bartle (2009) MSc
P ATIENT - SPECIFIC FSI FOR VASCULAR ACCESS IN HAEMODIALYSIS PATIENTS ๏ Resulting flow rates up to 30 times physiological norms ๏ Flow in vein significantly altered ๏ Repeated needling leads to pseudo aneurysms Objective: ๏ Develop a patient specific FSI model of arteriovenous vascular access configurations, verified by in vivo MRI data de Villiers (current) PhD ❖
FSI M ODEL Transverse isotropy Navier-Stokes in ALE setting Balance linear momentum (solid) ❖ W ICK (2011) F LUID - STRUCTURE INTERACTIONS USING Harmonic mesh motion DIFFERENT MESH MOTION TECHNIQUES . C OMPUTERS & S TRUCTURES ❖ W ICK (2013). S OLVING MONOLITHIC FLUID - STRUCTURE INTERACTION PROBLEMS IN A RBITRARY L AGRANGIAN E ULERIAN CO - ORDINATES WITH THE DEAL .II LIBRARY . A RCHIVE OF N UMERICAL S OFTWARE
V ALIDATION AND E XTENSIONS Turek flag Incompressible finite elasticity with transverse isotropy ๏ Extended model to 3D ๏ Parallel direct solver (SuperLU) ๏ Windkessel outlet boundary condition for physiologically meaningful flow split
C URRENT S TATE : CFD M ODEL simpleware + ANSA
S URFACE E LASTICITY ๏ Surfaces behave differently from the bulk ๏ broken bonds on surface, coatings, oxidation, … b d A surface energy Ψ ๏ Are surface effects significant? ∝ bulk energy Ψ d V ๏ Objective: capture surface effects within a continuum model ๏ accounting for surface using surface elasticity theory of G URTIN & M URDOCH (1975) ๏ solid and fluid-like surfaces ๏ fully nonlinear theory ๏ to provide details of numerical implementation ๏ generally restricted to linear theory surface tension H 2 0 carbon nanotubes ❖ J AVILI , M C B, S TEINMANN & R EDDY (2014), C OMP M ECH ❖ S URFACE E LASTICITY ON bitbucket.org ❖ Davydov, Javili, Steinmann, McB (2013), in Surface Effects in Solid Mechanics
K INEMATICS OF S URFACE E LASTICITY erse f := ∂ X / ∂ x y F := ∂ x / ∂ X b from di ff erential ations are denote y b f := ∂ c re b x / ∂ c X / ∂ b x F := ∂ b X ๏ Surface is coherent: ϕ = ϕ | ∂ B 0 b ๏ Surface deformation gradient is rank deficient ๏ inverse defined via relations: b F · b f = b f · b F = b b I =: I − N ⊗ N and i =: i − n ⊗ n .
G OVERNING E QUATIONS Strong form: balance of linear momentum balance of angular momentum - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Div P + b p F · P t = P · F t 0 = 0 in B 0 , d Div b P + b b p b F · b P t = b P · b F t 0 − P · N = 0 on S 0 , b b b b b Weak form: Z Z b P : d Grad δ b P : Grad δ ϕ d V + ϕ d A B 0 S 0 Z Z ϕ · b δ ϕ · b p b p ∀ δ ϕ ∈ H 1 ϕ ∈ H 1 δ b 0 ( B 0 ) , ∀ δ b 0 d V − 0 d A = 0 0 ( S 0 ) − S N B 0 0 surface divergence theorem superficial tensor Z Z Z b d • } · b b P · N = 0 Div { b • } d A = { b C { b • } · N d A N d L − S 0 C 0 S 0 twice mean surface curvature C = − d b Div N
C ONSTITUTIVE R ELATIONS bulk J : = Det F : Jacobian determinant µ , λ : Lam´ e constants 2 λ ln 2 J + 1 Ψ ( F ) = 1 2 µ [ F : F − 3 − 2 ln J ] Free energy P ( F ) = ∂ Ψ ∂ F = λ ln J f t + µ [ F − f t ] Piola stress surface J : = d b Det b µ , b b b F : Surface Jacobian determinant λ : Surface Lam´ e constants γ : Surface tension λ ln 2 b Ψ ( b b µ [ b F : b 2 b F − 2 − 2 ln b γ b F ) = 1 J + 1 2 b J ] + b Surface Free energy J F ) = ∂ b Ψ b P ( b = b λ ln b J b µ [ b F − b γ b J b f t + b f t ] + b f t Surface Piola stress ∂ b F
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