institutjeanlerondd'alembert … · mbrunetti,jshale?,sbordas?,cmaurini...

m brunetti∗, j shale?, s bordas?, cmaurini∗

∗ Institut Jean Le Rond d'Alembert · université pierre et marie curie? Research Unit in Engineering Science · université du luxembourg

FEniCS-shellsa UFL-based library for simulating thin structures

Imperial College London, 01/07/2015

FEniCS-shells2/28

m. brunetti, j. s. hale, s.bordas, c. maurini

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives andReferences

Outline



MITC implementation

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Future Perspectives and References

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Motivations

Why thin-structures?

I Shell, plate and beam (thin) structures are widely used in civil,mechanical and aeronautical engineering because they arecapable of carrying high loads with a minimal amount ofstructural mass.

I To our knowledge a unified open-source implementation of awide range of thin structural models is not yet available.

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MotivationsSome applications

Multistable shells [Coburn et al.,2013]

Stress focusing in elastic sheets

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Objectives

Why FEniCS-shells?

The UFL language provides an excellent framework for writingextensible, reusable and pedagogical numerical models of thinstructures.

FEniCS-shells is (will be) a library consisting of various thin structuralmodels and associated numerical techiniques expressed in the UFL.

I To have a solid and extensible open-source platform of qualitynumerical methods for thin structures.

I To link in a clear and direct way the continuous mathematicalmodel and its finite element solution.

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I Shells are three-dimensional elastic bodies which occupy a thinregion around a two-dimensional manifold situated inthree-dimensional space.

I A three-dimensional problem is reduced to a two-dimensionalproblem. Quantities of engineering relevance are computeddirectly.

I Non-trivial numerical problems arise also for flat shells (plates)and linear models.

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I Shearable theories (thick plates)

FLEXURE = BENDING + SHEARING

e.g. for the Reissner-Mindlin (RM) plate model

ERM =1

2

∫Ω

D∇sθ : ∇sθ +t−2

2

∫Ω

F |∇w − θ|2 − Le

Kinematic descriptors

w : transverse displacement

θ = θ1, θ2 : rotation

x1

x2

x3

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ERM =1

2

∫Ω


2

∫Ω

F |∇w − θ|2 − Le

Strain measures

K = ∇sθ : curvature

γ = ∇w − θ : shearing

x2

x3

θ2γ2

θ+2

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ERM =1

2

∫Ω


2

∫Ω

F |∇w − θ|2 − Le

I Bending theories (thin plates)

∇w = θ and FLEXURE = BENDING

e.g. for the Kirchhoff-Love (KL) plate model

EKL =1

2

∫Ω

D∇∇w : ∇∇w − Le

Remark: when t→ 0 the RM-model asymptotically converges to theKL-model with∇w = θ.

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FEniCS-shells. ModelsShearable Bending Linear Nonlinear

X Kirchhoff-Love plate model • •X Reissner-Mindlin plate model • •X Hierarchical plate models •X von Kármán plate model • •X Marguerre shallow shell model • •

Koiter shell model • •Naghdi shell model • •

I Bending models have been implemented by employing the C/DGalerkin formulation [Engel et al., 2002] in order to avoidH2(Ω)-finite elements

I Shearable models have been implemented by employing theMITC formulation [Dvorkin and Bathe, 1986] in order to avoidnumerical locking.

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MITC implementation

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Numerical lockingWe consider the sequence of problems in the thickness parameter t

minθ,w∈U

ß1

2a(θ,θ) +

t−2

2

∫Ω

|∇w − θ|2 − Le

™U = H1(Ω)×H1(Ω)

For t→ 0 we have the limit problem

minθ,w∈K

ß1

2a(θ,θ)− Le

™K = (θ, w) ∈ U : ∇w = θ

while the corresponding discrete problem has limit problem

minθh,wh∈Kh

ß1

2a(θh,θh)− Le

™Kh = K ∩ (Θh ×Wh)

IfKh is not large enough the basis functions cannot properly representthe Kirchhoff's constraint∇w = θ. The shear term doesn't vanish.

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0.02 0.05 0.10 0.20 0.5010-4

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ç : t = 10-1

á : t = 10-2

í : t = 10-4

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Locking cure

Since the problem relies in the shear termγ = t−2(∇w − θ) and forθ ∈H1(Ω),w ∈ H1(Ω)

∇w − θ ∈H(curl ,Ω)

it is possible to use the mixed formulation with penalty term

Find (θ, w) ∈ U andγ ∈H(curl ,Ω) such that

a(θ, θ) + (∇w − θ,γ) = (f, w) ∀(θ, w) ∈ U(∇w − θ, γ)− t2(γ, γ) = 0 ∀γ ∈H(curl ,Ω)

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MITC formulationThe idea: the discretization of the mixed formulation can betransformed into a displacement form

minθh,wh∈Uh

ß1

2a(θh,θh) +

t−2

2

∫Ω

|∇wh −Rhθh|2 − Le

™where the reduction operator

Rh : H1(Ω)→ Γ h ⊂H(curl ,Ω)

interpolates piecewise smooth functions into the shear spaceΓ h.

Main advantagesI It leads to systems of equations with positive definite matrices

and fewer unknownsI The action ofRh is local and the system matrix can be assembled

locally

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Elements in MITC family are different for the choice ofΘh,Wh,Γ h

and the tying, as expressed byRh, between the interpolation inΓh andthe shear strain as evaluated fromΘh,Wh,Γ h.

I Duran-Liberman [Duran, Liberman, 1992]

Θh

CG2

Wh

CG1

Γ h

NED1

Rh :

∫e

(θ −Rhθ) · t = 0 ∀e ∈ T

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FEniCS implementationFirst, we define the full mixed spaceUF

h = Θh ×Wh × Γ h

V_3 = FunctionSpace(mesh, "CG", 1)26 R = VectorFunctionSpace(mesh, "CG", 2)

RR = FunctionSpace(mesh, "N1curl", 1)28 U_F = MixedFunctionSpace([R, V_3, RR])

and the reduction/tying operator∫e

(γh · t)(θRh · t)

58 def R_e(gam, R_th_t, U):dSp = Measure(’dS’, metadata=’quadrature_degree’: 1)

60 dsp = Measure(’ds’, metadata=’quadrature_degree’: 1)n = FacetNormal(U.mesh())

62 t = as_vector((-n[1], n[0]))area = FacetArea(U.mesh())

64 return (area*inner(gam, t)*inner(R_th_t, t))("+")*dSp + \(area*inner(gam, t)*inner(R_th_t, t))*dsp

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We define the projection form a10

70 ga = lambda z1, z2: grad(z2) - z1g = ga(r, w)

72 g_t = ga(r_t, w_t)a_10 = R_e(g, rr_t, U_F) + R_e(g_t, rr, U_F)

the shear form a01

78 Pi_she = (eps**-2)*0.5*inner(T(ga(rr_, w_)), ga(rr_, w_))*dxF_she = derivative(Pi_she, u_, u_t)

80 a_01 = derivative(F_she, u_, u)

the primal mixed spaceUh = Θh ×Wh

U = MixedFunctionSpace([R, V_3])

and the unprojected bending form a00

102 Pi_ben = 0.5*inner(M(ep(r_)), ep(r_))*dxdPi_ben = derivative(Pi_ben, u_, u_t)

104 a_00 = derivative(dPi_ben, u_, u)

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Finally, we use a custom assembler to perform the projection at thelocal linear algebra level

114 A = mitc_assemble([a_00, a_01, a_10])

I C++ code to assemble the stiffness matrix

A = A00 +A10A01A10

A00 : unprojected bending term

A01 : projected shearing term

A10 : projection matrix

I JIT compilation with Instant

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Outline



MITC implementation

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RM plate. ConvergenceComparison with the analytical solution provided by [Lovadina, 1995].

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1.000

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von Kármán plate modelStarting with the scalings

u = O(ε2) w = O(ε) ε = ‖K‖

the vK plate model retains the minimal geometrical nonlinearities ableto catch the coupling between bending and membrane strains

E = ∇su+∇w ⊗∇w

2K = ∇∇w

whose integrability conditions correspond to the linearization of theGauss theorema egregium

curl curlE = detK

SinceEvK = EKL + t−2Em, whenever possibile a plate tends to bend ina developable surface (Em = 0 for detK = 0).

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Circular plate with lenticular crosssection subjected to a temperaturegradient through its thickness

x

y

Average curvatures bifurcation atcritical value [Mansfield, 1962]

κT =8

(1 + ν)3/2

0 2 4 6 8

0

2

4

6

8

inelastic curvature

ave

rage

curv

atu

res,

kx,k

y

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Outline



MITC implementation

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Future Perspectives

I to implement membrane locking-free element for the nonlinearKoiter shell model

I to implement membrane and shear locking-free element for thenonlinear Naghdi shell model

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Main ReferencesI Jeon, H.-M., Lee, P.-S. and Bathe, K. J., The MITC3 shell finite element enriched

by interpolation covers, Computers and Structures, 2014I Hale, J. S. and Baiz P., M., Towards effective shell modelling with the FEniCS

project, FEniCS Workshop 2013I Chapelle, D. and Bathe, K. J., The finite element analysis of shells, 2010.I Lovadina, C., A Brief Overview of Plate Finite Element Methods, Integral

Methods in Science and Engineering, 2010I Engel, G., et al. Continuous/discontinuous finite element approximations of

fourth-order elliptic problems in structural and continuum mechanics withapplications to thin beams and plates, and strain gradient elasticity, ComputerMethods in Applied Mechanics and Engineering, 2002

I Durán, R., and Liberman, E., On mixed finite element methods for theReissner-Mindlin plate model, Mathematics of computation, 1992

I Brezzi, F., Bathe, K .J., and Fortin, M., Mixed-interpolated elements forReissner-Mindlin plates, International Journal for Numerical Methods inEngineering, 1989

I Mansfield, E. H., Bending, buckling and curling of a heated thin plate,Proceedings of the Royal Society of London A, 1962

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