fenics-shells: a UFL-based library for simulating thin structures

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

∗ Institut Jean Le Rond d'Alembert · université pierre et marie curie

? Research Unit in Engineering Science · université du luxembourg

FEniCS-shells

a UFL-based library for simulating thin structures

Imperial College London, 01/07/2015

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2/28 m. brunetti, j. s. hale, s.

bordas, c. maurini

Motivations and Objectives Remarks on shell theories MITC implementation Results Future Perspectives and References

Outline

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives and References

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bordas, c. maurini

Outline

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

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 are capable of carrying high loads with a minimal amount of structural mass.

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

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Motivations

Some 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 writing extensible, reusable and pedagogical numerical models of thin structures.

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

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

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

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Outline

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives and References

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Remarks on shell theories

I Shells are three-dimensional elastic bodies which occupy a thin region around a two-dimensional manifold situated in three-dimensional space.

I A three-dimensional problem isreduced to a two-dimensional problem. Quantities of engineering relevance are computed directly.

I Non-trivial numerical problems arise also forflat shells (plates) andlinear 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

Z

Ω

D∇^sθ : ∇^sθ +t⁻² 2

Z

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

Z

Ω

D∇^sθ : ∇^sθ +t⁻² 2

Z

Ω

F |∇w − θ|²− Le

Strain measures

K = ∇^sθ :curvature γ = ∇w − θ : shearing

x2

x3

θ₂ γ2

θ⁺₂

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ERM= 1 2

Z

Ω

D∇^sθ : ∇^sθ +t⁻² 2

Z

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 Z

Ω

D∇∇w : ∇∇w − Le

Remark:whent → 0 the RM-model asymptotically converges to the KL-model with∇w = θ.

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FEniCS-shells. Models

Shearable 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 modelshave been implemented by employing theC/D Galerkin formulation [Engel et al., 2002]in order to avoid H²(Ω)-finite elements

I Shearable modelshave been implemented by employing the MITC formulation [Dvorkin and Bathe, 1986]in order to avoid numerical locking.

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Outline

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives and References

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Numerical locking

We consider the sequence of problems in the thickness parametert

min

θ,w∈U

ß1

2a(θ, θ) +t⁻² 2

Z

Ω

|∇w − θ|²− Le

™

U = H¹(Ω)×H¹(Ω)

Fort → 0 we have the limit problem min

θ,w∈K

ß1

2a(θ, θ) − Le

™

K = {(θ, w) ∈ U : ∇w = θ}

while the corresponding discrete problem has limit problem min

θ_h,w_h∈Kh

ß1

2a(θh, θh) −Le

™

Kh=K ∩ (Θh× Wh)

IfKhis not large enoughthe basis functions cannot properly represent the Kirchhoff's constraint∇w = θ. The shear term doesn't vanish.

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ç

á á á á

í í í í

0.02 0.05 0.10 0.20 0.50

10^-4 0.001 0.01 0.1 1

mesh size

relativeerror

ç: t = 10^-1 á: t = 10^-2 í: t = 10^-4

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Locking cure

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

∇w − θ ∈H(curl, Ω)

it is possible to use themixed formulationwith penalty term

Find(θ, w) ∈ U and γ ∈ 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 formulation

The idea:the discretization of the mixed formulation can be transformed into a displacement form

min

θh,wh∈Uh

ß1

2a(θh, θh) +t⁻² 2

Z

Ω

|∇wh−Rhθh|²− Le

™

where thereduction operator

Rh : H¹(Ω) → Γ_h⊂ H(curl , Ω)

interpolates piecewise smooth functions into the shear spaceΓh. Main advantages

I It leads to systems of equations withpositive definite matrices and fewer unknowns

I The action ofRhis local and the system matrixcan be assembled locally

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Elements in MITC family are different forthe choice ofΘ_h,Wh,Γ_h andthe tying, as expressed byRh, between the interpolation inΓhand the shear strain as evaluated fromΘh,Wh,Γh.

I Duran-Liberman [Duran, Liberman, 1992]

Θh

CG₂

Wh

CG₁

Γh

NED1

Rh : Z

e

(θ −Rhθ) · t = 0 ∀e ∈ T

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FEniCS implementation

First, we define thefullmixed spaceU_h^F = Θ_h× Wh× Γh 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 thereduction/tying operator Z

e

(γh· t)(˜θ^R_h · 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 theprojection forma10

70 ga = lambda z1, z2: grad(z2) - z1 g = 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)

theshear forma01

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

80 a_01 = derivative(F_she, u_, u)

theprimalmixed spaceUh= Θh× Wh U = MixedFunctionSpace([R, V_3])

and theunprojected bending forma00

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

104 a_00 = derivative(dPi_ben, u_, u)

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Finally, we use acustom assemblerto perform the projection at the local linear algebra level

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

I C++ codeto 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

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives and References

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RM plate. Convergence

Comparison with the analytical solution provided by [Lovadina, 1995].

ç

á

í

0.02 0.05 0.10 0.20 0.50

0.001 0.005 0.010 0.050 0.100 0.500 1.000

mesh size relativeerrorinrotationH1-norm

ç: t = 10^-2 á: t = 10^-4 í: t = 10^-6

ç

á

í

0.02 0.05 0.10 0.20 0.50

10^-6 10^-5 10^-4 0.001 0.01 0.1

mesh size

relativeerrorindisplacementsemi-norm

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von Kármán plate model

Starting with the scalings

u =O(²) w = O() = kKk

the vK plate model retainsthe minimal geometrical nonlinearitiesable to catchthe coupling between bending and membrane strains

E = ∇^su +∇w ⊗ ∇w

2 K = ∇∇w

whose integrability conditions correspond to thelinearization of the Gauss theorema egregium

curl curlE =det K

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

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Circular plate with lenticular cross section subjected to a temperature gradient through its thickness

x

y

Average curvatures bifurcation at critical value [Mansfield, 1962]

κT = 8

(1 +ν)^3/2

0 2 4 6 8

inelastic curvature averagecurvatures,kx,ky

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Outline

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives and References

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Future Perspectives

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

I to implement membrane and shear locking-free element for the nonlinearNaghdi shell model

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Main References

I Jeon, H.-M., Lee, P.-S. and Bathe, K. J., The MITC3 shell finite element enriched by interpolation covers, Computers and Structures, 2014

I Hale, J. S. and Baiz P., M., Towards effective shell modelling with the FEniCS project, FEniCS Workshop 2013

I 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, 2010

I Engel, G., et al. Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Computer Methods in Applied Mechanics and Engineering, 2002

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

I Brezzi, F., Bathe, K .J., and Fortin, M., Mixed-interpolated elements for Reissner-Mindlin plates, International Journal for Numerical Methods in Engineering, 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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fenics-shells: a UFL-based library for simulating thin structures

FEniCS-shells

a UFL-based library for simulating thin structures

Outline

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives and References

Outline

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives and References

Motivations

Motivations

Objectives

Outline

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives and References

Remarks on shell theories

FEniCS-shells. Models

Outline

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives and References

Numerical locking

Locking cure

MITC formulation

FEniCS implementation

Outline

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives and References

RM plate. Convergence

von Kármán plate model

Outline

Motivations and Objectives

Remarks on shell theories

MITC implementation

Results

Future Perspectives and References

Future Perspectives

Main References

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