Locking free elements for polyconvex

anisotropic material formulationsJulian Dietzsch1 and Michael Groß2

1Technische Universitat Chemnitz, Professorship of applied mechanics and dynamics

Reichenhainer Straße 70, D-09126 Chemnitz, julian.dietzsch@mb.tu-chemnitz.de2michael.gross@mb.tu-chemnitz.de

Short abstract

Our research activity takes place within the DFG project GR 3297/4-1.

The target is the development of a new locking free energy consistent time

integrator for a polyconvex anisotropic material formulation. In the first step,

we compare a non-standard mixed finite element [1] with standard methods

for tetrahedral and hexahedral elements for bodies with multiple material

domains and anisotropy directions within a static analysis. In the next steps,

we aim at the extension of this formulation to a dynamic system [3] and to

thermo-viscoelastic material behaviour.

Keywords Anisotropy, quasi-incompressibility, locking free finite elements.

1 Introduction

The considered finite element discretization is based on the Hu-Washizu

functional presented in [1]. This five-field functional with respect to the

reference configuration B0 reads

Π(x,H,B,Θ, p) =

∫B0

Ψ(C(x),H,Θ)dV +

∫B0p(J(x)−Θ)dV∫

B0B:(cof[C(x)]−H)dV ,

with the five independent variables x, H, B, Θ and p. The third invariant

of deformation gradient F and the right Cauchy-Green tensor are denoted

by J and C, respectively. The variables H and Θ approximates the cofactor

of C and J , respectively. The variables B and p represent the corresponding

Lagrange multiplier. The variation of the functional and the finite element

RCM 2017 - Research Challenges in Mechanics Julian Dietzsch

approximation of these fields lead to five nonlinear equations, which are

solved by the Newton-Raphson method. The resulting tangents condensate at

element level to a pure displacement formulation. We consider a hyperelastic,

isotropic, polyconvex strain energy Ψ = Ψiso as well as a transversely

isotropic, polyconvex strain energy Ψ = Ψiso + Ψti, given by

Ψiso(C,H,Θ) =ε12

(tr[C])2 +ε22

(tr[H])2 − ε3ln(Θ) + ε4(Θ2ε5 + Θ−2ε5 − 2)

Ψti(C,H,Θ) = ε6

(1

ε7 + 1(tr[CM ])ε7+1 +

1

ε8 + 1(tr[HM ])ε8+1 +

1

ε9Θ−2ε9

)which are presented in [1]. Here, M = a⊗ a denotes the structural tensor

and a the fiber direction in the reference configuration.

2 Numerical examples

As numerical example serves the well-known cook’s cantilever beam with a

quadratic distribution of an in-plane load on the Neumann boundary with

the prescribed pressure amplitude p and prescribed material parameters (see

Figure 3). We compare the non-standard mixed elements (TCo/HCo) with

the standard displacement elements (T/H), the selective reduced integrated

elements (SRI, hexahedral elements only) and the mixed element presented

in [2] (TP/HP) for linear tetrahedral and hexahedral elements as well as

the corresponding quadratic serendipity elements. Herein, we analyze the

spatial convergence with respect to the polynomial degree of the underlying

Lagrange polynomials and with respect to the level of mesh refinement. For

the TCoXYZ/HCoXYZ elements, X is the polynomial degree of x, Y is the

polynomial degree of H/B and Z is the polynomial degree of Θ/p. GPX spec-

ify the number X of Gaussian quadrature points. The elements SRI1/H1P0,

SRI2/H2P1, T1P0/T1 and T2P1/T2 provide equal or nearly equal solutions

for the presented cases. Expected equivalences with the TCo/HCo element

can be also found with e.g. the HCo222/H2, HCo220/H2P0, HCo110/H1P0

or TCo222/T2 element. Figure 1 shows the convergence of the y-coordinate

of some elements for an uniform fiber direction (see caption). Here, the

HCo210 element have the highest convergence rate, followed by the H2P0

element. For tetrahedral elements, the TCo210 element provides the fastest

convergence. Using a linear approximation, the HCo100 element have a fast

convergence. Figure 2 shows the convergence of the x-coordinate for two

different fiber directions (see caption). Here, the convergence rates of the

different elements behave in the same manner. The last example, shown in

Figure 4, have also different anisotropic directions (see caption). In this case,

the HCo210 and H2P0 elements converges from above. Nevertheless, both

show the highest convergence rate.

March 1-3, 2017 - Hannover

RCM 2017 - Research Challenges in Mechanics Julian Dietzsch

σVM

10 3 10 4 10 573

73.2

73.4

73.6

73.8

74

74.2

74.4

74.6

74.8

75

H1(GP8)

H2(GP27)

HCo210(GP27)

HCo100(GP8)

H1P0(GP8)

H2P1(GP27)

H2P0(GP27)

T1(GP5)

T2(GP11)

TCo210(GP11)

T2P0(GP11)

nDOFy-c

oord

inat

e

Figure 1: Deformed configuration Bt of the HCo210 method (nel = 6144)

and convergence of the y-coordinate for p = 1000 and a1 = a2 = [1 1 1].

σVM

10 3 10 4 10 5

30.5

31

31.5

32

32.5

33

33.5

34

H1(GP8)

H2(GP27)

HCo210(GP27)

HCo100(GP8)

H1P0(GP8)

H2P1(GP27)

H2P0(GP27)

T1(GP5)

T2(GP11)

TCo210(GP11)

T2P0(GP11)

nDOF

x-c

oord

inat

e

Figure 2: Deformed configuration Bt of the TCo210 method (nel = 7883) and

convergence of the y-coordinate for p = 1000 and a1 = [1 1 1],a2 = [1 1 − 1].

3 Outlook

For the anisotropic case, the CoFEM elements provide higher convergence

rates than the standard elements, but not in the same extent as in the isotropic

case. Nevertheless, in the next step, we aim at an extension of this formulation

to a dynamic system. We extend this to a continuous Galerkin formulation,

and then to an energy conserving formulation. Here, we investigate the

influence of the spatial approximation to the time discretization, because

independent spatial variables are also discretized independently in time.

March 1-3, 2017 - Hannover

RCM 2017 - Research Challenges in Mechanics Julian Dietzsch

y

x

z

44

10 48

44

16

A

p Materialparameters

ε1 ε2 ε342 84 1260

ε4 ε5 ε6100 10 3000

ε7 ε8 ε94 8 1

Figure 3: Geometry of the material domains (domain 1/domain 2) of the

cook cantilever beam (left) and the prescribed material parameters (right).

σVM

10 3 10 4 10 569

69.2

69.4

69.6

69.8

70

70.2

70.4

70.6

70.8

71

H1(GP8)

H2(GP27)

HCo210(GP27)

HCo100(GP8)

H1P0(GP8)

H2P1(GP27)

H2P0(GP27)

T1(GP5)

T2(GP11)

TCo210(GP11)

T2P0(GP11)

nDOF

y-c

oord

inat

e

Figure 4: Deformed configuration Bt of the HCo210 method (nel = 6144) and

convergence of the y-coordinate for p = 1000 and a1 = [1 0 0],a2 = [0 1 0].

References

[1] Schroder, J., Wriggers, P., Balzani, D., 2011. A new mixed finite element

based on different approximations of the minors of deformation tensors.

Computer Methods in Applied Mechanics and Engineering 49:3583–3600.

[2] Simo, J., Taylor, R., Pister., K., 1985. Variational and projection methods

for the volume constraint in finite deformation elasto-plasticity. Computer

Methods in Applied Mechanics and Engineering 1:177-208.

[3] Erler N. and Groß M., 2015. Energy-momentum conserving higher-order

time integration of nonlinear dynamics of finite elastic fiber-reinforced

continua. Computational Mechanics 55:921–942.

March 1-3, 2017 - Hannover