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Abstract This paper deals with volume bubble functions for mixed finite triangular elements in geometrically linear elasticity. In two different versions these functions are used in order to enrich the displacement field and the enhanced strain field, respectively. Appropriate conditions for satisfaction of the patch test are verified. In the numerical example, firstly the patch test is satisfied. Secondly, simulations of Cook’s membrane problem demonstrate, that the proposed formulations avoid locking and reduce stress oscillations for incompressible materials. Keywords: mixed finite element, volume bubble, incompatible modes, enhanced strains. 1 Introduction In many engineering fields such as automobile, energy and manufacturing finite el- ement meshes are generated with commercial software tools. These programs pre- fer triangular and tetrahedral elements rather than quadrilateral and brick elements in order to render robust discretizations also for complex two- and three-dimensional geometries. Additionally these elements facilitate more convenient manipulations in adaptive mesh refinement of h-type. The elements should be also of low order, which reduces the computational time. However it is well known that standard linear finite element formulations exhibit rather poor performance when extra physical constraints occur. Typical examples are the incompressibility constraint, leading to volume lock- ing and the shear constraint for bending dominated problems, which induces shear locking. Volumetric locking of finite elements can be reduced or eliminated by a variety of approaches. One approach is to use higher-order interpolation functions for the el- 1 Paper 285 Mixed Finite Element Formulations with Volume Bubble Functions for Triangular Elements I. Caylak and R. Mahnken Chair of Engineering Mechanics (LTM) University of Paderborn, Germany ©Civil-Comp Press, 2009 Proceedings of the Twelfth International Conference on Civil, Structural and Environmental Engineering Computing, B.H.V. Topping, L.F. Costa Neves and R.C. Barros, (Editors), Civil-Comp Press, Stirlingshire, Scotland

Mixed Finite Element Formulations with Volume Bubble Functions for Triangular Elements · 2009. 8. 12. · approaches. One approach is to use higher-order interpolation functions

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  • Abstract

    This paper deals with volume bubble functions for mixed finite triangular elements

    in geometrically linear elasticity. In two different versions these functions are used

    in order to enrich the displacement field and the enhanced strain field, respectively.

    Appropriate conditions for satisfaction of the patch test are verified. In the numerical

    example, firstly the patch test is satisfied. Secondly, simulations of Cook’s membrane

    problem demonstrate, that the proposed formulations avoid locking and reduce stress

    oscillations for incompressible materials.

    Keywords: mixed finite element, volume bubble, incompatible modes, enhanced

    strains.

    1 Introduction

    In many engineering fields such as automobile, energy and manufacturing finite el-

    ement meshes are generated with commercial software tools. These programs pre-

    fer triangular and tetrahedral elements rather than quadrilateral and brick elements

    in order to render robust discretizations also for complex two- and three-dimensional

    geometries. Additionally these elements facilitate more convenient manipulations in

    adaptive mesh refinement of h-type. The elements should be also of low order, whichreduces the computational time. However it is well known that standard linear finite

    element formulations exhibit rather poor performance when extra physical constraints

    occur. Typical examples are the incompressibility constraint, leading to volume lock-

    ing and the shear constraint for bending dominated problems, which induces shear

    locking.

    Volumetric locking of finite elements can be reduced or eliminated by a variety of

    approaches. One approach is to use higher-order interpolation functions for the el-

    1

    Paper 285 Mixed Finite Element Formulations with Volume Bubble Functions for Triangular Elements I. Caylak and R. Mahnken Chair of Engineering Mechanics (LTM) University of Paderborn, Germany

    ©Civil-Comp Press, 2009 Proceedings of the Twelfth International Conference on Civil, Structural and Environmental Engineering Computing, B.H.V. Topping, L.F. Costa Neves and R.C. Barros, (Editors), Civil-Comp Press, Stirlingshire, Scotland

  • ements. The six-node triangle and ten-node tetrahedron use quadratic interpolation

    functions and avoid many of the shortcomings of their constant strain counterparts.

    However, this method is accompanied by an increase of computational time. Con-

    cerning the development of triangluar or tetrahedral elements currently the following

    competing strategies can be distinguished:

    • Pressure stabilization: In CERVERA et al. [10] orthogonal sub-grid scales are

    used to stabilize the behavior of mixed linear/linear simplicial elements in in-

    compressible or nearly incompressible situations.

    • F-bar method: This method is proposed by DE SOUZA NETO et al. in [23]

    with the replacement of the deformation gradient with an assumed modified

    counterpart. In [24] DE SOUZA NETO et al. present an extension of the F-

    bar method, which allows the use of simplex finite element in the large strain

    analysis of nearly incompressible solids.

    • Average nodal pressures/strains: An assessment of the method is given in AN-

    DRADE PIRES et al. [4]. Uniform strain elements are introduced in DOHRMANN

    et al. [12] and show better results than standard three-node triangular and four-

    node tetrahedral elements, without introducing additional degrees of freedom.

    An extension to the large strain regime is provided in BONET [6]. In PUSO

    and SOLBERG [22] a stabilized nodally integrated tetrahedral element for the

    compressible regime is investigated. The disadvantage is that the nodal av-

    erage method is significantly more expensive than other tetrahedral and brick

    elements.

    • Mixed-enhanced elements: AURICCHIO et al. [5] propose a 2D finite-strain

    problem depending on a loading parameter. In the formulation by TAYLOR

    [35] the additional strains are derived from the symmetric gradient of a volume

    bubble. On this basis, in MAHNKEN et al. [20] volume and area bubble func-

    tions are used to damp significantly the oscillatory behaviour in bi-linear mixed

    finite elements for tetrahedra.

    In MAHNKEN et al. [20] and MAHNKEN and CAYLAK [21] area and volume bub-

    ble functions for stabilization of tetrahedral elements are introduced, where linear in-

    terpolation functions for the displacement field and linear interpolation functions for

    the pressure field are used. This paper concentrates on stabilization of mixed triangular

    elements in linear elastic regime. In particular we compare the stabilization effect with

    volume bubble functions the method of incompatible modes and the enhanced strain

    method. In the numerical example results for the displacement for the incompatible

    modes elements are slightly better for coarser meshes. Additionally we consider the

    stress distribution. In both cases for incompatible modes and enhanced strains, we ob-

    serve no oscillation in stress distribution. A paper considering area bubble functions

    is in preperation and will be submitted next.

    The structure of this paper is as follows: In Section 2 the construction of the volume

    bubble functions in the isoparametric domain is shown. Based on the strong and weak

    2

  • formulation in [20] and [21], here only the finite element matrix formulations for

    the method of incompatible modes and the enhanced strain method are presented.

    Furthermore issues related to the patch test and unique solution of the discrete finite

    element equation are addressed. Section 3 presents numerical examples: Firstly a

    numerical verification of the patch test is obtained. Furthermore, Cook’s membrane

    problem is investigated, in order to show the performance of the proposed triangular

    elements.

    2 Finite element formulation

    We consider a discretization of the domain B =⋃ne

    e=1 Be into ne triangular elementswith three nodes. Each element occupies a subdomain Be. In the isoparametric domain

    Be0, a local coordinate system with coordinates ξ, η is introduced, with the properties

    0 ≤ ξ, η ≤ 1. (1)

    The position x = [x, y]T for any ξ = [ξ, η]T within the real domain Be of each triangleis expressed as

    x(ξ) =3∑

    i=1

    N i(ξ)x̂i, (2)

    and the displacement u = [ux, uy]T as

    u(ξ) =3∑

    i=1

    N i(ξ)ûi, (3)

    where x̂i ∈ IR2×1, i = 1, 2, 3 summarizes the positions x̂i, ŷi at all 3 nodes of the

    element and ûi = [ûxi, ûyi]T , i = 1, 2, 3 are the displacements at node i of the element.

    The standard shape functions N i, i = 1, 2, 3: Be0 7→ Be of the isoparametric conceptfor the triangular elements are given as

    1. N1(ξ) = ξ

    2. N2(ξ) = η

    3. N3(ξ) = 1 − ξ − η = ϕ.

    (4)

    The derivatives are easily converted from the local coordinate system to the global

    coordinate system by using the Jacobian matrix

    ∂N

    ∂ξ

    ∂N

    ∂η

    = J

    ∂N

    ∂x

    ∂N

    ∂y

    and

    ∂N

    ∂x

    ∂N

    ∂y

    = J−1

    ∂N

    ∂ξ

    ∂N

    ∂η

    . (5)

    3

  • Here the Jacobian is expressed as

    J =

    x(ξ, η),ξ y(ξ, η),ξ

    x(ξ, η),η y(ξ, η),η

    =

    3∑

    i=1

    N i,ξx̂i3∑

    i=1

    N i,ξŷi

    3∑

    i=1

    N i,ηx̂i3∑

    i=1

    N i,ηŷi

    =

    x̂1 − x̂3 ŷ1 − ŷ3

    x̂2 − x̂3 ŷ2 − ŷ3

    ,

    (6)

    which for linear triangular elements has the important properties

    1. J = const. and j = det(J) = const.

    2. J is invertible and J−1 =1

    det(J)

    3∑

    i=1

    N i,ηyi −3∑

    i=1

    N i,ξyi

    3∑

    i=1

    N i,ηxi3∑

    i=1

    N i,ξxi

    .(7)

    2.1 Volume bubble functions for triangular elements

    In addition to the standard shape functions in Eq. 4, we use volume bubble functions,

    in order to eliminate the locking effects and to reduce oscillatory effects.

    N̄(ξ) = 27ξηϕ (8)

    This function is illustrated in Figure 1. It gives zero contribution along the three edges

    and has the value 1 at the center of the triangle, see e.g. [41].

    N

    2

    13ξ

    η

    P

    2

    13ξ

    ηa) b)

    Figure 1: Illustration of volume bubble function: a) points P̄ with maximum value ofthe volume bubble function; b) volume bubble N̄

    Remarks

    1. Compared to the standard shape functions in Eq.(4) volume bubble functions

    introduce additional terms ξη, ξ2η and ξη2, which can be readily ascertainedfrom the Pascal triangle in Figure 2.

    2. The derivatives of the volume bubble functions with respect to the local coordi-

    nations ξ, η are obtained from Eq. (8) as

    4

  • Figure 2: The Pascal triangle.

    1. N̄,ξ = 27(ηϕ − ξη)

    2. N̄,η = 27(ξϕ − ξη) (9)

    3. The integrals over the domain Be0 of the reference element in the isoparametric

    space have the properties

    Be0

    N̄,ξdB0 = 0 and∫

    Be0

    N̄,ηdB0 = 0, (10)

    where

    Be0

    (·)dB0 =∫ ∫

    (·)dξdη.

    2.2 Bubble functions for the mixed method of incompatible modes

    In the mixed method of incompatible modes the trial functions for the enriched dis-

    placement interpolation ũ ∈ IR2, the pressure field interpolation p ∈ IR and the strainfield interpolation ε ∈ IR3 within each element read

    1. ũ(ξ) =3∑

    i=1N i(ξ)ûi +

    nB∑

    i=1N̄ i(ξ)v̂i = u(ξ) + v(ξ)

    2. p(ξ) =3∑

    i=1N i(ξ)p̂i = N p̂

    3. ε(û, v̂) =3∑

    i=1Biu(ξ)ûi +

    nB∑

    i=1Biv(ξ)v̂i = Bu û + Bv v̂,

    (11)

    where u(ξ) and v(ξ) denote the compatible and incompatible part of the displacementfield, respectively. The unknowns in Eq.(11) are: ûi = [ûxi, ûyi]

    T , i = 1, 2, 3, p̂i,i = 1, 2, 3, v̂i = [v̂xi, v̂yi]

    T , i = 1, ..., nB , where nB = 1 for volume bubble functions.Additionally

    Bu =[

    B1u B2

    u B3

    u

    ]

    , N = [N1, N2, N3],

    û =[

    û1 û2 û3]T

    , v̂ =[

    v̂1]T

    , p̂ =[

    p̂1 p̂2 p̂3]T

    (12)

    5

  • with

    Biu =

    N i,x 0

    0 N i,yN i,y N

    i,x

    and Bv =

    N̄,x 0

    0 N̄,y

    N̄,y N̄,x

    are given. The derivatives of the shape functions with respect to the coordinates x, yare obtained as

    N i,xN i,y

    = J−1

    N i,ξN i,η

    and

    N̄,x

    N̄,y

    = J−1

    N̄,ξ

    N̄,η

    , (13)

    where the Jacobian J is defined in Eq.(6). A matrix notation of the weak form ([20]and [21]) renders

    1.ne∑

    e=1

    Be

    εT (δu)(Idevσ̃ + p1) dB =ne∑

    e=1

    Gext(δu) ∀ δu admissible

    2.ne∑

    e=1

    Be

    εT (δv)(Idevσ̃ + p1) dB = 0 ∀ δv admissible

    3.ne∑

    e=1

    Be

    δp(p −1

    31T σ̃) dB = 0 ∀ δp admissible

    (14)

    with the specific definitions for the matrices 1, Idev given in Appendix A. InsertingEq.(11) into the weak form (14) the following finite element residual equations are

    derived:

    1. Ru = Ae=1

    ne ∫

    Be

    BTu (Idevσ̃ + p1) dB −Ae=1

    ne

    F ext = 0

    2. Rev =∫

    Be

    BTv (Idevσ̃ + p1) dB = 0, e = 1, ..., ne.

    3. Rp = Ae=1

    ne ∫

    Be

    NT(

    1

    31T σ̃ − p

    )

    dB = 0,

    (15)

    where the stresses and the pressure are described as

    1. σ̃ = Cε(u, v) = C Buû + C Bvv̂,

    2. p = Np̂.(16)

    Additionally we make use of the assembly operator A introduced by Hughes[15].

    2.3 Bubble functions for the mixed method of enhanced strains

    In the mixed method of enhanced strains the displacement interpolation u ∈ IR2, thepressure field interpolation p ∈ IR and the strain field interpolation ε + α ∈ IR3 within

    6

  • each element read

    1. u(ξ) =3∑

    i=1N i(ξ)ûi

    2. p(ξ) =3∑

    i=1N i(ξ)p̂i = N p̂

    3. ε(û) + α(ξ) =3∑

    i=1Biu(ξ)ûi +

    nB∑

    i=1Gi(ξ)α̂i = Bu û + G α̂.

    (17)

    Here N i(ξ), i = 1, ..., 3 are the shape functions in Eq.(4) of the standard triangularelements. Furthermore in Eq.(17) the unknowns ûi = [ûxi, ûyi]

    T , i = 1, 2, 3, p̂i,i = 1, 2, 3, α̂i = [α̂xi, α̂yi]

    T , i = 1, ..., nB have been introduced, where nB = 1for volume bubble functions. Furthermore Bu is given in Eq.(12). The interpolationfunctions for the enhanced strains Gi(ξ), i = 1, ..., nB , acting in the physical domain,are derived in [20] following the element design procedure of SIMO and RIFAI [33],

    which by construction - leads to satisfaction of the patch test. In this way, firstly

    interpolation functions Ei, i = 1, ..., nB acting in the isoparametric domain are definedas

    1. Ei(ξ) =

    N̄ i,ξ 0

    0 N̄ i,ηN̄ i,η N̄

    i,ξ

    , i = 1, ..., nB =⇒ 2. E =[

    E1, ..., EnB]

    . (18)

    Secondly, the interpolation functions Gi, i = 1, ..., nB acting in the real domain areobtained as

    Gi = T 0 Ei, i = 1, ..., nB =⇒ G =

    [

    G1, ..., GnB]

    = T 0 E, (19)

    where T 0 is a transformation matrix from the isoparametric domain into the real do-main, see e.g. [21]. The specific formulation in terms of the Jacobian (6) is given

    as

    T 0 =

    [(J−1)11]2 [(J−1)21]

    2 (J−1)11(J−1)21

    [(J−1)12]2 [(J−1)22]

    2 (J−1)12(J−1)22

    2(J−1)11(J−1)12 2(J

    −1)21(J−1)22 (J

    −1)11(J−1)22 + (J

    −1)21(J−1)12

    .

    (20)

    A matrix notation of the weak form in the two dimensional case reads

    1.ne∑

    e=1

    Be

    εT (δu)(Idevσ̃ + p1) dB =ne∑

    e=1

    Gext(δu) ∀ δu admissible

    2.ne∑

    e=1

    Be

    δαT (Idevσ̃ + p1) dB = 0 ∀ δv admissible

    3.ne∑

    e=1

    Be

    δp(p −1

    31T σ̃) dB = 0 ∀ δp admissible.

    (21)

    7

  • By the relations(17) these equations result into the following finite element residuals

    1. Ru = Ae=1

    ne ∫

    Be

    BTu (Idevσ̃ + p1) dB −Ae=1

    ne

    F ext = 0

    2. Rev =∫

    Be

    GT (Idevσ̃ + p1) dAB = 0, e = 1, ..., ne

    3. Rp = Ae=1

    ne ∫

    Be

    NT(

    1

    31T σ̃ − p

    )

    dB = 0,

    (22)

    where by use of Eq.(17) the stresses and the pressure are obtained as

    1. σ̃ = Cε(u, α) = C Buû + C Gα̂,

    2. p = Np̂.(23)

    2.4 Bi-linear finite elements with enhanced stabilization

    In this section some remarks on solution of the finite element residual Eqs.(15) and

    (22) are given. In the sequel it suffices to concentrate on the finite element residual

    equations (15) for the method of incompatible modes. For further explanation we refer

    to [20]. The finite element residual equations for the method of assumed enhanced

    strains are obtained in the same manner, merely by substituting Bv with G.

    For the case of linear elasticity, by use of Eq.(15) and (16), assembling over all

    elements e = 1, ..., ne renders

    1. Ae=1

    ne{

    keuuû + keupp̂ + k

    euvv̂

    }

    = Ae=1

    ne

    F ext

    2.{

    kevuû + kevpp̂ + k

    evvv̂

    }

    = 0, e = 1, ..., ne

    3. Ae=1

    ne{

    kepuû + keppp̂ + k

    epvv̂

    }

    = 0.

    (24)

    8

  • with the element matrices

    1. keuu =∫

    Be

    BTu D Bu dB

    2. keuv =∫

    Be

    BTu D Bv dB = kevu

    T

    3. keup =∫

    Be

    BTu 1 N dB

    4. kevv =∫

    Be

    BTv D Bv dB

    5. kevp =∫

    Be

    BTv 1 N dB

    6. kepp = −∫

    Be

    NT N dB

    7. kepu =∫

    Be

    NT1

    31T CBu dB

    8. kepv =∫

    Be

    NT1

    31T Bv dB

    (25)

    and where

    D = Idev C Idev (26)

    with the specific definitions for the matrices C, Idev given in Appendix A. Since thenoncompatible Eq.(24.2) hold at the element level, we can eliminate the parameters v̂at the element level by static condensation as

    v̂ = − [kevv]−1

    [

    kevukevp

    ][

    ûp̂

    ]

    , e = 1, ..., ne (27)

    leading to

    Ae=1

    ne

    [

    keuu keup

    kepu kepp

    ]

    [

    keuvkepv

    ]

    [kevv]−1

    [

    kevukevp

    ]

    ︸ ︷︷ ︸

    ke∗

    [

    ûp̂

    ]

    = Ae=1

    ne [

    F ext0

    ]

    . (28)

    Prerequisites for the static condensation in Eq.(27) and further remarks are given in

    [20].

    3 Numerical examples

    The stabilized mixed finite element formulations developed in this paper have been

    implemented into a UEL user subroutine of the finite element program Abaqus [1].

    In this section we show results for two finite element examples. They are compared

    9

  • with those of different elements well known from the literature. All finite element

    formulations are distinguished with the following nomenclature:

    Linear displacements T1Quadratic displacements T2Quadratic displacements, linear pressure T2P1Linear displacements, reduced integration Q1RLinear displacements, linear pressure Q2P1Linear displacements, linear pressure T1P1Linear displacements, linear pressure, volume bubble for IM T1P1IM2STLinear displacements, linear pressure, volume bubble for ES T1P1ES2ST

    Here T1 and T2 are standard triangular elements with linear and quadratic interpola-tions for the displacement. T2P1 includes quadratic interpolations for the displace-ment and linear interpolations for the pressure. Q1R is a standard quadrilateral ele-ment with linear interpolation for the displacement and reduced integration. Q2P1is a standard quadrilateral element with quadratic interpolation for displacement and

    linear interpolations for the pressure. T1, T2, T2P1, Q1R and Q2P1 are availablein the commercial finite element program Abaqus[1]. T1P1 includes linear interpola-tions for the displacement and linear interpolations for the pressure. Furthermore the

    abbreviations IM and ES mean incompatible mode and enhanced strain, respectively,

    and ST means stabilization, as introduced in [21].

    3.1 Finite element patch test

    The first example is concerned with the so called Test A and Test C of the patch test

    introduced in TAYLOR et al. [36]. In particular it verifies correct implementation of

    the finite element formulation. Here Test A is a necessary conditions for convergence

    of finite element formulations, whereas Test C establishes a sufficient condition.

    a)

    (1,2)

    (2,1)

    (3,3)

    (4,0)(0,0)

    1

    2 3

    4

    5

    b)

    T1P1IM2ST , S11

    +2.00 e+00

    +2.00 e+00

    +2.00 e+00

    +2.00 e+00

    + e+002.00

    +2.00 e+00

    +2.00 e+00

    13

    2

    Figure 3: Finite element patch test: a), patch geometry, composed of 4 triangular

    elements; b), homogeneous stress distribution obtained with T1P1IM2ST

    The tests are performed on a distorted triangular patch composed of 4 triangular el-

    ements shown in Figure 3. All formulations, T1P1, T1P1IM2ST and T1P1ES2ST

    10

  • are tested. The related coordinates (x, y) at the 5 nodes of the patch are summarizedin Table 1. To verify Test A we start - in an analytical forward calculation - with a ho-

    mogeneous state of stress and strain with σx = 2 N/mm2, σy = 0, εz = 0. Assuming

    linear, isotropic elastic material with Youngs’modulus E = 1000N/mm2 and Poissonratio ν = 0.3 and ν = 0.4999, respectively, the equations

    εx =1

    E(σx − νσy − νσz)

    εy =1

    E(σy − νσx − νσz)

    σz =E

    (1 + ν)(1 − 2ν)(νεx + νεy + (1 − ν)εz)

    and

    u = εx x, v = εy y,

    are used to calculate the strain field and displacement field, respectively. The resulting

    values for the displacements (u, v) at all 5 nodes are given in Table 1 for ν = 0.3. Next,these displacements are used as boundary conditions for the finite element backward

    calculation with the triangular patch. As expected, the stress-strain state with σx =2N/mm2, σy = 0, εz = 0 is obtained at all Gaussian points and visualized in Figure3. Additional, in Table 1 the forces obtained from the FE-calculation are summarized

    for all five nodes at the x, y directions. For Test C nodes 2 and 3 are bounded in u-

    Coordinates Displacement Forces

    Node-No. x y u v Fx Fy

    1 1.0 2.0 0.00182 −0.00156 −3.0 0.02 0.0 0.0 0.0 0.0 −2.0 0.03 4.0 0.0 0.00728 0.0 3.0 0.04 3.0 3.0 0.00546 −0.00234 2.0 0.05 2.0 1.0 0.00364 −0.00078 0.0 0.0

    Table 1: Finite element patch test: coordinates, displacement and forces at five nodes

    for ν = 0.3

    direction and nodes 2 in v-direction. Forces are prescribed in the remaining nodes withvalues according to Table 1. As a result we obtain a stress-strain state with σx = 2N/mm2, σy = 0, εz = 0 as shown in Figure 3, which verifies the numerical patch test.

    3.2 Cook’s membrane problem

    In the second example Cook’s membrane problem is investigated in a two dimen-

    sional simulation. This example is one of the most popular benchmarks to investi-

    11

  • gate the finite element performance due to locking effects. The geometry is shown in

    Figure 4 a). The panel is fully constrained on the left hand side, and is loaded with a

    surface traction t̄ = 10 N/mm2 on the right hand side. We consider plane strain condi-tions for the compressible behavior with Poisson’s ratio ν = 0.33 and for the incom-pressible behavior with ν = 0.4999, with Young’s modulus E = 1000 N/mm2. For theconvergence study different discretizations with 6, 11, 22, 44, and 88 elements along

    the edge A-B of Figure 4 a) are examined. The various finite element discretizations

    are illustrated in Figue 4 b), c), d), e) and f).

    Subsequently we compare in Figure 5 a) the convergence behavior of literature

    elements elements for ν =0.4999, where Q1R could be regarded as the referencesolution due to the fastest convergence. As expected, the linear interpolated ele-

    ment T1 renders a large deviation from the quadrilateral element Q1R. T1P1 re-veals also inadequate performance for coarser meshes. In Figure 5 b) the proposed

    elements T1P1IM2ST and T1P1ES2ST , are compared with Q1R. All of the ele-ments reveal reasonable results. Furthermore, Figure 5 b) shows that the incompatible

    modes element T1P1IM2ST converges slightly faster than the enhanced strain ele-ment T1P1ES2ST .

    Additionally we consider the stress distribution S11 along the clamped edge A-B

    for the discretization with 22 elements along A-B. In Figure 6 a) the stress distri-

    butions are compared for element T1, T1P1, Q1R. It is obvious that the curve T1shows strong oscillations, with large deviation of the stress value at point A of Figure

    6 a) compared with the reference solution Q1R. T1P1 shows also evident oscilla-tions. Moreover, in Figure 6 b), the results of T1P1IM2ST , T1P1ES2ST are com-pared against the reference solution Q1R. It can be seen, that the incompatible mode

    version T1P1IM2ST provides slightly better results as the enhanced strain versionT1P1ES2ST at point A. In both cases for incompatible modes and enhanced strains,we observe no oscillation in stress distribution. The numerically calculated stress val-

    ues at point A for all tested elements are summarized in Table 2. Figure 7 show the

    Stress T1 T1P1 T2 T2P1 Q1RS11 -345.7060 -38.4567 -95.7416 -114.7210 -55.8120

    Stress Q2P1 T1P1IM2ST T1P1ES2STS11 -103.1860 -51.9067 -47.7949

    Table 2: Cook’s membrane problem: stress S11 in MPa at point A for all tested ele-ments for ν = 0.4999 with 22 elements along A-B

    stress contour plots S11. The contour plot of T1 shows evident irrational result, withstrong oscillations and non-smooth stress distributions, where the reference solution

    Q1R exhibits excellent results. T1P1IM2ST and T1P1ES2ST reveal nearly thesame stress distributions as Q1R.

    12

  • a)

    1

    2

    3

    44

    48

    t=1

    [mm]

    t-

    16

    A

    C

    B

    b)

    c) d)

    e) f)

    Figure 4: Cook’s membrane problem: geometry and discretization with 6, 11, 22, 44,

    88 elements along the edge A-B

    13

  • a) 1.5

    1.7

    1.9

    2.1

    2.3

    2.5

    2.7

    2.9

    3.1

    3.3

    0 22 44 66 88

    No. of elements along A-B

    Dis

    pla

    cem

    en

    tat

    C[m

    m]

    T1

    T1P1

    T2

    T2P1

    Q1R

    Q2P1

    b) 2.80

    2.85

    2.90

    2.95

    3.00

    3.05

    3.10

    3.15

    0 22 44 66 88

    No. of elements along A-B

    Dis

    pla

    cem

    en

    tat

    C[m

    m]

    Q1R

    T1P1IM2ST

    T1P1ES2ST

    Figure 5: Cook’s membrane problem: convergence study for the displacement at point

    C for T1, T1P1, T2, T2P1, Q1R, Q2P1, T1P1IM2ST , T1P1ES2ST , for ν =0.4999

    a)-100

    -80

    -60

    -40

    -20

    0

    20

    40

    60

    80

    100

    0 11 22 33 44

    Str

    ess-S

    11

    [MP

    a]

    T1

    T1P1

    Q1R b)-60

    -50

    -40

    -30

    -20

    -10

    0

    10

    0 11 22 33 44

    Str

    ess-S

    11

    [Mp

    a]

    Q1R

    T1P1IM2ST

    T1P1ES2ST

    Figure 6: Cook’s membrane problem: stress S11 along the clamped edge side A-B forT1, T1P1, Q1R, T1P1IM2ST , T1P1ES2ST , with ν = 0.4999

    14

  • a)

    T1, S11

    +2.00 e+01+6.67 e+00-6.67 e+00-2.00 e+01-3.33 e+01-4.67 e+01-6.00 e+02

    1

    2

    3

    b)

    Q1R, S11

    +2.00 e+01+6.67 e+00-6.67 e+00-2.00 e+01-3.33 e+01-4.67 e+01-6.00 e+02

    1

    2

    3

    c)

    T1P1IM2ST, S11

    +2.00 e+01+6.67 e+00-6.67 e+00-2.00 e+01-3.33 e+01-4.67 e+01-6.00 e+02

    1

    2

    3

    d)

    T1P1ES2ST, S11

    +2.00 e+01+6.67 e+00-6.67 e+00-2.00 e+01-3.33 e+01-4.67 e+01-6.00 e+02

    1

    2

    3

    Figure 7: Cook’s membrane problem, incompressible, contour plots: stress field S11for T1, Q1R, T1P1IM2ST , T1P1ES2ST

    15

  • 4 Conclusion

    This paper is concerned to two-dimensional linear elastic problems with mixed fi-

    nite element formulations for linear triangular elements within the framework of the

    method of incompatible modes and the enhanced strain method. For both mixed finite

    element formulations volume bubble functions are used to enrich the displacement

    field and the enhanced strain field, respectively. Furthermore, conditions for satisfac-

    tion of the patch test A and test C are verified in compressible and incompressible

    case, respectively.

    Cook’s membrane benchmark was investigated in order to compare the results of

    the proposed element formulations with elements from the literature. The conver-

    gence study have been executed for different discretizations. For the incompressible

    case the proposed formulations T1P1IM2ST and T1P1ES2ST converge fast to thereference solution. The proposed elements reveal accurate results of stress distribution

    compared to different elements known from the literature and show no locking in the

    incompressible limit.

    The results in this paper are concerned to physically and geometrically linear prob-

    lems. Therefore, future developments will be directed to nonlinearities.

    Acknowledgement

    The Deutsche Forschungsgemeinschaft (DFG) is acknowledged for its financial sup-

    port under grant MA 1979/8-1 within the joint research project: “Thermo-mechanical

    modeling and characterisation of the solid-liquid interactions in casting processes”.

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    Appendix A

    This part of the Appendix summarizes a matrix representation of vectors and tensors

    relevant for the finite element formulations in Section 2. For the plane strain case we

    obtain the elasticity matrix CPE

    CPE =E

    (1 + ν)(1 − 2ν)

    1 − ν ν ν 0ν 1 − ν ν 0ν ν 1 − ν 0

    0 0 0(1 − ν)

    2

    . (29)

    For the plane stress case we have

    CPS =E

    (1 − ν2)

    1 ν 0 0ν 1 0 0ν ν 0 0

    0 0 0(1 − ν)

    2

    . (30)

    The unit vector, unit matrix and deviatoric matrix are as follows:

    1 =

    1

    1

    1

    0

    , I =

    1 0 0 0

    0 1 0 0

    0 0 1 0

    0 0 0 1

    , Idev = I −1

    311T =

    1

    3

    2 −1 0 0

    −1 2 0 0

    −1 −1 2 0

    0 0 0 3

    .

    (31)

    Note that, the matrices above are used for the calculation of the stresses in the plane

    strain status. For the calculation of the stiffness matrices in Eq.(25) and the residuum

    19

  • in Eq.(15) and Eq.(22), corresponding terms of stresses and strains in z-direction,σz, εz, are deleted. Subsequently the matrices are redefined as

    1 =

    1

    1

    0

    , I =

    1 0 0

    0 1 0

    0 0 1

    , Idev = I −1

    311T =

    1

    3

    2 −1 0

    −1 2 0

    0 0 3

    (32)

    20