On the C1 continuous discretization of non-linear gradient elasticity: A comparison of NEM and FEM based on Bernstein–Bézier patches

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2010; 82:1282–1307Published online 7 December 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2802

On the C1 continuous discretization of non-linear gradientelasticity: A comparison of NEM and FEM based on

Bernstein–Bezier patches

P. Fischer, J. Mergheim and P. Steinmann∗,†

Chair of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstrasse 5, 91058 Erlangen, Germany

SUMMARY

In gradient elasticity, the appearance of strain gradients in the free energy density leads to the need ofC1 continuous discretization m ethods. In the present work, the performances of C1 finite elements andthe C1 Natural Element Method (NEM) are compared. The triangular Argyris and Hsieh–Clough–Tocherfinite elements are reparametrized in terms of the Bernstein polynomials. The quadrilateral Bogner–Fox–Schmidt element is used in an isoparametric framework, for which a preprocessing algorithm is presented.Additionally, the C1-NEM is applied to non-linear gradient elasticity. Several numerical examples areanalyzed to compare the convergence behavior of the different methods. It will be illustrated that theisoparametric elements and the NEM show a significantly better performance than the triangular elements.Copyright q 2009 John Wiley & Sons, Ltd.

Received 7 August 2009; Revised 22 October 2009; Accepted 24 October 2009

KEY WORDS: solids; finite element methods; elasticity; gradient elasticity; C1 finite elements; NaturalElement Method

1. INTRODUCTION

The modeling of materials displaying size effects is an issue of great interest. Size effects dependon the microstructure of the material and can be observed when the size of the specimen issufficiently small, i.e. approximately the size of the microstructure. At this length scale, specimenswith similar shape but different dimensions show different mechanical behaviors. Size effects havebeen observed for a wide variety of materials such as concrete [1], metals [2] and composites [3, 4],to mention just a view.

∗Correspondence to: P. Steinmann, Chair of Applied Mechanics, University Erlangen-Nuremberg, Egerlandstrasse 5,91058 Erlangen, Germany.

†E-mail: [email protected]

Contract/grant sponsor: Publishing Arts Research Council; contract/grant number: 98-1846389

Copyright q 2009 John Wiley & Sons, Ltd.

C1 CONTINUOUS NON-LINEAR GRADIENT ELASTICITY 1283

Within the classical Boltzmann theory such effects cannot be captured, since the theory does notcontain an internal length scale. In contrast, non-classical models like gradient elasticity, establishedby Mindlin [5, 6] and Toupin [7], incorporate higher-order strain gradients in the constitutiveframework. This implies that an internal length scale and size effects can be described sufficiently[8]. The introduction of higher-order strain gradients into the strain energy density (and thus intothe internal virtual work) implicates the need of higher regularity of the interpolation scheme inthe context of a Galerkin method. For a first-order strain gradient (Mindlin) continuum, global C1

continuity has to be ensured.The aim of this contribution is to present several locally defined C1-continuous interpolation

methods and compare their performances for non-linear gradient elasticity. The different interpo-lation methods are reparametrized in terms of the Bernstein–Bezier polynomials to simplify theconstruction of the shape functions and to unify their representation. Particular attention is paid tothe isoparametric concept, for which a geometric mesh optimization algorithm is proposed.

The construction of C1-continuous interpolation functions is far from a trivial task. Either thenumber of degrees of freedom is significantly increasing, which is due to the normal and higher-order derivatives necessary for the description of the shape functions or some kind of beyondnearest neighbor interaction is introduced between the nodes. The latter is the case for mesh-less methods [9–11], subdivision surfaces [12] or maximum-entropy approximation schemes [13].Different locally defined numerical methods that fulfill C1 continuity are the Hermite finiteelements [14] or the C1 Natural Element Method (NEM) [15]. In two dimensions, C1-continuouselements have been developed for the Kirchhoff thin plate theory in the 1960s. These finite elementsgo back to the approaches of Argyris et al. [16], Clough and Tocher (HCT element) [17] and Bogneret al. (BFS element) [18]. Meshfree methods like the EFG method seem to be straightforwardapproaches, since their shape functions are by definition of higher continuity. However, essen-tial boundary conditions cannot be applied directly. An efficient method that combines the easyachievement of the higher continuity with the direct application of essential boundary conditionsis the C1-NEM, originally developed for data interpolation [19].

Owing to the increased numerical complexity of C1 methods, many efforts have been under-taken to find alternative approaches to simulate gradient elasticity without the requirements ofC1-continuous interpolations. Therefore, implicit and operator split methods [20, 21], mixed formu-lations [22], discontinuous Galerkin formulations [23, 24] and micromorphic continuum formula-tions with Lagrange multipliers [25] or penalty parameters [26, 27] have been developed. However,these alternatives involve drawbacks, e.g. incompatibilities of the different approximation fieldsresulting in locking and unphysical results. Moreover, a high number of degrees of freedom hasto be used for the additional approximation fields.

Comparably few work has been done for the application of C1 methods to gradient continua.In gradient plasticity, C1-continuous polynomials were applied to approximate the plastic strainmeasure in [28, 29]. To discretize the displacement field, a nonconforming C1 element was usedin [30] for the investigation of crack tip fields in gradient plasticity. The EFG method was appliedto linear gradient elasticity in [10] and a meshless Petrov–Galerkin approach was applied in [11].Within a finite element framework, the Bell triangle [16] and the BFS element [31] were applied tolinear gradient elasticity in [32, 33] and to gradient plasticity in [34], and the results were comparedwith different formulations, based on the micromorphic theory [35, 36].

Another area where C1-continuous, piecewisely defined interpolation functions are necessaryis the field of computer-aided geometric design (CAGD), initiated by de Casteljau [37]. In thistheory the polynomial interpolation functions are given in a Bernstein–Bezier parameterization.

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2010; 82:1282–1307DOI: 10.1002/nme

1284 P. FISCHER, J. MERGHEIM AND P. STEINMANN

This description has the advantage that it allows for a consistent representation of the shapefunctions of the C1 finite elements and the C1-NEM. Additionally, this parameterization is basedon positions of control points, which include information on the continuity of the functions,instead of on coefficients that are hard to interpret. A parameterization of the Argyris elementby means of the Bernstein–Bezier curves is given in [38]. A good overview on the Bernstein–Bezier reparametrization of different elements can be found in [39, 40], whereby the latter includesalso higher-order continuity elements. These results, which were obtained for the construction ofsmooth surfaces in CAGD, can directly be transferred to the construction of smooth deformationfields in gradient elasticity.

In the present contribution, three C1 finite elements and the C1-NEM are compared. Thesemethods can be formulated within a unified Bernstein–Bezier framework and can be extendedanalogously to three dimensions [19, 41–43]. Alternative methods like the C1-NEM of Hiyoshi–Sugihara [44, 45] and maximum-entropy approaches [13] would definitely deserve furtherconsideration for the problem of gradient elasticity. But this goes beyond the scope of thiscontribution.

The triangular Argyris and the HCT element are used in a subparametric concept, i.e. thegeometry is approximated with polynomials of lower order than the unknown fields. The Argyriselement is a natural approach if a fully polynomial ansatz is chosen for the construction of C1

interpolation fields on triangular meshes in the plane. It has 21 degrees of freedom and incorporatesthe second derivatives at the three corner nodes as well as the normal derivatives at the mid-edgenodes. The HCT element, which is a plane triangular element as well, belongs to the class ofmacroelements. The ansatz functions are piecewise cubic polynomials on three subtriangles. Theresulting 30 degrees of freedom are condensed to 12 by enforcing C1 continuity in the inside ofthe element. The concept of quadrilateral elements is illustrated by means of the well-known BFSelement. It has 16 degrees of freedom, which are given by the primary unknowns, their derivativesand the mixed second-order derivatives at the nodes. Application of the subparametric concept forthe BFS element implicates high restrictions on the discretization [46], i.e. one would necessarilyend up with a mesh of parallelograms. Therefore, the isoparametric concept is used here for theBFS element, whereby the construction of the C1-continuous meshes requires particular attention[31]. A new algorithm to generate and optimize isoparametric meshes, based on a fictitious meshenergy, will be illustrated.

As a fourth interpolation scheme the C1-NEM is applied, which is a smooth extension of theNEM [47, 48], based on the Sibson interpolant [49]. In contrast to the finite element method, theNEM is based on a cloud of points. The shape functions are constructed by means of the first-orderand second-order Voronoi diagram, resulting in a set of functions being C1-continuous up to thenodal points. By embedding these functions locally into a Bernstein–Bezier patch, the desired C1

continuity at the nodes is achieved. The advantage of the C1-NEM is that the only degrees offreedom are the primary unknowns and their first derivatives at the nodes. Furthermore, essentialboundary conditions can easily be applied.

A main focus of the present work is on the consistent Bernstein–Bezier representation of thedifferent C1-continuous interpolation schemes. This links the Hermite finite elements to the C1-continuous NEM. The construction of the shape functions by means of the Bernstein–Bezier poly-nomials is briefly explained. Three numerical examples are considered to compare the performanceof the interpolation schemes for the application to non-linear gradient elasticity. The convergencerates of the methods are compared in terms of the L2, H1 and H2 error norms and local errordistributions are analyzed.



The paper is structured as follows. The basic settings of non-linear gradient elasticity are givenin Section 2. Section 3 deals with the discretization and linearization of the governing equations.Bernstein–Bezier patches are introduced in Section 4 and the Bernstein–Bezier representation ofthe C1-continuous finite elements as well as the construction of the isoparametric mesh for theBFS elements is given in Section 5. In Section 6, the main characteristics of the C1-NEM arerecapitulated and the Berstein–Bezier representation of its shape functions is given. Section 7outlines three numerical examples, to compare the performance of the different numerical methods.The paper closes with a summary and conclusions.

2. GOVERNING EQUATIONS

Before starting to describe the kinematic relations and balance equations, we define the non-standard mathematical operators, being used in the text. Let A and B be tensors of third order,we define the triple contraction A : ·B := AijkBijk. Furthermore, we define the transposition of athird-order tensor A by {AT}ijk := Akji .

Let N be the normal vector to a two-dimensional surface. The normal gradient of the arbitrarytensor C is then defined as

∇NXC :=[∇XC]·N (1)

and correspondingly the surface divergence ∇SX ·C is given by

∇SX ·C :=[∇XC] : [I−N⊗N]. (2)

In the following, the governing equations for gradient elasticity are viewed. Here, smooth boundarysurfaces are assumed for the sake of a simple representation.

2.1. Kinematic relations

We define the kinematic relations

x=u(X) and F :=∇Xu, (3)

with u being the deformation map of a point X in the material domain B0 to a point x in thespatial domain Bt and F being the deformation gradient. For the modeling of gradient elasticity,additionally the third-order tensor

G :=∇XF=∇X[∇Xu] (4)

is needed. The second derivative of the deformation map u is a measure for the curvature andthe stretch gradient. It is symmetric in the last two components by definition, i.e. Gijk :=�i, jk =�i,k j =Gikj.

2.2. Balance equations

In contrast to the classical (Boltzmann) continuum, the free energy density of the gradientcontinuum is not solely dependent on F but also on G, i.e.

W0 :=W0(F,G). (5)



The internal energy of the system is given as �int=∫B0W0(F,G)dV . Its variation can be written as

��int=∫B0

[�W0

�F:�F+ �W0

�G: ·�G

]dV =:

∫B0

[P :�F+Q : ·�G]dV . (6)

Therein, the Piola stress is denoted by P and is thermodynamically conjugate to the deformationgradient F, whereas Q, the so-called double stress, is thermodynamically conjugate to the straingradient G.

By applying the Gauss theorem twice and by the use of the surface divergence theorem, thevariation of the internal energy (6) can be transformed into the expression

��int=∫B0

[[∇X ·[∇X ·Q−P]]·�u]dV +∫

�B0

[tP ·�u−tQ ·∇NX�u]dA, (7)

whereby the tractions tP and the double tractions tQ are defined as

tP := [P−∇X ·Q]·N+[∇SX ·N]Q : [N⊗N]−∇S

X ·[Q ·N]T, (8)

tQ :=Q : [N⊗N]. (9)

The variation of the external energy ��ext is given in terms of prescribed tractions and bodyforces b by

��ext=−∫B0

b·�udV −∫BP

0

tP ·�udA+∫B

Q0

tQ ·∇NX�udA, (10)

whereby its structure is a direct consequence of (7). The expressions tP and tQ are the independentprescribed tractions on the boundaries �BP

0 and �BQ0 , respectively. The variation of the total

potential energy has to vanish and follows from (7) and (10) as

��=∫B0

[P :�F+Q : ·�G−b·�u]dV −∫

�BP0

tP ·�udA+∫

�BQ0

tQ ·∇NX�udA=0. (11)

Assuming the solution to be four times partially differentiable, the weak form (11) is equivalentto the fourth-order local form of the balance equation,

∇X ·(∇X ·Q−P)=b, (12)

subject to the following boundary conditions:

u=u on �Bu0 (13)

∇NXu= ∇N

Xu on �BF0 (14)

tP = tP on �BP0 (15)

tQ = tQ on �BQ0 . (16)

The two types of independent boundaries have the properties �BP0 ∩�Bu0 =�BQ

0 ∩�BF0 =∅ and

�BP0 ∪�Bu0 =�BQ

0 ∪�BF0 =�B0, compare with Figure 1.



Figure 1. Deformation measures and boundary conditions. The different line styles representthe different types of boundaries.

3. DISCRETIZATION

To solve the governing equations of gradient elasticity numerically, an appropriate discretizationscheme has to be derived. Owing to the occurring second-order derivatives of the variation of thedeformation map in (6), i.e. �G=∇2

X�u, the variation �u of the deformation map u is requiredto be in H2(B0), causing C1-continuous formulations. Thus, in the following appropriate C1-continuous discretization schemes are derived. In this section, the general formulation of thediscretized equations is stated; the specification of the particular schemes is given subsequently.

3.1. Spatial discretization

The usual approach in finite element methods to discretize the geometry is the isoparametricconcept, where the geometry X and the deformation field u are discretized by the same set ofshape functions. However, a common approach for triangular C1 elements and meshfree methodsis the subparametric concept. The deformation map u is split into the displacement field u andthe initial geometry X. The approximation Xh of the geometry X can be given in terms of thegeneralized coordinates YI and the appropriate shape functions �I as

Xh =∑IYI�I . (17)

These shape functions �I are linear for the triangular Argyris and HCT elements, and the gener-alized coordinates YI are the nodal coordinates. Within the NEM, a linear approximation of thegeometry by means of the Delaunay triangulation is considered. The BFS element is used inan isoparametric framework. Therefore, the shape functions �I are the cubic Hermitian shapefunctions and the generalized coordinates YI are composed of nodal coordinates and gradientsof the mesh lines. A detailed explanation of the isoparametric spatial discretization is given inSection 4.3.

Second derivative: In order to obtain the second derivative of the shape functions �2�I /�X2,one has to consider the following equations. Let

J := �X�n

(18)



denote the Jacobi mapping. The second derivative is then determined by means of the followingapplications of the chain rule:

�2�I

�X2= �

(��I

�n· �n�X

)/�X=�

(��I

�n·J−1

)/�X

= J−T · �2�I

�n2·J−1+ ��I

�n· �J

−1

�J: �2X

�n2·J−1. (19)

In case of a linear approximation of the geometry, it is obvious that the second part of (19) is zero,due to �2X/�n2=0.

3.2. Discrete displacement

The approximated displacement field uh is given in terms of nodal variables, denoted by aL , andassociated shape functions �L . The approximation of the deformation map is, thus, obtained interms of ansatz functions as

uh =Xh+uh =∑IYI�I +∑

LaL�L . (20)

For the approximation of the variation �u, the first summand of (20) vanishes, and

�uh =∑L

�aL�L (21)

does not depend explicitly on the approximation of the geometry. Please note that the generalizednodal variables aL do not necessarily represent nodal displacements but nodal values of gradientsof u. For the sake of clarity, the approximation of the deformation map variation can alternativelybe written as

�uh =∑I

�uI N I +∑J

�FJ ·MJ +∑K

�GK :LK . (22)

N I ,MJ and Lk are scalar-valued, vector-valued and tensor-valued ansatz functions that are summa-rized as �L in (20) and (21). As known from the classical finite elements �uI are the nodalvariations of the deformation map u, the quantities �FJ represent the nodal variations of thedeformation gradient, whereas �GK refer to the variations of the second derivatives of �u.

3.3. Discrete equations

The weak formulation (6) is discretized by the use of (20) and (21), resulting in∫Bh

[P(uh) :∇X�uh+Q(uh) : ·∇2X�uh]dV

−∫

�BPh

tP ·�uh dA+∫

�BQh

tQ ·∇NX�uh dA−

∫Bh

b·�uh dV =0. (23)



Equation (23) has to hold for all admissible variations �uh . This leads to a finite set of non-linearequations given by the discrete residuals

RI (uh) :=

∫Bh

[P(uh) ·∇X�I +Q(uh) :∇2X�I ]dV

−∫

�BPh

tP�I dA+∫

�BQh

tQ∇NX�I dA−

∫Bh

b�I dV =0. (24)

3.4. Stiffness matrix

The non-linear equation system (24) can be efficiently solved by a Newton–Raphson scheme. Tothis end, the linearization of the residuals is needed

Rk+1I =Rk

I +�RI =0, (25)

whereby the iteration increment �RI is given by

�RI = �RI

�aJ·�aJ =KIJ ·�aJ (26)

and the incremental update of the nodal variables is calculated by solving

KIJ ·�aJ =−RkI . (27)

The tangent stiffness matrix K is given by the composition of

KIJ = �RI

�aJ=∫Bh

∇X�I · �PT

�F·∇X�J dV +

∫Bh

∇X�I · �PT

�G:∇2

X�J dV

+∫Bh

∇2X�I : �QT

�F·∇X�J dV +

∫Bh

∇2X�I : �QT

�G:∇2

X�J dV, (28)

and is obviously symmetric.

4. BERNSTEIN–BEZIER PATCHES

The implementation of Hermite C1-continuous interpolation schemes leads to a significant increaseof complexity compared with standard C0 interpolation schemes. One difficulty is that functionvalues and the tangent vectors behave differently under affine transformations. Therefore, theshape functions cannot be defined on a reference element without the knowledge of the physicalconfiguration. The global construction of the shape functions requires usually the calculation of acoefficient matrix [50] and the solution of a possible badly conditioned system of equations. Here,the theory of Bernstein–Bezier polynomials is used to transform the abstract system of equationsto geometric constraints, necessary to achieve the desired inter-element continuity.

In the following, the notation of the Bernstein–Bezier polynomials is introduced and the construc-tion of the global basis functions in terms of control points is explained. For a more detaileddescription of the topic, the interested reader is referred to [51].



Let k :=(�1, . . . ,�N ) be barycentric coordinates with X1, ..,XN the corresponding set of points.a∈NN denotes a multi index with |a|=∑N

i=1 �i , a!=�Ni=1�i ! and ka :=�N

i=1[�i ]�i . A N -variateBernstein polynomial Ba of degree n :=|a| is then defined as

Ba := n!a!ka= n!a! [�1]

�1[�2]�2 . . . [�N ]�N . (29)

A Bernstein–Bezier patch bn with the scalar-valued Bezier coefficient ba is then calculated as

bn := ∑{a:|a|=n}

baBa. (30)

The Bezier functions and its coefficients can be linked to their control net. Each Bernstein–Bezierpatch bn can be described by the set of control points

Pa :=⎛⎜⎝

N∑i=1

1

n�iXi

ba

⎞⎟⎠ , (31)

with Pa∈RN . With this definition, the continuity of the Bernstein–Bezier patch along an edge canbe given directly by means of the geometrical position of its control points in the N -dimensionalspace.

The difficulty in constructing C1-continuous shape functions in two dimensions is to achievecontinuous derivatives across the boundary between two patches. Therefore, the following prop-erties of the Bernstein–Bezier patches are needed. In two dimensions, a function defined on twotriangular Bezier patches is C1-continuous if each neighboring pair of control net triangles betweenthe patches is coplanar. This statement is proved by Bohm et al. [39]. In Figure 3, two cubictriangular elements with their control points are sketched. The neighboring control net trianglesare shaded.

Example 1: For further illustration in Figure 2, three Bernstein polynomials of degree 5 ofthe bivariate case (1d) and their derivatives are presented. For the sake of clarity, their derivationfrom (29) is shortly demonstrated,

B50 = 5!5!0!�

51�

02=�51 (32)

B41 = 5!4!1!�

41�

12=5�41[1−�1]=5[�41−�51] (33)

B32 = 5!3!2!�

31�

22=10�31[1−�1]2=10[�31−2�41+�51]. (34)

Example 2: For the bivariate case, the piecewise linear function, connecting the control pointsis called the control polygon, compare with Figure 3. Considering only one dimension, simplecomputations result in the derivatives of the Bernstein–Bezier patches at the node X1, which ischaracterized by the local coordinate �1=1

�bn

��2

∣∣∣∣�1=1

=n[b[n−1]1−bn0], (35)



Figure 2. Bivariate Bernstein polynomials. The dashed lines show the directions of the tangent vector atX =0. Left: quintic Bernstein polynomials. Right: derivatives of the quintic Bernstein polynomials. The

remaining polynomials are omitted due to symmetry.

Figure 3. Left: control polygon for a bivariate quintic Bernstein–Bezier patch. Right: projection of twotriangular control nets to the physical space.

and the value of the derivative w.r.t. X is, therefore, given by

�bn

�X

∣∣∣∣�1=1

=n[b[n−1]1−bn0][X2−X1]. (36)

A direct consequence of (36) is that the first derivative across a shared point is continuous if thecontrol points in its direct neighborhood are coplanar, compare with Figure 3. Note that this resultdoes not depend on the degree of the polynomial.

The value of the second derivative w.r.t. �2 is given by

�2bn

��22

∣∣∣∣∣�1=1

=n[n−1][bn0−2b[n−1]1+b[n−2]2], (37)

i.e. the second derivative is zero at the point X1 if the control points Pa with N−�1�2 arecoplanar. For a continuous second derivative only the difference l1=b[n−2]2−[bn0−2b[n−1]1]is of interest, compare with Figure 3. When a new patch is introduced on [X0, X1] with



l0=b2[n−2]−[b0n−2b1[n−1]], it can be shown by straightforward computations that the followingequation holds:

[l0][l1] = [X1−X0]2

[X2−X1]2 (38)

if the function is globally C2-continuous at the shared point X1.

5. FINITE ELEMENT METHOD

In this contribution, we restrict ourselves to three different types of element formulations. Theseare the quintic Argyris element, the HCT, a cubic polynomial macro element and the bicubicBFS element, see Figure 4. The construction of the shape functions in terms of Bernstein–Bezierpatches is explained in detail. A finite element with area A has k vertices, in our case k is either 3or 4. The local vertices of the element are denoted by Xi , 1�i�k, whereas Ei =Xi+1−Xi denotethe edge vectors. We define Xk+1=X1 for the sake of convenience. For the construction of thenormal derivatives at the edges, which are required for globally C1-continuous interpolations, thefollowing definitions are introduced:

hi := A

‖Ei‖ ,

Si := −Ei ·Ei+1

‖Ei‖2 ,

T i := −Ei ·Ei−1

‖Ei‖2 ,

Vi, j := 12 [Ei ⊗E j +E j ⊗Ei ].

(39)

Additionally, a measure for the direction of the normal derivative is introduced as

ci :={sign(Ei

1) if Ei1 =0

sign(Ei2) else.

(40)

Figure 4. Left: Argyris element. Center: HCT element. Right: BFS element. The degrees of freedom arerepresented by n circles for the displacements and the complete [n−1]th order derivatives at the nodes,arrows for the mixed derivative and small lines for the normal derivatives at the center of the edges.



5.1. Argyris element

The Argyris element is constructed by the 21 parameters of the quintic polynomial function space.The shape functions are defined by prescribing their function values and the values of the deriva-tives up to second order at the three element vertices, which leads to 18 equations. The remainingthree values are determined by means of the normal derivatives at the center of the edges. Theresulting elementwise polynomial functions for the approximation are then globally C1-continuous.The Argyris element is used in a subparametric concept, i.e. the geometry is approximated linearly,whereas the displacements are approximated by C1-continuous piecewise quintic polynomialsplines. The geometric properties of the Bezier patches are used for the recursive construction ofthe shape functions.

For the sake of convenience, the expressions

Bia :=

n!a! [�i ]

�1 . . . [�i+N ]�N with �k =�k−N if k>N , (41)

and

bni := ∑{a:|a|=n}

baBia (42)

are introduced. They are equivalent to (29) and (30), but consider a cyclic permutation of thenodes involved. Using the latter definitions together with (39), we achieve the Bezier coefficients.The coefficients with max(a)�3 are uniquely defined by the values of the shape functions at thevertices of the triangle. At the vertex Xi , we obtain the following shape functions:

Ni = Bi500+Bi

410+Bi401+Bi

320+Bi302+Bi

311+CiN B

i221+Di

N Bi212,

Mi = 15 [Ei [Bi

410+2Bi320+Bi

311]−Ei−1[Bi401+2Bi

302+Bi311]]+Ci

MBi221+Di

MBi212,

Li = 120 [Vi,i Bi

320−Vi,i−1Bi311+Vi−1,i−1[Bi

302]+CiLB

i221+Di

LBi212.

(43)

The shape functions corresponding to the normal derivatives at the element edges are constructed by

MiN = 8

15ci hi Bi

221. (44)

The coefficients for the functions Ba, with max(a)=2, denoted by Ci� and Di

�, �∈{N ,M,L}, canbe constructed in two different ways in order to achieve C1-continuous shape functions. The usualapproach for the Argyris element is to enforce that all shape functions, including the ones relatedto the normal derivatives at the element edges are interpolating, i.e. the normal derivative at anelement edge is zero for all but one shape function. In order to obtain this property, the derivativesof the shape functions at the center of an element edge have to be computed at first without thefunctions Ba, with max(a)=2 and the shape functions Mi

N have to be added in an appropriate wayto turn the normal derivatives to zero. This results in the following definition of the coefficients:

CiN = Si , Di

N =T i−1. (45)

CiM = 1

30 [7Ei Si −5Ei+1], DiM=− 1

30 [7Ei−1T i−1−5Ei+1]. (46)

CiL = 1

60 [Vi,i Si −2V i,i+1], DiL= 1

60 [Vi−1,i−1T i−1−2V i−1,i+1]. (47)



However, usually one is only interested in having C1-continuous functions and the interpolatingcharacter of the shape functions Mi

N is of no importance. In this case only the coplanarity conditionof the neighboring triangles has to be satisfied, i.e. the coefficients bi221 depend only on the valuesbi320 and bi302, which makes the construction of the shape functions much easier. Therefore, thecoefficients are reduced to

CiN = Si , Di

N =T i−1, (48)

CiM = 2

5Ei Si , Di

M=− 25E

i−1T i−1, (49)

CiL = 1

20Vi,i Si , Di

L= 120V

i−1,i−1T i−1. (50)

5.2. HCT element

The degrees of freedom of the HCT element are the displacements and their derivatives at thevertices of the triangle. Again, the normal derivatives are described at the center of the edges.Contrary to the other two elements investigated in this contribution, the HCT element is a macroelement, i.e. it is composed of subelements with piecewise polynomial shape functions. The HCTelement is split into three subtriangles, which all share a common vertex Xc in the interior of themacro triangle. This point can be chosen arbitrarily. However, we chose Xc to be the barycenter ofthe triangle, i.e. Xc= 1

3

∑3i=1X

i , which is explicitly used in the construction of the shape functions.The approximation space is built of the 10 cubic polynomials on each subtriangle. The require-

ment of C1 continuity of the displacement field in the inside of the element reduces the totalnumber of degrees of freedom to 12. The mini triangles with the vertices Xk+1,Xk−1 and Xc

are denoted by the index k. The definitions (39) and (40) have to be defined on the subelements;therefore, ◦k represents the corresponding value on the subtriangle k=1,2,3.

In order to achieve C0 continuity, the following conditions have to be valid:

bi+1k0l =bi0kl , k+l=3. (51)

Additionally, the coplanarity condition of the control net, described in Section 4, necessary toachieve C1 continuity yields the following dependencies on the coefficients:

bi102 = 13 [bi−1

111 +bi111+bi201],

bi003 = 1

3

3∑j=1

b j102.

(52)

These dependencies are used in the following three equations (54)–(56) together with the definition

bic :=bi003Bi003+bi102B

i102+bi012B

i012. (53)

For the mini triangle i+2, the shape functions related to point i result in

Ni = B300+B210+B201+CiN B111+bi+2

c

Mi = 13 [E1

i+2B210−E3i+2B201+Ci

MB111]+bi+2c

MiN = c1i+2

13h

1i+2B111+bi+2

c .

(54)



For the mini triangle i+1, the shape functions are given by

Ni = B030+B120+B021+DiN B111+bi+1

c ,

Mi = 13 [E2

i+1B021−E1i+1B120+Di

MB111]+bi+1c ,

MiN = bi+1

c ,

(55)

and the remaining shape functions for the triangle opposite to the node i are simply given as

Ni = bic,

Mi = bic,

MiN = bic.

(56)

Again, similar to the Argyris element there are the two different ways to define the coefficientsCiN , Di

N , CiM and Di

M. For the usual approach, where all shape functions including the onesrelated to the normal derivatives at the element edges are interpolating, the coefficients have to bedefined as

CiN = S1i+2, Di

N =T 1i+1 (57)

CiM = 1

6 [E1i+2S

1i+2−E2

i+2], DiM=− 1

6 [E1i+1T

1i+1−E3

i+1]. (58)

When only the C1 continuity of the shape functions is required, we achieve

CiN = S1i+2, Di

N =T 1i+1, (59)

CiM = 1

3E1i+2S

1i+2, Di

M=− 13E

1i+1T

1i+1. (60)

5.3. BFS element

The BFS element is a four-noded bicubic quadrilateral element. The degrees of freedom are definedat the element nodes by the displacements u and its first derivatives �u/�Xi , i=1,2, as well asthe mixed derivative �2u/[�X1�X2]. In contrast to the Argyris and the HCT element, the shapefunctions of the BFS element are not C1-continuous on general bilinear subparametric meshes. C1

continuity is only guaranteed if the edges of the elements form even parallel lines across the wholebody. Therefore, the BFS element is used in an isoparametric framework. For the construction ofan isoparametric representation of the geometry, a bilinear mesh is used. The shape functions forthe C1 BFS element are constructed by means of a reference mesh, compare with the left-hand sideof Figure 5. It has to be guaranteed that the shape functions are globally C1-continuous definedon the reference mesh to end up with a C1-continuous approximation of the deformation map u.The calculation of the shape functions of one node I requires the consideration of all elementsadjoining this node. The connectivity is given by the connectivity of the bilinear reference mesh.

For the sake of simplicity, it is assumed that the local coordinates are all aligned in the same way,see Figure 5. The shape functions of the BFS element are a combination of cubic bipolynomials.Each direction of the coordinate system n is replaced by the 1d barycentric coordinates �1, �2



Figure 5. Left: reference mesh. Center: edge vectors defined by the bilinear reference mesh. Right: edgevectors defined at the boundary of the geometry.

or �1, �2. The corresponding Bernstein polynomials B� and B� in each direction are constructedanalogous to the 1d case. The shape functions of the reference element ei j are given as

N I = [Bi30+Bi

21][B j30+ B j

21],MI

1 = 13 B

i21[B j

30+ B j21],

MI2 = 1

3 [Bi30+Bi

21]B j21,

L I = 19 B

i21 B

j21,

(61)

where the indices i and j are used in a sense of (41) to indicate a cyclic permutation of themulti-index a.

It is important to realize that it is not necessary that there exists a global reference mesh witha unique global coordinate system. Instead, a semi-local coordinate systems n can be defined foreach patch of the elements adjacent to one node, compare with Figure 5. The local element-wisecoordinates n have to be related to the semi-local ones by means of a rotation map, to ensure C1

continuity. However, it is not possible to construct C1-continuous bicubic elements with more thanfour elements at a node, since the mixed derivatives have to be given in the two directions of thereference element edges.

Let eIij, i, j ∈{1,2}, denote the elements involved at the global node I , XI the coordinate of

node I in the bilinear mesh and XIij, i, j ∈{0,1,2}, the surrounding node, compare with Figure 5.

Defining the secant vectors

TI1 := 1

2 (XI20−XI

10),

TI2 := 1

2 (XI02−XI

01),(62)

the matrix DI :=[TI1 TI

2] is used for the geometry approximation

Xh =∑JYJ�J :=∑

IXI N I +DI ·MI +HI L I . (63)

For the construction of the matrix DI at the boundary, there exists different possibilities. Two ofthem are that either the vectors TI

1 and TI2 can be defined like presented in Figure 5 or the vectors

TI1 and TI

2 can be made perpendicular to each other. The second case can decrease the quality



Figure 6. Left: mesh, designed with the algorithm of Section 5.3, Center: mesh with geometric optimizationof Section 5.4, Right: mesh constructed by the Petera–Pitmann algorithm.

of the mesh; however, the applying of essential boundary conditions for the normal gradient isgetting easier.

The coefficients HI are taken to be zero if the mesh is of sufficient regularity. Otherwise a meshoptimization algorithm, see Section 5.4, should be applied to calculate the optimal value of HI .

5.4. Isoparametric optimization

The shape functions of the isoparametric BFS element depend on the geometry of the neighboringelements. This can lead to numerical errors when irregular reference meshes with differentlyoriented elements are used, e.g. see the mesh in Figure 6 on the left-hand side. When this meshis used in a simulation, the largest error is observed along the diagonal, where the disturbance ofthe mesh

G := �2X

�n2(64)

shows the highest values. This motivates the optimization of the mesh by minimizing G. Therefore,the following potential is defined:

�= 1

2

∫B�

G : ·GdVn, (65)

where dVn is an area increment in the local coordinate system of each reference element. Thispotential (65) has to be minimized with respect to the generalized coordinates, i.e. its variation�� has to vanish

��=∫B�

G : ·�GdVn=0. (66)

Owing to the quadratic nature of the potential, the variation (66) is a bilinear form on G and�Xh . The resulting linear system of equation gives the necessary change �YI of the geometryapproximation

�YI =K−1IJ RI , (67)

whereby the residual vector and the stiffness matrix are given by

RI =∫B�

G :∇2n�I dVn, (68)

KIJ =∫B�

∇2n�I :∇2

n�J dVn. (69)



The results of this mesh optimization are compared with non-optimal meshes, constructed asdescribed in Section 5.3 and with meshes, generated by the approach, given by Petera andPittmann [31]. The three resulting meshes for the quarter of a plate with a circular hole are shownin Figure 6. A convergence study for this example is done in Section 7.

6. NATURAL ELEMENT METHOD

The numerical performance of the presented C1 finite elements will be compared with a recentlydeveloped meshfree method, the NEM. The NEM is a numerical method, based on the Voronoitessellation and its dual Delaunay triangulation. The Delaunay triangulation defines the domain andthe Voronoi diagram is used for the construction of the interpolating shape functions. In contrast toother mesh free methods, the NEM has the advantage that essential Dirichlet boundary conditionsare exactly satisfied. In general, the method is restricted to convex domains, but different extensionsexist to overcome this limitation. A straightforward approach is the use of a-shapes [52] which isapplied in the present work. Further possible extensions are discussed in Cueto et al. [53].

The Voronoi diagram for a set of N points K :={X1, . . . ,XN } is defined as an decompositionof the space R2 into N sets T (XI ) in the way

T (XI )={X∈R2 :‖X−XI‖�‖X−XJ‖ ∀XJ ∈K}. (70)

The Sibson interpolant is constructed with the help of the second-order Voronoi diagram, whichis highlighted in Figure 7.

T I (X∗)={X∈R2 :‖X−X∗‖�‖X−XI‖�‖X−XJ‖ ∀XJ ∈K}. (71)

The Sibson interpolant �I at the point X∗ is then defined as

�I (X∗)= A(T I (X∗))∑J A(T J (X∗))

, (72)

with A(◦) the Lebesgue measure of a set ◦. The denominator in (72) is equal to the area of theVoronoi cell T (X∗) of the set K∪X∗. The Sibson interpolants �I are C∞ everywhere apart fromthe nodes and the boundary of their support, where they are C0 and C1, respectively. They form apartition of unity and are non-negative by definition. The Sibson interpolant for an interior pointis illustrated on the left hand side of Figure 8.

Figure 7. Voronoi diagram and Delaunay triangulation for a set of 8 points. Left: shaded area shows theVoronoi cell T (XI ). Right: the intersection of the Voronoi cell T (X∗) of the newly introduced point �

with T (XI) defining the value of the Sibson interpolant.



0

2

40

0.2

0.4

0.6

0.8

1

0

2

40

0.2

0.4

0.6

0.8

1

Figure 8. Left: Sibson interpolation function. It is clearly not C1-continuous at the nodal point. Right:C1-NEM shape functions for the displacement. The derivatives at the node are zero.

When we regard the Sibson interpolants as generalized barycentric coordinates, C1 continuitycan be achieved by locally embedding of the interpolants K in a cubic Bernstein–Bezier patch

�I (K)= ∑{a:|a|=3}

baBa(K). (73)

A geometric interpretation of this relation gives that N nonzero Sibson interpolants at a point X∗define the barycentric coordinates of an N−1 dimensional simplex. Since the Sibson interpolantshave local support, the number of nonzero interpolants varies for different points X∗. Therefore,the dimension N−1 of the barycentric coordinates is defined locally (Figure 8).

The coefficients ba, with max(a)=3, are directly defined by the nodal value. The derivativesat the node I define the expressions ba, with max(a)=2. The remaining Bezier coefficients withmax(a)=1 can be chosen arbitrarily and do not affect the C1 continuity of the resulting shapefunctions. Similar to Sukumar and Moran [15], we follow the approach of Farin [19] and choosethe remaining coefficients such that quadratic completeness of the approximation field is obtained.

The shape functions at a point X∗ are then given as

N I (X∗) = [�I ]3+∑a2I

Ba(K)+ 1

3

∑a1I

Ba(K),

MI (X∗) =∑a2I

[XJ −XI ]Ba(K)+ 1

3

∑a1I

[XJ +XK −2XI ]Ba(K),

(74)

where the following definition is used akI :={a :�I =k and maxJ =I (�J )=1}. An example of N I

is presented in Figure 8, right. The advantage of the C1-NEM is that the degrees of freedomare restricted to the field values and their first derivatives at each node. No additional degrees offreedom like edge normal derivatives or second derivatives are needed. Yet the drawback of theNEM is its high computational cost for the construction of the shape functions. The identificationof the active shape functions at each point of calculation as well as the construction and derivationof the different areas is more expensive than the evaluation of polynomials on finite elements.



7. NUMERICAL EXAMPLES

The performance of the three investigated C1 finite elements and the C1-NEM is now analyzedby means of three numerical benchmark problems. We focus on the performance of the numericalscheme and use a simple hyperelastic material model for all examples. The free energy densityW0=W0(F,G) is of the following additive type:

W0(F,G)=WH0 (F)+WG

0 (G), (75)

whereby a compressible Neo-Hookean model is chosen for WH0

WH0 = 1

2� ln2 J+ 1

2�[F :F−ndim−2ln J ] (76)

and a quadratic expression for WG0

WG0 = 1

2�l2G : ·G. (77)

For the material parameters of the non-gradient part, a Young’s modulus of 200kN/mm2 and thePoisson ratio of 0.3 are used. The parameter for the internal length scale is related to the size ofthe specimens and is defined as l= L .

In all three examples, homogeneous Neumann boundary conditions for the double tractionsare used, i.e. tQ=0. The imposition of various Dirichlet boundary conditions influences differentdegrees of freedom. In Table I, the prescribed degrees of freedom for four different settingsare summarized, whereby the indices n and t denote the normal and the tangential directions,respectively.

To the authors’ best knowledge, no analytical solution for any non-linear gradient elastic problemwith an inhomogeneous solution is known. Therefore, the error is measured in comparison withanother numerical overkill solution, which is determined with a significantly higher number ofdegrees of freedom. The rates of convergence are calculated with respect to the total numberof degrees of freedom to facilitate the comparison of the finite elements and the NEM. Sincethe applied meshes are quite regular, the convergence rates are expected to be half the rate ofconvergence calculated with respect to the element size h.

7.1. Uniaxial loading of plate with hole

In the first numerical example a stress concentration problem is considered, similar to a problemin [25, 26]. A rectangular specimen with length 3L and width L and a hole with radius 0.5L isclamped on the right-hand side in horizontal direction and stretched by u1=3L to 100% elongation.

Table I. Imposition of different boundary conditions. 1. homogeneous displacements, 2. constantdisplacements, 3. fixed normal direction 4. boundary is symmetry axis of the problem.

un ut �nun �nut �t un �t ut �2nun �2nut �2t un �2t ut �2nt un �2nt ut

1. 0 0 — — 0 0 — — 0 0 — —2. un ut — — 0 0 — — 0 0 — —3. 0 — — — 0 — — — 0 — — —4. 0 — — 0 0 — 0 — 0 — — 0



Figure 9. Geometries of the three examples investigated in this contribution: Left: plate with hole. Center:L-shaped domain. Right: rigid flat punch.

Figure 10. Relative error norms for the different mesh construction algorithms: No—without optimization,PePi—construction algorithm from [31], G-opt—presented geometric optimization algorithm.

Owing to symmetry reasons, only one quarter of the plate is simulated. A detailed description ofthe geometry and the loading conditions is presented in Figure 9.

In this example, the circular hole makes the construction of an isoparametric mesh for theBFS element difficult and can result in poorly designed elements. Therefore, the various meshconstruction algorithms of Sections 5.3, 5.4 and [31] are compared with each other. In Figure 10the L2, H1 and H2 error norms are presented for a refinement of the differently constructedisoparametric meshes, respectively. It is obvious that the geometric optimization algorithm revealsa much better absolute error together with an improved rate of convergence. In the followingcomparison of the different numerical schemes, these optimal isoparametric meshes are used.

In Figure 11, the relative error norms and semi-norms for the different numerical methods areshown. The rates of convergence and the absolute errors show only minor differences between thefour numerical schemes. However, considering the L2 and H1 norms the BFS element and theNEM have a slight advantage over the triangular elements. The poor behavior of the NEM for ahigh number of degrees of freedom is due to errors in the numerical integration and the accuracyof the algorithm for the computation of the shape functions. Here, a 19 point Gauss integrationscheme is used on the Delaunay triangulation. However, a stabilized conforming nodal integrationscheme, similar to [54], could improve the results. The non-smooth progression of the error normsrelated to the BFS element are induced by the mesh optimization algorithm. The Argyris elementand the HCT element show a similar performance except for the H2 norm, where the Argyriselement shows smaller errors due to its continuous second-order derivatives at the nodes.



Figure 11. Convergence behavior for the plate with hole.

Figure 12. Frobenius norm of local error ‖Fre f −Fh‖F for the plate with hole. Upper left: Argyris. Upperright: HCT element. Bottom left: BFS element with geometrical optimization. Bottom right: NEM.

The total error is composed of significantly different local error distributions for the variousnumerical methods. The local error of the deformation gradient F and the strain gradient G interms of the Frobenius norm‡ is displayed in Figures 12 and 13. For the Argyris and the HCTelement, the highest errors arise in strongly deformed regions, whereas the NEM shows a spuriouserror distribution. The major error for the BFS element results from the missing C1 continuity ofthe geometry at the upper left corner. However, this error becomes negligible for finer meshes anddoes not influence the positive performance of the BFS element.

7.2. L-shaped domain

In the second example, an L-shaped plate is considered, which was also simulated in [55, 56]. TheL-shaped panel is considered as one quarter of a double-symmetric frame, which is symmetricallyloaded on its vertical boundaries. Dirichlet boundary conditions associated with symmetry planesare imposed on the lower boundary and on the right-hand side, as illustrated in Figure 9. At the

‡The Frobenius norm for a second-order and third-order tensor is defined to be ‖Fref−Fh‖F :=[∑ij[F refij −Fh

ij ]2]1/2and ‖Gref−Gh‖F :=[∑ijk[Gref

ijk −Ghijk]2]1/2.



Figure 13. Frobenius norm of local error ‖Gre f −Gh‖F for the plate with hole. Upper left: Argyris. Upperright: HCT element. Bottom left: BFS element with geometrical optimization. Bottom right: NEM.

Figure 14. Convergence behavior for the L-shaped domain.

Neumann boundary, a constant load in the horizontal direction tP1 =−L−1kN/mm is applied. Aspresented in Figure 14, for this example, the investigated error norms of the four numerical methodsbehave in a similar manner although the convergence rates are lower than in the previous example.These low convergence rates, especially in the H2 seminorm, are due to the corner singularity inthe strain gradient, which is depicted in Figure 15. It seems that the NEM can deal better withthis kind of singularity and shows the best performance in the present example.

7.3. Rigid flat punch

As a third numerical example, a rigid flat punch is considered, which was also presented in [57, 58].The symmetric geometry and loading conditions are shown in Figure 16, only one half of thestructure is simulated. The deformation of a rectangular block in response to an indentation isstudied. The vertical displacement of the flat punch is increased up to 12.5% compression. Toavoid the use of a contact algorithm, the displacement at the punch is directly imposed on theupper surface. Usually this test is used to analyze the performance of incompressible elementformulations. Our focus is to test the different C1-continuous schemes for an example involvinga highly inhomogeneous deformation field. The domain is decomposed into a regular grid. Theelement size h is then asymptotically proportional to the square root of the total number of degrees



Figure 15. Deformed configuration of the L-shaped domain and the reference solution. Left: deformationgradient F11, Right: corner singularity in G112.

Figure 16. Convergence behavior for the rigid flat punch.

of freedom. Figure 16 shows the relative error in the L2 norm and the H1 and H2 semi-norms.Consistent with the previous examples, the NEM and the BFS element show the lowest absoluteerrors. The Argyris element exhibits the highest error values due to its high continuity at the nodes.In terms of the rates of convergence, all numerical schemes display similar results.

The local error distribution of F in term of the Frobenius norm is shown in Figure 17. For allmethods the major part of the error occurs at the corner of the punch, where the deformation ishighly inhomogeneous. The three finite element approaches show comparable results, whereby theNEM shows slight error concentrations at Voronoi points.

8. SUMMARY AND CONCLUSION

In the present contribution three C1-continuous finite elements, the Argyris, the HCT and the BFSelement, and additionally the C1-NEM were analyzed for the simulation of non-linear gradientelasticity. The formulation of the shape functions for the different numerical approaches was unified



Figure 17. Frobenius norm of local error ‖Fre f −Fh‖F for the rigid flat punch. Upper left: Argyris. Upperright: HCT element. Bottom left: BFS element. Bottom right: NEM.

by means of the Bernstein–Bezier patches. This leads to an intuitive construction of the shapefunctions. Additionally, an optimization algorithm for the design of isoparametric meshes has beenintroduced, which was shown to improve the quality of the result significantly.

The performance of the different C1-continuous methods was analyzed by means of threenumerical examples. The convergence rates in the L2, H1 and H2 semi-norms and the local errordistributions were compared. The rates of convergence of all methods do not differ significantly.However, the overall errors are quite more affected.

Considering the three analyzed numerical examples, the NEM and the BFS element show thebest performance comparing the absolute error to the number of degrees of freedom. But bothmethods have drawbacks: The computational costs for the calculation of the NEM shape functionsare comparatively high, and the results of the BFS element are strongly mesh dependent. Theconstruction of the isoparametric meshes becomes a complicated task for arbitrary geometries. Inthis case, the two investigated triangular elements, the Argyris and the HCT element, are preferred,which present reliable methods for any kind of domain. The performance of all four numericalschemes depends heavily on the smoothness of the solution: a singularity in the strain gradientor inhomogeneous deformation fields reduces the rates of convergence significantly. However,a conclusive evaluation of the four different methods in the framework of non-linear gradientelasticity requires further investigations, e.g. involving local mesh refinement. This will be thesubject of future work.

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On the C1 continuous discretization of non-linear gradient elasticity: A comparison of NEM and FEM based on Bernstein–Bézier patches