VEM for Inelastic Solids
R.L. Taylor and E. Artioli
1 Introduction
The virtual element method (VEM) is a generalization of the finite element method
recently introduced in [1–5]. It is capable to deal with general polygonal/polyhedral
meshes and to easily implement highly regular discrete spaces (see for instance [3]).
The VEM approach has experienced an increasing interest in the scientific commu-
nity, both from a mathematical and an engineering point of view.
By making use of non-polynomial shape functions, the VEM can easily handle
general polygons/polyhedrons without effort in integration at the element level. Poly-
tope meshes can be very useful in several instances, such as domains with cracks,
inclusions, and for the automatic use of hanging nodes, moving boundaries, and
adaptivity.
In addition to approaches cited above recent works on this topic have been applied
to many areas of scientific interest, including [6–15] to name a few. In the more
specific framework of structural mechanics, VEM has been introduced in [2] for
two dimensional linear elasticity at general “polynomial” order, in [11] for three
dimensional linear elasticity at the lowest order, in [6] for general two dimensional
elastic and inelastic problems in small deformation (lowest order), in [14] for contact
problems and in [16, 17] for applications to two-dimensional problems with elastic
and inelastic behavior.
Dedicated to Roger Owen on the occasioon of his 75th birthday.
R.L. Taylor (✉)
Department of Civil and Environmental Engineering, University of California,
Berkeley, USA
e-mail: [email protected]
E. Artioli
Department of Civil Engineering and Computer Science, University of Rome
‘Tor Vergata’, Rome, Italy
e-mail: [email protected]
© Springer International Publishing AG 2018
E. Oñate et al. (eds.), Advances in Computational Plasticity, Computational
Methods in Applied Sciences 46, DOI 10.1007/978-3-319-60885-3_18
381
382 R.L. Taylor and E. Artioli
The present contribution applies the VEM to the case of arbitrary order of accu-
racy in a general computational framework using the Finite Element Analysis Pro-
gram (FEAP) [18]. The goal is hence to develop a numerical tool that retains the
many features of an existing finite element platform for use in many problems with
existing algorithms for arbitrary order of accuracy. The key idea of the method is
to use a projection of the unknown field and relative gradient on a suitable polyno-
mial space base and to treat such an approximation as it would be done for stan-
dard isoparametric finite element technology. In particular, the aim of the paper is
to show that VEM structure can fit a general finite element coding setting and make
use of the many available features of a general purpose FEM platform, for arbitrary
“polynomial” order. Starting from this key point leads to use of classical tools of
linear/nonlinear finite element analysis as shown by representative benchmarks pre-
sented in the numerical test section of the work.
2 Theory for C𝟎 Elements
For the two-dimensional theory, we consider a general polygon with ne sides and
vertices. In addition we assume each of the sides may have a k-order interpolation
using nodes placed along the individual segments. For example for a quadratic order
we use the two vertices and one mid-side node to define the behavior of the edge; for
a cubic order we use the two end nodes and two nodes placed along the interior of the
edge. Accordingly, each i-edge may be interpolated between vertex i and vertex i + 1(assuming a numbering proceeding counter-clockwise around the element boundary)
as1:
𝐗 = N(1)a (𝜉) ��a ; a = i, i + 1, c1, c2,… (1)
where N(1)a are one-dimensional Lagrange interpolation functions expressed in terms
of the parent coordinate: −1 ≤ 𝜉 ≤ 1 [19]. The interpolations are based on spacing
of the nodal parent coordinates on the edge. These may be equally spaced, as is
common for traditional isoparametric elements, or at the Gauss-Lobatto points as
described in much of the VEM literature (e.g., the hitchhiker’s guide [4]).
The application of VEM is most easily implemented in existing finite element
programs (e.g.,FEAP [18]) by using a shape function form [19]. This allows for
reuse of much of the theory in finite element references [19–21] and existing program
modules for element residual and tangent operators without significant programming
effort. In the classical finite element method we express the interpolations over the
area (or volume) of an element in terms of two-dimensional shape functions which
may be defined as
1In this work the superscript on shape functions denotes the spatial dimension of the interpolation.
VEM for Inelastic Solids 383
v(𝐗, t) = 𝜑a(𝐗) va(t) (2)
where the shape functions 𝜑a(𝐗) satisfy the Kronecker property [19]
𝜑a(��b) ={
1 ; if a = b0 ; if a ≠ b (3)
For general polygonal shaped elements, however, it is difficult to find closed form
expressions for these shape functions. Instead in the VEM the displacement field is
expressed by an approximation which we denote as2
v(𝐗, t) ≈ N(2)a (𝐗) va (4)
which we assume also satisfy the Kronecker property. The above form is described by
a projection of the classical functions onto the space of VEM functions. In addition,
in VEM approximations to derivatives of the shape functions 𝜙a are obtained by a
separate projection operation, that is
𝜕v𝜕Xj
=𝜕𝜙a
𝜕Xjva ≈
𝜕N(2)a
𝜕Xjva (5)
3 The Local Virtual Element Space
Consider a two dimension domain Ω ⊂ ℝ2partitioned into a collection h of non-
overlapping polygons E, not necessarily convex: Ω = ∪E∈hE.
For each polygon E, we will denote by Vi (i = 1,… , ne) its vertices counterclock-
wise ordered, and by ei the edge connecting Vi to Vi+1. For each polygon E we define
a local finite element space Vk(E). The local virtual element space Vk(E) contains all
polynomials of degree k plus other functions whose restriction on each edge is a
polynomial of degree k. Any function vh ∈ Vk(E) satisfies the following properties:
∙ vh is a polynomial of degree k on each edge e of E, i.e., vh|e ∈ k(e);∙ vh on 𝜕E is globally continuous, i.e., vh|𝜕E ∈ C0(𝜕E);∙ Δvh is a polynomial of degree k − 2 in E, i.e., Δvh ∈ k−2(E).It is proved [4] that Vk(E) admits the following degrees of freedom:
∙ the value of vh at the vertices of E;
∙ the value of vh at k − 1 internal points on each edge e [viz. Eq. (1)];
∙ the moments of vh in E up to order k − 2 :
1|E| ∫E
vh m𝛼, 𝛼 = 1,… , nk−2
2The superscript on shape functions denotes the spatial dimension of the interpolation.
384 R.L. Taylor and E. Artioli
where the scaled monomials m𝛼 are defined in (8a) and nk−2 = dimk−2(E).
As a consequence, the dimension of Vk(E) is
dimVk(E) = ne + ne(k − 1) + nk−2 = nek +(k − 1)k
2.
4 Shape Function Approximation
We denote the approximate k-order shape functions of parameters in the VEM as
N(2)a = m(k)
i (Xj)P0ia (6a)
and those for the derivatives as
𝜕N(2)a
𝜕Xj= m(k−1)
i Pdija (6b)
In this regard, we again note that VEM differs from the usual isoparametric approach
where the derivatives are deduced directly from the shape functions by appropriate
use of the chain rule.
In the above we follow the practice of using scaled parameters to define the com-
plete set of k-order polynomials for m(k)i . Accordingly, we let
Xj =Xj − Xc
j
hd(7)
where Xcj is a suitably chosen point in the interior and hd is a measure of the
diameter of the VEM. In the present work Xcj is located at the centroid of the VEM
element which simplifies some of the expressions. Thus, for a k order VEM in two-
dimensions we use the set
𝐦(k) =[1 X1 X2 X2
1 X1X2 X22 … Xk
2
](8a)
for shape functions and for the derivatives the k − 1 set
𝐦(k−1) =[1 X1 X2 … Xk−1
2
](8b)
VEM for Inelastic Solids 385
4.1 Projector Development
A projection operator is introduced as follows:
P0E,k ∶ Vk(E) ⟶ k(E)
For the sake of simplicity, the subscripts E and/or k will be omitted when no confu-
sion can arise.
The operator P0is defined for every vh ∈ Vk(E) (up to a constant) by the following
orthogonality condition:
∫E∇pk ⋅ ∇
(P0vh − vh
)= 0 for all pk ∈ k(E). (9)
For the projectors we follow the developments presented in the hitchhiker’s guide
[4] and in Artioli et al. [16, 17]. We consider a VEM with ne vertices and boundary
segments.
The displacement projector, P0ia, may be computed from
P0ia = H−1
il Bla (10)
where
H1l =1Ve
ne∑a=1
mkl (��a) ; k = 1
H1l =1Ve ∫Ve
mkl d ; k > 1
(11a)
and
Hil =∫Ve
𝜕mki
𝜕Xj
𝜕mkl
𝜕Xj
dV (11b)
and
B1a =1ne
; a = 1, 2, 3 ; k = 1
B1a ={
0 ; a = 1, 2,… ,
1 ; a = (k + 1)(k + 2)∕2 ; k > 1
Bla =∫Ve
𝜕mkl
𝜕Xj
mki P
dija dV
(12)
Again details for the computation may be found in [4, 16].
Since neither the function vh nor its gradient are explicitly computable in the ele-
ment interior points, the method proceeds by introducing a projection operator PdE,k,
386 R.L. Taylor and E. Artioli
representing the approximated spatial gradient associated with the virtual displace-
ment, defined as:
PdE,k ∶ Vk(E) ⟶ k−1(E) (13)
Given vh ∈ Vk(E), such an operator Pdis defined as the unique function Pd(vh) ∈
k−1(E)2 that satisfies the condition:
∫EPd(vh) ⋅ pk−1 =
∫E∇(vh) ⋅ pk−1, ∀pk−1 ∈ k−1(E)2 . (14)
This operator represents the best approximation of the spatial gradient (in the square
integral norm) in the space of piecewise polynomials of degree (k − 1). Although the
functions in Vk(E) are virtual, the right hand side in (14) (and thus the operator Π)
turns out to be computable with simple calculations [16, 17].
The result for the derivative projector, Pdija, may be expressed by
Pdija = Gil Ps
lja (15)
where
Gij =∫Ve
mk−1i mk−1
j dV (16)
and for the boundary nodes
Pslja =
∑ne ∫
𝜕Vne
mk−1l nj N(1)
a (𝜉) dS (17)
where nj are components of the outward normal to the boundary and N(1)a are one-
dimensional shape functions that interpolate the boundary segment nodal vertex and
(for k > 1) mid-edge parameters. In addition for k > 1 there are internal parameters
required for these are computed as described in references [4, 16].
5 Quadratic VEM with Curved Edges
The use of VEM with interpolation along the edge permits the description of ele-
ments with curved sides. For example if we consider a quadratic VEM in two-
dimensions the edges between two adjacent vertices will have 3-nodes: 2-vertex
nodes and 1-node placed at the mid-side. The hexagonal VEM shown in Fig. 1a can
have the top edge curved as shown in Fig. 1b. The integration of (17) may be conve-
niently carried out using a one-dimensional isoparametric interpolation of the edge
using a Gauss quadrature [19].
VEM for Inelastic Solids 387
(a) Quadratic hexagonal VEM (b) Curved quadratic edge
Fig. 1 Quadratic VEM with curved edges
(a) Triangles for volume integration (b) Local numbering & coordinates.
Fig. 2 Volume integration of a VEM
Volume integrations of a VEM is often computed by dividing the element into
sub-triangles about a common node in the interior (viz. Fig. 2a) and using quadrature
in terms of barycentric coordinates. If the barycentric coordinates on the triangle are
denoted as La (Fig. 2b) and we let the position for L3 be 𝐗3 (local numbering) then
the interpolation for coordinates in the triangle may be described by
𝐗 = N1(La) ��1 + N2(La) ��2 + L3 ��3 + N4(La) ��4 (18a)
where
N4(La) = 4L2 L2 (19)
and the shape functions for the vertex nodes are given by
388 R.L. Taylor and E. Artioli
Fig. 3 Curved edge
boundary interpolation for
cubic VEM
N1(La) = L1 −12N4(La)
N2(La) = L2 −12N4(La)
N3(La) = L3
(20)
Grouping the terms together gives the interpolation
𝐗 = L1 ��1 + L2 ��2 + L3 ��3 + N4(La)[��4 −
12��1 −
12��2
](21)
A similar process may be used to define higher order VEMs with curved edges.
Figure 3 shows a cubic edge where the placements for the b1 may be at either equal
spaces as usually used in isoparametric finite element analysis or at Gauss-Lobatto
points as in some previous VEM presentations. Currently, curved edges are used
only at problem boundaries.
5.1 Element Development
The presentation of the shape functions in the previous sections and the procedure
to integrate over element volumes using regular subregions, allows for a straight for-
ward development of elements for small deformation problems in solid mechanics.
In this case we can define the strains in terms of the approximate shape function
derivatives given in (6b) as
ε =⎧⎪⎨⎪⎩𝜖11𝜖222 𝜖12
⎫⎪⎬⎪⎭=
⎡⎢⎢⎢⎢⎢⎢⎢⎣
𝜕N(2)a
𝜕X10
0𝜕N(2)
a
𝜕X2𝜕N(2)
a
𝜕X2
𝜕N(2)a
𝜕X1
⎤⎥⎥⎥⎥⎥⎥⎥⎦
{uava
}(22)
To account for near-incompressible constraints the three field mixed 𝐮-p-𝜃
approach may be used. [19, 20] For small strain problems in Cartesian coordi-
nates (e.g., plane strain and plane stress) the VEM and mixed strain are of identical
order and, consequently, no differences result between standard displacement and
the mixed approach can be appreciated.
If we consider problems with an axisymmetric geometry the strains are given by
VEM for Inelastic Solids 389
ε =⎧⎪⎨⎪⎩
𝜖rr𝜖zz𝜖θθ
2𝜖rz
⎫⎪⎬⎪⎭=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
𝜕N(2)a
𝜕r0
0𝜕N(2)
a
𝜕zN(2)a
r0
𝜕N(2)a
𝜕z𝜕N(2)
a
𝜕r
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
{uava
}(23)
where ua and va are nodal displacements in the radial and axial direction, respec-
tively. In this case we also need a projection for the shape functions N(2)a . For near
incompressible behavior, a 𝐮-p-𝜃 mixed form may be introduced to retain constant
volumetric behavior over each element. This may be introduced in the usual way as
Πv =∫E
p[𝜖v − 𝜖ii
]r dr dz (24)
in which we use
p = m(k)i pa
𝜖v = m(k)i 𝜖a
(25)
Introducing the above into the weak form of a solid mechanics problem leads
to the residual and tangent matrix for the consistent part of the formulation. In this
form of the formulation we may introduce whatever constitution we want in the for-
mulation and in the present work we utilize the material library from FEAP [18].
Note, however, that both consistency and stability are required for convergence of
any formulation.
5.2 Stabilization
Except for simplex elements the above development for a VEM results in rank defi-
cient tangent matrices and it is necessary to introduce proper stabilization terms.
Here we use the approach described by Artioli et al. [16] where the stabilization
tangent matrix for each component is given by
𝐊ab = 𝜏
[𝛿ab − Dai d−1ij Dbj
](26a)
where
dij = Dai Daj (26b)
and 𝜏 is a parameter dependent on the consistent tangent matrix part. The basic sta-
bilization matrix is for boundary nodes is given by
390 R.L. Taylor and E. Artioli
Dia = mi(Xa) (27a)
and for the internal node by
Dia =∫Ve
mi(X) dV (27b)
In previous presentations on elasticity the parameter 𝜏 is recommended to be
selected a one-half the trace of the consistent tangent matrix. In the present work, the
choice of the 𝜏 parameter for problems which may have near incompressible behav-
ior is computed from the trace of the consistent part of the stiffness scaled by the ratio
of the shear modulus to the bulk modulus of an isotropic material. Accordingly, we
use
𝜏 = 34tr(𝐊) 1 − 2𝜈
1 + 𝜈(28)
For inelastic material we replace the trace of the stiffness by that from the previous
time step to ensure that quadratic convergence occurs in our Newton algorithm.
6 Examples
The formulation for k order 1 and 2 VEM elements has been implemented as a mod-
ule in the general purpose finite element program FEAP [18]. Element formulations
for displacement and mixed 𝐮-p-𝜃 (viz. Zienkiewicz et al. [20]) are considered. The
implementation has access to the material library which permits the consideration
of elastic, visco-elastic and elasto-plastic models. All solution and graphics features
are also available so that mesh, displacement and stress contours may be displayed.
6.1 Example: Tension Strip with Slot
Stabilization Parameter VerificationTo verify the selection of our scaling of the stabilization parameter 𝜏 we consider an
axisymmetric behavior for the tension strips shown in Fig. 4 with all straight edges
of the slotted tensions strip quadrant restrained in the normal direction. The top edge
is then subjected to a displacement of 0.002 and the total reaction force measured.
The linear elastic properties of the material are selected to give a shear modulus (𝜇)
equal to 1000 and different Poisson ratios between 0.45 and 0.4999995 are consid-
ered. Table 1 presents the results for the two stabilizing criteria for a k = 1 VEM
and compare results to a mixed Q1P0 finite element solution. All subsequent VEM
analyses use the scaled Poisson ratio value.
VEM for Inelastic Solids 391
x
y
1.0 1.0
1.0
1.0
0.30.05 0.05
σy = 1
(a) Problem description (b) Mesh for quadrant
Fig. 4 Region and mesh used for slotted tension strip
Table 1 Verification on choice of stabilization parameter
𝜈 Total Load
12tr(𝐊) 3(1−2𝜈)
4(1+𝜈)tr(𝐊) Q1P0
0.4500000 1.0766E + 01 1.0733E + 01 1.0740E + 010.4950000 9.0359E + 01 8.2794E + 01 8.3322E + 010.4995000 7.6399E + 02 3.1542E + 02 3.2342E + 020.4999500 7.1749E + 03 4.4037E + 02 4.5616E + 020.4999950 7.1128E + 04 4.5855E + 02 4.7570E + 020.4999995 7.1058E + 05 4.6064E + 02 4.7775E + 02
Elasto-plastic Tension StripAs a second example we consider the solution of the tension strip shown in Fig. 4 for
both plane and axisymmetric models that employ an elasto-plastic material model
with isotropic linear hardening. For this analysis we set the properties to: E = 2500,
𝜈 = 0.25, 𝜎y = 5, Hiso = 5. The load-displacement curve for the two cases is shown
in Fig. 5a, b. Results were also verified by comparison with a solution using Q1P0
elements. Contours for the Mises stress are shown in Fig. 5c–f for the four cases con-
sidered. The stress contours for the VEM elements are computed using a local least
squares averaging scheme [22]. For the k = 1 models the stress over each element
is projected onto constants and averaged at the nodes. For the k = 2 elements the
stress over each sub-triangle of the element is assumed linear and these are averaged
at the nodes. As observed in Fig. 5 the results for the different schemes are smooth
considering the coarse nature of the mesh around the slot.
392 R.L. Taylor and E. Artioli
(a) Plane Strain (b) Axisymmetric
55
2
33
555
5
55
55
1
3
4
4
4
5
5
4
3
5
5
1
2
3
4
2
5
2
4
MISES STRESS
1 1.70675E+00
2 2.37301E+00
3 3.03927E+00
4 3.70553E+00
5 4.37179E+00
Time = 1.0000000E+00
55
5
2
5
5
5
5
5
5
33
3
5
2
2
3
4
3 44
1
5
4
2
1
55
4
2
5
5
5
4
5
4
5
MISES STRESS
1 1.60883E+00
2 2.32609E+00
3 3.04335E+00
4 3.76061E+00
5 4.47788E+00
Time = 1.0000000E+00
(c) Plane Strain k =1 (d) Plane Strain k =2
44
44
3
4
5
1
4
5
1
4
1
51
2
55
5
4
MISES STRESS
1 1.13637E+00
2 1.91251E+00
3 2.68866E+00
4 3.46481E+00
5 4.24096E+00
Time = 1.0000000E+00
44
44
3
4
5
1
4
5
1
4
1
51
2
55
5
4
MISES STRESS
1 1.13539E+00
2 1.91174E+00
3 2.68809E+00
4 3.46444E+00
5 4.24079E+00
Time = 1.0000000E+00
(e) Axisymmetric displ. (f) Axisymmetric mixed
Fig. 5 a–bLoad-displacement response for plane strain and axisymmetric, elasto-plastic strip with
a slot. c–f Mises stress contours
6.2 Contact of a Circular Disk on a Rigid Surface
As a last example we consider the behavior of k = 1 and k = 2 VEM formulations
for a circular disk with unit radius that is pressed into a rigid surface. One quadrant
of the disk is meshed for k = 1 and k = 2 VEM elements and has an initial gap of
0.002 units. The material properties are elasto-plastic with the same values as for
the slotted strip. A uniform displacement at the top of the disk quadrant is applied in
VEM for Inelastic Solids 393
(a) Deformed mesh for quadrant (b) Load-Displacement
Fig. 6 Region and mesh used for slotted tension strip
equal steps to a maximum value of 0.1 units. The final deformed mesh for the k = 2discretization is shown in Fig. 6a and the load-displacement result for both cases in
Fig. 6b. The contact constraint is enforced at each node using a Lagrange multiplier
approach (this solution method is valid since the master contact surface is rigid). The
use of quadratic elements smooths the result, primarily due to the added number of
nodes on the boundary but also due to curvature of the edge which gives a more
accurate modeling of the approach.
7 Closure
In this presentation we have formulated the two-dimensional virtual element method
in terms of shape functions that permit the straight forward coding of the consis-
tent part stiffness and residual for small strain inelastic behavior. The approach also
allows for curved edges on elements with quadratic and higher order forms. The
approach has been implemented in the general purpose finite element analysis sys-
tem FEAP and allows for solution of both steady state and transient analyses. The
methodology has been demonstrated for elasto-plastic behavior of plane strain and
axisymmetric problems including a simple case of contact on a rigid surface. With
a shape function approach it is possible to easily extend the solution to problems in
other areas, including for example thermal, seepage, and fluid forms which utilize
C0interpolations.
394 R.L. Taylor and E. Artioli
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