Virtual Elements for Elasticity - unicas.it · Virtual Elements for Elasticity L. Beirão da Veiga Department of Mathematics University of Milan in collaboration with: F. Brezzi and

Virtual Elements for Elasticity

L. Beirão da VeigaDepartment of Mathematics

University of Milan

in collaboration with:

F. Brezzi and D. Marini

CASSINO, February 2015

L. Beirão da Veiga (Univ. of Milan) VEM for Elasticity CASSINO - 2015 1 / 34

The Virtual Element Method

The Virtual Element Method (VEM) is a generalization of the FiniteElement Method that takes inspiration from modern Mimetic FiniteDifference schemes.

VEM allows to use very general polygonal and polyhedral meshes,also for high polynomial degrees and guaranteeing the patch test.In addition, it may lead to additional advantages, such as “easy tocode” highly regular spaces.


Why polygons/polyhedrons?

The interest (and use in commercial codes?) for polygons/polyhedra isrecently growing.

Immediate combination of tets and hexahedrons

Easier/better meshing of domain (and data) features

Automatic inclusion of “hanging nodes”

Adaptivity: more efficient mesh refinement/coarsening

Generate meshes with more local rotational simmetries

Robustness to distortion

.......

? for example CD-ADAPCO and ANSYS.


Some polytopal methods

Mimetic F.D. Shashkov, Lipnikov, Brezzi, Manzini, BdV, ....

HMM: Eymard, Droniou, ...

Polygonal FEM: Sukumar, Paulino, ...

Weak Galerkin FEM: Wang, ....

HHO: Ern, di Pietro

Polygonal DG: Cangiani, Houston, Georgoulis, ...

VEM: this talk !!!

..........


The linear elasticity problem

We consider the linear elasticity problem in two dimensions: − div(

2µ ε(u) + λ div(u) I)

= f in Ω

u = 0 on ∂Ω,

whereΩ ⊂ R2 is a polygonal domain;λ and µ are the Lamé coefficients in R+;the loading f is assumed component-wise in L2(Ω).

NOTE: When the ratio λ/µ >> 1 we fall into the range ofalmost-incompressible materials ...


Variational formulation

The variational formulation of the problem reads:Find u ∈ V such that

a(u,v) =< f ,v > ∀v ∈ V

where the spaceV := (H1

0 (Ω))2

and the bilinear forms

a(v ,w) := 2µ∫

Ωε(v) : ε(w) dx + λ

∫Ω

div v div w dx

=: 2µ aµ(v ,w) + λ aλ(v ,w).

NOTE: for given µ, λ the form a(·, ·) is symmetric, continuous andcoercive on V. The problem above is well posed.


A Virtual Element Method

We will build a discrete problem in following formfind uh ∈ Vh such thatah(uh,vh) =< f h,vh > ∀vh ∈ Vh,

whereVh ⊂ V is a finite dimensional space;ah(·, ·) : Vh ×Vh → R is a discrete bilinear form approximating thecontinuous form a(·, ·);< f h,vh > is a right hand side term approximating the load.

Let k ≥ 1 be a fixed integer index. Such index will represent thedegree of accuracy of the method.


A Virtual Element Method

We will build a discrete problem in following formfind uh ∈ Vh such thatah(uh,vh) =< f h,vh > ∀vh ∈ Vh,

whereVh ⊂ V is a finite dimensional space;ah(·, ·) : Vh ×Vh → R is a discrete bilinear form approximating thecontinuous form a(·, ·);< f h,vh > is a right hand side term approximating the load.

Let k ≥ 1 be a fixed integer index. Such index will represent thedegree of accuracy of the method.


The local spaces Vh|E

Let Th be a simple polygonal mesh on Ω. This can be anydecomposition of Ω in non overlapping polygons E with straight faces.

The space Vh will be defined element-wise, by introducinglocal spaces Vh|E ;the associated local degrees of freedom.

For all E ∈ Th:

Vh|E =

vh ∈ [H1(E) ∩ C0(E)]2 : −∆vh ∈ [Pk−2(E)]2,

vh|e ∈ [Pk (e)]2 ∀e ∈ ∂E.

the functions vh ∈ Vh|E are known explicitly on ∂E ;the functions vh ∈ Vh|E are unknown inside the element E !

it holds [Pk (E)]2 ⊆ Vh|E





For all E ∈ Th:

Vh|E =

vh ∈ [H1(E) ∩ C0(E)]2 : −∆vh ∈ [Pk−2(E)]2,

vh|e ∈ [Pk (e)]2 ∀e ∈ ∂E.







For all E ∈ Th:

Vh|E =

vh ∈ [H1(E) ∩ C0(E)]2 : −∆vh ∈ [Pk−2(E)]2,

vh|e ∈ [Pk (e)]2 ∀e ∈ ∂E.




Degrees of freedom for Vh|E

Let E ∈ Th and vh ∈ Vh|E .

pointwise values vh(ν) at all corners ν of E(k − 1) pointwise values on each edge:

vh(xei ) , xe

i k−1i=1 distinct points on edge e ;

volume moments:∫E

vh · pk−2 ∀pk−2 ∈ [Pk−2]2(E)

The dimension of the space Vh|E is

dim(Vh|E ) = 2Nk + k(k − 1),

with N number of edges of E .


Degrees of freedom for Vh|E

Let E ∈ Th and vh ∈ Vh|E .

pointwise values vh(ν) at all corners ν of E(k − 1) pointwise values on each edge:

vh(xei ) , xe

i k−1i=1 distinct points on edge e ;

volume moments:∫E

vh · pk−2 ∀pk−2 ∈ [Pk−2]2(E)

The dimension of the space Vh|E is

dim(Vh|E ) = 2Nk + k(k − 1),

with N number of edges of E .


Depiction of the degrees of freedom for Vh|E

m=1 m=2 m=3

Green dots stand for vertex pointwise valuesRed squares represent pointwise values on edgesBlue squares represent internal (volume) moments


The space Vh and the form ah(·, ·)

The global space Vh is built by assembling the local spaces Vh|E asusual:

Vh = v ∈ V : v |E ∈ Vh|E ∀E ∈ Th.

The bilinear form ah(·, ·) is built element by element

ah(vh,wh) =∑

E∈Th

aEh (vh,wh) ∀vh, wh ∈ Vh,

whereaE

h (·, ·) : Vh|E × Vh|E −→ R

are symmetric bilinear forms that approximate

aEh (·, ·) ' a(·, ·)|E .


The space Vh and the form ah(·, ·)

The global space Vh is built by assembling the local spaces Vh|E asusual:

Vh = v ∈ V : v |E ∈ Vh|E ∀E ∈ Th.

The bilinear form ah(·, ·) is built element by element

ah(vh,wh) =∑

E∈Th

aEh (vh,wh) ∀vh, wh ∈ Vh,

whereaE

h (·, ·) : Vh|E × Vh|E −→ R

are symmetric bilinear forms that approximate

aEh (·, ·) ' a(·, ·)|E .


Building the local bilinear forms

We recall the exact local form

aE (v ,w) := 2µ∫

Eε(v) : ε(w) + λ

∫E

div(v) div(w)

=: 2µ aEµ (v ,w) + λ aE

λ (v ,w).

Analogously we will introduce

aEh (vh,wh) := 2µ aE

h,µ(vh,wh) + λ aEh,λ(vh,wh),

for all vh,wh in Vh|E .

NOTE: we do not have a completely explicit expression for thefunctions in Vh|E !


Bilinear form aEh,λ(·, ·)

Note that ∀vh ∈ Vh|E and ∀p ∈ Pk−1(E) we can compute∫E

div(vh) p = −∫

Evh ·∇p +

∫∂E

(vh · nE )p .

Therefore we can compute the L2 projection of div(vh) on Pk−1(E):Π0

k−1(div(vh)

)∈ Pk−1(E)∫

E

(Π0

k−1div(vh))

p =

∫E

div(vh) p ∀p ∈ Pk−1(E).

We simply define, for all vh,wh ∈ Vh|E ,

aEh,λ(vh,wh) :=

∫E

Π0k−1(div(vh)

)Π0

k−1(div(wh)

).




div(vh) p = −∫

Evh ·∇p +

∫∂E

(vh · nE )p .


k−1(div(vh)

)∈ Pk−1(E)∫

E

(Π0

k−1div(vh))

p =

∫E



aEh,λ(vh,wh) :=

∫E

Π0k−1(div(vh)

)Π0

k−1(div(wh)

).




div(vh) p = −∫

Evh ·∇p +

∫∂E

(vh · nE )p .


k−1(div(vh)

)∈ Pk−1(E)∫

E

(Π0

k−1div(vh))

p =

∫E



aEh,λ(vh,wh) :=

∫E

Π0k−1(div(vh)

)Π0

k−1(div(wh)

).


Bilinear form aEh,µ(·, ·)

Note that ∀vh ∈ Vh|E and ∀p ∈ [Pk (E)]2 we can compute

aEµ (vh,p) =

∫Eε(vh) : ε(p) = −

∫E

vh · div(ε(p)

)+

∫∂E

vh ·(ε(p)nE

).

Thus we can compute the “aµ” projection of vh on [Pk (E)]2:Πa

k (vh) ∈ [Pk (E)]2

aEµ (Πa

k (vh),p) = aEµ (vh,p) ∀p ∈ [Pk (E)]2

Fix the kernel part

For instance, we can fix the kernel

span(

10

),

(01

),

(−yx

)by using an euclidean scalar product on the d.o.f. values.



Note that ∀vh ∈ Vh|E and ∀p ∈ [Pk (E)]2 we can compute

aEµ (vh,p) =

∫Eε(vh) : ε(p) = −

∫E

vh · div(ε(p)

)+

∫∂E

vh ·(ε(p)nE

).

Thus we can compute the “aµ” projection of vh on [Pk (E)]2:Πa

k (vh) ∈ [Pk (E)]2

aEµ (Πa

k (vh),p) = aEµ (vh,p) ∀p ∈ [Pk (E)]2

Fix the kernel part

For instance, we can fix the kernel

span(

10

),

(01

),

(−yx

)by using an euclidean scalar product on the d.o.f. values.



It clearly holds, for all vh,wh in Vh|E :

aEµ (vh,wh) = aE

µ (Πak (vh),Πa

k (wh)) + aEµ (vh − Πa

k (vh),wh − Πak (wh)) .

We define

aEh,µ(vh,wh) := aE

µ (Πak (vh),Πa

k (wh)) + sE (vh − Πak (vh),wh − Πa

k (wh))

where the bilinear form sE (·, ·) is only required to scale correctly.

For instance, for all vh,wh in Vh|E ,

sE (vh,wh) = 2µ#dofs∑i=1

dofi(vh)dofi(wh).




aEµ (vh,wh) = aE

µ (Πak (vh),Πa



We define

aEh,µ(vh,wh) := aE

µ (Πak (vh),Πa


k (wh))




dofi(vh)dofi(wh).




aEµ (vh,wh) = aE

µ (Πak (vh),Πa



We define

aEh,µ(vh,wh) := aE

µ (Πak (vh),Πa


k (wh))




dofi(vh)dofi(wh).



The bilinear form satisfies the following.

Consistency:

aEh,µ(p,vh) = aE

µ (p,vh) ∀vh ∈ Vh|E , ∀p ∈ [Pk (E)]2 .

Stability:There exist α?, α? ∈ R+ such that

α?aEµ (vh,p) ≤ aE

h,µ(vh,p) ≤ α?aEµ (vh,p) ∀vh ∈ Vh|E .

NOTE: in order for α?, α? to be uniform (element-independent) somemesh regularity assumptions are needed (eg. star-shaped with respectto a ball, minimal edge length condition).



The bilinear form satisfies the following.

Consistency:

aEh,µ(p,vh) = aE

µ (p,vh) ∀vh ∈ Vh|E , ∀p ∈ [Pk (E)]2 .

Stability:There exist α?, α? ∈ R+ such that

α?aEµ (vh,p) ≤ aE

h,µ(vh,p) ≤ α?aEµ (vh,p) ∀vh ∈ Vh|E .

NOTE: in order for α?, α? to be uniform (element-independent) somemesh regularity assumptions are needed (eg. star-shaped with respectto a ball, minimal edge length condition).


The discrete load term

We consider k ≥ 2 first. Let, for all E ∈ Th, the approximated load f h|Ebe the L2-projection of f |E on Pk−2(E) .

Then(f h,vh)h :=

∑E∈Th

∫E

f hvh

that is computable due to the internal dofs of Vh|E .

In the case k = 1 a simple integration rule based on the vertex valuesof the polygon can be used, for instance

(f h,vh)h :=∑

E∈Th

(∫E

f) 1

NE

∑ν∈∂E

vh(ν).

Note: for k = 2 better choices can be made.


A λ−uniform convergence result

Mesh assumptions: let every element E be star-shaped with respect toa ball of radius ≥ ρhE , and let every two vertexes be at least ρhEdistance apart (ρ fixed for the whole mesh family).

Theorem: [BdV, Brezzi, Marini, 2013]It holds the error estimate

||u − uh||H1(Ω) ≤ Chs|u|Hs+1(Ω) , 1 ≤ s ≤ k ,

with C independent of λ if k ≥ 2.

NOTE: and inf-sup condition holds for k ≥ 2, granting theindependence from λ. In the case k = 1 there exists a class of meshesfor which the inf-sup holds (see [BdV, Lipnikov, 2010]).

NOTE: it immediately extends to the Stokes problem!


A λ−uniform convergence result

Mesh assumptions: let every element E be star-shaped with respect toa ball of radius ≥ ρhE , and let every two vertexes be at least ρhEdistance apart (ρ fixed for the whole mesh family).

Theorem: [BdV, Brezzi, Marini, 2013]It holds the error estimate

||u − uh||H1(Ω) ≤ Chs|u|Hs+1(Ω) , 1 ≤ s ≤ k ,

with C independent of λ if k ≥ 2.

NOTE: and inf-sup condition holds for k ≥ 2, granting theindependence from λ. In the case k = 1 there exists a class of meshesfor which the inf-sup holds (see [BdV, Lipnikov, 2010]).NOTE: it immediately extends to the Stokes problem!


Construction of the stiffness matrix for aEh,µ(·, ·)

It is immediate to check that ∀ vh,wh ∈ Vh|E

aEµ (vh,wh) = aE

µ (Πakvh,Π

akwh) + aE

µ ((I − Πak )vh, (I − Πa

k )wh).

Then the bilinear form

aEh,µ(vh,wh) = aE

µ (Πakvh,Π

akwh) + sE ((I − Πa

k )vh, (I − Πak )wh)

is consistent and stable, provided the positive bilinear formsE : Vh|E × Vh|E → R scales like the original bilinear form a.

Let now the local stiffness matrix ME

MEij := aE

h,µ(φi , φj) ∀i , j = 1,2, ...,N

with φi the canonical basis associated to the degrees of freedom.


Construction of the stiffness matrix for aEh,µ(·, ·)

It is immediate to check that ∀ vh,wh ∈ Vh|E

aEµ (vh,wh) = aE

µ (Πakvh,Π

akwh) + aE

µ ((I − Πak )vh, (I − Πa

k )wh).

Then the bilinear form

aEh,µ(vh,wh) = aE

µ (Πakvh,Π

akwh) + sE ((I − Πa

k )vh, (I − Πak )wh)

is consistent and stable, provided the positive bilinear formsE : Vh|E × Vh|E → R scales like the original bilinear form a.

Let now the local stiffness matrix ME

MEij := aE

h,µ(φi , φj) ∀i , j = 1,2, ...,N

with φi the canonical basis associated to the degrees of freedom.


Construction of the stiffness matrix

We introduce also mαnα=1 a basis for the polynomial space

[Pk ]2 = spanmαnα=1.

Since Pm ⊆ Vh|E , the matrix

D =

dof1(m1) dof1(m2) . . . dof1(mn)dof2(m1) dof2(m2) . . . dof2(mn)

......

. . ....

dofN(m1) dofN(m2) . . . dofN(mn)

.expresses the mα in terms of the Vh|E basis.



We have also a matrix

B =

~Pkerφ1 . . . ~PkerφN

aE (m2, φ1) . . . aE (m2, φN)...

. . ....

aE (mn, φ1) . . . aE (mn, φN)

expressing (in terms of the bases) the right hand side in the definition

aE (Πvh,p) = aE (vh,p) ∀p ∈ Pm(E)/RP0(Πvh) = P0(vh)

NOTE: such matrix is computable! (integration by parts and d.o.f.sdefinition)



Let

G = BD =

~Pkerm1

~Pkerm2 . . . ~Pkermn0

(∇m2,∇m2

)0,P . . .

(∇m2,∇mn

)0,P

......

. . ....

0(∇mn,∇m2

)0,P . . .

(∇mn,∇mn

)0,P

.

Compute the matrices corresponding to the projection operator:

Π? = (G)−1B , Π = DΠ?.

Finally compute the local stiffness matrix

ME = ΠT? GΠ? + (I−Π)T (I−Π),



Let

G = BD =

~Pkerm1


(∇m2,∇m2

)0,P . . .

(∇m2,∇mn

)0,P

......

. . ....

0(∇mn,∇m2

)0,P . . .

(∇mn,∇mn

)0,P

.Compute the matrices corresponding to the projection operator:

Π? = (G)−1B , Π = DΠ?.


ME = ΠT? GΠ? + (I−Π)T (I−Π),



Let

G = BD =

~Pkerm1


(∇m2,∇m2

)0,P . . .

(∇m2,∇mn

)0,P

......

. . ....

0(∇mn,∇m2

)0,P . . .

(∇mn,∇mn

)0,P

.Compute the matrices corresponding to the projection operator:

Π? = (G)−1B , Π = DΠ?.


ME = ΠT? GΠ? + (I−Π)T (I−Π),


Low order numerical tests

We consider only k = 1 and two family of meshes on Ω = [0,1]2:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

We set λ = µ = 1 and the loading f in accordance with the solution

u(x , y) =(

sin (πx) sin (πy) , sin (πx) sin (πy)).


Low order numerical tests

Convergence plots for max-error at nodes and discrete energy error.

10−2

10−1

100

10−4

10−3

10−2

10−1

Max−norm error

Energy error

10−2

10−1

100

10−4

10−3

10−2

10−1

Max−norm error

Energy error


An additional example

Consider a [0,1]2 domain of elastic material (µ = λ = 1) with a stiffinclusion (µ = λ = 104).The domain is clamped on the left side and subjected to a verticalforce on the right hand side.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Very little happens in the inclusion, but it cannot be ignored: FEM mustmesh the inclusion while VEM can use a single element.



Sample of meshes (F2 and V2) used, after deformation:

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Coarser and finer meshes where used (F1,F2,F3 and V1,V2,V3).



Vertical displacement at upper right corner: -0.083818847125168(reference with a finer FEM mesh)

mesh type dofs displacementmesh F1 179 −0.080290340968924mesh V1 123 −0.080289101597552

mesh F2 673 −0.082401721109781mesh V2 428 −0.082400518411353

mesh F3 2609 −0.083363990429520mesh V3 1584 −0.083362795720282


Nonlinear elasticity and inelasticity

Together with C. Lovadina and D. Mora, we are currently developingthe VEM method for small deformation problems in structuralmechanics.

Without showing the details of the construction, we here present somepreliminary test for k = 1 and the following models:

Henky-Mises nonlinear elasticityVon Mises plasticity with linear isotropic and kinematic hardening

NOTE: the method is built in such a way to include (automatically)general “black-box” constitutive algorithms, that are applied only onceper element (similar to single Gauss point integration).

uh|E −→ Π0Eε(uh) −→ σE


Henky-Mises elasticity

The lamé constants λ, µ depend on the (point-wise) norm of the strain.

We consider a problem with known regular trigonometric solution on[0,1]2 and two family of meshes:

We report the maximum displacement error at the nodes and the errorin an H1 discrete norm

|||v|||21,h :=∑e∈Eh

he ‖∂v/∂te‖20,e


Henky-Mises elasticity

The results are as follows:

Mesh 1/h N Eh0,∞ R0,∞ Eh

1,2 R1,2

380 1.2225e-2 – 1.9750e-1 –1560 2.2164e-3 2.42 8.4773e-2 1.20

T 2h 3292 1.1218e-3 1.82 5.8722e-2 0.98

5664 6.7588e-4 1.87 4.5076e-2 0.989084 3.9767e-4 2.25 3.4684e-2 1.11

12816 2.8737e-4 1.89 2.9372e-2 0.9717188 2.1964e-4 1.83 2.5529e-2 0.9622848 1.5928e-4 2.26 2.1811e-2 1.10

4 50 1.5401e-1 – 1.0516e-0 –8 162 3.3021e-2 2.62 5.3972e-1 1.14

T 5h 16 578 7.1005e-3 2.42 2.7525e-1 1.06

32 2178 1.6650e-3 2.19 1.3832e-1 1.0464 8450 4.1133e-4 2.06 6.9382e-2 1.02128 33282 9.0462e-5 2.21 3.2452e-2 1.05


Von Mises plasticity

We consider the classical stretching test of a plastic strip with a hole,using a return map algorithm to solve the von Mises plasticity model:

E = 70, ν = 0.2, r0 = 0.8, Hkin = Hiso = 10.

Plastic parameter γ:

0 2 4 6 8 10 120

2

4

6

8

10

12

14

16

18

20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Von Mises plasticity

Vertical displacements at point (0,5) and horizontal displacement atpoint (5,0), maximum stress and “total stress” for the four Voronoimeshes and for a reference value obtained with a fine triangular mesh.

Mesh # elements Displ. A Displ. B σmax σTV1 64 0.7839 -0.3181 3.3842 244.2324V2 256 0.8173 -0.3928 4.1354 240.1062V3 1024 0.8253 -0.4212 4.4266 238.7653V4 4096 0.8277 -0.4300 4.7755 238.3688

Reference 45312 0.8284 -0.4334 4.9891 238.2631


A note on the literature

Basic paper on VEM: [BdV, Brezzi, Cangiani, Manzini, Marini,Russo: M3AS, 2013]Guide to coding: [BdV, Brezzi, Marini, Russo: M3AS, 2014]Linear elasticity: [BdV, Brezzi, Marini: SINUM, 2013]Small deformations (nonlinear): [BdV, Lovadina, Mora: inpreparation]NOT CITED: diffusion problem in mixed form, Stokes problem,Kirchhoff plates, wave propagation, DFN simulation, eigenvalueproblems, topology optimization, contact problems, relation withother methods... [B. Ayuso, S. Berrone, S. Bordas, G. Gatica, D.Mora, G. Paulino, I. Perugia, R. Rodriguez, C. Talischi, P.Wriggers, ...]


Conclusions

We presented a Virtual Element Method for linear elasticity:

(local) discrete spacesdegrees of freedombuilding the discrete bilinear forms

The discrete bilinear forms where shown to be k-consistent andstable

The method was shown to be convergent and free of volumetriclocking.

We have shown first extensions to the nonlinear case.

Virtual Elements is a very recent technology, that we believe to befull of potential!


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Virtual Elements for Elasticity - unicas.it · Virtual Elements for Elasticity L. Beirão da Veiga Department of Mathematics University of Milan in collaboration with: F. Brezzi and