Virtual Elements for plates and shells

Virtual Elements for plates and shells Claudia Chinosi Dipartimento di Scienze e Innovazione Tecnologica, Università del Piemonte Orientale, Alessandria, Italia E-mail: claudia.chinosi@unipmn.it eywords: Finite Element Method, Virtual Method, plates and shells. SUMMARY. General polygonal and polyhedral meshes naturally arise in the treatment of complex solution domains and heterogeneous materials and are particularly suited to moving meshes techniques as well as to adaptive mesh refinement and de-refinement. The Virtual Element Methods represent a new perspective for finite elements on polygons and polyhedra. INTRODUCTION In a variety of engineering applications it has to deal with complex shell structures. The Finite Element method is considered a fundamental numerical procedure to solve shell mathematical models and over the years various finite element schemes have been developed for the analysis of general shell structures. In recent times in order to solve a great variety of problems on fairly irregular decompositions, a Mimetic Finite Difference approach has been introduced (see e.g. []). This approach could be considered as an extension of the Finite Element spaces of lowest order to rather general element geometries. Recently some attempts have been made to extend this methodology to include higher order approximation in order to gain better accuracy in the numerical results. The analysis of these extensions gave rise to a new interpretation of Mimetic Finite Differences and to a subsequent new approach much closer to Finite Elements that is called Virtual Element Method (VEM) (see e.g. [2], [3], [4]). The basic idea of this new method can be summarized as follows. The Virtual Element spaces are just like the usual Finite Element spaces with the addition of suitable non-polynomial functions. The novelty here is that the non-polynomial part does not need to be known in detail but only through its degrees of freedom. Therefore the construction of the local stiffness matrix is simple and can be done for much more general geometries and higher order continuity conditions. In this paper we present the Virtual Element Method on the two-dimensional Laplace equation. We preserve the generality of the shape of the elements in the decomposition of the computational domain and the generality in the degree k of accuracy that we require to the method. After we generalize the Virtual Element approach to the linear plate bending problems in the irchhoff-love formulation. The performance of the new method is tested solving benchmark problems whose results are compared with the ones obtained using classical Finite Element schemes. Very good agreement with the theoretical results is observed. 2 THE POISSON PROBLEM We consider the problem u = f in Ω u = on Γ = Ω () where Ω R 2 is a polygonal domain with boundary Ω and f L 2 (Ω). The variational formulation reads { Find u V := H (Ω) suchthat (2) a(u,v) = (f,v) v V

with (, ) is the scalar product in L 2, a(u,v) = ( u, v), v = a(v,v). It is well known that problem (2) has a unique solution (see e.g. [5]). 3 THE DISCRETE PROBLEM. ABSTRACT FRAMEWOR Let{T h } h be a sequence of decompositions ofωinto disjoint open polygons, and let{e h } h be the set of edgeseof T h. The bilinear forma(, ) and the norm can be split as a(u,v) = T h a (u,v) u,v V, v = ( T h v 2,) /2 v V. (3) We observe that the seminorm is a norm on H (Ω), equivalent to the usual H (Ω) norm. For functions in the spaceh ( T h ) := T h H () we define a brokenh seminorm: A - We assume to have, for eachh, a spacev h V ; v h, = ( T h v 2, )/2 v V. (4) a symmetric bilinear forma h fromv h V h to R which can be split as we did forain (3): an elementf h V h a h (u h,v h ) = 3. An abstract convergence theorem T h a h (u h,v h ) u h,v h V h ; (5) A2 - In addition to the above hypothesis, we assume that there exists an integer k such that for all h, and for all in T h, we havep k () V h, (P k () is the space of polynomials of degree k on) and k Consistency: for allp P k () and for all v h V h, a h (p,v h ) = a (p,v h ) (6) Stability: there exist two positive constantsα andα, independent ofhand of, such that v h V h α a (v h,v h ) a h (v h,v h ) α a (v h,v h ) (7) Under the previous assumptions we can state the following theorem (see [2] for the proof). Theorem Under the assumptions A-A2 the discrete problem: { Find uh V h suchthat a h (u h,v h ) =< f h,v k > v h V h. (8) 2

has a unique solution u h. Moreover, for every approximation u I V h of u and for every approximationu π of u that is piecewise in P k, we have u u h C( u u I + u u π h, + F h ), (9) where C is a constant depending only on α and α, and for any h, F h is the smallest constant such that (f,v h ) < f h,v h > F h v h v h V h. () 4 DISCRETIZATION We construct the space V h and the bilinear form a h such that the assumptions of the Theorem are satisfied 4. Construction of V h For every decomposition T h of Ω into simple polygons with n edges, barycenter x and diameterh, we define for each element and fork the spaces: B k ( ) := {v C ( ) : v e P k (e) e } () V,k = {v H () : v B k ( ), v P k 2 ()} (2) It is easy to prove that the dimension ofv,k is given by N dim V,k = nk +k(k ) (3) We consider a set of degrees of freedom that is unisolvent forv,k (see [2]): V,k - The values ofv h at the vertices. E,k - Fork >, the values ofv h at k uniformly spaced points on each edgee. P,k - Fork >, the moments m(x)v h(x)dx m M k 2 (), where M k 2 () is the set of(k 2 k)/2 monomials {( ) s x x M k 2 =, s k 2}, (4) h with s := s +s 2 andx s := x s xs2 2. We can now define the space V h on the wholeω: V h = {v V : v B k ( ), v P k 2 (), T h }. (5) The dimension ofv h is given by N tot dimv h = N V +N E (k )+N P k(k )/2, (6) where N V, N E and N p are respectively the total number of internal vertices, internal edges and elements in T h. In agreement with the local choice of degrees of freedom we choose the following degrees of freedom that are unisolvent forv h : V - The values ofv h at the internal vertices. E - Fork >, the values ofv h atk uniformly spaced points on each internal edgee. P - Fork >, the moments m(x)v h(x)dx m M k 2 (), in each element. 3

4.2 Construction of a h We observe that the choice of the local degrees of freedom allows us to compute exactlya (p,v h ), p P k (), v h V,k without knowingv h in the interior of. Indeed we have a p (p,v h ) = p v h dx = pv h dx+ n v hds (7) Now, for any T h and for any sufficiently regular functionϕwe set ϕ := n n ϕ(v i ), V i = verticesof. (8) i= Next, we define the operatorπ k : V,k P k () V,k as the solution of: { a (Π k v h,q) = a (v h,q) q P k () Π k v h = v h (9) for all v h V,k. Then we choose a symmetric positive definite bilinear forms (, ) that verify c a (v h,v h ) S (v h,v h ) c a (v h,v h ) v h V,k with Π k v h = (2) for some positive constantc, c independent of andh. Finally we define a h (u h,v h ) = a (Π k u h,π k v h )+S (u h Π k u h,v h Π k v h ) u h,v h V,k. (2) The following theorem (see [2]) holds: Theorem 2 The bilinear form (2) satisfies the consistency property (6) and the stability property (7). 4.3 Choice ofs Here we illustrate a simple choice of S that ensures that the condition (2) is satisfied. Let χ i, i =,..N be the operator that associates to a smooth enough functionϕthei th local degree of freedomχ i (ϕ), we define the canonical basis{ϕ j } j=,..n as χ i (ϕ j ) = δ ij, i,j =,...N (22) We assume that there exists a γ > such that for all h and T h the distance between any two vertices of is γh,then in order to satisfy (2) it will be sufficient to choose: S (ϕ i Π k ϕ i,ϕ j Π k ϕ j) = χ r (ϕ i Π k ϕ i)χ r (ϕ j Π k ϕ j) (23) 4.4 Construction of the right-hand side N r= In the case k = we approximatef by a piecewise constant. Let Pk be the L2 ()-projection onto the spacep k, we define < f h,v h >= P f v h dx = P f v h, (24) T h T h 4 k

while fork 2 we set f h = Pk 2 f on each T h. Consequently the right-hand side becomes: < f h,v h > Pk 2fv h dx = T h fpk 2v h dx (25) 4.5 Error estimates T h In order to get the error estimates we give additional hypothesis on the decomposition T h (see e.g. [6]). We suppose that exists an integer N and a positive real number r such that for every h and for every T h we have the number of edges of is N the ratio between the shortest edge and the diameterh of is bigger thanr is star-shaped with respect to every point of a ball of radiusrh With these assumptions, according to the classical theory (see for instance [7]), the following proposition holds. Proposition There exists a constant C, depending only on k and r, such that for every s with s k + and for everyω H s () there exists aω π P k () such that ω ω π, +h ω ω π, Ch s ω s,. (26) Moreover, if ω H s () with 2 s k +, there exists aω I V,k such that ω ω I, +h ω ω I, Ch s ω s,. (27) Finally the following error estimate in thel 2 -norm can be derived (see [4]): Lemma Let the domain Ω be convex. Under the same assumptions of Theorem, for k 2, for every approximationu I of u inv h and for every approximationu π of u that is piecewise inp k, there exists a constant C independent of h such that ) u u h,ω Ch ( u u h + u u h h, +hˆk f f h,ω (28) withhˆk = if k = 2,hˆk = otherwise. Remark By applying the Theorem, the Proposition and the previous Lemma, if u is sufficiently regular, it follows that if k 3 u u h,ω Ch k+ u k+,ω (29) ifk =, assuming that the integration rule for the approximation of the load is at least of first order, it holds u u h,ω Ch 2 u 2 2,Ω + /2 (3) T h f 2, if k = 2 the optimal O(h 3 ) convergence rate is reached if a more accurate approximation of the right-hand side is used (see [4]). 5

5 THE PLATE PROBLEM Now we briefly illustrate the application of the Virtual Elements to the linear plate bending problem in the irchhoff-love formulation. 5. The continuous problem LetΩbe a convex polygonal domain occupied by the plate, let Ω be its boundary andf L 2 (Ω) be a transversal load acting on the plate. The irchhoff-love model for thin plates (see e.g. [8]) is to find the transversal displacementw, solution of D 2 w = f in Ω (3) et where D = 3 2( ν 2 ) is the bending rigidity, t is the thickness, E the Young modulus and ν the Poisson s ratio. Assuming the plate to be clamped, the boundary conditions become w = w n = on Ω, (32) where n is the outward normal direction. The variational formulation of (3)-(32) is: { Find w V := H 2 (Ω) suchthat a(w,v) = (f,v) v V (33) where the bilinear form a(, ) is now defined by: [ ] a(w,v) = D ( ν) (w / v / +2w /2 v /2 +w /22 v /22 dx+ν w v dx Ω Ω It is well known that problem (2) has a unique solution (see e.g. [5]). (34) 5.2 The discrete problem To discretize the plate problem we follow the same procedure as for the Poisson problem. Let {T h } h be a sequence of decompositions of Ω into disjoint open polygons, and let { E h } h be the set of edges e of T h. As in the case of the Poisson problem the idea is to construct a finite dimensional space V h V, a bilinear form a h (, ) : V h V h R and an element f h V h such that the discrete problem: { Find wh V h suchthat (35) a h (w h,v h ) =< f h,v k > v h V h, has a unique solution w h and good approximation properties hold. We will not explain here the construction ofv h, a h (, ), f h but we refer to the paper ([3])for details. 6 NUMERICAL RESULTS In this section we consider two test problems: a Poisson problem and a clamped plate problem. In both cases we apply the Virtual Element Method and we exploit its behavior by comparing the results with the Finite Element Method. 6

. X elements, hmean=6 2 X 2 elements, hmean=.73 4 X 4 elements, hmean=.365..... Figure : From left to right. Sequence of slightly distorted meshes X elements, hmean= 2 X 2 elements, hmean=.854 4 X 4 elements, hmean=.43...... Figure 2: From left to right. Sequence of strongly distorted meshes 6. The Poisson problem We consider the problem u = in Ω u = on Γ = Ω (36) where Ω is the unit square. The simple shape of the domain enables the solution to be explicitly represented. Specifically, using separation of variables it can be shown that u(x,y) = x2 6 2 π 3 k=,k odd { sin(kπ(+x)/2) k 3 sinh(kπ) } (sinh(kπ(+y)/2)+sinh(kπ( y)/2)) (37) We approximate problem (36) applying the Virtual Element of orderk =. We use uniform meshes, quadrilateral meshes with slightly distorted elements and quadrilateral meshes with strongly distorted elements. In order to make a comparison with the Finite Element Method we consider the quadrilateral finite element of degree one (Q (), if the element is a rectangle, see e.g.[8]). We employ meshes with N N elements with N =, 2, 4, that in the quadrilateral cases give the sequences shown in figures and 2. We evaluate the errors in the L 2 discrete norm. In figure 3 we report the errors against the mean value of the mesh size h in the three cases: uniform, slightly and strongly distorted meshes. We observe that in the cases of uniform and slightly distorted meshes the error curves of the Virtual Method and the Finite Element Method are very close and show a slope equal to the theoretical convergence order. In the case of strongly distorted mesh the Virtual Method 7

is in agreement with the optimal convergence order while the Finite Element Method shows a very bad behavior. L 2 error 2 u u h 3 FEM strongly distorted FEM slightly distorted FEM uniform VEM strongly distorted VEM slightly distorted VEM uniform h 2 4 2 log h Figure 3: Convergence curves for different meshes. 6.2 The plate problem We consider the problem D 2 w = f in Ω w = w n = on Ω (38) where Ω is the unit square and f is chosen in order to obtain as exact solution the function w = x 2 (x ) 2 y 2 (y ) 2. We approximate problem (38) with the Virtual Element of lowest degree of accuracy k = 2. This element is caracterized by the following local space V,k = {v H 2 () : v e P 3 (e), (v /n ) e P (e) e, 2 v = } (39) with the related degrees of freedom: value of v, v /, v /2 at the vertices of. This element is the extension to polygonal domains of two finite elements for plate: the Hsieh-Clough-Tocher reduced triangle (see [8]) and the finite element of lowest degree of the family of the assumed stresses hybrid methods (see [9]). We compare the results obtained by the Virtual Element (VEM3) and the Finite Elements CTR and HYB3 on a sequence of uniform meshes ofn N elements,n = 4,8,6,32. In figure 4 we report the relative errors in L 2, H and H 2 norm respectively against the mesh size h. As expected the results are in agreement with the theoretical predictions. 7 CONCLUSIONS We presented the features of the Virtual Element Method in the simple case of the Poisson problem. We applied the lowest element to a benchmark Poisson problem and we observed the capability 8

Figure 4: From left to right. Relative errors inl 2, H andh 2 norm of the method to perform well also in the case of very irregular mesh as opposed to what happens with the corresponding finite element. We tested the behaviour of the method also in the case ofc approximations as is the case of the plate bending problem. Also in this case we applied the lowest element to a benchmark plate problem on a uniform mesh and we observed the good agreement of the results with the theoretical predictions. The next development will be to apply the method to plate problems with polygonal domains and then to deal with complex structures such as shells for which the method seems to be particularly suitable. References [] Brezzi, F., Lipnikov,. and Simoncini, V., A family of mimetic finite difference methods on polygonal and polyedral meshes, Math. Models Methods Appl. Sci., 5, 533-553 (25). [2] Beirao da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D. and Russo, A., Basic principles of Virtual Element Methods, Math. Models Methods. Appl. Sci., 23(), 99-24 (23). [3] Brezzi, F. and Marini, L.D., Virtual Element Method for plate bending problems,comput. Methods Appl. Mech. Engrg., 253, 455-462 (23). [4] Beirao da Veiga, L., Brezzi, F. and Marini, L.D., Virtual Elements for linear elasticity problems,siam J. Numer. Anal. 5(2), 794-82 (23). [5] Lions J.L. and Magenes E., Problème aux limites non homogène et applications. Vol I, Dunod, Paris (968) [6] Brezzi F., Buffa A. and Lipnikov. Mimetic finite differences for elliptic problems, M2AN:Math.Model.Numer.Anal., 43, 277-295 (29) [7] Brenner, S.C. and Scott, R.L., The mathematical theory of finite element methods, Texts in Applied Mathematics, 5, Springer Verlag, New York (28). [8] Ciarlet P.G., The Finite Element Method for Elliptic Problems, North-Holland (978). [9] Brezzi F. and Marini L.D., On the numerical solution of plate bending problems by hybrid methods, R.A.I.R.O., R-3, 9, 5-5 (975). 9