The Virtual Element Method: an introduction with focus on fluid flows

Transcription

1 The Virtual Element Method: an introduction with focus on fluid flows L. Beirão da Veiga Dipartimento di Matematica e Applicazioni Università di Milano-Bicocca VEM people (or group representatives): P. Antonietti, B. Ayuso, S. Berrone, S. Bordas, F. Brezzi, A. Cangiani, R. Falk, G. Gatica, C. Lovadina, G. Manzini, D. Marini, D. Mora, G. Paulino, P. Pietra, R. Rodriguez, S. Scacchi, A. Russo, G. Vacca, M. Verani, P. Wriggers,... L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

2 The Virtual Element Method The Virtual Element Method (VEM) is a generalization of the Finite Element Method. VEM allow to use very general polygonal and polyhedral meshes, also for high polynomial degrees and without overwhelming complications. The flexibility of VEM is not limited to the mesh: an example will be shown later. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

3 Why polygons/polyhedrons? The interest (and use in commercial codes ) for polytopes is growing. Immediate combination of tets and hexahedrons Easier/better meshing of domain and data features (cracks, sliding meshes, inclusions...) Automatic use 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. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

4 Virtual elements: current range of problems Poisson problem (initial paper) Diffusion-convection-reaction in primal and mixed form Stokes problem Plate bending problems Linear elasticity, nonlinear elasticity and inelasticity Discrete fracture networks Cahn-Hilliard equation Contact problems Eigenvalue problems Topology optimization Time dependent problems... L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

5 The model problem We consider the Poisson problem in two dimensions u = f in Ω, u = 0 on Ω, where Ω R 2 is a polygonal domain; the loading f is assumed in L 2 (Ω). Variational formulation: where a(v, w) = find u V := H0 1 (Ω) such that a(u, v) = f v dx v V, Ω Ω v w dx, v, w V. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

6 A Virtual Element Method [BdV,Brezzi,Cangiani,Manzini,Marini,Russo, M3AS 2013] We will build a discrete problem in following form { find uh V h such that where a h (u h, v h ) =< f h, v h > v h V h, V h V is a finite dimensional space; a h (, ) : V h V h R is a discrete bilinear form approximating the continuous form a(, ); < f h, v h > is a right hand side term approximating the load L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

7 A Virtual Element Method [BdV,Brezzi,Cangiani,Manzini,Marini,Russo, M3AS 2013] We will build a discrete problem in following form { find uh V h such that where a h (u h, v h ) =< f h, v h > v h V h, V h V is a finite dimensional space; a h (, ) : V h V h R is a discrete bilinear form approximating the continuous form a(, ); < f h, v h > is a right hand side term approximating the load Let m 1 be a fixed integer index. Such index will represent the degree of accuracy of the method. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

8 The local spaces V h E Let Ω h be a simple polygonal mesh on Ω. This can be any decomposition of Ω in non overlapping polygons E with straight faces. The space V h will be defined element-wise, by introducing local spaces V h E ; the associated local degrees of freedom. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

9 The local spaces V h E Let Ω h be a simple polygonal mesh on Ω. This can be any decomposition of Ω in non overlapping polygons E with straight faces. The space V h will be defined element-wise, by introducing local spaces V h E ; the associated local degrees of freedom. For all E Ω h : V h E = { v H 1 (E) : v P m 2 (E), v e P m (e) e E }. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

10 The local spaces V h E Let Ω h be a simple polygonal mesh on Ω. This can be any decomposition of Ω in non overlapping polygons E with straight faces. The space V h will be defined element-wise, by introducing local spaces V h E ; the associated local degrees of freedom. For all E Ω h : V h E = { v H 1 (E) : v P m 2 (E), v e P m (e) e E }. the functions v V h E are continuous (and known) on E; the functions v V h E are unknown inside the element E! it holds P m (E) V h E L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

11 Depiction of the degrees of freedom for V h E m=1 m=2 m=3 Green dots stand for vertex pointwise values Red squares represent edge pointwise values (m 2 per edge) Blue squares represent internal (volume) moments v h p m 2 p m 2 P m 2 (E). E L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

12 The space V h The global space V h is built by assembling the local spaces V h E as usual: V h = {v H 1 0 (Ω) : v E V h E E Ω h } The total d.o.f.s are one per internal vertex, m 1 per internal edge and m(m 1)/2 per element. The choice of degrees of freedom guarantees the global continuity of the functions in V h. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

13 The bilinear form a h (, ) The bilinear form a h (, ) is built element by element a h (v h, w h ) = E Ω h a E h (v h, w h ) v h, w h V h, where a E h (, ) : V h E V h E R are symmetric bilinear forms that mimic a E h (, ) a(, ) E by satisfying stability and consistency conditions. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

14 A projection operator We introduce the following energy projector Π : V h E P m (E) defined by, for all v h V h E, { a E (Πv h, p) = a E (v h, p) p P m (E)/R P 0 (Πv h ) = P 0 (v h ) (projection on constants). L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

15 A projection operator We introduce the following energy projector Π : V h E P m (E) defined by, for all v h V h E, { a E (Πv h, p) = a E (v h, p) p P m (E)/R P 0 (Πv h ) = P 0 (v h ) (projection on constants). An integration by parts gives a E (p, v h ) = p v h dx E = ( p) v h dx + E E ( p n E ) v h ds. for all p P m (E), v h V h E. Thus Π is COMPUTABLE. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

16 The local bilinear forms It is immediate to check that v h, w h V h E a E (v h, w h ) = a E (Πv h, Πw h ) + a E ((I Π)v h, (I Π)w h ). L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

17 The local bilinear forms It is immediate to check that v h, w h V h E a E (v h, w h ) = a E (Πv h, Πw h ) + a E ((I Π)v h, (I Π)w h ). We introduce the discrete bilinear form ah E (v h, w h ) = a E (Πv h, Πw h ) + s E ((I Π)v h, (I Π)w h ) where the symmetric bilinear form s E : V h E V h E R is positive definite; scales (in h) like the original bilinear form s E (v h, v h ) a E (v h, v h ) v h V h E, Πv h = 0. NOTE: the above discrete form is exact for polynomials in P m. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

18 The discrete load term We consider m 2 first. Let, for all E Ω h, the approximated load f h E be the L 2 -projection of f E on P m 2 (E). Then (f h, v h ) h := E Ω h E f h v h dx that is computable due to the internal dofs of V h E. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

19 The discrete load term We consider m 2 first. Let, for all E Ω h, the approximated load f h E be the L 2 -projection of f E on P m 2 (E). Then (f h, v h ) h := E Ω h E f h v h dx that is computable due to the internal dofs of V h E. In the case m = 1 a simple integration rule based on the vertex values of the polygon can be used, for instance (f h, v h ) h := ( ) 1 f dx v h (ν) dx. N E E Ω h E ν E Note: for m = 2 better choices can be made [Ahmad, Alsaedi, Brezzi, Marini, Russo, CMA 2013], [BdV, Brezzi, Marini, SINUM 2013]. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

20 A convergence result Let the sequence {Ω h } h satisfy the following mesh assumptions: each element E in Ω h is star-shaped with respect to a ball of uniform radius ( or suitable union of ) ; for each element E in Ω h, the lenght of all edges is comparable with its diameter h E ( not needed by paying (1 + log(he /h e )) ). Then the following holds [... volley team..., M3AS 2013]. Theorem Let the stability and consistency assumptions hold. Then, if f H s (Ω h ) and u H s+1 (Ω h ), we have u u h H 1 (Ω) C ( hs u H s+1 (Ω h ) + f ) H s (Ω h ) for 0 s m and with C independent of h. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

21 k-plain div(k u)+αu = f exact solution: ue(x, y) =y x +log y 3 + x +1 xy xy 2 + x 2 y + x 2 + x 3 +sin(5x) sin(7y) 1 diffusion: K(x, y) =1 zero-order term: α(x, y) =1 right-hand-side: f(x, y) = log y 3 + x +1 1 y 5 x + (y 3 + x +1) y 4 (y 3 + x +1) 2 xy xy2 + x 2 y + x 2 + x 3 6 y +75sin(5x) sin(7y) 3 y 3 + x +1 L2 norm of the exact solution: ue 0,Ω = H1 seminorm of the exact solution: ue 1,Ω = Generated by genue on 09-Nov :08:59

22 0 h=hmean slope: Π uh u e 0,Ω = 4.011, Π u h u e 0,Ω = h 5 h 4 h 3 h 2 0 Π uh u e 0,Ω Π u h u e 0,Ω 0 h=hmean slope: Π uh u e 0,Ω = 3.010, Π u h u e 0,Ω = h 5 h 4 h 3 h 2 0 Π uh u e 0,Ω Π u h u e 0,Ω 10 6 h h L 2 error k =3 H 1 error triangle polygons, h max = , h mean = triangle polygons, h max = , h mean = triangle polygons, h max = , h mean = triangle polygons, h max = , h mean = triangle polygons, h max = , h mean =

23 0 h=hmean slope: Π uh u = 5.009, Π u u = e 0,Ω h e 0,Ω h=hmean slope: Π uh u e 0,Ω = 4.003, Π u h u e 0,Ω = Π 0 u h u e 0,Ω Π 0 u h u e 0,Ω 10 5 h 5 Π u h u e 0,Ω 10 3 h 5 Π u h u e 0,Ω 10 6 h h 4 h 3 h h 2 h h 1 h L 2 error k =4 H 1 error triangle polygons, h max = , h mean = triangle polygons, h max = , h mean = triangle polygons, h max = , h mean = triangle polygons, h max = , h mean = triangle polygons, h max = , h mean =

24 0 h=hmean slope: Π uh u e 0,Ω = 5.928, Π u h u e 0,Ω = h=hmean slope: Π uh u e 0,Ω = 5.020, Π u h u e 0,Ω = Π 0 u h u e 0,Ω Π 0 u h u e 0,Ω 10 6 Π u h u e 0,Ω 10 5 h 5 Π u h u e 0,Ω 10 7 h h h 4 h h 3 h h h h L 2 error k =5 H 1 error triangle polygons, h max = , h mean = triangle polygons, h max = , h mean = triangle polygons, h max = , h mean = triangle polygons, h max = , h mean = triangle polygons, h max = , h mean =

25 Convection-diffusion: an example We consider the problem on Ω = [0, 1] 2 ε u + b u = 0 in Ω, u(0, y) = (1 cos (2πy))/2 y [0, 1], u(1, y) = 0 y [0, 1], homogeneous Neumann BCs in remaining boundary, where ε = 10 2 and b = ( 1, 1/3). The data on the left edge will be transported (with some amplitude loss) and generate a layer near the right edge. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

26 Convection-diffusion: an example Triangular mesh with VEM: joined L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

27 Convection-diffusion: an example Solution plot (polynomial degree one): VEM solution k= L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

28 The three dimensional case In principle the 3D case is analogous; the degrees of freedom are one point value per vertex (m 1) point values per edge M0 f, Mf 1,.., Mf m 1 moments per face M0 E, ME 1,.., ME m 2 moments per element. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

29 The three dimensional case In principle the 3D case is analogous; the degrees of freedom are one point value per vertex (m 1) point values per edge M0 f, Mf 1,.., Mf m 1 moments per face M0 E, ME 1,.., ME m 2 moments per element. A more efficient formulation with less degrees of freedom per face M0 f, Mf 1,.., Mf m 2 moments per face can be built using the L 2 projector in [Ahmed, Alsaedi, Brezzi, Marini, Russo, CMA 2013]. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

30 The Stokes problem Find (u, p) such that ν u p = f in Ω, div u = 0 in Ω, u = 0 on Ω, VARIATIONAL FORM: Find (u, p) V Q, such that ν u : v + p div v = f v for all v V, Ω Ω Ω q div u = 0 for all q Q, Ω where the velocity and pressure spaces V = [H 1 0 (Ω)]2, Q = L 2 0 (Ω). L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

31 A first VEM approximation where Find (u h, p h ) V h Q h, such that a h (u h, v h ) + p h div v h = (f, v) h for all v h V h, Ω q h div u h = 0 for all q h Q h, Ω V h = [V h ] 2 (of degree m), Q h = { q L 2 0 (Ω) : q } E P m 1 E Ω h, a h (v h, w h ) ν v h : w h (vector version of the previous one!) Ω NOTE: The second form is computable (integrate by parts); the inf-sup condition holds! (m 2) L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

32 A new virtual velocity space (m 2) Let W h = { v [H0 1 } (Ω)]2 : v E W h E E Ω h where W h E := { v [H 1 (E)] 2 : v e [P m (e)] 2 e E, { ν v s [Pm 2 (E)] 2, s L 2 0 (E), div v P m 1 (E). The functions in W h E are solution of a local Stokes problem with polynomial data; in particular, the divergence of functions in W h E are polynomials of degree m 1 (known explicitly); the dimension of W h E is the same as that of V h E. (!!) L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

33 Degrees of freedom We take as degrees of freedom for W h E : vertex pointwise values; (m-2) pointwise values for every edge e E; L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

34 Degrees of freedom We take as degrees of freedom for W h E : vertex pointwise values; (m-2) pointwise values for every edge e E; divergence moments (div v h ) p m 1 p m 1 P m 1 (E)/R; E L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

35 Degrees of freedom We take as degrees of freedom for W h E : vertex pointwise values; (m-2) pointwise values for every edge e E; divergence moments (div v h ) p m 1 p m 1 P m 1 (E)/R; E orthogonal moments (x = [ x 2, x 1 ] ) v h x p m 3 p m 3 P m 3 (E). E L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

36 New VEM approximation Let [BdV, Lovadina, Vacca, ArXiv and submitted] Find (u h, p h ) W h Q h, such that a h (u h, v h ) + p h div v h = (f, v) h for all v h W h, Ω q h div u h = 0 for all q h Q h, Ω By using that [P m 2 ] 2 = P m 1 x P m 3, it can be shown that we can still compute v h : p m for all p m [P m (E)] 2, E and thus build the form a h (, ) (loading term analogous). L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

37 Advantages The second row of the discrete problem implies p m 1 div u h = 0 E Ω h, p m 1 P m 1 (E). E Moreover we remind that THEREFORE: (divu h ) E P m 1 (E) E Ω h. the solution u h is pointwise divergence free; the second row of the problem implies, in particular, that all the divergence moments degrees of freedom of u h vanish! L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

38 Reduced problem Let the reduced problem Find (u h, p h ) Ŵh Q h, such that a h (u h, v h ) + p h div v h = (f, v) h for all v h Ŵh, Ω q h div u h = 0 for all q h Q h, where Ω Ŵ h W h with all the divergence moment DoFs set to zero; Q h are piecewise constant functions. NOTE: the solution u h is the same as that of the original problem but many degrees of freedom (both for velocities and pressures) are saved. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

39 A convergence result (m 2) Let the sequence {Ω h } h satisfy the following mesh assumptions: each element E in Ω h is star-shaped with respect to a ball of uniform radius; for each element E in Ω h, the lenght of all edges is comparable with its diameter h E. Then the following holds [LBdV, Lovadina, Vacca, ArXiv and submitted]. Theorem If the right hand side makes sense, we have u u h H 1 (Ω) + p p h L 2 (Ω) C ( hs u H s+1 (Ω h ) + f ) H s (Ω h ) + p H s (Ω h ) for 0 s m and with C independent of h. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

40 Numerical tests TEST: Ω = [0, 1] 2 and loading in accordance with ( ) 1 u(x, y) = 2 cos2 (x) cos(y) sin(y) 1 p(x, y) = sin(x) sin(y). 2 cos2 (y) cos(x) sin(x) We consider the error quantities δ(u) := ui u h 1,h u I and δ(p) := pi p h L 2 1,h p I. L 2 and the mesh families L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

41 Numerical tests: Voronoi meshes L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

42 Numerical tests: triangular meshes L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

43 Numerical tests: quad meshes L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

44 Conclusions The Virtual Element Method is a novel generalization of FEM (2013) that allows for general polytopal meshes The flexibility of the method is reflected also in other aspects, such as the divergence-free scheme shown Virtual elements need (more development and) to approach and be tested on more complex problems (Navier-Stokes,...) and benchmarks We believe the idea is valuable and opens up various interesting research directions both for theoretical and applicative people. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

45 Construction of the stiffness matrix We introduce the following energy projector defined by, for all v h V h E, Π : V h E P m (E) { a E (Πv h, p) = a E (v h, p) p P m (E)/R P 0 (Πv h ) = P 0 (v h ) The operator P 0 is a projection on constants, that for m 2 is simply the average, and for m = 1 the vertex value average. NOTE: due to the consistency assumption, the projection operator above is computable (more later). L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

46 Construction of the stiffness matrix It is immediate to check that v h, w h V h E a E (v h, w h ) = a E (Πv h, Πw h ) + a E ((I Π)v h, (I Π)w h ). Then the bilinear form a E h (v h, w h ) = a E (Πv h, Πw h ) + s E ((I Π)v h, (I Π)w h ) is consistent and stable, provided the positive bilinear form s E : V h E V h E R scales like the original bilinear form a. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

47 Construction of the stiffness matrix It is immediate to check that v h, w h V h E a E (v h, w h ) = a E (Πv h, Πw h ) + a E ((I Π)v h, (I Π)w h ). Then the bilinear form a E h (v h, w h ) = a E (Πv h, Πw h ) + s E ((I Π)v h, (I Π)w h ) is consistent and stable, provided the positive bilinear form s E : V h E V h E R scales like the original bilinear form a. Let now the local stiffness matrix M E M E ij := a E h (ϕ i, ϕ j ) i, j = 1, 2,..., N with {ϕ i } the canonical basis associated to the degrees of freedom. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

48 Construction of the stiffness matrix We introduce also {m α } n α=1 a basis for the polynomial space P m = span{m α } n α=1. Since P m V h E, the matrix dof 1 (m 1 ) dof 1 (m 2 )... dof 1 (m n ) dof 2 (m 1 ) dof 2 (m 2 )... dof 2 (m n ) D = dof N (m 1 ) dof N (m 2 )... dof N (m n ) expresses the {m α } in terms of the V h E basis. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

49 Construction of the stiffness matrix We have also a matrix P 0 ϕ 1... P 0 ϕ N a E (ϕ 1, m 2 )... a E (ϕ N, m 2 ) B =..... a E (ϕ 1, m n )... a E (ϕ N, m n ) expressing (in terms of the bases) the right hand side in the definition { a E (Πv h, p) = a E (v h, p) p P m (E)/R P 0 (Πv h ) = P 0 (v h ) NOTE: such matrix is computable! (integration by parts and d.o.f.s definition) L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

50 Construction of the stiffness matrix Let P 0 m 1 ( P 0 m 2... P 0 m n 0 m2, m 2 )0,P... ( ) m2, m n G = BD =..... (. 0 mn, m 2 )0,P... ( ) mn, m n 0,P 0,P. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

51 Construction of the stiffness matrix Let P 0 m 1 ( P 0 m 2... P 0 m n 0 m2, m 2 )0,P... ( ) m2, m n G = BD =..... (. 0 mn, m 2 )0,P... ( ) mn, m n Compute the matrices corresponding to the projection operator: Π = (G) 1 B, Π = DΠ. 0,P 0,P. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31

52 Construction of the stiffness matrix Let P 0 m 1 ( P 0 m 2... P 0 m n 0 m2, m 2 )0,P... ( ) m2, m n G = BD =..... (. 0 mn, m 2 )0,P... ( ) mn, m n Compute the matrices corresponding to the projection operator: Π = (G) 1 B, Π = DΠ. Finally compute the local stiffness matrix M E = Π T G Π + (I Π) T (I Π). 0,P 0,P. L. Beirão da Veiga (Univ. of Milan-Bicocca) The Virtual Element Method PARIS / 31