A Two-field Finite Element Solver for Poroelasticity on Quadrilateral Meshes

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1 A Two-field Finite Element Solver for Poroelasticity on Quadrilateral Meses Graam Harper 1, Jiangguo Liu 1, Simon Tavener 1, and Zuoran Wang 1 Colorado State University, Fort Collins, CO 80523, USA {arper,liu,tavener,wangz}@mat.colostate.edu Abstract. Tis paper presents a finite element solver for linear poroelasticity problems on quadrilateral meses based on te displacementpressure two-field model. Tis new solver combines te Bernardi-Raugel element for linear elasticity and a weak Galerkin element for Darcy flow troug te implicit Euler temporal discretization. Te solver does not use any penalty factor and as less degrees of freedom compared to oter existing metods. Te solver is free of nonpysical pressure oscillations, as demonstrated by numerical experiments on two widely tested bencmarks. Extension to oter types of meses in 2-dim and 3-dim is also discussed. Keywords: Bernardi-Raugel elements Locking-free Poroelasticity Raviart-Tomas spaces Weak Galerkin (WG). 1 Introduction Poroelasticity involves fluid flow in porous media tat are elastic and can deform due to fluid pressure. Poroelasticity problems exist widely in te real world, e.g., drug delivery, food processing, petroleum reservoirs, and tissue engineering [6, 7, 19] and ave been attracting attention from te scientific computing community [10, 16, 17, 23] (and references terein). Some recent work can be found in [9 11, 20, 24]. Matematically, poroelasticity can be modeled by coupled Darcy and elasticity equations as sown below { (2µε(u) + λ( u)i) + α p = f, (1) t (c 0 p + α u) + ( K p) = s, ( were u is te solid displacement, ε(u) = ) 1 2 u + ( u) T is te strain tensor, λ, µ (bot positive) are Lamé constants, f is a body force, p is te fluid pressure, K is a permeability tensor (tat as absorbed fluid viscosity for notational convenience), s is a fluid source or sink (treated as negative source), α (usually close G. Harper, J. Liu, and Z. Wang were partially supported by US National Science Foundation under grant DMS

2 2 G.Harper et al. to 1) is te Biot-Williams constant, c 0 0 is te constrained storage capacity. Appropriate boundary and initial conditions are posed to close te system. An early complete teory about poroelasticity was formulated in Biot s consolidation model [3]. A more recent rigorous matematical analysis was presented in [18]. It is difficult to obtain analytical solutions for poroelasticity problems. Terefore, solving poroelasticity problems relies mainly on numerical metods. According to wat variables are being solved, numerical metods for poroelasticity can be categorized as 2-field: Solid displacement, fluid pressure; 3-field: Solid displacement, fluid pressure and velocity; 4-field: Solid displacement and stress, fluid pressure and velocity. Te simplicity of te 2-field approac is always attractive and ence pursued by tis paper. Continuous Galerkin (CG), discontinuous Galerkin (DG), mixed, nonconforming, and weak Galerkin finite element metods all ave been applied to poroelasticity problems. A main callenge in all tese metods is te poroelasticity locking, wic often appears in two modes [24]: (i) Nonpysical pressure oscillations for low permeable or low compressible media [8, 15], (ii) Poisson locking in elasticity. Based on te displacement-pressure 2-field model, tis paper presents a finite element solver for linear poroelasticity on quadrilateral meses. Te rest of tis paper is organized as follows. Section 2 discusses discretization of planar linear elasticity by te 1st order Bernardi-Raugel elements on quadrilaterals. Section 3 presents discretization of 2-dim Darcy flow by te novel weak Galerkin finite element metods, in particular, WG(Q 0, Q 0 ; RT [0] ) on quadrilateral meses. In Section 4, te above two types of finite elements are combined wit te first order implicit Euler temporal discretization to establis a solver for poroelasticity on quadrilateral meses, wic couples te solid displacement and fluid pressure in a monolitic system. Section 5 presents numerical tests for tis new solver to demonstrate its efficiency and robustness (locking-free property). Section 6 concludes te paper wit some remarks. 2 Discretization of Elasticity by Bernardi-Raugel (BR1) Elements In tis section, we consider linear elasticity in its usual form { σ = f(x), x Ω, u Γ D = u D, ( σn) Γ N = t N, (2) were Ω is a 2-dim bounded domain occupied by a omogeneous and isotropic elastic body, f is a body force, u D, t N are respectively Diriclet and Neumann

3 A Two-field Finite Element Solver for Poroelasticity on Quadrilateral Meses 3 data, n is te outward unit normal vector on te domain boundary Ω = Γ D Γ N. As mentioned in Section 1, u is te solid displacement, is te strain tensor, and ε(u) = 1 2 ( u + ( u) T ) (3) σ = 2µ ε(u) + λ( u)i, (4) is te Caucy stress tensor, were I is te order two identity matrix. Note tat te Lamé constants λ, µ are given by λ = Eν (1 + ν)(1 2ν), µ = E 2(1 + ν), (5) were E is te elasticity modulus and ν is Poisson s ratio. In tis section, we discuss discretization of linear elasticity using te first order Bernardi-Raugel elements (BR1) on quadrilateral meses. Te Bernardi- Raugel elements were originally developed for Stokes problems [2]. Tey can be applied to elasticity problems wen combined wit te reduced integration tecnique [4, 5, 24]. In tis context, it means use of less quadrature points for te integrals involving te divergence term. Te BR1 element on a quadrilateral can be viewed as an enricment of te classical bilinear Q 2 1 element, wic suffers Poisson locking wen applied directly to elasticity. Let E be a quadrilateral wit vertices P i (x i, y i )(i = 1, 2, 3, 4) starting at te lower-left corner and going counterclockwise. Let e i (i = 1, 2, 3, 4) be te edge connecting P i to P i+1 wit te modulo convention P 5 = P 1. Let n i (i = 1, 2, 3, 4) be te outward unit normal vector on edge e i. A bilinear mapping from (ˆx, ŷ) in te reference element Ê = [0, 1]2 to (x, y) in suc a generic quadrilateral is establised as follows { x = x1 + (x 2 x 1 )ˆx + (x 4 x 1 )ŷ + ((x 1 + x 3 ) (x 2 + x 4 ))ˆxŷ, (6) y = y 1 + (y 2 y 1 )ˆx + (y 4 y 1 )ŷ + ((y 1 + y 3 ) (y 2 + y 4 ))ˆxŷ. On te reference element Ê, we ave four standard bilinear functions ˆφ 4 (ˆx, ŷ) = (1 ˆx)ŷ, ˆφ 1 (ˆx, ŷ) = (1 ˆx)(1 ŷ), ˆφ3 (ˆx, ŷ) = ˆxŷ, ˆφ2 (ˆx, ŷ) = ˆx(1 ŷ). (7) After te bilinear mapping defined by (6), we obtain four scalar basis functions on E tat are usually rational functions of x, y: φ i (x, y) = ˆφ i (ˆx, ŷ), i = 1, 2, 3, 4. (8) Tese lead to eigt node-based local basis functions for Q 1 (E) 2 : [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] φ1 0 φ2 0 φ3 0 φ4 0,,,,,,,. (9) 0 φ1 0 φ2 0 φ3 0 φ4

4 4 G.Harper et al. Furtermore, we define four edge-based scalar functions on Ê: ˆψ 1 (ˆx, ŷ) = (1 ˆx)ˆx(1 ŷ), ˆψ 3 (ˆx, ŷ) = (1 ˆx)ˆxŷ, ˆψ2 (ˆx, ŷ) = ˆx(1 ŷ)ŷ, ˆψ4 (ˆx, ŷ) = (1 ˆx)(1 ŷ)ŷ. (10) Tey become univariate quadratic functions on respective edges of Ê. For a generic convex quadrilateral E, we utilize te bilinear mapping to define ψ i (x, y) = ˆψ i (ˆx, ŷ), i = 1, 2, 3, 4. (11) Ten we ave four edge-based local basis functions, see Figure 1 (left panel): b i (x, y) = n i ψ i (x, y), i = 1, 2, 3, 4. (12) Finally te BR1 element on a quadrilateral is defined as BR1(E) = Q 1 (E) 2 + Span(b 1, b 3, b 3, b 4 ). (13) Fig. 1. Left panel: Four edge bubble functions needed for te 1st order Bernardi-Raugel element on a quadrilateral; Rigt panel: WG(Q 0, Q 0; RT [0] ) element on a quadrilateral. On eac quadrilateral, tere are totally twelve vector-valued basis functions. Teir classical gradients are calculated ad oc and so are te strains. Clearly, teir classical divergences are not constants. Te elementwise averages of divergence or te local projections into te space of constants are calculated accordingly. Let V be te space of vector-valued sape functions constructed from te BR1 elements on a sape-regular quadrilateral mes E. Let V 0 be te subspace of V consisting of sape functions tat vanis on Γ D. Let u V and v V 0. Ten te bilinear form in te strain-div formulation utilizing te Bernardi- Raugel elements reads as A SD (u, v) = ( ) 2µ ε(u ), ε(v) + λ( u, v) E, (14) E E E

5 A Two-field Finite Element Solver for Poroelasticity on Quadrilateral Meses 5 were te overline bar indicates te elementwise averages of divergence. Te linear form for discretization of te body force is simply F (v) = E E (f, v) E, v V 0. (15) Now tere are two sets of basis functions: node-based and edge-based. Compatibility among tese two types of functions needs to be maintained in enforcement or incorporation of boundary conditions. (i) For a Diriclet edge, one can directly enforce te Diriclet condition at te two end nodes and set te coefficient of te edge bubble function to zero; (ii) For a Neumann edge, integrals of te Neumann data against te tree basis functions (two linear polynomials for te end nodes, one quadratic for te edge) are computed and assembled accordingly. 3 Discretization of Darcy Flow by WG(Q 0, Q 0 ; RT [0] ) Elements In tis section, we consider a 2-dim Darcy flow problem prototyped as { ( K p) + c p = f, x Ω, p Γ D = p D, (( K p) n) Γ N = u N, (16) were Ω is a 2-dim bounded domain, p te unknown pressure, K a conductivity matrix tat is uniformly SPD, c a known function, f a source term, p D a Diriclet boundary condition, p D a Neumann boundary condition, and n te outward unit normal vector on Ω, wic as a nonoverlapping decomposition Γ D Γ N. As an elliptic boundary value problem, (16) can be solved by many types of finite element metods. However, local mass conservation and normal flux continuity are two most important properties to be respected by finite element solvers for Darcy flow computation. In tis regard, continuous Galerkin metods (CG) are usable only after postprocessing. Discontinuous Galerkin metods (DG) are locally conservative by design and gain normal flux continuity after postprocessing. Te enanced Galerkin metods (EG) [21] are also good coices. Te mixed finite element metods (MFEM) ave bot properties by design but result in indefinite discrete linear systems tat need specially designed solvers. Te recently developed weak Galerkin metods [12, 22], wen applied to Darcy flow computation, ave very attractive features in tis regard: Tey possess te above two important properties and result in symmetric positive linear systems tat are easy to solve [12 14]. Te weak Galerkin metods [22] rely on novel concepts to develop finite elements for differential equations. Discrete weak basis functions are used separately in element interiors and on interelement boundaries (or mes skeleton). Ten discrete weak gradients of tese basis functions are computed via integration by

6 6 G.Harper et al. parts. Tese discrete weak gradients can be establised in certain known spaces, e.g., te local Raviart-Tomas spaces RT 0 for triangles, te standard RT [0] for rectangles, and te unmapped RT [0] for quadrilaterals [14]. Tese discrete weak gradients are used to approximate te classical gradient in variational forms. Recall tat for a quadrilateral element E, te local unmapped Raviart- Tomas space as dimension 4 and can be generated by tese four basis functions RT [0] (E) = Span(w 1, w 2, w 3, w 4 ), (17) were w 1 = [ ] [ ] [ ] [ ] 1 0 X 0, w 0 2 =, w 1 3 =, w 0 4 =, (18) Y and X = x x c, Y = y y c are te normalized coordinates using te element center (x c, y c ). For a given quadrilateral element E, we consider 5 discrete weak functions φ i (0 i 4) as follows: φ 0 for element interior: It takes value 1 in te interior E but 0 on te boundary E ; φ i (1 i 4) for te four sides respectively: Eac takes value 1 on te i-t edge but 0 on all oter tree edges and in te interior. Any suc function φ = {φ, φ } as two independent parts: φ is defined in E, wereas φ is defined on E. Ten its discrete weak gradient w,d φ is specified in RT [0] (E) via integration by parts [22] (implementationwise solving a size-4 SPD linear system): ( w,d φ) w = E φ (w n) E φ ( w), E w RT [0] (E). (19) Wen a quadrilateral becomes a rectangle E = [x 1, x 2 ] [y 1, y 2 ], we ave w,d φ 0 = 0w 1 + 0w ( x) w ( y) w 2 4, w,d φ 1 = 1 x w 1 + 0w ( x) 2 w 3 + 0w 4, w,d φ 2 = 1 x w 1 + 0w ( x) 2 w 3 + 0w 4, w,d φ 3 = 0w y w 2 + 0w ( y) 2 w 4, w,d φ 4 = 0w y w 2 + 0w ( y) 2 w 4, were x = x 2 x 1, y = y 2 y 1. (20) Let E be a sape-regular quadrilateral mes. Let Γ D be te set of all edges on te Diriclet boundary Γ D and Γ N be te set of all edges on te Neumann boundary Γ N. Let S be te space of discrete sape functions on E tat are degree 0 polynomials in element interiors and also degree 0 polynomials on edges. Let S 0 be te subspace of functions in S tat vanis on Γ D. For (16), we seek

7 A Two-field Finite Element Solver for Poroelasticity on Quadrilateral Meses 7 p = {p, p } S suc tat p Γ D = Q (p D) (te L 2 -projection of Diriclet boundary data into te space of piecewise constants on Γ D) and were and A (p, q) = F(q), q = {q, q } S 0, (21) A (p, q) = E E E F(q) = E E K w,d p w,d q + E fq γ Γ N γ E E E c p q (22) u N q. (23) As investigated in [14], tis Darcy solver is easy to implement and results in a symmetric positive-definite system. More importantly, it is locally massconservative and produces continuous normal fluxes. 4 Coupling BR1 and WG(Q 0, Q 0 ; RT [0] ) for Linear Poroelasticity on Quadrilateral Meses In tis section, te Bernardi-Raugel elements (BR1) and te weak Galerkin WG(Q 0, Q 0 ; RT [0] ) elements are combined wit te implicit Euler temporal discretization to solve linear poroelasticity problems. Assume a given domain Ω is equipped wit a sape-regular quadrilateral mes E. For a given time period [0, T ], let 0 = t (0) < t (1) <... < t (n 1) < t (n) <... < t (N) = T be a temporal partition. Denote t n = t (n) t (n 1) for n = 1, 2,..., N. Let V and V 0 be te spaces of vector-valued sape functions constructed in Section 2 based on te first order Bernardi-Raugel elements. Let S and S 0 be te spaces of scalar-valued discrete weak functions constructed in Section 3 based on te WG(Q 0, Q 0 ; RT [0] ) elements. Let u (n), u(n 1) V be approximations to solid displacement at time moments t (n) and t (n 1), respectively. Similarly, let p (n), p(n 1) S be approximations to fluid pressure at time moments t (n) and t (n 1), respectively. Note tat te discrete weak trail function as two pieces: p (n) = {p (n),, p (n), }, (24) were p (n), lives in element interiors and p (n), lives on te mes skeleton. Applying te implicit Euler discretization, we establis te following timemarcing sceme, for any v V 0 and any q S0, ( ) 2µ ε(u (n) ), ε(v) + λ( u (n), v) α(p(n),, v) = (f (n), v), ( c 0 p (n),, q ) ( ) + t n K p (n), q + α( u (n), q ) (25) ( = c 0 p (n 1),, q ) ( + t n s (n), q ) + α( u (n 1), q ),

8 8 G.Harper et al. for n = 1, 2,..., N. Te above two equations are furter augmented wit appropriate boundary and initial conditions. Tis results in a large monolitic system. Tis solver as two displacement degrees of freedom (DOFs) per node, one displacement DOF per edge, one pressure DOF per element, one pressure DOF per edge. Let N d, N e, N g be te numbers of nodes, elements, and edges, respectively, ten te total DOFs is 2 N d + N e + 2 N g, (26) wic is less tan tat for many existing solvers for poroelasticity. 5 Numerical Experiments In tis section, we test tis novel 2-field solver on two widely used bencmarks. In tese test cases, te permeability is K = κi, were κ is a piecewise constant over te test domain. Finite element meses align wit te aforementioned pieces. Rectangular meses (as a special case of quadrilateral meses) are used in our numerics. Fig. 2. Example 1: A sandwiced low permeability layer. Example 1 (A sandwiced low permeability layer). Tis problem is similar to te one tested in [8, 9], te only difference is te orientation. Te domain is te unit square Ω = (0, 1) 2, wit a low permeability material (κ = 10 8 ) in te middle region 1 4 x 3 4 being sandwiced by te oter material (κ = 1). Oter parameters are λ = 1, µ = 1, α = 1, c 0 = 0. Listed below are te boundary conditions. For solid displacement: For te left side: Neumann (traction) σn = t N = (1, 0); For te bottom-, rigt-, and top-sides: omogeneous Diriclet (rigid) u = 0;

9 A Two-field Finite Element Solver for Poroelasticity on Quadrilateral Meses 9 Fig. 3. Sandwiced low permeability layer: Numerical displacement and dilation, numerical pressure and velocity, numerical pressure contours for = 1/32 and t = Left column for time T 1 = 0.001; Rigt column for time T 2 = Furter srinking in solid, drop of maximal fluid pressure, and pressure front moving are observed.

10 10 G.Harper et al. For fluid pressure: For te left side: omogeneous Diriclet (free to drain) p = 0; For 3 oter sides: omogeneous Neumann (impermeable) ( K p) n = 0. Te initial displacement and pressure are assumed to be zero. See Figure 2 for an illustration. For tis problem, we examine more details tan wat is sown in te literature. We use a uniform rectangular mes wit = 1/32 for spatial discretization and t = for temporal discretization. Tis way, t 2. Sown in Figure 3 are te numerical displacement and dilation (div of displacement), te numerical pressure and velocity, and te numerical contours at time T 1 = and T 2 = 0.01, respectively. As te process progresses and te fluid drains out from te left side (x = 0), it is clearly observed tat (i) Te maximal pressure is dropped from around to around ; (ii) Te pressure front moves to te rigt and becomes more concentrated around te interface x = 0.25; (iii) Te solid is furter srunk: maximal srinking (negative dilation) magnitude increases from around to around Fig. 4. Example 2: Cantilever bracket problem. Example 2 (Cantilever bracket problem). Tis bencmark is often used to test spurious pressure oscillations [1, 17, 24]. Te domain is te unit square Ω = (0, 1) 2. A no-flux (Neumann) condition is prescribed on all four sides for te fluid pressure. Te left side is clamped, tat is, a omogeneous Diriclet condition u = 0 is posed for te solid displacement for x = 0. A downward traction (Neumann condition) σn = t N = (0, 1) is posed on te top side, wereas te rigt and bottom sides are traction-free. Te initial displacement and pressure are assumed to be zero. See Figure 4. Here we follow [1, 24] to coose te following parameter values E = 10 5, ν = 0.4, κ = 10 7, α = 0.93, c 0 = 0.

11 A Two-field Finite Element Solver for Poroelasticity on Quadrilateral Meses 11 Fig. 5. Example 2: Cantilever bracket problem. Numerical pressure at time T = is obtained by combining Bernardi-Raugel and weak Galerkin elements troug te implicit Euler discretization wit t = 0.001, = 1/32. We set t = 0.001, = 1/32. Sown in Figure 5 are te numerical pressure profile and contours at T = 0.005, obtained from using te Bernardi-Raugel elements and weak Galerkin elements. Clearly, te numerical pressure is quite smoot and tere is no nonpysical oscillation. Tis new finite element solver as been added to our Matlab code package DarcyLite. Te implementation tecniques presented in [13] are used. It is demonstrated in [24] tat nonpysicial pressure oscillations occur wen te classical continuous Galerkin metod is used for elasticity discretization. 6 Concluding Remarks In tis paper, we ave developed a finite element solver for linear poroelasticity on quadrilateral meses based on te two-field model (solid displacement and fluid pressure). Tis solver relies on te Bernardi-Raugel elements [2] for discretization of te displacement in elasticity and te novel weak Galerkin elements [14] for discretization of pressure in Darcy flow. Tese spatial discretizations are combined wit te backward Euler temporal discretization. Te solver does not involve any nonpysical penalty factor. It is efficient, since less unknowns are used, compared to oter existing metods. Tis new solver is robust, free of poroelasticity locking, as demonstrated by experiments on two widely tested bencmarks. Tis paper utilizes te unmapped local RT [0] spaces for discrete weak gradients, wic are used to approximate te classical gradient in te Darcy equation. Convergence can be establised wen quadrilateral are asymptotically parallelograms [14]. Tis type of new solvers are simple for practical use, since any

12 12 G.Harper et al. polygonal domain can be partitioned into asymptotically parallelogram quadrilateral meses. Efficient and robust finite element solvers for Darcy flow and poroelasticity on general convex quadrilateral meses are currently under our investigation and will be reported in our future work. Our discussion focuses on quadrilateral meses, but te ideas apply to oter types of meses. For triangular meses, one can combine te Bernardi-Raugel elements on triangles for elasticity [2] and te weak Galerkin elements on triangles for Darcy [12]. Tis combination provides a viable alternative to te tree-field solver investigated in [24]. Similar 2-field solvers can be developed for tetraedral and cuboidal exaedral meses. Tis is also a part of our current researc efforts for developing efficient and accessible computational tools for poroelasticity. References 1. Berger, L., Bordas, R., Kay, D., Tavener, S.: Stabilized lowest-order finite element approximation for linear tree-field poroelasticity. SIAM J. Sci. Comput. 37, A2222 A2245 (2015) 2. Bernardi, C., Raugel, G.: Analysis of some finite elements for te Stokes problem. Mat. Comput. 44, (1985) 3. Biot, M.: General teory of tree-dimensional consolidation. J. Appl. Pys. 12, (1941) 4. Brenner, S., Sung, L.Y.: Linear finite element metods for planar linear elasticity. Mat. Comput. 59, (1992) 5. Cen, Z., Jiang, Q., Cui, Y.: Locking-free nonconforming finite elements for planar linear elasticity. Disc. Cont. Dyn. Sys. suppl pp (2005) 6. Ceng, A.H.: Poroelasticity, Teory and Applications of Transport in Porous Media, vol. 27. Springer (2016) 7. Cowin, S., Doty, S.: Tissue Mecanics. Springer (2007) 8. Haga, J., Osnes, H., Langtangen, H.: On te causes of pressure oscillations in low permeable and low compressible porous media. Int. J. Numer. Aanl. Met. Geomec. 36, (2012) 9. Hu, X., Mu, L., Ye, X.: Weak Galerkin metod for te Biot s consolidation model. Comput. Mat. Appl. p. In press (2018) 10. Hu, X., Rodrigo, C., Gaspar, F., Zikatanov, L.: A nonconforming finite element metod for te Biot s consolidation model in poroelasticity. J. Comput. Appl. Mat. 310, (2017) 11. Lee, J., Mardal, K.A., Winter, R.: Parameter-robust discretization and preconditioning of Biot s consolidation model. SIAM J. Sci. Comput. 39, A1 A24 (2017) 12. Lin, G., Liu, J., Mu, L., Ye, X.: Weak galerkin finite element metdos for Darcy flow: Anistropy and eterogeneity. J. Comput. Pys. 276, (2014) 13. Liu, J., Sadre-Marandi, F., Wang, Z.: Darcylite: A Matlab toolbox for Darcy flow computation. Proc. Comput. Sci. 80, (2016) 14. Liu, J., Tavener, S., Wang, Z.: Te lowest-order weak Galerkin finite element metod for te Darcy equation on quadrilateral and ybrid meses. J. Comput. Pys. p. In press (2018) 15. Pillips, P.: Finite element metods in linear poroelasticity: Teoretical and computational results. P.D. tesis, University of Texas at Austin (2005)

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