Stabilized and Coupled FEM/EFG Approximations for Fluid Problems

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1 COMPUTATIONAL MECHANICS WCCM VI in conjunction with APCOM4, Sept. 5-, 24, Beijing, China c 24 Tsinghua University Press & Springer-Verlag Stabilized and Coupled FEM/EFG Approximations for Fluid Problems T.P. Fries, H.G. Matthies Institute for Scientific Computing, Technische Universität Braunschweig Hans-Sommer-Str. 65, D 386 Braunschweig, Germany t.fries@tu-bs.de Abstract Coupled FEM/EFG approximations are employed for the solution of fluid problems, taking advantage of the individual features of meshfree and meshbased methods. Standard coupling approaches are modified in order to obtain shape functions that may be stabilized reliably. The coupled fluid solver can be applied straightforward to problems including moving and rotating obstacles. Key words: meshfree, meshless, coupling, fluid, Navier-Stokes INTRODUCTION Stabilized finite element methods have developed to be standard tools for the simulation of flow problems [5]. However, in the presence of rotating and moving obstacles, the maintenance of a conforming mesh may be almost impossible. Therefore, a comparably new and innovative class of methods will be used which enables the approximation of partial differential equations based on a set of nodes without the need for an additional mesh. However, these meshfree methods [, 7] are comparably time-consuming which limits their usefulness in practice. Throughout this paper the Element Free Galerkin (EFG) method [2] is considered as a popular member of meshfree methods. Coupling meshbased FEM approximations and meshfree EFG approximations enables one to use meshfree methods only in small parts of the domain, where a mesh is difficult to maintain, and standard meshbased methods in the rest of the domain, thereby combining the advantages of both methodologies. The coupling approaches of Huerta [] and Belytschko [3] are discussed. They are modified in order to obtain shape functions that are more suited for stabilization [8, 9]. The resulting formulation is successfully applied to a number of test cases. The plan of the paper is as follows: The next section gives an outline of the Element Free Galerkin (EFG) method [2], which employs shape functions built by the Moving Least Squares (MLS) concept [5]. Then, the approaches of Huerta [] and Belytschko [3] for coupling FEM and EFG approximations are discussed and modified to allow a stabilization of the coupled scheme. The following section compares the coupling approaches for the solution of the one-dimensional advection-diffusion equation, and a convergence test is carried out. Results from the coupled fluid solver are presented, and the advantage of this approach for the solution of complex flows with moving and rotating obstacles is displayed. All test cases show that the coupled approximations have the same order of convergence as pure FEM calculations, and that reliable and accurate solutions are obtained with the modified coupling approaches. OUTLINE OF THE ELEMENT FREE GALERKIN (EFG) METHOD Throughout this paper, we focus in particular on the Element Free Galerkin (EFG) method [2], however, most conclusions can be applied to other meshfree methods as well. The EFG in its original

2 version employs Moving Least Squares (MLS) approximations in a Bubnov-Galerkin formulation of a problem, i.e. test and shape functions of a weak form are identical. Also, a certain treatment of essential boundary conditions has been published as being part of the method [2], but this aspect is not considered here. Moving Least Squares Approximations The construction of meshfree approximations with the MLS methodology, introduced in [5], shall only be briefly outlined at this point, see [, 7] for a more detailed discussion. In the MLS, the domain is discretized by a set of particles I with corresponding dilatation parameters ρ i, which determine the supports Ω i of the resulting meshfree shape functions. One defines an approximation at an arbitrary fixed point x Ω of ũ (x, x) = p T (x) a (x), where p is a complete basis vector which in general consists of monomials. The length k of this vector depends on the dimension of the problem and the desired order of consistency of the resulting approximations. The vector a is the vector of unknown coefficients of the approximation at x. These unknowns are determined by a minimization of the following weighted error functional: J (a) = i I w (x x i ) [u (x i ) ũ (x i )] 2 = i I w (x x i ) [ u i p T (x i ) a (x) ] 2, which leads to a k k system of equations for a. The weighting function w (x x i ) is in general a bell-shaped function e.g. an approximation of the Gaussian function and is non-zero only in the corresponding support Ω i. It ensures the locality of the approximation. The resulting system of equation is w (x x i ) p (x i ) p T (x i ) a (x) i I = w (x x i ) p (x i ) u i, i I M (x) a (x) = B (x) u, hence a (x) = [M (x)] B (x) u. In words, the minimization of the error functional enables one to find a connection between the unknown coefficients of the approximation a with the nodal unknowns u. At this stage, the so-called moving procedure is applied [5], which in fact results into a replacement of x by x. Inserting a into the approximation gives ũ (x) = p T (x) [M (x)] B (x) u = p T (x) [M (x)] i I w (x x i ) p (x i ) u i, where the meshfree shape functions N i (x) = p T (x) [M (x)] w (x x i ) p (x i ) may directly be extracted. In the EFG method, these shape functions are as well taken as test functions in the weak form of a problem. EFG Approximations for Stabilized Weak Forms A straightforward usage of numerical methods based on the Bubnov-Galerkin principle may result in severe numerical problems and stabilization is required. Stabilized methods have developed to be standard tools in the numerical world [5]. They all share the common property of perturbing the test function of a weak form and multiply this modification with the residual of the differential equation under consideration, thereby maintaining the consistency of the formulation [8]. The perturbation of the test function leads to Petrov-Galerkin methods, i.e. test and trial functions are no longer identical. Stabilization for Advection-Dominated Problems It is a well-known fact that Bubnov-Galerkin formulations result for advection-dominated problems as they frequently arise in the context of fluid dynamics, in heavy oscillations, polluting the solution inside the whole domain, see e.g. [4]. These oscillations are apparent in classical meshbased methods, such as the Finite Element method, and occur analogously in meshfree methods. The Streamline-Upwind/Petrov-Galerkin (SUPG) [4] and Galerkin/Least-Squares (GLS) [4] stabilization are the most popular stabilization schemes to smooth oscillations in advection-dominated problems.

3 Ω el = Ω FEM Ω Ω FEM Ω Ω EFG FEM Γ EFG Γ Fig. Decomposition of the domain into Ω FEM, Ω EFG and Ω. Stabilization of Equal-order Interpolations Most desirable from a computational viewpoint are equal-order interpolations for the velocities and pressure. Then, every node has the same degrees of freedom, which are approximated by the same shape functions. The (most restrictive) governing stability criterion of equal-order interpolations for variational problems with constraints such as the incompressible Navier-Stokes equations is the Babuška-Brezzi condition [6]. Stabilization ideas often circumvent this condition (rather than satisfying it). It can be shown that the GLS stabilization may be used for the stabilization of equal-order interpolations as well [2]. However, SUPG stabilized formulations still require stabilization in order to enable equal-order interpolations. In practice, this may be done with the Pressure-Stabilizing/Petrov-Galerkin (PSPG) stabilization [3, 8]. Consequences Numerical studies for the solution of the incompressible Navier-Stokes equations with meshfree equal-order approximations show that SUPG/PSPG stabilization leads to less diffusive, i.e. slightly superior results than GLS [9]. It has also been shown that it is important for the success of the stabilization to employ meshfree shape functions with small dilatation parameters, i.e. small support sizes [8, 9]. A reliable stabilization of non-linear problems cannot be guaranteed with large supports of the meshfree shape functions. This fact is an important motivation for the modification of the existing coupling approaches in the following section. COUPLING EFG AND FEM The EFG method has been successfully applied to a large number of problems, where the maintenance of a conforming mesh may be almost impossible. However, the integration of the weak form is very time-consuming compared with standard meshbased methods such as the FEM. Therefore, EFG and FEM shall be coupled to employ EFG only in small areas, where a mesh is hardly to maintain and FEM in the rest of the domain. In the literature, there are most importantly two coupling approaches: the one introduced by Belytschko et al. in [3] and the other of Huerta et al. []. Preliminaries For a coupling of EFG and FEM, the domain Ω is decomposed into disjoint domains Ω el and Ω EFG, with the common boundary Γ EFG. The domain Ω el is discretized with standard quadrilateral finite elements. The union of all elements along Γ EFG is called the transition area Ω, so that Ω el may further be decomposed into the disjoint domains Ω FEM and Ω, connected by a boundary labeled Γ FEM ; clearly Ω FEM Ω EFG =. This situation can be depicted from Figure. Throughout this paper, consistency of first order is fulfilled by the set of meshbased, meshfree and coupled shape functions. This results in the ability of reproducing linear solutions exactly. Standard bilinear shape functions are chosen in the finite elements. The complete basis vector for the meshfree shape functions is p (x) = (, x) T in one dimension and p (x) = (, x, y) T in two dimensions. Coupling Approach of Huerta The coupling approach of Huerta [] considers the contributions of the meshbased FEM shape functions in the computation of the MLS shape functions by modified consistency conditions. The resulting, coupled set of shape functions is consistent up to the desired order.

4 original Huerta coupling modified Huerta coupling function values N * N EFG I FEM I * I EFG function values N * N EFG I FEM I * I EFG Ω FEM Γ FEM Ω * Γ EFG Ω EFG domain Ω Ω FEM Γ FEM Ω * Γ EFG Ω EFG domain Ω Fig. 2 Shape functions of the coupling approach of Huerta in the original and modified version. In the original approach, FEM nodes are placed in the standard way in the elements inside Ω FEM, however not in Ω. The corresponding meshbased shape functions of the FEM nodes remain unchanged, the coupling is considered only in the particles shape functions. EFG particles with corresponding supports Ω i may be arbitrarily distributed in Ω EFG and Ω. Then, the shape functions for the nodes and particles are computed as follows: FEM : N i = Ni FEM EFG : N i = N ( i EFG coupled : N i = p T (x) ) j I FEM j (x) p T (x j ) i I FEM i I EFG [M (x)] w (x x i ) p (x i ) i I, with I FEM = { i xi Ω } { } { FEM, I EFG = i Ω i Ω EFG and I } = i Ω i Ω el. In words, I EFG is the set of particles whose supports are fully inside Ω EFG, and I is the set of particles that have supports overlapping with elements. Ni FEM are the standard bilinear finite element shape functions, and Ni EFG are the standard MLS functions, defined in the previous section. Figure 2 shows the sets I FEM, I and I EFG and displays the resulting shape functions of this approach in a section of a one-dimensional domain with a regular node/particle distribution around the transition area Ω. Modification Instead of keeping the FEM shape function unchanged inside the transition area as in the original approach, one may additionally place EFG particles at the FEM node positions along { Γ FEM and superimpose the two shape functions at these nodes. That is, I FEM reduces to i xi Ω FEM \ Γ FEM}, and for the nodes along Γ FEM we define coupled : N i = ( p T (x) ) j I FEM j (x) p T (x j ) [M (x)] w (x x i ) p (x i ) + Ni FEM i : x i Γ FEM. Figure 2 shows the resulting shape functions of this approach. The important advantage of this modification is that smaller dilatation parameters are possible (although in this figure ρ i = 2.9 x has been taken for both approaches). For example, in the original approach and a regular distribution of particles and nodes in one dimension, one finds that for the regularity of the matrix M (x), dilatation parameters of ρ i > 2. x are required []. In contrast, with the modified approach ρ i >. x are sufficient. This holds analogously in multi-dimensional domains. The advantage can most importantly be realized for the solution of stabilized weak forms of non-linear partial differential equations. Numerical studies show for stabilized flow computations that sufficient and reliable stabilization is often only possible for about ρ i.7 x in regular node/particle distributions [9]. For the role of small dilatation parameters in the solution of stabilized problems see also [8].

5 original Belytschko coupling modified Belytschko coupling function values N * N EFG I FEM I * I EFG function values N * N EFG I FEM I * I EFG Ω FEM Γ FEM Ω * Γ EFG Ω EFG domain Ω Ω FEM Γ FEM Ω * Γ EFG Ω EFG domain Ω Fig. 3: Shape functions of the coupling approach of Belytschko in the original and modified version. Coupling Approach of Belytschko In the approach of Belytschko [3], meshfree and meshbased shape functions for the nodes i are computed and defined independently and coupled with help of a ramp function R (x). FEM nodes are placed in Ω el, in contrast to the coupling approach of Huerta, where they are placed in Ω FEM only. EFG particles with supports Ω i are distributed arbitrarily in Ω Ω EFG and may also be included inside Ω FEM. The latter will only affect the shape function inside Ω Ω EFG, i.e. particles with Ω i Ω FEM have no influence at all. Belytschko et al. define the ramp function as follows: R (x) =, x Ω FEM, x Ω EFG i I i (x), x Ω, I = { i xi Γ EFG }, i.e. it varies monotonically between and in Ω. The linear consistency of the resulting set of meshbased, meshfree and coupled shape functions is maintained [3]. The shape functions are defined as follows: FEM : N i = Ni FEM i I FEM EFG : N i = Ni EFG i I EFG coupled : N i = [ R (x)] Ni FEM + R (x) Ni EFG i I, { with I FEM = i x i Ω FEM, Ω { } { i Ω }, FEM I EFG = i Ω i Ω EFG and I = i Ω i Ω el }. The resulting shape functions are shown in Figure 3 and have been used with ρ i = 2.5 x in [3] for some test cases. One may note that EFG particles inside Ω FEM have an undesirable influence in Ω Ω EFG, leading to a numerically awkward form of the shape function (dashed line in Figure 3). Especially in the context of stabilization, it is neither clear from a mathematical viewpoint nor from an empirical viewpoint how to choose suitable stabilization parameters for these shape functions [8]. Modification In order to avoid these numerically awkward shape functions, this approach is modified slightly. The EFG particles are restricted to the area Ω Ω EFG. The resulting shape functions are then (for sufficiently small dilatation parameters) suitable to motivate the usage of standard stabilization parameters [8]. Consequently, a reliable stabilization is obtained. The resulting shape functions of the modified coupling approach of Belytschko is shown in the right part of Figure 3. NUMERICAL RESULTS One-dimensional Advection-Diffusion Equation The aim of this study is to show that the same order of convergence is obtained with all coupling approaches. The obtained convergence of order 2 in the L 2 -norm is equivalent to purely meshbased

6 n=2 n=35 n=49 L 2 error 3 convergence of the coupling schemes 2 Belyt., orig. Belyt., mod. Huerta, orig. Huerta, mod. pure FEM 4 Ω FEM Ω Ω EFG 5 2 node number Fig. 4 Convergence test of the coupling approaches. computations with bi-linear finite elements only. The test case is defined as follows. Approximate the stabilized one-dimensional advection-diffusion equation ( w + τc w ) ( c ũ ) x x K 2 ũ x f dω =, x (, ), 2 Ω with f = 2cπ cos (2πx) + 4Kπ 2 sin (2πx), c =, K = and boundary conditions u() = u() =. The exact solution is u (x) = sin (2πx). Throughout the convergence test, the ratio of the domains is kept constant at Ω F EM : Ω : Ω EF G = 6 : : 6, which may be seen in the left part of Figure 4. The right part shows the convergence results for the two original and modified coupling approaches of Huerta and Belytschko respectively. The rate of convergence remains the same than in pure FEM computations. The higher rates of convergence of a pure EFG computation cannot be reproduced, which is in agreement to [3] and []. Two-dimensional Incompressible Navier-Stokes Equations The SUPG/PSPG stabilized weak form of the instationary, incompressible Navier-Stokes equations is ( ) ui w i σ ij n j dγ = w i ϱ Γ Ω t + u u i w i j f i dω + σ ij dω + q u i dω x j Ω x j Ω x i ( u i + τ u k + ) [ ( ) q ui ϱ x k ϱ x i t + u u i j f i σ ] ij dω, x j x j ( u i x j with σ ij = pδ ij + µ + u j being the stress tensor. u i and p are approximations of the velocities and pressure, ϱ stands for the density, the external forces are f i = in the test cases presented herein. The Navier-Stokes equations are given in Arbitrary Lagrangian Eulerian (ALE) form, to consider possible mesh movements, such as in the rotor test case shown below. The first line of the NS equations is the Bubnov-Galerkin part, the second is the SUPG/PSPG stabilization, realized by a modification of the test function multiplied with the residual of the momentum equations. The boundary conditions are u i = g on the Dirichlet part and σ ij n j = h i on the Neumann part of the boundary. The stabilization parameters of the FEM shape functions are computed with τ = / ( ) t x i ) ( ) 2 ( ) 2 2 ui 4µ +, h el h 2 el as suggested in [7, 9]. This formula has also been applied to the meshfree and coupled shape functions, where the element length h el is replaced by the support length ρ. For details, see [8, 9].

7 a) b) c) u=, v= Ω Ω EFG Ω EFG Ω Ω EFG Ω Ω EFG.8.6 y vertical center velocity profile, Re=.4 reference coupled, 2x2 Ω coupled, 4x4 coupled, 6x6.2 pure FEM, 2x2 Ω FEM pure FEM, 4x4 pure FEM, 6x u Fig. 5: a) Driven cavity test case with domain decomposed into Ω FEM, Ω EFG and Ω, b) discretization with 4 4 node/particle distribution, c) convergence against reference solution. Driven Cavity The driven cavity test case is a standard benchmark for fluid problems. Reference solutions are given in []. A flow inside a quadratic domain Ω = (, ) (, ) with no-slip boundary conditions on the left, right and lower wall develops under a shear flow applied on the upper boundary until a stationary solution is reached. Figure 5 gives an outline of the problem and shows a discretization with 4 4 nodes/particles. The modified coupling approaches are applied, in order to enable the use of small dilatation parameters of ρ i =.3 x. For a Reynolds number of Re = convergence may not be reached for dilatation parameters ρ i 2. x, underlining the importance of the modified coupling versions. The results of both approaches are almost identical for this test case and therefore, only the result for Huerta coupling is shown. In Figure 5c) the convergence against the reference solution along the vertical center velocity profile is shown and coupled results are compared with pure FEM solutions. The Ω EFG holes are placed such that the center profile directly cuts through them. In between the EFG particles, linear interpolation has been applied for simplicity. One may see that coupling does not adversely affect the solution. Cylinder Flow The channel flow around a cylinder has been developed as a test case by Turek in [6]. The cylinder is placed slightly unsymmetrically in y-direction of the channel. For sufficiently high Reynolds numbers the well-known Kármán vortex street develops behind the cylinder. A quasi-stationary solution is obtained. Turek gives reference solutions for the lift and drag coefficients c L and c D of the cylinder. Figure 6 shows a sketch of the Kármán vortex street, together with the development of c D in time until a periodical solution is obtained. In the right part, the mesh around the cylinder is shown, together with the Ω EFG part, where particles are distributed. In the left part of Figure 7, the results for the drag coefficient obtained with the modified approach of Huerta and Belytschko and the pure FEM computation are compared ( t =.5). The horizontal lines show the limits, in which the exact value for the maximum of c D lies [6]. One may again see that the results are quite close together. The drag coefficient is slightly improved with the coupled approaches, the coupling approach of Huerta achieves somewhat better results than the approach of Belytschko. Both are slightly better than the pure FEM computation. Results for lift coefficient and pressure difference between the most left and right point on the cylinder surface are not shown here, because the drag coefficient turned out to be the most sensitive. For c L and p, the results between the coupled approaches and pure FEM are almost indistinguishable. The right part of Figure 7 shows the dependency of the drag coefficient on the time step t. The Strouhal number St = D v mt, with the diameter D =. of the cylinder, the average inflow from the left with v m = and the time T for 2 periods of c D (equals period of c L ), is displayed in the figure

8 a).4 y.2 Kármán vortex street c) b) x 3.3 drag coefficient 3.2 c D time [s] Fig. 6: Cylinder test case at Re =, a) the Kármán vortex street, b) development of the drag coefficient in time, c) discretization around the cylinder. drag coefficient drag coefficient c 3.22 D coupled Belytschko coupled Huerta pure FEM time [s] c 3.22 D Belyt., t=., St=.299 Belyt., t=.5, St=.2932 Belyt., t=.25, St=.2947 Huerta, t=., St=.299 Huerta, t=.5, St=.2932 Huerta, t=.25, St= time [s] Fig. 7: The left part compares the different modified coupling approaches and the pure FEM solution for t =.5, the right part shows the convergence in time of the coupled approaches. as well. A clear convergence against the reference Strouhal number of.295 St.35 may be seen and the amplitudes of the drag coefficient improve. Flow Around a Rotating Obstacle The previously described test cases verified the coupled fluid solver. However, they did not take advantage of the beneficial properties of this approach. The following test case replaces the cylinder of the previous test case by a rotating obstacle. Standard meshbased methods fail to give results due to the distortion of the mesh which must follow the rotation. However, this is no problem with the coupled fluid solver, where the rotating inner mesh and the stationary outer mesh are separated by a meshfree area. Figure 8 shows the discretization around the meshfree area and the resulting momentum around the rotor s center in dependence of the angle α of the inner mesh. Rather than giving all details of the test case settings, in here only the easiness with which the proposed coupled FEM/EFG approximations solve this complicated flow problem shall be shown. CONCLUSION In this paper, coupling approaches of meshfree and meshbased methods for the solution of flow problems are discussed. Meshfree methods are introduced considering the EFG method in particular. However, most conclusions can be applied to other meshfree methods as well. Standard coupling procedures are reviewed and modified such that the resulting shape functions are more suited for stabilization. This is a crucial aspect, because meshfree methods for stabilized, non-linear problems

9 moment around rotor center x 3.5 moment M angle α Fig. 8: Discretization around the rotating obstacle and resulting moment around the center of the rotor in dependence of the angle α. require certain attention [8, 9]. A convergence test for a one-dimensional test case shows that the coupled approaches achieve the same order of convergence than purely meshbased FEM calculations. The coupling approaches are then used for the solution of the two-dimensional incompressible Navier-Stokes equations. The coupled fluid solver is verified with standard test cases and is employed to solve a flow around a moving and rotating obstacle, showing the straightforward usability of this approach to complex flow problems. We conclude that coupled FEM/EFG approximations are a very promising tool for the simulation of complex flow problems. REFERENCES [] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P.: Meshless Methods: An Overview and Recent Developments. Comp. Methods Appl. Mech. Engrg., 39, 3 47, 996. [2] Belytschko, T.; Lu, Y.Y.; Gu, L.: Element-free Galerkin Methods. Internat. J. Numer. Methods Engrg., 37, , 994. [3] Belytschko, T.; Organ, D.; Krongauz, Y.: A Coupled Finite Element Element-free Galerkin Method. Comput. Mech., 7, 86 95, 995. [4] Brooks, A.N.; Hughes, T.J.R.: Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comp. Methods Appl. Mech. Engrg., 32, , 982. [5] Donea, J.; Huerta, A.: Finite Element Methods for Flow Problems. John Wiley & Sons, Chichester, 23. [6] Franca, L.P.; Hughes, T.J.R.: Two classes of mixed finite element methods. Comp. Methods Appl. Mech. Engrg., 69, 89 29, 988. [7] Fries, T.P.; Matthies, H.G.: Classification and Overview of Meshfree Methods. Informatikbericht-Nr. 23-3, Technical University Braunschweig, ( Brunswick, 23. [8] Fries, T.P.; Matthies, H.G.: A Review of Petrov-Galerkin Stabilization Approaches and an Extension to Meshfree Methods. Informatikbericht-Nr. 24-, Technical University of Braunschweig, ( Brunswick, 24.

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